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Context for question 2

Curve

C

is defined by the polar equation

r (θ) = 2 sin^{2} θ

for

0 \leq θ \leq π

. Curve

C

and the semicircle

r = \frac{1}{2}

for

0 \leq θ \leq π

are shown in the

x y

-plane.

(Note: Your calculator should be in radian mode.)

2(A)Parametric and Polar CurvesEasy1 mark

Find the rate of change of

r

with respect to

θ

at the point on curve

C

where

θ = 1.3

.

2(B)Parametric and Polar CurvesMedium3 marks

Find the area of the region that lies inside curve

C

but outside the graph of the polar equation

r = \frac{1}{2}

.

2(C)Parametric and Polar CurvesHard3 marks

It can be shown that

\frac{d x}{d θ} = 4 sin θ cos^{2} θ - 2 sin^{3} θ

for curve

C

. For

0 \leq θ \leq \frac{π}{2}

, find the value of

θ

that corresponds to the point on curve

C

that is farthest from the

y

-axis. Justify your answer.

2(D)Optimisation and Related RatesEasy2 marks

A particle travels along curve

C

so that

\frac{d θ}{d t} = 15

for all times

t

. Find the rate at which the particle's distance from the origin changes with respect to time when the particle is at the point where

θ = 1.3

. Show the setup for your calculations.