Calculus Notes 3.4: Rates of Change in the Natural and Social Sciences.

Answers:1. All of the examples involve expressing quantities as an average rate of

change, and then using the idea of the derivative to compute an instantaneous rate of change.

2. Derivative:

Start up: 1. This section discusses many different kinds of examples. What is

the main idea underlying them all?2. A particle moves along the y-axis so that its position at time t is

given by . For what value of t is the velocity of the particle zero? 0A 1B 2C 3D 4E

2 4 3y t t t

' 2 4y t t 0 2 4t 4 2t2 t

2C

Calculus Notes 3.4: Rates of Change in the Natural and Social Sciences.

'find C x

210000 5 0.01given C x x x

' 5 0.02C x x ' 500find C

Example 1: Suppose C(x) is the total cost that a company incurs in producing x units of a certain commodity. The function C is called the cost function. The instantaneous rate of change of cost with respect to the number of items produced, is called the marginal cost by economists.

501 500C C

' 500 5 0.02 500 15C

15015.01 15000

: ' 500 501 500Note C C C

15.01

What is the actual cost of producing the 501st item?

Calculus Notes 3.4: Rates of Change in the Natural and Social Sciences.

3 2' 4 24 36s t t t t

4 3 28 18s t t t t

s s s s1 11; 2 24; 3 27; 6 216

Example 2: A particle moves according to a law of motion s=f(t), t≥0, where t is measured in seconds and s in feet.

24 6 9 0t t t 3 2' 4 24 36 0s t t t t

@ 0 & 3t t

d. When is the particle at rest?

b. Find the velocity at time t.

a. Find the position at t=1, t=2, t=3, and t=6.

c. Find the velocity at t=2 and t=4.

3 2' 2 4 2 24 2 36 2s 3 2

' 4 4 4 24 4 36 4s ' 2 8s ' 4 16s

4 3 3 0t t t

Calculus Notes 3.4: Rates of Change in the Natural and Social Sciences.

0t

4 3 28 18s t t t t

3 2' 0 4 24 36 0s t t t t

Example 2: A particle moves according to a law of motion s=f(t), t≥0, where t is measured in seconds and s in feet.

1 3t

f. Find the total distance traveled on the intervals [0,1], [0,2], [0,3], [0,6]

e. When is the particle moving forward (that is, in the positive direction)?

g. When is the particle speeding up? Slowing down?

1 0 11 0 11s s

1& 3t t

4 3 3 0t t t

2 0 24 0 24s s

3 0 27 0 27s s

6 0 216 0 216s s

Slowing down:

Speeding up:

PS 3.4 pg.166 #2, 3, 8, 12, 15, 18, 20, 22, 29, 30 (10)