Tangents, Velocities, and Other Rates of Change

Rates of Change
1
Remarks
We will refer to two rates throughout the rest of our
course:
ARC = Average Rate of Change
This is the slope of the secant line through two
points
IRC = Instantaneous Rate of Change
This is the slope of the tangent line at a single point
2
Average Rate of Change (ARC)
The ARC is computed by finding the slope
m
y2
x2
y1
x1
f x2
x2
f x1
x1
Note that two points are required to find
the ARC
This is the slope of secant line – so, we
begin by defining the secant
3
1
se·cant (sknt, -knt)
n. Abbr. sec
1.
a. A straight line intersecting a curve at two or
more points.
b. The straight line drawn from the center
through one end of a circular arc and
intersecting the tangent to the other end of
the arc.
c. The ratio of the length of this line to the
length of the radius of the circle.
2. The reciprocal of the cosine of an angle in a
right triangle.
4
Secant Line
Secant Line – A line passing
thorough two points on a
graph of a function. The slope
of the secant line is the
average rate of change (ARC)
Q
f(x1) P
x1
f(x2)
x2
f ( x2 ) f ( x1 )
x2 x1
msec
5
Remarks
The function which describes the secant line is
computed by first finding the slope
m
y2
x2
y1
x1
Then use the point-slope form of the linear
equation
y
y1
m x x1
6
2
Example
Suppose s (t) = 2t 3 represents the position of a Dodge
Tomahawk V-10 motorcycle along a straight track,
measured in feet from the starting line at time t
seconds. What is the average rate of change of s (t)
from t = 2 to t = 3?
s
t
s2 s1
t2 t1
2 3
3
2 2
3 2
3
54 16
1
f
s
38
7
Instantaneous Rate of Change (IRC)
The IRC is the slope of the tangent line to a curve at
single point.
It has many practical applications, and can be used to
describe how an object travels through the air, in
space, or across the ground. The changes in the speed
of an airplane, a space shuttle, and a car all may be
described using the instantaneous rate of change
concept. When describing motion, this concept is also
referred to as velocity.
Now we will examine the tangent
8
tan·gent (t n j nt)
• adj.
• 1. Making contact at a single point or along a line;
touching but not intersecting.
• 2. Irrelevant.
• n.
• 1. A line, curve, or surface meeting another line,
curve, or surface at a common point and sharing a
common tangent line or tangent plane at that point.
►
"[A]nd I dare say that this is not only the most useful and
general [concept] in geometry, that I know, but even that I
ever desire to know." Descartes (1637)
9
3
Tangent
f(x1)
P
Tangent Line - Makes
contact at a single point on a
function; touching but does
not intersect the function
x1
The slope of the tangent line to the graph of a function
at a point is called the instantaneous rate of change
(IRC) of the function at that point.
The IRC is the limit of the function that describes the
ARC
10
Tangent Line
A tangent line to a function at a point is a line that just
touches the graph of the function at the point in
question and is “parallel” (in some way) to the graph at
that point.
11
Secant Line  Tangent Line as P  Q
Let Q approach P (Q → P). The secant line
becomes the tangent line as Q and P coincide.
12
4
Velocity
velocity (və-lŏs'ĭ-tē) n., pl. -ties.
Rapidity or speed of motion; swiftness.
Physics. A vector quantity whose magnitude is a
body's speed and whose direction is the body's
direction of motion.
The rate of speed of action or occurrence.
The rate at which money changes hands in an
economy.
[Middle English velocite, from Old French, from Latin
vēlōcitās, from vēlōx, vēlōc-, fast.]
13
Velocity is the slope on
distance – time graph
14
Secant Line  Tangent Line
15
5
The Slope of the Tangent Line
at the Point x = a
Using the
difference quotient
lim
h
a
f ( a h)
h
lim
h
0
f (a)
f ( h) f ( a )
h a
lim
Alternate form
h
a
f ( a ) f ( h)
a h
16
f ( a h)
h
f (a)
f ( a h)
f (a )
a
a h
17
f ( a ) f ( h)
a h
f ( h)
f (a )
a
h
18
6
Compare
ARC
f ( x2 ) f ( x1 )
x2 x1
y
x
The average rate
of change
Slope of the secant line
IRC
The instantaneous
rate of change
y
x
lim
x
0
lim
x
0
f ( x2 ) f ( x1 )
x2 x1
Slope of the tangent line
19
Average Velocity
v
displacement
time
f (a h)
h
f ( a)
The slope of the secant line
Instantaneous Velocity
v lim
h
0
f ( a h)
h
f (a)
The slope of the tangent line
Example
20
Estimate the IRC at x=0
10
0
21
7
Estimate the IRC at x=0
Example
20
10
20 10
4 0
0
10
4
2.5
4
22
Estimate the IRC at x=0
Example
18
18 10
3 0
10
0
8
3
2.7
3
23
Estimate the IRC at x=0
Example
16
16 10
2 0
10
0
6
2
3
2
24
8
Estimate the IRC at x=0
Example
14
14 10
1 0
10
4
1
4
0 1
As x2x1 secant line  tangent line
25
Example
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.
The first thing that we need to do is get a
formula for the average rate of change of the
volume. In this case this is,
ARC
V t
V 5
t 5
t 3 6t 2 35 10
t 5
t 3 6t 2 25
t 5
To estimate the instantaneous rate of change of the
volume at t = 5 we just need to pick values of t that
are getting closer and closer to t = 5.
26
We create a table of values of t and the average rate
of change for those values.
t
6
5.5
5.1
5.01
5.001
5.0001
ARC
25.0
19.75
15.91
15.0901
15.009001
15.00090001
t
4
4.5
4.9
4.99
4.999
4.9999
ARC
7.0
10.75
14.11
14.9101
14.991001
14.99910001
From this table it looks like the ARC is approaching
15 and so we can estimate that the IRC is 15 at
this point.
27
9
Example
t
(sec)
x
(ft)
The position of a car is given by the
values in the following table
0
1
2
3
4
5
0
13
37
72
113
176
Estimate the instantaneous velocity when t = 2 by
averaging velocities for the periods [1,2] and [2,3]
28
t
(sec)
x
(ft)
0
1
2
3
4
5
0
13
37
72
113
176
37 13
2 1
24
[1, 2] m
[2,3] m
24 35
2
72 37
3 2
29.5
35
feet
second
29
Other Rates of Change
If y
f ( x) and x changes from x1 to x2
then the change in x called the
increment of x or delta-x is
x
and y
y2
x2
x1
y1 is the corresponding change in y
30
10
Five Steps to IRC at x = a
Step 1
f ( a h)
Step 2
f (a )
Step 3
f ( a h)
f a
f ( a h) f a
h
f ( a h) f a
Step 5 lim
h 0
h
Step 4
Hint: Simplify at each step
Example
31
Find the equation of the tangent line
y
2 x 1 at 4,3
We first use five steps to find the slope at a=4,
then find the equation of the tangent line
step 1
f (a h)
f (4
h)
2(4
)h 1
9 2h
step 2
f (a)
f (4) 3
32
step 3
step 4
f (a h ) f (a )
f (a h ) f (a )
h
9 2h 3
h
2h
h
9 2h 3
9 2h 3
9 2h 3
h
9 2h 3
9 2h 3
9 2h
h
9
9 2h 3
2
9 2h 3
33
11
step 5
m lim
h
0
lim
h
0
f (a h)
h
f ( a)
2
9 2h 3
2
6
1
3
The equation of the tangent line
y
y`
y 3
m x x1
1
x 4
3
x 3y
5
34
The displacement (in feet) of a certain particle
Example
moving in a straight line is given by s=t 3 6
where t is measured in seconds. Find the
instantaneous velocity when t
step 1
f ( a h)
step 2
f (a)
step 3
f a h
4 h
6
4
6
4
3
3
f a
4 h
6
3
4
6
3
4 h
6
3
43
35
Example
The displacement (in feet) of a certain particle
moving in a straight line is given by s=t 3 6
where t is measured in seconds. Find the
instantaneous velocity when t
4
3
step 4
f a h f a
4 h
43
h
6h
3
2
3
4 48h 12h h 43
6h
48h 12h2 h3
48 12h h 2
6h
6
h2
8 2h
6
36
12
Example
The displacement (in feet) of a certain particle
moving in a straight line is given by s=t 3 6
where t is measured in seconds. Find the
instantaneous velocity when t
step 5
h2
6
lim8 2h
h
0
8
4
v 4
8
ft
s
37
Example
The displacement (in feet) of a certain particle
moving in a straight line is given by s=t 3 6
where t is measured in seconds. Find the
instantaneous velocity when t
4
Alternate Method
lim
h
a
f (a) f (h)
a h
lim
h
4
f (4) f (h)
4 h
4 h 4 2 4h h 2
1
lim
6h 4
4 h
1
16 16 16
6
Example
8
1
43 h 3
lim
6h 4 4 h
1
lim 42 4h h 2
6h 4
ft
s
38
The displacement (in feet) of a certain particle
moving in a straight line is given by s=t 3 6
where t is measured in seconds. Find the
instantaneous velocity when t
4
39
13
Find equation of tangent line
Example
2
at x
x
to f x
step 1
f 3 h
step 2
f 3
step 3
f 3 h
3
2
3 h
2
3
f 3
2
2
3
3 h
2h
3 3 h
40
Example
Find equation of tangent line
2
at x
x
to f x
f 3 h
h
step 4
step 5
lim
h
0
f 3 h
h
2h
3 3 h
h
f 3
f 3
3
2
3 3 h
2
3 3
2
9
The equation of the tangent line
y
2
3
Example
2
x 3
9
y
2
4
x
9
3
41
Find equation of tangent line
to f x
2
at x
x
3
42
14
If a ball is thrown into the air with a velocity
Practice
of 58 ft/s, its height (in feet) after t seconds
58t 11t 2 . Find the velocity
is given by h
when t
step 1
f a h
4 seconds.
58 4 h
11 4 h
2
58 4 58h 11 16 11 8h 11 h2
step 2
f a
step 3
f a h
58 4
11 4
2
58 4 11 16
f a
58 4 58h 11 16 11 8h 11 h2 58 4 11 16
58h 11 8h 11 h2
43
Practice
If a ball is thrown into the air with a velocity
of 58 ft/s, its height (in feet) after t seconds
is given by h
when t
step 4
58t 11t 2 . Find the velocity
4 seconds.
f a h f a
58h 11 8h 11 h2
h
h
58 11 8 11 h
step 5 lim
h
0
f a h
h
f a
v 4
30
58 11 8
30
ft
s
44
Summary
Slope of the Tangent line
at a point
Slope of the Secant line
through two points
Instantaneous Rate of
Change (IRC)
at a Point
Average Rate of Change
(ARC)
between two points
Calculated by taking the
Limit as x  a point
of the
Difference Quotient
Calculated using the
equation for slope
m = Δy / Δx
45
15

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