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2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts
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2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

Apr 01, 2015

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Page 1: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

2.1A: Rates of Change & Limits

Created by Greg Kelly, Hanford High School, Richland, WashingtonRevised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

Page 2: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

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

Then your average speed is:mi

200 mi 4 hr 50 hr

If you look at your speedometer at some time during this trip, it might read 65 mph. This is your instantaneous speed at that particular instant.

average speed = change in position = Δy elapsed time Δt

Page 3: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

A rock falls from a high cliff…

The position (measured from the cliff top) is given by:

216y t

Position at 0 sec:216 2 64y

average velocity from t=0 to t=2:

2 2

avg

[16(2) ] [16(0) ] ft32

(2) (0) sec

yV

t

What is the instantaneous velocity at 2 seconds?

216 0 0y Position at 2 sec:

Page 4: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

instantaneous0

limt

yV

t

for some very small change in t

where h = some very small change in t

First, we can move toward a value for this limit expression for smaller and smaller values of h (or Δt)…

)2()2(

)2(16)2(16lim

22

0

h

hh

Page 5: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

instantaneous0

limh

yV

t

2 2

0

16 2 16 2limh

h

h

hy

t

1 80

0.1 65.6

0.01 64.16

0.001 64.016

0.0001 64.0016

0.00001 64.0002

We can see that the velocity limit approaches 64 ft/sec as h becomes very small.

We say that near 2 seconds (the change in time approaches zero), velocity has a limit value of 64.

(Note that h never actually became zero in the denominator, so we dodged division by zero.)

Evaluate this expression with shrinking h values of:1, 0.1, 0.01, 0.001, 0.0001, 0.00001

Page 6: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

2

0

16 2 64limh

h

h

The limit as h approaches zeroanalytically:

2

0

16 4 4 64limh

h h

h

2

0

64 64 16 64limh

h h

h

0lim 64 16h

h

64

=

=

=

64 16(0)=

Page 7: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

Consider:sin x

yx

What happens as x approaches zero?

Graphically:

sin /y x x

WINDOW

Y=

Page 8: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

sin /y x x

Looks like y→1

from both sides as x→0 (even though there’s a gap in the graph AT x=0!)

Page 9: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

sin /y x x

Numerically:

TblSet

You can scroll up or down to see more values.

TABLE

Page 10: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

sin /y x x

It appears that the limit of is 1, as x approaches zero

sin x

x

Page 11: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

Limit notation: limx c

f x L

“The limit of f of x as x approaches c is L.”

So:0

sinlim 1x

x

x

Page 12: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

The limit of a function is the function value that is approached

as the function approaches an x-coordinate from left and right

(not the function value AT that x-coordinate!)

)(lim2

xfx

= 2

Page 13: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

Properties of Limits:

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

(See page 58 for details.)

For a two-sided limit to exist, the function must approach the same height value from both sides.

One-sided limits approach from only the left or the right side.

Page 14: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

The limit of a function refers to the function valueas the function approaches an x-coordinate

from left and right (not the function value AT that x-coordinate!)

2

lim 2x

f x

(not 1!)

)(lim2

xfx

)(lim2

xfx

Page 15: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

1 2 3 4

1

2

Near x=1:

limit from the left

limit from the right

1

limx

f x

does not exist because

the left- and right-hand limits do not match!

)(lim1

xfx

)(lim1

xfx

= 0

= 1

Page 16: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

Near x=2:

2lim 1x

f x

2

lim 1x

f x

limit from the left

limit from the right

2

lim 1x

f x

because the left and right hand limits match.

1 2 3 4

1

2

Page 17: 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

Near x=3: 3

lim 2x

f x

3

lim 2x

f x

left-hand limit

right-hand limit

3

lim 2x

f x

because the left- and right-hand limits match.

1 2 3 4

1

2