2.4 rates of change and tangent lines quick review in exercises 1 and 2, find the increments dx and...
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2.4
Rates of Change and Tangent Lines
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Quick Review
5 ,3 ,2 ,5 .1 BA baBA , ,3 ,1 .2
8D x 3D y
7
4
3
2
In Exercises 1 and 2, find the increments Dx and Dy from point A to point B.
1 ,5 ,3 ,2 .3 BA 3 ,3 ,1 ,3 .4 BA
In Exercises 3 and 4, find the slope of the line determined by the points.
1D ax 3D by
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Quick Review
2
3 slope with 3 ,2 through .5
1 ,4 and 6 ,1 through .6
62
3 xy
4
19
4
3 xy
In Exercises 5 – 9, write an equation for the specified line.
24
3 toparallel and 4 ,1 through .7 xy
3
25
3
7 xy
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Quick Review
24
3 lar toperpendicu and 4 ,1 through .8 xy
532 toparallel and 3 ,1 through .9 yx3
8
3
4 xy
3
19b
In Exercises 5 – 9, write an equation for the specified line.
3 ,2 through line theof slope the will of alueFor what v .10 b3
7
3
2 xy
?3
5 be ,4 and b
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What you’ll learn about Average Rates of Change Tangent to a Curve Slope of a Curve Normal to a Curve Speed Revisited
Essential QuestionHow does the tangent line determine the direction of abody’s motion at every point along its path?
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Average Rates of ChangeThe average rate of change of a quantity over a period of time is the amount of change divided by the time it takes.
In general, the average rate of change of a function over an interval is the amount of change divided by the length of the interval.
Also, the average rate of change can be thought of as the slope of a secant line to a curve.
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Example Average Rates of Change1. Find the average rate of change of f (x) = 3x
2 – 8 over the interval [1, 3].
13 ff 13
813833 22
13
91
52 2
24 12
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Example Instantaneous Rate of Change2. Suppose that the amount of air in a balloon after t hours is given by
V (t) = t 3 – 6t
2 + 35. Estimate the instantaneous rate of change of the volume after 5 hours.
55lim
0
VtVt
55 t
3556535565lim
2323
0
ttt t
t
tttt
159lim
23
0
159lim 2
0
tt
t
15 hourper feet
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Tangent to a CurveIn calculus, we often want to define the rate at which the value of a function y = f(x) is changing with respect to x at any particular value x = a to be the slope of the tangent to the curve y = f(x) at x = a.
The problem with this is that we only have one point and our usual definition of slope requires two points.
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Tangent to a CurveThe process becomes:1. Start with what can be calculated, namely, the slope of a
secant through P and a point Q nearby on the curve.
2. Find the limiting value of the secant slope (if it exists) as Q approaches P along the curve.
3. Define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope.
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Slope of a CurveTo find the tangent to a curve y = f (x) at a point P (a, f (a))calculate the slope of the secant line through P and a point Q (a+h, f (a+h)). Next, investigate the limit of the slope as h→0.
If the limit exists, it is the slope of the curve at P and we define the tangent at P to be the line through P with this slope.
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Slope of a Curve at a Point
0
The at the point , is the number
lim
provided the limit exists.
The at is the line through with this slope.
h
y f x P a f a
f a h f am
h
P P
slope of the curve
tangent line to the curve
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Example Tangent to a Curve3. Find the slope of the parabola f (x) = 2 x
2 – 8 at the point P (2, 0). Write the equation for the tangent to the parabola at this point.
h
fhfh
22lim
0
h
hh
822822lim
22
0
h
hhh
82lim
2
0
82lim0
hh
8
280 xy
168 xy
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Slope of a Curve
0
All of the following mean the same:
1. the slope of ( ) at
2. the slope of the tangent to ( ) at
3. the (instantaneous) rate of change of ( ) with respect to at
4. limh
y f x x a
y f x x a
f x x x a
The expression is the of at .
f a h f a
hf a h f a
f ah
difference quotient
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a. Find the slope of the curve at x = a.
h
afhafh
0
lim h
ahah
35
35
lim0
haha
haah
1
33
3535lim
0
haha
hh
1
33
5lim
0
23
5
a 33
5lim
0
ahah
Slope of a Curve
3
5Let .4
xxf
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b. Where does the slope equal -5/16?
16
5
3
52
a
163 2 a
3a43 a
7a
Slope of a Curve
3
5Let .4
xxf
4
1 ,
4
5 ,7
4
5 ,1 and
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c. What happens to the tangent to the curve at the point (a, 5/(a – 3 )) for the different values of a?
Slope of a Curve
3
5Let .4
xxf
The slope – 5/(a – 3)2 is always negative.
approaches slope the,3 As asteep.ly increasing becomes tangent theand
.3 asagain thissee We a
As a moves away from 3 in either direction, the slope approaches 0 and the tangent becomes increasingly horizontal.
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Normal to a CurveThe normal line to a curve at a point is the line perpendicular to the tangent at the point.
The slope of the normal line is the negative reciprocal of the slope of the tangent line.
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Example Normal to a Curve5. Write an equation for the normal to the curve y = x
2 – 2x – 3 at x = – 1 .
h
fhfh
11lim
0
h
hhh
31213121lim
22
0
h
hhh
4lim
2
0
4lim
0
h
h4
14
10 xy
4
1
4
1 xy
normalm4
1
,1 0
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Speed Revisited2The function 16 is an object's . An object's average speed
along a coordinate axis for a given period of time is the average rate of change
of its position ( ).
It's
y t
y f t
position function
instanta
0
at any time is the of
position with respect to time at time , or lim . h
t
f t h f tt
h
neous speed instantaneous rate of change
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Speed Revisited6. Wile E. Coyote drops an anvil from the top of a cliff. Find the
instantaneous rate of speed at 4 seconds.
h
fhfh
44lim
0
h
hh
22
0
416416lim
h
hhh
12816lim
2
0
12816lim
0
h
h
128 secondper feet
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Pg. 92, 2.4 #2-40 even