MAT1051 Midterm Exam

[ Group A ]
November 25, 2023
Name: ID:

Section: Signature:

• You will find 7 multiple-choice questions and 3 open-ended questions on the follow-
ing pages.

• Important Information about the multiple choice questions:

– Use erasable pencils. For each question there is exactly one correct answer. Mark
your answers into the optical “Answer Sheet”. If you do NOT mark your answer
into the “Answer Sheet” it will NOT be graded!
– Be sure that you fill in your information in the “Answer Sheet” correctly, other-
wise your exam paper may NOT be graded.
– There are two different “Question Booklets” (Group A and Group B). Do NOT
forget to mark the “exam paper group” in your “Answer Sheet”.
– Below each multiple-choice question, there is enough space for you to carry all
calculations before marking your answer in the optical "Answer Sheet."
– For each wrong answer in the the multiple-choice questions, 1 point will be de-
ducted.

• Important Information about the open-ended questions:

– For the open-ended questions, write your solutions clearly in the exam paper.
– Show all your work in all open-ended questions. No credit will be given for cor-
rect final answers without justification.

• Calculators are not allowed.

• Exam duration: 90 minutes.

Problem Score Points

MCQ 35
Q8 10
Q9 15
Q10 10
Total 70
MAT1051 Bahçeşehir University 2023-2024 Fall

2
1. (5 points) Let f (x) = p and g (x) = 1 − x 2 . The domain of ( f ◦ g )(x) is
x
A. (−2, 2) B. [−1, 1] C. (−1, 1) D. [0, ∞) E. (−∞, ∞)

2. (5 points) Only one of the following statements is true about the function
x + 2|x|
f (x) = .
x
A. The range of the function is {−1, 3}.
B. The function intersects the x-axis at x = 2.
C. f (x) is an even function.
D. The function is differentiable at x = 0.
E. There is exactly one y-intercept for the function.

p
3. ³(5 points)
´ The equation of the tangent line to the curve y = cos( x) + 2 at the point
π2
4
, 2 is
³ 2
´ ³ 2
´ ³ 2
´
A. y = π x − π4 +2 B. y = π1 x − π4 +2 C. x = π2 D. y = − π1 x − π4 +2 E. y = 2

1
MAT1051 Bahçeşehir University 2023-2024 Fall

4. (5 points) p p
9 − x2 − 3 − x
lim p =
x→3− x 3 − 7x 2 + 12x
p
A. 2 − p1 B. 0 C. Does not exist D. p1 E. ∞
3 3

5. (5 points) Let

25x−5 + 5,

 x ≤1
2
f (x) = 6x Ax+2x+10
2 +2 , 1 < x < 2, where A and B are positive constants.

log (B x) + 1, x ≥ 2

2

Find all values of A and B such that f (x) is continuous at x = 1 and x = 2.

A. A = 1 and B = 2
7
B. A = 0 and B = 2 6
13
C. A = 1 and B = 2 3
D. A = 0 and B = 2
E. There are no such A and B that can make f continuous at x = 1 and x = 2.

2
MAT1051 Bahçeşehir University 2023-2024 Fall

6. (5 points) The solutions to the equation

tan sin−1 (x) = 2x

¡ ¢

in the interval (−1, 1) are

A. x = ±1
B. x = 0 and x = ±1
C. x =0
D. x = 0 and x = p1
2p
E. x = 0 and x = ± 23

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MAT1051 Bahçeşehir University 2023-2024 Fall

7. (5 points) Suppose that f (x) and g (x) are differentiable functions defined for all x, and
satisfy the following properties:

• f (−3) = g (−3)
• g 0 (x) < f 0 (x) for all x.

Which of the following statements are true?

(i) The graphs of f and g do not intersect.

(ii) g (x) < f (x) for all x > −3.
(iii) The graphs of f and g intersect at exactly one point.
(iv) g (x) < f (x) for all x < −3.
(v) The graphs of f and g intersect at more than one point.

A. (ii) only B. (ii) and (iii) C. None of the above D. (ii) and (v) E. (i) and (iv)

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MAT1051 Bahçeşehir University 2023-2024 Fall

dy
8. (10 points) Find .
dx
x
a) e y = x − y

x cos(x) tan−1 (x 2 )
b) y =
ln(x)

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MAT1051 Bahçeşehir University 2023-2024 Fall

9. (15 points)

a) (10 pts) Let f (x) = sin |x|. Using the limit definition of the derivative, find the values of x
for which f (x) is differentiable, and find a formula for f 0 .

2x + e x ¢0
. Find f −1 (1) .
¡
b) (5 pts) Let f (x) = x −x
3 +3

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MAT1051 Bahçeşehir University 2023-2024 Fall

10. (10 points) Determine whether the following statements are true or false. If true, justify
your answer and if false, provide a counterexample.

a) The equation
10
ln(x) − =0
x
has exactly one solution on the interval 12 , e 4 .
£ ¤

True False

b) Let the function f (x) be defined for all real numbers except x = −4. Furthermore,
f (x) satisfies the following inequality:
p
4e x − 11 (πx)2 + 1
µ ¶
−1
tan (x) ≤ f (x) ≤ .
2e x x +4

Then, lim f (x) = 2.

x→∞
True False