Grade 12 Calculus Questions with Worked Solutions

The rules you use most

• Power rule: if f(x) = axⁿ then f'(x) = anxⁿ⁻¹.
• Rewrite before differentiating: √x becomes x^(1/2), and 1/x² becomes x⁻².
• Stationary points: solve f'(x) = 0.
• Point of inflection of a cubic: solve f''(x) = 0.
• Gradient of a tangent at x = a is f'(a).

Worked example: sketching a cubic

Let f(x) = x³ − 3x² − 9x + 5. Then f'(x) = 3x² − 6x − 9 = 3(x − 3)(x + 1), so stationary points are at x = 3 and x = −1. f(−1) = 10 gives a local maximum and f(3) = −22 gives a local minimum.

f''(x) = 6x − 6 = 0 at x = 1, so the point of inflection is (1, −6). Show the y-intercept (0, 5), both turning points and the inflection point on your sketch.

Worked example: optimisation

A rectangular box with a square base and no lid must have a volume of 32 cm³. Let the base side be x and the height h, so x²h = 32 and h = 32/x². Surface area S = x² + 4xh = x² + 128/x.

S' = 2x − 128/x² = 0 gives 2x³ = 128, so x³ = 64 and x = 4. Then h = 2 and S = 48 cm². Always end by answering the question asked, with units.

Rates of change and exam tips

• 'Rate of change' means differentiate; 'average rate' means use the gradient between two points.
• For distance, speed is the first derivative and acceleration is the second.
• When asked for the equation of a tangent, find the gradient with f'(a) and then use y − y₁ = m(x − x₁).
• Show the derivative even if you can see the answer — the derivative line carries marks.