Level 3 Integration Walkthrough

2024 NCEA Level 3 Integration Question 2(c)

2024 Paper

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Question

Consider the differential equation \[ \frac{dy}{dx}=12y^2e^{3x}. \]

Given that \(y=0.5\) when \(x=0\), find the value of \(y\) when \(x=\frac{1}{3}\).

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

First walkthrough idea

Hint to try first

This equation is separable because the \(y\)-part and the \(x\)-part can be moved onto opposite sides.

Step 1

Separate the variables

This puts all the \(y\)-terms on one side and all the \(x\)-terms on the other.

Show the first step’s working

This puts all the \(y\)-terms on one side and all the \(x\)-terms on the other.

Key result

\[ y^{-2}\,dy=12e^{3x}\,dx \]

Walkthrough overview

What this question practises

This 2024 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Separating variables with an exponential model.

This is Question 2(c) from the 2024 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 separating variables with an exponential model. 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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