2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington...
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Transcript of 2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington...
2.1A: Rates of Change & Limits
Created by Greg Kelly, Hanford High School, Richland, WashingtonRevised 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
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:
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
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
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)=
Consider:sin x
yx
What happens as x approaches zero?
Graphically:
sin /y x x
WINDOW
Y=
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!)
sin /y x x
Numerically:
TblSet
You can scroll up or down to see more values.
TABLE
sin /y x x
It appears that the limit of is 1, as x approaches zero
sin x
x
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
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
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.
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
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
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
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