Level 3 Integration Walkthrough

2025 NCEA Level 3 Integration Question 2(d)

2025 Paper

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Question

A particle's acceleration can be modelled by the equation \[ a(t)=2.4e^{-0.3t}-1, \] where \(a(t)\) is the acceleration of the particle, in \(\text{m s}^{-2}\), and \(t\) is the time, in seconds, from the start of timing.

Initially, at a fixed point \(P\), the particle had a velocity of \(6\text{ m s}^{-1}\).

How far from the point \(P\) is the particle \(3\) seconds after timing started?

You must use calculus and show the results of any integration needed to solve the problem.

First walkthrough idea

Hint to try first

Integrate acceleration to get velocity, and keep the \(+C\).

Step 1

Integrate acceleration to get velocity

Both terms integrate neatly, and the constant of integration must stay.

Show the first step’s working

Both terms integrate neatly, and the constant of integration must stay.

Key result

\[ v(t)=-8e^{-0.3t}-t+C \]

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Building velocity and displacement from acceleration with initial conditions.

This is Question 2(d) from the 2025 NCEA Level 3 Integration paper for AS91579 — Apply integration methods in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91579.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise building velocity and displacement from acceleration with initial conditions. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

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