Calculus practice questions.

A population is modelled by the logistic differential equation dP/dt = 0.2P(1 - P/600). Its non-zero equilibrium (long-term) population is

• 0.2
• 120
• 600
• 3000

Answer: 600

Equilibria occur where dP/dt = 0: P = 0 or 1 − P/600 = 0, giving P = 600 — the carrying capacity. 0.2 is the growth rate, not a population value.

The derivative of y = 5x⁴ is

• 5x³
• 20x³
• 20x⁴
• 4x³

Answer: 20x³

Power rule: d/dx(axⁿ) = anxⁿ⁻¹, so d/dx(5x⁴) = 5 × 4 × x³ = 20x³. Multiply by the old power, then reduce the power by 1.

Let f(x) = 2x² − 5x + 1. Determine f′(x).

• 2x − 5
• 4x − 5
• 4x − 5x
• 4x² − 5

Answer: 4x − 5

Differentiate term by term: d/dx(2x²) = 4x, d/dx(−5x) = −5, and the constant differentiates to 0, giving 4x − 5.

The derivative f′(a) of a function f at x = a is best described as which of the following?

• The value of the function at x = a
• The gradient of the tangent at x = a
• The area under the curve up to x = a
• The average value of f between 0 and a

Answer: The gradient of the tangent at x = a

The derivative at a point equals the gradient (slope) of the tangent line to the curve at that point, which is also the instantaneous rate of change.

Given f(x) = 5x⁴, determine f′(x).

• 5x³
• 20x³
• 20x⁴
• 5x⁵/5

Answer: 20x³

Using the power rule, f′(x) = 5 × 4 × x³ = 20x³.

A curve has equation y = 3x² − 12x + 5. Determine the value of x where the gradient of the curve is 0.

• x = 4
• x = 2
• x = −2
• x = 6

Answer: x = 2

dy/dx = 6x − 12; setting 6x − 12 = 0 gives x = 2.