Find the coordinates of A.

For the graph of \(f\), write down the amplitude.

For the graph of \(f\), write down the period.

Hence, write \(f\left( x \right)\) in the form \(p\,\,{\text{cos}}\,\left( {x + r} \right)\).

Find the maximum speed of the ball.

Find the first time when the ball’s speed is changing at a rate of 2 cm s⁻².

−0.394791,13

A(−0.395, 13)      A1A1 N2

[2 marks]

13      A1 N1

[1 mark]

\({2\pi }\), 6.28      A1 N1

[1 mark]

valid approach      (M1)

eg recognizing that amplitude is p or shift is r

\(f\left( x \right) = 13\,\,{\text{cos}}\,\left( {x + 0.395} \right)\)   (accept p = 13, r = 0.395)     A1A1 N3

Note: Accept any value of r of the form \(0.395 + 2\pi k,\,\,k \in \mathbb{Z}\)

[3 marks]

recognizing need for d ′(t)      (M1)

eg  −12 sin(t) − 5 cos(t)

correct approach (accept any variable for t)      (A1)

eg  −13 sin(t + 0.395), sketch of d′, (1.18, −13), t = 4.32

maximum speed = 13 (cms⁻¹)      A1 N2

[3 marks]

recognizing that acceleration is needed      (M1)

eg   a(t), d "(t)

correct equation (accept any variable for t)      (A1)

eg  \(a\left( t \right) =  - 2,\,\,\left| {\frac{{\text{d}}}{{{\text{d}}t}}\left( {d'\left( t \right)} \right)} \right| = 2,\,\, - 12\,\,{\text{cos}}\,\left( t \right) + 5\,\,{\text{sin}}\,\left( t \right) =  - 2\)

valid attempt to solve their equation   (M1)

eg  sketch, 1.33

1.02154

1.02      A2 N3

[5 marks]