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2.1 Rates of Change and Limits 2.1 Rates of Change and Limits

2.1 Rates of Change and Limits - PowerPoint Presentation

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2.1 Rates of Change and Limits - PPT Presentation

Greg Kelly Hanford High School Richland Washington Photo by Vickie Kelly 2007 Grand Teton National Park Wyoming Suppose you drive 200 miles and it takes you 4 hours Then your average speed is ID: 465662

hand limit limits approaches limit hand approaches limits left function speed small change match values table instantaneous scroll smaller

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Slide1

2.1Rates of Change and Limits

Greg Kelly, Hanford High School, Richland, Washington

Photo by Vickie Kelly, 2007

Grand Teton National Park, WyomingSlide2

Suppose you drive 200 miles, and it takes you 4 hours.

Then your average speed is:

If you look at your speedometer during this trip, it might read 65 mph. This is your

instantaneous speed

.Slide3

A rock falls from a high cliff.

The position of the rock is given by:

After 2 seconds:

average

speed:

What is the

instantaneous

speed at 2 seconds?Slide4

for some very small change in

t

where

h

= some very small change in

t

We can use the TI-89 to evaluate this expression for smaller and smaller values of

h

.Slide5

1

80

0.1

65.6

.01

64.16

.001

64.016

.0001

64.0016

.00001

64.0002

We can see that the velocity approaches 64 ft/sec as

h

becomes very small.

We say that the velocity has a

limiting value

of 64 as

h

approaches zero

.

(Note that

h

never actually becomes zero.)Slide6

The limit as

h

approaches zero:

0

Since the 16 is unchanged as

h

approaches zero, we can factor 16 out.Slide7

Consider:

What happens as

x

approaches zero?

Graphically:

WINDOW

Y=

GRAPHSlide8

Looks like y=1Slide9

Numerically:

TblSet

You can scroll down to see more values.

TABLESlide10

You can scroll down to see more values.

TABLE

It appears that the limit of as

x

approaches zero is 1Slide11

Limit notation:

“The limit of

f

of

x

as

x

approaches

c

is

L

.”

So:Slide12

The

limit

of a function refers to the value that the function

approaches

,

not

the actual value (if any).

not 1Slide13

Properties of Limits:

Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power.

(See your book for details.)

For a limit to exist, the function must approach the

same value

from both sides.

One-sided limits

approach from either the left or right side only.Slide14

1

2

3

4

1

2

At x=1:

left hand limit

right hand limit

value of the function

does not exist because the left and right hand limits do not match!Slide15

At x=2:

left hand limit

right hand limit

value of the function

because the left and right hand limits match.

1

2

3

4

1

2Slide16

At x=3:

left hand limit

right hand limit

value of the function

because the left and right hand limits match.

1

2

3

4

1

2Slide17

The Sandwich Theorem:

Show that:

The maximum value of sine is 1, so

The minimum value of sine is -1, so

So:Slide18

By the sandwich theorem:

Y=

WINDOWSlide19

p