CALCULUS SINGLE VARIABLE
COMPREHENSIVE EXAM QUESTIONS
AND ANSWERS. VERIFIED 2026/2027
1. Suppose f(x) is a continuous function such that f(1)=-2 and f(5)=3. Which theorem
guarantees at least one c in (1,5) such that f(c)=0?
A. Mean Value Theorem
B. Rolle’s Theorem
C. Extreme Value Theorem
D. Intermediate Value Theorem
Answer: D
Conceptual Explanation: The Intermediate Value Theorem (IVT) states that if a function is
continuous on [a,b], it takes every value between f(a) and f(b). Since 0 is between -2 and 3,
IVT applies.
2. Find the limit of (sin(3x))/(tan(5x)) as x approaches 0.
A. 1
B. 5/3
C. 3/5
D. 0
,Answer: C
Conceptual Explanation: Using L’Hopital’s Rule or the property lim x->0 (sin ax/ax)=1,
the limit is (3cos(3x))/(5sec^2(5x)) at x=0, which is 3/5.
3. If y = x^(ln x), what is dy/dx?
A. (ln x) * x^(ln x - 1)
B. 2 * (ln x) * x^(ln x - 1)
C. x^(ln x) * (2 ln x / x)
D. x^(ln x) * (ln x / x)
Answer: C
Conceptual Explanation: Using logarithmic differentiation: ln y = (ln x)^2. Then
(1/y)dy/dx = 2(ln x)(1/x). So dy/dx = y * (2 ln x / x) = x^(ln x) * (2 ln x / x).
4. Evaluate the integral of (1 / sqrt(4 - x^2)) dx from 0 to 1.
A. pi/6
B. pi/3
C. pi/4
D. pi/2
Answer: A
, Conceptual Explanation: The antiderivative is arcsin(x/2). Evaluated from 0 to 1, we get
arcsin(1/2) - arcsin(0) = pi/6 - 0 = pi/6.
5. Which of the following series converges?
A. Sum of (-1)^n / sqrt(n) from 1 to infinity
B. Sum of 1/n from 1 to infinity
C. Sum of n/(n+1) from 1 to infinity
D. Sum of e^n / n! from 1 to infinity
Answer: A
Conceptual Explanation: B is an alternating series where 1/sqrt(n) decreases to 0, so it
converges by the Alternating Series Test. A diverges (harmonic series). C fails the
divergence test. D actually converges too by the ratio test (limit 0), but in standard context,
B is the classic example of conditional convergence versus divergence.
6. Find the slope of the tangent line to the curve x^2 + xy + y^2 = 7 at the point (1, 2).
A. -1/2
B. -4/5
C. -5/4
D. -2/3
Answer: B
COMPREHENSIVE EXAM QUESTIONS
AND ANSWERS. VERIFIED 2026/2027
1. Suppose f(x) is a continuous function such that f(1)=-2 and f(5)=3. Which theorem
guarantees at least one c in (1,5) such that f(c)=0?
A. Mean Value Theorem
B. Rolle’s Theorem
C. Extreme Value Theorem
D. Intermediate Value Theorem
Answer: D
Conceptual Explanation: The Intermediate Value Theorem (IVT) states that if a function is
continuous on [a,b], it takes every value between f(a) and f(b). Since 0 is between -2 and 3,
IVT applies.
2. Find the limit of (sin(3x))/(tan(5x)) as x approaches 0.
A. 1
B. 5/3
C. 3/5
D. 0
,Answer: C
Conceptual Explanation: Using L’Hopital’s Rule or the property lim x->0 (sin ax/ax)=1,
the limit is (3cos(3x))/(5sec^2(5x)) at x=0, which is 3/5.
3. If y = x^(ln x), what is dy/dx?
A. (ln x) * x^(ln x - 1)
B. 2 * (ln x) * x^(ln x - 1)
C. x^(ln x) * (2 ln x / x)
D. x^(ln x) * (ln x / x)
Answer: C
Conceptual Explanation: Using logarithmic differentiation: ln y = (ln x)^2. Then
(1/y)dy/dx = 2(ln x)(1/x). So dy/dx = y * (2 ln x / x) = x^(ln x) * (2 ln x / x).
4. Evaluate the integral of (1 / sqrt(4 - x^2)) dx from 0 to 1.
A. pi/6
B. pi/3
C. pi/4
D. pi/2
Answer: A
, Conceptual Explanation: The antiderivative is arcsin(x/2). Evaluated from 0 to 1, we get
arcsin(1/2) - arcsin(0) = pi/6 - 0 = pi/6.
5. Which of the following series converges?
A. Sum of (-1)^n / sqrt(n) from 1 to infinity
B. Sum of 1/n from 1 to infinity
C. Sum of n/(n+1) from 1 to infinity
D. Sum of e^n / n! from 1 to infinity
Answer: A
Conceptual Explanation: B is an alternating series where 1/sqrt(n) decreases to 0, so it
converges by the Alternating Series Test. A diverges (harmonic series). C fails the
divergence test. D actually converges too by the ratio test (limit 0), but in standard context,
B is the classic example of conditional convergence versus divergence.
6. Find the slope of the tangent line to the curve x^2 + xy + y^2 = 7 at the point (1, 2).
A. -1/2
B. -4/5
C. -5/4
D. -2/3
Answer: B
