WGU C958 CALCULUS 1 COMPREHENSIVE EXAM – QUESTIONS AND ANSWERS |
VERIFIED AND WELL DETAILED ANSWERS | PLUS RATIONALES |DOWNLOAD
AND PASS | LATEST EXAM UPDATE 2026/2027
Core Domains
Limits and Continuity
Differentiation Rules and Techniques
Applications of Derivatives
Integration Techniques
The Fundamental Theorem of Calculus
Applications of Integration
Sequences and Series
Parametric, Polar, and Vector Functions
Introduction
This comprehensive examination is designed to assess a student's mastery of the
fundamental concepts and techniques of single-variable calculus. The assessment
covers a wide range of topics, from the intuitive understanding of limits to the
powerful applications of integration. It evaluates both conceptual knowledge and
procedural fluency through a series of multiple-choice and scenario-based questions.
The exam emphasizes real-world application, critical thinking, and effective decision-
making, requiring candidates to not only compute answers but also to select and
apply the appropriate mathematical tools to solve complex problems. This ensures a
thorough evaluation of readiness for advanced mathematical study.
SECTION ONE: QUESTIONS 1–50
Question 1
What is the value of the limit lim (x→3) (x² - 9) / (x - 3)?
,A. 0
B. 3
C. 6
D. The limit does not exist.
🟢 Correct Answer: C. 6
🔴 Explanation: The expression is indeterminate of the form 0/0. By factoring the
numerator as (x - 3)(x + 3) and canceling the common factor (x - 3), the limit
becomes lim (x→3) (x + 3). Substituting x = 3 gives 3 + 3 = 6.
Question 2
Which of the following conditions is NOT required for a function f(x) to be
continuous at a point x = c?
A. f(c) is defined.
B. lim (x→c) f(x) exists.
C. f(c) = lim (x→c) f(x).
D. f'(c) exists.
🟢 Correct Answer: D. f'(c) exists.
🔴 Explanation: Continuity at a point requires the function to be defined, the limit
to exist, and the function value to equal the limit. Differentiability (the existence of
the derivative) implies continuity, but continuity does not imply differentiability. A
function can be continuous at a point but not differentiable there, such as f(x) = |x|
at x = 0.
Question 3
What is the derivative of f(x) = sin(x²)?
A. cos(x²)
B. 2x cos(x²)
,C. -2x cos(x²)
D. 2x sin(x²)
🟢 Correct Answer: B. 2x cos(x²)
🔴 Explanation: This requires the chain rule. The derivative of the outer function
sin(u) is cos(u), and the derivative of the inner function u = x² is 2x. Multiplying
these gives 2x cos(x²).
Question 4
Evaluate the integral ∫ (3x² + 2x + 1) dx.
A. x³ + x² + x + C
B. 6x + 2 + C
C. x³ + x² + x
D. 3x³ + 2x² + x + C
🟢 Correct Answer: A. x³ + x² + x + C
🔴 Explanation: Using the power rule for integration, ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, we
integrate each term separately: ∫3x² dx = x³, ∫2x dx = x², and ∫1 dx = x. Combining
these and adding the constant of integration C gives x³ + x² + x + C.
Question 5
A particle's position is given by s(t) = t³ - 6t² + 9t. At what time(s) is the particle at
rest?
A. t = 1 and t = 3
B. t = 0 and t = 2
C. t = 3 only
D. t = 1 only
🟢 Correct Answer: A. t = 1 and t = 3
, 🔴 Explanation: The particle is at rest when its velocity is zero. Velocity is the
derivative of position: v(t) = s'(t) = 3t² - 12t + 9. Setting v(t) = 0 gives 3(t² - 4t + 3)
= 0, which factors to 3(t - 1)(t - 3) = 0. Thus, the particle is at rest at t = 1 and t = 3.
Question 6
What is the value of the definite integral ∫ from 0 to π/2 of cos(x) dx?
A. 0
B. 1
C. -1
D. π/2
🟢 Correct Answer: B. 1
🔴 Explanation: The antiderivative of cos(x) is sin(x). Evaluating from 0 to π/2 gives
sin(π/2) - sin(0) = 1 - 0 = 1.
Question 7
If f(x) = e^(2x), what is f''(x)?
A. 2e^(2x)
B. 4e^(2x)
C. e^(2x)
D. 2e^(x)
🟢 Correct Answer: B. 4e^(2x)
🔴 Explanation: The first derivative is f'(x) = 2e^(2x) (using the chain rule). The
second derivative is the derivative of f'(x), which is f''(x) = 2 * (2e^(2x)) = 4e^(2x).
Question 8
Find the limit: lim (x→0) (sin(5x)) / x.
VERIFIED AND WELL DETAILED ANSWERS | PLUS RATIONALES |DOWNLOAD
AND PASS | LATEST EXAM UPDATE 2026/2027
Core Domains
Limits and Continuity
Differentiation Rules and Techniques
Applications of Derivatives
Integration Techniques
The Fundamental Theorem of Calculus
Applications of Integration
Sequences and Series
Parametric, Polar, and Vector Functions
Introduction
This comprehensive examination is designed to assess a student's mastery of the
fundamental concepts and techniques of single-variable calculus. The assessment
covers a wide range of topics, from the intuitive understanding of limits to the
powerful applications of integration. It evaluates both conceptual knowledge and
procedural fluency through a series of multiple-choice and scenario-based questions.
The exam emphasizes real-world application, critical thinking, and effective decision-
making, requiring candidates to not only compute answers but also to select and
apply the appropriate mathematical tools to solve complex problems. This ensures a
thorough evaluation of readiness for advanced mathematical study.
SECTION ONE: QUESTIONS 1–50
Question 1
What is the value of the limit lim (x→3) (x² - 9) / (x - 3)?
,A. 0
B. 3
C. 6
D. The limit does not exist.
🟢 Correct Answer: C. 6
🔴 Explanation: The expression is indeterminate of the form 0/0. By factoring the
numerator as (x - 3)(x + 3) and canceling the common factor (x - 3), the limit
becomes lim (x→3) (x + 3). Substituting x = 3 gives 3 + 3 = 6.
Question 2
Which of the following conditions is NOT required for a function f(x) to be
continuous at a point x = c?
A. f(c) is defined.
B. lim (x→c) f(x) exists.
C. f(c) = lim (x→c) f(x).
D. f'(c) exists.
🟢 Correct Answer: D. f'(c) exists.
🔴 Explanation: Continuity at a point requires the function to be defined, the limit
to exist, and the function value to equal the limit. Differentiability (the existence of
the derivative) implies continuity, but continuity does not imply differentiability. A
function can be continuous at a point but not differentiable there, such as f(x) = |x|
at x = 0.
Question 3
What is the derivative of f(x) = sin(x²)?
A. cos(x²)
B. 2x cos(x²)
,C. -2x cos(x²)
D. 2x sin(x²)
🟢 Correct Answer: B. 2x cos(x²)
🔴 Explanation: This requires the chain rule. The derivative of the outer function
sin(u) is cos(u), and the derivative of the inner function u = x² is 2x. Multiplying
these gives 2x cos(x²).
Question 4
Evaluate the integral ∫ (3x² + 2x + 1) dx.
A. x³ + x² + x + C
B. 6x + 2 + C
C. x³ + x² + x
D. 3x³ + 2x² + x + C
🟢 Correct Answer: A. x³ + x² + x + C
🔴 Explanation: Using the power rule for integration, ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, we
integrate each term separately: ∫3x² dx = x³, ∫2x dx = x², and ∫1 dx = x. Combining
these and adding the constant of integration C gives x³ + x² + x + C.
Question 5
A particle's position is given by s(t) = t³ - 6t² + 9t. At what time(s) is the particle at
rest?
A. t = 1 and t = 3
B. t = 0 and t = 2
C. t = 3 only
D. t = 1 only
🟢 Correct Answer: A. t = 1 and t = 3
, 🔴 Explanation: The particle is at rest when its velocity is zero. Velocity is the
derivative of position: v(t) = s'(t) = 3t² - 12t + 9. Setting v(t) = 0 gives 3(t² - 4t + 3)
= 0, which factors to 3(t - 1)(t - 3) = 0. Thus, the particle is at rest at t = 1 and t = 3.
Question 6
What is the value of the definite integral ∫ from 0 to π/2 of cos(x) dx?
A. 0
B. 1
C. -1
D. π/2
🟢 Correct Answer: B. 1
🔴 Explanation: The antiderivative of cos(x) is sin(x). Evaluating from 0 to π/2 gives
sin(π/2) - sin(0) = 1 - 0 = 1.
Question 7
If f(x) = e^(2x), what is f''(x)?
A. 2e^(2x)
B. 4e^(2x)
C. e^(2x)
D. 2e^(x)
🟢 Correct Answer: B. 4e^(2x)
🔴 Explanation: The first derivative is f'(x) = 2e^(2x) (using the chain rule). The
second derivative is the derivative of f'(x), which is f''(x) = 2 * (2e^(2x)) = 4e^(2x).
Question 8
Find the limit: lim (x→0) (sin(5x)) / x.
