NSAA NSAA Mathematics - Calculus

Question 1

What is the derivative of f(x) = 5x³ - 2x + 7?

Accepted Answer

15x² - 2

Answer

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

Accepted Answer

x⁴ - x² + C

Answer

∫4x³dx = x⁴ and ∫-2x dx = -x²; result is x⁴ - x² + C.

Question 3

Find the gradient of y = 3x² - x at x = 2.

Accepted Answer

11

Answer

dy/dx = 6x - 1; at x = 2: 6(2) - 1 = 12 - 1 = 11.

Question 4

A function has a stationary point where:

Accepted Answer

The first derivative equals zero

Answer

A stationary point occurs where dy/dx = 0, meaning the gradient of the curve is zero.

Question 5

What is ∫₀² 3x² dx?

Accepted Answer

8

Answer

∫3x² dx = x³; evaluated from 0 to 2: 2³ - 0³ = 8.

NSAA - Natural Sciences Admissions Assessment Practice Test