AP CALCULUS ABChapter 3:DerivativesSection 3.4:

Velocity and Other Rates of Change

What you’ll learn aboutInstantaneous Rates of changeMotion Along a LineSensitivity to ChangeDerivatives in Economics

… and why

Derivatives give the rates at which things change in the world.

Instantaneous Rates of Change

0

The instantaneous rate of changeof with respect to at

is the derivative

lim

provided the limit exists.

When we say rate of change, we mean instantaneous rate of change.

h

f x a

f a h f af a

h

Example Instantaneous Rates of Change

2If the area of a circle as a function of the radius is ,

a Find the rate of change of the area with respect to the radius .

b Evaluate the rate of change of when 4.

A r

A r

A r

a The rate of change is the derivative 2

b The rate of change when 4 84 is 2

dAr

drr

Example Instantaneous Rates of Change

Ex: Find the rate of change of the Area of a square with respect to the length of a side when s = 4cm.

842 ,4at

2

2

ds

dAs

sds

dA

ssA

Motion Along a Line

Suppose that an object is moving along a coordinate line so that

we know its position s on that line as a function of time :

The of the object over the time interval from

to is

t

s f t

t t t

displacement

.

The of the object over that time interval is

displacement.

travel timeav

s f t t f t

f t t f tsv

t t

average velocity

Instantaneous Velocity

0

The instantaneous velocity is the derivative of the position

function with respect to time. At time the velocity is

lim .t

s f t t

f t t f tdsv t

dt t

Speed

Speed is the absolute value of velocity.

Speed ds

v tdt

Acceleration

2

2

Acceleration is the derivative of velocity with respect to time.

If a body's velocity at time is then the body's

acceleration at time is .

dst v t

dt

dv d st a t

dt dt

Free-fall Constants (Earth)

2 22

2 22

ft 1English Units 32 , 32 16

sec 2m 1

Metric Units 9.8 , 9.8 4.9sec 2

g s t t

g s t t

Section 3.4 – Velocity and Other Rates of Change Position of an object (neglecting air resistance)

under the influence of gravity, can be represented by

where s0= original height of the object

v0= original velocity of the object

g = acceleration due to gravity

002

2

1stvgtts

Example Finding Velocity

2

A projectile is shot upward from the surface of the earth and reaches

a height of 4.9 120 meters after seconds.

Find the velocity after 5 seconds.

s t t t

9.8 120

when 5, 9.8 5 120 71 meters per second

dsv tdtt v

Sensitivity to ChangeWhen a small change in x produces a large change in the value of a function f(x), we say that the function is relatively sensitive to changes in x.

The derivative f’(x) is a measure of this sensitivity.

This has applications in Biology and Physics.

Derivatives in Economics Economists have a specialized vocabulary for rates of change and

derivatives. They call them . In a manufacturing operation,

the is a function of , the number of unitx

marginals

cost of production c(x) s

produced. The is the rate of change of

cost with respect to the level of production, so it is .

Sometimes the marginal cost of production is loosely defined to be

th

dc

dx

marginal cost of production

e extra cost of producing one more unit.

The is the rate of change of revenue with respect

to the level of production, so it is .dr

dx

marginal revenue

Example Derivatives in Economics

2

Suppose that the dollar cost of producing washing machines

is 2000 100 0.1 .

Find the marginal cost of producing 100 washing machines.

x

c x x x

2The marginal cost is 2000 100 0.1 100 0.2

The marginal cost of producing 100 machines is 100 0.2 100 $80

dc x x x x

dx