TMUA Calculus 2 — Questions and Answers
Question 1: Find the derivative of f(x) = x·ln(x) − x.
- ln(x) (Correct answer)
- ln(x) + 1
- 1/x − 1
- x/ln(x)
Correct answer: ln(x)
Using the product rule, d/dx[x·ln(x)] = ln(x) + 1, so f′(x) = ln(x) + 1 − 1 = ln(x).
Question 2: Evaluate the definite integral ∫₀¹ x·eˣ dx.
- 1 (Correct answer)
- e − 1
- e
- 2 − e
Correct answer: 1
Integrating by parts with u = x, dv = eˣ dx gives [x·eˣ − eˣ]₀¹ = (e − e) − (0 − 1) = 1.
Question 3: A function g satisfies g′(x) = 3x² − 6x and g(0) = 4. What is g(2)?
- −4
- 0 (Correct answer)
- 4
- 8
Correct answer: 0
Integrating gives g(x) = x³ − 3x² + 4; then g(2) = 8 − 12 + 4 = 0.
Question 4: Which of the following best describes lim_{x→0} (sin x)/x?
- The limit does not exist
- The limit is 0
- The limit is 1 (Correct answer)
- The limit is ∞
Correct answer: The limit is 1
The standard result lim_{x→0} (sin x)/x = 1, provable by L'Hôpital's rule or the squeeze theorem.
Question 5: The curve y = x³ − 3x has a local maximum at x = ?
- x = −1 (Correct answer)
- x = 1
- x = 0
- x = −3
Correct answer: x = −1
Setting y′ = 3x² − 3 = 0 gives x = ±1; y″(−1) = −6 < 0 confirms a local maximum at x = −1.
Question 6: ∫ sec²(x) dx = ?
- sin(x) + C
- −csc²(x) + C
- tan(x) + C (Correct answer)
- 2 sec(x)tan(x) + C
Correct answer: tan(x) + C
The derivative of tan(x) is sec²(x), so the antiderivative of sec²(x) is tan(x) + C.
Question 7: If f(x) = x⁴ − 2x² + 1, what is f′(x)?
- 4x³ − 4x (Correct answer)
- 4x³ − 2x
- x³ − x
- 4x⁴ − 2x²
Correct answer: 4x³ − 4x
Applying the power rule term by term: f′(x) = 4x³ − 4x.
TMUA Calculus 2
Find the derivative of f(x) = x·ln(x) − x.