Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley...
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Transcript of Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley...
![Page 1: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/1.jpg)
Section 2.6Tangents, Velocities and Other Rates of Change
AP CalculusSeptember 18, 2009Berkley High School, D2B2
![Page 2: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/2.jpg)
Calculus, Section 2.6 2
Slope of the tangent line
Definition 1: The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with the slope…
ax
afxfm
ax
)()(lim
![Page 3: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/3.jpg)
Calculus, Section 2.6 3
![Page 4: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/4.jpg)
Calculus, Section 2.6 4
Example
Find an equation of the tangent line to the parabola y=x2 at the point P(1,1)
2
11
)1(lim1
)1)(1(lim
1
1lim
1
)1()(lim
)()(lim
:1 Def Using
1
1
22
1
1
xx
xxx
x
x
fxfax
afxfm
x
x
x
x
ax
![Page 5: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/5.jpg)
Calculus, Section 2.6 5
Example
Find an equation of the tangent line to the parabola y=x2 at the point P(1,1)
12
1
21
121
b. find toneedbut ,2
xy
b
b
b
bmxy
m
![Page 6: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/6.jpg)
Calculus, Section 2.6 6
Slope of the tangent line
Definition 2: The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with the slope…
h
afhafm
h
)()(lim
0
![Page 7: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/7.jpg)
Calculus, Section 2.6 7
![Page 8: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/8.jpg)
Calculus, Section 2.6 8
Example
Find an equation of the tangent line to the hyperbola y=3/x at the point P(3,1).
3
1
03
1
3
1lim
1
3lim
1
3
)3(3lim
33
33
lim
13
3
lim
)3()3(lim
)()(lim
:2 Def Using
0
0
0
0
0
0
0
h
hh
hhh
hh
hh
h
hh
h
fhfh
afhafm
h
h
h
h
h
h
h
![Page 9: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/9.jpg)
Calculus, Section 2.6 9
23
1
2
11
33
11
b. find toneedbut ,3
1
xy
b
b
b
bmxy
m
Example
Find an equation of the tangent line to the hyperbola y=3/x at the point P(3,1).
![Page 10: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/10.jpg)
Calculus, Section 2.6 10
Velocity
As we discussed earlier, if the equation describes position, then the slope represents velocity.
The secant line (between two points on the curve) represents average velocity.
The tangent line represent instantaneous velocity.
![Page 11: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/11.jpg)
Calculus, Section 2.6 11
Velocity
The tangent line represent instanteous velocity.
h
afhafav
h
)()(lim)(
0
![Page 12: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/12.jpg)
Calculus, Section 2.6 12
In general…
The secant line (between two points on the curve) represents average rate of change.
The tangent line represent instantaneous rate of change.
Look at what they are asking for in the problems, and use the appropriate definition.
![Page 13: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/13.jpg)
Calculus, Section 2.6 13
Assignment
Section 2.6, 1-17, odd (exclude 5)
![Page 14: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/14.jpg)
Calculus, Section 2.6 14
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Calculus, Section 2.6 15
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Calculus, Section 2.6 16
![Page 17: Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.](https://reader036.fdocuments.us/reader036/viewer/2022083005/56649f1c5503460f94c32036/html5/thumbnails/17.jpg)
Calculus, Section 2.6 17