NSAA Mathematics - Calculus Flashcards

If dy/dx > 0 for all x in an interval, the function is:

Answer: Increasing

A positive first derivative indicates the function is increasing over that interval.

The chain rule states that if y = f(g(x)), then dy/dx =

Answer: f'(g(x)) · g'(x)

The chain rule: dy/dx = f'(g(x)) · g'(x).

Find the area under y = x² between x = 0 and x = 3.

Answer: 9

∫₀³ x² dx = [x³/3]₀³ = 27/3 - 0 = 9.

What is the derivative of e^(3x)?

Answer: 3e^(3x)

Using chain rule: d/dx(e^(3x)) = 3e^(3x).

At a maximum point, the second derivative f''(x) is:

Answer: Negative

At a local maximum, f''(x) < 0, indicating the curve is concave downward.

What is the derivative of ln(x)?

Answer: 1/x

The derivative of ln(x) is 1/x.