Summary in Graph

Exam Summary (GO Classes DA Test Series 2026 | Calculus

and Optimization | Limit, Continuity | Topic Wise Test 1)

Qs. Attempted: 11 Correct Marks: 9

6+5 3+6

Correct Attempts: 6 Penalty Marks: 2.33

3+3 1 + 1.33

Incorrect
Attempts:
5 Resultant Marks: 6.67
3+2 2 + 4.67

Total Questions: 15
10 + 5

Total Marks: 20
10 + 10

Exam Duration: 45 Minutes

Time Taken: 26 Minutes

EXAM RESPONSE EXAM STATS FEEDBACK

Technical

Q #1 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

4x2 −kx
Consider the function f (x) = 2 , where k is a constant.
x +12x+32
Find the value of k for which limx→−4 f (x) exists.

A. k = 16
B. k = −64
C. k = −16
D. k = 64

Your Answer: A Correct Answer: C Incorrect Time taken: 00min 49sec Discuss

Q #2 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

If a function f (x) is not defined at x = a, which of the following is a true statement?

A. limx→a f (x) cannot exist.

B. limx→a− f (x) ≠ limx→a+ f (x)
C. limx→a f (x) must approach infinity.
D. limx→a f (x) might be equal to zero.
 Your Answer: B Correct Answer: D Incorrect Time taken: 00min 43sec Discuss

Q #3 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Determine the following limit:

lim
x→∞
( √x9 2
+ x − 3 x)

Which of the following is the correct evaluation of the limit?

A. 1
3

B. 3

C. 0

D. 1
6

Your Answer: D Correct Answer: D Correct Time taken: 00min 48sec Discuss

Q #4 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Determine the following limit:

lim ( x2 − ln(1x) )
x→1−

Which of the following options correctly describes the limit?

A. The limit is −∞.

B. The limit is finite and equal to 2 .
C. The limit is ∞.
D. The limit is 0 .

Your Answer: Correct Answer: C Not Attempted Time taken: 04min 27sec Discuss

Q #5 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

What values of m and b make the following function continuous:

⎧x − 7 2
if x < −2
f (x) = ⎨ mx + b
⎩5 if − 2 ≤ x ≤ 2
if x > 2

A. m = 2, b = 1
B. m = 1, b = 2
C. m = 2, b = −1
D. m = 1, b = −2
 Your Answer: A Correct Answer: A Correct Time taken: 00min 53sec Discuss

Q #6 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

What is the value of limx →∞ ( √x 2

+ x − √x 2
− x) ?

A. 0
B. 1
C. 2
D. ∞

Your Answer: B Correct Answer: B Correct Time taken: 00min 46sec Discuss

Q #7 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Consider the function h(x) . What is limx h ( x) ?

6 x+5
= →−∞
√4 x 2
−9

A. −∞

B. 0

C. 3

D. −3

Your Answer: C Correct Answer: D Incorrect Time taken: 00min 34sec Discuss

Q #8 Multiple Select Type Award: 1 Penalty: 0 Calculus

You are given the following information about the function f (x)
(i) The domain of f (x) is the interval [−3, 1]
(ii) The function f (x) is continuous in the intervals [−3, −1) and (−1, 1] and it is not continuous at x = −1 .
(iii) f (−2) = −1.5

using this information. Which, if any, of the following limits must exists?

A. limx →−2 f ( x)
B. limx →−2
+f ( x)
C. limx →−1
+f ( x)
D. limx →−1 f ( x)

Your Answer: Correct Answer: A;B Not Attempted Time taken: 00min 57sec Discuss

Q #9 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

x−2
What is the value of lim ( x − 5) x −6
?
x→6 −
 A. 0

B. e −4

C. e 4

D. 1

Your Answer: Correct Answer: C Not Attempted Time taken: 02min 05sec Discuss

Q #10 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

fx
Suppose limx→6 | ( )| = 2. Which of the following statements must be true about limx→6 ( ) ? fx
fx
A. limx→6 ( ) does not exist.
B. limx f (x) = 2.
C. limx f (x) exists and is equal to either 2 or −2 .
→6

D. There is not enough information about f (x) to determine whether limx f (x) exists.
→6

→6

Your Answer: Correct Answer: D Not Attempted Time taken: 01min 09sec Discuss

Q #11 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

Suppose that a function f is defined by

⎧ 3x − 2 0<x<4
f (x) = ⎨⎩ 18 x=4
x − 5x + 6 x > 4
2

Let M = limx f (x) and N = limx f (x). Find 3M + 5N .

→4
−
→4
+

A. 12

B. 40

C. 56

D. 144

Your Answer: B Correct Answer: B Correct Time taken: 00min 49sec Discuss

Q #12 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

Consider the discontinuous function

f (x) = { 10 xx =≠ 00
 gx
and let ( ) = x . Compute limx f (g(x)).
2
→0

Which of the following is correct?

fgx
A. limx→0 ( ( )) = 0
B. limx→0 f (g(x)) = 1
C. limx→0 f (g(x)) does not exist
D. limx→0 f (g(x)) = −1

Your Answer: B Correct Answer: A Incorrect Time taken: 06min 44sec Discuss

Q #13 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

Evaluate the limit:

x 2
x
− cos( ) − e x+
1
3
3 4
3
lim
x→0 ln(x + 1)

Select the correct answer:

A. −1
B. 0
C. 1
D. ∞

Your Answer: B Correct Answer: A Incorrect Time taken: 02min 48sec Discuss

Q #14 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

a√x+3−b
If limx→1 x−1 =
1
4
, then what are the values of a and b ?
A. a = 1; b = 2
B. a = 2; b = 1
C. a = 4; b = 2
D. a = 2; b = 1

Your Answer: A Correct Answer: A Correct Time taken: 02min 05sec Discuss

Q #15 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

Consider the following statements:

fx gx hx x fx
S1: If ( ) ≤ ( ) ≤ ( ) for all with limx→a ( ) = −1 and limx→a ( ) = 1, then limx→a ( ) exists hx gx
with −1 ≤ limx→ a g ( x) ≤ 1 .
fgh
S2: Let , , be functions with the same domain A and f (x) ≤ g(x) ≤ h(x) for all x ∈ A. If
limx→ c f (x) = L = limx c h(x) for c some limit point of A, then limx c g(x) = L as well.
→ →

Which of the following options correctly interprets the provided statements?

A. Statement S1 is true, and statement S2 is false.

B. Statement S1 is false, and statement S2 is true.
S
C. Both statements 1 and 2 are true. S
S
D. Both statements 1 and 2 are false.S
Your Answer: B Correct Answer: B Correct Time taken: 00min 26sec Discuss

Copyright & Stuff

Summary in Graph

Exam Summary (GO Classes DA Test Series 2026 | Calculus

and Optimization | Differentiability, maxima and minima |
Topic Wise Test 2)

Qs. Attempted: 6 Correct Marks: 3

6+0 3+0

Correct Attempts: 3 Penalty Marks: 0.33

3+0 0.33 + 0

Incorrect
Attempts:
3 Resultant Marks: 2.67
3+0 2.67 + 0

Total Questions: 15
10 + 5

Total Marks: 20
10 + 10

Exam Duration: 45 Minutes

Time Taken: 18 Minutes

EXAM RESPONSE EXAM STATS FEEDBACK

Technical

Q #1 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

fx
Let ( ) = AxB ln(x), where A and B are unspecified constants. Suppose that (e , 10) is a point of local 5

extremum for f (x). Calculate the values of A and B.

A.A = 2e , B = − 1
5

B. A = 2e, B = 1
5

C. A = e, B = −5
D. A = e, B = 5

Your Answer: Correct Answer: A Not Attempted Time taken: 00min 39sec Discuss

Q #2 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

If a and b are positive numbers, what is the maximum value of f (x) = xa (1 − x)b for 0 ≤ x ≤ 1 ?
A.
a a bb
( a + b) a
( + )b

a b
( + ) ( a + b)
B. a a bb
 a
C. a+ b

a b a ( + )b
D.
( + )
a s bρ

Your Answer: Correct Answer: A Not Attempted Time taken: 02min 05sec Discuss

Q #3 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

fx
Consider the function ( ) = ( − 5)( + 10)2 = x x x 3
+ 15 x 2
− 500.
f
Find where is concave up and find where is concave down. f
Which of the following statements is correct about the function ( ) ? fx

A.f is concave down on [−5, ∞), concave up on (−∞, −5].

B. f is concave down on (−∞, −5], concave up on [−5, ∞).
C. f is concave down on (−∞, 5], concave up on [5, ∞).
D. f is concave down on [5, ∞), concave up on (−∞, 5].

Your Answer: B Correct Answer: B Correct Time taken: 04min 52sec Discuss

Q #4 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

The figure below shows the graphs of , f f , and f

′ ′′
. Identify which graph is that of f ′′
.

f x C (x),
A. ( ) = f (x) = A(x),
′
f ′′
x B ( x)
( )=
B. f (x) = B(x), f (x) = C (x),
′
f ′′
( x) = A( x)
C. f (x) = A(x), f (x) = B(x),
′
f ′′
( x) = C ( x)
D. f (x) = C (x), f (x) = B(x),
′
f ′′
( x) = A( x)

Your Answer: Correct Answer: C Not Attempted Time taken: 00min 15sec Discuss

Q #5 Multiple Select Type Award: 1 Penalty: 0 Calculus

. Suppose f is differentiable everywhere. Which of the following formulas are equal to f (a), for every a ? ′
 f (h)−f (a)
A. limh→0 h −a

f (x)−f (a)
B. limx→a x− a

f (a+h)−f (h)
C. limh→0 a

f (a+h)−f (a)
D. limh→0 h

Your Answer: A;B;C;D Correct Answer: B;D Incorrect Time taken: 00min 35sec Discuss

Q #6 Multiple Select Type Award: 1 Penalty: 0 Calculus

Let f (x) be continuous on [1, 5] and differentiable on (1, 5). Suppose that f ′ (x) > 3 for all x, and f (1) = 1.
According to the Mean Value Theorem (MVT), what can be concluded about f (5) ?

A. f (5) > 13
B. f (5) < 13
C. f (5) = 13
D. f (5) ≥ 12

Your Answer: A Correct Answer: A;D Incorrect Time taken: 02min 27sec Discuss

Q #7 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Which of the following completes the statement of the Mean Value Theorem applied to the function
x
f (x) = ex on the interval [1, 3] ?
"There is a point c in the interval (1, 3) such that ..."

xex −ex
1
ec −e
A. =
3

x2 2

cec −ec e3
B. = 9
c 2

c
1
e 3 −e
C. ec =
3

3−1

ec (c−1) e3 e
D. = 6 − 2
c2

Your Answer: Correct Answer: D Not Attempted Time taken: 02min 30sec Discuss

Q #8 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

If f is a continuous function on the closed interval [a, b], which of the following must be true?

A. There is a number c in the open interval (a, b) such that f (c) = 0.

B. There is a number c in the open interval (a, b) such that f (a) < f (c) < f (b)
C. There is a number c in the closed interval [a, b] such that f (c) ≥ f (x) for all x in [a, b].
D. There is a number c in the open interval (a, b) such that f ′ (c) = 0.

Your Answer: D Correct Answer: C Incorrect Time taken: 01min 17sec Discuss
Q #9 Numerical Type Award: 1 Penalty: 0 Calculus

Suppose that f is continuous and differentiable on the interval [1, 6]. Also suppose that f (1) = −8 and
f ′ (x) ≤ 4 for all x in the interval [1, 6]. What is the largest possible value for f (6) ?

Your Answer: 12 Correct Answer: 12 Correct Time taken: 01min 17sec Discuss

Q #10 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Let f (x) be a smooth function with exactly two inflection points at x = −1 and x = 2, and a local maximum
at x = 0. Which of the following must be true?

A. f ′′ (x) ≥ 0 on (−1, 2)
B. f ′′ (x) ≤ 0 on (−1, 2)
C. f (x) ≥ 0 on (−1, 2)
D. f (x) ≤ 0 on (−1, 2)

Your Answer: B Correct Answer: B Correct Time taken: 01min 30sec Discuss

Q #11 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

Consider the function y = g(x).

Using the graph of y = g(x), determine the sign of g ′′ (−3) and h′ (x) for h(x) = [g(x)]2 .

A. g ′′ (−3) > 0 and h′ (−4) > 0

B. g ′′ (−3) < 0 and h′ (−4) > 0
C. g ′′ (−3) > 0 and h′ (−4) < 0
D. g ′′ (−3) < 0 and h′ (−4) < 0

Your Answer: Correct Answer: D Not Attempted Time taken: 00min 03sec Discuss

Q #12 Multiple Select Type Award: 2 Penalty: 0 Calculus

Consider the table below:

Suppose l(x) is a linear function of x, with l(4) = 0, and l′ (4) < f ′ (4). Which of the following statements
about l(x) is true? (Mark all that apply)

A. l(x) > 0 for x > 4.

B. l(x) < 0 for x > 4.
C. l(x) is increasing for all x.
D. l(x) is decreasing for all x.

Your Answer: Correct Answer: B;D Not Attempted Time taken: 00min 09sec Discuss

Q #13 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

S1: If f : (a, b) → R is a real-valued function differentiable at p ∈ (a, b), then

f ( p + h) − f ( p − h)
lim = f ′ (p)
h→0 2h

S2: If a differentiable function f : [a, b] → R is not one-to-one, i.e., there exist points p, q ∈ [a, b] with p ≠ q
such that f (p) = f (q). By the Mean Value Theorem, there exists a point t between p and q such that
f ′ ( t) = 0.
Which of the following option is correct?

A. Statement S1 is true, and statement S2 is false.

B. Statement S1 is false, and statement S2 is true.
C. Both statements S 1 and S 2 are true.
D. Both statements S 1 and S 2 are false.

Your Answer: Correct Answer: C Not Attempted Time taken: 00min 02sec Discuss

Q #14 Multiple Select Type Award: 2 Penalty: 0 Calculus

Which of the following statements is/are NOT CORRECT?

A. If f is concave up on [0, 1] and concave down on [1, 2] then 1 is an inflection points.

B. If a function f is differentiable on [−1, 1], then there is a point x in that interval where f ′ (x) = 0.
C. If f ′′ (−2) > 0 then f is concave up at x = −2.
D. An inflection point is a point, where the function f ′ (x) changes sign.

Your Answer: Correct Answer: B;D Not Attempted Time taken: 00min 03sec Discuss

Q #15 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

Let f (x) be de defined for all x. Suppose f (x) is continuous and differentiable whenever x ≠ 0 and satisfies

f (0) = 1 lim f ( x) = 1
x→0+
lim f ( x) = a lim f ′ ( x) = b
x→0 −
x→0 +
Under what conditions on the constants a and b does f (0) guarantee to be a local maximum?

A. a < 1 and b > 0

B. a < 1 and b < 0
C. a > 1 and b < 0
D. a > 1 and b > 0

Your Answer: Correct Answer: B Not Attempted Time taken: 00min 06sec Discuss

Copyright & Stuff

Summary in Graph

Exam Summary (GO Classes DA Test Series 2026 | Calculus

and Optimization | Taylor Series | Topic Wise Test 3)

Qs. Attempted: 5 Correct Marks: 4

5+0 4+0

Correct Attempts: 4 Penalty Marks: 0.33

4+0 0.33 + 0

Incorrect
Attempts:
1 Resultant Marks: 3.67
1+0 3.67 + 0

Total Questions: 15
10 + 5

Total Marks: 20
10 + 10

Exam Duration: 45 Minutes

Time Taken: 18 Minutes

EXAM RESPONSE EXAM STATS FEEDBACK

Technical

Q #1 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Let f be a function having derivatives of all orders for all real numbers. The third-degree Taylor polynomial for
f about x = 2 is given by:
T (x) = 7 − 9(x − 2)2 − 3(x − 2)3

Which of the following statements are correct?

A. f (2) = 0 and f ′′ (2) = −18

B. f (2) = 7 and f ′′ (2) = −9
C. f (2) = 7 and f ′′ (2) = −18
D. f (2) = 0 and f ′′ (2) = −9

Your Answer: C Correct Answer: C Correct Time taken: 01min 29sec Discuss

Q #2 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

x
Consider the function g(x) = 2 + x2 + .
(4+x2 )
2

Find the degree 3 Taylor polynomial of g(x) around x = 0.

 x x3
A. 2 + x2 − 16 + 32
x x3
B. 2 + x2 + 16
+ 32
x x3
C. 2 + x −2
16
− 32
x
D. 2 + x2 + 16 x3
− 32

Your Answer: Correct Answer: D Not Attempted Time taken: 01min 32sec Discuss

Q #3 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Let f : R → R be a function such that f (0) = 0 for some points but not for all points. Suppose that f and its
first derivative exist and are differentiable at all points. Suppose also that the second derivative f ′′ (x) = 0 for
all x ∈ R.
Which of the following statements is true?

A. f (x) is a polynomial of degree 2 .

B. f (x) is a constant function.
C. f (x) is a linear function but not constant.
D. f (x) is a cubic function.

Your Answer: C Correct Answer: C Correct Time taken: 00min 57sec Discuss

Q #4 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Selected values of a function f and its first three derivatives are indicated in the table above. What is the
third-degree Taylor polynomial for f about x = 1 ?

A. 2 − 3x + x2 − 13 x3
3
2
B. 2 − 3(x − 1) + 32 (x − 1)2 − 13 (x − 1)3
C. 2 − 3(x − 1) + 32 (x − 1)2 − 23 (x − 1)3
D. 2 − 3(x − 1) + 3(x − 1)2 − 2(x − 1)3

Your Answer: Correct Answer: B Not Attempted Time taken: 02min 11sec Discuss

Q #5 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Suppose the functions f (x) and g(x) have Taylor series expansions at zero, up to second-degree terms, given
by

f ( x) = a 0 + a 1 x + a 2 x2 + …
g ( x) = b 0 + b 1 x + b 2 x2 + …

Find the Taylor series for h(x) = f (x)g(x) at zero, up to second-degree terms.

A. h(x) = a0 b0 + (a0 b1 + a1 b0 ) x + (a0 b2 + a1 b1 + a2 b0 ) x2 + …

 B. h(x) ab
= 0 0 +( ab 0 1 + a b ) x + (a b
1 1 0 2 + a b )x 2 0
2
+…

C. h(x) = a b 0 0 + (a b 1 0 + a b ) x + (a b
0 1 1 1 + a b )x 0 2
2
+…

D. h(x) = a b 0 0 + (a b 1 0 + a b ) x + (a b
0 1 1 2 + a b )x 2 0
2
+…

Your Answer: Correct Answer: A Not Attempted Time taken: 02min 49sec Discuss

Q #6 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Let f (x) = √ x. The second Taylor polynomial p (x) of f (x) at x = 1 is given. Find p 2 2 (3) . (Choose the
nearest answer.)

A. 1.1
B. 1.2
C. 1.3
D. 1.5

Your Answer: Correct Answer: D Not Attempted Time taken: 01min 42sec Discuss

Q #7 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Which of the following is NOT a Taylor series for f (x) ?

f ( x) f ( x)
A. f (x + h) f ( x) + f ( x) h + h h
′′ ′′′
′ 2 3
= + +⋯
2! 3!
f (x f (x
B. f (x)
′′ ′′′

f (x ) + f (x ) (x − x ) +
= 0
′
0 0
2!
0)
( x−x 0)
2
+
3!
0)
( x−x 0)
3
+⋯
f f
C. f (x) = f (0) + f (0)x + x + x
′′ ′′′
(0) (0)
′ 2 3
+⋯
2! 3!
f (x f (x
D. f (x)
′′ ′′′

= f (x 0) + f (x ) x +
′
0
2!
0)
x 2
+
3!
0)
x 3
+⋯

Your Answer: C Correct Answer: D Incorrect Time taken: 00min 21sec Discuss

Q #8 Multiple Choice Type Award: 1 Penalty: 0.33 Engineering Mathematics

Which is the Taylor series for the function ln(x) at the point a = 1 ?

A. (x − 1) − x − 1) x − 1) x − 1)
1 2 1 3 1 4
( + ( − ( +⋯
2 3 4

B. (x − 1) − (x − 1)2 + 2( x − 1) − 6(x − 1) + ⋯ 3 4

C. ln(x) + x1 (x − 1) − (x − 1) + (x − 1) − (x − 1) + ⋯
1 2 2 3 6 4

x 2
x x 3 4

D. ln(x) + x1 (x − 1) − 2x
(x − 1) +
1
2
x
(x − 1) −
x
(x − 1) + ⋯
2

3
1
3
3 1

4 4
4

Your Answer: Correct Answer: A Not Attempted Time taken: 00min 54sec Discuss

Q #9 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Let T5 (x) = 3 x x2
−5
3
+7 x 4
+3 x 5
be the fifth-degree Taylor polynomial for the function f about x = 0.

What is the value of f ′′′

(0) ?

A. -30
B. -15
C. -5
D. − 56
 Your Answer: A Correct Answer: A Correct Time taken: 00min 30sec Discuss

Q #10 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

The function f (x) is approximated near x = 0 by the second-degree Taylor polynomial T2 (x) =
5−7 x + 8x2 . Which of the following statements is true?
A. f (0) = 5, f ′ (0) = −7, f ′′ (0) = 8, f ′′′ (0) = 0
B. f (0) = 5, f ′ (0) = 7, f ′′ (0) = −8, f ′′′ (0) = 0
C. f (0) = 5, f ′ (0) = −7, f ′′ (0) = 16, f ′′′ (0) = 0
D. f (0) = 5, f ′ (0) = 7, f ′′ (0) = 8, f ′′′ (0) = 0

Your Answer: C Correct Answer: C Correct Time taken: 00min 26sec Discuss

Q #11 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

The graph of y = f (x) is given below. Assume that f is infinitely differentiable everywhere. The Taylor series
for f (x) about x = 0 is given by:

f ( x) = ∑c x
∞

n=0
n
n= c 0 + c 1 x + c 2 x2 + c 3 x3 + ⋯

Based on the graph, determine the signs of c0 , c1 , and c2 respectively:

A. Negative, Positive, Positive

B. Positive, Negative, Negative
C. Zero, Positive, Negative
D. Negative, Zero, Positive

Your Answer: Correct Answer: A Not Attempted Time taken: 00min 16sec Discuss

Q #12 Multiple Select Type Award: 2 Penalty: 0 Calculus

Let f be a twice differentiable function at x = 3. Let P1 (x) be the local linearization of f at x = 3 and
P2 (x) = 1 − 2(x − 3) + 2(x − 3)2 be the second Taylor polynomial of f at x = 3.
Which of the following statements MUST be true?

A. P1 (x) = 1 − 2(x − 3)
 B. f ′′ (3) = 2
C. f (3) < P1 (3)
D. f (3) = P1 (3)

Your Answer: Correct Answer: A;D Not Attempted Time taken: 00min 58sec Discuss

Q #13 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

Let f be a real-valued function defined on R. Suppose that f has continuous first and second derivatives, and
at some point x = a, the first derivative f (a) is zero, and the second derivative f ′′ (x) is positive for all x.
′

Which of the following statements is true?

A. If x ≠ a, then f (x) = f (a).

B. If x ≠ a, then f (x) < f (a)
C. If x ≠ a, then f (x) > f (a).
D. If x ≠ a, then f (x) ≤ f (a)

Your Answer: Correct Answer: C Not Attempted Time taken: 00min 13sec Discuss

Q #14 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

The polynomial P2 (x) = 1 + 3(x − a) − 2(x − a)2 is the second degree Taylor polynomial approximating
the function f for x near a. The graph of f is given in the figure. Which of the points A, B, C, or D on the x-
axis has a as its x-coordinate?

A. A
B. B
C. C
D. D

Your Answer: Correct Answer: D Not Attempted Time taken: 00min 31sec Discuss

Q #15 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

Given that the Taylor polynomial of degree four for f (x) about x = 0 is

2−3 x + 5 x3 + 7 x4

f (x2 )−2
what is the Taylor polynomial of degree five for g(x) = x about x = 0 ?

A. 5x5 − 3x
B. −3x + 5x3 − 3x5
C. 5x5 − 3x2
D. 5x5 − 3x + 2x2
Your Answer: Correct Answer: A Not Attempted Time taken: 02min 36sec Discuss

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Summary in Graph

Exam Summary (GO Classes CS Test Series 2026 | Calculus |

Topic Wise Test)

Qs. Attempted: 20 Correct Marks: 25

10 + 10 7 + 18

Correct Attempts: 16 Penalty Marks: 0.33

7+9 0.33 + 0

Incorrect
Attempts:
4 Resultant Marks: 24.67
3+1 6.67 + 18

Total Questions: 20
10 + 10

Total Marks: 30
10 + 20

Exam Duration: 60 Minutes

Time Taken: 40 Minutes

EXAM RESPONSE EXAM STATS FEEDBACK

Technical

Q #1 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Evaluate the limit

lim 1 − cos x
x→0 sin2 x

A. 1
B. 1
2
C. 2
D. 0

Your Answer: B Correct Answer: B Correct Time taken: 00min 30sec Discuss

Q #2 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Determine the value of following limit

lim ( √ x2 + 4 x + 1 − x)
x→∞

A. 2
 B. 4
C. 1
2

D. 3

Your Answer: A Correct Answer: A Correct Time taken: 00min 49sec Discuss

Q #3 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

The function f (x) = x4 − 6x2 is increasing on the intervals

A. (0, √3) only

B. (√3, ∞) only
C. (−∞, −√3) and (0, √3) only
D. (−√3, 0) and (√3, ∞) only

Your Answer: D Correct Answer: D Correct Time taken: 03min 21sec Discuss

Q #4 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

The function f (x) = cos x − x

A. is an even function
B. is an odd function
C. is neither an even nor an odd function
D. None of these

Your Answer: C Correct Answer: C Correct Time taken: 00min 08sec Discuss

Q #5 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Which of the following functions satisfy the conditions of Rolle's Theorem on the interval [−1, 1]?

f (x) = 1 − x2/3
g ( x) = x3 − 2 x2 − x + 2
π
h(x) = cos( (x + 1))
4

Rolle's Theorem applies to:

A. both f and g
B. both g and h
C. g only
D. h only

Your Answer: C Correct Answer: C Correct Time taken: 03min 36sec Discuss

Q #6 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Suppose that the derivative of a function h is given by:

h′ (x) = x(x − 1)2 (x − 2)

On what interval(s) is h increasing?

A. (−∞, 0)
B. (−∞, 0) and (2, ∞)
C. (0, 2)
D. (0, 1) and (2, ∞)

Your Answer: D Correct Answer: B Incorrect Time taken: 01min 13sec Discuss

Q #7 Multiple Select Type Award: 1 Penalty: 0 Calculus

Let q(x) be a continuous function which is defined for all real numbers. A portion of the graph of q ′ (x), the
derivative of q(x), is shown below.

On which of the following interval(s) is q(x) increasing?

A. (0, 2)
B. (2, 4)
C. (7, 9)
D. None of these

Your Answer: A;B Correct Answer: B;C Incorrect Time taken: 00min 23sec Discuss

Q #8 Multiple Select Type Award: 1 Penalty: 0 Calculus

Choose the CORRECT statement -

A. The function f (x) = exp(−x2 ) − 1 has the root x = 0.

B. If a function f is differentiable on [−1, 1], then there is a point x in that interval where f ′ (x) = 0.
C. If 1 is a root of f , then f ′ (x) changes sign at 1.
D. If f ′′ (0) < 0 and f ′′ (1) > 0 then there is a point in (0, 1), where f has an inflection point.

Your Answer: A Correct Answer: A;D Incorrect Time taken: 01min 47sec Discuss

Q #9 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Evaluate y ′′ (1) where y = ex + xe .

A. 0
B. 1
C. e2
D. e

Your Answer: C Correct Answer: C Correct Time taken: 04min 09sec Discuss

Q #10 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Consider the following statements:

I. f (x) is continuous on [a, b]

II. f (x) is differentiable on (a, b)
III. f (a) = f (b)

Which of the above statements are required in order to guarantee a c ∈ (a, b) such that
f ′ (c)(b − a) = f (b) − f (a)?

A. I only
B. I and II only
C. I, II, and III
D. I and III only

Your Answer: B Correct Answer: B Correct Time taken: 00min 57sec Discuss

Q #11 Multiple Select Type Award: 2 Penalty: 0 Calculus

Let I = (a, b) be an open interval and let f be a function which is differentiable on I . Which of the followings
is/are true statements -

A. If f ′ (x) = 0 for all x ∈ I , then there is a constant r such that f (x) = r for all x ∈ I .
B. If f ′ (x) > 0 for all x ∈ I , then f (x) is strictly increasing on I .
C. If f ′ (x) < 0 for all x ∈ I , then f (x) is strictly decreasing on I .
D. If f ′ (x) > 0 for all x ∈ I , then f (x) is strictly decreasing on I .

Your Answer: A;B;C Correct Answer: A;B;C Correct Time taken: 00min 31sec Discuss

Q #12 Multiple Select Type Award: 2 Penalty: 0 Calculus

Which of the following is/are FALSE?

1
A. The absolute maximum value of f (x) = on the interval [2, 4] is 2.
x
B. If f (x) is a continuous function and f (3) = 2 and f (5) = −1, then f (x) has a root between 3 and 5.
C. The function g(x) = 2x3 − 12x + 5 has 5 real roots.
D. If h(x) is a continuous function and h(1) = 4 and h(2) = 5, then h(x) has no roots between 1 and 2.

Your Answer: A;C Correct Answer: A;C;D Incorrect Time taken: 02min 03sec Discuss
Q #13 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

Suppose g(x) is a polynomial function such that g(−1) = 4 and g(2) = 7. Then there is a number c between
−1 and 2 such that

A. g(c) = 1
B. g ′ (c) = 1
C. g(c) = 0
D. g ′ (c) = 0

Your Answer: B Correct Answer: B Correct Time taken: 01min 30sec Discuss

Q #14 Multiple Select Type Award: 2 Penalty: 0 Calculus

Which of the following limit is/are correct?

A. lim √
x
x=1
x→∞
B. lim √
x
x=e
x→∞
x
2
C. lim
x→∞
(1 + x ) = e2
x
2
D. lim
x→∞
(1 + x ) =e

Your Answer: A;C Correct Answer: A;C Correct Time taken: 00min 49sec Discuss

Q #15 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

Suppose f is twice differentiable with

f ′′ (x) = 7x − 2, f ′ (−2) = 0, and f (−2) = −2.

Find f (0).

A. −337/6
B. −74/3
C. 23/9
D. 37/4

Your Answer: B Correct Answer: B Correct Time taken: 11min 08sec Discuss

Q #16 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

The sum of three positive numbers is 12 and two of them are equal. Find the largest possible product.

A. 86
B. 64
C. 48
D. 72

Your Answer: B Correct Answer: B Correct Time taken: 01min 54sec Discuss
Q #17 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

fx egx g
If ( ) = x ( ), (0) = 2 and g (0) = 1, then f (0) is
′ ′

A. 1
B. 3
C. 2
D. 0

Your Answer: B Correct Answer: B Correct Time taken: 00min 56sec Discuss

Q #18 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

Let f be differentiable for all x. If f (1) = −2 and f (x) ≥ 2 for x ∈ [1, 6], then
′

fA. (6) ≥ 8
B. f (6) < 8
C. f (6) < 5
D. f (6) = 5

Your Answer: A Correct Answer: A Correct Time taken: 01min 21sec Discuss

Q #19 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

The equation x 5
+ x + 1 = 0 has a solution in the interval
A. [0, 1]
B. [−1, 0]
C. [−2, −1]
D. [1, 2]

Your Answer: B Correct Answer: B Correct Time taken: 01min 24sec Discuss

Q #20 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

Which of the following expression evaluates to given integral

x
ln(ln )
∫
x ln x dx
ln x
A.
x + C

B. 1
2
x
(ln ln )2 + C
x
C. (ln )2 + C
D. (ln ln ) + x C

Your Answer: B Correct Answer: B Correct Time taken: 01min 29sec Discuss
You're doing Great!

Copyright & Stuff

Summary in Graph

Exam Summary (GO Classes GATE CS/DA | Calculus |

Weekly Quiz 1)

Qs. Attempted: 12 Correct Marks: 9

6+6 5+4

Correct Attempts: 7 Penalty Marks: 0

5+2 0+0

Incorrect
Attempts:
5 Resultant Marks: 9
1+4 5+4

Total Questions: 30
15 + 15

Total Marks: 45
15 + 30

Exam Duration: 120 Minutes

Time Taken: 26 Minutes

EXAM RESPONSE EXAM STATS FEEDBACK

Technical

Q #1 Numerical Type Award: 1 Penalty: 0 Calculus

Let f be a continuous function defined for all real numbers. Suppose

3 2 6 18

∫ f (x)dx = 2, ∫ f (x)dx = 7, ∫ f (x)dx = −5, and ∫ f (x)dx = −3

0 1 3 9

6
1
Find ∫ f ( t) dt
3
3

Your Answer: Correct Answer: 21 Not Attempted Time taken: 00min 38sec Discuss

Q #2 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

e ln(x)
Which of the following represents the value of ∫ dx?
1
x

A. 1
B. 2
C. 1/2
D. e
 Your Answer: Correct Answer: C Not Attempted Time taken: 00min 02sec Discuss

Q #3 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Find ∫ xe x dx
−3

1
A.
2
e −3 x (x − 3) + C

1
B. −
9
e −3 x (3x + 1) + C

1
C. −
6
xe 2 −3 x+C

1
D. −
3
e −3 x (x + 1) + C

Your Answer: Correct Answer: B Not Attempted Time taken: 00min 53sec Discuss

Q #4 Numerical Type Award: 1 Penalty: 0 Calculus

fx
Below is a portion of the graph of an even function ( ), which has domain (−∞, ∞) even though the
fx
graph below only shows the function on the interval [0, 5]. Note that ( ) has a vertical asymptote at x = 1.

Find

f (1.5 + h) − f (1.5)
lim
h→0 h

Your Answer: Correct Answer: 4 Not Attempted Time taken: 00min 03sec Discuss

Q #5 Multiple Select Type Award: 1 Penalty: 0 Calculus

Some information about the derivative p (x) and the second derivative p (x) of a function p(x) is provided
′ ′′

in the table below.

 x −4 −3 −2 −1 0 1 2
p ( x)
′
1 0 −2 0 −1 0 2
p ( x)
′′
−1 0 0 0 0 2 1

At which of the following values of x must p(x) have a local minimum?

A. x = −3
B. x = −2
C. x = −1
D. x = 1

Your Answer: Correct Answer: D Not Attempted Time taken: 00min 03sec Discuss

Q #6 Numerical Type Award: 1 Penalty: 0 Calculus

Consider the piecewise function

⎧⎪⎪ 7ex C + 3xx− 2

−
x<0
⎪
q ( x) = ⎨
⎪⎪⎪⎩ 6 + 5x
2 + 3x + 4x
x≥0
where C is a constant. Find xlim q(x)
→∞

Your Answer: Correct Answer: 0 Not Attempted Time taken: 00min 08sec Discuss

Q #7 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

gx
The function ( ) is given by the equation

g(x) = { ax x≤12

b − ln(3x) x > 1

a b
where and are constants.
a
Find the value of such that function is differentiable at x = 1.
A. 1/2
B. −1/2
C. 2
D. −2

Your Answer: Correct Answer: B Not Attempted Time taken: 03min 41sec Discuss

Q #8 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Consider the family of functions

b
g(x) = a ln(x) + x

defined for x > 0, where a and b are positive constants.

Any function g(x) in this family has only one critical point. In terms of a and b, what is the x-coordinate of
that critical point?

A. x = b/a is a critical point where the function shows local minima

B. x = b/a is a critical point where the function shows local maxima
C. x = a/b is a critical point where the function shows local minima
D. x = a/b is a critical point where the function shows local maxima

Your Answer: A Correct Answer: A Correct Time taken: 01min 28sec Discuss

Q #9 Numerical Type Award: 1 Penalty: 0 Calculus

The following are tables of values for two differentiable functions f (x) and g(x) and their derivatives.
Missing values are denoted by a “?". Assume that each of these functions is defined for all real numbers, that
f ′ (x) and g ′ (x) are continuous.

x 0 2 3 6 9 x −1 1 3 7 11
f (x) −1 ? 0 −2 ? g(x) −4 1 2 6 7
f ′ (x) 1 4 −1 ? 1 g ′ ( x) 7 ? 3 4 ?

Let z(x) = f (g(x)). Find z ′ (3)

Your Answer: Correct Answer: 12 Not Attempted Time taken: 00min 00sec Discuss

Q #10 Multiple Select Type Award: 1 Penalty: 0 Calculus

Which of the following is/are TRUE?

f ( x)
A. If lim f (x) = 0 and lim g(x) = 0, then lim does not exist.
x→5 x→5 x→5 g(x)
f
B. If f , g, are any two functions which are continuous for all x, then is continuous for all x.
g
C. It is possible that functions f and g are not continuous at a point x0 , but f + g is continuous at x0 .
D. If f ′ (c) = 0 then f (x) has a local maximum or a local minimum at x = c.

Your Answer: A;C;D Correct Answer: C Incorrect Time taken: 00min 51sec Discuss

Q #11 Multiple Select Type Award: 1 Penalty: 0 Calculus

Suppose f is a function such that f ′ (x) = 4x3 and f ′′ (x) = 12x2 .

Which of the following is /are true?

A. f has a local maximum at x = 0 by the first derivative test

B. f has a local minimum at x = 0 by the first derivative test
C. f has a local maximum at x = 0 by the second derivative test
D. f has a local minimum at x = 0 by the second derivative test

Your Answer: Correct Answer: B Not Attempted Time taken: 00min 07sec Discuss

Q #12 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

The derivative of ( ) = gx e √ x is

A. √ xe √ −1x

B. 2 e xx
√ −0.5

C.
0.5 ex √

√x

D. e √ x

Your Answer: C Correct Answer: C Correct Time taken: 00min 21sec Discuss

Q #13 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

The function

f (x) = { emx + b x≤1

x if
if x>1

is continuous and differentiable at x = 1.

Find the value of m − b?
A. e
B. − e
C. e − 1
D. 1 − e

Your Answer: A Correct Answer: A Correct Time taken: 01min 20sec Discuss

Q #14 Numerical Type Award: 1 Penalty: 0 Calculus

Given F (x) = (f (g(x))) , g(1) = 2, g (1) = 3, f (2) = 4, and f (2) = 5, find F (1)
2 ′ ′ ′

Your Answer: 120 Correct Answer: 120 Correct Time taken: 01min 31sec Discuss

Q #15 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Find the values of A and B that make

f ( x) = { x + 1 x≥0
2
if
A sin x + B cos x if x<0

differentiable at x = 0.
A. A = 0, B = 1
B. A = 1, B = 0
C. A = 0, B = −1
D. A = −1, B = 0
 Your Answer: A Correct Answer: A Correct Time taken: 01min 21sec Discuss

Q #16 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

bx2 + 15x + 15 + b
Is there a number b such that lim exists? If so, find the value of b and the value of the
x→−2 x2 + x − 2
limit.

A. −1
B. −2
C. 1
D. There is no such b for that above limit exist

Your Answer: Correct Answer: A Not Attempted Time taken: 01min 00sec Discuss

Q #17 Numerical Type Award: 2 Penalty: 0 Calculus

1
lim (x + sin x) x
x→∞

Your Answer: 1 Correct Answer: 1 Correct Time taken: 00min 45sec Discuss

Q #18 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

If g is continuous (but not differentiable) at x = 0, g(0) = 8, and f (x) = xg(x), find f ′ (0).

A. 0
B. 8
C. 1
D. f (x) is also not differentiable at x = 0.

Your Answer: B Correct Answer: B Correct Time taken: 03min 19sec Discuss

Q #19 Numerical Type Award: 2 Penalty: 0 Calculus

Suppose that f (x) and g(x) are differentiable functions and that h(x) = f (x)g(x). You are given the
following table of values:

h(1) 24
g(1) 6
f ′ (1) −2
h′ (1) 20

Using the table, find g ′ (1).

Your Answer: Correct Answer: 8 Not Attempted Time taken: 01min 48sec Discuss

Q #20 Multiple Select Type Award: 2 Penalty: 0 Calculus

Suppose f (x) is continuous and diffrentiable for all x. And −1 ≤ f ′ (x) ≤ 3 fo all x. Which of the following
is/are ALWAYS true?

A. f (5) ≤ f (3) + 6
B. f (5) ≥ f (3) − 2
C. f (5) ≤ f (3) + 10
D. f (5) ≥ f (3) − 10

Your Answer: Correct Answer: A;B;C;D Not Attempted Time taken: 00min 52sec Discuss

Q #21 Numerical Type Award: 2 Penalty: 0 Calculus

Define

−6x + 2 x≤2
g ( x) = {
x−2 x>2
4

Find ∫ g(x)dx.
1

Your Answer: -11 Correct Answer: -5 Incorrect Time taken: 01min 53sec Discuss

Q #22 Multiple Select Type Award: 2 Penalty: 0 Calculus

Which of the following is/are TRUE?

5 5

A. ∫ (ax + bx + c) dx = 2 ∫
2
(ax2 + c) dx
−5 0

B. If f and g are continuous and f (x) ⩾ g(x) for a ⩽ x ⩽ b, then

b b
∫ f (x)dx ⩾ ∫ g(x)dx
a a

C. If f and g are differentiable, then

d
[f (g(x))] = f (g(x))g (x)
′ ′

dx
d f ′ ( x)
D. If f is differentiable, then √ f ( x) = .
dx 2 √ f ( x)

Your Answer: A;B;C;D Correct Answer: A;B;C Incorrect Time taken: 00min 57sec Discuss

Q #23 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

Shown below are portions of the graphs of y = f (x), y = f ′ (x), and y = f ′′ (x).
Determine which graph is which.
 A. A − f (x), B − f (x), C − f (x)
′ ′′

B. A − f (x), B − f (x), C − f (x)

′ ′′

C. A − f (x), B − f (x), C − f (x)

′′ ′

D. A − f (x), B − f (x), C − f (x)

′ ′′

Your Answer: Correct Answer: B Not Attempted Time taken: 00min 09sec Discuss

Q #24 Multiple Select Type Award: 2 Penalty: 0 Calculus

fx
The function ( ) is defined as follows:

⎧ x x+ 1 x ≤ 0
f ( x) = ⎨
⎩ ? x>0
2

Note that the formula for ( ) for fx x > 0 is unknown.

fx
However, it is known that ( ) is differentiable at each point in its domain (−∞, ∞), and that f (x) > 0 for
′

all x ≥ 0.
Which of the following option is/are TRUE?

A. x = 0 has global minima

B. x = 0 has global maxima
C. x = −1 has global maxima
D. x = −1 has global minima

Your Answer: Correct Answer: D Not Attempted Time taken: 00min 19sec Discuss

Q #25 Multiple Select Type Award: 2 Penalty: 0 Calculus

hx hx
Suppose ( ) is a function such that ( ) has exactly three critical point. Two of which are shown in the table
below.
Assume that both ( ) and hx h (x) are differentiable on (−∞, ∞)
′

x 0 3 5 7
h ( x) 2 ? 4 4
h ( x)
′
−1 0 0 ?

Further using Lagrange mean value theorem in the interval [5, 7], we can determine the interval of the third
critical point.
On which of the following intervals must h(x) be increasing on the entire interval?

A. (0, 3)
B. (3, 5)
C. (5, 6)
D. (6, 7)

Your Answer: Correct Answer: B Not Attempted Time taken: 00min 03sec Discuss

Q #26 Multiple Select Type Award: 2 Penalty: 0 Calculus

Let q(x) be a continuous function which is defined for all real numbers. A portion of the graph of q ′ (x), the
derivative of q(x), is shown below.

On which of the following interval(s) is q ′′ (x) positive?

A. (0, 2)
B. (2, 4)
C. (7, 9)
D. (5, 7)

Your Answer: Correct Answer: A;B;D Not Attempted Time taken: 00min 03sec Discuss

Q #27 Numerical Type Award: 2 Penalty: 0 Calculus

We consider a function f (x) defined for all real numbers. We suppose that the first and second derivatives
f ′ (x) and f ′′ (x) are also defined for all real numbers. Below we show the graph of the second derivative of f
. You may assume that f ′′ (x) is decreasing outside of the region shown.
Suppose that f (0) = 5. How many critical points does f have?
′

Your Answer: Correct Answer: 2 Not Attempted Time taken: 00min 10sec Discuss

Q #28 Numerical Type Award: 2 Penalty: 0 Calculus

fx
Consider a continuous function ( ), and suppose that ( ) and its first derivative fx f (x) are differentiable
′

fx
everywhere. Suppose we know the following information about ( ) and its first and second derivatives.

On the interval (−∞, −2), we have ( ) = 2−x . fx

xlim f ( x) = 6
f (2) = −5, f (3) = 7, and f (4) = 8
→∞

f (x) is equal to 0 at x = −1, 2, 4, and not at any other x-values.

′

f (x) < 0 on the intervals −1 < x < 0 and 3 < x < 5, and not on any other interval.
′′

Find the global minimum of f (x) on (−∞, ∞)? (minimum value of f (x))

Your Answer: Correct Answer: -5 Not Attempted Time taken: 00min 02sec Discuss

Q #29 Numerical Type Award: 2 Penalty: 0 Calculus

fx
If ( ) = (1 + ) (1 + x x ) (1 + x ) (1 + x ) , then f (0) =?
2 3 4 ′

Your Answer: 2 Correct Answer: 1 Incorrect Time taken: 01min 24sec Discuss

Q #30 Numerical Type Award: 2 Penalty: 0 Calculus

Let M = xlim (ex + 3x)

→0
+
1/ x . Find the value of loge M .

Your Answer: 3 Correct Answer: 4 Incorrect Time taken: 00min 37sec Discuss
Copyright & Stuff
Summary in Graph

Exam Summary (GO Classes| DA | Calculus | Weekly Quiz

3)

Qs. Attempted: 4 Correct Marks: 3

4+0 3+0

Correct Attempts: 3 Penalty Marks: 0.33

3+0 0.33 + 0

Incorrect
Attempts:
1 Resultant Marks: 2.67
1+0 2.67 + 0

Total Questions: 15
10 + 5

Total Marks: 20
10 + 10

Exam Duration: 45 Minutes

Time Taken: 8 Minutes

EXAM RESPONSE EXAM STATS FEEDBACK

Technical

Q #1 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Two 3D surface graphs and two contour plots are shown below. Each graph corresponds to one contour plot.
Match each surface graph (A or B) with the correct contour plot (A or B).

Graph B
Graph A

Contour Plot A Contour Plot B

Which of the following matchings is correct?

A. Graph A → Contour A, Graph B → Contour B

B. Graph A → Contour B, Graph B → Contour A
C. Graph A → Contour A, Graph B → Contour A
D. Graph A → Contour B, Graph B → Contour B

Your Answer: Correct Answer: A Not Attempted Time taken: 00min 07sec Discuss

Q #2 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

You are applying gradient descent with a fixed learning rate γ. The following table shows the function values
at each step of gradient descent over 10 iterations for three different functions.

Match each function in Column 1 with the most appropriate explanation in Column 2.

Column 1 Column 2

(Function Values at each step of Gradient Descent) (Learning Rate Behavior)

f 1 : 100, 99, 98, 97, 96, 95, 94, 93, 92, 91 (i) Too high

f 2 : 100, 50, 75, 60, 65, 45, 75, 110, 90, 85 (ii) Too low

f 3 : 100, 80, 65, 50, 40, 35, 31, 29, 28, 27.5, 27.3 (iii) About right

Choose the correct matching:

A. f 1 − (ii), f
2 − (i), f 3 − (iii)

B. f 1 − (i), f 2 − (ii), f 3 − (iii)

C. f 1 − (iii), f 2 − (ii), f 3 − (i)

D. f 1 − (ii), f 2 − (iii), f 3 − (i)

Your Answer: Correct Answer: B Not Attempted Time taken: 00min 25sec Discuss

Q #3 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

At the optimal point for the following constrained optimization problem:

Minimize f ( x, y )
Subject to g(x, y) = 0

Which of the following statements is true?

A. The gradient of f is equal to the gradient of g

B. The gradients of f and g are orthogonal

C. The gradients of f and g are parallel

D. The gradients of f and g are zero vectors

Your Answer: Correct Answer: C Not Attempted Time taken: 00min 08sec Discuss
Q #4 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Which of the following is a necessary and sufficient condition for a twice differentiable function f : R → R to
be convex?

A. f (x) > 0 for all x ∈ R

B. f ′ (x) > 0 for all x ∈ R
C. f ′′ (x) ≥ 0 for all x ∈ R
D. f ′′ (x) = 0 for all x ∈ R

Your Answer: Correct Answer: C Not Attempted Time taken: 00min 03sec Discuss

Q #5 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Let f : Rn → R be a convex function. Which of the following is always true?

A. Every local minimum of f is a global minimum

B. ∇f (x) = 0 at every point

C. f has only one critical point

D. The Hessian of f is always the zero matrix

Your Answer: Correct Answer: A Not Attempted Time taken: 00min 11sec Discuss

Q #6 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Let f (x, y) = x2 + y 2 . What is the direction of steepest descent at point (1, −1) ?

A. (2, −2)

B. (−2, 2)

C. (−1, 1)

D. (1, −1)

Your Answer: D Correct Answer: B Incorrect Time taken: 01min 10sec Discuss

Q #7 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Let f : R 2
→ R be a twice continuously differentiable function. Suppose:
∂f ∂f
= y and = x2
∂x ∂y

Which of the following statements is correct?

 A. Such a function f exists but is not continuous on R. 2

B. No such function f exists.

C. The function f exists, but is not differentiable.

D. None of the above.

Your Answer: Correct Answer: B Not Attempted Time taken: 00min 05sec Discuss

Q #8 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Let

f ( x) = e x

What is the second-degree Taylor polynomial of f (x) centered at x = 0 ?

A. 1 + x +
x2
2

B. 1 + x + x3
6

C. 1 + x2
2

D. x + x2
2

Your Answer: A Correct Answer: A Correct Time taken: 01min 19sec Discuss

Q #9 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Let f (x) e−x , and consider its Taylor polynomial of degree 4 centered at x = 0.
=

What is the coefficient of x4 ?

A. − 4!
1

B. 1

4!

C. − 12

D.
1

Your Answer: B Correct Answer: B Correct Time taken: 01min 50sec Discuss

Q #10 Multiple Choice Type Award: 1 Penalty: 0.33 Calculus

Let

2 x + y2
∇ (f x, y ) = [ ]
2 y + 2xy

Then the Hessian matrix of f (x, y) is:

 2 0
A. [ ]
0 2

2 2y
B. [ ]
2y 2 + 2x

2 2y
C. [ ]
2x 2 + 2x

2 y
D. [ ]
y 2

Your Answer: B Correct Answer: B Correct Time taken: 00min 40sec Discuss

Q #11 Multiple Select Type Award: 2 Penalty: 0 Calculus

Which of the following are true about gradient descent? (select all statements that are true.)

A. After each iteration, we modify the weight vector in the direction of the gradient.

B. We have to choose a non-variable learning rate.

C. After each iteration, we modify the weight vector in the direction of the negative gradient.

D. In the gradient descent algorithm, each update of the weight vector depends on all the training
examples.

Your Answer: Correct Answer: C;D Not Attempted Time taken: 00min 18sec Discuss

Q #12 Multiple Select Type Award: 2 Penalty: 0 Calculus

Consider a function f (x, y) with gradient

x−1
∇f ( x, y ) = [ ]
2y + x

Starting at the point ( x = 1, y = 2 ), and using a learning rate γ = 1, perform one step of gradient descent.
What is the updated point (x′ , y ′ ) ?

A. (1, −3)

B. (0, −4)

C. (2, −2)

D. (1, 0)

Your Answer: Correct Answer: A Not Attempted Time taken: 00min 44sec Discuss

Q #13 Multiple Select Type Award: 2 Penalty: 0 Calculus

Consider the function

f ( x, y ) = x2 + 2 y 2
subject to the constraint

x 2
+ y 2
≤ 4.

Which of the following statements is/are correct?

fxy
A. The global minimum of ( , ) on the region occurs at (0, 0) with value f = 0.
B. The global maximum occurs on the boundary at points (0, 2) and (0, −2).

C. The Lagrange multiplier method gives the candidate points (±2, 0), (0, ±2).

D. The point (2, 2) is a feasible critical point inside the region.

Your Answer: Correct Answer: A;B;C Not Attempted Time taken: 00min 07sec Discuss

Q #14 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

For what values of the parameters , α β ∈ R is the function

f (x , x ) = x + 2αx x
1 2
2
1 1 2 + βx 2
2

a convex function?

A. β≥α 2

B. β>α 2

C. β≥α
D. β ≥ 0 and α = 0

Your Answer: Correct Answer: A Not Attempted Time taken: 00min 12sec Discuss

Q #15 Multiple Choice Type Award: 2 Penalty: 0.67 Calculus

Consider the function

f (x, y, z) = 3xy + z 2

Compute the rate of change of f at the point (1, −2, 2) in the direction of the vector
v→ = ⟨−1, 2, −2⟩
A. 2/3

B. 4/3

C. 8/3

D. 2

Your Answer: Correct Answer: B Not Attempted Time taken: 00min 04sec Discuss
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