Level 3 Differentiation Walkthrough

2023 NCEA Level 3 Differentiation Question 3(e)

2023 Paper

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Question

A power line hangs between two poles. The equation of the curve \(y=f(x)\) that models the shape of the power line can be found by solving the differential equation

\[ a\frac{d^2y}{dx^2}=\sqrt{1+\left(\frac{dy}{dx}\right)^2} \]

Use differentiation to verify that the function

\[ y=\frac{a}{2}\left(e^{x/a}+e^{-x/a}\right) \]

satisfies the above differential equation, where \(a\) is a positive constant.

First walkthrough idea

Hint to try first

Differentiate once to find \(y'\), then again to find \(y''\).

Step 1

Differentiate once

The factors of \(a\) cancel when differentiating each exponential term.

Show the first step’s working

The factors of \(a\) cancel when differentiating each exponential term.

Key result

\[ y'=\frac{e^{x/a}}{2}-\frac{e^{-x/a}}{2} \]

Walkthrough overview

What this question practises

This 2023 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Verifying a differential equation from a catenary model.

This is Question 3(e) from the 2023 NCEA Level 3 Differentiation paper for AS91578 — Apply differentiation 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 AS91578.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise verifying a differential equation from a catenary 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

← Question 3(d)