Differentiation of trigonometric functions and differentiation rules — Question 110

Original QCE Vault practice · 5 marks

Q110 · Practice questionTechnology-activeComplex unfamiliar5 marks

QUESTION 110 (5 marks)

A particle has displacement

s(t)=t\sin t

for

0\le t\le\pi

. Determine, to three decimal places, the time at which the velocity is maximised and the corresponding maximum velocity.

WORKED SOLUTION

5 marks

ANSWER

t\approx1.077

and

v_{\max}\approx1.391

Worked solution

The velocity is

v(t)=\sin t+t\cos t

. Its stationary points satisfy

v'(t)=2\cos t-t\sin t=0

. Numerical solution on

[0,\pi]

gives

t\approx1.076874

. Checking the endpoints and the sign change of

v'

shows this is the maximum, with

v\approx1.391008

Determines

v(t)=\sin t+t\cos t

[1 mark]

Forms

v'(t)=2\cos t-t\sin t

[1 mark]

Solves the nonlinear equation numerically.

[1 mark]

Justifies the maximum.

[1 mark]

Determines the maximum velocity.

[1 mark]

Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.

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