TMUA Calculus 5 — Questions and Answers
Question 1: The volume of the solid formed by rotating y = √x about the x-axis from x = 0 to x = 4 is:
- 4π
- 8π (Correct answer)
- 16π
- 2π
Correct answer: 8π
V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π[x²/2]₀⁴ = 8π.
Question 2: Which of the following is an antiderivative of 1/(1 + x²)?
- arcsin(x)
- arctan(x) (Correct answer)
- ln(1 + x²)
- −1/(1 + x²)²
Correct answer: arctan(x)
d/dx[arctan(x)] = 1/(1 + x²), so arctan(x) + C is the antiderivative.
Question 3: Using implicit differentiation, find dy/dx if x² + y² = 25.
- y/x
- −x/y (Correct answer)
- x/y
- −y/x
Correct answer: −x/y
Differentiating: 2x + 2y·(dy/dx) = 0, so dy/dx = −x/y.
Question 4: lim_{h→0} [f(x+h) − f(x)]/h defines:
- The definite integral of f
- The average rate of change of f on [0, x]
- The derivative f′(x) (Correct answer)
- The second derivative f″(x)
Correct answer: The derivative f′(x)
This is the limit definition of the derivative f′(x) at a point x.
Question 5: f(x) = x⁴ − 8x² + 3 has local minima at:
- x = 0 only
- x = 2 and x = −2 (Correct answer)
- x = 1 and x = −1
- x = 4 and x = −4
Correct answer: x = 2 and x = −2
f′(x) = 4x(x² − 4) = 0 at x = 0, ±2; f″(±2) = 32 > 0 confirms minima at x = ±2.
Question 6: The product rule for differentiating u(x)·v(x) gives:
- u′·v′
- u′·v + u·v′ (Correct answer)
- u·v′ − v·u′
- (u′·v − u·v′)/v²
Correct answer: u′·v + u·v′
The product rule states d/dx[u·v] = u′·v + u·v′.
Question 7: ∫ (2x + 3)⁵ dx = ?
- (2x + 3)⁶/12 + C (Correct answer)
- (2x + 3)⁶/6 + C
- (2x + 3)⁶/10 + C
- 5(2x + 3)⁴ + C
Correct answer: (2x + 3)⁶/12 + C
With u = 2x + 3, du = 2 dx; ∫ u⁵ du/2 = u⁶/12 + C = (2x + 3)⁶/12 + C.
TMUA Calculus 5
The volume of the solid formed by rotating y = √x about the x-axis from x = 0 to x = 4 is: