Unit 10

8 marks · Easy

The function is shown below.

(a)

Differentiate with respect to .

[2 marks]

(b)

Sketch the graph of for . On your graph, clearly label the maximum and minimum points and the value of at which .

[3 marks]

(c)

For the function in part (a), find the value of the derivative when .

[1 mark]

(d)

State the gradient of the graph when .

[2 marks]

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

Show answer

(a)

(b) Graph with max at , min at , zero at · must show curve shape, labels 0°, 90°, 180°, 270°, 360°, y-values at those points

(d) Gradient is 0 at

Must show: sine curve shape from 0° to 360°, labels at 0°, 180°, 360° (where sine is zero), labels at max/min

Common mistakes: wrong power rule; wrong graph shape; calculating tan or cos instead of sin gradient

Show mark scheme and worked solution

Marking points

M1 (a) · 1 mark · Attempts to differentiate each term

A1 (a) · 1 mark · Correct result:

B1 (b) · 1 mark · Sketches curve with right general shape

B2 (b) · 1 mark · Labels max at 90°, min at 270°

B3 (b) · 1 mark · Shows that zeros are at 0°, 180°, 360°

A2 (c) · 1 mark · Correct value: 31 (follow-through)

B4 (d) · 2 marks · States/deduces gradient is 0 at 90°, with correct reasoning.

Worked solution

a) Differentiate: gives .

b) Sine curve from 0° to 360°: At 0°,180°,360° value is 0. Maximum at 90° (value 1), minimum at 270° (value -1).

c) Substitute : .

d) At the maximum or minimum of a smooth curve, the gradient is 0. At , has a gradient of 0.