Level 3 Differentiation Walkthrough
2022 NCEA Level 3 Differentiation Question 1(c)
2022 Paper - Normal line and x-intercept
Question
The graph below shows the function \(y=\sqrt{x+2}\), and the normal to the function at the point where the function intersects the \(y\)-axis.
Find the coordinates of point \(P\), the \(x\)-intercept of the normal.
You must use calculus and show any derivatives that you need.
The diagram is shown in its initial state. JavaScript adds any interactive controls and later walkthrough visuals.
First walkthrough idea
Tip to try first
Start at the point where the curve meets the \(y\)-axis, since the normal is taken there.
Step 1
Find the point on the curve when \(x=0\)
Reveal the explanation and working for this step.
Show the first step’s working
The function meets the \(y\)-axis where \(x=0\).
Key result
When \(x=0\), \(y=\sqrt{0+2}=\sqrt{2}\), so the point is \((0,\sqrt{2})\).
Walkthrough overview
What this question practises
This 2022 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.
Method: The normal line and its x-intercept.
This is Question 1(c) from the 2022 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 the normal line and its x-intercept. 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.