Level 3 Differentiation Walkthrough

2017 NCEA Level 3 Differentiation Question 1(c)

2017 Paper

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Question

The normal to the parabola

\[y=\frac12(x-3)^2+2\]

at the point \((1,4)\) intersects the parabola again at the point \(P\).

Parabola and normal line The upward-opening parabola has vertex at three comma two. A rising normal line passes through one comma four and meets the right branch again at the point P. 48 4812 (1, 4) P x y

Find the \(x\)-coordinate of point \(P\).

You must use calculus and show any derivatives that you need to find when solving this problem.

First walkthrough idea

Focus to try first

Find the tangent gradient at the known point, convert it to the perpendicular normal gradient, then solve where that line meets the parabola.

Step 1

Find the tangent gradient

Differentiate the parabola, then substitute the x-coordinate of the known point.

Show the first step’s working
\[\frac{dy}{dx}=\frac12\cdot2(x-3)=x-3\] \[m_{\text{tangent}}=1-3=-2\]

Walkthrough overview

What this question practises

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

Method: Finding a normal and its second intersection with a parabola.

This is Question 1(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 finding a normal and its second intersection with a parabola. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

A normal gradient is the negative reciprocal of the tangent gradient, not simply its negative.

Continue practising

← Question 1(b) Question 1(d) →