WGU C958 — CALCULUS I-COMPREHENSIVE
EXAM QUESTIONS AND CORRECT ANSWERS
(VERIFIED ANSWERS) PLUS RATIONALES 2026
Q&A | INSTANT DOWNLOAD PDF.
• CORE DOMAINS *
•
1. Functions, Graphs, and Models *
•
2. Limits and Continuity *
•
3. Definition of the Derivative *
•
4. Differentiation Rules and Techniques *
•
5. Applications of Derivatives *
•
6. Antiderivatives and Indefinite Integration *
•
7. Definite Integrals and the Fundamental Theorem of Calculus *
•
8. Applications of Integration *
•
, 9. Exponential and Logarithmic Functions in Calculus *
•
10. Problem Solving and Mathematical Reasoning *
• INTRODUCTION *
• The WGU C958 Calculus I Comprehensive Examination assesses a
candidate's mastery of differential and integral calculus of one variable. It
evaluates conceptual understanding and applied problem-solving across
limits, continuity, derivatives, and integration. The examination uses multiple-
choice and scenario-based questions requiring analytical reasoning and
mathematical decision-making. Emphasis is placed on real-world
applications including rates of change, optimization, related rates, and area
accumulation. Successful performance demonstrates readiness to apply
calculus principles in science, engineering, and quantitative reasoning
contexts. *
SECTION ONE: QUESTIONS 1–100
Question 1
Evaluate the limit: lim (x→3) (x² − 9)/(x − 3)
A. 0
B. 6
C. 3
D. Does not exist
RATIONALE: Factor the numerator as (x − 3)(x + 3). Cancel the common factor
(x − 3), leaving x + 3. Substitute x = 3 to obtain 6.
Question 2
Evaluate the limit: lim (x→∞) (4x² + 3x)/(2x² − 5)
A. ∞
B. 0
, C. 2
D. 4
RATIONALE: Divide numerator and denominator by x², yielding (4 + 3/x)/(2 −
5/x²). As x → ∞, terms with x in the denominator approach 0, leaving 4/2 = 2.
Question 3
Which of the following represents 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
RATIONALE: Both the difference quotient with h approaching 0 and the
difference quotient with x approaching a are valid definitions of the derivative.
Question 4
Evaluate the limit: lim (x→0) sin(4x)/x
A. 1
B. 4
C. 0
D. 1/4
RATIONALE: Use the standard trigonometric limit lim (u→0) sin(u)/u = 1. Rewrite
sin(4x)/x as 4 · sin(4x)/(4x). As x → 0, the expression approaches 4 · 1 = 4.
Question 5
A function f is continuous at x = c if which of the following conditions hold?
A. f(c) is defined
B. lim (x→c) f(x) exists
C. lim (x→c) f(x) = f(c)
D. All of the above
, RATIONALE: Continuity at a point requires that the function is defined at that
point, the limit exists at that point, and the limit equals the function value.
Question 6
Which type of discontinuity occurs when lim (x→c⁻) f(x) and lim (x→c⁺) f(x) both
exist but are not equal?
A. Removable discontinuity
B. Jump discontinuity
C. Infinite discontinuity
D. Oscillating discontinuity
RATIONALE: A jump discontinuity occurs when the left-hand and right-hand
limits exist but are not equal, creating a "jump" in the graph.
Question 7
Find the derivative of f(x) = x³ using the limit definition.
A. 3x
B. 3x²
C. x²
D. 3x² + 1
RATIONALE: f'(x) = lim (h→0) [(x + h)³ − x³]/h = lim (h→0) [3x²h + 3xh² + h³]/h =
lim (h→0) [3x² + 3xh + h²] = 3x².
Question 8
Find the derivative of f(x) = 5x⁴ − 3x² + 7.
A. 20x³ − 6x
B. 20x³ − 6x + 7
C. 20x³ − 6x
D. 5x³ − 3x
RATIONALE: Apply the power rule term by term: d/dx[5x⁴] = 20x³, d/dx[−3x²] =
−6x, and d/dx[7] = 0.
EXAM QUESTIONS AND CORRECT ANSWERS
(VERIFIED ANSWERS) PLUS RATIONALES 2026
Q&A | INSTANT DOWNLOAD PDF.
• CORE DOMAINS *
•
1. Functions, Graphs, and Models *
•
2. Limits and Continuity *
•
3. Definition of the Derivative *
•
4. Differentiation Rules and Techniques *
•
5. Applications of Derivatives *
•
6. Antiderivatives and Indefinite Integration *
•
7. Definite Integrals and the Fundamental Theorem of Calculus *
•
8. Applications of Integration *
•
, 9. Exponential and Logarithmic Functions in Calculus *
•
10. Problem Solving and Mathematical Reasoning *
• INTRODUCTION *
• The WGU C958 Calculus I Comprehensive Examination assesses a
candidate's mastery of differential and integral calculus of one variable. It
evaluates conceptual understanding and applied problem-solving across
limits, continuity, derivatives, and integration. The examination uses multiple-
choice and scenario-based questions requiring analytical reasoning and
mathematical decision-making. Emphasis is placed on real-world
applications including rates of change, optimization, related rates, and area
accumulation. Successful performance demonstrates readiness to apply
calculus principles in science, engineering, and quantitative reasoning
contexts. *
SECTION ONE: QUESTIONS 1–100
Question 1
Evaluate the limit: lim (x→3) (x² − 9)/(x − 3)
A. 0
B. 6
C. 3
D. Does not exist
RATIONALE: Factor the numerator as (x − 3)(x + 3). Cancel the common factor
(x − 3), leaving x + 3. Substitute x = 3 to obtain 6.
Question 2
Evaluate the limit: lim (x→∞) (4x² + 3x)/(2x² − 5)
A. ∞
B. 0
, C. 2
D. 4
RATIONALE: Divide numerator and denominator by x², yielding (4 + 3/x)/(2 −
5/x²). As x → ∞, terms with x in the denominator approach 0, leaving 4/2 = 2.
Question 3
Which of the following represents 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
RATIONALE: Both the difference quotient with h approaching 0 and the
difference quotient with x approaching a are valid definitions of the derivative.
Question 4
Evaluate the limit: lim (x→0) sin(4x)/x
A. 1
B. 4
C. 0
D. 1/4
RATIONALE: Use the standard trigonometric limit lim (u→0) sin(u)/u = 1. Rewrite
sin(4x)/x as 4 · sin(4x)/(4x). As x → 0, the expression approaches 4 · 1 = 4.
Question 5
A function f is continuous at x = c if which of the following conditions hold?
A. f(c) is defined
B. lim (x→c) f(x) exists
C. lim (x→c) f(x) = f(c)
D. All of the above
, RATIONALE: Continuity at a point requires that the function is defined at that
point, the limit exists at that point, and the limit equals the function value.
Question 6
Which type of discontinuity occurs when lim (x→c⁻) f(x) and lim (x→c⁺) f(x) both
exist but are not equal?
A. Removable discontinuity
B. Jump discontinuity
C. Infinite discontinuity
D. Oscillating discontinuity
RATIONALE: A jump discontinuity occurs when the left-hand and right-hand
limits exist but are not equal, creating a "jump" in the graph.
Question 7
Find the derivative of f(x) = x³ using the limit definition.
A. 3x
B. 3x²
C. x²
D. 3x² + 1
RATIONALE: f'(x) = lim (h→0) [(x + h)³ − x³]/h = lim (h→0) [3x²h + 3xh² + h³]/h =
lim (h→0) [3x² + 3xh + h²] = 3x².
Question 8
Find the derivative of f(x) = 5x⁴ − 3x² + 7.
A. 20x³ − 6x
B. 20x³ − 6x + 7
C. 20x³ − 6x
D. 5x³ − 3x
RATIONALE: Apply the power rule term by term: d/dx[5x⁴] = 20x³, d/dx[−3x²] =
−6x, and d/dx[7] = 0.
