NSAA NSAA Mathematics - Calculus 1 — Questions and Answers
Question 1: What is the derivative of f(x) = 5x³ - 2x + 7?
- 15x² + 2
- 15x² - 2 (Correct answer)
- 5x² - 2
- 15x³ - 2
Correct answer: 15x² - 2
Using power rule: d/dx(5x³) = 15x², d/dx(-2x) = -2, d/dx(7) = 0; so f'(x) = 15x² - 2.
Question 2: Evaluate the integral ∫(4x³ - 2x) dx
- x⁴ - x² + C (Correct answer)
- 12x² - 2 + C
- x⁴ - x + C
- 4x⁴ - 2x² + C
Correct answer: x⁴ - x² + C
∫4x³dx = x⁴ and ∫-2x dx = -x²; result is x⁴ - x² + C.
Question 3: Find the gradient of y = 3x² - x at x = 2.
- 5
- 10
- 11 (Correct answer)
- 12
Correct answer: 11
dy/dx = 6x - 1; at x = 2: 6(2) - 1 = 12 - 1 = 11.
Question 4: A function has a stationary point where:
- The second derivative equals zero
- The function value equals zero
- The first derivative equals zero (Correct answer)
- The function reaches its global maximum
Correct answer: The first derivative equals zero
A stationary point occurs where dy/dx = 0, meaning the gradient of the curve is zero.
Question 5: What is ∫₀² 3x² dx?
- 4
- 6
- 8 (Correct answer)
- 12
Correct answer: 8
∫3x² dx = x³; evaluated from 0 to 2: 2³ - 0³ = 8.
Question 6: What is the second derivative of f(x) = x⁴ - 3x² + 5?
- 4x³ - 6x
- 12x² - 6 (Correct answer)
- 4x - 3
- 12x³ - 6x
Correct answer: 12x² - 6
f'(x) = 4x³ - 6x; f''(x) = 12x² - 6.
NSAA NSAA Mathematics - Calculus 1
What is the derivative of f(x) = 5x³ - 2x + 7?