Level 3 Differentiation Walkthrough

2017 NCEA Level 3 Differentiation Question 2(c)

2017 Paper

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Question

The tangent to the curve

\[y=\sqrt{x}\]

is drawn at the point \((4,2)\).

Square-root curve and tangent The curve y equals square root of x begins at the origin. Its tangent at four comma two extends left to meet the x-axis at Q. (4, 2) Q x y

Find the coordinates of the point \(Q\) where the tangent intersects the \(x\)-axis.

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

First walkthrough idea

Focus to try first

Differentiate the square-root curve, find the tangent gradient at the stated point, then set y to zero to locate the x-intercept.

Step 1

Differentiate the curve

Rewrite the square root as a power before applying the power rule.

Show the first step’s working
\[y=x^{1/2}\] \[\frac{dy}{dx}=\frac12x^{-1/2}=\frac{1}{2\sqrt{x}}\]

Walkthrough overview

What this question practises

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

Method: A tangent to a square-root curve and its x-intercept.

This is Question 2(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 a tangent to a square-root curve 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

Use the derivative for the gradient and the original curve for the point before forming the tangent equation.

Continue practising

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