A-level · Differentiation

Differentiating trigonometric functionsExam-style questions

How fast a sine or cosine is changing. Why the gradient of sin x traces out cos x; proving it from first principles with the small-angle results; why x must be in radians; the other four derivatives from the quotient rule, and the minus every co-function carries; the chain rule on sin kx and sin²x; and using trig derivatives for rates, tangents and stationary points.

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19 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
In this question you must show detailed reasoning.

It is given that $f(x) = \sin 2x + 2\sin^2 x$ , where $0 \le x \le \pi$ .
(a) (i) Find the values of $x$ for which $f'(x) = 0$ .

(ii) Find the values of $x$ for which $f''(x) = 0$ .

(7 marks)

The graph of $y = f'(x)$ , for $0 \le x \le \pi$ , is shown in the diagram. It has a maximum point at $A$ and a minimum point at $C$ , and it meets the $x$ -axis at $B$ and $D$ .(b) Hence find the coordinates of $A$ , $B$ , $C$ and $D$ .

(3 marks)
(c) (i) State the feature of the graph of $y = f'(x)$ that identifies the values of $x$ for which $f(x)$ is increasing.

(ii) Write down the values of $x$ , in the interval $0 \le x \le \pi$ , for which $f(x)$ is increasing. Give your answer in set notation.

(3 marks)
Level 5 · 13 marksTry it