Level 3 Differentiation Walkthrough

2017 NCEA Level 3 Differentiation Question 3(e)

2017 Paper

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Question

For the function

\[y=e^x\cos(kx):\]

(i) Find \(\dfrac{dy}{dx}\) and \(\dfrac{d^2y}{dx^2}\).

(ii) Find all the value(s) of \(k\) such that the function satisfies

\[\frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y=0\]

for all values of \(x\).

First walkthrough idea

Focus to try first

Use the product rule twice. Then substitute \(y\), \(y'\), and \(y''\) into the differential equation and collect like trigonometric terms.

Step 1

Find the first derivative

Apply the product rule and use \(\frac{d}{dx}\cos(kx)=-k\sin(kx)\).

Show the first step’s working
\[\frac{dy}{dx}=e^x\cos(kx)+e^x\bigl(-k\sin(kx)\bigr)\] \[\frac{dy}{dx}=e^x\bigl(\cos(kx)-k\sin(kx)\bigr).\]

Walkthrough overview

What this question practises

This 2017 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Second derivatives and a differential-equation parameter.

This is Question 3(e) from the 2017 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 second derivatives and a differential-equation parameter. 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)