Level 3 Differentiation Walkthrough

2018 NCEA Level 3 Differentiation Question 2(c)

2018 Paper

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Question

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

Piecewise graph of y equals f of x A decreasing ray has holes at negative seven comma two and negative three comma negative two. A separate horizontal piece runs from a filled point at negative three comma two to an open point at one comma two. An upper curve begins at a filled point at one comma five, peaks smoothly at three comma nine, and ends open at five comma five. A horizontal piece continues to seven comma five, where an increasing ray begins. −10−9−8−7−6−5−4−3−2−112345678910 −3−2−112345678910 x y

For the function above:

(i) What is the value of \(f(1)\)? State clearly if the value does not exist.

(ii) For what value(s) of \(x\) does \(f(x)\) not have a limit?

(iii) Find all values of \(x\) that meet each condition:

\[\text{(1) }f'(x)>0\] \[\text{(2) }f'(x)=0\text{ and }f''(x)<0\] \[\text{(3) }f(x)\text{ is continuous but not differentiable}\]

First walkthrough idea

Focus to try first

Keep four ideas separate: a filled dot gives the function value, a limit follows nearby values, \(f'>0\) means increasing, and differentiability requires a smooth join.

Step 1

Read the function value

At a given \(x\), the filled point gives \(f(x)\); an open circle is excluded.

Show the first step’s working

At \(x=1\), the filled point is at \(y=5\), while the point at \(y=2\) is open.

\[f(1)=5\]

Walkthrough overview

What this question practises

This 2018 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Function values, limits, derivatives, and continuity from a graph.

This is Question 2(c) from the 2018 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 function values, limits, derivatives, and continuity from a 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

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