Level 3 Differentiation Walkthrough
2022 NCEA Level 3 Differentiation Question 2(e)
2022 Paper - Tangent line and distance between intercepts
Question
The curve with equation \((y-5)^2=16(x-2)\) has a tangent of gradient \(1\) at point \(P\).
This tangent intersects the \(x\)- and \(y\)-axes at points \(R\) and \(S\) respectively.
You must use calculus and show any derivatives that you need when solving this problem.
Both implicit differentiation and rearranging to the upper branch \(y=5+4\sqrt{x-2}\) are valid methods here.
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First walkthrough idea
Tip to try first
Differentiate the curve equation implicitly to get an expression for \(\frac{dy}{dx}\).
Step 1
Differentiate implicitly
Reveal the explanation and working for this step.
Show the first step’s working
Differentiate both sides of \((y-5)^2=16(x-2)\) with respect to \(x\).
Key result
Implicit differentiation gives \(2(y-5)\frac{dy}{dx}=16\).
Walkthrough overview
What this question practises
This 2022 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.
Method: Implicit differentiation and tangent-line distance.
This is Question 2(e) 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 implicit differentiation and tangent-line distance. Use the hints to plan the method, then repeat the question without hints and check each step.
Common mistake to avoid
Use the derivative for the gradient and the original curve for the point before forming the tangent equation.