MAT 1221Survey of Calculus

Section 2.3

Rates of Change

http://myhome.spu.edu/lauw

Expectations

Use equal signs Show formula steps Show individual derivatives steps Double check the algebra

HW

WebAssign HW 2.3 There is a hint on problem 1 at the end

of your HO. Additional HW listed at the end of the

handout (need to get done, but no need to turn in)

Fact: Slope of Tangent Line

Volume

30 /slope ml s

time

30ml/s

4.5at t s

What is “Rate of Change”?

We are going to look at how to understand and how to find the “rate of change” in terms of functions.

(The connection between derivatives, slope of tangent lines and the rates of change.)

Two Worlds and Two Problems

Real World Abstract World

The Velocity Problem The Tangent Problem

?

Two Worlds and Two Problems

Real World Abstract World

The Velocity Problem The Tangent Problem

0

( ) ( )limh

f x h f xf x

h

( )y f x

The Velocity Problem

y = distance dropped (ft)t = time (s)Displacement Function(Positive Downward)

Find the velocity of the ball at t=2.

2( ) 16y f t t

2t

The Velocity Problem

Again, we are going to use a limiting process.

Find the average velocity of the ball from t=2 to t=2+h by the formula

2t

2t h

(2 ) (2)f h f

h

distance traveledAverage velocity

time elapsed

The Velocity Problem

2t

2t h

t h Average Velocity (ft/s)

2 to 3 1

2 to 2.1 0.1

2 to 2.01 0.01

2 to 2.001

0.001

2( ) 16f t t(3) (2)

1

f f

(2.1) (2)

0.1

f f

(2.01) (2)

0.01

f f

(2.001) (2)

0.001

f f

The Velocity Problem

2t

2t h

We “see” from the table that velocity of the ball at t=2 should be ____ft/s.

The Velocity Problem

2t

2t h

We “see” from the table that velocity of the ball at t=2 should be ____ft/s.

The instantaneous velocity at t=2 is _____ ft/s.

(The ball is traveling at____ ft/s 2 seconds after it dropped.)

Limit Notations

When h is approaching 0, is approaching 64.

We say as h0,

Or,

(2 ) (2)64

f h f

h

64)2()2(

lim0

h

fhfh

(2 ) (2)f h f

h

Definition

For the displacement function , the instantaneous velocity at time t is

if it exists.

0

( ) ( )limh

f t h f tf t

h

( )y f t

Two Worlds and Two Problems

Real World Abstract World

The Velocity Problem The Tangent Problem

( )y f x2t

( )y f t

0

( ) ( )limh

f t h f tf t

h

0

( ) ( )limh

f x h f xf x

h

Remarks

In the context of moving objects, the independent variable is time t. We use the following notations

Distance function Velocity function Acceleration function : rate of change of

the velocity function

( )s t

( ) ( )v t s t

( ) ( )a t v t

Example 2

Given

where s is in meters and t is in seconds, find

(a) v(t)

(b) a(t)

(c) The velocity and acceleration at t=2s

(d) The time when the velocity is 5m/s.

3( ) 2 5s t t t

?units

Remarks

When units are given, you answers in (c) and (d) should have units.

The wonderful design of the notations helps you to get the units easily.

dsv

dt dv

adt

Example 2

Suppose we model the amount of certain drug inside a patient’s body by mg after t hours of injection.

(a) Find

(b) Explain the meaning of the answer in (a)

)(tQ

3( ) 40 0.5 0.1Q t t t

(3)Q

?units

Definition

For y=f(t), the (instantaneous) rate of change at t is

0

( ) ( )( ) lim

h

f t h f tf t

h

Expectations

Show the substitution step. Units are required for some of the

answers. Use equal signs