NCEA Calculus 2023 TWO(c - Solution & Explanation

Question

A curve is defined by the equation

f(x)=\dfrac{e^x}{x^2+2x}

.

Find the

x

-value(s) of any point(s) on the curve where the tangent to the curve is parallel to the

x

-axis.

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

Official Answer

f'(x)=\dfrac{(x^2+2x)e^x-e^x(2x+2)}{(x^2+2x)^2}

=\dfrac{e^x\left((x^2+2x)-(2x+2)\right)}{(x^2+2x)^2}

=\dfrac{e^x(x^2-2)}{(x^2+2x)^2}

f'(x)=0\Rightarrow e^x(x^2-2)=0

e^x\ne 0

x^2-2=0

x=\pm\sqrt{2}\text{ or }x=\pm 1.41

Grading Criteria

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Video Explanation

2023 NCEA L3 Calculus Exam Walkthrough by infinityplusone(starts at 33:55)

2023 TWO(c)