NZ Level 8 (NZC) Level 3 (NCEA) [In development] Basic derivatives of sine and cosine

## Interactive practice questions

Consider the graph of $y=\sin x$y=sinx.

a

Which of the following best describes the graph of $y=\sin x$y=sinx?

It is constantly increasing.

A

It is constantly decreasing.

B

The graph increases and decreases periodically.

C

It is constantly increasing.

A

It is constantly decreasing.

B

The graph increases and decreases periodically.

C
b

Which of the following best describes the nature of the gradient of the curve?

Select all the correct options.

Between points where the gradient is $0$0, the gradient is always negative.

A

The gradient of the curve is $0$0 once every $2\pi$2π radians.

B

Between points where the gradient is $0$0, the gradient is always positive.

C

The gradient of the curve is $0$0 once every $\pi$π radians.

D

Between points where the gradient is $0$0, the gradient is positive and negative alternately.

E

Between points where the gradient is $0$0, the gradient is always negative.

A

The gradient of the curve is $0$0 once every $2\pi$2π radians.

B

Between points where the gradient is $0$0, the gradient is always positive.

C

The gradient of the curve is $0$0 once every $\pi$π radians.

D

Between points where the gradient is $0$0, the gradient is positive and negative alternately.

E
c

Select all the intervals in which the gradient of $y=\sin x$y=sinx is positive.

$\frac{\pi}{2}π2<xπ A$\pi\le x<\frac{3\pi}{2}$πx<3π2 B$\frac{3\pi}{2}3π2<x2π

C

$0\le x<\frac{\pi}{2}$0x<π2

D

$\frac{\pi}{2}π2<xπ A$\pi\le x<\frac{3\pi}{2}$πx<3π2 B$\frac{3\pi}{2}3π2<x2π

C

$0\le x<\frac{\pi}{2}$0x<π2

D
d

Select all the intervals in which the gradient of $y=\sin x$y=sinx is negative.

$\frac{3\pi}{2}3π2<x2π A$0\le x<\frac{\pi}{2}$0x<π2 B$\frac{\pi}{2}π2<xπ

C

$\pi\le x<\frac{3\pi}{2}$πx<3π2

D

$\frac{3\pi}{2}3π2<x2π A$0\le x<\frac{\pi}{2}$0x<π2 B$\frac{\pi}{2}π2<xπ

C

$\pi\le x<\frac{3\pi}{2}$πx<3π2

D
e

The gradient function $y'$y is to be graphed on the axes below. The plotted points correspond to where the gradient of $y=\sin x$y=sinx is $0$0.

Given that the gradient at $0$0 is $1$1, graph the gradient function $y'$y.

f

Which of the following is the equation of the gradient function $y'$y graphed in the previous part?

$y'=-\cos x$y=cosx

A

$y'=-\sin x$y=sinx

B

$y'=\sin x$y=sinx

C

$y'=\cos x$y=cosx

D

$y'=-\cos x$y=cosx

A

$y'=-\sin x$y=sinx

B

$y'=\sin x$y=sinx

C

$y'=\cos x$y=cosx

D
Easy
Approx 6 minutes

Consider the graph of $y=\cos x$y=cosx.

Consider the graphs of $y=\sin x$y=sinx and its derivative $y'=\cos x$y=cosx below. A number of points have been labelled on the graph of $y'=\cos x$y=cosx.

Consider the graphs of $y=\cos x$y=cosx and its derivative $y'=-\sin x$y=sinx below. A number of points have been labelled on the graph of $y'=-\sin x$y=sinx.

### Outcomes

#### M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

#### 91578

Apply differentiation methods in solving problems