NCEA Level 3 Calculus 91578 Derivatives free practice questions with answers

Question 1 (13 marks) [Differentiation rules and tangents]

(a) A curve has the equation y = xe^(−2x) + 3. Find the equation of the tangent to the curve at x = 1. Give the gradient and intercept to 3 significant figures.

(a) Answer: y = −0.135x + 3.27

• Evidence level: Merit (differentiate with the appropriate rule, then use the point and gradient).
• dy/dx = (−2x + 1)e^(−2x).
• At x = 1: gradient m = −0.135 and y = 3.14.
• Tangent: y − 3.14 = −0.135(x − 1.00), so y = −0.135x + 3.27.

Tangent equation: the derivative gives the gradient at the given x; the curve gives the point; then y − y₁

= m(x − x₁), keeping full calculator accuracy until the end.

(b) Differentiate f(x) = ln(x² + 4x + 3) using the chain rule, and simplify your answer. Hence find f’(1), correct to 3 significant figures.

(b) Answer: f’(x) = 2

• Evidence level: Achieved (choose and apply the correct differentiation rule, then simplify).
• Identify the rule: f(x) is a function of a function, so use the chain rule.
• f’(x) = 2
• Substitute x = 1: f’(1) = 0.750.

Differentiation rules: product (uv)’ = u’v + uv’; quotient (u/v)’ = (u’v − uv’)/v²; chain d/dx[g(u)] = g’(u)u’ — write down the derivative of each piece before combining.

(c) Let f(x) = 2x + sin(x), where x is in radians. (i) Find f’(x), and the gradient of the curve at x = significant figures). (ii) Find the number of stationary points of f in the interval 0 ≤ x ≤ 2π, and give their x-values to 3 significant figures.

(c) Answer: (i) f’(x) = cos(x) + 2; gradient 2.00; (ii) 0 stationary points

• Evidence level: Merit (differentiate trigonometric functions, then solve a trig equation in an interval).
• (i) f’(x) = cos(x) + 2. At x = • (ii) Stationary points need cos(x) + 2 = 0, which means cos(x) = −2; this value is outside −1 to 1, so there are none.
• Solutions in 0 ≤ x ≤ 2π: none.
• Number of stationary points = 0.

Trig derivatives: d/dx sin(bx) = b cos(bx); for stationary points set the derivative to zero and find ALL solutions of the trig equation inside the interval. x + 2 x² + 4x + 3 ; f’(1) = 0.750 x + 2 x² + 4x + 3 . π

2 (3 π

2 : f’ = 2.00.