WGU C958 — CALCULUS I-COMPREHENSIVE EXAM QUESTIONS AND CORRECT ANSWERS
WITH RATIONALES.PDF
Core Domains
Functions, Graphs, and Precalculus Review
Limits and Continuity
Derivatives and Differentiation Rules
Applications of Derivatives
Antiderivatives and Indefinite Integration
Definite Integrals and the Fundamental Theorem of Calculus
Integration Techniques and Applications
Differential Equations and Modeling
Introduction
The WGU C958 Calculus I Comprehensive Examination evaluates the candidate's mastery of
fundamental differential and integral calculus concepts required for advanced mathematical
study. The assessment measures understanding of limits, continuity, differentiation rules,
applications of derivatives, antiderivatives, definite integrals, and the Fundamental Theorem of
Calculus. Content emphasizes analytical problem-solving, graphical interpretation, and real-
world modeling through rates of change and accumulation. This comprehensive test bank
challenges the candidate to evaluate limits algebraically and graphically, compute derivatives
using power, product, quotient, and chain rules, apply differentiation to optimization and related
rates problems, evaluate integrals using substitution and the Fundamental Theorem, and model
physical phenomena using differential equations, preparing the candidate for success on the
WGU C958 Objective Assessment.
SECTION ONE: QUESTIONS 1–100
1. Evaluate the limit: lim(x→3) (x² − 9)/(x − 3)
A. 0
B. 3
C. 6
D. Does not exist
,🟢 C. 6
🔴 RATIONALE: Direct substitution yields 0/0, an indeterminate form. Factor the numerator as
(x − 3)(x + 3), cancel the common factor (x − 3), leaving x + 3. Substituting x = 3 gives 3 + 3 =
6.
2. Evaluate the limit: lim(x→0) sin(5x)/x
A. 0
B. 1
C. 5
D. Does not exist
🟢 C. 5
🔴 RATIONALE: Use the standard trigonometric limit lim(u→0) sin(u)/u = 1. Rewrite sin(5x)/x as
5 · sin(5x)/(5x). As x→0, 5x→0, so the expression approaches 5 · 1 = 5.
3. Evaluate the limit: lim(x→∞) (4x² − 3x + 1)/(2x² + 7)
A. 0
B. 1
C. 2
D. ∞
🟢 C. 2
🔴 RATIONALE: Divide numerator and denominator by the highest power x²: (4 − 3/x +
1/x²)/(2 + 7/x²). As x→∞, all terms with x in the denominator approach 0, leaving 4/2 = 2.
4. Evaluate the limit: lim(x→2⁻) |x − 2|/(x − 2)
A. −1
B. 0
C. 1
D. Does not exist
🟢 A. −1
,🔴 RATIONALE: For x < 2, x − 2 < 0, so |x − 2| = −(x − 2). The ratio becomes −(x − 2)/(x − 2) =
−1. The left-hand limit is −1.
5. Evaluate the limit: lim(x→0) (1 − cos x)/x²
A. 0
B. 1/2
C. 1
D. 2
🟢 B. 1/2
🔴 RATIONALE: Multiply numerator and denominator by (1 + cos x) to obtain sin²x/(x²(1 + cos
x)). Since lim(x→0) sin x/x = 1, this becomes 1/(1 + cos 0) = 1/2.
6. Which of the following is the limit definition of the derivative?
A. lim(h→0) [f(x + h) − f(x)]/h
B. lim(x→a) [f(x) − f(a)]/(x − a)
C. Both A and B
D. lim(x→∞) [f(x + h) − f(x)]/h
🟢 C. Both A and B
🔴 RATIONALE: Both forms are valid definitions of the derivative. Form A uses the difference
quotient with h, and form B uses the difference quotient with x approaching a. Both represent
the instantaneous rate of change.
7. Find the derivative of f(x) = 5x³ − 2x + 7.
A. 15x² − 2
B. 5x² − 2
C. 15x³ − 2
D. 15x² + 7
🟢 A. 15x² − 2
🔴 RATIONALE: Apply the power rule: d/dx[5x³] = 15x², d/dx[−2x] = −2, and d/dx[7] = 0. The
derivative is 15x² − 2.
, 8. Find the derivative of f(x) = ln(x² + 3x + 2).
A. (2x + 3)/(x² + 3x + 2)
B. 1/(x² + 3x + 2)
C. (2x + 3)
D. ln(2x + 3)
🟢 A. (2x + 3)/(x² + 3x + 2)
🔴 RATIONALE: Using the chain rule, let u = x² + 3x + 2, then du/dx = 2x + 3. The derivative of
ln(u) is (1/u) · du/dx = (2x + 3)/(x² + 3x + 2).
9. Find dy/dx for y = arctan(2x).
A. 2/(1 + 4x²)
B. 1/(1 + 4x²)
C. 2/(1 + 2x²)
D. 2x/(1 + 4x²)
🟢 A. 2/(1 + 4x²)
🔴 RATIONALE: The derivative of arctan(u) is u′/(1 + u²). Here u = 2x, u′ = 2. So the derivative is
2/(1 + (2x)²) = 2/(1 + 4x²).
10. Find the derivative of f(x) = √(3x + 1).
A. 3/(2√(3x + 1))
B. 1/(2√(3x + 1))
C. 3/√(3x + 1)
D. 6/√(3x + 1)
🟢 A. 3/(2√(3x + 1))
🔴 RATIONALE: Rewrite as (3x + 1)^(1/2). Chain rule: (1/2)(3x + 1)^(−1/2) · 3 = 3/(2√(3x + 1)).
11. Find the derivative of f(x) = x² sin x.
A. 2x sin x + x² cos x
B. 2x cos x
WITH RATIONALES.PDF
Core Domains
Functions, Graphs, and Precalculus Review
Limits and Continuity
Derivatives and Differentiation Rules
Applications of Derivatives
Antiderivatives and Indefinite Integration
Definite Integrals and the Fundamental Theorem of Calculus
Integration Techniques and Applications
Differential Equations and Modeling
Introduction
The WGU C958 Calculus I Comprehensive Examination evaluates the candidate's mastery of
fundamental differential and integral calculus concepts required for advanced mathematical
study. The assessment measures understanding of limits, continuity, differentiation rules,
applications of derivatives, antiderivatives, definite integrals, and the Fundamental Theorem of
Calculus. Content emphasizes analytical problem-solving, graphical interpretation, and real-
world modeling through rates of change and accumulation. This comprehensive test bank
challenges the candidate to evaluate limits algebraically and graphically, compute derivatives
using power, product, quotient, and chain rules, apply differentiation to optimization and related
rates problems, evaluate integrals using substitution and the Fundamental Theorem, and model
physical phenomena using differential equations, preparing the candidate for success on the
WGU C958 Objective Assessment.
SECTION ONE: QUESTIONS 1–100
1. Evaluate the limit: lim(x→3) (x² − 9)/(x − 3)
A. 0
B. 3
C. 6
D. Does not exist
,🟢 C. 6
🔴 RATIONALE: Direct substitution yields 0/0, an indeterminate form. Factor the numerator as
(x − 3)(x + 3), cancel the common factor (x − 3), leaving x + 3. Substituting x = 3 gives 3 + 3 =
6.
2. Evaluate the limit: lim(x→0) sin(5x)/x
A. 0
B. 1
C. 5
D. Does not exist
🟢 C. 5
🔴 RATIONALE: Use the standard trigonometric limit lim(u→0) sin(u)/u = 1. Rewrite sin(5x)/x as
5 · sin(5x)/(5x). As x→0, 5x→0, so the expression approaches 5 · 1 = 5.
3. Evaluate the limit: lim(x→∞) (4x² − 3x + 1)/(2x² + 7)
A. 0
B. 1
C. 2
D. ∞
🟢 C. 2
🔴 RATIONALE: Divide numerator and denominator by the highest power x²: (4 − 3/x +
1/x²)/(2 + 7/x²). As x→∞, all terms with x in the denominator approach 0, leaving 4/2 = 2.
4. Evaluate the limit: lim(x→2⁻) |x − 2|/(x − 2)
A. −1
B. 0
C. 1
D. Does not exist
🟢 A. −1
,🔴 RATIONALE: For x < 2, x − 2 < 0, so |x − 2| = −(x − 2). The ratio becomes −(x − 2)/(x − 2) =
−1. The left-hand limit is −1.
5. Evaluate the limit: lim(x→0) (1 − cos x)/x²
A. 0
B. 1/2
C. 1
D. 2
🟢 B. 1/2
🔴 RATIONALE: Multiply numerator and denominator by (1 + cos x) to obtain sin²x/(x²(1 + cos
x)). Since lim(x→0) sin x/x = 1, this becomes 1/(1 + cos 0) = 1/2.
6. Which of the following is the limit definition of the derivative?
A. lim(h→0) [f(x + h) − f(x)]/h
B. lim(x→a) [f(x) − f(a)]/(x − a)
C. Both A and B
D. lim(x→∞) [f(x + h) − f(x)]/h
🟢 C. Both A and B
🔴 RATIONALE: Both forms are valid definitions of the derivative. Form A uses the difference
quotient with h, and form B uses the difference quotient with x approaching a. Both represent
the instantaneous rate of change.
7. Find the derivative of f(x) = 5x³ − 2x + 7.
A. 15x² − 2
B. 5x² − 2
C. 15x³ − 2
D. 15x² + 7
🟢 A. 15x² − 2
🔴 RATIONALE: Apply the power rule: d/dx[5x³] = 15x², d/dx[−2x] = −2, and d/dx[7] = 0. The
derivative is 15x² − 2.
, 8. Find the derivative of f(x) = ln(x² + 3x + 2).
A. (2x + 3)/(x² + 3x + 2)
B. 1/(x² + 3x + 2)
C. (2x + 3)
D. ln(2x + 3)
🟢 A. (2x + 3)/(x² + 3x + 2)
🔴 RATIONALE: Using the chain rule, let u = x² + 3x + 2, then du/dx = 2x + 3. The derivative of
ln(u) is (1/u) · du/dx = (2x + 3)/(x² + 3x + 2).
9. Find dy/dx for y = arctan(2x).
A. 2/(1 + 4x²)
B. 1/(1 + 4x²)
C. 2/(1 + 2x²)
D. 2x/(1 + 4x²)
🟢 A. 2/(1 + 4x²)
🔴 RATIONALE: The derivative of arctan(u) is u′/(1 + u²). Here u = 2x, u′ = 2. So the derivative is
2/(1 + (2x)²) = 2/(1 + 4x²).
10. Find the derivative of f(x) = √(3x + 1).
A. 3/(2√(3x + 1))
B. 1/(2√(3x + 1))
C. 3/√(3x + 1)
D. 6/√(3x + 1)
🟢 A. 3/(2√(3x + 1))
🔴 RATIONALE: Rewrite as (3x + 1)^(1/2). Chain rule: (1/2)(3x + 1)^(−1/2) · 3 = 3/(2√(3x + 1)).
11. Find the derivative of f(x) = x² sin x.
A. 2x sin x + x² cos x
B. 2x cos x
