3.4 Velocity and Other Rates of Change

Today we will study and understand the derivative as a rate of change of a function.

What other words can be used for derivative?

Slope of tangent Instantaneous rate of changeVelocitySpeed Increase or decrease of a quantity

with respect to another quantity

Enlarging Circles (increase of a quantity with respect to another quantity)

Evaluate the rate of change of when 5 in. and at 10 in.A r r

Find the rate of change of the area of a circle with respect to its radius A r

What are the units?

2

2

A r

dAr

dr

When r=5, 10

When r=10, 20

dA

drdA

dr

2 (This looks strange, but the change is in

square inches of area per inch of radius

do not reduce to " " because this would

change the meaning!)

inin

in

Notation for particle motion:In position functions, our textbook often uses ( ) to

denote the position of an object at any time

( ) the p

. S

ositi

ome

other

on of an o

common notations inc

bject moving horizontally along

l :

h

e

t

ud

s t

t

x t

( ) the position of an object moving vertically along the axis at any time

( ) the position of a falling object moving vertically at any time

e axis at

In thi

any ti

s class, all o

me

f the o

m

y t

x

y t

h t t

t

tion problems will involve motion along a straight line.

For motion on a line, what is the relationship between velocity and speed?

where ( ) is the position at any time

Speed is the absolute value of velocity:

Speed= ( ) '( )

s t t

dsv t s t

dt

Studying Particle Motion – without a calculator

3 2( ) 3 9x t t t t A particle moves along the x-axis so that at time t its position is given by

a. What is the velocity of the particle at any time t?

b. During what time intervals is the particle moving to the left? Justify your answer.

c. At what time on [0, 3] is the particle moving fastest? Justify your answer.

d. At what time on [0, 4] is the particle moving fastest? Justify your answer.

Studying Particle Motion – on a calculator…

2 3( ) 4 2x t t t A particle moves along the x-axis so that at time t its position is given by

a. What is the velocity of the particle at any time t?

b. During what time intervals is the particle moving to the left? Justify your answer.

c. At what time on [0, 1.4] is the particle moving fastest? Justify your answer.

d. At what time on [0, 2.5] is the particle moving fastest? Justify your answer.

3,3 15,5

0,1.4 15,5 0,2.5 5,15

Assignment: 3.4A: p.135: 1, 19, 23, 27

(read Example on p. 134)

Study for Quiz #5, Sections 3.1-3.3