University of North Carolina — MATH 231 Calculus of Functions of One Variable I — Final Exam With Questions and answers with rationales

University of North Carolina — MATH
231 Calculus of Functions of One
Variable I — Final Exam With
Questions And Answers With
Rationales


COURSE TOPICS COVERED

Topic Description

Definition of limit, limit laws, one
Limits &
sided limits, continuity, IVT,
Continuity
asymptotes

Definition of derivative,
Derivatives
differentiation rules, chain rule,

,Topic Description

implicit differentiation, logarithm
differentiation

Related rates, optimization, linear
Applications of
approximations, MVT, curve
Derivatives
sketching, L'Hôpital's Rule

Riemann sums, definite integrals,
Integrals indefinite integrals, FTC,
substitution

Applications of Area between curves, volumes,
Integrals average value, particle motion

Transcendental Exponential, logarithmic, and
Functions inverse trigonometric functions

,PART I: LIMITS & CONTINUITY (Questions 1-
30)


1. Evaluate lim𝑥→2 (3𝑥 2 − 2𝑥 + 1).
• A) 7
• B) 9
• C) 11
• D) 13
• E) 15
Answer: B
Rationale: Since the function is a polynomial,
it is continuous everywhere. Direct
substitution: 3(2)2 − 2(2) + 1 = 12 − 4 +
1 = 9.

, sin(3𝑥)
2. Evaluate lim𝑥→0 .
𝑥

• A) 0
• B) 1
• C) 2
• D) 3
• E) Does not exist
Answer: D
Rationale: Using the standard
sin 𝑢 sin(3𝑥)
limit lim𝑢→0 = 1: lim𝑥→0 =3⋅
𝑢 𝑥
sin(3𝑥)
lim𝑥→0 = 3(1) = 3.
3𝑥



4𝑥 2 +3𝑥
3. Evaluate lim𝑥→∞ .
2𝑥 2 −5

• A) 0
• B) 1