Mathematıcs I (ENG) - İşletme (İngilizce)

Ünite 1

Soru 1

A = {0, 1, -1, -2, 2, 4, -4}, B = {1, -2, -3, -4, 5}, C={0, 2, 4, 6}. (A \ B) ∪ C = ?

{-1, 0, 2, 4, 5, 6}

{-1, 0, 2, 4, 6}

{-4, -3, -2, -1, 0, 1, 2, 4, 5, 6}

{-4, -2, 0, 1, 2, 4, 6}

{-3, -1, 0, 2, 4, 6}

Açıklama:

x = A \ B = {-1, 0, 2, 4} ; x ∪ C = {-1, 0, 2, 4, 6}. Correct answer is B.

Doğru Cevap: B

Soru 2

The total number of students in a class is 87, the number of students taking Music course is 55, the number of students taking Dance course is 45. If the number of students taking neither Music nor Dance is 26. What is the number of students taking both Music and Dance courses ?

Açıklama:

55 + 45 + 26 - x = 87 ; x = 126 - 87 = 39. Correct answer is D.

Doğru Cevap: D

Soru 3

(5⁹ - 5⁷) / (25² - 25³) = ?

-25

-625

-125

125

Açıklama:

x = (5⁷ (5² - 1)) / (5⁴ (1 - 5²)) = -125. Correct answer is C.

Doğru Cevap: C

Soru 4

((5^-1 + 2^-1)^{- 1} / (5^-1 - 2^-1))^{- 1} = ?

-3 / 7

-7 / 3

3 / 7

7 / 3

21 / 10

Açıklama:

x = ((1 / 5) + (1 / 2))^-1 / ((1 / 5) - (1 / 2))^-1 = -3 / 7. Correct answer is A.

Doğru Cevap: A

Soru 5

(35^1/2 / 5^1/2) . (700)^1/2 = ?

Açıklama:

x = ((7 . 5)^1/2 / 5^1/2) . (7 . 100)^1/2 = 70. Correct answer is E.

Doğru Cevap: E

Soru 6

Which one of these is the minimum : 1 / 3, 2 / 5, 3 / 6, 3 / 10, 7 / 15 ?

3 / 10

7 / 5

1 / 3

3 / 6

2 / 5

Açıklama:

x = min {1 / 3, 2 / 5, 3 / 6, 3 / 10, 7 / 15} = min {10, 12, 15, 9, 14} / 30 = 3 / 10. Correct answer is A.

Doğru Cevap: A

Soru 7

Which one of these is an irrational number : 49^1/2, -(64)^1/3, -(25)^1/2, 0, 3^1/2 ?

-(64)^1/3

-(25)^1/2

49^1/2

3^1/2

Açıklama:

x = 3^1/2 = 1.732.... . Correct answer is E.

Doğru Cevap: E

Soru 8

[-3, 5) ∩ (4, 7] = ?

[4, 5)

[4, 5]

(4, 5)

(4, 5]

[-3, 4)

Açıklama:

x = (4, 5). Correct answer is C.

Doğru Cevap: C

Soru 9

A = {a, b, c, d}, B = {b, c, e, f}, C = {c, d, e, g}. {b} = ?

(A ∪ B) \ C

(A ∩ B) \ C

A ∩ B ∩ C

C \ (A ∩ B)

B ∩ C

Açıklama:

x = (A ∩ B) \ C. Correct answer is B.

Doğru Cevap: B

Soru 10

a, b, c : real numbers ; a > b > c. |b - a| + |c - b| + |a - c| = ?

2c - 2b

2b - 2c

2c - 2a

2a - 2c

Açıklama:

x = a - b + b - c + a - c = 2a - 2 c. Correct answer is E.

Doğru Cevap: E

Soru 11

A = {x| x is a natural number between 30 and 40}
A = {x: x is a natural number between 30 and 40}
A = {x| x is a natural number}
The empty set is a subset of the set A.
A is its own subset.

Which of the above are implicit representations of the set A= {31, 32, 33,…39}?

I and II

I and IV

II and IV

I, II and III

III, IV and V

Açıklama:

A set can also be defined by giving a rule that determines whether an element is a member or not. For example, the expression
B = {x: x is a natural number between 10 and 20}
means that x is the collection of all natural numbers greater than 10 and less than 20. The set B above equivalently can be written as
B = {x| x is a natural number between 10 and 20}
or
B = {11, 12, ..., 19}.
Obviously, 15 ∈ B, 10 ∉ B, 25 ∉ B. The notations {x: …} and {x| …} are called implicit representations of sets.
As also understood from the information given, “A = {x| x is a natural number between 30 and 40" and “ A = {x: x is a natural number between 30 and 40}” are the implicit representations of the set A= {31, 32, 33,…39}, so the correct answer is A.
The expression “A = {x| x is a natural number}” is not well-defined, so it is not a set.
The expression “A is its own subset.” in the option IV explains the notion of set, but it is not the implicit representation of the set A.

Doğru Cevap: A

Soru 12

A ∩ B = {2, 3}
A ∪ B = {1, 2, 2, 3, 3, 5, 8, 10, 11}
A \ B = {1, 5, 8}
B \ A = {10, 11}

If A = {1, 2, 3, 5, 8} and B = {2, 3, 10, 11}, then which of the above are correct?

I and II

II and IV

I, II and IV

I, III and IV

II, III and IV

Açıklama:

Operations on sets are somewhat similar to operations of addition, multiplication and subtraction of numbers.
Let A and B be two sets.
The set of elements that are in either A or B or both is called the union of the sets A and B and is denoted by A ∪ B, i.e.,
A ∪ B = {x| x ∈ A or x ∈ B}
The set of all elements of the sets A and B is called the union of the sets A and B and is denoted by
A ∪ B.
Union is the act of combining two sets together into a single set.
Example
A = {1, 3, 5, 8}, B = {1, 3, 7}. Then A ∪ B = {1, 3, 5, 7, 8}.
If an element appears in both sets then we only list it once in the new set.
Example
A = {x| x is a city in Turkey with population greater than 1 million}, B = {x| x is a city in Turkey with population less than 500 000}. Then A ∪ B ={x| x is a city in Turkey with population greater than 1 million or less than 500 000}. The set of elements A which are not in B is called the difference between A and B and is denoted by A \ B.
A \ B = {x| x ∈ A and x ∉ B}
Example
A = {3, 5, 8, 10}, B = {4, 5, 9}. Then A \ B = {3, 8, 10}.
Example
A = {0, 1, 2, 3, 4, …}, B = {1, 3, 5, 7, …}. Then A \ B = {0, 2, 4, 6, ...}.
Usually the sets that we deal with are subsets of some ambient set. Such a set is called a universal set and is denoted by U. In other words, U is the universal set if all the sets under examination are subsets of U. The difference U \ A is called the complement of A and is denoted by A^c . That is,
A^c = U \ A = {x| x ∈ U and x ∉ A}
Example
U = {1, 2, …, 10}, A = {9, 10}. Then U \ A = {1, 2, …, 8}.
The intersection of two sets A and B, written A ∩ B is the set consisting of the elements of both A and B. Thus, A ∩ B = {x| x ∈ A and x ∈ B}
Example A = {1, 2, 3, 5, 8}, B = {2, 3, 10, 11}. Then
A ∩ B = {2, 3}. x ∈ A ∩ B if and only if x ∈ A and x ∈ B.
Example
A = {x| x is a city in Turkey with population less than 1 million},
B = {x| x is a city in Turkey with population greater than 500 000}.
Then A ∩ B ={x| x is a city in Turkey with population between 500 000 and 1 million}.
As also understood from the information given,
If A = {1, 2, 3, 5, 8} and B = {2, 3, 10, 11}, the expressions in the options;
I “A ∩ B = {2, 3}”
III “A \ B = {1, 5, 8}”
IV “B \ A = {10, 11}” are correct, so the correct answer is D.
If an element appears in both sets then we only list it once in the new set, so the expression in the option II “A ∪ B = {1, 2, 2, 3, 3, 5, 8, 10, 11}” is not correct. It is written as;
A ∪ B = {1, 2, 3, 5, 8, 10, 11}.

Doğru Cevap: D

Soru 13

The sets A = {4, 8, 11, 15} and B = {8, 15, 4 ,11} are the same.
The empty set Ø is a subset of any set.
The equality A = B is equivalent to two inclusions: A ⊂ B and B ⊂ A.
The set of elements that are in either A or B or both is called the union of the sets A and B and is denoted by A ∪ B.
x ∈ A ∩ B if and only if x ∈ A and x ∈ B.
If A ∩ B = Ø then the sets A and B are called disjoint sets.
Universal set is unique.

Which of the statements above regarding to sets are correct?

I, II and IV

II, IV, V and VI

III, IV, V, VI and VII

I, II, III, IV, V and VI

II, III, IV, V, VI and VII

Açıklama:

The statements regarding to sets in the options;
I “The sets A = {4, 8, 11, 15} and B = {8, 15, 4 ,11} are the same.”
II “The empty set Ø is a subset of any set.”
III “The equality A = B is equivalent to two inclusions: A ⊂ B and B ⊂ A.”
IV “The set of elements that are in either A or B or both is called the union of the sets A and B and is denoted by A ∪ B.”
V “x ∈ A ∩ B if and only if x ∈ A and x ∈ B.”
VI “If A ∩ B = Ø then the sets A and B are called disjoint sets.” are correct, so the correct answer is D.
The statement in the option VII “Universal set is unique.” is not correct. Universal set is not unique. It varies depending on the problem.

Doğru Cevap: D

Soru 14

Ø
{a}
{b}
{c}
{a, b}
{a, c}
{b, c}
{a, b, c}

Let A = {a, b, c}. Which of the above are the subsets of A?

II, III and IV

III, IV, V and VI

V, VI, VII and VIII

I, II, III, IV, V, VI and VII

I, II, III, IV, V, VI, VII and VIII

Açıklama:

Recommendation for Correction:
Page 9
Example
E = {1, 2, …, 10}, A = {1, 2, 3, 4}, B = {3, 4, 5}. Find A ∪ B, A ∩ B, A \ B, B \ A, A^c and B^c .
Example
A = {1, 2, 3, 4}, B = {3, 4, 5}. Find A ∪ B, A ∩ B, A \ B, B \ A, A^c and B^c .
If A is a finite set and n is the number of its elements, that is s(A) = n, then the number of all subsets of A is 2ⁿ.
For example, if A = {1, 2, 3} then the subsets of A
are Ø, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}.
The set A has 2³ = 8 subsets. Recall that, by convention, every set is its own subset, that is A ⊂ A. The empty set Ø is a subset of any set.
As also understood from the information given the subsets of A are
Ø, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}, so the correct answer is E.

Doğru Cevap: E

Soru 15

s(A ∩ B) = 13
s(A) + s(B) - s(A ∪ B) = s(A ∩ B)
s(B \ A) = 13
s(A \ B) = 7

For given sets A and B assume that s(A) = 20, s(B) = 26 and s(A ∪ B) = 33. Which of the statements regarding to the sets A and B are correct?

I and II

I and III

II and IV

I, III and IV

I, II, III and IV

Açıklama:

Example
For given sets A and B assume that s(A) = 10, s(B) = 16 and s(A ∪ B) = 23. Find s(B \ A).
Solution
s(A ∩ B) = s(A) + s(B) - s(A ∪ B) = 10 + 16 - 23 = 3. Therefore s(A \ B) = 7 and s(B \ A) = 13
As also understood form the solution given for the example, all of the statements in the options are correct, so the correct answer is E.
s(A ∩ B) = s(A) + s(B) - s(A ∪ B) = 20 + 26 - 33 = 13. Therefore s(A \ B) = 7 and s(B \ A) = 13.

Doğru Cevap: E

Soru 16

Every natural number is an integer. Every integer is a rational number.

Every rational number has an infinite number of representations by fractions.

Every rational number has a finite periodical representation.

There is one-to-one correspondence between the points of the real line and the set of real numbers.

Which of the statements above regarding to the number sets and the real line are correct?

I and II

II and IV

III and IV

I, II and IV

I, II, III and IV

Açıklama:

Every natural number is an integer. Every integer is a rational number. We do not consider the zero as a natural number.

Every rational number has an infinite number of representations by fractions.

Every rational number has a finite or infinite periodical representation.

The term “number” means real number.

There is one-to-one correspondence between the points of the real line and the set of real numbers.

As also understood from the information given, the correct answer is D. The statements regarding to the number sets and the real line in the options I “Every natural number is an integer. Every integer is a rational number.”, II “Every rational number has an infinite number of representations by fractions.” and IV “There is one-to-one correspondence between the points of the real line and the set of real numbers.”are correct. The statement in the option III “Every rational number has a finite periodical representation.” is not correct because of the fact that every rational number has a finite or infinite periodical representation.

Doğru Cevap: D

Soru 17

325 of them like at least one of these sports.

275 of them like none of these sports.

105 of them like only football.

35 of them like football and basketball but not volleyball.

135 of them like basketball or volleyball but not football.

Which of the statements above are correct according to the case given below?
"600 men are interviewed about which sports they like. It is found that 190 like football, 140 like basketball, 120 like volleyball, 65 like football and basketball, 50 like football and volleyball, 40 like basketball and volleyball, 30 like all three sports."

I and III

I, III and IV

II, III and V

I, II, III and IV

I, II, III, IV and V

Açıklama:

Example
600 men are interviewed about which sports they like. It is found that 190 like football, 140 like basketball, 120 like volleyball, 65 like football and basketball, 50 like football and volleyball, 40 like basketball and volleyball, 30 like all three sports.

How many of them like at least one of these sports?

How many of them like none of these sports?

How many of them like only football?

How many of them like football and basketball but not volleyball?

How many of them like basketball or volleyball but not football?

Solution
Define
F = The set of all men who like football,
B = The set of all men who like basketball,
V = The set of all men who like volleyball.
Then s(F) = 190, s(B) = 140, s(V) = 120, s(F ∩ B) = 65, s(F ∩ V) = 50, s(B ∩ V) = 40,
s(F ∩ B ∩ V) = 30 where, as defined above, the symbol s(.) is used for the number of elements. Below, we give the Venn diagram solution of this problem where the numbers on the regions they are sitting indicate the number of elements of the corresponding set.

For example, the number 30 indicates that he number of elements of the intersection F ∩ B ∩ V is 30.
We can easily count the required numbers in the questions:

105 + 35 + 30 + 20 + 65 +10 + 60 = 325

600 - 325 = 275,

105,

65 - 30 = 35,

60 + 10 + 65 = 135.

As also undersood from the information given, all statements in the options are correct according to the case given, so the correct answer is E.

Doğru Cevap: E

Soru 18

The set of all elements of the sets A and B is called the union of A and B

The set of the elements that the set A and the set B have in common is called the intersection of A and B

The set of all elements of A which are not in B is called the difference of A and B

The difference of U and A is called the complement of U.

Which of the statements above regarding to operations on sets are correct?

I and II

I and III

II and IV

I, II and III

I, II, III and IV

Açıklama:

If all the sets under examination are subsets of some set U then U is called a universal set. Given two sets A and B:

The set of all elements of the sets A and B is called the union of A and B

The set of the elements that the set A and the set B have in common is called the intersection of A and B.

The set of all elements of A which are not in B is called the difference of A and B

The difference of U and A is called the complement of A.

As also understood from the information given, the correct answer is D. The statements in the options I “The set of all elements of the sets A and B is called the union of A and B “, II “The set of the elements that the set A and the set B have in common is called the intersection of A and B.”, and III “The set of all elements of A which are not in B is called the difference of A and B. are correct. The statement in the option IV “The difference of U and A is called the complement of U.” is not correct because of the fact that the difference of U and A is called the complement of A not U.

Doğru Cevap: D

Soru 19

a⁰ = 1 for all nonzero numbers a.

If a = 0, a⁰ is called power of a.

∞ (infinity) is not a number. It represents an infinitely large quantity.

The whole real line is an interval denoted by (-∞, ∞).

Powers are used when we multiply a real number by itself repeatedly.

Which of the statements above regarding to numbers are correct?

I and II

I, II, III and V

I, III, IV and V

II, III, IV and V

I, II, III, IV and V

Açıklama:

a⁰ = 1 for all nonzero numbers a.

If a = 0, a⁰ is called indeterminate.

∞ (infinity) is not a number. It represents an infinitely large quantity.

The whole real line is an interval denoted by (-∞, ∞).

Powers are used when we multiply a real number by itself repeatedly.

As also understood from the information given, the correct answer is C.
The statements in the options
I. a⁰ = 1 for all nonzero numbers a.”,
III. “∞ (infinity) is not a number. It represents an infinitely large quantity.”,
IV. “The whole real line is an interval denoted by (-∞, ∞).” and
V. “Powers are used when we multiply a real number by itself repeatedly.”
are correct.
The statement in the option
II. “If a = 0, a⁰ is called power of a.” is not correct because of the fact that if a = 0, a⁰ is called indeterminate.

Doğru Cevap: C

Soru 20

A ∪ B = (-1, 5)

A ∩ B = (2, 3)

A \ B = (-1, 2]

Let A = (-1, 3), B = (2, 5) be open intervals. Which of the statements above are correct?

I and II

I and III

II and III

I, II and III

Açıklama:

Let A = (-1, 3), B = (2, 5) be open intervals. Find A ∪ B, A ∩ B, and A \ B.
A ∪ B = (-1, 5), A ∩ B = (2, 3), A \ B = (-1, 2]
As also understood from the information given, the correct answer is E. All of the statements in the options are correct.

Doğru Cevap: E

Soru 21

Given the sets A = {a, b, c, d}, B = {a, c, f, g}
and C= {c, d, g, h, i}. Find (A ∪ B) ∩ C.

{a, c, d}

{c, d, g}

{c, d, h}

{g, h, i}

{a, d, g}

Açıklama:

A ∪ B = {a, b, c, d, f, g}.
Intersection of this set with C is
(A ∪ B) ∩ C = {c, d, g}.

Doğru Cevap: B

Soru 22

In a school, the total number of students are 250. The number of the members of the Mathematics Club is 153, and the number of the members of the Sports Club is 110. Given that there are only two clubs in the school, and the number of the students who are neither a member of the Mathematics Club nor the Sports club is 17.
Find the number the students who are a member of both the Sports Club and the Mathematics Club.

Açıklama:

Denote by M and S the set of all students who are members of the Mathematics and Sports clubs, respectively.
Then, s(M ∪ S) = 250-17 = 233.
Using the formula (M ∪ S) = s(M) + s(S) - s(M ∩ S),
we have s(M ∩ T) = 233 = 153 + 110 - x, where x is the required number.
Then we find that x = 153 + 110 - 233.
Thus, x equals to 30.

Doğru Cevap: C

Soru 23

Evaluate (2^{-1 +} 3^-2) / 3^-2

11/2

13/2

15/2

17/2

19/2

Açıklama:

Since
2^-1is 1/2¹ which equals to 1/2;
3^-1is 1/3¹ which equals to 1/3; and 3^-2is 1/3² which equals to 1/9,
what we have in hand is ( (1/2) + (1/3) ) / (1/9). (1/2) + (1/3) equals to 11/18,
and if we divide it by 1/9, this equals to 11/2.

Doğru Cevap: A

Soru 24

Evaluate the product (27^-2 / 9^-2) . 3²

-3

-1

Açıklama:

27^-2 / 9^-2equals to (27/9)^-2 = 3^-2.
When we do the operation 3^-2. 3², we find 3^-2+2 which equals to 3⁰. 3⁰equals to 1.

Doğru Cevap: D

Soru 25

Among the numbers 7/12, 3/5, 11/15, 14/20, and 23/30 which is the greatest number?

7/12

3/5

11/15

14/20

23/30

Açıklama:

Extend the given fractions to the common denominator 60.
Then we find that
7/12 =35/60,
3/5 = 36/60,
11/15 = 44/60,
14/20 = 42/60, and
23/30 = 46/60.
The greatest number among these is, therefore, 23/30.

Doğru Cevap: E

Soru 26

Which of the following is the intersection of intervals (-3, 7), and [4, 9)?

[4, 7]

(4, 7]

[4, 7)

(-3, 9)

[4, 9)

Açıklama:

The interval consisting of the elements of both intervals (-3, 7), and [4, 9) is [4, 7).

Doğru Cevap: C

Soru 27

Given that a < b < c and all a, b, and c are real numbers, which of the following equals to the expression |b - a| + |a - c| ?

b + c

-2a +b + c

a + c

2a - b + c

b - c

Açıklama:

From the definition of the absolute value
|b - a| = b - a since a < b
and
|a - c| = c - a since a < c.
Therefore |b - a| + |a - c| = b - a + c - a = -2a + b + c.

Doğru Cevap: B

Soru 28

Which of the following is an half-opened interval?

(-1, +1)

[0, 5]

(-∞, ∞)

[-7, -2]

(3, 5]

Açıklama:

According to the definition, intervals like (a, b], [a, b) are half-open intervals. (3, 5] is the only one fits to this definition.

Doğru Cevap: E

Soru 29

Let A = [0, 5], B = [1, 7), and C = (2, 8). Find C \ (A ∪ B).

(0, 7)

(0, 8)

(7,8)

[7, 8)

(0, 7]

Açıklama:

Given these intervals, (A ∪ B) is [0, 7).
Then, (2, 8) \ [0, 7) is the required result.
It is [7, 8), and D is the correct answer.

Doğru Cevap: D

Soru 30

Let A = (-∞, ∞), B = (-∞, 0] and C = (-2, 2). Find C \ (A ∪ B).

(0, 2)

[0, 2)

(0, 2]

(-∞, ∞)

Açıklama:

Since A represents the infinite whole real line interval, so does A ∪ B.
Then, C \ (A ∪ B) must be an empty set.

Doğru Cevap: C

Soru 31

A = {2, -3, -4, -5, 6}, B = {1, 3, 5, 7}, C = {1, 2, -2, -3, 3, 5, -5}.
(C \ B) ∩ A = ?

{2, -3, -5}

{2, -3, 6}

{-5, 6, 1}

{2, -3, -4, 6}

{-4, -5}

Açıklama:

x = C \ B = {2, -2, -3, -5} ; x ∩ A = {2, -3, -5}. pg. 4. Correct answer is A.

Doğru Cevap: A

Soru 32

The number of students taking Piano course is 45, the number of students taking Flute course is 22. The number of students taking neither Piano nor Flute courses is 17. The number of students taking both Piano and Flute courses is 15. What is the total number of students in this class ?

Açıklama:

P = {x| x is a student who taking Piano course}
F = {x| x is a student who taking Flute course}
The number of students taking Piano or Flute courses is
s(P ∪ F) = s(P) + s(F) -s(P ∩ F).
s(P ∪ F) = 45 + 22 - 15 = 67 - 15 = 52
Additionally, 17 students are taking neither Piano, nor Flute courses. Therefore, the total number of students is 52 + 17 = 69.
Correct answer is C.

Doğru Cevap: B

Soru 33

(2⁸ + 2⁶) / (4³ + 4⁴) = ?

0.25

Açıklama:

x = (2² (2⁶ + 1)) / (4 (1 + 4³)) = 1. pg. 14. Correct answer is B.

Doğru Cevap: B

Soru 34

(4^-1 - 3^-1)^{- 1} / (4^-1 + 3^-1)^-1 = ?

-1 / 7

-7 / 4

-3 / 7

-7 / 3

-7

Açıklama:

x = ((1 / 4) - (1 / 3))^-1 / ((1 / 4) + (1 / 3))^-1 = -7 . pg. 11. Correct answer is E.

Doğru Cevap: E

Soru 35

(80^1/2 / 25^1/2) . (20)^1/2 = ?

Açıklama:

x = ((16 . 5)^1/2 / (5 . 5)^1/2) . (4 . 5)^1/2 = 8 . pg. 12 . Correct answer is D.

Doğru Cevap: D

Soru 36

Which one of these is the maximum : 2 / 5, 1 / 4, 3 / 8, 3 / 10, 7 / 16 ?

3 / 10

3 / 8

1 / 4

2 / 5

7 / 16

Açıklama:

Let's combine the denominators in common 80
x = max {2 / 5, 1 / 4, 3 / 8, 3 / 10, 7 / 16}
= max {32/80, 20/80, 30/80, 24/80, 35/80}
= 35/80 = 7 / 16.
pg. 12. Correct answer is E.

Doğru Cevap: E

Soru 37

Which one of these is an irrational number : 625^1/2, (-64)^1/3, -(25)^1/2, 0, 27^1/2 ?

625^1/2

-(25)^1/2

(-64)^1/3

27^1/2

Açıklama:

〖625〗^(1⁄2) 〖=(〖25〗^2)〗^(1⁄2)=25 is not an irrational number
-〖64〗^(1⁄3)= - (4^3 )^(1⁄3)= -4 is not an irrational number
-(〖25)〗^(1⁄2)= -(〖5^(2×)〗^(1⁄2))= -5 is not an irrational number
0 is not an irrational number
〖27〗^(1⁄2) 〖=(3^3)〗^(1⁄2)=3^(3⁄2)=3√3 is an irrational number

Correct answer is D.

〖625〗^(1⁄2) 〖=(〖25〗^2)〗^(1⁄2)=25 is not an irrational number
-〖64〗^(1⁄3)= - (4^3 )^(1⁄3)= -4 is not an irrational number
-(〖25)〗^(1⁄2)= -(〖5^(2×)〗^(1⁄2))= -5 is not an irrational number
0 is not an irrational number
〖27〗^(1⁄2) 〖=(3^3)〗^(1⁄2)=3^(3⁄2)=3√3 is an irrational number

Correct answer is D.

Doğru Cevap: D

Soru 38

A = (-4, 6] and B = [3, 8)
Find A ∪ B.

(-4, 3]

(-4, 8)

[-4, 3]

[-4, 8)

[6, 8)

Açıklama:

A = (-4, 6] = {x| x ∈ R and -4 < x ≤ 6},
B = [3, 8) = {x| x ∈ R and 3 ≤ x < 8},
A ∪ B = {x| x ∈ A or x ∉ B}. Then A ∪ B= {x| -4 < x < 8 } = (-4, 8)
Correct answer is B.

Doğru Cevap: B

Soru 39

The sets A = {a, b, c, d, e}, B = {b, c, e, f, g}, C = {c, d, f, g} are given. Which of the following is the set {c} ?

(A ∪ C) \ B

(B ∩ C) \ A

A ∩ B ∩ C

A \ (C ∩ B)

B ∩ A

Açıklama:

The corresponding diagram is

A ∩ B ∩ C = {c}
Correct answer is C

Doğru Cevap: C

Soru 40

a, b, c : real numbers ; c > a > b. |b - c| + |a - b| + |c - a| = ?

2c - 2b

2a - 2c

2c - 2a

2b - 2c

Açıklama:

c > b. then |b - c| = -b + c
a > b then |a - b| = a - b
c > a then |c - a| = c - a
Then |b - c| + |a - b| + |c - a| = -b + c + a - b + c - a
= -2b + 2c
= 2c - 2b
Answer is A.

Doğru Cevap: A

Soru 41

Which of the following is not a subset of the set A={0,1,2,3,a,b}?

{0,1,2,3,a,b}

Empty set

{0,1,2,3}

{a,b}

{0,4,a,b}

Açıklama:

Since "4" is not an element of set A, the set {0,4,a,b} is not a subset of set A.

Doğru Cevap: E

Soru 42

Which of the following statements is false?

If A is subset of B and B is subset of A, then A=B.

Empty set is subset of any set.

A set is subset of itself.

If A is subset of B and C is subset of B, then intersection of A and C can not be empty set.

If A is subset of B and B is subset of C, then A is subset of C.

Açıklama:

Assume that B={1, 3, 5, 7}, A={1, 3} and C={5, 7}. Both A and C are subsets of set B, but their intersection is empty set. All the other statements are true.

Doğru Cevap: D

Soru 43

Assume that
A={x| 0}
and
B={x| 0}.
What is the intersection of A and B?

{6}

Empty set

{6,12,18}

{2,3}

{2,4,6,12,18}

Açıklama:

Since x is both divisible by 2 and 3, it must be divisible to 6 for intersection. Then the intersection set contains 6, 12 and 18.

Doğru Cevap: C

Soru 44

Which of the following defines the shaded area below?

[A \ (B∪C)] ∪ [(B∩C) \ A]

[A \ (B∪C)] ∪ [(B∪C)\A]

[A \ (B∪C) ] ∪ (B∩C)

[A ∩ (B∪C) ] \ (B

A ∩ [(B∩C)\A]

Açıklama:

[A \ (B∪C)]

[(B∩C) \ A]

The shaded are is defined in A.

Doğru Cevap: A

Soru 45

Given that A={0,1,2,3,a,b}, how many subsets of A contains both letters ("a" and "b")?

Açıklama:

Let's combine a and b to single element that we call *.
Then our set will become A={0,1,2,3,*}.
Since this set has 5 elements it has 32 subsets (2⁵=32).
Out of these 32 sets, there are 16 sets that doesn't contain *. (If we delete *, there remains a set containing 4 elements which has 2⁴=16 subsets).
A subset will either contain * or it will not contain *.
Thus 32-16=16 sets contain *, namely a and b.

Doğru Cevap: C

Soru 46

Which of the following statements is true?

Every real number is a rational number also.

Every integer is a rational number.

Every integer is a natural number.

Every rational number is an integer.

Every rational number is a natural number.

Açıklama:

√2 is a real number but it is not rational (It is impossible to write √ 2 as a/b where a and b are integers), so A is false.
-3 is an integer but it is not a natural number, so C is false also.
5/7 is a rational number but it is not an integer so D is false.
5/7 is rational but it is not a natural number so E is false also.
Only statement in B is true.

Doğru Cevap: B

Soru 47

Which of the following rational numbers is the biggest?

700/701

701/702

702/703

703/704

704/705

Açıklama:

The difference between the numerator and denominator is 1 in all given numbers. In this case the value of the ratio increases as the numerator (or denominator increases) For a simple example think of 1/2 and 2/3. 1/2=0.5 and 2/3=0.67. Thus the answer is E.

Doğru Cevap: E

Soru 48

What is the intersection of the intervals (-∞, 4] and (2, ∞)

Empty set

(2, 4]

(2, 4)

[2, 4)

(-∞, ∞)

Açıklama:

The intersection interval is (2, 4] because 2 is open in the second interval and 4 is closed in both intervals.

Doğru Cevap: B

Soru 49

Evaluate (3²-3^-2)/(3²+3^-2)

40/41

41/40

28/26

26/28

9/8

Açıklama:

3²=9 3^-2=1/9
So, (3²-3^-2)/(3²+3^-2)=(9-1/9)/(9+1/9)=(80/9)/(82/9)=80/82=40/41

Doğru Cevap: A

Soru 50

64^2/3=?

Açıklama:

64=2⁶ so 64^2/3=(2⁶)^2/3=2^6*2/3=2⁴=16

Doğru Cevap: D

Soru 51

Given the sets A={2, 4, 6, 7}, B={1, 2, 3, 4} and C={8, 7,4}. Find the (AUBUC).

{1,2,3,4,6,8}

{1,2,3,4,6,7,8}

{3,4,6,7,8}

{1,2,3,7,8}

{2,3,4,6,7}

Açıklama:

A={2, 4, 6, 7}, B={1, 2, 3, 4} and C={8, 7,4}. AUB={1,2,3,4,6,7} and AUBUC={1,2,3,4,6,7,8}. The answer is B.

Doğru Cevap: B

Soru 52

What is the value of

Açıklama:

The answer is D.

Doğru Cevap: D

Soru 53

Evaluate

15/37

27/64

-36/27

-27/64

-64/27

Açıklama:

The answer is E.

Doğru Cevap: E

Soru 54

What is the product

Açıklama:

The answer is A.

Doğru Cevap: A

Soru 55

Among the numbers 3, -3/5, 7/2 and 4/5 which is the greatest number?

4/5

3/5

7/2

-3/5

Açıklama:

Extend the given fractions to the common denominator 10.

The greatest fraction is 7/2. The answer is D.

Doğru Cevap: D

Soru 56

Which of the following is the intersection of intervals [-3, 5) and [-2, 0)?

[-2, 0]

[-2, 0)

[-5, 0)

[-2, 1)

(-2, 0)

Açıklama:

The interval consisting of the elements of both [-3, 5) and [-2, 0) is [-2, 0). The answer is B.

Doğru Cevap: B

Soru 57

What is the value of

3/5

1/5

6/5

7/5

Açıklama:

. The answer is C.

Doğru Cevap: C

Soru 58

Which of the following is the intersection of intervals (-1/3, 4] and [-1, 7/2)?

(-1/3, 7/2)

[-1, 7/2)

[-1, 4)

[0, 3)

[-1/3, 7/2)

Açıklama:

The interval consisting of the elements of both intervals (-1/3, 4] and [-1, 7/2) is (-1/3, 7/2). The answer is A.

Doğru Cevap: A

Soru 59

Given the sets A={a, 1, d, -3}, B={e, -3, 0} and C={a, d, e, 0}. Find (AUC)\B.

{d, 1}

{a, d, e, 1}

{a, d, 1}

{a, d, 1, -3}

{a, d, 1, e}

Açıklama:

A={a, 1, d, -3}, B={e, -3, 0} and C={a, d, e, 0}.(AUC)={a, d, e, 0, 1, -3}. The set of elements AUC which are not in B is {a, d, 1}. Therefore (AUC)\B={a, d, 1}. The answer is C.

Doğru Cevap: C

Soru 60

What is the value

51/5

49/5

Açıklama:

The answer is B.

Doğru Cevap: B

Soru 61

Given the sets A={1,3,4,9,12}, B={3,4,7,10} and C={3,9,10}. Find(A∪B)\C.

{1,3,5,8,9}

{1,3,9,12}

{1,4,7,12}

{3,4,7,10}

{1,3,4,9,12}

Açıklama:

A ∪ B = {1,3,4,7,9,10,12} The set of elements A ∪ B which are not in C is {1,4,7,12}. Therefore (A ∪ B) \ C = {1,4,7,12}.

Doğru Cevap: C

Soru 62

The total number of students in a class is 45, the number of students passing Mathematics test is 35, the number of students passing Turkish test is 40. If the number of students passing neither Mathematics nor Turkish is 5, find the number of students passing both Mathematics and Turkish tests.

Açıklama:

Since the total number of students is 45 then s(M ∪ T) = 45 - 5 = 40.
Using the formula (M ∪ T) = s(M) + s(T) - s(M ∩ T),
we have s(M ∩ T) = 35+ 40 - 40 = 35 which is the required number.
Correct answer is 35.

Doğru Cevap: E

Soru 63

Wich one of the following describes green area?

(A ∪ B) ∩ (A ∪ C)

(A ∪ B) / C

(A ∪ B) ∩ C

(A ∪ C) ∩ B

(B ∪ C) ∩ (A ∪ C)

Açıklama:

Venn diagram showing A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Correct answer is A.

Doğru Cevap: A

Soru 64

In a touristic group, 10 tourists can speak English, 15 tourists can speak German, 9 tourists can speak both English and German, and 3 tourists can speak neither English nor German. Find the number of tourists in this group.

Açıklama:

E ∪ G = {x| x is a tourist who can speak English or German}
E ∩ G = {x| x is a tourist who can speak both English and German}
Then
s(E ∪ G) = s(E) + s(G) - s(E ∩ G)= 10 + 15 - 9 = 16.
Additionally, 3 tourists can speak neither English, nor German. Therefore, the total number of tourists is 16 + 3 = 19.
Correct answer is D.

Doğru Cevap: D

Soru 65

Wich one of the following irregular number?

-16

Açıklama:

A real number which is not a rational is an irrational number
Correct answer is B.

Doğru Cevap: B

Soru 66

A = [1, 3], B = [2, 5). Find A \ B.

[1, 3)

[1, 2]

[1, 2)

(1, 2)

(1, 2]

Açıklama:

A = [1, 3] = {x| x ∈ R and 1 ≤ x ≤ 3},
B = [2, 5) = {x| x ∈ R and 2 ≤ x < 5},
A \ B = {x| x ∈ A and x ∉ B}. Then A \ B = {x| 1 ≤ x < 2 } = [1, 2)
Correct answer is C.

Doğru Cevap: C

Soru 67

Wich one of the following numbers is greater?

Açıklama:

6/23= (6 x 12)/(23 x 12)= 72/276
9/32= (9 x 8)/(32 x 8)= 72/256
3/13= (3 x 24)/(13 x 24)= 72/312
2/5= (2 x 36)/(5 x 36)= 72/180
24/99= (24 x 3)/(99 x 3)= 72/297
Then, the greatest number is 2/5= (2 x 36)/(5 x 36)= 72/180

Correct answer is D.

6/23= (6 x 12)/(23 x 12)= 72/276
9/32= (9 x 8)/(32 x 8)= 72/256
3/13= (3 x 24)/(13 x 24)= 72/312
2/5= (2 x 36)/(5 x 36)= 72/180
24/99= (24 x 3)/(99 x 3)= 72/297
Then, the greatest number is 2/5= (2 x 36)/(5 x 36)= 72/180

Correct answer is D.

Doğru Cevap: D

Soru 68

Write in increasing order.

Açıklama:

a= -3/4= -9/12 and b= - 2/3= - 8/12 then aOn the other hand c = 2/3 and 0Then a
Correct answer is A.

Doğru Cevap: C

Soru 69

100

Açıklama:

4 x 25= 100
Correct answer is E.

Doğru Cevap: E

Soru 70

Which of the following is the intersection of intervals [-5, 7], and (3, 9]?

(3, 7]

(3, 7)

(3, 5]

[-5, 9]

[3, 7)

Açıklama:

The interval consisting of the elements of both intervals [-5, 7] and (3, 9] is (3, 7]. Correct answers is A.

Doğru Cevap: A

Soru 71

Given sets A = {1, 5, 7, 9}, B = {2, 5, 7}. Find the difference A \ B?

{1, 9}

{2, 9}

{5, 7}

{1, 2, 5}

{5, 7, 9}

Açıklama:

A = {1, 5, 7, 9}, B = {2, 5, 7}.
A \ B = {x| x ∈ A and x ∉ B} then A \ B = {1, 9}.

Doğru Cevap: A

Soru 72

There are 22 students in a class. 16 of them has blue pens. 18 of them has red pens. How many of them has both blue and red pens?

Açıklama:

s(B)=16
s(R)=18
s(B ∪ R)=22
s(B ∩ R) = s(B) + s(R) - s(B ∪ R)=16+18-22=34-22=12

Doğru Cevap: C

Soru 73

E = {0, 1, 2, …, 10}, A = {1, 2, 3}, B = {3, 4, 5}. Which one is equal to {4,5}?

A ∪ B

A ∩ B

A \ B

B \ A

A^c

Açıklama:

A ∪ B = {1, 2, 3, 4, 5},
A ∩ B = {3},
A \ B = {1, 2},
B \ A = {4,5},
A^c = {5, 6, 7, 8, 9, 10},
B^c = {1, 2, 6, 7, 8, 9, 10}.
correct answer is D.

Doğru Cevap: D

Soru 74

Let A = {a, b, c, d}. Find the number of all nonempty subsets of A?

Açıklama:

n = s(A) = 4, 2ⁿ = 2⁴ = 16.
A has only one empty subset Ø, therefore the number of nonempty subsets is 16 - 1 = 15.

Doğru Cevap: C

Soru 75

For given sets A and B assume that s(A) = 16, s(B) = 20 and s(A ∪ B) = 26. Find s(B \ A).

Açıklama:

s(A ∩ B) = s(A) + s(B) - s(A ∪ B)
= 16 + 20 - 26
= 10
The corresponding diagram is

where the numbers 6, 10 and 10 indicate the numbers of elements in the corresponding subsets.
Therefore s(B \ A) = 10.

Doğru Cevap: B

Soru 76

Which one is one of the infinite representation of 16/20?

1/2

2/3

3/4

4/5

5/6

Açıklama:

m/n= p/q => m . q = n . p
Therefore, every rational number has an infinite number of representations.
4/5 = 8/10 = 12/15 = 16/20 = ...

Doğru Cevap: D

Soru 77

Which one is an irrational number?

3/4

0.5

Açıklama:

A real number which is not a rational is an irrational number. π=3.14 ... in an irrational numbers.

Doğru Cevap: D

Soru 78

Write a = -3, b = -1/2, c = - 5/3, d = 3/2 in increasing order?

a < b < c < d.

a < c < b < d.

b < a < c < d.

c < a < d < b.

d < b < c < a.

Açıklama:

a=-3, c=-1,6666..., b=-0.5, d=1.5 Then,
a < c < b < d.

Doğru Cevap: B

Soru 79

Simplify the following number 16^-1/4?

1/2

Açıklama:

Doğru Cevap: B

Soru 80

Given intervals A = [-1, 3], B = [0, 5). Find A \ B?

[-1, 5)

[0, 3]

(0, 3)

[-1, 0)

(3, 5)

Açıklama:

A \ B = [-1, 0)

Doğru Cevap: D

Ünite 2

Soru 1

f : R → R, f(x) = 5 x³ + x - 2. f(-1) = ?

-8

-5

-2

Açıklama:

y = -5 - 1 - 2 = -8 . Correct answer is A.

Doğru Cevap: A

Soru 2

f : R → R, f(x) = 3 x² - 5. f ^-1(x) = ?

(3 x² - 5)^-1

(-3 x² + 5)^-1

((x + 5) / 3)^1/2

((-x + 5) / 3)^-1/2

(-3 x² + 5)^-1/2

Açıklama:

x = ((y + 5) / 3)^1/2 . Correct answer is C.

Doğru Cevap: C

Soru 3

f : R → R, f(x) = 2 + 3 x³ ; g : R → R , g(x) = -x + 1. (f ∘ g)(-1) = ?

-1

-3

Açıklama:

y = 2 + 3 (-x +1)³ = 2 + 3 (1 + 1)³ = 26 . Correct answer is B.

Doğru Cevap: B

Soru 4

f : R → R, f(x) = 2 + 3 x³ ; g : R → R , g(x) = -x + 1. (g ∘ f)(-1) = ?

-1

-3

Açıklama:

y = -(2 + 3 x³) + 1 = 2 . Correct answer is E.

Doğru Cevap: E

Soru 5

f : R → R, f(x) = 2 + 3 x³ ; g : R → R , g(x) = -x + 1. (f . g)(-1) = ?

-5

-2

Açıklama:

y = -2 x + 2 - 3 x⁴ + 3 x³ = -2 . Correct answer is D.

Doğru Cevap: D

Soru 6

f : R → R, f(x) = x³ + 2. f ^-1(1) = ?

3^1/3

1 / 3

-1

Açıklama:

x = (y - 2)^1/3 = -1 . Correct answer is E.

Doğru Cevap: E

Soru 7

f : R → R, f(x) = 2 + 3 x³ ; g : R → R , g(x) = -x + 1. (f + g)(-1) = ?

-1

Açıklama:

x = 2 + 3 x³ - x + 1 = 1 . Correct answer is D.

Doğru Cevap: D

Soru 8

f : R → R, f(x) = (64 - 8x)^1/3 ; what is the largest domain of x = ?

(-∞, 8]

(-∞, ∞)

[4, ∞)

[8, ∞)

[4, 8]

Açıklama:

x = 64 - 8 x ≥ 0 ; 8 ≥ x . Correct answer is A.

Doğru Cevap: A

Soru 9

f : R → R, f(x) = -x + 3, if x ≤ 4 ; f(x) = -1, if x > 4. 3 f(-2) + 4 f(16) = ?

-1

-6

Açıklama:

y = 3 (-(-2) + 3) + 4 (-1) = 15 - 4 = 11. Correct answer is A.

Doğru Cevap: A

Soru 10

How many different inverse functions can be defined from a set of 10 elements to a set of 2 elements ?

Açıklama:

Constant functions map every element from their domain to the same element of the range. We may therefore define a constant function for every element in the range. Since there are 10 different elements in the range we may write 10 different constant functions and inverse functions. Correct answer is D.

Doğru Cevap: D

Soru 11

Given the function f: R→R, f(x)=7x²+12, what is the value of f(2)?

Açıklama:

f(x)=7x²+12, f(2)=7.2²+12 ,then f(2)=7.4+12=40

Doğru Cevap: C

Soru 12

For the functions f: R→R, f(x)=3x-8 and g: R→R, g(x)=(x+12)/4, what is the value of f(3).g(8)?

-4

-16

Açıklama:

f(x)=3x-8 → f(3)=3.3-8=1
g(x)=(x+12)/4 → g(8)=(8+12)/4=20/4=5
f(3).g(8)=1.5=5

Doğru Cevap: B

Soru 13

Given the functions f: R→R, f(x)=x-5 and g: R→R, g(x)=2x+9, calculate the value of (fog)(-1)?

-3

-1

Açıklama:

g(x)=2x+9 → g(-1)=-1.2+9=7
f(x)=x-5 →f(7)=7-5=2 (we used the output of the function g(x), as the input of the function f(x))

Doğru Cevap: E

Soru 14

Let the functions f: R→R, f(x)=2x+7 and g: R→R, g(x)=3x-2, what is the value of (gof)(x)?

6x-3

6x+3

6x-19

6x+19

Açıklama:

f(x)=2x+7 and g(x)=3x-2 →(gof)(x)=3.(2x+7)-2=6x+21-2=6x+19

Doğru Cevap: E

Soru 15

Let the functions m: R→R, m(x)=(3/x)+7 and k: R→R, k(x)=3/(x-2), what is the value of m(k(1))?

-3

-6

Açıklama:

k(x)=3/(x-2)→k(1)=3/(1-2)=-3
m(x)=(3/x)+7→m(-3)=(3/-3)+7=6
m(k(1))=(mok)(1)=6

Doğru Cevap: B

Soru 16

The function f: R→R is given by,

What is the value of 2f(6)-3f(3)+6f(-1)?

-5

-1

Açıklama:

f(6)=2-6=-4 hence 6>5
f(3)=3.3=9 hence 1<3<5
f(-1)=(-1)²+4=5 hence -1<1
so, 2f(6)-3f(3)+6f(-1)=2.(-4)-3.9+6.5=-8-27+30=-5

Doğru Cevap: A

Soru 17

What is the inverse of the function f(x)=6x-5 ?

6x-5

(2x-5)/12

(2x+5)/12

(x-5)/6

(x+5)/6

Açıklama:

6x-5=y→(y+5)/6=x→f^-1(x)=(x+5)/6

Doğru Cevap: E

Soru 18

Let f: R→R, f(x)=7x+1 and g: R→R, g(x)=x²-1 be given. What is the value of f/g(2)?

14/3

15/2

Açıklama:

f(2)=7.2+1=15
g(2)=2²-1=4-1=3
f/g(2)=15/3=5

Doğru Cevap: C

Soru 19

For the functions f: R→R, f(x)=x-1 and g: R→R, g(x)=(x+2)/4, what is the value of (f.g)(3)?

-3/2

3/2

5/2

Açıklama:

f(3)=3-1=2 and g(3)=(3+2)/4=5/4
(f.g)(3)=f(3).g(3)→2.5/4=5/2

Doğru Cevap: D

Soru 20

According to the graph of f(x) is given above, what is the value of f(-4)+3f(5)-2f(3) ?

Açıklama:

By using the graph which is given in the question;
f(-5)=1, f(-4)=8, f(3)=-6 and f(5)=6 then,
f(-4)+3f(5)-2f(3)=8+(3.6)-(2.-6)=8+18-(-12)=38

Doğru Cevap: B

Soru 21

Let f : R → R, f (x) = 5x² +10x + 7 given. What is the value of f (-1)?

Açıklama:

Substituting x for -1, we find 5.(-1²) + 10.(-1) + 7 = 5 . 1 + (-10) + 7 = 5 -10 + 7 = 2.

Doğru Cevap: B

Soru 22

Given that f : R→ R, f (x) = 3x + 5 which of the following equals to f^-1(2)?

Açıklama:

It is clear that this function has an inverse. Hence, for finding f^-1(x), we write y = 3x + 5. What we find is x = (5 - y) / 3. Then, f^-1(x) = (5 - y) / 3. Substituting 2 with x, we find 1.

Doğru Cevap: A

Soru 23

For the functions f : R→ R , f (x) = 3x - 7 and g : R→ R , g(x) = -x + 3, what is the value of the composition (f ∘ g)(2)=?

-1

-2

-3

-4

-5

Açıklama:

(f ∘ g)(2) equals to f ( g(2) ). g(2), substituting 2 with x in the g function is -2 +3 = 1. Then, (f ∘ g)(2) = f (1) = 3 . 1 -7 = 3 - 7 = -4.

Doğru Cevap: D

Soru 24

For the functions f : R→ R , f (x) = x² -5 and g : R→ R , g(x) = x³ - 2, what is the value of the composition (g ∘f)(-1)=?

-22

-44

-66

-88

-111

Açıklama:

(g ∘f)(-1) = g ( f(-1) ) = g (-4) = -66

Doğru Cevap: C

Soru 25

What is the largest domain of the function f(x) = (2x - 4) / (2x - 10)?

R \ {0, 2, 5}

R \ {2, 5}

R \ {2}

R \ {5}

Açıklama:

This function makes sense if its denominator is not zero, i.e. the condition of 2x - 10 ≠ 0 should be satisfied. Hence, if x ≠ 5, this condition is satisfied. Therefore, the largest domain of the function is R \ {5}.

Doğru Cevap: E

Soru 26

How many different constant functions can be defined from a set of 5 elements to a set of 9 elements?

5⁹

9⁵

Açıklama:

Constant functions map every element from their domain to the same element of the range. We may therefore define a constant function for every element in the range. Since there are 9 different elements in the range we may write 9 different constant functions.

Doğru Cevap: B

Soru 27

A correspondence assigning each element of the set A, one and only one element of the set B is called a function from the set A into the set B.
In this context, the set A is called the domain or the departure set of the function.
In this context, the set B is called the range or terminal set.
For a given function f and an element a from its domain A, the element corresponding to a under the rule f is called the image of the element a, and is denoted by f (a).
A function defined from the set A to the set B is denoted by or

Let A and B be two sets different from the empty set. Which of the statements above regarding to functions are correct?

I and II

I, III and IV

II, IV and V

I, III, IV and V

I, II, III, IV and V

Açıklama:

Let A and B be two sets different from the empty set. A correspondence assigning each element of the set A, one and only one element of the set B is called a function from the set A into the set B. Functions are generally denoted by the lower case letters such as f, g, h. In this case a function defined from the set A to the set B is denoted by
f:

.
In this context, the set A is called the domain or the departure set of the function, and the set B is called the range or terminal set. For a given function f and an element a from its domain A, the element corresponding to a under the rule f is called the image of the element a, and is denoted by f (a).
As also understood from the information given, all of the statements regarding to functions in the options are correct, so the correct answer is E.

Doğru Cevap: E

Soru 28

If we denote this real number by x, and the function by f we may express it as
It is customary to also denote this function by y = f (x).
In the expression y = f (x), x is called the independent variable.
In the expression y = f (x), y is called the dependent variable.
In the equation y = f (x)=3x+4, the dependent variable y is a function of the independent variable x.

Consider the function that assigns to each real number three times itself and four. Which of the statements above is/are correct?

I and II

I and IV

I, III and IV

I, II, III and IV

Açıklama:

Consider the function that assigns to each real number three times itself and four. If we denote this real number by x, and the function by f we may express it as

It is customary to also denote this function by y = f (x).
In the expression y = f (x), x is called the independent variable, whereas y is called the dependent variable.
In the expression y is called the dependent variable.
In the equation y = f (x)=3x+4, the dependent variable y is a function of the independent variable x.
As also understood from the information given, the correct answer is E. All of the statements in the options are correct.

Doğru Cevap: E

Soru 29

Let

be given. What is the value of f (3)?

-11

Açıklama:

Substituting 3 for x we find f (3) = 5.3 - 4 = 15 - 4 = 11, so the correct answer is A.

Doğru Cevap: A

Soru 30

Given that

which of the following is f ^-1(x)?

3x - 6

6x+3

3x-6

Açıklama:

From the equation 3x+6=y we find x = f ^-1(y) =

. Thus f ^-1(x) =

, so the correct answer is B.

Doğru Cevap: B

Soru 31

If a rule defining a function is given but the domain has not been specified explicitly, then the largest set which makes the rule meaningful is understood.
This set is denoted by D_ffor the rule y=f (x) and is called the natural domain of the function.
The function given by the rule f(x) =makes sense if its denominator is not zero.
For the function given by the rule f(x) =, x-1≠0 should be satisfied.

I and III

II and III

I, III and IV

I, II, III and IV

Açıklama:

If a rule defining a function is given but the domain has not been specified explicitly, then the largest set which makes the rule meaningful is understood.
This set is denoted by D_f for the rule y=f (x) and is called the natural domain of the function.
The function given by the rule f(x) =

makes sense if its denominator is not zero, i.e. x-1≠0 should be satisfied.
As also understood from the information given, the correct answer is E.

Doğru Cevap: E

Soru 32

Consider the function f:

. For every x₁, x₂A if x₁ ≠ x₂implies
f(x₁) ≠ f(x₂) then the function f is called …

one-to-one

surjective

bijective

constant

identity

Açıklama:

Consider the function f:

.
For every x₁, x₂A if x₁ ≠ x₂implies
f(x₁) ≠ f(x₂)
then the function f is called one-to-one (or injective), so the correct answer is A.
Although the range and the image of a function are two concepts that need to be distinguished, we still know that they are closely related. The image is always a subset of the range. Yet, there are functions for which the range and the image sets are exactly the same. Thus, given a function f :

if the image is equal to the range, i.e. f (A)=B the function f is called surjective (onto). Equivalently, for every element b of the set B if there can be found an element a of A such that f (a)=b, then f is called onto.
If a function which is both one-to-one and onto is called a bijection.
Constant function and identity function are the concepts related to types of functions.
A function assigning each element from its domain a single element in its range is called a constant function.
Given A ≠

, a function defined on the set A and assigning every element of A to itself is called the identity function.

Doğru Cevap: A

Soru 33

Consider the function . For every x₁, x₂A if x₁ ≠ x₂implies f(x₁) ≠ f(x₂) then the function f is called one-to-one (or injective).
Given a function if the image is equal to the range, i.e. f (A)=B the function f is called surjective (onto).
If a function which is both one-to-one and onto is called a bijection.
Constant function and identity function are the concepts related to types of functions. A function assigning each element from its domain a single element in its range is called a constant function.
Given A ≠, a function defined on the set A and assigning every element of A to itself is called the identity function.

Which of the statements above regarding to properties and types of functions is/are correct?

III

II and III

I, III and IV

I, II, III and V

I, II, III, IV and V

Açıklama:

Consider the function

. For every x₁, x₂A if x₁ ≠ x₂implies
f(x₁) ≠ f(x₂) then the function f is called one-to-one (or injective).
Given a function

if the image is equal to the range, i.e. f (A)=B the function f is called surjective (onto).
If a function which is both one-to-one and onto is called a bijection.
Constant function and identity function are the concepts related to types of functions. A function assigning each element from its domain a single element in its range is called a constant function.
Given A ≠

a function defined on the set A and assigning every element of A to itself is called the identity function.
As also understood from the information all of the statements regarding to properties and types of functions in the options are correct, so the correct answer is E.

Doğru Cevap: E

Soru 34

Functions, which are represented by different formulas on different subsets of its domain are called … functions.

constant

identity

piecewise defined

surjective

bijective

Açıklama:

Functions, which are represented by different formulas on different subsets of its domain are called piecewise defined functions, so the correct answer is C. Definitions for the other concepts regarding to functions in the other options are as follows:
Constant function and identity function are the concepts related to types of functions. A function assigning each element from its domain a single element in its range is called a constant function.
Given A ≠

, a function defined on the set A and assigning every element of A to itself is called the identity function.
Given a function

if the image is equal to the range, i.e. f (A)=B the function f is called surjective (onto).
If a function which is both one-to-one and onto is called a bijection.

Doğru Cevap: C

Soru 35

Given that x is a real number and f is given by the rule

, which of the following sets is the largest domain of definition of f ?

(-∞, 3]

(6, ∞)

(9, 3)

(1, 6)

(1, ∞)

Açıklama:

Recommended Revision
Page 44
The question 8 on the page 44 (-∞,2]
The answer for the question 8 on the page 45D_f=[-∞, 2).
The inequality

should be satisfied. Hence, D_f = (-∞, 3], so the correct answer is A.

Doğru Cevap: A

Soru 36

Two real lines intersecting perpendicularly at both their zeros constitute the Cartesian coordinate system.
The Cartesian coordinate system is a tool which enables us to view the graphs of functions.
In the Cartesian coordinate system, the horizontal real line is called the x-axis, or abscissa,
In the Cartesian coordinate system, the vertical real line is called the y-axis, or ordinate.
We call the point of intersection of the number lines as the origin in the Cartesian coordinate system.

Which of the statements above regarding to graphs of functions is/are correct?

I and III

II, IV and V

I, II, III and V

I, II, III, IV and V

Açıklama:

Now is the time to plot the graph of functions in order to make them visible to the eye. We will talk about the Cartesian coordinate system, a tool which enables us to view the graphs of functions.
Let us get acquainted with the Cartesian coordinate system. Two real lines intersecting perpendicularly at both their zeros constitute the Cartesian coordinate system. In this system, the horizontal real line is called the x-axis, or abscissa, and the vertical real line is called the y-axis, or ordinate. We call the point of intersection of the number lines as the origin.
As also understood from the information given, the correct answer is E. All of the statements regarding to graphs of functions in the options are correct.

Doğru Cevap: E

Soru 37

Given the functions f : R → R , f (x)= x-2 and g : R → R , g (x)= 2x-9, determine (4f -2g)(x).

Açıklama:

(4f -2g)(x) = 4(x-2) - 2(2x-9) = 4x - 8 - 4x + 18 = 10

Doğru Cevap: E

Soru 38

Given that x is a real number and f is given by the rule f (x) = (2x - 12)^1/2 , which of the following sets is the largest domain of the definition of f ?

(-∞, -1]

(-∞, -6]

(-∞, 6]

[6, ∞)

[1, ∞)

Açıklama:

In order for the square root to make sense, the argument (the terms inside the square root) should not be negative. (2x - 12)^1/2is the square root of 2x - 12. Therefore, the inequality 2x-12 ≥ 0 should be satisfied. This means that x ≥ 6 must be satisfied. Hence, the largest domain of definition of f is [6, ∞).

Doğru Cevap: D

Soru 39

Let f : R → R , f (x)= 2x + 6 and g : R → R g(x)=x²+3 be given. Calculate f / g (-1).

Açıklama:

Since g cannot be zero, f / g is meaningful, and the answer can be calculated.
The answer is, then, (-2+6) / ((-1)²+3) = 4 / 4 ) = 1

Doğru Cevap: A

Soru 40

The functions f : R→ R , f (x) = 2x + 9 and g : R→ R , g(x) = 12 - 3x are given. What is the value of (4f - 3g)(2)?

Açıklama:

4(4 + 9) - 3(12-6) = 52 - 18 = 34

Doğru Cevap: C

Soru 41

Let f:R→R f(x)=x³-x²+5x-12 be given. What is the value of f(5)?

102

113

127

Açıklama:

f(x)=x³-x²+5x-12 if we put 5 instead of x then,
f(5)=5³-5²+25-12=125-25+25-12=113

Doğru Cevap: D

Soru 42

Given that f : R→R, f (x) = 7x - 3 which of the following is f ^-1(x) ?

(x-3)/7

(x+3)/7

7x+3

7x-3

(x+7)/3

Açıklama:

f (x) = 7x - 3 and f ^-1(x)=(x+3)/7

Doğru Cevap: B

Soru 43

How many different functions can be defined from a set of 8 elements to a set of 11 elements?

Açıklama:

Constant functions map every element from their domain to the same element of the range. We may therefore define a constant function for every element in the range. Since there are 11 different elements in the range we may write 11 different constant functions.

Doğru Cevap: C

Soru 44

Let f :R→R, f (x) = 9x - 1. Which of the following is the value of f ^-1 (16)?

11/5

17/9

21/3

Açıklama:

f ^-1 (x)=(x+1)/9→f ^-1 (16)=16+1/9=17/9

Doğru Cevap: D

Soru 45

For the functions f :R→R, f (x) = 5x³ and g :R→R, g(x) = x - 1, what is the value of the composition (f ∘g)(1)=?

125

625

Açıklama:

(f ∘g)(1)=f(g(1)) hence g(1)=0 f(0)=5.(0)³=0

Doğru Cevap: A

Soru 46

Given the functions f :R→R, f (x) = 1 - x and g :R→R, g(x) = x - 2 find (gof)(7)=?

-2

-8

Açıklama:

(gof)(x)=g(f(x)) then (gof)(7)=g(f(7)) by the way f(7)=1-7=-6 (gof)(7)=g(-6)=-6-2=-8

Doğru Cevap: E

Soru 47

Let f :R→R, f (x) = 1 + 3x and g :R→R, g(x) = x +12 be given. What is the value of (f+g)(1) =?

Açıklama:

(f+g)(1)=f(1)+g(1)=1+3.1+1+12=17

Doğru Cevap: E

Soru 48

Let f :R→R, f (x) = x^-1 and g :R→R, g(x) = 2x -1 be given. What is the value of (f.g)(3) =?

-3

5/3

7/3

-8/3

Açıklama:

(f.g)(3) =f(3).g(3)=3^-1.2.3-1=3^-1.5=5/3

Doğru Cevap: C

Soru 49

The graph of f :R→R is given below. According to this, what is the value of 2f (0) ?

-2

-4

Açıklama:

for x=0 y=-2 then 2f(0)=2.-2=-4

Doğru Cevap: D

Soru 50

The graph of f :R→R is given above. According to this, what is the value of 2f (5/2)-f (-3) ?

-20

-10

Açıklama:

From the graph we see that f(5/2)=-6 and f(-3)=8, so 2f (5/2)-f (-3)=(2.-6)-8=-20

Doğru Cevap: A

Soru 51

Assume that the function f(x)=x²-1 is defined over the domain set A={1, 2, 3, 4}. Which of the following is the image set of function f?

{1, 2, 3, 4}

{0, 1, 2, 3}

{0, 3, 8, 15}

{2, 5, 10, 17}

{1, 4, 9, 16}

Açıklama:

Since f(x)=x²-1, let's find the image of each element of set A.
1²-1=0
2²-1=3
3²-1=8
4²-1=15
Thus, the image set is {0, 3, 8, 15}

Doğru Cevap: C

Soru 52

For which of the functions below 1 is element of the natural domain?

f(x)=x²/(x²-1)

f(x)=(2+x³)/(x³-1)

f(x)=(1+5x+x²)/(x²+2x-3)

f(x)=(x²-1)/(x²-x-2)

f(x)=(x²+4x-1)/(x²+4x-5)

Açıklama:

The denominator for the functions in A, B, C and E equals to zero when x=1. Since a/0 is undefined 1 is not in the natural domain for these functions. Thus the answer is D, since f(x)=(x²-1)/(x²-x-2) is defined for x=1 (altough f(1)=0 !)

Doğru Cevap: D

Soru 53

Which of the following functions is not one-to-one?

f(x)=x²-1

f(x)=x-1

f(x)=x+1

f(x)=-x-1

f(x)=x

Açıklama:

If f(a)=f(b) then a=b for one to one functions. Consider the function in A.
f(x)=x²-1
f(2)=3
f(-2)=3 but 2 is not equal to -2. Thus this function is not one-to-one. All the other given functions are linear and hence one-to-one. Note that f(x)=x³ is not linear but one-to-one also. Thus, all linear functions are one-to-one but not all one-to-one functions are linear.

Doğru Cevap: A

Soru 54

Which of the following functions (which are defined on natural numbers to natural numbers) is onto(surjective)?

f(x)=x²

f(x)=5+x²

f(x)=x+5

f(x)=x³

f(x)=x

Açıklama:

is onto (surjective)if every element of

is mapped to by some element of

. In other words, nothing is left out.

Let all these functions defined from A to B (where both sets are set of natural numbers).It is clear that the function f(x)=x² is not onto because for instance 5 can not be square of a natural number. By the same logic the function in B is not onto since x² +5 can not be equal to for instance 4 if x is natural number. The functions in C and D are also not onto. Only f(x)=x is onto.

Doğru Cevap: E

Soru 55

Suppose f(x)=x²-1 and g(x)=3x-1. Find f(g(x)).

3x²-1

9x²-1

9x²-2

9x²-6x

9x²-6x-2

Açıklama:

f(g(x)=(3x-1)²-1=(9x²-6x+1)-1=9x²-6x

Doğru Cevap: D

Soru 56

What is the inverse of function f(x)=(3x-2)/4?

(4x+2)/3

(4x-2)/3

(3x-4)/3

(4x-1)/3

(3x+4)/3

Açıklama:

suppose y=(3x-2)/4
We have to leave x alone on the right side and rewrite x as a function of y.
So:
4y=3x-2
4y+2=3x
(4y+2)/3=x
Thus f^-1(x)=(4x+2)/3

Doğru Cevap: A

Soru 57

Which of the following graphs belongs to the function f(x)=x²+4?

Açıklama:

since f(x)=x²+4, f(0)=4. Thus the graphs in A and B are eliminated. Also since x²+4 can not be equal to zero for real values of x, the graph can not cross the y-axis at y=0. Thus, E and D are eliminated also. Therefore the answer is C. Besides see that the coefficient of x² is positive. Therefore the function has a positive slope for x>0 and a negative slope for x<0.

Doğru Cevap: C

Soru 58

Suppose f(x)= x+2 and g(x)=x². Find f(g(2)).

Açıklama:

f(g(x))=f(x²)=2+x². Thus for x=2, f(g(2))=6

Doğru Cevap: B

Soru 59

Find the inverse of the function f(x)=(2x-3)/(4x+1).

(x-3)/(2-4x)

(3x-2)/(2+4x)

(4x+3)/(2-x)

(x-3)/(2+4x)

(x+3)/(2-4x)

Açıklama:

Let's try to leave x alone from the expression f(x)=(2x-3)/(4x+1)=y
(2x-3)/(4x+1)=y
2x-3=y(4x+1)
2x-3=4xy+y
2x-4xy=y+3
x(2-4y)=y+3
x=(y+3)/(2-4y)
Thus f^-1(x)=(x+3)/(2-4x)

Doğru Cevap: E

Soru 60

Suppose that f(x)=3x-1 and g(x)=x²-1. Find f/g(3).

1/3

8/3

10/8

Açıklama:

Since f(x)=3x-1 and g(x)=x²-1, then f/g(x)=(3x-1)/(x²-1).
For x=3, f/g(3)=8/8=1

Doğru Cevap: B

Ünite 3

Soru 1

Açıklama:

Doğru Cevap: C

Soru 2

Açıklama:

Doğru Cevap: C

Soru 3

Açıklama:

Doğru Cevap: D

Soru 4

Açıklama:

Doğru Cevap: B

Soru 5

Açıklama:

Doğru Cevap: B

Soru 6

Açıklama:

Doğru Cevap: E

Soru 7

2, 3

-2, -3

1, 4

2, -3

-2, 3

Açıklama:

Doğru Cevap: B

Soru 8

When you divide a number by 3 and then add 2, the result is the same as when you multiply the same number by 2 then subtract 23. What is the number?

Açıklama:

Doğru Cevap: B

Soru 9

Which of the following linear function has slope 2?

Açıklama:

Doğru Cevap: C

Soru 10

Which of the options given is a quartic polynomial?

Açıklama:

Doğru Cevap: E

Soru 11

Which of the following is a polynomial function, with degree 3?

f(x)= x³+3x-5

f(x)= x²+3x-2

f(x)= x²+3x-2x

f(x)=3x

f(x) =3x-2x

Açıklama:

The degree of f(x)= x³+3x-5 is 3.

Doğru Cevap: A

Soru 12

Which of the following is not a point on the line y=2x+1?

(0,1)

(-1/2,0)

(0,-1/2)

(1,3)

(2,5)

Açıklama:

(0,-1/2) does not satisfy the equivalence of y=2x+1.

Doğru Cevap: C

Soru 13

Which of the following is a solution set of 2x-2=0?

Açıklama:

Doğru Cevap: D

Soru 14

Which of the following functions is quadratic and has y-intercept -1?

f(x)= x²-2x-1

f(x)= x²-x+1

f(x)= x²-x

f(x)= x-1

f(x)=-1

Açıklama:

f(x)= x²-2x-1 is quadratic and has y-intercept -1.

Doğru Cevap: A

Soru 15

What is the solution set of the inequality x-2>0?

(2,0)

Açıklama:

x-2>0
x>2

Doğru Cevap: A

Soru 16

Which of the following quadratic functions has not x-intercept on naturel numbers?

y=x²-x-2

y=x²-x+2

y=x²-x-1

y=2x²-x-1

y=3x²-3x-3

Açıklama:

There is not any naturel numbers satisfy the equation of 0=x²-x+2

Doğru Cevap: B

Soru 17

Which of the following is not a polynomial function?

f(x)=x²-(1/2)x+2

f(x)=(1/2)x-1/2

f(x)=(-1/2)x

f(x)=2

f(x)=x²-x^-1/2+2

Açıklama:

-1/2; the power of x^-1/2
is not a naturel number.

Doğru Cevap: E

Soru 18

Which of the following function' s root is 0?

f(x)=4x-5

f(x)=4

f(x)=4x²

f(x)=4x²-4

f(x)=4x-4

Açıklama:

For y=0, x=0 for the equality of f(x)=4x²

Doğru Cevap: C

Soru 19

Which of the following equation is identity?

y=2x-2

x+x=2x

x-2=0

x²=2x

x²=-x²

Açıklama:

All real numbers satisfy the equality of x+x=2x.

Doğru Cevap: B

Soru 20

Which of the following linear equality's graph does not intersect the y-axis?

y=x

y=2x+1

y=2

x=2

x=2y-2

Açıklama:

In order to intersect the y-axis, the equalty of x=0 should be satisfied for x=2. But it can not be.

Doğru Cevap: D

Soru 21

Which of the following functions are polynomial?
I- f(x)=4x²-5x+3
II- g(x)=2x^-3-5x-1
III- h(x)=0.6x³-0.5x
IV- p(x)=7x^3/2-2x+4

I and IV

I and II

II and III

I and III

III and IV

Açıklama:

g(x) and p(x) are not in polynomial for because the powers of the elements must be natural numbers. Since -3 and 3/2 are not natural numbers, only f(x) and h(x) are polynomials.

Doğru Cevap: D

Soru 22

Which of the following functions is(are) a polynomial(s) of degree 1?
I- f(x)=2
II- g(x)=2x
III- h(x)=2/x
IV- k(x)=x/2

I and III

II and IV

II and III

I and IV

I and II

Açıklama:

h(x) is not a polynomial because, power of x in expression is -1, which is not a natural number. Besides f(x)=2 is a constant function, which means its degree is zero. Thus only g(x)=2x and k(x)=x/2 (which is equal to 0.5x) are polynomials of degree 1, namely they are linear functions.

Doğru Cevap: B

Soru 23

Which of the following polynomial equations have a natural number root?

3x+4=0

4x+3=0

3x+5=0

5x+5=0

5x-5=0

Açıklama:

-4/3, -3/4,-5/3 and -1 are not natural numbers, but 1 is. So the answer is E.

Doğru Cevap: E

Soru 24

Which of the following sets contain the roots of the polynomial equation
f(x)=x²-5x+6=0?

(2,3)

(-2,3)

(2,-3)

(1,6)

(-6,1)

Açıklama:

x²-5x+6=(x-2)(x-3)=0
Thus either x-2=0, which means x=2, or x-3=0, which means x=3. So the solution set is (2,3).

Doğru Cevap: A

Soru 25

Which of the following quadratic equations has a root x=0?

4x²-15x+3=0

x²-4x+3=0

x²-9x=0

3x²-7x-5=0

3x²+12=0

Açıklama:

When you substitute x with zero, only the equation in C is true.

Doğru Cevap: C

Soru 26

Which of the following expression's graphs passes through the origin?
I- f(x)=3
II- g(x)=3x
III- h(x)=3x²
IV- p(x)=3+3x²

I and II

II and III

I and III

II and IV

I and IV

Açıklama:

For a function's graph to pass through the origin, it must satisfy f(x)=0 for x=0. This condition is satisfied only for II and III

Doğru Cevap: B

Soru 27

Which of the following parabolas cross the x axis?

f(x)=x²-4x+5

f(x)=x²-2x+3

f(x)=x²-2x+10

f(x)=x²+4x+5

f(x)=x²-5x+4

Açıklama:

One parabola cross x axis only if it has real roots, which means in turn, the discirminant of parabola equation is greater than or equal to zero. This condition is satisfied only for the equation x²-5x+4

Doğru Cevap: E

Soru 28

Which of the following parabolas have a positive y intercept?

f(x)=x²-5x-4

f(x)=x²+5x-1

f(x)=-x²-6x-3

f(x)=x²-5x+4

f(x)=x²

Açıklama:

For a parabola to have a positive y intercept f(0) must be positive. This is true for only f(x)=x²-5x+4 for x=0 (since f(0)=4).

Doğru Cevap: D

Soru 29

What is the solution set for the inequality x²-5x+4<0

x<1

-4

Açıklama:

Firstly we have to find the roots of the quadratic expression f(x)=x²-5x+4=0
x²-5x+4=(x-1)(x-4)=0 is satisfied only if x=1 or x=4.
Thus we can divide the real numbers line to 3 regions:
1-)x<1
2-)1
3-) 4
In the first region (where x<1), f(x)=x²-5x+4>0. You can see this by substituting arbitrary numbers for x. (For instance think of x=0).
In the second region (where 12-5x+4<0. You can see this by substituting arbitrary numbers for x. (For instance think of x=2).
In the third region (where 42-5x+4>0. You can see this by substituting arbitrary numbers for x. (For instance think of x=5).
Thus only for the values of x in the second region, f(x) takes a negative value.

Doğru Cevap: A

Soru 30

What is the solution set for the inequality x²-6x-16<0?

-8

-2

x<-2

Açıklama:

Firstly we have to find the roots of the quadratic expression f(x)=x²-6x-16=0
x²-6x-16=(x-8)(x+2)=0 is satisfied only if x=-2 or x=8.
Thus we can divide the real numbers line to 3 regions:
1-)x<-2
2-)-2
3-) 8
In the first region (where x<-2), f(x)=x²-6x-16>0. You can see this by substituting arbitrary numbers for x. (For instance think of x=-3).
In the second region (where -22-6x-16<0. You can see this by substituting arbitrary numbers for x. (For instance think of x=0).
In the third region (where 82-6x-16>0. You can see this by substituting arbitrary numbers for x. (For instance think of x=10).
Thus only for the values of x in the second region, f(x) takes a negative value.

Doğru Cevap: C

Soru 31

(1,5)

(4,5)

Açıklama:

Doğru Cevap: A

Soru 32

1 and 4

-1 and 4

-4 and 1

1 and 5

-5 and 1

Açıklama:

This equation the sum of the roots is 5 and the product of the roots is 4. Therefore, the roots
should be 1 and 4.

Doğru Cevap: A

Soru 33

Which of the following is a polynomial function, with degree 7?

Açıklama:

Doğru Cevap: C

Soru 34

Which of the following is a solution set of 7x-21=0?

Açıklama:

if 7x-21=0 7x=21 and x=3.

Doğru Cevap: A

Soru 35

What is the solution set of the inequality 4x-128<0?

(-∞, 48)

(4,32)

(-∞, 24).

(32,∞).

Açıklama:

if 4x-128<0 4x<128 and x<32
ÇK=(-∞, 32).

Doğru Cevap: C

Soru 36

Which of the following linear function has slope -5?

8y=6x+3

y=2x-3

2y=-10x+4

y=4-3x

2y=5x+7

Açıklama:

slope =a for equation y = ax+b

if 2y=-10x+4 y=-5x+2 and slope=-5

Doğru Cevap: C

Soru 37

Which of the following is the point of intersection of the lines y=3x-14 and y=-x-2?

(3,4)

(2,3)

(-5,3)

(3,5)

(3,-5)

Açıklama:

We find the common
solution. For this, 3x-14=-x-2
3x+x=14-2
4x=12
x=3
Now we write x=3 in the equation y=-x-2=-3-2=-5 . So, (3,-5) is the intersection point of the lines

Doğru Cevap: E

Soru 38

Which of the following the slope of the line passing through the points (1, 2) and (-2, -1)?

-2

-1

1/2

Açıklama:

Doğru Cevap: C

Soru 39

Which of the following the line equation passing through the points (1, 2) and (-2, -1) ?

y=-2x-1

y=2x+2

y=-x+2

y=3x-2

y=x+1

Açıklama:

Doğru Cevap: E

Soru 40

(1,3)

(-1,2)

(3,0)

(-3,3)

(3,6)

Açıklama:

Doğru Cevap: C

Ünite 4

Soru 1

100

1000

Açıklama:

Doğru Cevap: E

Soru 2

ln(x²) - ln(4 e² x) = -2. x = ?

1 / e

2 / (1 - e)

e²

Açıklama:

2 ln x - In 4 - 2 In e - In x = -2 ; In x = In 4 ; x = 4. Correct answer is B.

Doğru Cevap: B

Soru 3

ln(1 / (x - 1))² ) = -4. x = ?

e² + 1

-4 / (e - 1)

2 e

4 e³

e - 4

Açıklama:

-2 ln(x - 1) = -4 ; ln(x - 1) = 2 = ln e² ; x - 1 = e² ; x = e² + 1. Correct answer is A.

Doğru Cevap: A

Soru 4

a = log₂5, b = log₂ 10. 2^{(a - b)} log₉ 3 = ?

0.25

1.50

Açıklama:

c = log₂ (5 / 10) = log₂ 2^-1 ; log₂ c = 2^-1 ; 3 = 9^1/2 ; x = 2^-1 1/2 = 2^-2 = 0.25 . Correct answer is A.

Doğru Cevap: A

Soru 5

log₂9 . log₅8 . log₉25 = ?

Açıklama:

change of base : x = (ln 9 / ln 2) (ln 8 / ln 5) (ln 25 / ln 9) = (1 / ln 2) (3 In 2 / In 5) (2 In 5) = 6. Correct answer is C.

Doğru Cevap: C

Soru 6

Which power of 2 is between 500 and 600 ?

Açıklama:

512 = 2⁹ . Correct answer is D.

Doğru Cevap: D

Soru 7

e^-2x + e^-x = 0. x = ?

-e

no solution

1 / e

-1 / e

Açıklama:

for all x : e^-2x > 0, e^-x > 0 ; no solution. Correct answer is C.

Doğru Cevap: C

Soru 8

A bacteria population increases exponentially. At t = 0, its population is 50. At t = 1 hour, its population is 250. At t = 4 hours, what is its population ?

3.125

10.000

6.500

2.500

7.750

Açıklama:

P(t) = P₀ e^{k t} ; P(0) = P₀ = 50 ; P(1) = 50 e^{k 1} = 250 ; e^k = 5 ; ln e^k = ln 5 ; k = ln 5 ; x = 50 e^{(ln5) 4} ; a = ln 5⁴ ; x = 50 e^a = 50 5⁴ = 10 5⁵ = 3.125. Correct answer is A.

Doğru Cevap: A

Soru 9

According to the Richter scale, earthquake magnitude 7.0 is how many times greater than 5.0 ?

500

1.000

100

Açıklama:

7 = log₁₀ 10⁷ ; 6 = log₁₀ 10⁵ ; x = 10⁷ / 10⁵ = 100. Correct answer is E.

Doğru Cevap: E

Soru 10

What is the half-life of a radioactive substance if 90% of the initial amount remains at the end of 25 years ?

50 (ln 0.5) ln 0.9

75 (ln 0.9) / ln 0.5

25 (ln 0.5) / ln 0.9

100 ln (0.5 / 0.9)

150 ln (0.9 / 0.5)

Açıklama:

Q(t) = Q₀ e^{k t} ; 0.9 Q0 = Q⁰ e^{k 25} ; ln 0.9 = k 25 ; k = (ln 0.9) / 25 ; 0.5 Q₀ = Q₀ e^{((ln 0.9) / 25) t} ; ln 0.5 = ((ln 0.9) / 25) t ; t = 25 (ln 0.5) / ln 0.9 . Correct answer is C.

Doğru Cevap: C

Soru 11

Evaluate the expression log₂16/ln2.

ln(e⁴+2)

1/2

ln(e⁴/2)

ln(e⁴-2)

Açıklama:

log₂16=log₂2⁴=4
(4/ln2)=(4lne)/ln2=lne⁴/ln2=ln(e⁴/2)

Doğru Cevap: D

Soru 12

-2/3

-8/9

2/5

3/8

7/2

Açıklama:

log3(1/(x+1))=2→ 3²=1/(X+1)→9=1/(X+1) then 9x+9=1→x=-8/9

Doğru Cevap: B

Soru 13

What is the solution of the equation ln(81x⁴)=20?

e³/3

e²/5

e⁵/3

Açıklama:

We write 81x⁴ = (3x)⁴ and use the law
ln ab = b ln a,
then, 4ln(3x)=20→ln(3x)=5→e⁵=3x→e⁵/3=x

Doğru Cevap: E

Soru 14

log₅2 . log₄3 . log₉5=?

1/6

1/2

Açıklama:

using the change-of-base formula, log₅2=ln2/ln5
log₄3=ln3/ln4=ln3/ln2²=ln3/2ln2
log₉5=ln5/ln9=ln5/3ln3
then,log₅2 . log₄3 . log₉5=(ln2/ln5) . (ln3/2ln2) . (ln5/3ln3)=1/6

Doğru Cevap: A

Soru 15

Which power of 4 between 4000-4100?

Açıklama:

we can calculate that: 4²=16, 4³=64, 4⁴=256, 4⁵=1024, 4⁶=4096 Then the power of 4, which lies between 4000 and 4100, is 6.

Doğru Cevap: C

Soru 16

Solve the exponential equation; 5^2x+3=1/25

-5/2

-2

5/2

Açıklama:

Note that one can rewrite;
5^2x+3=1/25=5^2x+3=5^-2
Since the exponential function 5^x is one-to one, we have
2x+3=-2 then x=-5/2

Doğru Cevap: A

Soru 17

What is one of the solution of the exponential equation e^2x - 2e^x -15=0

ln2

ln3

ln4

ln5

ln6

Açıklama:

e^2x - 2e^x -15=0 → let e^x=a then w can write the equation as, a²-2a-15=0
→ (a-5)(a+3)=0 → e^x=a=5, e^x=a=-3 then the solution is; ln5 and ln-3.

Doğru Cevap: D

Soru 18

In a certain culture, the number of cells increases at a rate proportional to the number of cells present. If there are 150 cells present initially and 600 cells in one hour, find the number of cells in 4 hours?

8100

16200

38400

55000

102400

Açıklama:

Let the function P(t) denote the number of the cells at time t in this culture.
P(t) = P₀e^kt
P(0)=P₀=150
P(1)=P₀e^k.1→ P(1)=150e^k =600 → e^k=4 then k=ln4 =2ln2
The equation is, P(t) = 150e^2ln2t
for t=4→P(4)=150e^2(ln2).4
=150.2⁸=38400.

Doğru Cevap: C

Soru 19

There are 2 earthquakes are given below;
1^st earthquake's magnitude is M=7.1 and the 2^nd earthquake's magnitude is M=5.2 on the Richter scale.
How many times 1^st earthquake has a magnitude greater than 2^nd earthquake about?
( On the Richter scale, the magnitude M of an earthquake of intensity I is expressed by the formula: M = log₁₀I.)

125

Açıklama:

For the first earthquake, the intensity is; M = log₁₀I→ 7.1=log₁₀I→ I=10^7.1
For the second earthquake, the intensity is; M = log₁₀I→ 5.2=log₁₀I→ I=10^5.2
on the Richter scale yields an intensity change by a factor of
10^7.1_{/10^{5.2 =}10^1.9=79.43}
^{Strictly speaking; the fist earthquake had a magnitude about 80 times greater than second earthquake.}

Doğru Cevap: D

Soru 20

What is the half-life of a radioactive substance if 95% of the initial amount remains at the end of 10 years?

23,12

34,67

76,98

113,34

135,13

Açıklama:

Let Q(t) be the amount of the radioactive substance at time t and A be the amount of initial radioactive substance. Since decay of the amount of radioactive substance is exponential, the function Q(t) has the following form:
Q(t) = Q₀e^kt
for t=0→A= Q₀ then Q(10)=0,95A=A.e^k10
ln 0,95=10k and k=(ln0,95)/10
To find the half-life of this radioactive material we should solve the equation;
0,5A=A.e^{((ln0,95)/10).t}
ln0.5=(ln0,95/10)).t →135,13

Doğru Cevap: E

Soru 21

Which of the following is not an exponential function?

f(x)=x³

f(x)=4^x

f(x)=(0.5)^x

f(y)=(5/2)^y

f(x)=10^x

Açıklama:

The form of the exponential functions is f(x)=a^x for all values of a other than 1. So f(x)=x³ is not an exponential function.

Doğru Cevap: A

Soru 22

Which of the following is not true for an exponential function f(x)=a^xif 0

Domain of the exponential function f(x)=a^x is (-∞, ∞)

Range of the exponential function f(x)=a^x is (0, ∞)

The exponential function f(x)=a^x is increasing

As x goes to minus infinity, f(x)=a^x goes to infinity

As x goes to infinity, f(x)=a^x goes to zero

Açıklama:

f(x)=a^x is not an increasing function for 0

Doğru Cevap: C

Soru 23

Which of the following properties are true for the exponential functions?
I.a^x+y=a^xa^y
II. a^x-y=a^x/a^y
III. (a^x)^y=a^x+y
IV.(ba)^y=b^ya^y

I, II and III

I, II and IV

II, III and IV

I, III and IV

I and IV

Açıklama:

(a^x)^y=a^xy , so other than III all the properties hold for exponential functions.

Doğru Cevap: B

Soru 24

If 10.000 TL is invested for 4 years at an annual rate of interest of 20%, find the value of the investment at the end of 4 years if interest is compounded annually.

8000

14000

18000

20736

22462

Açıklama:

Total value at the end=10.000(1.2)⁴=2.0736*10000=20736

Doğru Cevap: D

Soru 25

Which of the following are true for the function log_ax when a>1?
I. Domain of the logarithmic function log_ax is (0, ∞)
II. Range of the logarithmic function log_ax is (-∞, ∞)
II. log_ax is a decreasing function

Only I

Only II

Only III

I and III

I and II

Açıklama:

log_ax is an increasing function if a>1. So III is false.

Doğru Cevap: E

Soru 26

How many of the following arguments are true if x>0 and y>0.

log_a(xy)= log_ax+log_ay,
log_a(x/y)= log_ax-log_ay

log_a(1/y)= -log_ay
log_a(x)^y=y+log_ax

Açıklama:

Only the last argument is false. log_a(x)^y=ylog_ax. So the remaining 3 arguments are true.

Doğru Cevap: B

Soru 27

Find the solution set of the equation 0=6+e^2x- 5e^x

(2,3)

(ln2, 3)

(2, ln3)

(ln2, ln3)

(ln1, ln4)

Açıklama:

0=6+e^2x- 5e^x
0=(e^x- 3)(e^x- 2)
So the system has 2 roots:
e^x- 3=0 then e^x=3 so x=ln3
e^x- 2=0 then e^x=2 so x=ln2
Therefore solution set is (ln2, ln3)

Doğru Cevap: D

Soru 28

If the half-life of a radioactive material is 3000 years, how much of the 100 grams will remain after 5000 years?

15.6

22.7

26.8

28.6

31.5

Açıklama:

The quantity at time t is given by the formula Q(t)=Q(0)*e^kt where Q(0) is the initial amount.
Since the half life is 3000 years, we can write:
0.5Q(0)=Q(0)*e^3000k which means that:
0.5=e^3000k
ln0.5=lne^3000k
ln0.5=3000k
ln0.5/3000=k
Thus, after finding k we can calculate:
Q(t=5000)=100*e^{5000*ln0.5/3000}
=100*e^5*ln0.5/3
=100*e^-1.155=31.49

Doğru Cevap: E

Soru 29

The population of a bacteria doubles in 4 hours. How many bacteria would we have in 10 hours given that we have 1000 bacteria now?

5657

3568

2563

1896

567

Açıklama:

Given the initial bacteria population Q(0), the population at time t is given by Q(t)=Q(0)e^kt.
Since we know that population doubles in 4 hours, we can write:
2Q(0)=Q(0)e^4k
2=e^4k
ln2=4k, so k=ln2/4=0.1733
Then Q(t=10)=1000e^10k
=1000.e^1.733
=5657 (approximately)

Doğru Cevap: A

Soru 30

A radioactive material losses 20 % of its weight after 10 years. What is the half-life of this material approximately?

Açıklama:

Since 20 percent is lost, 80 percent is remaining at the end of 10 years. Thus we can write:
0.8Q=Q.e^10k
ln0.8/10=k=-0.0223
So we can find the half life by:
0.5Q=Q.e-^0.0223t
ln0.5=-0.0223t, then t=31.06 years

Doğru Cevap: A

Soru 31

Which power of 4 is between 250 and 650?

Açıklama:

250<4^x<650
4*4*4*4=4⁴=256
x=4

Doğru Cevap: B

Soru 32

What is solution of 5⁰?

500

Açıklama:

5⁰=1
The correct answer is 1.

Doğru Cevap: B

Soru 33

Suppose that we have 2000 TL to invest at an annual rate of interest of 10% for 4 years in an account that pays simple interest. What is the value of the investment (total amount) at the end of second year?

2200

2420

2500

2928,2

3000,5

Açıklama:

2000.(1+(10/100))⁴=2000.(1,1)⁴=2928,2
The correct answer is 2928,2 ₺.

Doğru Cevap: D

Soru 34

What is the solution of log₂16?

Açıklama:

log₂16=log₂2⁴=4
The correct answer is 4.

Doğru Cevap: C

Soru 35

If the half-life of 100 grams of radioactive material is 2000 years, how much of the 100 grams will remain after 1000 years?

50.15

65.52

68.78

70.71

75.75

Açıklama:

The correct answer is 70,71 gr.

Doğru Cevap: D

Soru 36

Which power of 6 is between 1200 and 1500?

Açıklama:

1200<6^x<1500
6*6*6*6=6⁴=1296
The correct answer is 4.

Doğru Cevap: A

Soru 37

What is the solution of log₃27?

Açıklama:

log₃27=log₃3³=3
The correct answer is 3.

Doğru Cevap: B

Soru 38

Which power of 7 is between 2000 and 2500

Açıklama:

2000<7^x<2500
7*7*7*7=2401; so 7⁴=2401;
2000<7⁴<2500
The correct answer is 4.

Doğru Cevap: E

Soru 39

Which power of 2 is between 125 and 250?

Açıklama:

125<2^x<250
2⁷=128
The correct answer is 7.

Doğru Cevap: D

Soru 40

What is the solution of log₇49?

Açıklama:

log₇49=log₇7²=2
The correct answer is 2.

Doğru Cevap: D

Soru 41

What is the solution of the logarithmic equation

Açıklama:

. We get

and

. Hence, we find the solution to be x=-1 and x=3.
The definition of logarithmic function cannot be x = -1. The solution is x=3.

Doğru Cevap: C

Soru 42

What is the solution of the equation

1/2

Açıklama:

(y=lnx if and only if

)

We find the solution as x-1=1

x=2.

Doğru Cevap: A

Soru 43

What is the solution set of the exponential equation

Açıklama:

.
We find the solutions x=3 or x=-1.

Doğru Cevap: B

Soru 44

What is the solution set of the equation

Açıklama:

We write

and use the ln1=0

.
We find the solution as x=-2 or x=3.

Doğru Cevap: C

Soru 45

What is the solution of the equality

Açıklama:

.
We use the law

. We write 6.1=x

x=6.

Doğru Cevap: D

Soru 46

Çukurhisar (Eskişehir) was devastated by an earthquake which was measured 9.1 on the Richter scale on May 21th 1910. Nearly 101 years later,Gölçük (Kocaeli) was destroyed by an earthquake which is measured 7.4 on the Richter scale on August 13th 2011. How many times Çukurhisar earthquake hada magnitude greater than the Gölçük earthquake?

1,7

45,8

50,1

Açıklama:

We know that, on the Richter scale, the magnitude M of an earthquake of intensity I is expressed by the formula M = log10 I.
For Çukurhisar earthquake, because M = 9,1 we have 9,1 = log10 I and

For Gölçük earthquake, because M = 7,4 we have 7,4 = log10 I and

One can compare that an increase of 9,1 - 7.4 = 1,7 units on the Richter scale yields an intensity change by a factor of

Doğru Cevap:

Soru 47

Evaluate the expression

-3

-4

Açıklama:

It is known that

Doğru Cevap: B

Soru 48

Which power of 4 is between 200 and 300?

Açıklama:

We can calculate that

and

. Then the power of 4, which lies between
200 and 300, is 4.

Doğru Cevap: C

Soru 49

What is the solution of the equation

Açıklama:

We consider the equation as

This equation can be written as the product of factors

Hence

. From the inverse relation of ln and the natural exponent, we obtain that the
solution set of this equation is x=ln2 and x=ln3.

Doğru Cevap: E

Soru 50

What is the solution set of the exponential equation

-1

-2

Açıklama:

To solve the exponential equation

Divide both sides by 6 and use the power rule to give

Hence the solution of this equation is x = 1 from the equation

Doğru Cevap: A

Soru 51

What is the solution of the logarithmic equation

Açıklama:

To solve the equation

we rewrite

, then the solution is

Doğru Cevap: B

Soru 52

What is the solution of

Açıklama:

It is known that;

is obtained.

Doğru Cevap: A

Soru 53

What is the solution of

Açıklama:

Note that one can rewrite

Since the exponential function

is one-to-one, we have

This is a quadratic function an we express it as

Then

must be zero. Hence, we obtain the solution as

Doğru Cevap: B

Soru 54

If 5.000 TL invested for 3 years at an annual rate of interest 10%, find the value of the investment at the end of 3 years if interest is compounded annually?

5650 TL

5955 TL

6250 TL

6655 TL

6950 TL

Açıklama:

At the end of 3th year, we have

Doğru Cevap: D

Soru 55

What is the solution of

Açıklama:

Note that we can rewrite

then

is obtained.

Doğru Cevap: D

Soru 56

then what is the solution of

-2

-1

Açıklama:

Then

is obtained.

Doğru Cevap: C

Soru 57

What is the solution set of

Açıklama:

The equation can be written as

This equation can be written as the product of factors

Then

, or

should be satisfied.
Hence the solution set of this equation is

Doğru Cevap: E

Soru 58

What is the solution of the equation

Açıklama:

is obtained.

Doğru Cevap: A

Soru 59

Which of the following is not represent an exponential function?

Açıklama:

An exponential function moves multiplicative. However

increases arithmetically. Hence the solution is B.

Doğru Cevap: B

Soru 60

In normal conditions, a bacteria increases at a rate of proportionally. If there is 30 bacterias exist initially and 180 bacterias in one hour, find the number of bacterias in four hour?

18.810

21.180

28.320

34.200

38.880

Açıklama:

The proportion is 180/30 = 6
Then the increase 30 . 6 . 6 . 6 . 6 = 38.880

Doğru Cevap: E

Soru 61

Which of the following is not a exponential function?

3^x

0.5^x

2^x

(4/3)^x

x²

Açıklama:

Exponential functions are in the form f(x)=a^x where a is any positive real number other than 1. Thus x² is not an exponential function because x is in base, not in power.

Doğru Cevap: E

Soru 62

Find x if 2^x+3*5^x+1=4000.

0.5

Açıklama:

2^x+3*5^x+1=4000
2³*2^x*5*5^x=4000.
40*10^x=4000
10^x=100
so x=2

Doğru Cevap: C

Soru 63

1000 TL is deposited in a bank account which yields 20 % interest rate on annual base. What will be the interest earning over this account after 3 years?

1728

728

600

440

200

Açıklama:

The total amout will be:
1000*1.2*1.2*1.2=1728 TL
So the interest earning is 1728-1000=728

Doğru Cevap: B

Soru 64

I. log_a(xy) = log_ax * log_ay
II. log_a(x/y) = log_ax / log_ay
III. log_a(1/y) = −log_ay
IV. log_a(x^y) = ylog_ax
Which of the above arguments are true?

III and IV

II and III

I and II

I and IV

I and III

Açıklama:

I and II are false. Because:
log_a(xy) = log_ax + log_ay
log_a(x/y) = log_ax - log_ay

Doğru Cevap: A

Soru 65

Find x if e^2x + 5e^x - 6 = 0

-6

-1

Açıklama:

If we factorize e^2x + 5e^x - 6 = 0, we get:
(e^x - 1)*(e^x + 6)=0
so either e^x - 1=0 or e^x + 6=0
If e^x - 1=0 then e^x = 1 which means x=0.
If e^x + 6=0 then e^x = -6 which can not be the solution because it is not in domain of the natural nase function e^x .
Thus x=0.

Doğru Cevap: C

Soru 66

Find the solution set of e^2x - 7e^x + 12= 0.

(ln2, ln6)

(1, ln2)

(1, ln6)

(ln3, ln4)

(ln3, ln6)

Açıklama:

e^2x - 7e^x + 12= 0 can be rewritten as:
(e^x - 4)(e^x - 3)=0
Thus the equation has two roots:
for e^x - 4=0, e^x = 4 so x=ln4
for e^x - 3=0, e^x = 3 so x=ln3

Doğru Cevap: D

Soru 67

If ln 2=a, what will be the value of ln 0.5?

1/4 a

1/2 a

-a

-2a

Açıklama:

ln 0.5= ln 1/2 = ln1-ln2
Since we know that ln1=0,
ln 0.5=0-a=-a

Doğru Cevap: D

Soru 68

If ln 2=x and ln 5=y what will be the value of ln 200 in terms of x and y?

3x+2y

2x+3y

3x-2y

5x+2y

5x-2y

Açıklama:

ln 200 = ln 2³*5² = ln 2³ + ln 5² = 3 ln 2 + 2 ln 5=3x+2y

Doğru Cevap: A

Soru 69

There are 100 bacteria initially in a bottle. After 2 hours the number of bacteria increases to 900. How many bacteria will there be in the bottle after 10 hours?

100*3⁸

900*3⁵

900*3¹⁰

100*3⁵

100*3¹⁰

Açıklama:

The population of the bacteria can be modelled by the equation P(t)=P(0)e^kt where P(0) is the initial poulation.
Thus P(t=2)=P(0)e^2k . So:
900=100e^2k
9=e^2k=(e^k)²
3²=(e^k)²
3=e^k
The population after 10 hours will be:
P(t=10)=100e^10k=100*(e^k)¹⁰=100*3¹⁰

Doğru Cevap: E

Soru 70

Half-life of the radioactive element x is 400 years. How many grams of x will remain after 600 years if the inital amount is 1 kg?

256

354

482

518

646

Açıklama:

Since half-life is 400 years, we can model the amount of element x as:
0.5 Q= Q.e^400k
Thus 0.5 = e^400k so ln 0.5=ln e^400k =400k so k=ln 0.5 /400
Q(t=600)=1000*e^600k
let e^600k= b,
then 600k = ln b and since we found that k = ln 0.5/400
600* ln 0.5 /400= ln b=1.5 ln 0.5
thus b=(0.5)^1.5= 0.3535
Then Q(t=600)=1000*0.3535=353.5 gr=354 gr (approximately)

Doğru Cevap: B

Ünite 5

Soru 1

Which of the following is the value of

does not exist.

Açıklama:

we need to factorise the ratio which follows as;
f(x)=(x²-1)/(x-1)=(x-1)(x+1)/(x-1)=x+1
and if we put "1" instead of x, then the answer will be 1+1=2.

Doğru Cevap: C

Soru 2

Which of the following is the value of

0,6

0,36

0,18

0,09

Açıklama:

Since the constant function takes the same value for all x∈R, we have

=0,36.

Doğru Cevap: B

Soru 3

Which of the following is the value of lim x→1 (2x³+x-12)/(5x-9)

1/2

5/2

9/4

Açıklama:

From the quotient rule for limits together with limx→1(5x-9)≠0 we obtain limx→1 2x³+x-12=limx→1(2x³+x-12)/limx→1(5x-9) =-9/-4= 9/4

Doğru Cevap: E

Soru 4

Find the lim x→2 (5x-2)(3x²+1).

104

144

Açıklama:

From the product rule for limits, we have, (lim x→2 (5x-2).lim x→2 (3x²+1)) ,so (lim x→2 (5x-2)=8 and lim x→2 (3x²+1)=13 from the product rule again we have, 8*13=104

Doğru Cevap: D

Soru 5

What is the value of the

does not exist

Açıklama:

by fractionating the function f(x)=(x-2)(x+2)/(x-2)=x+2 then lim x→2 x+2= 4

Doğru Cevap: B

Soru 6

+∞

-∞

does not exist

Açıklama:

The function is a rational function. Since the degree of the numerator is less than the degree of the denominator, the following result is true:

Doğru Cevap: C

Soru 7

∞

does not exist

-1

Açıklama:

The function is a rational function. Both the numerator and denominator approach infinity as x becomes large and their highest degrees are the same. By the way, the constants before the x² are same, 1/1=1

Doğru Cevap: A

Soru 8

-1

∞

does not exist

Açıklama:

The function is a rational function. Since the degree of the numerator is greater than the degree of the denominator, the following result is;

Doğru Cevap: D

Soru 9

Açıklama:

Doğru Cevap: B

Soru 10

If f: R→R,

then, which of the following is true?

The function has removable discontinuity at x=0

The limit of f(x) does not exist at x=0

The value limit of the function is 2 at x=-1

The value limit of the function is 6 at x=7

The function is discontinuous at x=-5

Açıklama:

Doğru Cevap: A

Soru 11

lim_x_→-1 5^-1 = ?

-1

0.2

-0.2

Açıklama:

0.2 ; limit of a constant is itself. pg. 107. Correct answer is C .

Doğru Cevap: C

Soru 12

While approaching from the right lim_x_→2 (x² - x - 2) / (x - 2) = ?

does not exist

∞

Açıklama:

(x - 2) (x + 1) / (x - 2) = x + 1 ; x ≠ 2 . pg. 108. Correct answer is E.

Doğru Cevap: E

Soru 13

lim_x_→2 (x - x^-1) (x² - 1) = ?

4,5

1,5

2,5

0,5

5,5

Açıklama:

(2 - 1/2) (4 - 1) = (3/2)/3=(3/2).(1/3)=1/2 =0,5. pg. 110. Correct answer is D.

Doğru Cevap: D

Soru 14

f(x) = x² - 2 for x < 3 ; f(x) = x³ - 2 for x > 3 ; while approaching from the left lim_x_→3 f(x) = ?

Açıklama:

y = 9 - 2 = 7. pg. 109. Correct answer is D.

Doğru Cevap: D

Soru 15

While approaching from the right lim_x_→-1 (x + 1) / (x³ + 2 x² - 1) = ?

-1

∞

-1 / 3

does not exist

Açıklama:

(x + 1) / (x + 1) (x² + x - 1) = 1 / (x² + x - 1) ; 1 / (1 - 1 - 1) = -1 ; x ≠ -1 . pg. 108 . Correct answer is B.

Doğru Cevap: B

Soru 16

While approaching from the left lim_x_→1 |x - 1| / (x - 1) = ?

∞

-1

does not exist

Açıklama:

(x - 1) / (x - 1) = -1 . pg. 109. Correct answer is D.

Doğru Cevap: D

Soru 17

lim_x_→_∞ (x³ + x² + 1) / (x⁴ + x - 1) = ?

∞

-∞

does not exist

Açıklama:

0 ; Since the degree of the numerator is less than the degree of the denominator. pg. 114. Correct answer is E.

Doğru Cevap: E

Soru 18

What is the set of continuity of f(x) = 2 x / (x³ - 1) ?

(-∞, ∞)

R \ {1}

R \ {-1, 1}

R \ {-1}

Açıklama:

R \ {1} ; (x³ - 1) = 0 ; x = 1. pg. 117. Correct answer is B.

Doğru Cevap: B

Soru 19

lim_x_→_∞ (x⁹⁹ - 1) / (-x⁹⁸ + x⁹⁷ + 1) = ?

∞

does not exist

-∞

-1

Açıklama:

-∞ ; Since the degree of the numerator is greater than the degree of the denominator. pg. 114. Correct answer is D.

Doğru Cevap: D

Soru 20

lim_x_→-_∞ (8 x⁵⁵⁵ - 1) / (4 x⁵⁵⁵ + 2 x⁵⁵⁴ + 1) ?

does not exist

-∞

-2

∞

Açıklama:

2 ; Since the degree of the numerator is equal to the degree of the denominator. pg. 114. Correct answer is C.

Doğru Cevap: C

Soru 21

Which of the following is the value of lim_x→57?

Açıklama:

Since the constant function takes the same value for all x ∈ R, we have lim_x→57 = 7.

Doğru Cevap: D

Soru 22

Which of the following is the value of lim_x→7[(x²-2x - 35) / (x - 7)]

Açıklama:

The function is a rational function. Because the limits of the numerator and the denominator are equal to 0, we need to factorise the ratio which follows as
[(x²-2x - 35) / (x - 7)] = [(x - 7) (x + 5)] / x-7 = x + 5.
Following the limit rules, we find lim_x→7x + 5 = 7 + 5 = 12.

Doğru Cevap: C

Soru 23

Which of the following is the value of lim_x→1[(x²+5x -12) / (2x - 8)]

Açıklama:

From the quotient rule for limits together with lim_x→1(2x - 8) ≠ 0, we obtain
(1² + 5 . 1 - 12) / (2 . 1 - 8) = -6 / -6 = 1.

Doğru Cevap: A

Soru 24

Which of the following is the value of lim_x→2(x + 3) (2x - 5)?

-10

-5

Açıklama:

From the product rule for limits, we have
lim_x→2(x + 3) (2x - 5) = lim_x→2(x + 3) . lim_x→2(2x - 5)
= (2 + 3) . (4 - 5)
= -5

Doğru Cevap: B

Soru 25

3x²+ 5, x < -3
If f : R - {-3} → R, f (x ) = ⎨-x + 12, x > -3
then which of the following is the the value of lim_x→-3⁺f(x)?

-6

-12

Açıklama:

Because x approaches -3 from the right side x is more than -3 and for x > -3, f is given by the rule f (x)=-x+12. Hence, the result is as follows.
lim_x→-3⁺f(x) = -(-3) + 12 = 3 + 12 = 15

Doğru Cevap: E

Soru 26

Evaluate lim_x→2 ( x / |x-2| ).

-2

-∞

+∞

The limit does not exist.

Açıklama:

Since the denominator of the expression at x = 2 is 0, we must look at the one-sided limits in order to find the limit at x = 2 if it exists. Then we have,
lim_x→2⁺ ( x / |x-2| ) = lim_x→2⁺ ( 2 / 0⁺ ) = +∞
lim_x→2^- ( x / |x-2| ) = lim_x→2^- ( 2 / 0⁺ ) = +∞
Hence, since the one-sided limits show the same behavior at x = 2, the limit exist at x =2, and it is +∞.

Doğru Cevap: D

Soru 27

What is the limit of the function f(x) = ( x-7 / x - 5) at x = 5?

+∞

-∞

does not exist.

Açıklama:

lim_x→5^-( x-7 / x - 5) = -2 / 0^-= +∞.
lim_x→5⁺( x-7 / x - 5) = -2 / 0⁺= -∞.
The limit of the function at x = 5 does not exist due to the different behaviour of one-sided limits.

Doğru Cevap: E

Soru 28

If f (x ) = [(x²+ 3x + 1) / (x⁵ + 5)], then what is the value of lim_x→∞ f(x)?

5/6

6/5

_+∞

_-∞

Açıklama:

The function is a rational function. Since the degree of the numerator is less than the degree of the denominator, the value of the limit is 0.

Doğru Cevap: A

Soru 29

Determine the set of continuity of the function f (x) = x²-1.

(-∞, 0)

R \ {1}

R \ {-1,1}

R - [-1,1]

Açıklama:

f (x) = x²-1 is a polynomial function. Hence, it is a continuous function, and its set of continuity is R.

Doğru Cevap: B

Soru 30

Determine the set of continuity of the function f(x) = [(x - 3) / (x³ - 27)].

R \ {-3, 3}

R \ {3}

R \ {0,3}

[3, ∞)

Açıklama:

Since a rational function f is continuous on the domain of its definition, f is continuous on its domain. This function is continuous for all real numbers except 3 which makes its denominator 0. Hence, the set of continuity of this function is R \ {3}.

Doğru Cevap: C

Soru 31

Which of the following is the value of

Açıklama:

From the quotient rule for limits together with

we obtain

The answer is B.

Doğru Cevap: B

Soru 32

Which of the following is the value of

-1/5

-2/3

-3/5

-2

Açıklama:

From the quotient rule for limits together with

we obtain

The answer is D.

Doğru Cevap: D

Soru 33

-66

-13

Açıklama:

From the product rule for limits, we have

. The answer is B.

Doğru Cevap: B

Soru 34

then which of the following is the value of

-2

Açıklama:

Because x approaches 1 from the left side x is less than 1 and for x<1, f is given by the rule f(x)=-2x+5. So, the result follows as

. The answer is E.

Doğru Cevap: E

Soru 35

then which of the following is the value of

-8

-15

-3

Açıklama:

Becuase x approaches 3 from the right side x is greater than 3 and for x>3, f is given by the rule

. So, the result follows as

. The answer is A.

Doğru Cevap: A

Soru 36

What is the limit of the function

at x=0?

-1

-2

-5

Açıklama:

The answer is B.

Doğru Cevap: B

Soru 37

-1

-3

1/3

-1/3

Açıklama:

Since x approaches 0 from the left side, x is negative and for x<0, |3x|=-3x is true. So, we get following result.

. The answer is C.

Doğru Cevap: C

Soru 38

then

∞

-1

Açıklama:

The function is a rational function. Since the degree o the numerator is greater than the degreeof the denominator, the following result is true:

. The answer is A.

Doğru Cevap: A

Soru 39

then

-1

Açıklama:

The function is a rational function. Since the degree of the numerator is less than the degree of the denominator, the following result is true:

. The answer is E.

Doğru Cevap: E

Soru 40

Determine the set of continuity of the function

(-1, 1)

R\{-1, 1}

[-1, 1]

(0, 1)

(-1, 0)

Açıklama:

Since a rational function f is continuous on the domain of its definition, f is continuous on its domain. the is continuous for all real numbers x except -1 and 1. Thus, f is continuous on R\{-1, 1}. The answer is B.

Doğru Cevap: B

Soru 41

lim_x_→0 0⁰ = ?

∞

does not exist

-1

Açıklama:

1 ; limit of a constant is itself. pg. 107. Correct answer is B

Doğru Cevap: B

Soru 42

While approaching from the left lim_x_→1 (x³ - 1) / (x - 1) = ?

∞

does not exist

Açıklama:

(x - 1) (x² + x + 1) / (x - 1) = x² + x + 1 = 3 ; x ≠ 1 . pg. 108. Correct answer is D.

Doğru Cevap: D

Soru 43

lim_x_→-1 (x³ - x⁰) (x - 1) = ?

-1

Açıklama:

(-1 - 1) (-1 - 1) = 4. pg. 110. Correct answer is E.

Doğru Cevap: E

Soru 44

f(x) = x^-1 + 2 for x < -2 ; f(x) = x^-2 + 2 for x > -2 ; while approaching from the right lim_x_→-2 f(x) = ?

1.50

2.25

2.50

1.75

Açıklama:

y = 0.25 + 2 = 2.25. pg. 109. Correct answer is C.

Doğru Cevap: C

Soru 45

While approaching from the right lim_x_→1 (x² - 1) / (x⁴ - 1) = ?

0.5

does not exist

∞

-1

Açıklama:

(x² - 1) / (x² - 1) (x² + 1) = 1 / (x² + 1) ; 1 / (1 + 1) = 0.5 ; x ≠ 1 . pg. 108 . Correct answer is A.

Doğru Cevap: A

Soru 46

While approaching from the right lim_x_→-2 |x + 2| / (x + 2) = ?

-1

does not exist

∞

Açıklama:

(x + 2) / (x + 2) = 1 . pg. 109. Correct answer is C.

Doğru Cevap: C

Soru 47

lim_x_→-_∞ (x⁴ + x³ - 1) / (x⁵ + x² + 1) = ?

does not exist

∞

-∞

Açıklama:

0 ; Since the degree of the numerator is less than the degree of the denominator. pg. 114. Correct answer is D.

Doğru Cevap: D

Soru 48

What is the set of continuity of f(x) = x² / (4 - x²) ?

R \ {-2, 2}

R \ {-2}

(-∞, ∞)

R \ {2}

Açıklama:

R \ {1} ; (4 - x²) = 0 ; x = {-2, 2}. pg. 117. Correct answer is A.

Doğru Cevap: A

Soru 49

lim_x_→-_∞ (-x²⁶ + 1) / (x²⁵ + x²⁴ - 1) = ?

-1

∞

does not exist

-∞

Açıklama:

∞ ; Since the degree of the numerator is greater than the degree of the denominator. pg. 114. Correct answer is C.

Doğru Cevap: C

Soru 50

lim_x_→-_∞ (10 x³⁹⁷ + 1) / (-2 x³⁹⁷ + 2 x³⁹⁶ - 1) = ?

∞

-5

-∞

does not exist

Açıklama:

-5 ; Since the degree of the numerator is equal to the degree of the denominator. pg. 114. Correct answer is B.

Doğru Cevap: B

Soru 51

Assume that a bus travels 2 hours and covers a distance of 140 km and travels for 2 hours more and covers a total distance of 300 km. What would be the average velocity of this bus in its last 2 hours of travel?

Açıklama:

Let's denote the average velocity by V_a, denote the distance by X and denote time by t.
Then V_a=(X₂-X₁)/(t₂-t₁)=(300-140)/(4-2)=80

Doğru Cevap: A

Soru 52

Find the limit of the function f(x)=(x²-5x+6)/(x²-9) for x=3.

1/6

1/5

2/3

1/3

Açıklama:

f(x)=(x²-5x+6)/(x²-9) = 0/0 which is indefinite for x=3. Thus we have to factorize the denominator and nominator of the function.
f(x)=(x²-5x+6)/(x²-9)=((x-3)(x-2))/((x+3)(x-3))=(x-2)/(x+3)
thus for x=3 lim f(x)=(3-2)/(3+3)=1/6

Doğru Cevap: B

Soru 53

For which values of x is the function f(x)=(x²-5x+6)/(x²-9) is discontinuous?

2 and 3

-2 and -3

1 and 5

-1 and -5

3 and -3

Açıklama:

A function would be discontinuous when the value of function is undefined. In this case the values of x which make the denominator (x²-9) equal to zero will make the functiıon discontinuous. Thus for x=3 or x=-3, the denominator x²-9=0

Doğru Cevap: E

Soru 54

Find the limit of the following function for x=2.

-2

-4

Açıklama:

We can take limits from left and right sides.
From left: lim f(x)=4-2*2=0
From right lim f(x)=2-2=0
So limit exists and it is equal to zero. This is obvious from the graph of the function:

Doğru Cevap: C

Soru 55

Assume that for x=2, the left limit of f(x) is "a" for x=2, and the right limit of f(x) is "b". Then what will be a-b?

Açıklama:

For x=2:
the left limit of f(x)=9-x²=9-4=5=a
the right limit of f(x)=x-2=2-2=0=b
Then a-b=5-0=5
One can better understand from the graph of the function:

As you can see the function is discontinuous at x=2. So the left and right limits at that point are not equal to each other.

Doğru Cevap: D

Soru 56

For which of the following functions the limit is equal to zero as x goes to infinity?

f(x)=(2x-1)/(5x+3)

f(x)=(3x²-1)/(x+2)

f(x)=(x³-1)/(4x²+5)

f(x)=(x⁴-10)/(x²-5)

f(x)=(x³-1)/(4x⁷+15)

Açıklama:

The biggest power in denominator must be higher than the biggest power in nominator for limit being equal to zero if x goes to infinity. So the answer is E as 3<7.

Doğru Cevap: E

Soru 57

Which of the following functions' limit is infinite for x=3?

f(x)=(x²+3x+2)/((x²-x-6)

f(x)=(x²-9)/((x²-3x)

f(x)=(x²+2x-15)/((x²-2x-3)

f(x)=(x²+x-12)/((x²+3x-18)

f(x)=(x²-3x)/((x²-9)

Açıklama:

lim f(x)=lim (x²+3x+2)/((x²-x-6)=lim (x+1)(x+2)/(x-3)(x+2)=lim (x+1)/(x-3)=4/0=infinity
So the answer is A. In all other functions limit goes to a finite number.

Doğru Cevap: A

Soru 58

Which of the following functions is continious at x=2?

f(x)=x/(x-2)

f(x)=x³/(x²-4)

f(x)=(x³-8)/(x²-4)

f(x)=(x³-2x-6)/(x²-4)

f(x)=(5-x)/(3x-6)

Açıklama:

(x³-8)/(x²-4)=(x-2)(x²+2x+4)/(x-2)(x+2)=(x²+2x+4)/(x+2)=19/4 for x=2. All other expressions are in the form of a/0 for x=2; which equals to infinity, and therefore not continuous. So the answer is C.

Doğru Cevap: C

Soru 59

In which domain is the function f(x)=(x³-8)/(x²-4) defined?

R-{-2}

R-{2}

R-{-2,2}

R-{-2, 4}

Açıklama:

(x³-8)/(x²-4)=(x-2)(x²+2x+4)/(x-2)(x+2)=(x²+2x+4)/(x+2) which is infinite for only x=-2. Therefore the domain is R-{-2}

Doğru Cevap: B

Soru 60

Which of the following functions is continuous at x=0?

f(x)=1/x

f(x)=1/x²

f(x)=3/x³

f(x)= |3x|

f(x)=(x²-4)/(x²-2x)

Açıklama:

Other than f(x)= |3x| all the functions goes to infinity for x=0. Therefore they are not continuous. So the answer is D because lim f(x)=f(x)=0 for x=0 when f(x)= |3x|

Doğru Cevap: D

Ünite 6

Soru 1

For the function

, find the derivative

at the point x = 2.

Açıklama:

The answer is D.

Doğru Cevap: D

Soru 2

For the function

, find the derivative

at the point x=0.

e + 1

Açıklama:

The answer is A.

Doğru Cevap: A

Soru 3

Which of the following is the slope of the graph of

at the point (2, 2)?

136

100

180

Açıklama:

12.8+21.4=180
The answer is E.

Doğru Cevap: E

Soru 4

Which of the following is the slope of the graph of

at the point (0, 0)?

-2

-1

Açıklama:

The answer is E.

Doğru Cevap: E

Soru 5

Given the function

, find the derivative

-5120

-1280

-101

-1

Açıklama:

5.(1+3)^4.(2-3)=-1280
The answer is C.

Doğru Cevap: C

Soru 6

For the function

, find the second derivative

-1

25e

Açıklama:

The answer is E.

Doğru Cevap: A

Soru 7

Given function

, find the second derivative

1/4

3/4

-1/27

-2/27

Açıklama:

The answer is D.

Doğru Cevap: D

Soru 8

At which point on the graph of the function

the slope is zero?

1/3

-2/3

-1/3

-1

Açıklama:

The answer is D.

Doğru Cevap: D

Soru 9

Find the interval of decrease for the function

(-1, 1)

(0, 1)

[-1, 0)

(0, 1]

(-2, 0)

Açıklama:

The answer is A.

Doğru Cevap: A

Soru 10

For the function

, find the derivative function

Açıklama:

The answer is B.

Doğru Cevap: B

Soru 11

What is the slope of the tangent line to the curve y=3x²-1 at point (2, 11)?

Açıklama:

Let denote the slope with m. Then:

Doğru Cevap: E

Soru 12

What is the slope of the tangent line to the curve y=x³ at point (-1, -1)?

-3

-2

Açıklama:

If we denote the slope of the curve with m, then:

Doğru Cevap: A

Soru 13

For the function f(x)=-3x+11, find the derivative f'(x) at the point x=0?

-3

-5

Açıklama:

f'(x)=(-3x+11)'=-3, f'(0)=-3. So, the answer is A.

Doğru Cevap: A

Soru 14

For the function

, find the derivative f'(x), at the point x=0?

-11

Açıklama:

The answer is B.

Doğru Cevap: B

Soru 15

For the function

, find the derivative function f'(x)?

Açıklama:

The answer is D.

Doğru Cevap: D

Soru 16

Which of the following gives the average velocity of a particle in time interval (t₁, t₂) if its' position at time t is defined by f(t)?

(f(t₂)+f(t₁))/(t₂+t₁)

(f(t₂)-f(t₁))/(t₂-t₁)

(f(t₂)-f(t₁))/(t₂+t₁)

(f(t₂)+f(t₁))/(t₂-t₁)

f(t₂)/t₂-f(t₁)/t₁

Açıklama:

Average velocity could be found by dividing the position difference to the time difference.

Doğru Cevap: B

Soru 17

The position of a particle moving along the x-axis at time t is given by f(t) = 3t² -2t (metre). Find the average velocity over the interval [4, 6]

5.6

13.6

Açıklama:

f(t) = 3t² -2t
so:
f(6)=3*6² -2*6=96
f(4)=3*4² -2*4=40
Average velocity(4,6)=(f(6)-f(4))/(6-4)=(96-40)/2=28

Doğru Cevap: E

Soru 18

Which of the following gives the instantaneous velocity of a particle at time t if its' position at time t is defined by f(t)?

lim_h_→0 (f(t+h)+f(t))/h

lim_h_→0 (f(t+h)-f(t))/t

lim_h_→0 (f(t+h)-f(t))/h

lim_h_→0 (f(t-h)-f(h))/h

lim_h_→0 (f(h)-f(t))/h

Açıklama:

The velocity at time t is the derivative of function f(t) at time t which is defined by the limit given in C.

Doğru Cevap: C

Soru 19

The motion of a particle is given by the function f(t) = 2t² - t Find its velocity at the instant t = 3.

Açıklama:

velocity at time t=lim_h_→0 (f(t+h)-f(t))/h
since f(t) = 2t² - t:
lim_h_→0 (f(t+h)-f(t))/h=((2(t+h)²-(t+h))-(2t²-t))/h
=lim_h_→0 (2t²+2h²+4th-t-h-2t²+t)/h
=lim_h_→0 (2h²+4th-h)/h
=h(2h+4t-1)/h
=2h+4t-1
And for t=3 and h=0;
2h+4t-1=11

Doğru Cevap: D

Soru 20

What is the derivative of f(x)=axⁿ-bx+c?

ax-b

a-b

ax+c

anx^(n-1)-b

anx-bx

Açıklama:

df(x)/dx=f '(x)=a*n*x^(n-1)-b=anx^(n-1)-b

Doğru Cevap: D

Soru 21

What is the derivative of f(x)=(3x²+5x)/(2x-1)

(6x²-6x-5)/(4x²-4x+1)

(6x²+6x-5)/(4x²+4x-1)

(12x²-6x-5)/(4x²-4x+1)

(12x²-4x-5)/(4x²-4x+1)

1/(4x²-4x+1)

Açıklama:

If f(x)=h(x)/g(x) then f '(x)=(h '(x)g(x)-g '(x)h(x))/g²(x) from the Quitent rule.
So if f(x)=(3x²+5x)/(2x-1),
then:
f '(x)=(h '(x)g(x)-g '(x)h(x))/g²(x)=((6x+5)*(2x-1)-2(3x²+5x))/(2x-1)²
=(12x²+4x-5-6x²-10x)/(2x-1)²
=(6x²-6x-5)/(4x²-4x+1)

Doğru Cevap: A

Soru 22

Which of the following is the slope of the graph of

at the point (1, 1)?

Açıklama:

The answer is D.

Doğru Cevap: D

Soru 23

A particle moves along the x-axis in such a way that its position at time t is

. Find the average velocity over the interval [1, 3].

Açıklama:

The answer is B.

Doğru Cevap: B

Soru 24

Given the function

, find the derivative

at the point x=0.

-1

-3

Açıklama:

The answer is C.

Doğru Cevap: C

Soru 25

Given the function f(x)=ln(2x), find the second derivative

at the point x=3.

1/6

1/3

Açıklama:

The answer is E.

Doğru Cevap: E

Soru 26

At which point on the graph of the function

the slope is zero?

(-3/2, -7)

(-3/2, -37/4)

(-32/4, -3/2)

(0 ,-5)

(3, 8)

Açıklama:

So, the slope of the tangent line to the graph is zero at (-3/2, -37/4). The answer is B.

Doğru Cevap: B

Soru 27

A particle moves along the x-axis in such a way that its position at time t is

. Find the average velocity over the interval [1, 4].

124/3

111/2

110/3

Açıklama:

The answer is A.

Doğru Cevap: A

Soru 28

Given the function

find the second derivative

at the point x=1.

Açıklama:

The answer is C.

Doğru Cevap: C

Soru 29

Find the 3rd derivative of f(x)=x³+lnx. (f'''(x)=?)

6+2/x³

6-2/x³

6x-(1/x)²

6x+(1/x)²

6-1/x

Açıklama:

f(x)=x³+lnx
f '(x)=3x²+1/x
f ''(x)=6x-(1/x)²
f '''(x)=6+2/x³

Doğru Cevap: A

Soru 30

Which of the following functions is monotone increasing in interval (-∞, +∞) | x ≠0?

f(x)=x⁶

f(x)=2x³

f(x)=-2x⁴

f(x)=5x²

f(x)=10x⁸

Açıklama:

f(x)=x³
f '(x)=3x²>0 for all values of x other than x=0.
For all other functions f '(x)<0 for (-∞,0) and f '(x)>0 for x>0.

Doğru Cevap: B

Soru 31

For the function f(x)=-x²+13x-27 find the derivative at the point x=5.

Açıklama:

f(x)=-x²+13x-27
f'(x)=-2x+13
f'(5)=-2.5+13=3.

Doğru Cevap: B

Soru 32

What is the derivative of the function which is given below?
f(x)=3x.e^-5x

e^-5x

9e^-5x

-12e^-5x

(1+5x)e^-5x

(3-15x)e^-5x

Açıklama:

f(x)=3x.e^-5x
f'(x)=(3x)'.e^-5x+3x.(e^-5x)'
f'(x)=3.e^-5x+3x.(-5e^-5x)
f'(x)=(3-15x)e^-5x

Doğru Cevap: E

Soru 33

Which of the following is the slope of the graph of y=x²+7x-8 at the point (2,10)?

Açıklama:

f(x)=x²+7x-8
f'(x)=2x+7
f'(2)=11

Doğru Cevap: D

Soru 34

What is the derivative of the function which is given below?
f(x)=(x²-7)^1/2

x/(x²-7)^1/2

1/(x²-7)^1/2

2/(x²-7)^1/2

1/(2(x²-7)^1/2)

(x²-7)^1/2

Açıklama:

By using the chain rule,
f'(x)=1/2.(x²-7)-^1/2.(x²-7)'
f'(x)=1/2.(x²-7)-^1/2.2x
f'(x)=x.(x²-7)-^1/2
f'(x)=x/(x²-7)^1/2

Doğru Cevap: A

Soru 35

What is the derivative of the function f(x)=(x³-7) / (3x²+1) at the point x=1?

Açıklama:

f(x)=(x³-7) / (3x²+1)
f'(x)=[(x³-7)'.(3x²+1) - (x³-7)(3x²+1)'] / (3x²+1)²
f'(x)=[(3x²).(3x²+1) - (x³-7)(6x)] / (3x²+1)²
f'(1)=[(3).(3+1) - (1-7)(6)] / (3+1)² at x=1
f'(1)=[(12) - (-36)] / (4)²
f'(1)=3

Doğru Cevap: C

Soru 36

What is the second derivative of the function f(x)=e^3x ?

e^3x

3e^2x

6e^2x

6e^x

9e^3x

Açıklama:

f(x)=e^3x
f'(x)=3e^3x
f(x)=9e^3x

Doğru Cevap: E

Soru 37

Given function f(x) = (x+1)(3-x), find the local maximum value for this function

Açıklama:

f(x) = (x+1)(3-x)
f'(x) = (x+1)'(3-x)+ (x+1)(3-x)'
f'(x) = (3-x)+(-x-1)
f'(x) = 2-2x
f'(x) =2-2x=0 → x=1
f(1) = 4

Doğru Cevap: D

Soru 38

For the function f(x)=x²-4x+12 which of the followings is correct?

f'(x)=2x

f''(x)=0

The function has local minimum at (2,8)

The function has local maximum at (2,8)

The function has no local minimum or local maximum point

Açıklama:

f(x)=x²-4x+12
f'(x)=2x-4
f'(x)=2x-4=0 →x=2
f''(x)=2, 2>0 local minimum.
f(2)=4-8+12=8

Doğru Cevap: C

Soru 39

Three sides of a rectangular enclosure having one side along a wall must be fenced. Assume that 500 m of fence is available. Find the largest possible area of the enclosure.

7500 m²

15625 m²

25000 m²

31250 m²

50000 m²

Açıklama:

Since the perimeter is x + 2y and the total fence available is 500 m we write x + 2y = 500. From this, x is found as,
x=500-2y and the area of this rectengular is x.y=(500-2y).y = 500y-2y²
f(y)=500y-2y²
f'(y)=500-4y
f'(y)=500-4y=0 (for local maximum value)
y=125m and x=250 m
max area is x.y=125m*250m=31250 m²

Doğru Cevap: D

Soru 40

At which point on the graph of the function y = x² - 6x - 1 the slope is zero?

(3,-10)

(3,10)

(-3,-10)

(-3,10)

(0,3)

Açıklama:

f(x)=x² - 6x - 1
f'(x)=2x - 6
for x=3, f'(x)= 0
and the output of the function at x=3 is f(3)=(3)² -6(3)- 1=9 - 18 - 1 = -10
The point is (3,-10)

Doğru Cevap: A

Soru 41

What is the slope (f'(x)) of function f(x)=2x³-3x²+4x-3 at the point x=2?

Açıklama:

If f(x)=2x³-3x²+4x-3, then f'(x)=6x²-6x+4. And for x=2 f'(x)=16

Doğru Cevap: B

Soru 42

The position of a vehicle at time t is defined by the function f(t)=5t²-2t+1. What is the average average velocity of vehicle between time interval [1,3]?

15/4

Açıklama:

The average velocity is defined by v_av=(f(t₂)-f(t₁))/(t₂-t₁). So we have to calculate f(t) for t=1 and for t=3.
f(t)=5t²-2t+1
f(3)=45-6+1=40
f(1)=5-2+1=4
Then, v_av=(f(t₂)-f(t₁))/(t₂-t₁)=(40-4)/(3-1)=18 for interval [1,3].

Doğru Cevap: C

Soru 43

Which one can be said true for derivatives?

The derivative of a differentiable function's second derivative can be called the third derivative of the function.

The derivative of a differentiable function can be called the second derivative of the function.

For the identity function f(x) = x, the derivative is x, that is, (x)' = x.

The slope of the secant line to a curve y = f(x) at (x₀, f (x₀ )) is m = f ' (x₀).

A derivative of a constant is equal to that constant.

Açıklama:

If a function y = f(x) is differentiable, its derivative f '(x) is a function of x.
If the derivative f '(x) is differentiable, the derivative of f '(x) is called the second derivative of y = f(x) and is denoted by y˝=f ˝(x).
The derivative of the second derivative, i.e., f ˝(x) is called the third derivative which is denoted by y'''= f ''' (x).
For a constant function f(x) = c the derivative is zero, that is, (c)' = 0.
The slope of the tangent line to a curve y = f(x) at (x0, f (x0 )) is m = f ' (x0).
For the identity function f(x) = x, the derivative is 1, that is, (x)' = 1. The answer is A.

Doğru Cevap: A

Soru 44

Wthat is he slope of the tangent line to the curve y = x³ - 2x + 5 at the point (0, 4) equal to?

-2

-4

Açıklama:

The derivative of the function f is equal to 3x² - 2. When the point is plug into the derivative it yields to 3*(0)² - 2 = -2. The answer is -2, choice A.

Doğru Cevap: A

Soru 45

A car moves along a road in such a way that its position at time t is f(t) = t³ + t² + t (metre). What is the average velocity over the interval [1, 5] equal to?

38,5

39,5

Açıklama:

For the car given, t₁=1 and t₂=5. The values of f at these points are f(t₁) = f(1) = 1³+1²+1 = 3,
f(t₂) = f(5) = 5³ + 5² +5 = 155. So, the average velocity is equal to (155 - 3)/(5 - 1) = 152/4 = 38 metres/seconds. The answer is A.

Doğru Cevap: A

Soru 46

A car moves along a road in such a way that its position at time t is f(t) = t³ + t² + t (metre). What is the velocity of the car at the instant t=2 equal to?

Açıklama:

In order to find the instantenous velocity the derivative of the function must be taken. This is equal to 3t² + 2t + 1. The velocity at this instant is equal to the value found when t=2 is plugged in: (3*2²)+ (2*2) + 1 = 12 + 4 + 1= 17(metre/seconds). The answer is B.

Doğru Cevap: B

Soru 47

What is the derivative of f(x) = 2x³ + 5x + 3 at the point x = 3 equal to?

Açıklama:

The derivative of f(x) = 2x³ + 5x + 3 is equal to f'(x) = 6x² + 5. At point x=3, f'(3) = 6*(3)² + 5 = 54 + 5 = 59. The answer is C.

Doğru Cevap: C

Soru 48

For the function f(x) = 2x³ + 3x² + 5x what is the sum of f'(2) and f(1) equal to?

Açıklama:

The derivative of the function f(x) = 2x³ + 3x² + 5x is equal to 6x² + 6x + 5. f'(2) = 6*(2)² + 6*2 + 5 = 24+12+5=41 and f(1) = 2*(1)³ + 3*(1)² + 5*1 = 2+3+5=10. the sum of these two values 41+10= 51. The answer is C.

Doğru Cevap: C

Soru 49

What is the derivative of (2x + 1)³equal to?

2(2x + 1)²

3(2x + 1)²

6(2x + 1)²

6(2x + 1)³

Açıklama:

The Chain Rule: The derivative of composite function f (g(x)) is f'(g(x)) * g' (x). For this question our g(x) = 2x+1. The derivative of (2x + 1)³yields to 3(2x + 1)²* 2 which is equal to 6(2x + 1)². The answer is D.

Doğru Cevap: D

Soru 50

Which of the statements given is true for the local extreme points of the function x³ + 2x² - 4x ?

x=-2 is a local minimum

x=2/3 is a local maximum

x=-2 is a local minimum and x=2/3 is a local maximum

x=-2 is a local maximum and x=2/3 is a local minimum

x=-2 is a local maximum, x=2/3 is a local minimum and x=2 is a local minimum

Açıklama:

For the function x³ + 2x² - 4x the derivative is equal to 3x² + 4x - 4 and the second derivative is equal to 6x + 4. If we equal the derivative functions to zero we would have 3x² + 4x - 4=0 is (3x-2)(x+2)=0 x will be equal to -2 and 2/3, these are the critical points. When we plug each one to the second derivative we will find 6(-2)+4=-8 and 6(2/3)+4=8. -8<0 which makes x=-2 a local maximum and 8>0 which makes x=2/3 a local minimum. The answer is D.

Doğru Cevap: D

Soru 51

If f(x) is differentiable on an interval I,
I. f '(x) < 0 for all x ∈ (a, b) then f(x) is called monotone increasing
II. f '(x) > 0 is satisfied for all x ∈ (a, b) then f(x) is called monotone increasing
III. f '(x) > 0 is satisfied for all x ∈ (a, b) then f(x) is called monotone decreasing
IV. f '(x) < 0 for all x ∈ (a, b) then f(x) is called monotone decreasing
which of the given statements is true about monotonicity?

I and II

I and III

I and IV

I, II and III

I, II and IV

Açıklama:

The monotonicity property of a function can be investigated by the sign of the derivative. If the
inequality f '(x) > 0 is satisfied for all x ∈ (a, b) then f(x) is called monotone increasing on the
interval (a, b); likewise, if f '(x) < 0 for all x ∈ (a, b) then f(x) is called monotone decreasing on
the interval (a, b). The answer is C.

Doğru Cevap: C

Soru 52

For the given f(x) =2x² + 2x + 2 function which of the statements are true?

This function has a local maximum and minimum point.

The critical value is equal to zero.

The critical value is a positive number.

This function has a local maximum point.

This function has a local minimum point.

Açıklama:

Second derivative test for local extrema: if a is a critical point and the second derivative f ˝(a) is
positive, i.e. f ˝(a) > 0, then a is a local minimum point. If f ˝(a) is negative, i.e. f ˝(a) < 0, then a
is a local maximum point. For f(x) the derivative is equal to f'(x)=4x+2 and the second derivative is equal to f''(x)=4 this value is a positive value which indicates a local minimum point. If we equal the derivative to zero, f'(x)=4x+2=0 the x value would be equal to -1/2. The answer is E.

Doğru Cevap: E

Soru 53

What is the first derivative of the function f(x)=x²+2x+lnx at point x=1?

3+ln2

Açıklama:

f(x)=x²+2x+lnx
f'(x)=2x+2+(1/x) and for x=1 f'(1)=2+2+1=5

Doğru Cevap: D

Soru 54

At which (x,y) point is the value of the function
y=f(x)=x²-6x+5 is minimum?

(0,5)

(3,0)

(6,1)

(2,-3)

(3,-4)

Açıklama:

The first derivative must be equal to zero and the second derivative must be positive for a minimum value of a function. Thus:
f(x)=x²-6x+5
f'(x)=2x-6=0 so 2x=6, x=3 and when x=3 y=f(x)=9-18+5=-4
The second derivative is:
f''(x)=2>0
Thus for given function, point (3,-4) is the minimum.

Doğru Cevap: E

Soru 55

What is the second derivative(f''(x)) of the function f(x)=x²+lnx-1?

2-(1/x²)

2x+(1/x)

2x-(1/x)

2+(1/x²)

2x-1

Açıklama:

f(x)=x²+lnx-1
f'(x)=2x+(1/x)
f''(x)=2-(1/x²)

Doğru Cevap: A

Soru 56

What is the largest possible area of a rectangle if its perimeter is 80?

800

600

400

300

200

Açıklama:

Let x and y be the side lengths of the rectangele. If the perimeter is 80, then 2x+2y=80
x+y=40
y=40-x
The are of the rectangle is A=x*y=x*(40-x)=40x-x²
Since it is a maximization problem the first derivative of the area function (A) must be equal to zero and the second derivative must be negative.
A''(x)=-2<0 (Second derivative rule satisfied)
A'(x)=40-2x=0, 40=2x, 20=x, y=40-x=20 So the area A=20*20=400

Doğru Cevap: C

Soru 57

Given that g(x)=2x+1 and f(x)=x^1/2what is the derivative of f(g(x))?

(2x+1)^-1/2

x^-1/2

x^1/2

Açıklama:

f(g(x))=(2x+1)^1/2
From the chain rule we know that (f(g(x)))'=f'(g(x))g'(x)
Applying the chain rule we get (f(g(x)))'=(1/2)(2x+1)^-1/22=(2x+1)^-1/2

Doğru Cevap: A

Soru 58

What is the equation for the tangent line to the function f(x)=x²-2x+1 at point (2,1)?

y=x-1

y=2x-3

y=3x-5

y=x+1

y=x

Açıklama:

The equation of tangent line is given by the formula:
y - f(x₀) = f'(x₀)(x - x₀)
x₀=2 and f(x₀)=1 are given.
Since f(x)=x²-2x+1 then f'(x)=2x-2 and f'(x₀)=2*2-2=2 for x₀=2.
Thus the equation for the tangent line will be:
y-1=2(x-2)
y-1=2x-4
y=2x-3

Doğru Cevap: B

Soru 59

Which of the following is an extreme point for the function f(x)=x³-4x?

x=0

x=2

x=-2

x= (√ 2)/ 3

x=1

Açıklama:

For the extreme points the first derivative must be equal to zero. Since f(x)=x³-4x The first derivative of function f(x) will be:
f'(x)=3x²-4
Then we find the roots of the f'(x) as following:
3x²-4=0
3x²=4
x= -√ 2/3 and x= √ 2/3 are the roots of this equation. So the answer is D.

Doğru Cevap: D

Soru 60

On which interval is the function f(x)=x³-12x increasing?

(-∞, -2)U(0,2)

(-∞, -3)U(3,∞)

(-∞, -4)U(4,∞)

(-∞, -1)U(2,∞)

(-∞, -2)U(2,∞)

Açıklama:

A function is increasing where its first derivative is positive. Thus, we have to find the intervals where the first derivative of function is positive.
f(x)=x³-12x
f'(x)=3x²-12
So the function is increasing where f'(x)>0
Namely:
3x²-12>0
3x²>12
x²>4
This is possible when x>2 and when x<-2
Thus the function is increasing in the interval (-∞, -2)U(2,∞)

Doğru Cevap: E

Soru 61

What is the slope of the tangent line to the function f(x)=x²+3x at point x=2?

Açıklama:

We have to find the derivative of f(x) for x=2.
Let's find it by the help of limit:

So the answer is A.

Doğru Cevap: A

Soru 62

What is the average velocity of a particle in time interval [4,6] whose time(t) related position is given by the function f(t)=t²+4t?

Açıklama:

average velocity in time interval [x,y]=(f(y)-f(x))/(y-x)
So for given function average velocity in [4,6]=(f(6)-f(4))/(6-4)=((6²+4*6)-(4²+4*4))/(6-4)=28/2=14

Doğru Cevap: E

Soru 63

The motion of a particle is given by the function f(t) = 2t²-4t+3. Find its velocity at the instant t = 3.

Açıklama:

The velocity at time t is given by the first derivative of the motion function. So we havee to find f '(t) for f(t) = 2t²-4t+3.
f '(t)=4t-4
Thus for t=3 f '(t)=4t-4=4*3-4=8

Doğru Cevap: D

Soru 64

What is the first derivative of the function f(x)=(2/x)+3x²

6x+2

3x+2

6x-(2/x²)

6x+(2/x²)

3x-(2/x²)

Açıklama:

Remember the derivative rule (x^a)'=ax^a-1. So, applying this rule:
f(x)=2x^-1+3x²
f '(x)=2*(-1)x^-1-1+3*2x=-2x^-2+6x=6x-(2/x²)

Doğru Cevap: C

Soru 65

Which of the functions is not differentiable for x=0?

f(x)=3x²

f(x)=2x

f(x)=6

f(x)= |4x|

f(x)=3^x

Açıklama:

Only the absolute value function is not differentiable at zero because the left limit and right limit are not equal.

Doğru Cevap: D

Soru 66

What is the first derivative of function f(x)=x²e^x

2xe^x

2xe^x+x²e^x

2xe^x-x²e^x

2x²e^x

2xe^x+x²lnx

Açıklama:

We have to appy chain rule here.
let f(x)=h(x)g(x) where g(x)=e^x and h(x)=x²
f '(x)=h'(x)g(x)+h(x)g'(x)
Since h'(x)=2x and g'(x)=e^x
h'(x)=2xe^x+x²e^x

Doğru Cevap: B

Soru 67

Find the second derivative of f(x)=2x³+(1/x)

12x+(2/x³)

12x-(2/x³)

12x-(1/x³)

12x+lnx

12x-lnx

Açıklama:

f(x)=2x³+(1/x)
f '(x)=6x²-(1/x²)
f ''(x)=12x+(2/x³)

Doğru Cevap: A

Soru 68

For which values of x does the function f(x)=0.5x⁴-4x² has extremum values?

0, 2 and 4

-2 and 0

0 and 2

-2 and 2

-2,0 and 2

Açıklama:

f(x)=0.5x⁴-4x²
f '(x)=2x³-8x=2x(x²-4)
f '(x)=0 for x=-2, x=2, and x=0

Doğru Cevap: E

Soru 69

A rectangular land plot has a perimeter of 200 metres. What is the possible largest are of this land plot in squaremeters?

2000

2500

3000

3200

3500

Açıklama:

Let x and y be the length of sides of the land plot. Then 2x+2y=200, so x+y=100
We can rewrite this equation as y=100-x
The area of rectangle is A=xy=x(100-x)=100x-x²
A is maximum when dA/dx=A'(x)=0 and A''(x)<0.
Thus A'(x)=100-2x=0 so x=50 and y=100-x=50.
So the area A=50*50=2500
Note that A''(x)=-2<0. Thus it is a maximum point.

Doğru Cevap: B

Soru 70

In which interval is the function f(x)=x²-4x monotone increasing?

x<2

Açıklama:

For a function to be monotone increasing the first derivative of it must be positive.
Thus for f(x)=x²-4x the first derivative is f '(x)=2x-4
Since 0<2x-4
4<2x
2

Doğru Cevap: C

Ünite 7

Soru 1

The demand function of a product is given by p=100-4x, and the supply function is p=1/2x-10. Which of the following is the equilibrium price?

11/9

20/9

10/9

Açıklama:

Writing the demand function for x as x=25-1/4p, and writing the supply as x=2p+20 the equilibrium price may be found as
25-1/4p=2p+20
which is written as
25-20=2p+1/4p
and the price is found as
p=20/9. Therefore, the answer is D.

Doğru Cevap: D

Soru 2

If the demand function of a particular commodity is given by p(q)=350-(q/5) what is the price when the demand is of unit elastic?

175

150

275

135

275

Açıklama:

Let us write the demand function for q
q=1750-5p
Taking the derivative with respect to p we get
dq/dp = -5
The price elasticity of demand is

From this equation we can easily write the price for the unit elasticity which is
1=5p/(1750-5p)
and the sought for price is p=175. Therefore, the answer is A.

Doğru Cevap: A

Soru 3

If the demand function of a particular commodity is given by p(q)=900-q/3 what is the price when demand is of unit elastic?

900

175

450

270

350

Açıklama:

Let us write the demand function for q
q=2700-3p
Taking the derivative with respect to p we get
dq/dp = -3
The price elasticity of demand is

From this equation we can easily write the price for the unit elasticity which is
1=3p/(2700-3p)
and the sought for price is p=450. Therefore, the answer is C.

Doğru Cevap: C

Soru 4

A company produces a certain item whose demand function is p=-4x+150, and supply function is p=6x-275. For what values of x is there a market shortage?

x>42,5

x<42,5

x<21

x>21

x<47,5

Açıklama:

We first find the equilibrium point by equating the supply and demand functions. Therefore, the equality
-4x+150=6x-275, x=42,5. Because there will be a shortage in the market, to the left of this point the correct is x<42,5. Therefore, the answer is B.

Doğru Cevap: C

Soru 5

Given that the demand function is p=60-x/3, the fixed cost 120 TL, and the variable cost for each item produced is 3 TL, what is the maximum profit?

450,5

550,75

2317,86

1536

2316,75

Açıklama:

We know that the total revenue function is given by R=p*x,

Therefore, the answer is E.

Doğru Cevap: E

Soru 6

The bus company BusAway determines that when a return ticket between Eskişehir and Ankara costs p TL (0

6,5

10,8

12,8

Açıklama:

We need to equate the demand of elasticity to one,

The answer is D.

Doğru Cevap: D

Soru 7

The fixed costs of a product is 2750 TL, variable cost of one unit is 1,5 TL, and the selling price of the product is 3 TL. Which of the following is the break-even point?

1850

1833,33

1733

1633,33

750,35

Açıklama:

Break-even point is the point where the total cost and the total revenue are equal C(x)=R(x).
For our question, R(x)=p*x=3x and C(x)=v*x+a=1,5x+2750.
1,5x+2750=3x, 1,5x=2750, x=1833,33. The answer is B.

Doğru Cevap: B

Soru 8

Let p denote the price and q denote the quantity of a product. If the demand to this product is given by q(p)=450-2(p²), what is the price elasticity of demand when p=10?

1,6

1,2

1,5

2,6

2,5

Açıklama:

Once again price elasticity of demand is

The answer is A.

Doğru Cevap: A

Soru 9

If the total cost function of a product is given by C(x)=4x+750, and the demand is p=10-(2x/4) which of the following is the maximum profit obtained through the sales of this product?

Açıklama:

The total revenue function for the product is,
R(x)=p*x=10-(2x/4)*x=10x-2(x^2)/4
Since the total profit is the difference between the total revenue and the total cost function we have

The answer is C.

Doğru Cevap: C

Soru 10

A company produces a certain item whose demand function is p=-5x+500, and supply functionis p=2x-200. For what values of x is there a market shortage?

x>25

x>120

x<120

x>100

x<100

Açıklama:

We first find the equilibrium point by equating the supply and demand functions. Therefore, the equality
-5x+500=2x-200
gives
7x=700, x=100
Since there will be a shortage in the market, to the left o this point the correct answer x<100. The answer is E.

Doğru Cevap: E

Soru 11

________ tells us the desire or willingness of a customer to pay a certain price for a particular item she wants to buy.

Supply

Market

Demand

Equilibrium

Shortage

Açıklama:

in the simplest terms the definition of demand tells us the desire or willingness of a customer to pay a certain price for a particular item she wants to buy. Suppose that the seasonal price of a pair of sneakers is 180 TL. The same sneakers cost 100 TL in the discount season. A customer therefore may prefer to buy it at the discount season, and even maybe buys two pairs instead of one pair. Therefore, demand is proportional to the price of the goods.

Doğru Cevap: C

Soru 12

The point where the quantity demanded equals the supply provided is called ________.

Supply

Demand

the Equilibrium Point

Shortage

the Break-Even Point

Açıklama:

The point where the quantity demanded equals the supply provided is called the equilibrium point.

Doğru Cevap: C

Soru 13

_______ is the point at which cost or expenses and revenue are equal.

Surplus

the Equilibrium Point

Shortage

the Break-Even Point

the Total Revenue

Açıklama:

The break-even point is the point at which cost or expenses and revenue are equal: there is no net loss or gain, and one is said to have “broken even.”

Doğru Cevap: D

Soru 14

The additional cost needed to produce or purchase one more unit of a good or service is called________.

the Marginal Analysis

the Marginal Cost

the Marginal Revenue

the Marginal Profit

the Margianl Risk

Açıklama:

The additional cost needed to produce or purchase one more unit of a good or service is called the marginal cost. It corresponds to the derivative of the total cost function C(x).

Doğru Cevap: B

Soru 15

A hardware manufacturer company sells its new product for 130 TL per item. Total cost consists of a fixed cost of 4400 TL, and the production cost of 50 TL per item. How many items must the manufacturer sell to gain a profit of 2500 TL?

80 units

82,65 units

86,25 units

87,75 units

85,50 units

Açıklama:

If the company wants to make a profit of 2500TL then the profit function must be equal to this quantity. In other words,
P(x)= 2500 ⇒ 80 x−4400=2500⇒80 x=6900 so that
x=86,25 units.

Doğru Cevap: C

Soru 16

If demand is elastic, which of the followings is correct?

|E_p|<1, revenue R increases as price p increases.

|E_p|>1, revenue R(p)=p q(p) decreases as price p increases.

|E_p|<1, revenue R decreases as price p increases.

|E_p|>1, revenue R(p)=p q(p) increases as price p increases.

|E_p|=1, revenue is unaffected by a small change in price.

Açıklama:

If demand is elastic, i.e. |E_p|>1, revenue R(p)=p q(p) decreases as price p increases.
If demand is inelastic, i.e. |E_p|<1, revenue R increases as price p increases.
If demand is of unit elasticity, i.e. |E_p|=1, revenue is unaffected by a small change in
price.

Doğru Cevap: B

Soru 17

A hardware manufacturer company sells its new product for 130 TL per item. Total cost consists of a fixed cost of 4400 TL, and the production cost of 50 TL per item. What is the break-even point?

Açıklama:

Let x denote the number of items produced and sold. The total revenue is
R (x) = 130 x
and the total cost is
C (x) = 50 x + 4400
To find the break-even point we equate total cost to total revenue, so
C(x)=R(x)⇒ 50 x+4400=130 x⇒80 x=4400
so that
x =55
Thus, the manufacturer has to sell at least 55 units to break even, i.e. no profit, or no loss.

Doğru Cevap: C

Soru 18

The mobile phone manufacturer BuyMe&UseMe predicts that the demand to their brand new smartphone will be 2000 units if its price is set to 1200 TL, and the demand will be 3000 units if the price is reduced 200 TL per item. What is the demand function for this new product?

p=-x/4 +1600

p=-x/5-1600

p=-x/5 +2000

p=-x/5 +1600

p=x/6 +1600

Açıklama:

let x denote the demand and p the price of the product. We are given (x₁,p₁)=2000,1200) and (x2, p2)=(3000,1000), since there is TL discount in the price. The demand line that passes through these points has slope m=-1/5 and its equation is p=-x/5 +1600

Doğru Cevap: D

Soru 19

If demand is of unit elasticity,which of the following is true?

E_p=0

E_p=1

E_p>1

E_p<1

E_p<0

Açıklama:

If demand is of unit elasticity, i.e. |E_p|=1, revenue is unaffected by a small change in price.

Doğru Cevap: B

Soru 20

The private electric company EsLeki supplying electric to the citizens of Eskişehir has a monthly demand of p(x)=200-3x, where x corresponds to one kilowatt hour. The cost function of the company is given by C(x)=75+80x-x2, 0≤x≤40. What is the value of x ?

Açıklama:

The profit function we would like to maximise is P(x)=R(x)−C(x)
R(x)=x p(x)=x ⋅(200−3x)=200x −3x²
so that
P(x)=−2x ² +120x−75
To find the maximum price, let us take the derivative of P(x) and equate it to zero:
P'(x)=−4x+120 = 0 ⇒ x=30

Doğru Cevap: C

Soru 21

The demand function of A is p = 1.200 - 3 x, and the supply function of A is p = 5 x - 400. Which of the following is the equilibrium price ?

600

500

400

300

200

Açıklama:

p : price ; x : quantity ; at equilibrium : 1.200 - 3 x = 5 x - 400 ; 1.600 = 8 x ; x = 200 ; p = 600. pg. 160. Correct answer is A.

Doğru Cevap: A

Soru 22

The demand function of A is p(q) = 515 - (q / 5). What is the price when the demand is of unit elastic ?

555.0

812.5

257.5

1.110.0

55.5

Açıklama:

E_p = |(p / q) (dq / dp)| = 1 for unit elastic ; q = 2.575 - (5 p) ; dq / dp = -5 ; E_p = |(p / (2.575 - (5 p))) (-5)| = 1 ; p / (2.575 - (5 p)) (-5) = 1 ; 5 p / (2.575 - 5 p) = 1 ; 10 p = 2.575 ; p = 257.5 . pg. 166. Correct answer is C.

Doğru Cevap: C

Soru 23

The demand function of A is p = 60 - 3 x / 4, the fixed cost is 120 TL, and the variable cost is 10 TL for each units produced ; which of the following is the profit function ?

-0.75 x² - 60

-0.75 x² + 50 x - 180

-0.75 x² + 70 x - 120

-0.75 x² + 50 x - 120

-0.75 x² - 50 x - 120

Açıklama:

P(x) = R(x) - C(x) ; R(x) = (60 - (3 / 4) x) x ; C(x) = 120 + 10 x ; P(x) = -0.75 x² + 50 x - 120 . pg. 161. Correct answer is D.

Doğru Cevap: D

Soru 24

The demand function of A is p = 40 - (x / 2), the fixed cost 48 TL, and the variable cost for each item produced is 4 TL ; what is the maximum profit ?

448

860

600

1.022

1.296

Açıklama:

P(x) = R(x) - C(x) ; R(x) = (40 - (x / 2)) x ; C(x) = 48 + 4 x ; P(x) = -0.5 x² + 36 x - 48 ; P'(x) = -x + 36 = 0 ; x = 36 ; P(36) = 600 . pg. 164. Correct answer is C.

Doğru Cevap: C

Soru 25

The demand function of A is p = -3 x + 3333, and supply function is p = 5 x + 333. For what values of x is there a surplus for A ?

< 403

> 330

< 220

> 111

> 375

Açıklama:

-3 x + 3.333 = 5 x + 333 ; 8 x = 3.000 ; for surplus : x > 375 . pg. 160 . Correct answer is E.

Doğru Cevap: E

Soru 26

The demand for A is q = 600 - 0.02 p² , (0 ≤ p ≤ 250). Which of the following is the value of p for which the demand is of unit elastic ?

100

Açıklama:

E_p = |(p / q) (dq / dp)| = 1 for unit elastic ; dq / dp = -0.04 p ; Ep = |(p / (600 - (0.02 p²))) (-0.04 p)| = 1 ; p / (600 - (0.02 p2)) (-0.04 p) = 1 ; 0.04 p² / (600 - 0.02 p²) = 1 ; 0.06 p² = 600 ; p² = 10.000 ; p = 100 . pg. 166. Correct answer is C.

Doğru Cevap: C

Soru 27

The fixed cost of a product is 4.000 TL, variable cost for one unit is 0.4 TL and the selling price of the product is 4.4 TL. Which of the following is the break-even point ?

1.400

900

800

1.200

1.000

Açıklama:

R(x) = p x = 4.4 x ; C(x) 4.0000 + 0.4 x ; 4.4 x = 4.000 + 0.4 x ; 4 x = 4.000 ; x = 1.000 . pg. 161. Correct answer is E.

Doğru Cevap: E

Soru 28

The demand function of A is q (p) = 492 - 0.03 p² ; what is the price elasticity of demand when p = 20 ?

0.05

0.25

0.6

1.2

1.5

Açıklama:

E_p = |(p / q) (dq / dp)| = ; dq / dp = -0.06 p ; E_p = |(p / (492 - (0.03 p²))) (-0.06 p)| = 0.06 p² / (492 - (0.03 p²)) ; p = 20 ; E_p = 0.06 x 400 / (492 - 0.03 x 400) = 24 / (492 - 12) = 24 / 480 = 0.05 . pg. 160. Correct answer is A.

Doğru Cevap: A

Soru 29

The total cost function of A is C(x) = 3 x + 600, and the demand is p = 30 - (x / 4) ; which of the following is the maximum profit obtained through the sales of this product ?

100

129

140

118

144

Açıklama:

P(x) = R(x) - C(x) ;
R(x) = (30 - (x / 4)) x ;
P(x) = -0.25 x² + 27 x - 600 ;
P'(x) = -0.5 x + 27 = 0 ; x = 54 ;
P(54) = 27² - 600 = 729 - 600 = 129. pg. 164. Correct answer is B.

Doğru Cevap: B

Soru 30

The total cost of A is C (x) = 2.400 + (2 x² − 120 x) / 3 , where x represents the quantity produced. Which of the following is the production quantity that makes the cost minimum ?

Açıklama:

Local minimum : C'(x) = 4 x - 120 = 0 ; x = 30 ; C'' (30) = 4 > 0 . pg. 164. Correct answer is A.

Doğru Cevap: A

Soru 31

120

240

360

480

Açıklama:

Market Equilibrium

Doğru Cevap: E

Soru 32

100

200

250

300

400

Açıklama:

ELASTICITY

Doğru Cevap: B

Soru 33

Açıklama:

MARGINAL ANALYSIS

Doğru Cevap: A

Soru 34

Açıklama:

ELASTICITY

Doğru Cevap: C

Soru 35

1563

1598

1635

1673

1796

Açıklama:

MARGINAL ANALYSIS

Doğru Cevap: D

Soru 36

Açıklama:

MARGINAL COST

Doğru Cevap: C

Soru 37

Açıklama:

MARGINAL ANALYSIS

Doğru Cevap: E

Soru 38

0,8

0,6

0,5

0,3

Açıklama:

ELASTICITY OF DEMAND

Doğru Cevap: A

Soru 39

Açıklama:

MARKET EQUILIBRIUM

Doğru Cevap: E

Soru 40

1120

1250

1280

1310

1320

Açıklama:

BREAK-EVEN ANALYSIS

Doğru Cevap: A

Soru 41

The mobile phone manufacturer BestPhone predicts that the demand to their brand new smartphone will be 5000 units if its price is set to 3000 TL, and the demand will be 6000 units if the price is reduced 500 TL per item. What is the slope of demand function?

1/2

-1/2

1/4

-1/4

Açıklama:

The slope of the demand function is

So, we find the slope as (2500-3000)/(6000-5000)=-1/2.

Doğru Cevap: B

Soru 42

Suppose that the olive oil firm Zeytindali supplies 300 bottles onto the market when the price is 100 TL They provide 500 bottles when the price goes upto 150 TL. What is the slope of the supply function?

1/2

-1/2

1/4

-1/4

Açıklama:

The slope of the demand function is

If x denotes the quantity of supply, and p is the price then we can
write (x1, p1)=(300,100) and (x2,p2)=(500,150). So, the slope is equal to 50/200=1/4.

Doğru Cevap: D

Soru 43

A smart TV brand SmartTV has a new product and they predict their demand function as a;

and ther supply function as a;

What is the market price of this product?

100

125

150

Açıklama:

The price of a product at the intersection point, provided that it is in the first quadrant, is called market price. We may easily find that the market price of the ne TV in question.

x is found as 50. When we put it on the demand or supply equations, we find market price as 125.

Doğru Cevap: D

Soru 44

A company has a new product and they predict their demand function for this product as a;

and supply function as a;

What is the market price for this product would be?

135

180

240

Açıklama:

The price of a product at the intersection point, provided that it is in the first quadrant, is called market price. Equate the supply and demand function giving;

x is found as 30. When we put it on the demand or supply equation, we find market price as 135.

Doğru Cevap: C

Soru 45

A furniture manufacturer company sells its new product for 500 TL per item. Total cost consists of a fixed cost of 12000 TL, and the
production cost of 300 TL per item. What is the break-even point?

120

150

180

240

300

Açıklama:

The break-even point is the point at which cost or expenses and revenue are equal: so;
Cost: 300x+12000
Revenue : 500x
300x+12000=500x, then x=120.

Doğru Cevap: A

Soru 46

A company produces a certain item whose demand function is

and supply function is

For what values of x is there a market shortage?

Açıklama:

We first find the equilibrium point by equating the supply and demand functions. Therefore, the equality

gives

Since there will be a shortage in the market, to the left of this point the correct answer is

Doğru Cevap: E

Soru 47

The fixed costs of a product is 8000 TL, variable cost for one unit is 3 TL and the selling price of the product is 7 TL. Which of the following is the break-even point?

400

800

1200

1600

2000

Açıklama:

Break-even point is the point where the total cost and the total revenue are equal (C(x)=R(x)).

Doğru Cevap: E

Soru 48

Let p denote the price and q denote the quantity of a product. If the demand to this product is given by

What is the price elasticity of demand when p=10?

Açıklama:

Price elasticity of demans is

Doğru Cevap: C

Soru 49

A company has a new product which has a demand function,

and supply function,

What is the market price?

1500

1600

1700

1800

1900

Açıklama:

The price of a product at the intersection point, provided that it is in the first quadrant, is called market price.

When we put it on the demand or supply equations, we find market price as 1700.

Doğru Cevap: C

Soru 50

A company produces a certain item whose demand function is

and supply function is

For what values of x is there a market surplus?

Açıklama:

We first find the equilibrium point by equating the supply and demand functions. Therefore, the equality

gives

Since there will be a surplus in the market, to the right of this point the correct answer is

Doğru Cevap: E

Soru 51

(4,5)

(3,5)

(2,5)

(1,5)

(4,4)

Açıklama:

Doğru Cevap: A

Soru 52

Açıklama:

The equilibrium point of given demand and supply functions is the point (4,5). Therefore, we must look for this point for graphs. Correct answer is B.

Doğru Cevap: B

Soru 53

18000

19000

20000

21000

22000

Açıklama:

Doğru Cevap: C

Soru 54

3400

3600

3800

4000

4200

Açıklama:

Doğru Cevap: D

Soru 55

Açıklama:

Doğru Cevap: E

Soru 56

Açıklama:

Doğru Cevap: A

Soru 57

Açıklama:

Doğru Cevap: B

Soru 58

Açıklama:

Doğru Cevap: A

Soru 59

Açıklama:

Doğru Cevap: E

Soru 60

Açıklama:

Doğru Cevap: C

Soru 61

Suppose the total cost function is 200+2x²+4x and total revenue function is 350-x²+19x, where x denotes the production amount for a company. For what value of x is the break even point?

x=12

x=10

x=7

x=5

x=3

Açıklama:

For break even point total revenue is equal to total cost. Thus:
200+2x²+4x = 350-x²+19x
3x²-15x-150=0
and dividing both sides by 3 we get:
x²-5x-50=0 so:
(x-10)(x+5)=0
Therefore x=10 and x=-5 are the roots of this equation. Since production amount can not be negative, the answer is x=10.

Doğru Cevap: B

Soru 62

Assume that total cost function is 200+3x²+5x. For which level of production is the marginal cost is equal to 95?

x=5

x=10

x=15

x=18

x=19

Açıklama:

Marginal cost is the first derivative of the total cost. Thus since TC=200+3x²+5x,
MC=dTC/dx=6x+5
So:
6x+5=95
6x=90
x=15

Doğru Cevap: C

Soru 63

Assume that total revenue function of a factory producing trucks is given by 2x²+x and the total cost function is given by 100+x²+17x. What is the profit maximizing level of production?

x=20

x=15

x=12

x=10

x=8

Açıklama:

The profit is the diiferecnce between total revenue and total cost.
Thus profit function P(x)=TR(x)-TC(x)=2x²+x - (100+x²+17x)=x²-16x-100
For maximum profit first derivative must be equal to zero. Thus:
dP(x)/dx=2x-16=0
Thus x=8

Doğru Cevap: E

Soru 64

Suppose that the demand function is p=100-2x for quantity produced (x) in a factory, where p denotes the price. What will be the marginal revenue for 6th product?

Açıklama:

Total Revenue=p*x=(100-2x)x=100x-2x²
Since marginal revenue is the first derivative of the total revenue:
MR=dTR/dx=100-4x
and for x=6, MR=100-4*6=76

Doğru Cevap: A

Soru 65

Assume that the demand function (price) for a good is defined as p(x)=100-2x. What will be the price elasticity of demand for p=40?

-3/2

-4/3

-3/4

-2/3

-1/2

Açıklama:

price elasticity of demand=(dx/dp)*(p/x)
so we first have to write x as a function of p.
Therefore:
p=100-2x
2x=100-p
x=50-0.5p
Thus dx/dp=-0.5
And for p=40, x=50-0.5*40=30
Thus for x=30 and p=40;
price elasticity of demand=(dx/dp)*(p/x)=-0.5*40/30=-20/30=-2/3

Doğru Cevap: D

Soru 66

Assume that labor demand function of a company is defined as L=300-2w where w denotes the hourly wage rate and L denotes the amount of labor hour demanded. For what level of wage rate will the labor cost be maximum for a company?

150

300

Açıklama:

Labor cost(LC)=L*w=(300-2w)w=300w-2w²
The labor cost will be maximum when the first derivative is equal to zero. Thus:
dLC/dw=300-4w=0
w=75

Doğru Cevap: C

Soru 67

For which of the following total cost functions, is the marginal cost constant?

TC=3x²+180

TC=3x+180

TC=3x³+180

TC=3x²+6x+180

TC=180/x

Açıklama:

For only total cost functions in the form of "ax" or "ax+b" the marginal cost is constant and equal to "a". The one in B satisfies this condition.

Doğru Cevap: B

Soru 68

Assume that the total cost function for a factory is defined as TC=1900+x³-3x².
For which interval of x is the marginal cost decreasing?

x<1

2>x

Açıklama:

Since TC=1900+x³-3x².
MC=dTC/dx=3x²-6x
For MC be decreasing, its first derivative, namely dMC/dx must be negative.
Thus:
dMC/dx=6x-6
So for 6x-6<0
6x<6
x<1

Doğru Cevap: A

Soru 69

Suppose that supply function for a brand of chocolate is defined as S(p)=5+4p whereas demand function for it is defined as D(p)=35-6p where p denotes its price. What will be the equilibrium price for this chocolate?

Açıklama:

Market equilibrium exists where supply is equal to demand.
Thus 5+4p=35-6p
10p=30
p=3

Doğru Cevap: E

Soru 70

Suppose the supply and demand functions for crackets are defined as S(p)=30+20p and D(p)=150-10p where p denotes its price. For which of the following price there exists an excess demand?

Açıklama:

When price is below equilibrium level there exists an excess demand since demand is higher than supply at that point. So we have to find the equilibrium price first.
S(p)=30+20p = D(p)=150-10p
120=30p
4=p
Since only 3 is less than 4 among the answers, excess demand occurs only when price is equal to 3.

Doğru Cevap: D

Ünite 8

Soru 1

lim_x_{→0, y→0} (x² + xy + y²) / (x + xy + y) = ?

does not exist

Açıklama:

(x² + x k x + k² x²) / (x + x k x + k x) = (x² (1 + k + k²)) / (x (k x + 1 + k)) = (x (1 + k + k²)) / (k x + 1 + k) = 0. pg. 182. Correct answer is B

Doğru Cevap: B

Soru 2

f(x, y) = e^y + yx² - x - 1 ; ∂f / ∂y = ?

e^y + x² - x

e^y + 2 x

e^y + 2 y x²

e^y + x²

e^y + y x²

Açıklama:

e^y + x² . pg. 183. Correct answer is D.

Doğru Cevap: D

Soru 3

f(x, y) = e^xy + xy² - xy + 2 ; ∂²f / ∂y² = ?

x e^xy + 2 x y - x

e^xy + 2 x

x² e^xy + x

x e^xy + x

x² e^xy + 2 x

Açıklama:

∂f / ∂y = x e^xy + 2xy - x ; ∂²f / ∂y² = x² e^xy + 2x . pg. 183. Correct answer is E.

Doğru Cevap: E

Soru 4

f (x, y, z) = xyz - xy - xz + yz ; grad f (-1, 2, -3) = ?

(-3, -4, -1)

(4, 2, 6)

(-5, 1, 1)

(2, -1, 4)

(1, 5, -6)

Açıklama:

grad f (x, y, z) = (∂f / ∂x , ∂f / ∂y , ∂f / ∂z) = (yz - y - z, xz - x + z, xy - x + y) = (-6 - 2 + 3 , 3 + 1 - 3 , -2 + 1 + 2) = (-5, 1, 1). pg. 187. Correct answer is C.

Doğru Cevap: C

Soru 5

z = f(x, y) is defined implicitly by xz + yz - 1 = 0 ; ∂f / ∂x (-1, 2, 1) = ?

-1

-2

Açıklama:

Doğru Cevap: A

Soru 6

z = f(x, y) is defined implicitly by xz - yz + 3 = 0 ; Which of the following ones below is the equation of the tangent plane to the graph of z = f (x, y) around the point (1, -2, -1) ?

-x + 2 y + 3 z + 8 = 0

2 x - y + 2 z - 7 = 0

-x - 3 y + 2 z + 6 = 0

x + y - z + 5 = 0

-3 x + 3 y + z + 4 = 0

Açıklama:

at (1, -2, -1) : ∂F / ∂x = z = -1 ; ∂F / ∂y = -y = 2 ; ∂F / ∂z = x - y = 3 ; -1 (x - 1) + 2 (y - (-2)) + 3 (z - (-1)) = -(x - 1) + 2 (y + 2) + 3 (z + 1) = -x + 1 + 2 y + 4 + 3 z + 3 = -x + 2 y + 3 z + 8 = 0 . pg. 187. Correct answer is A.

Doğru Cevap: A

Soru 7

What is the shortest distance between the point P(0, 2) and the line y = x ?

2^-1/2

2^3/2

2^-1

2^1/2

Açıklama:

2² = (d1)² + (d2)² = ((x1)² + (y1)²)+ ((x1)² + (2 - y1)²) = ((x1)² + (x1)²) + ((x1)² + (2 - x1)²) = (2 (x1)²) + ((x1)² + 4 - 4 (x1) + (x1)²) = 4 (x1)² + 4 - 4 (x1) ; 4 (x1)² - 4 (x1) = 0 ; x1 = 1
y1 = 1 ; d2 = 2^1/2 . pg. 193. Correct answer is E.

Doğru Cevap: E

Soru 8

z = x² - y ; x = u² v ; y = u² + v² ; at u = v = -1, ∂z / ∂u = ?

-2

-1

-4

Açıklama:

z = u⁴ v² - u² - v² ; ∂z / ∂u = 4 u³ v² - 2 u ; at u = v = -1, ∂z / ∂u = -2 . pg. 188. Correct answer is A.

Doğru Cevap: A

Soru 9

Which of the following is the local minimum point of the function f (x, y) = x² + 2 y² + 2 x + 3 y ?

(3, 5)

(-1, -4)

(3 / 5, 1)

(-2, -3 / 4)

(1, 2 / 3)

Açıklama:

∂f / ∂x = 2x + 2 = 0 ; x = -2 ; ∂f / ∂y = 4 y + 3 = 0 ; y = -3 / 4 ; a = ∂²f / ∂x² = 2 > 0 ; b = ∂²f / ∂y² = 2 ; c = ∂²f / ∂x ∂y = 0 ; D = a . b - c² = 2 . 2 - 0 = 4 > 0 ; pg. 192. Correct answer is D.

Doğru Cevap: D

Soru 10

What is the positive stationary vaule of the function f (x, y) = x - y, satisfying the condition x² - 2 y² = 10 ?

2^1/2

2^-1

3^-1

5^1/2

3^1/2

Açıklama:

L(x, y, λ )= x - y + λ (x² - 2 y² − 10) ; ∂L / ∂x = 1 + 2 λ x = 0 ; λ = -1 / (2 x) ; ∂L / ∂y = -1 - 4 λ y = 0 ; λ = -1 / (4 y) ; 2 x = 4 y ; x = 2 y ; ∂L / ∂λ = 0 ; x² - 2 y² = 10 ; (2 y)² - 2 y² = 10 ; 2 y² = 10 ; y = {-5^1/2 , 5^1/2}; x = {-2 5^1/2 , 2 5^1/2} ; f(x, y) = x - y = 5^1/2 > 0. pg. 193. Correct answer is D.

Doğru Cevap: D

Soru 11

What is the domain of the function f(x,y)=√(x²-4y²)

(-1,0)

(0,1)

(0,2)

Açıklama:

Since negative numbers do not have square roots,x²-4y² must be non-negative. Thus:
x²-4y²≥0
(x-2y)(x+2y)≥0
This can be true only if both (x-2y) and (x+2y) are non-negative or both are non-positive.
I. First we take the first case (both are non-negative).
Thus:
x-2y≥0 which means x≥2y
x+2y≥0 which means x≥-2y
Thus the solution set is x≥2y ∩ x≥-2y
II. Now we take the second case (both are non-positive)
Thus:
0≥x-2y which means 2y≥x
0≥x+2y which means -2y≥x
This can only be true when x=y=0, so the answer is

, the answer is A

Doğru Cevap: A

Soru 12

For x→0 and y→0 find lim(x²+3y²)/(x²-3y²).

1/3

-2

The limit does not exit

Açıklama:

If we substitute x=0 and y=0 the result is 0/0 which is undefined. So we will arbitrarirly take y=kx.
Then:
(x²+3y²)/(x²-3y²)=(x²+3k²x²)/(x²-3k²x²)=x²(1+3k²)/x²(1-3k²)=(1+3k²)/(1-3k²)
The result depends on the value of k. For instance if k=2 result is -13/11, but if k=4 result is -49/47.
Therefore the limit does not exit.

Doğru Cevap: E

Soru 13

What is the partial derivative f_x of the function f(x, y)=5x²+xy-y²?

5x+y

5x-2y

10x+y

10x-2y

5+xy

Açıklama:

f(x, y)=5x²+xy-y²
f_x=10x+y (since the derivative of y² is zero)

Doğru Cevap: C

Soru 14

If f(x,y)=x²+5xy-3y² find f_xy-f_yx

2x-5y

-3x+11y

3x-11y

Açıklama:

There is no need to calculation. Since f_xy=f_yx, f_xy-f_yx=0

Doğru Cevap: D

Soru 15

Given that z=(x+y)², y=u³ x=u²-u find the partial derivative dz/du

9u^5-12u⁴+5u³+6u²-2u

9u⁵+12u⁴-5u³-6u²+2u

6u⁵-5u³-6u²+2u

6u⁵+10u⁴-4u³-6u²+2u

u⁵+12u⁴-u³-6u²+2

Açıklama:

dz/du=(dz/dx)*(dx/du)+(dz/dy)*(dy/du)
Since z=(x+y)², dz/dx=2x+2y and dz/dy=2x+2y
Since y=u³, dy/du=3u²
x=u²-u, dx/du=2u-1
By substitution
dz/du=(2x+2y)(2u-1)+(2x+2y)3u²=(2x+2y)(3u²+2u-1)=(2(u²-u+3u³)(3u²+2u-1))=(2u²-2u+3u³)(3u²+2u-1)=6u⁵+10u⁴-4u³-6u²+2u

Doğru Cevap: D

Soru 16

The function z=f(x, y) is implicitly defined by x²y-3y²+2zx=0. Find dz/dx

-y-z/x

-z-y/x

-x-z/y

-x-2y

x+y

Açıklama:

x²y-3y²+2zx=0
2xydx+x²dy-6ydy+2xdz+2zdx=0
Since we are interested in dz/dx we can take dy=0 (y doesnt is constant).
So 2xydx+2xdz+2zdx=0 then 2xdz=-2xydx-2zdx and divind both sides to dx we obtain 2xdz/dx=-2xy-2z
so dz/dx=(-2xy-2z)/2x=-y-z/x

Doğru Cevap: A

Soru 17

The utility function of a consumer is defined by U(x,y)=x^0.5y^0.5 who is constrained by the budget 120=2x+4y (He has 120 liras to spend on x and y, whose prices are 2 and 4 liras respectively). What are the amounts of (x,y) that maximizes his utility given that he spends all his budget?

(30,20)

(30,15)

(60,0)

(0,30)

(40,10)

Açıklama:

We can decrease the number of unknown variables to one by substitution.
Since 120=2x+4y, y=(120-2x)/4=30-0.5x
Then we can rewrite his utility function as U(x,y)=x^0.5y^0.5=U(x)=x^0.5(30-0.5x)^0.5=(x(30-0.5x))^0.5=(30x-0.5x²)^0.5
For the maximum utility the first derivative of utility with respect to x must be equal to zero. Thus:
Ux=0.5(30-x)(30x-0.5x²)^-0.5=0
30-x=0, which means x=30. Substituting this in y=30-0.5x=15. Thus the utility maximizing combination of (x,y)=(30,15)

Doğru Cevap: B

Soru 18

How many of the following points are stationary points for the function f(x,y)=x⁴+x²y²-2x²+2y²?
I.(0,0)
II.(1,0)
III.(1,1)
IV. (-1,0)
V.(0,√2)

Açıklama:

f(x,y)=x⁴+x²y²-2x²+2y²?
We have to find the points simultaneously satisfy fx=0 and fy=0
fx=4x³+2xy²-4x=0, so x(4x²+2y²-4)=0
fy=2x²y+4y=0, so y(2x²+4)=0
Points given in III and V doesnt satisfy these conditions simultaneously. So the remaining 3 points are stationary points.

Doğru Cevap: C

Soru 19

What can be the maximum area of a rectengular field whose perimeter is 40 metres?

100

Açıklama:

Let one side be x and other be y. Then 2x+2y=40 if the perimeter is 40. We want to find the maximum of f(x,y)=xy given perimeter is 40. We can decrease the number of unknowns by substitution using the perimeter equation.
2y+2x=40, y=20-x
f(x,y)=g(x)=x(20-x)=20x-x² will be the function of one variable to be maximized.
g_x=20-2x=0 then x=10, since y=20-x=10. So the area will be 10*10=100

Doğru Cevap: E

Soru 20

What is the shortest distance from point (3,5) to the x=y line?

0.5

0.8

√2

√3

Açıklama:

(3,5)noktası ile x=y doğrusu arasındaki uzaklık: U= √((x-3)^2+ (x-5)^2 )
en kısa uzaklık için türevi alıp 0^' a eşitleyelim:
U^'= (2 (x-3)+2 (x-5))/(2√((x-3)^2+ (x-5)^2 ))=0
U^'=0 için x=4 olur.
aradaki uzaklık:U= √2 bulunur.

Doğru Cevap: D

Soru 21

What is the value of the limit;
lim_{(x,y)→(0,0)} (5x²+y²) / (x²+y²)=?

-5

does not exist.

Açıklama:

y=kx,
(5x²+k²x²) / (x²+k²x²)=x²(5+k²) / x²(1+k²)=(5+k²) / (1+k²).
By the way we have an answer depending on k, the limit does not exist.

Doğru Cevap: E

Soru 22

If f(x,y)=5+e^-3y+x.lnx
What is the partial derivative ∂f/∂x= ?

e^-3x

e^-3x / lnx

lnx+1

-3y.lnx

-3e^-3y+x.lnx

Açıklama:

f(x,y)=5+e^-3y+x.lnx
∂f/∂x=(x.lnx)'=x'.lnx+x.lnx'
=lnx+1

Doğru Cevap: C

Soru 23

If f(x,y)=3x²y+e^xy+lny
What is the partial derivative ∂²f /∂x² ?

y.e^xy

xy.e^xy

6xy + ye^xy

6y + y²e^xy

6x+ e^xy

Açıklama:

If f(x,y)=3x²y+e^xy+lny
∂f /∂x=6xy+ye^xy
^{∂²f /∂x²=6y+y²}e^xy

Doğru Cevap: D

Soru 24

Given that f(x,y)=(3lnx+1)/(e^-3x) + (y²x²)/(x+y)
What is the value of the partial derivative of ∂f / ∂y at the point (1,2)=?

-1/2

1/2

8/9

Açıklama:

f(x,y)=(3lnx+1)/(e^-3x) + (y²x²)/(x+y)
∂f / ∂y = [(2x²y).(x+y)-(x²y²).1] / (x+y)²
By the way x=1,y=2
∂f / ∂y = [(4).(3)-(4).1] / (3)²=8/9

Doğru Cevap: D

Soru 25

Find the gradient of the function f(x,y,z)=xy+zy²+xyz at the point (1,-1,3)?

(-2,-2,2)

(-4,-2,0)

(4,2,1)

(-1,1,3)

(3,0,-1)

Açıklama:

grad f=[(y+yz),(x+2zy+xz),(y²+xy)] at (1,-1,3)=(-4,-2,0)

Doğru Cevap: B

Soru 26

Find the equation of the tangent plane to z=ln⁡(2x+y)">z=ln(2x+y) at (−1,3)">(−1,3).

z=4x+2y-2

z=x+2y-3

z=2x+y-1

z=-2x+2y-3

z=3x+y-3

Açıklama:

z=ln⁡(2x+y)">f(x,y)=ln(2x+y) z₀=ln1=0
z=ln⁡(2x+y)">∂f/∂x =2/z=ln⁡(2x+y)">(2x+y) =2
z=ln⁡(2x+y)">z=ln⁡(2x+y)">∂f/∂y=1/z=ln⁡(2x+y)">(2x+y)=1
z=ln⁡(2x+y)">z=ln⁡(2x+y)">then, z-0=2(x-(-1))+1(y-3)
z=ln⁡(2x+y)">z=ln⁡(2x+y)">z=2(x+1)+y-3
z=ln⁡(2x+y)">z=ln⁡(2x+y)">z=2x+y-1

Doğru Cevap: C

Soru 27

For z = x²+xy, and x=u²-v and y=-uv, find the partial derivative ∂z / ∂v at u=2, v=-1.

-22

-12

Açıklama:

∂z / ∂v = (∂z / ∂x) . (∂x / ∂v)+ (∂z / ∂y) . (∂y / ∂v)
∂z / ∂v = (2x+y) . (-1) + (x) . (-u)
Hence, x=u²-v and y=-uv
∂z / ∂v = (2(u²-v)+(-uv)) . (-1) + (u²-v) . (-u)
By the way u=2, v=-1
∂z / ∂v = (2(5)+(2)) . (-1) + (5) . (-2)= -22

Doğru Cevap: A

Soru 28

For z = x²+ylnx, and x=u³+v² and y=v, find the partial derivative ∂z / ∂v at u=1, v=-3.

-7

Açıklama:

For z = x²+ylnx, and x=u³+v² and y=v, find the partial derivative ∂z / ∂v at u=1, v=0.
∂z / ∂v = (∂z / ∂x) . (∂x / ∂v)+ (∂z / ∂y) . (∂y / ∂v)
∂z / ∂v = (2x+(y/x)) . (2v) + (lnx) . (1)
Hence, x=u³+v² and y=v
∂z / ∂v = (2(u³+v²)+(v/(u³+v²))) . (2v) + (ln(u³+v²)) . (1)
On substitution u=1, v=0
∂z / ∂v = (2(1)+(0)) . (0) + (0) . (1)= 0

Doğru Cevap: B

Soru 29

Which of the following is the local minimum point of the function f(x,y)=2x²+2xy+2y²-6x ?

(2,-1)

(2,-2)

(-2,-1)

(-2,0)

(0,2)

Açıklama:

∂f / ∂x = 4x+ 2y-6 ∂f / ∂y=2x+4y
for the critical points;
4x+ 2y-6=0 and 2x+4y=0, so x=2, y=-1.
∂²f / ∂x² =4 ∂²f / ∂y² =4 ∂f / ∂xy =2
D=4.4-2²=12.
Since D>0, the function has a local minimum at (2,-1).

Doğru Cevap: A

Soru 30

A rectangular farming area of 6050 m² needs to be designed. The area is bounded from one side by the wall and requires fencing from the other three parts (one length and two widths). What is the minimum length of the fencing required.

175 m

210 m

220 m

250 m

275 m

Açıklama:

xy=6050
L(x, y, λ) = x+2y+ λ(xy-6050).
∂L/∂x =0, ∂L/∂y =0, ∂L/∂λ =0
1+λy=0, 2+λx=0, xy-6050=0
Hence, λ=-1/y=-2/x
x=2y
2y²=6050
y=55, x=110
x+2y=220 m.

Doğru Cevap: C

Soru 31

What is the value of the limit

-5

-3

-1

Açıklama:

The correct choice is A.

Doğru Cevap: A

Soru 32

What is the value of the limit

-2

-3

-5

Açıklama:

The correct choice is D.

Doğru Cevap: D

Soru 33

what is the partial derivative

-2y

Açıklama:

The answer is C.

Doğru Cevap: C

Soru 34

Given that

, what is the value of the partial derivative

at the point (2, 1)?

-11

-21

-10

-20

Açıklama:

The correct answer is E.

Doğru Cevap: E

Soru 35

Find the gradient of the function

at the point (-1, 2, 3)?

(-18, 17, -3)

(18, 11, -6)

(-18, -17, 3)

(18, -17, -3)

(18, 17, 3)

Açıklama:

The correct answer is B.

Doğru Cevap: B

Soru 36

Find the gradient of the function

at the point (1, 2, 3)?

(114, 117, 35)

(-35, -117, -114)

(3, 118, 116)

(35, 117, 104)

(-35, 117, 114)

Açıklama:

The correct answer is D.

Doğru Cevap: D

Soru 37

-2

-3

Açıklama:

The correct answer is A

Doğru Cevap: A

Soru 38

What is the shortest distance between the point P(2, 0) and a line y=3x.

9/10

3/10

Açıklama:

The answer is B.

Doğru Cevap: B

Soru 39

-5

Açıklama:

The answer is D.

Doğru Cevap: D

Soru 40

Given that

, what is the value of the partial derivative

(2, -2)?

-11

-21

-44

Açıklama:

The answer is E.

Doğru Cevap: E

Soru 41

lim_x_{→0, y→0} (x³ + xy + y²) / (x + xy + y²) = ?

does not exist

Açıklama:

(x³ + x k x + k² x²) / (x + x k x + k² x²) = (x² (x + k + k²)) / (x (1 + k x + k² x)) = (x (1 + k + k²)) / ((k² + k) x + 1) = 0. pg. 182. Correct answer is A

Doğru Cevap: A

Soru 42

f(x, y) = e^y + yx² - x - 1 ; ∂f / ∂x = ?

e^y + x² - x

y + 2 x

2 y x²

x e^y + x²

2 y x - 1

Açıklama:

2 y x - 1 . pg. 183. Correct answer is E

Doğru Cevap: E

Soru 43

f(x, y) = e^xy + xy² - xy + 2 ; ∂²f / ∂x² = ?

y e^xy + 2 x y - x

e^xy + 2 y

y² e^xy

x e^xy + 2 x

x² e^xy + y²

Açıklama:

∂f / ∂y = y e^xy + y² - y ; ∂²f / ∂y² = y² e^xy . pg. 183. Correct answer is C

Doğru Cevap: C

Soru 44

f (x, y, z) = xyz + xy + xz - yz ; grad f (-3, 1, -2) = ?

(-4, -3, -1)

(5, 5, -8)

(-6, -1, -1)

(3, -2, -4)

(-3, 5,-7)

Açıklama:

correct answer is E.

Doğru Cevap: E

Soru 45

z = f(x, y)= xz + yz + 1 so ∂f / ∂y (1, -2, -1) = ?

-2

-1

Açıklama:

∂f / ∂y = z so at (1,-2,-1) ∂f / ∂y(1,-2,-1)=-1 pg. 190 . Correct answer is D

Doğru Cevap: D

Soru 46

z = f(x, y) is defined implicitly by xz + yz - 3 = 0 ; Which of the following ones below is the equation of the tangent plane to the graph of z = f (x, y) around the point (-1, 2, 1) ?

x - 3 y + 2 z + 6 = 0

3 x - 2 y + 2 z - 5 = 0

-x - 3 y + z + 3 = 0

x + y + z - 2 = 0

-2 x - 3 y + z - 4 = 0

Açıklama:

at (-1, 2, 1) : ∂F / ∂x = z = 1 ; ∂F / ∂y = z = 1 ; ∂F / ∂z = x + y = 1 ; 1 (x - (-1)) + 1 (y - 2) + 1 (z - 1) = (x + 1) + (y - 2) + (z - 1) = x + 1 + y - 2 + z - 1 = x + y + z - 2 = 0 . pg. 187. Correct answer is D

Doğru Cevap: D

Soru 47

What is the shortest distance between the point P(0, -4) and the line y = x ?

2^1/2

2^-1/2

2^3/2

2^-1

Açıklama:

4² = (d1)² + (d2)² = 2 (d1)² ; d1 = 2^3/2 . pg. 193. Correct answer is D

Doğru Cevap: D

Soru 48

z = x + y² ; x = u - v ; y = u v² ; at u = v = -1, ∂z / ∂v = ?

-5

-3

Açıklama:

z = u - v + u² v⁴ ; ∂z / ∂v = -1 + 4 u² v³ ; at u = v = -1, ∂z / ∂v = -5 . pg. 188. Correct answer is C

Doğru Cevap: C

Soru 49

Which of the following is the local minimum point of the function f (x, y) = 3 x² + y² - 2 x - 4 y ?

(3, -4)

(1 / 3, 2)

(2 / 5, -1)

(2, -3 / 4)

(-1, 4 / 3)

Açıklama:

∂f / ∂x = 6 x - 2 = 0 ; x = 1 / 3 ; ∂f / ∂y = 2 y - 4 = 0 ; y = 2 ; a = ∂²f / ∂x² = 6 > 0 ; b = ∂²f / ∂y² = 2 ; c = ∂²f / ∂x ∂y = 0 ; D = a . b - c² = 6 . 2 - 0 = 12 > 0 ; pg. 192. Correct answer is B

Doğru Cevap: B

Soru 50

What is the positive stationary value of the function f (x, y) = x + 2 y, satisfying the condition 2 x² - y² = -14 ?

Açıklama:

L(x, y, λ )= x + 2 y + λ (2 x² - y² + 14) ; ∂L / ∂x = 1 + 4 λ x = 0 ; λ = -1 / (4 x) ; ∂L / ∂y = 2 - 2 λ y = 0 ; λ = 1 / y ; 4 x = y ; ∂L / ∂λ = 0 ; 2 x² - y² = -14 ; 2 x² - (4 x)² = -14 ; 14 x² = 14 ; x = {-1 , 1}; 4 = {-4, 4} ; f(x, y) = x + 2 y = 9 > 0. pg. 193. Correct answer is D

Doğru Cevap: D

Soru 51

Which of the following (x,y) point is in the domain of the function

where x and y are real numbers?

(3,4)

(0,5)

(5,1)

(2,3)

(-4,-3)

Açıklama:

The expression in the squareroot must be non-negative because negative numbers do not have real number roots. Thus 0=<16-x²-y². Only the point in D satisfies this condition.

Doğru Cevap: D

Soru 52

Which of the following functions is not continuous at point (1,2)?

f(x,y)=5-x²-y²

f(x,y)=(5-x²-y²)/2x-3y

f(x,y)=(2x-y)/(5-x²-y²)

f(x,y)=(4x-1)/(3x-2y)

f(x,y)=(2x-2)(y-2)

Açıklama:

For the functions in A, B and E; f(x,y)=0 for x=1 and y=2. For the function in D f(1,2)=3/-1=-3 but for the function in C f(x,y) is not defined because the denominator of the expression is equal to zero.

Doğru Cevap: C

Soru 53

What is the limit of the function f(x,y)=(x²-y²)/(5x-5y) for x=1 and y=1?

2/5

Undefined

Infıinity

Açıklama:

f(x,y)=(x²-y²)/(5x-5y) for x=1 and y=1 is equal to 0/0 which is undefined. But we have to simplify the expression to check whether it is still undefined or not.
Thus:
f(x,y)=(x²-y²)/(5x-5y)=(x-y)(x+y)/5(x-y)=(x+y)/5=2/5 for x=1 and y=1. So the answer is B.

Doğru Cevap: B

Soru 54

What is the partial derivative df(x,y)/dx for f(x,y)=e^xy?

e^xy

xlny

ylnx

xe^xy

ye^xy

Açıklama:

Suppose z=f(x,y).
Then we can rewrite the function as z=e^xy?
Taking the natural logarithm of both sides we have:
lnz=lne^xy=xy
Taking the derivatives of the both sides we have:
(1/z)dz=ydx+xdy
When we take the partial derivative we behave all the other variables as constant. Thus here dy=0. Therefore we can rewrite the last expression as:
(1/z)dz=ydx
Thus dz/dx=yz=ye^xy since z=e^xy

Doğru Cevap: E

Soru 55

What is the second degree partial derivative d²f(x,y)/dxdy for f(x,y)=e^xy?

(1+xy)e^xy

xye^xy

xlny+ylnx

lnx/lny

(1+lnx)lny

Açıklama:

According to Clairaut’s theorem d²f(x,y)/dxdy=d²f(x,y)/dydx. Thus it makes no difference which variable is partially differentiated first. Let's start with x.
Assume that z=f(x,y)
So z=e^xy
lnz=lne^xy=xy
Taking derivatives of both sides we get:
(1/z)dz=xdy+ydx and for dz/dx we consider dy=0
Thus dz/dx=yz=ye^xy
Let's assume that t=dz/dx. We now have to find dt/dy according to chain rule. So:
t=ye^xy?
dt=e^xydy+yxe^xydy (from the rule if a=bc then a'=cb'+bc')
So:
dt=(1+xy)e^xydy
dt/dy=(1+xy)e^xy
So the answer is A.

Doğru Cevap: A

Soru 56

What is the equations of a tangent plane for a function f (x, y) = x²+3y² at a point (x, y) = (1, 1)?

z=2x-6y-4

z=2x+6y-4

z=6x+2y-4

z=x+y-4

z=3x+4y-8

Açıklama:

In order to find the equation for the tangent plane we have to first find the first partial derivatives f_x and f_y (which are df(x,y)/dx and df(x,y)/dy respectively).
Thus for f (x, y) = x²+3y² :
f_x =2x=2 for x=1
f_y =6y=6 for y=1
and z=f(x,y)=4 for (x,y)=(1,1)
The equation for the tangent plane is given by z=f(a,b)+f_x(x-a)+f_y(y-b)
Therefore the equation for the tangent plane for a=1, b=1 is given by
z=4+2(x-1)+6(y-1)=2x+6y-4

Doğru Cevap: B

Soru 57

What is the equations of a normal line for a function f (x, y) = x²+3y² at a point (x, y) = (1, 1)?

12-6z=3x-3=2y-2

4-6z=2x-3=y-1

24-6z=3x-3=y-1

6z=3x-3=y-1

z=3x-3=y-1

Açıklama:

In order to find the equation for the normal line we have to first find the first partial derivatives f_x and f_y (which are df(x,y)/dx and df(x,y)/dy respectively).
Thus for f (x, y) = x²+3y² :
f_x =2x=2 for x=1
f_y =6y=6 for y=1
and z=f(x,y)=4 for (x,y)=(1,1)
The equation for the tangent line is given by z=f(a,b)+f_x(x-a)+f_y(y-b)
Therefore the equation for the tangent plane for a=1, b=1 is given by
z=4+2(x-1)+6(y-1)=2x+6y-4
therefore the equation for the normal line is
(z-4)/-1=(x-1)/2+(y-1)/6
so 4-z=(x-1)/2+(y-1)/6
which means:
24-6z=3x-3=y-1

Doğru Cevap: C

Soru 58

Assume that z=x²+xy where x=u²v and y=2uv. What is the partial derivative dz/du for u=1 and v=2?

Açıklama:

dz/du=(dz/dx)*(dx/du)+(dz/dy)*(dy/du) from the chain rule.
Given that z=x²+xy where x=u²v and y=2uv:
dz/dx=2x+y
dx/du=2vu
dz/dy=x
dy/du=2v
So for u=1 and v=2; x=u²v=2 and y=2uv=4
dz/dx=2x+y=2u²v+2uv=4+4=8
dx/du=2vu=4
dz/dy=x=2
dy/du=2v=4
Finally:
dz/du=(dz/dx)*(dx/du)+(dz/dy)*(dy/du)=8*4+2*4=40

Doğru Cevap: E

Soru 59

Which of the following (x,y) points is not a critical one (either maximum, minimum or saddle point) for the function f(x,y)=x²y²-x²-y²?

(0,0)

(-1,1)

(1,1)

(1,0)

(-1,-1)

Açıklama:

For critical points the first derivatives df(x,y)/dx=f_x and df(x,y)/dy=f_y must be equal to zero simultaneously. Namely f_x=f_y=0.
Since f(x,y)=x²y²-x²-y²?
f_x=2xy²-2x=0 so 2x(y²-1)=0 which means either x=0 or y=1 or y=-1
f_y=2yx²-2y=0 so 2y(x²-1)=0 which means either y=0 or x=1 or x=-1
But since both equations must be satisfied simultaneously we have 5 critical points:
(0,0), (-1,1), (1,-1), (1,1) and (-1,-1).
As you can see the point (1,0) (namely x=1 and y=0) satisfies the second equation (fy=0) but it does not satisfy the first one. Thus the answer is D.

Doğru Cevap: D

Soru 60

Which of the following (x,y) points is a local maximum for the function f(x,y)=x²y²-x²-y²?

(0,0)

(-1,1)

(1,-1)

(1,1)

(-1,-1)

Açıklama:

Firstly, for critical points the first derivatives df(x,y)/dx=f_x and df(x,y)/dy=f_y must be equal to zero simultaneously. Namely f_x=f_y=0.
Since f(x,y)=x²y²-x²-y²?
f_x=2xy²-2x=0 so 2x(y²-1)=0 which means either x=0 or y=1 or y=-1
f_y=2yx²-2y=0 so 2y(x²-1)=0 which means either y=0 or x=1 or x=-1
But since both equations must be satisfied simultaneously we have 5 critical points:
(0,0), (-1,1), (1,-1), (1,1) and (-1,-1).
And also for a local maximum the following conditions must be satisfied simultaneously
I. f_xx<0
II.0xx*f_yy-f_xy*f_yx
Thus we now have to find the second partial derivatives for an evaluation:
f_xx=2y²-2
f_yy=2x²-2
f_xy=f_yx=4yx
Among the critical points for only point (0,0) the first condition is satisfied (namely f_xx<0). Lets check whether this point also satisfies the second condition:
f_xx*f_yy-f_xy*f_yx=(2y²-2)*(2x²-2)-16x²y²=(-2)*(-2)-0=4
since 0<4 the second condition is also satisfied. Therfore point (0,0) is a local maximum.

Doğru Cevap: A

Mathematıcs I (ENG) - Tüm Sorular

Ünite 1

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