Level 3 Differentiation Walkthrough

2016 NCEA Level 3 Differentiation Question 2(c)

2016 Paper

← Back to paper

Question

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

Graph of the piecewise function y equals f of x A curved left branch has a smooth minimum at negative four and a corner at negative two. A descending line has an open point at negative one comma one while a filled point is at negative one comma five, and it ends open at one comma negative five. A filled point at one comma negative three starts a concave-down curve with a maximum at three comma five; the curve meets a horizontal ray at x equals four. -6-4-2246 5 -5 x y

For the function \(y=f(x)\) above:

Find the value(s) of \(x\) that meet the following conditions:

(a) \(f\) is not continuous.

(b) \(f\) is not differentiable.

(c) \(f'(x)=0\).

(d) \(f''(x)<0\).

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

First walkthrough idea

Focus to try first

Read open and filled points, corners, stationary points, concavity, and limits as separate graph features.

Step 1

Find the discontinuities

Compare the value shown by a filled point with the values approached by the nearby branches.

Show the first step’s working

At \(x=-1\), the descending line has an open point at height \(1\), while the filled point gives a different function value. At \(x=1\), the line approaches the open point at \(y=-5\), but the next branch begins at the filled point \((1,-3)\).

\[\boxed{x=-1\text{ and }x=1}\]

Walkthrough overview

What this question practises

This 2016 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Continuity, differentiability, stationary points, concavity, and limits from a graph.

This is Question 2(c) from the 2016 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.

Page updated .

Learning summary

Review the method, not only the answer

This walkthrough helps you practise continuity, differentiability, stationary points, concavity, and limits from a graph. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

After solving the derivative condition, check the point's nature and answer the conclusion the question actually asks for.

Continue practising

← Question 2(b) Question 2(d) →