NCEA Calculus 2020 THREE(c - Solution & Explanation

Question

The normal to the graph of the function

y = \sqrt{2x + 1}

at the point

(4,3)

intersects the

x

-axis at point P.

Loading diagram...

Find the

x

-coordinate of point P.

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

Official Answer

y=(2x+1)^{\frac{1}{2}}

\frac{dy}{dx}=\frac{1}{2}(2x+1)^{-\frac{1}{2}}\times 2

=(2x+1)^{-\frac{1}{2}}

=\frac{1}{\sqrt{2x+1}}

At

x=4\ \frac{dy}{dx}=\frac{1}{3}

Normal gradient

=-3

y-3=-3(x-4)

y=-3x+15

x

-intercept

\Rightarrow y=0

x=5

Grading Criteria

Correct solution with correct derivative.
Must have correct gradient of normal to justify $x=5$

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Video Explanation

NCEA Level 3 Calculus Differentiation 2020 NZQA Exam - Worked Answers by infinityplusone(starts at 44:17)

2020 THREE(c)