CALCULUS SINGLE VARIABLE
COMPREHENSIVE ASSESSMENT
QUESTIONS AND ANSWERS. VERIFIED
2026/2027
1. If f(x) = e^(g(x)) and g(1) = 0, g’(1) = 5, what is f’(1)?
A. 0
B. 1
C. 5
D. e^5
Answer: C
Conceptual Explanation: Using the chain rule, f’(x) = e^(g(x)) * g’(x). At x=1, f’(1) =
e^(g(1)) * g’(1) = e^0 * 5 = 1 * 5 = 5.
2. Which of the following describes the behavior of the function f(x) = x^3 - 3x^2 at its critical
points?
A. Local max at x=0, local max at x=2
B. Local min at x=0, local max at x=2
C. Inflection point at x=0, local max at x=2
D. Local max at x=0, local min at x=2
,Answer: D
Conceptual Explanation: f’(x) = 3x^2 - 6x = 3x(x-2). Critical points are 0 and 2. f’‘(x) = 6x -
6. f’‘(0) = -6 (max), f’’(2) = 6 (min).
3. Evaluate the limit as x approaches infinity of (1 + 2/x)^x.
A. 1
B. 2
C. e
D. e^2
Answer: D
Conceptual Explanation: This is a standard limit form where lim(x to inf) (1 + a/x)^(bx) =
e^(ab). Here a=2 and b=1, so the limit is e^2.
4. The Mean Value Theorem guarantees a point c in (1, 3) for f(x) = x^2 such that f’(c) equals:
A. 2
B. 4
C. 8
D. 16
Answer: B
Conceptual Explanation: MVT states f’(c) = (f(3) - f(1)) / (3 - 1) = (9 - 1) / 2 = = 4.
, 5. What is the derivative of arcsin(3x)?
A. 3 / sqrt(1 - 9x^2)
B. 1 / sqrt(1 - 9x^2)
C. 3 / (1 + 9x^2)
D. -3 / sqrt(1 - 9x^2)
Answer: A
Conceptual Explanation: The derivative of arcsin(u) is u’ / sqrt(1 - u^2). Here u = 3x and
u’ = 3.
6. If the integral from 1 to 5 of f(x) dx = 12 and the integral from 4 to 5 of f(x) dx = 3, what is
the integral from 1 to 4 of f(x) dx?
A. 15
B. 36
C. 4
D. 9
Answer: D
Conceptual Explanation: By property of integrals, Int(1 to 5) = Int(1 to 4) + Int(4 to 5). So
12 = Int(1 to 4) + 3, meaning Int(1 to 4) = 9.
7. Find the slope of the tangent line to x^2 + y^2 = 25 at the point (3, 4).
A. 3/4
COMPREHENSIVE ASSESSMENT
QUESTIONS AND ANSWERS. VERIFIED
2026/2027
1. If f(x) = e^(g(x)) and g(1) = 0, g’(1) = 5, what is f’(1)?
A. 0
B. 1
C. 5
D. e^5
Answer: C
Conceptual Explanation: Using the chain rule, f’(x) = e^(g(x)) * g’(x). At x=1, f’(1) =
e^(g(1)) * g’(1) = e^0 * 5 = 1 * 5 = 5.
2. Which of the following describes the behavior of the function f(x) = x^3 - 3x^2 at its critical
points?
A. Local max at x=0, local max at x=2
B. Local min at x=0, local max at x=2
C. Inflection point at x=0, local max at x=2
D. Local max at x=0, local min at x=2
,Answer: D
Conceptual Explanation: f’(x) = 3x^2 - 6x = 3x(x-2). Critical points are 0 and 2. f’‘(x) = 6x -
6. f’‘(0) = -6 (max), f’’(2) = 6 (min).
3. Evaluate the limit as x approaches infinity of (1 + 2/x)^x.
A. 1
B. 2
C. e
D. e^2
Answer: D
Conceptual Explanation: This is a standard limit form where lim(x to inf) (1 + a/x)^(bx) =
e^(ab). Here a=2 and b=1, so the limit is e^2.
4. The Mean Value Theorem guarantees a point c in (1, 3) for f(x) = x^2 such that f’(c) equals:
A. 2
B. 4
C. 8
D. 16
Answer: B
Conceptual Explanation: MVT states f’(c) = (f(3) - f(1)) / (3 - 1) = (9 - 1) / 2 = = 4.
, 5. What is the derivative of arcsin(3x)?
A. 3 / sqrt(1 - 9x^2)
B. 1 / sqrt(1 - 9x^2)
C. 3 / (1 + 9x^2)
D. -3 / sqrt(1 - 9x^2)
Answer: A
Conceptual Explanation: The derivative of arcsin(u) is u’ / sqrt(1 - u^2). Here u = 3x and
u’ = 3.
6. If the integral from 1 to 5 of f(x) dx = 12 and the integral from 4 to 5 of f(x) dx = 3, what is
the integral from 1 to 4 of f(x) dx?
A. 15
B. 36
C. 4
D. 9
Answer: D
Conceptual Explanation: By property of integrals, Int(1 to 5) = Int(1 to 4) + Int(4 to 5). So
12 = Int(1 to 4) + 3, meaning Int(1 to 4) = 9.
7. Find the slope of the tangent line to x^2 + y^2 = 25 at the point (3, 4).
A. 3/4
