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2.4 Rates of Change & Tangent Lines. Average Rate of Change. The average rate of change of a quantity over a period of time is the slope on that interval of time. Ex.: Find the average rate of change of f(x) = x 3 – x over the interval [1, 3]. Secant & Tangent Lines. - PowerPoint PPT Presentation
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2.4RATES OF CHANGE &
TANGENT LINES
Average Rate of Change
The average rate of change of a quantity over a period of time is the slope on that interval of time.Ex.: Find the average rate of change of f(x) = x3 – x over the interval [1, 3].
x
ym
change of rate avg.
13
)1()3(
ff
2
024 12
Secant & Tangent Lines
Secant lines touch a graph at two points. The slope of a secant line represents the
AVERAGE RATE OF CHANGE of a function over a given interval.
(1, 1)
(5, 7)
2
3
15
17
x
ym
Secant & Tangent Lines
A tangent line touches a graph at one point only. Tangent lines determine the direction of a
body’s (graph’s) motion at every point along its path.
Tangent lines represent the INSTANTANEOUS RATE OF CHANGE. (the slope at an actual point, not over an interval)
Tangent Lines
The more secant lines you draw, the closer you are getting to a tangent line. SOUND FAMILIAR TO SOME CONCEPT WE’VE
DONE???
Tangent Lines
Example: Consider the function f(x) = x2 for x ≥ 0. What is the slope of the curve at (1, 1)?
The slope at (1, 1) can be approximated by a secant line through (4, 16).
14
116
x
ym
3
15
5
Tangent Lines
Example: Consider the function f(x) = x2 for x ≥ 0. What is the slope of the curve at (1, 1)?
The slope at (1, 1) can be better approximated by a secant line through (3, 9).
13
19
x
ym
2
8
4
Tangent Lines
Example: Consider the function f(x) = x2 for x ≥ 0. What is the slope of the curve at (1, 1)?
The slope at (1, 1) can be even better approximated by a secant line through (2, 4).
12
14
x
ym
1
3
3
Tangent Lines
Example: Consider the function f(x) = x2 for x ≥ 0. What is the slope of the curve at (1, 1)?
An even better approximation for the slope at (1, 1) would be to use a secant line through (1.1, 1.21).
11.1
121.1
x
ym
1.0
21.0
1.2
How long could we continue to do this?
Tangent Lines
Example: Consider the function f(x) = x2 for x ≥ 0. What is the slope of the curve at (1, 1)?
What about using the point (1+h, (1+h)2) to find the slope at (1, 1)? (where h is a small change)
1)1(
1)1( 2
h
h
x
ym
h
hh 1)21( 2
h
hh 22
h
hh )2(
h2
If h is a small change, I can say h 0. Therefore the slope of a tangent line at (1, 1) is 2.
Tangent Lines
The slope of the curve at the point is: y f x ,P a f a
0
lim h
f a h f am
h
• The slope of a curve at a point is the same as the slope of the tangent line to the curve at that point.
Tangent Lines
Example: Find the slope of the parabola y = x2 at the point where x = 2. Then, write an equation of the tangent line at this point.
h
fhfm
h
)2()2(lim
0
h
hh
22
0
)2()2(lim
h
hhh
4)44(lim
2
0
h
hhh
2
0
4lim
h
hhh
)4(lim
0
)4(lim0
hh
4
Tangent Lines
Example: Find the slope of the parabola at the point (2, 4). Then, write an equation of the tangent line at this point.
The slope of a line tangent to the parabola at (2, 4) is m = 4.To find the equation of the tangent line, use y = mx + b b )2(44
b84b 4
Since m = 4 and b = -4, the equation of the tangent line is y = 4x – 4
Tangent Lines
Example:Let . Find the slope of the curve at x = a. x
xf1
)(
0
lim h
f a h f am
h
haha
h
11
lim0
(get common denominator)
hhaahaa
h
)()(
lim0
hhaahaa
h
)(lim
0
)(lim
0 haha
hh
)(
1lim
0 haah
0 2
1
a
Tangent Lines
Example:Let . Where does the slope equal ?x
xf1
)( 4
1
We just found that the slope at any point a of f(x) is 2
1
am
Therefore, when does ?4
112
a
24 a24 a
a2
Substituting in these a values into x in the original function, we see the graph has a slope of -1/4 at (2, 1/2) and (2, -1/2)
Tangent Lines
The following statements mean the same thing: The slope of y = f(x) at x = a The slope of the tangent line to y = f(x) at x
= a The Instantaneous rate of change of f(x)
with respect to x at x = ah
afhafh
)()(lim
0
x
afxafx
)()(lim
0
Normal Lines
The normal line to a curve at a point is the line that is perpendicular to the tangent line at that point.Example: Write an equation for the normal
to the curve f(x) = 4 – x2 at x = 1.
Slope of tangent line:
h
fhfm
h
)1()1(lim
0
h
hh
))1(4())1(4(lim
22
0
h
hhh
)3()21(4(lim
2
0
h
hhh
3214lim
2
0
h
hhh
2
0
2lim
h
hhh
)2(lim
0
)2(lim0
hh
2(Slope of Tangent line)
Normal Lines
The normal line to a curve at a point is the line that is perpendicular to the tangent line at that point.Example: Write an equation for the normal
to the curve f(x) = 4 – x2 at x = 1.
2Slope of tangent line 2
1Slope of normal
lineNormal Line: bmxy
b )1(2
13
b2
5
Normal Line:
2
5
2
1 xy
Wrapping it Back Together
Problem at the beginning of Chapter 2:A rock breaks loose from the top of a tall cliff. What is the speed of the rock at 2 seconds?
Speed/Velocity is an INSTANTANEOUS RATE OF CHANGE.
Free fall equation: y = 16t2
h
fhfh
)2()2(limChange of Rate Inst.
0
h
hh
22
0
)2(16)2(16lim
h
hhh
64)44(16lim
2
0
Wrapping it Back Together
h
hhh
64)44(16lim
2
0
h
hhh
64166464lim
2
0
h
hhh
2
0
1664lim
h
hhh
)4(16lim
0
)4(16lim0
hh
0 64
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