NCEA Calculus 2020 ONE(d - Solution & Explanation

Question

A curve has the equation

y = x^2 \cos x

.

Show that the equation of the tangent to the curve at the point

(\pi, -\pi^2)

is

y + 2\pi x = \pi^2

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

Official Answer

\text{(d)}

\dfrac{dy}{dx}=x^2\cdot -\sin x+2x\cos x

At

x=\pi

\dfrac{dy}{dx}=\pi^2\cdot(-\sin\pi)+2\pi\cos\pi

=-2\pi

At

x=\pi

y=-\pi^2

Tangent equation

y-y_1=m(x-x_1)

y+\pi^2=-2\pi(x-\pi)

y+\pi^2=-2\pi x+2\pi^2

y+2\pi x=\pi^2

Grading Criteria

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Video Explanation

NCEA Level 3 Calculus Differentiation 2020 NZQA Exam - Worked Answers by infinityplusone(starts at 6:54)

2020 ONE(d)