Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley...
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Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2
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Transcript of Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley...
Section 2.6Tangents, Velocities and Other Rates of Change
AP CalculusSeptember 18, 2009Berkley High School, D2B2
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
Calculus, Section 2.6 3
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
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
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
Calculus, Section 2.6 7
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
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).
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.
Calculus, Section 2.6 11
Velocity
The tangent line represent instanteous velocity.
h
afhafav
h
)()(lim)(
0
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.
Calculus, Section 2.6 13
Assignment
Section 2.6, 1-17, odd (exclude 5)
Calculus, Section 2.6 14
Calculus, Section 2.6 15
Calculus, Section 2.6 16
Calculus, Section 2.6 17
