Level 3 Differentiation Walkthrough

2017 NCEA Level 3 Differentiation Question 2(d)

2017 Paper

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Question

Find the coordinates of the point \(P(x,y)\) on the curve

\[y=\sqrt{x}\]

that is closest to the point \((4,0)\).

Point on the square-root curve closest to four comma zero A point P lies on y equals square root of x. A dashed segment joins P to the fixed point four comma zero on the x-axis. 246810 24 P (x, y) (4, 0) x y

You do not need to prove that your solution is the minimum value.

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

First walkthrough idea

Focus to try first

Use the curve equation to turn the squared distance into a function of x alone, then locate its stationary value.

Step 1

Write the squared distance

The distance formula contains a square root. Its square has the same minimising point and is simpler to differentiate.

Show the first step’s working
\[d=\sqrt{(x-4)^2+(y-0)^2}\] \[D=d^2=(x-4)^2+y^2\]

Because \(d\ge0\), the squaring function is increasing: making \(D=d^2\) smallest also makes \(d\) smallest.

Walkthrough overview

What this question practises

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

Method: Minimising squared distance from a point to a curve.

This is Question 2(d) 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 minimising squared distance from a point to a curve. 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

← Question 2(c) Question 2(e) →