NSAA Mathematics - Calculus Flashcards

The product rule for differentiation states: d/dx[f(x)g(x)] =

Answer: f'(x)g(x) + f(x)g'(x)

The product rule: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x).

Evaluate ∫ sin(x) dx

Answer: -cos(x) + C

∫ sin(x) dx = -cos(x) + C.

A particle's position is given by s(t) = t³ - 6t. What is the velocity at t = 2?

Answer: 6

v(t) = ds/dt = 3t² - 6; v(2) = 3(4) - 6 = 12 - 6 = 6.

Find the x-coordinate of the stationary point of y = x² - 6x + 9.

Answer: x = 3

dy/dx = 2x - 6 = 0 → x = 3.

The integral ∫ 1/x dx equals:

Answer: ln|x| + C

∫ 1/x dx = ln|x| + C.

Using implicit differentiation, find dy/dx if x² + y² = 25.

Answer: -x/y

Differentiating: 2x + 2y(dy/dx) = 0 → dy/dx = -x/y.