11) Which of the following is the domain of the function 14) Assume that 𝑥 ≠ 0 and 𝑎, 𝑏 ∈ ℝ .

If the equation of
𝑏
3 the normal line of the curve 𝑓(𝑥) = 𝑎𝑥 + 𝑥 at the point
𝑦 = √𝑎𝑟𝑐𝑠𝑖𝑛(3 − 𝑥) + 𝑥 2 . 𝐼𝑛(3 − 𝑥) ?
𝑃(2,3) is 𝑦 = 7 − 2𝑥 then which of the following is
A) (2,3) B) (3,4) C) (3,4] D) [2,3) E) None of them
𝑏−𝑎?

A)9 B) 2 C) 5 D) 1 E) 7

12) lim− 𝑐𝑜𝑡𝑥 . ln(1 + 𝑠𝑖𝑛𝑥) = ?

𝑥→𝜋

A)−∞ B) −1 C) 0 D) 1 E) ∞

15) A function 𝑓 defined by 𝑦 = 𝑓(𝑥) is given as providing

the equality (𝑥 2 + 𝑥)𝑦 = 𝑦 𝑥+1 . Then, which of the
𝒅𝒚
following is the value of |
𝒅𝒙 𝑷(𝟏,𝟐)
?

A) 𝑙𝑛2−3
B)
𝑙𝑛2−4 𝑙𝑛2−5
C) 𝑙𝑛2−1 D)
𝑙𝑛2−6
E)
𝑙𝑛2−7
𝑙𝑛2−1 𝑙𝑛2−1 𝑙𝑛2−1 𝑙𝑛2−1

2
e6x +7x+9 .(cos 2x)4
13) If 𝑓(x) = is defined on
(1+x)6 .(1+x4 )
𝜋
the interval 0 ≤ 𝑥 < 4 , then which of the following is
the value of 𝑓′(0) ?
A) 9e9 B) 7 e9 C) 5 e9 D) 7e9 E) e9

16) If 𝑓(𝑥) = ln(𝑥 + 5) , then which of the following is

the value of 𝑓 (55) (0) ?

54! 54! 55! 55!
A) − B) C) D) − E) Hiçbiri
555 555 554 554
 Yildiz Technical University Department of Mathematics
MAT1071 Mathematics 1 Midterm Exam A The 9th article of Student Disciplinary
Regulations of Higher Education
Council (YÖK) Law No. 2547 states that
Name-Surname Group people who are “cheating or helping
Student Number Place to cheat or attempt to cheat in exams”
Department Signature will be punished by suspension of one
or two semesters.
Date 24.11.2021 Duration 75 min

1) In which of the following are the continuity states of the functions f , g and h defined as
1−𝑐𝑜𝑠𝑥
1
𝑥 2 . 𝑠𝑖𝑛 2 , 𝑥 ≠ 0
1
. 𝑠𝑖𝑛𝑥 3 , 𝑥 ≠ 0 , 𝑥≠0
𝑥2
𝑓 (𝑥 ) = { 𝑥 , 𝑔 (𝑥 ) = { 𝑥3 , ℎ (𝑥 ) = { 1
0 , 𝑥=0 0 , 𝑥=0 , 𝑥=0
2

at the point x=0 correctly given?

A) All three are continuous B) All three are discontinuous C) f and g are discontinuous , h is continuous

D) f and g are continuous, h is discontinuous E) g is discontinuous, f and h are continuous

dy
2) Which of the following is the value of the derivative | for the function y = f(x) implicitly given by the
dx ( π , π )
4 2
equation x 𝑠𝑖𝑛(2𝑦) = 𝑦 𝑐𝑜𝑠(2𝑥) ? A) 1 B) 4 C) 3 D) 2 E) 6

3) In which of the following is the continuity and differentiability state of the f function defined as
tan(𝑠𝑖𝑛𝑥) , 𝑥 ≥ 0
𝒇(𝒙) = { 1 at the point x=0 correctly given?
𝑥2
. 𝑠𝑖𝑛𝑥 3 , 𝑥 < 0

A) Continuous at x=0 ; but non-differentiable B) Both continuous and differentiable at x=0

C) Discontinuous at x=0 ; but differentiable D) Both discontinuous and non-differentiable at x=0 E) None of them

1 1
4) If 𝑓(𝑥) = arcsin(ln𝑒 𝑥 ) + 𝑎𝑟𝑐𝑡𝑎𝑛2𝑥 − 𝑎𝑟𝑐𝑐𝑜𝑠(𝑥 − 2) , then which of the following is the value of 𝑓 (2) ?

𝜋 3𝜋 7𝜋 5𝜋
A) − 12
B) 4
C) 12
D) 12
E) 0
5) 7) Which of the following is the derivative of the function
𝑦 = 2𝑥 − 𝑠𝑖𝑛3 𝑥 + 𝑐𝑜𝑠√𝑥 ?

A) 𝑦 ′ = 𝑥 𝑙𝑛2 − 3𝑠𝑖𝑛2 𝑥 − 𝑠𝑖𝑛√𝑥

B) 𝑦 ′ = 2𝑥 𝑙𝑛2 − 3𝑠𝑖𝑛2 𝑥 𝑐𝑜𝑠𝑥 − √𝑥 𝑠𝑖𝑛√𝑥

𝑠𝑖𝑛√𝑥
C) 𝑦 ′ = 2𝑥 𝑙𝑛2 − 3𝑠𝑖𝑛2 𝑥 𝑐𝑜𝑠𝑥 − 2√𝑥

1 𝑠𝑖𝑛 √𝑥
D) 𝑦 ′ = 𝑥 𝑙𝑛2 − 3𝑠𝑖𝑛2 𝑥 −
√𝑥

𝑠𝑖𝑛√𝑥
E) 𝑦 ′ = 2𝑥 𝑙𝑛2 − 3𝑠𝑖𝑛2 𝑥 −
2√𝑥

8) Which of the following is the derivative of the function

𝑦 = arcsin (𝑥. 𝑙𝑛𝑥) ?
Which of the following statement is true for the function f
whose graph of the first derivative is given? A) 𝑦 ′ =
𝑙𝑛𝑥+1
B) 𝑦 ′ =
𝑙𝑛𝑥
1+𝑥 2 𝑙𝑛2 𝑥 √1−𝑥 2 𝑙𝑛2 𝑥
A) The function f is decreasing on the interval (𝑘, 𝑑)
𝑙𝑛𝑥−1 𝑙𝑛𝑥+1
B) The function f is decreasing on the interval (𝑐, 𝑒) C) 𝑦 ′ = D) 𝑦 ′ =
1+𝑥 2 𝑙𝑛2 𝑥 √1−𝑥 2 𝑙𝑛2 𝑥

C) The function f is increasing on the interval (n, m) E) 𝑦 ′ =

𝑥+𝑙𝑛𝑥
√1−𝑥 2 𝑙𝑛2 𝑥
D) The function f is increasing on the interval (c, k)

E) The function f is increasing on the interval (𝑑, 𝑚) 9) For the function 𝑓(𝑥) = 𝑒 5𝑥 + 𝑥 3 + 2 , what is
(𝑓 −1 )′ (3) =?
1 1 1 1
6) Which of the following statement is true for the A) B) C) D) 1 E)
3 5𝑒 15 +27 7 5
function 𝑓(𝑥) whose graph is given?

10) lim ( 𝑥 2 + 𝑥√𝑥 2 − 1 ) = ?

𝑥→ −∞

A) 0 B) 2 C) 1
D) −
1
E) ∞
2 2
A) 𝑓(𝑥) has 1 essential discontinuity, 1 jump
discontinuity, 2 non-differentiable points

B) 𝑓(𝑥) has 1 essential discontinuity, 1 removable

discontinuity, 1 non-differentiable point

C) 𝑓(𝑥) has 1 jump discontinuity, 1 removable

discontinuity, 3 non-differentiable points

D) 𝑓(𝑥) has 1 removable discontinuity, 1 jump

discontinuity, 2 non-differentiable points

E) 𝑓(𝑥) has 1 essential discontinuity, 1 removable

discontinuity, 3 non-differentiable points
17) lim+ (𝑡𝑎𝑛𝑥)𝑠𝑖𝑛𝑥 = ? A) −1 B) 1 C) 𝑒 −1 D) 𝑒 E) 𝑒 2
𝑥→0

𝑐 − 𝑑. arctan(𝑥 + 1) , 𝑥 < 0
18) It is known that the function f(x) which is defined as 𝑎𝑟𝑐𝑠𝑖𝑛𝑥
𝒇(𝒙) = { , 0≤𝑥≤1 is
𝑐 + 𝑑. arccot(2 − 𝑥) , 𝑥 > 1
continuous for ∀ 𝑥 ∈ ℝ. Then, which of the following is the correct values for the numbers c and d?
𝜋 𝜋 𝜋
A) 𝑐 =
𝜋
4
𝑑=1 B) 𝑐 = 1 𝑑 = 4
C) 𝑐 = −1 𝑑 = 4
D) 𝑐 = 4
𝑑 = −1 E) None of them

19) Suppose that 𝑓 is a differentiable function and . If

and the equation of tangent line to the graph of 𝑔(𝑥) at is then

what is A) −2 B) −1 C) 1 D) 2 E) −3

20) Let 𝑓(𝑥) and 𝑔(𝑥) be differentiable functions at the point 𝑥 = 1 ; 𝑓(1) = 3, 𝑓 ′ (1) = −1, 𝑔(1) = 5 and
𝑑 𝑓(𝑥) 2
( )| = 25 . Then which of the following is the linearization of the function 𝑔(𝑥) at the point 𝑥 = 1?
𝑑𝑥 𝑔(𝑥) 𝑥=1
7 7 7 7
A) 𝐿(𝑥) = 5 − 3 (𝑥 − 1) B) 𝐿(𝑥) = 5 + 3 (𝑥 − 1) C) 𝐿(𝑥) = 3 + 5(𝑥 − 1) D) 𝐿(𝑥) = − 3 + 5(𝑥 − 1) E) None of them