New 2026 Level 3 Differentiation prediction exam is out: a full paper with worked answers, free. Try it

L3 Differentiation

Question Bank

Basic Derivatives Gradients & Rates of Change Limits & Continuity Tangents & Normals Stationary & Inflection Points Implicit Differentiation Parametric Related Rates Optimisation Kinematics Verifying Differential Equations
Achieved

A blue hot-air balloon lifting off from a misty paddock, its burner flame glowing

A hot-air balloon lifts off from a field. Its height above the ground, in metres, $t$ seconds after lift-off is modelled by

$ h = 12\sqrt{t + 25} - 60 $

for $0 \le t \le 300$.

Find the vertical velocity of the balloon 39 seconds after lift-off.

You must use calculus and show any derivatives that you need to find when solving this problem.

Achieved

A sprinter's distance from the start line, in metres, $t$ seconds after the starting gun is modelled by

$ s = 10(t - 1 + e^{-t}) $

for $0 \le t \le 10$.

Find the sprinter's acceleration 1.5 seconds after the gun. Give your answer correct to 3 significant figures.

You must use calculus and show any derivatives that you need to find when solving this problem.

Achieved

A ball is thrown straight up from a balcony. Allowing for air resistance, its height above the point of release, in metres, $t$ seconds after it is thrown is modelled by

$ h = 80(1 - e^{-0.5t}) - 20t $

for $0 \le t \le 3$.

Find the greatest height the ball reaches above the point of release. Give your answer correct to 3 significant figures.

You must use calculus and show any derivatives that you need to find when solving this problem.

You may assume that your answer is a maximum.

Achieved

A glass lift in the open atrium of an apartment building with timber balconies and hanging plants

A lift in an apartment building starts from rest at the ground floor. Its upward velocity, in m/s, $t$ seconds after it starts is modelled by

$ v = 1.5te^{-0.5t} $

for $0 \le t \le 15$.

Find the time when the acceleration of the lift is zero.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

A particle moves along a straight line. Its displacement from a fixed point $O$, in centimetres, $t$ seconds after it starts moving is

$ s = t^2 - 10t + 16\ln(t + 1) $

for $0 \le t \le 6$.

Find the times when the particle changes direction, and its displacement from $O$ at each of those times. Give the displacements correct to 3 significant figures.

You must use calculus and show any derivatives that you need to find when solving this problem.

Merit

A green buoy in a harbour with ripples spreading around it

After a large wave passes, a navigation buoy in a harbour bobs up and down. Its height above its rest position, in metres, $t$ seconds later is modelled by

$ y = 0.6e^{-0.2t}\sin 2t $

for $t \ge 0$.

Find the first time after $t = 0$ that the buoy is momentarily at rest, and its height above its rest position at that moment. Give your answers correct to 3 significant figures.

You must use calculus and show any derivatives that you need to find when solving this problem.