**Instantaneous Rates of Change**

Starter

This graph shows the height of a ball that is thrown in the air, against the time it is in the air.

Example

The graph shows the concentration of reactant present in a solution during a chemical reaction as time increases.

Work out the rate of reactant loss after 7 seconds.

Instantaneous rates of change

What is the velocity of the object after 2 mins?

After 3 mins?

**L.O. - Use a tangent to find the rate of change of a graph at an instant.**

(a) Explain why the velocity of ball is always changing.

(b) Explain how you know that the velocity of the ball is positive at the beginning and negative at the end.

(c) Explain why there must be an instant when the velocity of the ball is 0.

Starter

This graph shows the height of a ball that is thrown in the air, against the time it is in the air.

(a) Explain why the velocity of ball is always changing.

Graph is curved

(b) Explain how you know that the velocity of the ball is positive at the beginning and negative at the end.

Height gets bigger over time at beginning and smaller over time at the end.

(c) Explain why there must be an instant when the velocity of the ball is 0.

There is a point when the ball stops travelling up and starts travelling down.

Instantaneous rates of change

What is the velocity of the object after 2 mins?

After 3 mins?

Instantaneous rates of change

What is the velocity of the object after 2 mins?

62 - 40

2.25 - 0

After 3 mins?

= 2.7777...

= -2.7777...

Example

The graph shows the concentration of reactant present in a solution during a chemical reaction as time increases.

Work out the rate of reactant loss after 7 seconds.

0.125 - 0.07 0.55

8.5 - 4.5 4

= = 0.135

Activity

Complete the worksheet on tangents and gradients.

Activity

Complete the worksheet on tangents and gradients.

1) (a) (i) gradient = 2 (ii) gradient = -2

(b) gradient = 0

2) Gradients are 2, 4, 6, -2, -4 respectively. The gradient can be found by doubling the x-coordinate.

3) Gradients are 5, 7, 9, 1, -1. 2x + 3 gives the gradient of the curve at any point.

4) Gradient = 11. Gradient is -11 at x = -3.5.

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