NCEA Calculus 2023 ONE(c - Solution & Explanation

Question

The graph shows the curve

y = \dfrac{2}{(x+1)^3}

, along with the tangent to the curve drawn at

x = 1

.

Loading diagram...

A second tangent to this curve is drawn which is parallel to the first tangent shown in the diagram.

Find the

x

-coordinate of the point where this second tangent touches the curve.

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

Official Answer

\dfrac{dy}{dx}=-6(x+1)^{-4}

=\dfrac{-6}{(x+1)^4}

When

x=1

,

\dfrac{dy}{dx}=\dfrac{-3}{8}

or

-0.375

\dfrac{-6}{(x+1)^4}=\dfrac{-3}{8}

3(x+1)^4=48

(x+1)^4=16

x+1=\pm 2

x=1

or

-3

\therefore

Second tangent touches the curve when

x=-3

Grading Criteria

Achievement (u)

Correct derivative for $\dfrac{dy}{dx}$ .

AND

Correct gradient of $\dfrac{3}{8}$ found.

Merit (r)

Finds the correct value of $x$ for the second tangent, with evidence of derivative.

Excellence T1

-

Excellence T2

-

Video Explanation

2023 NCEA L3 Calculus Exam Walkthrough by infinityplusone(starts at 5:47)

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2023 ONE(c)

Achievement (u)

Merit (r)

Excellence T1

Excellence T2