TMUA Calculus 3 — Questions and Answers
Question 1: For f(x) = e^(x²), what is f′(x)?
- e^(x²)
- 2x·e^(x²) (Correct answer)
- x²·e^(x²−1)
- 2·e^(x²)
Correct answer: 2x·e^(x²)
By the chain rule, d/dx[e^(x²)] = e^(x²) · 2x.
Question 2: The area enclosed between y = x² and y = x over [0, 1] is:
- 1/6 (Correct answer)
- 1/3
- 1/2
- 1/4
Correct answer: 1/6
∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6.
Question 3: If h(x) = sin(3x), what is h″(x)?
- −9 sin(3x) (Correct answer)
- 9 sin(3x)
- −3 sin(3x)
- 3 cos(3x)
Correct answer: −9 sin(3x)
h′(x) = 3 cos(3x) and h″(x) = −9 sin(3x).
Question 4: Which theorem guarantees that a continuous function on a closed interval [a, b] attains a maximum and a minimum?
- Mean Value Theorem
- Rolle's Theorem
- Extreme Value Theorem (Correct answer)
- Intermediate Value Theorem
Correct answer: Extreme Value Theorem
The Extreme Value Theorem states that a continuous function on a closed, bounded interval attains its maximum and minimum.
Question 5: Evaluate lim_{x→∞} (3x² + 1)/(x² − 5).
- 0
- 3 (Correct answer)
- ∞
- −1/5
Correct answer: 3
Dividing numerator and denominator by x² gives (3 + 1/x²)/(1 − 5/x²) → 3 as x → ∞.
Question 6: Using the substitution u = x² + 1, evaluate ∫ 2x(x² + 1)⁴ dx.
- (x² + 1)⁵/5 + C (Correct answer)
- (x² + 1)⁵ + C
- 2(x² + 1)⁵ + C
- (x² + 1)⁵/10 + C
Correct answer: (x² + 1)⁵/5 + C
With u = x² + 1, du = 2x dx, so the integral becomes ∫ u⁴ du = u⁵/5 + C = (x² + 1)⁵/5 + C.
Question 7: A particle moves with velocity v(t) = 6t − t². Its acceleration at t = 2 is:
- 8
- 4
- 2 (Correct answer)
- −2
Correct answer: 2
a(t) = v′(t) = 6 − 2t; at t = 2, a(2) = 6 − 4 = 2.
TMUA Calculus 3
For f(x) = e^(x²), what is f′(x)?