Level 3 Differentiation Walkthrough

2017 NCEA Level 3 Differentiation Question 3(c)

2017 Paper

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Question

The graph below shows the function \(y=f(x)\).

Piecewise graph of y equals f of x A horizontal branch at y equals one meets a rising line at negative two comma one. The line has an open point at negative one comma two and a filled point at negative one comma one, then ends filled at zero comma three. A separate curve begins open at zero comma two, falls to a corner at one comma zero, rises to a maximum at two comma five, and then falls through three comma zero. −4−3−2−11234 −2−1123456 x y

For the function above:

(i) Find the value(s) of \(x\) that meet each condition:

\[\text{(1) }f'(x)=0\]

(2) \(f(x)\) is continuous but not differentiable

(3) \(f(x)\) is not continuous

\[\text{(4) }f''(x)<0\]

(ii) What is the value of \(\displaystyle\lim_{x\to-1}f(x)\)? State clearly if the value does not exist.

First walkthrough idea

Focus to try first

Read filled and open points carefully. Then keep limits, continuity, differentiability, gradient, and concavity as separate ideas.

Step 1

Find where the gradient is zero

Look for horizontal pieces or smooth turning points, but exclude corners and endpoints where the derivative does not exist.

Show the first step’s working

On the interval shown, the left branch is horizontal for \(-4<x<-2\). At \(x=-2\) the graph has a corner, so there is no single gradient there.

The right-hand arch has a smooth maximum at \(x=2\), so its tangent is horizontal.

\[f'(x)=0\quad\text{for}\quad -4<x<-2\quad\text{and}\quad x=2.\]

Common error

A corner may have a horizontal branch on one side, but it is not differentiable unless the gradients from both sides agree.

Walkthrough overview

What this question practises

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

Method: Derivatives, continuity, limits, and concavity from a piecewise graph.

This is Question 3(c) 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 derivatives, continuity, limits, and concavity from a piecewise graph. 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(b) Question 3(d) →