3.7 Rates of ChangeObjectives:

1. Find the average rate of change of a function over an interval.2. Represent average rate of change geometrically as the slope of a secant line.3. Use the difference quotient to find a formula for the average rate of change of a function.

Distance Traveled by a Falling Object

d(t) 16t2

Where d(t) is the distance traveled (in feet) and t is the time in seconds.

distance traveled average speed time interval

average speed =distance traveled

time interval=

d(b) d(a)b a

from t a to t b

Find the average speed of the falling rock

A. From t = 2 to t = 5

B. From t = 0 to t = 3.5

Example #1Average Speed Over a Given Interval

d(t) 16t2

ft/sec112

3

336

3

64400

3

216516

25

)2()5( 22

dd

ft/sec56

3

196

5.3

0196

5.3

0165.316

05.3

)0()5.3( 22

dd

Average Rate of Change of a Function

change in f(x)change in x

f(b) f(a)

b a

A cone-shaped tank is being filled with water.

The approximate volume of water in the tank in

cubic meters is , where x is the height

of water in the tank.

Find the average rate of change of the volume of

water as the height increases from 1 to 3 meters.

Example #2Rates of Change of Volume

V(x) x 3

4

3

33

m25.38

26

2

1

4

26

24

1

4

27

24

1

4

3

13

)1()3(

VV

A manufacturing company makes toy cars. The cost (in dollars) of producing x cars is given by the function

Find the average rate of change of the cost:

A. From 0 to 10 cars

Example #3Manufacturing Costs

c(x) 4x 23 8

carper

cc74.0$

10

400

10

8048104

010

)0()10( 33 23 2

Find the average rate of change of the cost:

B. From 10 to 25 cars

C. From 25 to 50 cars

Example #3Manufacturing Costs

c(x) 4x 23 8

carper

cc41.0$

15

4002500

15

81048254

1025

)10()25( 333 23 2

carper

cc32.0$

25

2500000,10

25

82548504

2550

)25()50( 333 23 2

Example #4Rates of Change from a Graph

4 8 12

10

12

14

16

18

20

22

24

26

28

30

The graph left shows the weekly sales (in hundreds of dollars) of magazine subscriptions made during a 12-week sales drive. The sales in any single week is s(x), where x is the number of weeks since the sales drive began.

What is the average rate of change in sales:

A. From week 2 to week 4

B. From week 6 to week 11Weeks

Sale

s(H

und

reds

of

Dolla

rs)

5.1

2

3

2

1512

24

24

ss

Sales decrease $150 per week

6.1

5

8

5

1422

611

611

ss

Sales increase $160 per week

Using the previous graph and two points located on the curve we can see the geometric interpretation for the average rate of change.

Geometric Interpretation of Average Rate of Change

(3, 13)

(8,18)

4 8 12

10

12

14

16

18

20

22

24

26

28

30

The slope of a secant line connecting two points on the curve represents the average rate of change for the interval from weeks 3 to 8.

The distance traveled by a dropped object (ignoring wind resistance) is given by the function d(t) = 4.9t2, with distance d(t) measured in meters and time t in seconds. Find a formula for the average speed of a falling object from time x to time x + h. Use the formula to find the average speed from 2.8 to 3 seconds.

Example #5Computing Average Speed Using a Formula

hxh

hxh

h

xhxhx

h

xhxhx

h

xhx

xhx

xdhxd

9.48.99.48.99.49.48.99.4

9.429.49.49.4

2222

22222

2.08.23

8.2

h

x m/s 42.282.09.48.28.9

Find the difference quotient of

and use it to find the average rate of change of V as h changes from 2 to 2.1 meters.

Example #6Using a Rate of Change Formula

V(x) x3

4

22322

3322333

3333

44

33

44

hxhxh

hxhhx

h

xhxhhxx

h

xhx

xhx

xVhxV

1.021.2

2

h

x /mm 61.12

1.01.023233

22