Math 232 Calculus 2 - Fall 2018
§6.1 - VELOCITY AND NET CHANGE
§6.1 - Velocity and Net Change
After completing this section, students should be able to:
• Explain the difference between displacement and distance traveled.
• Estimate displacement and distance traveled from a graph of position over time,
or from a graph of velocity over time.
• Compute displacement and distance traveled from an equation of position as a
function of time, or from an equation of velocity over time.
• Explain how to calculate the net change of a quantity from the rate of change of
that quantity over time.
• Find an equation for velocity and position from an equation for acceleration plus
initial conditions.
• Find an equation for the amount of a quantity from an equation for its rate of
change plus an initial condition.
2
§6.1 - VELOCITY AND NET CHANGE
Example. A squirrel is running up and down a tree. The height of the squirrel from
the ground over time is given by the function s(t) graphed below, where t is in seconds
and s(t) is height in meters.
A. After 5 seconds, how far is the squirrel from its original position?
B. How far has the squirrel run in the first 5 seconds?
3
§6.1 - VELOCITY AND NET CHANGE
Definition. Displacement means ...
Definition. Distance traveled means ...
Example. If I get in a 25 meter long pool on the shallow end, and swim 5 laps, what is
my displacement and what is my distance traveled?
4
§6.1 - VELOCITY AND NET CHANGE
Example. A swimmer is swimming left and right in a long narrow pool. Her velocity
over time is given by the following graph, where velocity v(t) is in meters per second
and time t is in seconds.
Here, distance is measured from the left end of the pool, so a positive velocity means
and a negative velocity means
.
A. Describe the swim. Was the swimmer swimming at a constant speed? When was
the swimmer swimming left vs. right? At what time(s) did the swimmer turn
around?
5
§6.1 - VELOCITY AND NET CHANGE
B. What is the displacement of the swimmer between time 0 and time 12?
C. How far did the swimmer swim in the first 3 seconds?
D. the first 9 seconds?
E. the first 12 seconds?
6
§6.1 - VELOCITY AND NET CHANGE
Note. Suppose f (t) represents the velocity of an object.
• The displacement of the object between time t = a and time t = b is given by ...
• The distance traveled by the object between time t = a and time t = b is given by ...
7
§6.1 - VELOCITY AND NET CHANGE
Example. The velocity function for a particle moving left and right is given by v(t) =
t2 − 2t − 3, where v(t) is in meters per second and t is in seconds.
1. When does the particle turn around?
2. Find the displacement of the particle between time t = 1 and t = 4.
3. Find the total distance traveled between t = 1 and t = 4.
4. If the particle starts at position 2, give a formula for the position of the particle at
time t.
8
§6.1 - VELOCITY AND NET CHANGE
Example. Suppose f (t) represents the rate of change of a quantity over time (e.g. the
rate of water flowing out of a resevoir). Then
Z b
•
f (t) dt represents ...
a
Z
• If F(0) is the amount of the quantity at time 0, then F(0) +
f (t) dt represents ...
a
Z
b
| f (t)| dt represents ...
•
a
9
b
§6.1 - VELOCITY AND NET CHANGE
Example. The population of bacteria is changing at a rate of f (t) = e−t − 1/e. What is
the net change in population between time t = 0 and time t = 2?
10
§6.1 - VELOCITY AND NET CHANGE
Extra Example. The acceleration of a particle moving up and down is given by a(t) =
3π sin(πt), where a(t) is given in m/s2 and t is given in seconds. Suppose that v(0) = 2
and s(0) = −1. Find the velocity and position functions. What is its displacement in
the first 2 seconds? How much total distance did it travel in the first 2 seconds.
11
§6.2 - AREA BETWEEN CURVES
§6.2 - Area Between Curves
After completing this section, students should be able to
• Use an integral to compute the area between two curves.
• Decide if it is easier to integrate with respect to x or with respect to y when
computing the area between two curves.
• Calculate the area between multiple curves by dividing it into several pieces.
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§6.2 - AREA BETWEEN CURVES
Recall: to compute the area below a curve y = f (x), between x = a and x = b, we can
divide up the region into rectangles.
The area of one small rectangle is
The approximate area under the curve is
The exact area under the curve is
13
§6.2 - AREA BETWEEN CURVES
To compute the area between the curves y = f (x) and y = g(x), between x = a and
x = b, we can divide up the region into rectangles.
The area of one small rectangle is
The approximate area between the two curves is
The exact area between the two curves is
This formula works as long as f (x)
g(x).
14
§6.2 - AREA BETWEEN CURVES
Example. Find the area between the curves y = x2 + x and y = 3 − x2
15
§6.2 - AREA BETWEEN CURVES
Review. The area between two curves y = f (x) and y = g(x) between x = a and x = b
is given by:
16
§6.2 - AREA BETWEEN CURVES
Review. The area between the curves y = 2x + 1 and y = 5 − 2x2 is given by:
Z 1
A.
2x + 1 − 5 + 2x2 dx
−2
Z 1
5 − 2x2 − 2x + 1 dx
B.
−2
1
Z
5 − 2x2 − 2x − 1 dx
C.
−2
5
Z
5 − 2x2 + 2x + 1 dx
D.
−3
E. None of these.
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§6.2 - AREA BETWEEN CURVES
Example. The shaded area between the curves y = cos(5x), y = sin(5x), x = 0, and
x = π4 is given by:
Z π/4
A.
sin(5x) − cos(5x) dx
0
Z
π/4
cos(5x) − sin(5x) dx
B.
0
C. Both of these answers are correct.
D. Neither of these answers are correct.
18
§6.2 - AREA BETWEEN CURVES
Extra Example. Set up the integral to find the shaded area bounded by the three curves
in the figure shown.
• f (x) = x2 − x − 6
• g(x) = x − 3
• h(x) = −x2 + 4
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§6.2 - AREA BETWEEN CURVES
Note. The area between two curves x = f (y) and x = g(y) between y = c and y = d is
given by:
This formula works as long as f (y)
g(y).
20
§6.2 - AREA BETWEEN CURVES
To compute the area between the curves x = f (y) and x = g(y), between y = c and
y = d, we can again divide up the region into rectangles.
The area of one small rectangle is
The approximate area between the two curves is
The exact area between the two curves
21
§6.2 - AREA BETWEEN CURVES
p
y2 36 + y3
Example. Find the area between the curves f (y) = sin(y)+5, g(y) =
, y = −2,
6
and y = 2.
22
§6.2 - AREA BETWEEN CURVES
Example. The area between the curves y = x2 and y = 3x2, and y = 4 is given by:
Z 2
A.
x2 − 3x2 dx
0
Z
2
3x2 − x2 dx
B.
0
Z
2
C.
√
r
y
dy
3
r
y
dy
3
y−
0
Z
4
D.
√
y−
0
Z
E.
0
4
r
y √
− y dy
3
23
§6.2 - AREA BETWEEN CURVES
Extra Example. In the year 2000, the US income distribution was: (data from World
Bank, see http://wdi.worldbank.org/table/2.9)
Income Category
Fraction of
Population
Fraction of
Total Income
Bottom 20%
2nd 20%
3th 20%
4th 20%
Next 10%
Highest 10%
0.20
0.20
0.20
0.20
0.10
10
0.05
0.11
0.16
0.22
0.16
0.30
Cumulative
Fraction of
Population
0.20
0.40
0.60
0.80
0.90
1.00
Cumulative
Fraction of
Income
0.05
0.16
0.32
0.54
0.70
1.00
The Lorenz curve plots the cumulative fraction of population on the x-axis and the
cumulative fraction of income received on the y-axis.
The Gini index is the area between the Lorenz curve and the line y = x, multiplied by
2.
Estimate the Gini index for the US in the year 2000 using the midpoint rule.
24
§6.2 - AREA BETWEEN CURVES
Extra Example. Find the area between the curves 2x = y2 − 4 and y = −3x + 2 that lies
above the line y = −1
25
§6.2 - VOLUMES
§6.2 - Volumes
After completing this section, students should be able to
• Calculate a volume by integrating the cross-sectional area.
• Calculate the volume of a solid of revolution using the disk / washer method.
• Identify the parts of the formula for the volume of a solid of revolution that
correspond to cross-sectional area and thickness.
• Use calculus to derive fomulas for familar shapes such as pyramids and cones.
26
§6.2 - VOLUMES
If you can break up a solid into n slabs, S1, S2, . . . Sn, each with thickness ∆x, then
Volume of solid ≈
The thinner the slices, the better the approximation, so
Volume of solid =
27
§6.2 - VOLUMES
x2 y2
Example. Find the volume of the solid whose base is the ellipse +
= 1 and whose
4
9
cross sections perpendicular to the x-axis are squares.
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§6.2 - VOLUMES
Volumes found by rotating a region around a line are called solids of revolution.
For solids of revolution, the cross sections have the shape of a
shape of a
.
or the
The area of the cross-sections can be described with the formulas
The volume of a solid of revolution can be described with the formulas:
When the region is rotated around the x-axis, or any other horizontal line, then we
integrate with respect to
.
When the region is rotated around the y-axis, or any other vertical line, then we
integrate with respect to
.
29
§6.2 - VOLUMES
√
Example. Consider the region bounded by the curve y = 3 x, the x-axis, and the line
x = 8. What is the volume of the solid of revolution formed by rotating this region
around the x-axis?
30
§6.2 - VOLUMES
√
Example. Consider the region in the first quadrant bounded by the curves y = 3 x and
y = 14 x. What is the volume of the solid of revolution formed by rotating this region
around the x-axis? The y-axis?
END OF VIDEO
31
§6.2 - VOLUMES
Review. Suppose a 3-dimensional solid can be sliced perpendicular to the x-axis and
the slice at position x has area given by the function A(x). Then the volume is given
by:
Review. If the volume is a solid of revolution, then the volume is given by:
Question. Which of the following is NOT a solid of revolution?
A. a bowl of soup
B. a watermelon
C. a square cake
D. a bagel
32
§6.2 - VOLUMES
Example. The region between the curves y = ex, x = 0, and y = e3 is rotated around
the x-axis, to make a solid of revolution. When computing the volume, what are the
cross-sections and which variable do we integrate with respect to?
A. cross-sections are disks, integrate with respect to dx
B. cross-sections are disks, integrate with respect to dy
C. cross-sections are washers, integrate with respect to dx
D. cross-sections are washers, integrate with respect to dy
Set up an integral to calculate the volume.
33
§6.2 - VOLUMES
Example. The region between the curve y = ex, x = 0, and y = e3 is rotated around
the y-axis, to make a solid of revolution. When computing the volume, what are the
cross-sections and which variable do we integrate with respect to?
A. cross-sections are disks, integrate with respect to dx
B. cross-sections are disks, integrate with respect to dy
C. cross-sections are washers, integrate with respect to dx
D. cross-sections are washers, integrate with respect to dy
Set up an integral to calculate the volume.
34
§6.2 - VOLUMES
Set up an integral to calculate the volume if this region is rotated around the line x = 5
instead of the y-axis.
35
§6.2 - VOLUMES
Extra Example. Consider the region bounded by y =
6
,
x2
x = 1, x = 2, and the x-axis.
Set up an integral to compute the volume of the solid obtained by rotating this region
about the line x = 12 .
36
§6.2 - VOLUMES
√
Example. Find the volume of the solid whose base is the region between y = x, the
x-axis, and the lines x = 1 and x = 5, and whose cross sections perpendicular to the
x-axis are equilateral triangles.
37
§6.2 - VOLUMES
√
Example. Find the volume of the solid whose base is the region between y = x, the
x-axis, and the lines x = 1 and x = 5, and whose cross sections perpendicular to the
y-axis are equilateral triangles.
38
§6.2 - VOLUMES
Extra Example. Find the volume of a pyramid with a square base of side length a and
height h.
39
§6.2 - VOLUMES
Extra Example. Find the volume of a cone with a circular base of radius a and height
h.
40
§6.2 - VOLUMES
Extra Example. Set up an integral to find the volume of a bagel, given the dimensions
below.
41
§6.5 - ARCLENGTH
§6.5 - Arclength
After completing this section, students should be able to:
• Explain the relationship between the formula for arc length and the distance formula.
• Calculate the arclength of a curve of the form y = f (x).
42
§6.5 - ARCLENGTH
Example. Find the length of this curve.
43
§6.5 - ARCLENGTH
Note. In general, it is possible to approximate the length of a curve y = f (x) between
x = a and x = b by dividing it up into n small pieces and approximate each curved
piece with a line segment.
Arclength is given by the formula:
44
§6.5 - ARCLENGTH
Example. Find the arclength of y = x3/2 between x = 1 and x = 4.
END OF VIDEO
45
§6.5 - ARCLENGTH
Review. For a curve y = f (x), the arclength of the curve between x = a and x = b is
given by the formula:
√
Example. Set up an integral to calculate the arc length of the curve y = x between
x = 0 and x = 3.
46
§6.5 - ARCLENGTH
Example. Find a function a(t) that gives the length of the curve y =
and x = t.
47
ex +e−x
2
between x = 0
§6.5 - ARCLENGTH
Note. Although arc length integrals are usually straightforward to set up, the square
root sign makes them notoriously difficult to evaluate, and sometimes impossible to
evaluate.
48
§6.6 - SURFACE AREA
§6.6 - Surface Area
After completing this section, students should be able to:
• Identify the components of the formula for the area of a surface of revolution that
correspond to circumference and slant height.
• Compute the area of a surface of revolution.
49
§6.6 - SURFACE AREA
How could you calculate the surface area
p of a surface of revolution?
Example. Find the surface area of y = (x), rotated around the x-axis, between x = 0
and x = 2.
50
§6.6 - SURFACE AREA
To find the surface area of a surface of revolution, imagine approximating it with pieces
of cones.
We will need a formula for the area of a piece of a cone.
51
§6.6 - SURFACE AREA
The area of this piece of a cone is
A = 2πr`
r1 + r2
where r =
is the average radius and ` is the length along the slant. (See textbook
2
for derivation.)
52
§6.6 - SURFACE AREA
Use the formula for the area of a piece of cone A = 2πr` to derive a formula for surface
area.
53
§6.6 - SURFACE AREA
Formulas:
If we rotate the curve y = f (x) between x = a and x = b around the x-axis,
surface area =
If we rotate the curve y = f (x) around the y-axis, what will the corresponding formulas
be?
54
§6.6 - SURFACE AREA
Example. Find the surface area of the surface of revolution formed by rotating about
√
the x-axis the curve y = x between x = 0 and x = 2.
55
§6.6 - SURFACE AREA
Example. Find the surface area when the curve y =
rotated around the y=axis.
56
√
x between x = 0 and x = 2 is
§6.6 - SURFACE AREA
Example. Prove that the surface area of a sphere is 4πR2.
57
§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
§6.4 - Work as an Integral and Other Applications
After completing this section, students should be able to:
• Use integration to calculate the work done when a varying force, given by a
function, moves an object over a distance.
• Set up and solve problems involving the work done to pull up a rope.
• Set up and solve problems involving the work done to empty a tank.
• Solve problems the use Hooke’s Law to find the work done in stretching a spring.
• Use integration to find the mass of a wire with varying density.
• Use integration to find the force on a dam.
58
§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
Definition. If if a constant force F is applied to move an object a distance d, then the
work done to move the object is defined to be
Question. What are the units of force? What are the units of work?
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§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
Note. Suppose a Calculus book is 2 pounds (US units), which is 0.9 kg (metric units).
The pounds is a unit of
. The kg is a unit of
.
The force on the book is
in US units, or
in metric units.
Example. How much work is done to lift a 2 lb book off the floor onto a shelf that is 5
feet high?
Example. How much work is done to lift a 0.9 kg book off the floor onto a shelf that is
1.5 meters high?
60
§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
Example. A particle moves along the x axis from x = a to x = b, according to a force
f (x). How much work is done in moving the particle? (Note: the force is not constant!)
61
§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
Example. How much work is required to lift a 1000-kg satellite from the earth’s surface
to an altitude of 2 · 106 m above the earth’s surface?
GMm
The gravitational force is F =
, M is the mass of the earth, m is the mass of the
r2
satellite, and r is the distance between the satellite and the center of the earth, and G is
the gravitational constant.
The radius of the earth is 6.4·106 m, its mass is 6·1024 kg, and the gravitational constant,
G, is 6.67 · 10−11.
Reference https://www.physicsforums.com/threads/satellite-and-earth.157112/
62
§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
Z
Review. In the expression W =
b
F(x) dx what do W, F(x), and dx represent?
a
Review. Which of the following statements are true:
A) If you are told that an object is 5 kg, and you want the force due to gravity (in
metric units), you need to multiply by g = 9.8m/s2.
B) If you are told that an object is 5 lb, and you want the force due to gravity (in
English units), you need to multiply by 32 f t/s2.
C) Both.
D) Neither.
63
§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
Example. In the alternative universe of the Golden Compass, the souls of humans and
their animal companions, called daemons, are closely tied. Suppose that the force
2
needed to separate a human and its daemon is given by f (x) = 10xe−x /1000 pounds,
where x represents the distance between the human and the daemon in feet. Lyra and
her daemon are currently 5 feet apart. How much work will it take to separate them
an additional 5 feet?
64
§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
Example. A 200-kg cable is 300 m long and hangs vertically from the top of a tall
building. How much work is required to lift the cable to the top of the building?
What if we just needed to lift half the cable?
65
§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
Example. An aquarium has a square base of side length 4 meters and a height of 3
meters. The tank is filled to a depth of 2 m How much work will it take to pump the
water out of the top of the tank through a pipe that rises 0.5 meters above the top of
the tank?
66
§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
Extra Example. A bowl is shaped like a hemisphere with radius 2 feet, and is full of
water. How much work will it take to pump the water out of the top of the bowl? Use
the fact that water weights 62.5 pounds per cubic foot.
67
§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
Springs follow Hooke’s law: the force required to stretch them a distance x past their
equilibrium position is given by f (x) = kx, where k is a constant that depends on the
spring.
Example. A spring with natural length 15 cm exerts a force of 45 N when stretched to
a length of 20 cm.
1. Find the spring constant
2. Set up the integral/s needed to find the work done in stretching the spring 3 cm
beyond its natural length.
3. Set up the integral/s needed to find the work done in stretching the spring from a
length of 20 cm to a length of 25 cm.
68
§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
Other Applications - Mass from Density
Example. Find the mass of a wire that lies along the x-axis if the density of the wire at
1
for 0 ≤ x ≤ 3.
position x is given by ρ(x) =
4−x
69
§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
Other Applications - Force on a Dam
If you are standing under water, the pressure from a column of water above your head
is:
This pressure is the same in all directions, so the pressure on a vertical wall of the
swimming pool is:
The force of water on a strip of a vertical dam is given by:
70
§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
The force of water on a vertical dam is give by:
71
§6.4 - WORK AS AN INTEGRAL AND OTHER APPLICATIONS
Example. Find the total force on the face of this vertical dam, assuming that the water
level is at the top of the dam.
72
§6.5 - AVERAGE VALUE OF A FUNCTION
§6.5 - Average Value of a Function
To find the average of a list of numbers q1, q2, q3, . . . , qn, we sum the numbers and divide
by n:
For a continuous function f (x) on an interval [a, b], we could estimate the average
value of the function by sampling it at a bunch of evenly spaced x-values c1, c2, . . . , cn,
which are spaced ∆x apart:
average ≈
The approximation gets better as n → ∞, so we can define
average = lim
n→∞
73
§6.5 - AVERAGE VALUE OF A FUNCTION
The resulting formula is analogous to the formula for an average of a list of numbers,
since taking an integral is analogous to
, and dividing by the length of the
interval b − a is analogous to dividing by
.
1
Example. Find the average value of the function g(x) =
on the interval [2, 5].
1 − 5x
Is there a number c in the interval [2, 5] for which g(c) equals its average value? If so,
find all such numbers c. If not, explain why not.
74
§6.5 - AVERAGE VALUE OF A FUNCTION
Question. Does a function always achieve its average value on an interval?
Theorem. (Mean Value Theorem for Integrals) For a continuous function f (x) on an interval
Rb
f (x)dx
a
[a, b], there is a number c with a ≤ c ≤ b such that f (c) =
.
b−a
Proof:
75
§6.5 - AVERAGE VALUE OF A FUNCTION
Review. The average value of a function f (x) on the interval [a, b] is defined as:
and the Mean Value Theorem for Integrals says that:
76
§6.5 - AVERAGE VALUE OF A FUNCTION
Example. For the function f (x) = sin(x),
a) Find its average value on the interval [0, π].
b) Find any values c for which f (c) = fave. Give your answer(s) in decimal form.
c) Graph the curve f (x) = sin(x) and on your graph draw a rectangle of area equal to
the area under the curve from 0 to π.
77
§6.5 - AVERAGE VALUE OF A FUNCTION
Z
g(x) dx = 12. Which of the
Example. Suppose g(x) is a continuous function and
2
following are necessarily true?
A. For some number x between 2 and 5, g(x) = 3.
B. For some number x between 2 and 5, g(x) = 4.
C. For some number x between 2 and 5, g(x) = 5.
D. All of these are necessarily true.
E. None of these are necessarily true.
78
5
§6.5 - AVERAGE VALUE OF A FUNCTION
Extra Example. The temperature on a July day starts at 60◦ at 8 A.M., and rises (without
ever falling) to 96◦ at 8 P.M.
1. Why can’t you say with certainty that the average temperature between 8 A.M.
and 8 P.M. was 78◦?
2. What can you say about the average temperature during this 12-hour period?
3. Suppose you also know that the average temperature during this period was 84◦.
Is it possible that the temperature was 80◦ at 6 P.M.? At 4 P.M.?
79
§8.1 - INTEGRATION REVIEW
§8.1 - Integration Review
After completing this section, students should be able to:
R
1. Compute an integral of a function, like sec2(x) dx, by recalling the antiderivative
of the function.
2. Rewrite or simplify an integrand in order to compute an integral.
3. Recognize when u-substitution is useful and apply it to compute an integral.
R √
4. Use u-substitution for integrals like x 1 + x dx in which x must be rewritten in
terms of u.
80
§8.1 - INTEGRATION REVIEW
Z
cos4(θ) sin(θ) dθ
Example.
Z
Example.
2
3
dy
5 − 3y
81
§8.1 - INTEGRATION REVIEW
Z
1
dx
x−1 + 1
Z
1+x
dx
4 + x2
Example.
Example.
82
§8.2 - INTEGRATION BY PARTS
§8.2 - Integration by Parts
After completing this section, students should be able to:
• Use Rintegration by parts to compute an integral that is a product of two factors,
like xex dx
• Identify factors that are good candidates for u vs dv
• Use integration by parts more than one time if necessary.
R
• Use integration by parts to compute integrals like arctan(x) dx, by using 1dx as
du
R
• Use integration by parts to compute integrals like sin(x)ex dx, in which the integrands cycle around, and it possible to solve for the integral without ever fully
computing it.
83
§8.2 - INTEGRATION BY PARTS
Recall: the Product Rule says:
Rearranging and integrating both sides gives the formula:
Note. This formula allows us to rewrite something that is difficult to integrate in terms
of something that is hopefully easier to integrate. Integrating using this method is
called:
84
§8.2 - INTEGRATION BY PARTS
Example. Find
R
xex dx.
85
§8.2 - INTEGRATION BY PARTS
Review.
Z
u dv =
R
Example. Integrate t sec2(2t) dt using integration by parts. What is a good choice for
u and what is a good choice for dv?
86
§8.2 - INTEGRATION BY PARTS
Example. Find
R
x(ln x)2dx
87
§8.2 - INTEGRATION BY PARTS
Example. Integrate
R2
1
arctan(x)dx.
88
§8.2 - INTEGRATION BY PARTS
Example. Find
R
e2x cos(x)dx.
89
§8.2 - INTEGRATION BY PARTS
Question. How do we decide what to call u and what to call dv?
Question. Which of these integrals is a good candidate for integration by parts? (More
than one answer is correct.)
R
A. x3 dx
R
B. ln(x) dx
R
C. x2ex dx
R
2
D. xex dx
Z
ln y
E.
√ dy
y
90
§8.3 - INTEGRATING TRIG FUNCTIONS
§8.3 - Integrating Trig Functions
After completing this section, students should be able to:
• Compute integrals of powers of sine and cosine that include at least one odd power
by converting sines to cosines or vice versa and using u-substitution.
• Compute integrals of even powers of sine and cosine using the trig identities
1 1
1 1
cos( θ) = + cos(2θ) and sin( θ) = − cos(2θ)
2 2
2 2
• Compute some powers of sec and tan by converting them to sine and cosine, or by
applying u-substitution.
R
R
• Compute sec(x) dx and csc(x) dx.
R
R
2
• Compute sec (x) dx and tan2(x) dx
91
§8.3 - INTEGRATING TRIG FUNCTIONS
Note. Here are some useful trig identities for the next few sections.
1. Pythagorean Identity:
2. Converted into tan and sec:
3. Converted into cot and csc:
4. Even and Odd:
5. Angle Sum Formula: sin(A + B) =
6. Angle Sum Formula: cos(A + B) =
7. Double Angle Formula: sin(2θ) =
8. Double Angle Formulas: cos(2θ) =
9.
10.
11. cos2(θ) =
12. sin2(θ) =
92
§8.3 - INTEGRATING TRIG FUNCTIONS
Z
sin4(x) cos(x) dx
Example. Find
Z
Example. Find
sin4(x) cos3(x) dx
93
§8.3 - INTEGRATING TRIG FUNCTIONS
Z
Example. Find
sin5(x) cos2(x) dx
94
§8.3 - INTEGRATING TRIG FUNCTIONS
Example. Consider
Z
cos2(x)dx
.
According to a TI-89 calculator
Z
sin(x) cos(x) x
cos2(x) dx =
+
2
2
.
According to the table in the back of the book,
Z
1
1
cos2(x) dx = x + sin 2x
2
4
.
Are these answers the same?
95
§8.3 - INTEGRATING TRIG FUNCTIONS
Compute
R
cos2(x) dx by hand. Hint: cos2(x) =
Example. Compute
R
sin2(x) dx by hand.
96
1 + cos(2x)
2
§8.3 - INTEGRATING TRIG FUNCTIONS
Example. Compute
R
sin6(x) dx by hand.
97
§8.3 - INTEGRATING TRIG FUNCTIONS
Z
Review. What tricks can be used to calculate
98
cos7(5x) sin4(5x) dx?
§8.3 - INTEGRATING TRIG FUNCTIONS
Which of these integrals can be attacked in the same way, using the identity
sin2(x) + cos2(x) = 1 and u-substitution?
Z
p
Z
3
D.
sin (2x) cos(2x) dx
A.
sin3(x) cos4(x) dx
Z
B.
3
√
Z
cos ( x)
dx
√
x
Z
C.
tan3(x) dx
E.
Z
cos2(x) sin4(x) dx
F.
99
sin2(x) dx
§8.3 - INTEGRATING TRIG FUNCTIONS
Even powers of sine and cosine.
Review. What trig identities are most useful in evaluating
Example. Compute
R
cos2(x) sin4(x) dx by hand.
100
R
cos2(x) sin4(x) dx?
§8.3 - INTEGRATING TRIG FUNCTIONS
Conclusions:
Z
To find
sinm(x) cosn(x) dx,
if m is odd and n is even:
if n is odd and m is even:
if both m and n are odd:
if both m and n are even:
101
§8.3 - INTEGRATING TRIG FUNCTIONS
Note. Often the answers that you get when you integrate by hand do not look identical
to the answers you will see if you use your calculator, Wolfram Alpha, or the integral
table in the back of the book. Of course, the answers should be equivalent. Why do
you think the answers look so different?
102
§8.3 - INTEGRATING TRIG FUNCTIONS
These integrals have their own special tricks.
Z
Example.
tan2(x) dx
Z
Example.
sec(x) dx
103
§8.4 - TRIG SUBSTITUTIONS
§8.4 - Trig Substitutions
After completing this section, students should be able to:
• Decide if an integral might be appropriate for computing using trig substitution.
• Determine what trig substitution should be used.
• Perform trig substitution to compute an integral, including converting back to
original variables using a triangle and / or trig identities as needed.
104
§8.4 - TRIG SUBSTITUTIONS
The following three trig identities are useful for doing trig substitutions to solve some
kinds of integrals with square roots in them.
sin2(x) + cos2(x) = 1
tan2(x) + 1 = sec2(x)
105
cot2(x) + 1 = csc2(x)
§8.4 - TRIG SUBSTITUTIONS
Example. According to Wolfram Alpha,
Z
√
1
x2
−1 x
dx = 49 sin
− x 49 − x2
√
2
7
49 − x2
Let’s see where that answer comes from using a trig substitution.
END OF VIDEO
106
§8.4 - TRIG SUBSTITUTIONS
Review. To compute
R
2
√x
49−x2
dx, which substitution is most useful?
A. u = 49 − x2
B. x = sin(θ)
C. x = 7 sin(θ)
D. x = tan(θ)
E. x = 49 tan(θ)
F. x = 7 sec(θ)
107
§8.4 - TRIG SUBSTITUTIONS
Z
Example. Find
1
dx. (Assume a is positive.)
√
2
2
x +a
108
§8.4 - TRIG SUBSTITUTIONS
Z
2/3
Example. Compute the integral
1/3
√
9x2 − 1
dx
x
109
§8.4 - TRIG SUBSTITUTIONS
Which trig substitutions for which problems?
110
§8.4 - TRIG SUBSTITUTIONS
What trig substitutions would be most useful for these integrals?
Z
2
1.
dx
√
2
4+x
Z
(100x2 − 1)3/2 dx
2.
r
Z
x
3.
x2
4 − dx
9
Z
4.
(25 − x2)2 dx
Z √
5.
−x2 − 6x + 7 dx
111
§8.4 - TRIG SUBSTITUTIONS
Extra Example. Use calculus to find the volume of a torus with dimensions R and r as
shown.
112
§8.5 - INTEGRALS OF RATIONAL FUNCTIONS
§8.5 - Integrals of Rational Functions
After completing this section, students should be able to:
1. Recognize whether an integral is a good candidate for the method of partial fractions.
2. Rewrite a rational expression as a sum of appropriate partial fractions, performing
long division first if the numerator has degree greater or equal to the denominator.
3. Compute an integral using the method of partial fractions.
113
§8.5 - INTEGRALS OF RATIONAL FUNCTIONS
Example. According to Wolfram Alpha,
Z
5
7
3x + 2
dx
=
ln
|1
−
x|
+
ln |x + 3| + C
4
4
x2 + 2x − 3
Let’s see where this answer came from.
114
§8.5 - INTEGRALS OF RATIONAL FUNCTIONS
Z
Review. True or False:
1
dx = ln |2x2 − 7x − 4| + C
2
2x − 7x − 4
115
§8.5 - INTEGRALS OF RATIONAL FUNCTIONS
Z
Example. Find
2x2 + 7x + 19
dx
x2 − 5x + 6
116
§8.5 - INTEGRALS OF RATIONAL FUNCTIONS
Example. How would you set up partial fractions to integrate this?
Z
5x + 7
dx
(x − 2)(x + 5)(x)
117
§8.5 - INTEGRALS OF RATIONAL FUNCTIONS
Z
Example. How would you set up partial fractions to integrate this?
4x2 + 3x + 7 A
B
=
A.
+
x x−2
x(x − 2)2
4x2 + 3x + 7 A
B
=
B.
+
x (x − 2)2
x(x − 2)2
4x2 + 3x + 7 A
B
C
C.
=
+
+
x x − 2 (x − 2)2
x(x − 2)2
4x2 + 3x + 7 A Bx + C
D.
= +
x (x − 2)2
x(x − 2)2
118
4x2 + 3x + 7
dx
x3 − 4x2 + 4x
§8.6 INTEGRATION STRATEGIES
§8.6 Integration Strategies
After completing this section, students should be able to:
• Choose an appropriate integration strategy for a given integral.
• Express the limitations of the integration techniques that we have learned, as well
as the limitations of all integration techniques known to humankind.
119
§8.6 INTEGRATION STRATEGIES
For each integral, indicate what technique you might use to approach it and give the
first step. You do not need to finish any of the problems.
Z
Z
sin(x)
6.
dx
2
1.
x3 ln x dx
3 + sin (x)
Z
Z
x3
7.
dx
2.
cos2(x) dx
25 − x2
Z
Z
3
x
dx
dx
8.
√
3.
2
25 − x
x ln(x)
Z √
Z
9.
e x dx
4.
arcsin(x) dx
Z 2
x +1
5.
√ dx
x
120
PHILOSOPHY ABOUT INTEGRATION
Philosophy about Integration
Definition. (Informal Definition) An elementary function is a function that can be built
up from familiar functions, like
• polynomials
• trig functions
• exponential and logarithmic functions
using familiar operations:
• addition
• subtraction
• multiplication
• division
• composition
Example. Give an example an elementary function. Make it as crazy as you can.
121
PHILOSOPHY ABOUT INTEGRATION
Question. Is it always true that the derivative of an elementary function is an elementary function?
Question. Is it always true that the integral of an elementary function is an elementary
function?
122
PHILOSOPHY ABOUT INTEGRATION
Techniques of integration ... and their limitations.
123
§8.9 -IMPROPER INTEGRALS
§8.9 -Improper Integrals
After completing this section, students should be able to:
• Determine if an integral is improper and explain why.
• Explain how to calculate an improper integral or determine that it diverges by
taking a limit.
• Divide up an improper integral into several separate integrals in order to compute
it, when it is improper in several ways.
• Calculate improper integrals or determine that they diverge.
• Choose appropriate functions to compare with integrands, when using the Comparison Theorem.
• Use the Comparison Theorem to determine if integrals converge or diverge without
actually integrating.
• Give an example to show how failing to notice that an integral is improper and
computing it as if it were proper can lead to nonsense.
124
§8.9 -IMPROPER INTEGRALS
Here are two examples of improper integrals:
∞
Z
1
1
dx
x2
and
Z
π
2
tan(x) dx
0
Question. What is so improper about them?
Definition. An integral is called improper if either
(Type I)
or,
(Type II)
or both.
125
§8.9 -IMPROPER INTEGRALS
Type 1 Improper Integrals
To integrate over an infinite interval, we take the limit of the integrals over expanding
finite intervals Z ∞
1
Example. Find
dx
2
x
1
Z
∞
f (x) dx is defined as ...
Definition. The improper integral
a
Z
∞
f (x) dx converges if ...
We say that
a
and diverges if ...
126
§8.9 -IMPROPER INTEGRALS
Z
b
f (x) dx as ...
Definition. Similarly, we define
−∞
Z
b
f (x) dx converges if ...
and say that
−∞
and diverges ...
Z
−1
Example. Evaluate
−∞
1
dx and determine if it converges or diverges.
x
END OF VIDEO
127
§8.9 -IMPROPER INTEGRALS
Review. Which of the following are NOT improper integrals?
Z ∞
A.
e−x dx
1
Z
3
B.
0
Z
1
dx
x2
5
ln |x| dx
C.
−5
0
Z
4
dx
x
+
4
−∞
E. They are all improper
Z ∞ integrals.
1
Example. Evaluate
√ dx and determine if it converges or diverges.
x
1
D.
128
§8.9 -IMPROPER INTEGRALS
∞
Z
Question. For what values of p > 0 does
1
129
1
dx converge?
xp
§8.9 -IMPROPER INTEGRALS
Example. Find the area under the curve y = e3x−2 to the left of x = 2.
130
§8.9 -IMPROPER INTEGRALS
Type 2 Improper integrals
When the function we are integrating goes to infinity at one endpoint of an interval,
we take a limit of integrals over expanding sub-intervals.
Definition. If f (x) → ∞ or f (x) → −∞ as
Definition. If f (x) → ∞ or f (x) → −∞ as
x → a+, then
−
x → b , then
Z b
Z b
f (x) dx =
f (x) dx =
a
a
131
§8.9 -IMPROPER INTEGRALS
Example. Find the area under the curve y = √
x
x2
−1
between the lines x = 1 and x = 2.
1.5
1.4
1.3
1.2
1.1
1
END OF VIDEO
132
2
3
4
5
§8.9 -IMPROPER INTEGRALS
Review. True or False: If f (x) is continuous on (1, 2] and f (x) → ∞ as x → 1+, then
R2
f (x) dx diverges. (Hint: remember the pre-class video on Type 2 integrals.)
1
Z
Example. Find
1
10
4
dx .
(x − 3)2
4
blows up at x = 3, this integral must be computed as the sum of
(x − 3)2
two indefinite integrals.
Note. Since
If you compute it without breaking it up YOU WILL GET THE WRONG ANSWER!
133
§8.9 -IMPROPER INTEGRALS
∞
Z
Question. For what values of p > 0 does
1
Z
1
Question. For what values of p > 0 does
0
134
1
dx converge?
xp
1
dx converge?
xp
§8.9 -IMPROPER INTEGRALS
Theorem. Comparison Theorem for Integrals: Suppose 0 ≤ g(x) ≤ f (x) on (a, b) (where a or
b could be −∞ or ∞).
Z b
Z b
, then
g(x) dx
also.
(a) If
f (x) dx
a
a
Z
(b) If
b
Z
g(x) dx
a
, then
b
also.
f (x) dx
a
135
§8.9 -IMPROPER INTEGRALS
Z
Example. Does
2
∞
2 + sin(x)
dx converge or diverge?
√
x
136
§8.9 -IMPROPER INTEGRALS
Review. If 0 ≤ f (x) ≤ g(x) on the interval [a, ∞), then which of the following are true?
R∞
R∞
A. If a f (x) dx converges, then a g(x) dx converges.
R∞
R∞
B. If a f (x) dx converges, then a g(x) dx diverges.
R∞
R∞
C. If a f (x) dx diverges, then a g(x) dx converges.
R∞
R∞
D. If a f (x) dx diverges, then a g(x) dx diverges.
E. None of these are true.
137
§8.9 -IMPROPER INTEGRALS
Z
Example. Does
1
∞
cos(x) + 7
dx converge or diverge?
3
4x + 5x − 2
138
§8.9 -IMPROPER INTEGRALS
Z
Example. Does
7
∞
3x2 + 2x
dx converge or diverge?
√
6
x −1
139
§8.9 -IMPROPER INTEGRALS
∞
Z
Extra Example. Does
2
√
x2 − 1
dx converge or diverge?
x3 + 3x + 2
140
§8.9 -IMPROPER INTEGRALS
Z
∞
2
e−x dx converge or diverge?
Example. Does
0
141
§8.9 -IMPROPER INTEGRALS
Question. What are some useful functions to compare to when using the comparison
test?
Z ∞
1
1
1
Question. True or False: Since − ≤ 2 for 1 < x < ∞, and
dx converges, the
2
x xZ
x
1
∞
1
Comparison Theorem guarantees that
− dx also converges.
x
1
142
§8.9 -IMPROPER INTEGRALS
Comparison Test Practice Problems
Decide what function to compare to and whether the integral converges or diverges.
Z 5
Z ∞
cos(t) + 4
1
6.
dt
Hint: do a u√
1.
dt
−1
t+1
5t
1 e +2
substitution.
Z ∞ √ 2
Z ∞
x −1
5
2.
dx
7.
dz
3 + 3x + 2
z + 2z
x
e
2
1
Z ∞
Z
∞
x2
4 sin(x) + 5
3.
dx
dx
8.
√
2+4
x
3
1
7
x +x
Z ∞
Z 2 √
x+3
t+2
9.
dx
√
dt
4.
2
4
t
7
x −x
0
Z ∞
Z ∞
6
5
5.
dt
10.
dx
√
√
x
t−5
5
0
xe + 1
143
§8.9 -IMPROPER INTEGRALS
Z
∞
x cos(x2 + 1) dx
Example. Find
−∞
144
§8.9 -IMPROPER INTEGRALS
Z
∞
Z
t
f (x) dx = lim
Question. True or False:
−∞
t→∞
145
f (x) dx
−t
§10.1 - SEQUENCES AND SERIES INTRO
§10.1 - Sequences and Series Intro
After completing this section, students should be able to:
• Explain the difference between a sequence and a series.
• Use a recursive formula to write out the terms of a sequence.
• Use a closed form formula to write out the terms of a sequence.
• Translate a list of terms of a sequence into a recursive formula or a closed form
formula.
• Explain what it means for a sequence to converge or diverge.
• Write out partial sums for a series.
• Explain what it means for a series to converge or diverge.
• Use numerical evidence to make a guess about whether a sequence converges.
• Use numerical evidence from partial sums to make a guess about whether a series
converges.
146
§10.1 - SEQUENCES AND SERIES INTRO
Definition. A sequence is an ordered list of numbers.
Example. 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 9, . . .
, or {an}.
A sequence is often denoted {a1, a2, a3, . . .}, or {an}∞
n=1
Example. For each sequence, write out the first three terms:
(
)∞
3n + 1
1.
(n + 2)! n=1
(
2. (−1)k
k+3
3k
)∞
k=2
147
§10.1 - SEQUENCES AND SERIES INTRO
Definition. Sometimes, a sequence is defined with a recursive formula (a formula that
describes how to get the nth term from previous terms), such as
a1 = 2,
an = 4 −
1
an−1
Example. Write out the first three terms of this recursive sequence.
Note. Sometimes it is possible to describe a sequence with either a recurvsive formula
or a ”closed-form”, non-recursive formula.
148
§10.1 - SEQUENCES AND SERIES INTRO
Example. Write a formula for the general term an, starting with n = 1.
A. {7, 10, 13, 16, 19, · · · }
Definition. An arithmetic sequence is a sequence for which consecutive terms have
the same common difference.
If a is the first term and d is the common difference, then the arithmetic sequence has
the form:
(starting with n = 0)
An arithmetic sequence can also be written:
(with the index starting at n = 1.)
149
§10.1 - SEQUENCES AND SERIES INTRO
Example. For each sequence, write a formula for the general term an (start with n = 1
or with n = 0).
B. {3, 0.3, 0.03, 0.003, 0.0003, · · · }
C.
n
15 75 375 1875
2 , 4 , 8 , 16 , · · ·
o
D. {3, −2, 43 , − 98 , . . .}
Definition. A geometric sequence is a sequence for which consecutive terms have the
same common ratio.
If a is the first term and r is the common ratio, then a geometric sequence has the form:
(with the index starting at 0)
A geometric sequence can also be written:
(with the index starting at 1)
150
§10.1 - SEQUENCES AND SERIES INTRO
Example. For each sequence, write a formula for the general term an, starting with
n = 1.
4
8 16
, − 25
, 36 , . . .}
E. {− 29 , 16
F. {−6, 5, −1, 4, 3, 7, 10, 17, . . .}
END OF VIDEO
151
§10.1 - SEQUENCES AND SERIES INTRO
Review. A sequence is ...
Example. Consider the sequence {3, 7, 11, 15, 19, · · · }
1. What are the next three terms in this sequence?
2. What is a recursive formula for this sequence?
3. What is a explicit (closed form) formula for this sequence?
152
§10.1 - SEQUENCES AND SERIES INTRO
1 3
9 27
Example. Consider the sequence − , , − ,
,···
2 10 50 250
1. What are the next three two terms in this sequence?
2. What is a recursive formula for this sequence?
3. What is a explicit (closed form) formula for this sequence?
153
§10.1 - SEQUENCES AND SERIES INTRO
∞
n2 · 5
Example. Consider the sequence (−1)
n! n=0
1. What are the first three terms in this sequence?
2. What is a recursive formula for this sequence?
154
§10.1 - SEQUENCES AND SERIES INTRO
Definition. A sequence {an} converges if:
Otherwise, the sequence diverges. In other words, a sequence diverges if:
Example. Which of the following sequences converge?
A. {3, 7, 11, 15, 19, · · · }
1 3
9 27
,···
B. − , , − ,
2 10 50 250
C.
n
1 2 3 4
2, 3, 4, 5, · · ·
o
155
§10.1 - SEQUENCES AND SERIES INTRO
Definition. For any sequence {an}∞
, the sum of its terms a1 + a2 + a3 + · · · is a series.
n=1
Often this series is written as
∞
X
an
n=1
Example. Consider the sequence
series:
n o∞
1
2n n=1 .
If we add together all the terms, we get the
∞
X
1
=
2n
n=1
What does it mean to add up infinitely many numbers?
156
§10.1 - SEQUENCES AND SERIES INTRO
Definition. The partial sums of a series
∞
X
an are defined as the sequence {sn}∞
, where
n=1
n=1
s1 =
s2 =
s3 =
sn =
Definition. The series
∞
X
an is said to converge if :
n=1
Otherwise, the series diverges.
Note. Associated with any series
∞
X
an, there are actually two sequences of interest:
n=1
1.
2.
157
§10.1 - SEQUENCES AND SERIES INTRO
∞
X
1
, write out the first 4 terms and the first 4 partial
2+n
n
n=1
sums. Does the series appear to converge?
Example. For the series
158
§10.1 - SEQUENCES AND SERIES INTRO
Review. What is the difference between the following two things?
∞
1
• the sequence k
4 k=1
∞
X
1
• the series
k
4
k=1
∞
1
Question. What does it mean for the sequence k
to converge vs. diverge?
4 k=1
∞
X
1
to converge vs. diverge?
Question. What does it mean for the series
k
4
k=1
∞
X
1
Question. Does the series
converge or diverge?
k
4
k=1
159
§10.1 - SEQUENCES AND SERIES INTRO
Example. Using your calculator, Excel, or any other methods, compute several partial
sums for each of the following series and make conjectures about which series converge
and which diverge.
A. 4 + 0.2 + 0.02 + 0.002 + · · ·
∞
X
(−1) j
B.
j=1
C.
∞
X
k=1
k
k+1
160
10.2 SEQUENCES
10.2 Sequences
After completing this section, students should be able to:
• Define increasing, decreasing, non-decreasing, non-increasing, and monotonic.
• Define bounded.
• Use the first derivative to determine if sequences are increasing, decreasing and
whether they are bounded.
• Determine if a sequence converges and find its limit by evaluating the limit of a
function using Calculus 1 techniques.
• State the limit laws and use them to break apart limits and determine convergence.
• Recognize when limit laws don’t apply due to component sequences diverging.
• Find the first term and common ratio of a geometric sequence and use the common
ratio to determine if the sequence converges or diverges.
• State conditions involving boundedness and monotonic-ness that ensure that a
sequence converges, and use this condition to prove that sequences converge.
• Use the squeeze theorem to prove that a sequence converges.
• Use the idea of the squeeze theorem to prove that a sequence diverges to ∞ or −∞
161
10.2 SEQUENCES
Definition. A sequence {an} is bounded above if
A sequence {an} is bounded below if:
Example. Which of these sequences are bounded?
A. {3, 0.3, 0.03, 0.003, 0.0003, · · · }
B. {1, 2, 2, 3, 3, 3, 4, 4, 4, 4, · · · }
C. {3, −2, 34 , − 89 , . . .}
162
10.2 SEQUENCES
Definition. A sequence {an} is increasing if
A sequence {an} is non-decreasing if
A sequence {an} is decreasing if
A sequence {an} is non-increasing if
A sequence {an} is monotonic if it is
Example. Which of these sequences are monotonic?
A. {3, 0.3, 0.03, 0.003, 0.0003, · · · }
B. {1, 2, 2, 3, 3, 3, 4, 4, 4, 4, · · · }
C. {3, −2, 34 , − 89 , . . .}
D. {−6, 5, −1, 4, 3, 7, 10, 17, . . .}
END OF VIDEO
163
10.2 SEQUENCES
Question. What is the difference between increasing and non-decreasing? Decreasing
and non-increasing?
Review. Give an example of a sequence that is
• monotonically increasing and bounded
• monotonic non-increasing but not bounded
• not monotonic but bounded
• not monotonic and not bounded
164
10.2 SEQUENCES
n−5
Example. Is the sequence
n2
∞
monotonic? Bounded?
n=1
165
10.2 SEQUENCES
Review. Recall that a geometric sequence is a sequence that can be written in the form:
Here, r represents
and a represents
What is an example of a geometric sequence?
166
.
10.2 SEQUENCES
Example. Which of these are geometric sequences? Which of them converge?
)∞
(
(−1)n4n
•
5n+2 n=0
5 · 0.5n
•
3n−1
∞
2
9 27
• 4/3, 2, 3, , . . .
2 4
• {2, −4, 8, −16, 32, −64, . . .}
167
10.2 SEQUENCES
Question. For which values of a and r does {a · rn}∞
converge?
n=0
168
10.2 SEQUENCES
The following are some techniques for proving that a sequence converges:
(−1)tet−1
Example. Does
3t+2
(
)∞
converge or diverge?
t=0
Trick 1: Recognize geometric sequences
169
10.2 SEQUENCES
ln(1 + 2en)
Example. Does
n
(
)∞
converge or diverge?
n=1
Trick 2: Suppose an = f (n) where n = 1, 2, 3, . . ., for some function f defined on all
positive real numbers. If lim f (x) = L then ...
x→∞
So ... replace an with f (x) and use l’Hospital’s Rule or other tricks from Calculus 1 to
show that lim f (x) exists.
x→∞
170
10.2 SEQUENCES
cos(n) + sin(n)
Example. Does
n2/3
(
)∞
converge or diverge?
n=5
Trick 3: Use the Squeeze Theorem: trap the sequence between two simpler sequences
that converge to the same limit.
171
10.2 SEQUENCES
Example. {n + sin(n)}∞
n=0
172
10.2 SEQUENCES
Example. Does 0.1, 0.12, 0.123, 0.1234, . . . , 0.12345678910, 0.1234567891011, 0.123456789101112, . .
converge or diverge?
Trick 4: If {an} is
and
, then it converges.
173
10.2 SEQUENCES
Trick 5: Use the Limit Laws
The usual limit laws about addition, subtractions, etc. hold for sequences as well as
for functions. (See textbook.)
For example, if lim an = L and lim bn = M, then
n→∞
n→∞
lim (an + bn) =
n→∞
lim (anbn) =
n→∞
lim (can) =
(c is a constant)
n→∞
4 · πk
k2
+ k
Example. Does
2k2 − k
6
(
)∞
converge or diverge?
k=3
174
10.2 SEQUENCES
Question. Do the limit laws help establish the convergence of this sequence?
3 − 2n ∞
n+
2
n=2
175
10.2 SEQUENCES
True or False:
1. If {ak } converges, then so does {|ak |}.
2. If {|ak |} converges, then so does {ak }.
3. If {ak } converges to 0, then so does {|ak |}.
4. If {|ak |} converges to 0, then so does {ak }.
176
10.2 SEQUENCES
Example. Does
n (−1)n o
n2
converge or diverge?
177
10.2 SEQUENCES
True or False:
1. Suppose an = f (n) for some function f , where n = 1, 2, 3, . . .. If lim f (x) = L then
x→∞
lim an = L.
n→∞
2. Suppose an = f (n) for some function f . If lim an = L, then lim f (x) = L.
n→∞
178
x→∞
10.2 SEQUENCES
Additional problems if additional time:
Do the following sequences converge or diverge? Justify your answer.
(
)∞
cos( j)
1.
ln(j + 1) j=1
(
)
t t−1 ∞
(−1) 4
2.
32t
t=3
 √

∞
3

 k 

3. 
 ln(k) 

k=2
n ∞
3
4.
n! n=1
∞
n!
5. n
3 n=1
179
§10.3 - SERIES
§10.3 - Series
After completing this section, students should be able to:
• Determine if a geometric series converges or diverges.
• Recognize a telescoping series and use its partial sums to determine if it converges
or diverges.
• Determine if sums and scalar multiples of series converge or diverge based on the
convergence status of their component series.
180
§10.3 - SERIES
Definition. A geometric sequence is a sequence of the form ...
Definition. A geometric series is a series of the form ...
Example. Is
∞
X
5(−2)i
i=2
32i−3
a geometric series? If so, what is the first term and what is the
common ratio?
181
§10.3 - SERIES
Fact. A geometric sequence {arn}∞
converges to 0 when
n=0
when
and diverges when
.
Question. For what values of r does the geometric series
, converges to
∞
X
n=0
Stragegy:
k
1. Find a formula for the Nth partial sum sumN
k=0 a · r .
2. Take the limit of the partial sums.
182
arn converge?
§10.3 - SERIES
Conclusion: The geometric series
∞
X
arn converges to
n=0
The geometric series
∞
X
arn diverges when
n=0
Example. Does
∞
X
5(−2)i
i=2
32i−3
converge or diverge?
END OF VIDEO
183
.
when
.
§10.3 - SERIES
Tricks for determining when series converge:
Trick 1: Recognize geometric series.
Review. A geometric series is a series of the form:
Review. For what values of r does a geometric series converge?
Example. For what values of x does the series
∞
X
3xn−1
n=2
converge to (in terms of x)?
184
2n
converge? What does it
§10.3 - SERIES
Trick 2: Recognize telescoping series.
!
∞
X
k
ln
Example.
k+1
k=2
185
§10.3 - SERIES
Example.
∞
X
n=2
3
n2 − 1
186
§10.3 - SERIES
Trick 3: ∞
Use Limit Laws.
∞
X
X
an = A and
bn = B, then
Fact. If
n=1
n=1
∞
X
a n + bn =
n=1
∞
X
a n − bn =
n=1
∞
X
c · an =
n=1
where c is a constant.
187
§10.3 - SERIES
Example. Does the series converge or diverge? If it converges, to what?
∞
X
4 · 5n − 5 · 4n
n=1
6n
188
§10.3 - SERIES
Question. True or False: If
diverges.
Question. True or False: If
verges.
∞
X
an diverges and
n=1
∞
X
∞
X
∞
X
bn converges, then
(an + bn)
n=1
an diverges and
n=1
∞
X
n=1
189
n=1
∞
X
bn diverges, then
(an + bn) din=1
§10.3 - SERIES
Question. True or False: If
Question. True or False: If
∞
X
an converges, then so does
∞
X
n=1
n=5
∞
X
∞
X
an converges, then so does
n=5
n=1
190
an.
an.
§10.3 - SERIES
Question. True or False: If
∞
X
an = A and
n=1
Question. True or False: If
∞
X
n=1
∞
X
bn = B, then
n=1
an = A and
∞
X
n=1
191
∞
X
an · bn = A · B
n=1
bn = B, then
∞
X
a
n=1
n
bn
=
A
.
B
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
§10.4 - The Divergence Test and the Integral Test
After completing this section, students should be able to:
• State the Divergence Test and use it to prove that a series diverges.
• Explain why the Divergence Test cannot by used to prove that a series converges.
• Determine whether it is appropriate to use the integral test.
• Use the integral test, when appropriate, to prove that a series converges.
• Use the p-test to prove that a series converges.
• Identify the Harmonic Series.
• Use an integral, when appropriate, to find a bound on the remainder of a series
with positive terms after evaluating a partial sum, and to find bounds on the value
of the sum based on partial sums and integrals.
• Use an integral, when appropriate, to determine how many terms are needed to
approximate the sum of a series to within a specified level of accuracy.
192
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
Example. Does this series converge or or diverge?
∞
X
1
n2
n=1
193
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
Z ∞
∞
X
1
1
The series
is
closely
related
to
the
improper
integral
dx .
2
2
n
x
1
n=1
194
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
Example. Does this series converge or or diverge?
∞
X
1
√
x
n=1
195
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
Theorem. (The Integral Test) Suppose f is a continuous, positive, decreasing function
on [1, ∞) and an = f (n). Then
Z ∞
∞
X
1. If
an converges.
f (x) dx converges, then
1
Z
2. If
∞
f (x) dx diverges, then
1
n=1
∞
X
an diverges.
n=1
196
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
Example. Does
∞
X
ln n
n=1
n
converge or diverge?
END OF VIDEO
197
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
Example. Does
∞
X
k=1
k
converge or diverges?
k+1
Note. If the sequence of terms an do not converge to 0, then the series
Theorem. (The Divergence Test) If
then the series
∞
X
an diverges.
n=1
198
P
an ...
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
Example.
∞
X
t sin(1/t)
t=1
∞
X
Example.
(−1)n
t=1
Note. If the sequence of terms an do converge to 0, then the series
.
199
P
an
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
Z
Review. We know that
1
∞
1
dx converges to 1. Which of the following are true?
2
x
∞
X
1
converges.
A.
2
n
n=1
∞
X
1
B.
= 1.
2
n
n=1
C. Both of the above.
D. None of the above.
200
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
∞
X
n
Example. Does
converge or diverge?
en
n=1
201
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
Example. Does the following series converge or diverge?
1
1
1 1
+ +
+
+ ···
5 8 11 14
202
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
∞
X
1
Question. For what values of p does the p-series
converge?
np
n=1
203
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
Definition. The Harmonic Series is the series:
Question. Does the Harmonic Series converge or diverge?
204
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
Bounding the Error
∞
X
Definition. If
an converges, and sn is the nth partial sum, then for large enough n, sn
n=1
is a good approximation to the sum s∞ =
∞
X
ak . Define Rn be the error, or remainder:
k=1
Rn =
R∞
Use the pictures above to compare R2 to 2 f (x) dx and
positive, decreasing function drawn with an = f (n).
205
R∞
2
f (x) dx where f (x) is the
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
R∞
Use the pictures above to compare Rn to n f (x) dx and
positive, decreasing function drawn with an = f (n).
R∞
n+1
f (x) dx where f (x) is the
Note. If an = f (n) for a continuous, positive, decreasing function f (x),
≤ Rn ≤
This is called the Remainder Estimate for the Integral Test
206
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
Example. (a) Put a bound on the remainder when you use the first three terms to
∞
X
6
.
approximate
2
n
n=1
(b) Use the bound on the remainder to put bounds on the sum s∞. Hint: s∞ = s3 + R3.
(c) How many terms are needed to approximate the sum to within 3 decimal places?
Note: by convention, this means Rn < 0.0005.
207
§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST
Question. Which of the following are always true?
1. Suppose f is a continuous, positive, decreasing function on [1, ∞) and for n ≥ 1,
Z ∞
∞
X
an converges, if and only if
f (x) dx converges.
an = f (n). Then
1
n=1
2. Suppose f is a continuous, positive, decreasing function on [5, ∞) and for n ≥ 5,
Z ∞
∞
X
an = f (n). Then
an converges if and only if
f (x) dx.
5
n=1
3. Suppose f is a continuous, positive function on [1, ∞) and for n ≥ 1, an = f (n).
Z ∞
∞
X
Then
an converges if and only if
f (x) dx converges.
n=1
1
208
§10.5 - COMPARISON TESTS FOR SERIES
§10.5 - Comparison Tests for Series
After completing this section, students should be able to:
• For the (ordinary) comparison test, give conditions that will guarantee convergence
of a series and conditions that will guarantee divergence of a series, and justify
why these conditions make sense.
• For the limit comparison test, state what values of the limit of the ratio of terms
allows you to determine that a series converges or diverges, and what values are
inconclusive.
• Determine what series to compare another series to, when using the comparison
or limit comparison test.
• Identify situations that make it preferable to use the ordinary comparison test
instead of the limit comparison test and vice versa.
209
§10.5 - COMPARISON TESTS FOR SERIES
Theorem. (The Comparison Test for Series) Suppose that
0 ≤ an ≤ bn for all n.
1. If
converges, then
2. If
diverges, then
P∞
n=1 an
and
P∞
n=1 bn
are series and
converges.
diverges.
Note. The following series are especially handy to compare to when using the comparison test.
1.
which converges when
2.
which converges when
210
§10.5 - COMPARISON TESTS FOR SERIES
Example. Does
∞
X
n=1
3n
converge or diverge?
5n + n2
211
§10.5 - COMPARISON TESTS FOR SERIES
P
P
Theorem. (The Limit Comparison Test) Suppose an and bn are series with positive terms.
If
an
lim
=L
n→∞ bn
where L is a finite number and L > 0, then either both series converge or both diverge.
Example. Does
∞
X
n=1
3n
converge or diverge?
5n − n2
212
§10.5 - COMPARISON TESTS FOR SERIES
Review. The (Ordinary) Comparison Test for Series: Suppose that
are series with positive terms and 0 ≤ an ≤ bn for all n.
1. If
converges, then
2. If
diverges, then
converges.
diverges.
213
P∞
n=1 an
and
P∞
n=1 bn
§10.5 - COMPARISON TESTS FOR SERIES
P∞
P∞
Review. Suppose
an and
bn are series with positive terms. Which of the followP
ing will allow us to conclude that ∞ bn converges? (More than one answer may be
correct.)
∞
X
an converges.
A. lim an = lim bn and
n→∞
n→∞
P
an
= 0 and ∞ an converges.
n→∞ bn
P
an 1
C. lim
= and ∞ an converges.
n→∞ bn
3
P
an
D. lim
= 5 and ∞ an converges.
n→∞ bn
B. lim
P
P
Review. The Limit Comparison Test: Suppose an and bn are series with positive
terms. If
an
=L
lim
n→∞ bn
where L
,
then:
214
§10.5 - COMPARISON TESTS FOR SERIES
Advice on the Comparison Theorems:
Question. What series are especially handy to compare to when using the comparison
test?
Question. How to decide whether to use the Ordinary Comparison Test or the Limit
Comparison Test?
215
§10.5 - COMPARISON TESTS FOR SERIES
Example. Decide if the series converges or diverges.
∞
X
3n − 5
√
n3 + 2n
n=1
216
§10.5 - COMPARISON TESTS FOR SERIES
Example. Decide if
∞
X
n sin2(n)
n=3
n3 + 7n
converges or diverges.
217
§10.5 - COMPARISON TESTS FOR SERIES
Example. Decide if
∞
X
n sin2(n)
n=3
n3 − 7n
converges or diverges.
218
§10.5 - COMPARISON TESTS FOR SERIES
P
P
an
= 0, then the series an and bn have
n→∞ bn
Question. True or False: For an, bn > 0, if lim
the same convergence status.
an
= 0?
n→∞ bn
Can anything be concluded if lim
219
§10.5 - COMPARISON TESTS FOR SERIES
Question. Find the error: Consider the two series
∞
X
an = (−1) + (−2) + (−3) + (−4) + (−5) + (−6) . . .
n=1
and
∞
X
bn = 2 + (−1) + (1/2) + (−1/4) + (1/8) + (−1/16) + . . .
n=1
P∞
is a geometric series with ratio r = −1/2.
P
P
Since an ≤ bn for all n, and bn converges, an also converges, by the Ordinary
Comparison Test.
Note that
n=1 bn
220
§10.5 - COMPARISON TESTS FOR SERIES
Note. Orders of magnitude:
221
§10.5 - COMPARISON TESTS FOR SERIES
Note. Review of the convergence tests for series so far:
1.
2.
3.
4.
5.
6.
222
SECTION 10.6 - ALTERNATING SERIES
Section 10.6 - Alternating Series
After completing this section, students should be able to:
• Define an alternating series.
• Identify the conditions needed to guarantee that an alternating series converges.
• Bound the remainder when using a specified partial sum to approximate an alternating series.
• Determine how many terms are needed to approximate an alternating series within
a specified level of accuracy.
• Explain the relationship between convergent, absolutely convergent, and conditionally convergent.
∞
∞
X
X
• Prove that a series
an converges by showing that
|an| converges and using
n=1
n=1
the fact that absolutely convergent implies convergent.
223
SECTION 10.6 - ALTERNATING SERIES
Definition. An alternating series is a series whose terms are alternately positive and
negative. It is often written as
∞
X
(−1)k−1bk
k=1
where the bk are positive numbers.
Example. (The Alternating Harmonic Series)
224
SECTION 10.6 - ALTERNATING SERIES
Does the Alternating Harmonic Series converge? Hint: look at ”even” partial sums
and ”odd” partial sums separately.
225
SECTION 10.6 - ALTERNATING SERIES
Theorem. (Alternating Series Test) If the series
∞
X
(−1)n−1bn = b1 − b2 + b3 − b4 . . .
n=1
satisfies:
1.
2.
3.
then the series is convergent.
226
SECTION 10.6 - ALTERNATING SERIES
Example. Which of these series are guaranteed to converge by the Alternating Series
Test?
√5
3
+
√5
4
√5
5
+
√5
6
√5
7
+ ···
A.
√5
2
B.
2
2
− 12 + 23 − 13 + 42 − 14 + 25 − 51 + · · ·
C.
1
8
1
1
1
1
1
− 14 + 27
− 91 + 64
− 16
+ 125
− 25
+ ···
−
−
−
D. 2.1 − 2.01 + 2.001 − 2.0001 + 2.00001 · · ·
227
SECTION 10.6 - ALTERNATING SERIES
Question. Why is the condition lim bn = 0 necessary?
n→∞
Question. Why is the condition bn+1 ≤ bn for all large n necessary?
228
SECTION 10.6 - ALTERNATING SERIES
Example. Does the series converge or diverge?
∞
X
2
n n
(−1) 3
n −2
n=1
229
SECTION 10.6 - ALTERNATING SERIES
Example. Does the series converge or diverge?
∞
X
(−1)k (1 + k)1/k
k=1
230
SECTION 10.6 - ALTERNATING SERIES
Bounding the Remainder
For the same type of series:
• series is alternating
• limn→∞ bn = 0
• bn+1 ≤ bn
We want to put a bound on the remainder. Call the sum of the infinite series s∞ and
the nth partial sum sn.
1. Write an equation for the nth remainder Rn.
2. Find an upper bound on |Rn|:
|Rn| ≤
231
SECTION 10.6 - ALTERNATING SERIES
1
1
+ 25
− ···
Example. Consider the series − 14 + 19 − 16
If we add up the first 6 terms of this series, what is true about the remainder? (PollEv)
A. positive and < 0.01
B. positive and < 0.02
C. positive and < 0.05
D. negative with absolute value < 0.01
E. negative with absolute value < 0.02
F. negative with absolute value < 0.05
G. none of these.
232
SECTION 10.6 - ALTERNATING SERIES
Example. How many terms of the series
1
1
1 1
+
− ···
− + −
4 9 16 25
do we need to add up to approximate the limit to within 0.01?
233
SECTION 10.6 - ALTERNATING SERIES
Definition. A series
X
an is called absolutely convergent if
Example. Which of these series are convergent? Which are absolutely convergent ?
∞
X
1.
(−0.8)m
convergent
abs. convergent
m=0
∞
X
1
2.
√
k
k=1
∞
X
1
3.
(−1) j
j
convergent
abs. convergent
convergent
abs. convergent
j=5
234
SECTION 10.6 - ALTERNATING SERIES
Question. Is it possible to have a series that is convergent but not absolutely convergent?
Definition. A series
X
an is called conditionally convergent if
Question. Is it possible to have a series that is absolutely convergent but not convergent?
235
SECTION 10.6 - ALTERNATING SERIES
Review. Which of the following statements are true about a series
∞
X
an?
A. If the series is absolutely convergent, then it is convergent.
B. If the series is convergent, then it is absolutely convergent.
C. Both are true.
D. None of these statements are true.
Question. Which of the following Venn Diagrams represents the relationship between
convergence, absolute convergence, and conditional convergence?
236
SECTION 10.6 - ALTERNATING SERIES
Example. Does this series converge or diverge? If it converges, does it converge
absolutely or conditionally?
∞
X
cos(nπ/3)
n=1
n2
237
SECTION 10.6 - ALTERNATING SERIES
Example. Does the series converge or diverge?
∞
X
cos(n) + sin(n)
n=2
n3
238
§10.7 - RATIO AND ROOT TESTS
§10.7 - Ratio and Root Tests
After completing this section, students should be able to:
• Use the ratio test to determine if a series converges or diverges.
• Use the root test to determine if a series converges or diverges.
• Give an example of a series for which the ratio test and the root test are both
inconclusive.
239
§10.7 - RATIO AND ROOT TESTS
Recall: for a geometric series
P
arn
Theorem. (The Ratio Test) For a series
∞
X
an+1
1. If lim
= L < 1, then
an is
n→∞ an
P
an :
.
n=1
∞
X
an+1
an+1
2. If lim
= L > 1 or lim
= ∞, then
an is
n→∞ an
n→∞ an
.
n=1
∞
X
an+1
= 1, then
3. If lim
an
n→∞ an
.
n=1
240
§10.7 - RATIO AND ROOT TESTS
Example. Apply the ratio test to
∞
X
n2(−10)n
n=1
n!
241
§10.7 - RATIO AND ROOT TESTS
Review. In which of these situations can we conclude that the series
an+1
=0
n→∞ an
an+1
B. lim
= 0.3
n→∞ an
an+1
C. lim
=1
n→∞ an
an+1
= 17
D. lim
n→∞ an
an+1
E. lim
=∞
n→∞ an
P
Review. (The Ratio Test) For a series an :
∞
X
an+1
1. If lim
= L < 1, then
an is
n→∞ an
∞
X
an converges?
A. lim
.
n=1
∞
X
an+1
an+1
= L > 1 or lim
= ∞, then
an is
2. If lim
n→∞ an
n→∞ an
.
n=1
3. If lim
n→∞
an+1
= 1 or DNE , then
an
.
242
§10.7 - RATIO AND ROOT TESTS
Example. Apply the ratio test to
∞
X
(1.1)n
n=1
(2n)!
243
§10.7 - RATIO AND ROOT TESTS
Example. Apply the ratio test to the series
∞
X
n=2
3
n2 − n
244
§10.7 - RATIO AND ROOT TESTS
Extra Example. Apply the ratio test to the series
a1 = 1, an =
245
sin n
an−1
n
§10.7 - RATIO AND ROOT TESTS
Theorem. (The Root Test)
∞
X
p
p
n
n
an
1. If lim |an| = L > 1 or lim |an| = ∞, then
n→∞
n→∞
.
n=1
∞
X
p
n
an
2. If lim |an| = L < 1, then
n→∞
.
n=1
∞
X
p
n
3. If lim |an| = 1, then
an
n→∞
.
n=1
246
§10.7 - RATIO AND ROOT TESTS
Example. Determine the convergence of
∞
X
5n
n=1
nn
247
§10.7 - RATIO AND ROOT TESTS
Rearrangements
P
Definition. A rearrangement of a series an is a series obtained by rearranging its
terms.
P
P
Fact. If an is absolutely convergent with sum s, then any rearrangement of an also
has sum s.
P
But if an is any conditionally convergent series, then it can be rearranged to give a
different sum.
Example. Find a way to rearrange the Alternating Harmonic Series so that the rearrangement diverges.
Example. Find a way to rearrange the Alternating Harmonic Series so that the rearrangement sums to 2.
248
§10.8 - STRATEGY FOR CONVERGENCE TESTS FOR SERIES
§10.8 - Strategy for Convergence Tests for Series
After completing this section, students should be able to:
• Identify appropriate tests to use to prove that a given series converges or diverges.
• Compare and contrast the conditions needed to apply particular convergence tests.
249
§10.8 - STRATEGY FOR CONVERGENCE TESTS FOR SERIES
List as many convergence tests as you can. What conditions have to be satisfied?
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
250
§10.8 - STRATEGY FOR CONVERGENCE TESTS FOR SERIES
Question. The limit comparison test and the ratio test both involve ratios. How are
they different?
251
§10.8 - STRATEGY FOR CONVERGENCE TESTS FOR SERIES
Example. Which convergence test would you use for each of these examples? Carry
out the convergence test if you have time.
∞
X
2n
1.
n3
n=1
∞
X
ln n
2.
(−1)n
n+3
3.
4.
5.
6.
n=1
∞
X
n=1
∞
X
n=1
∞
X
n=1
∞
X
n=1
1
√
3
n2 + 6n
1
1
− n
n! 2
n2
en2
3
n ln n
252
§11.1 - APPROXIMATING SERIES WITH POLYNOMIALS
§11.1 - Approximating Series with Polynomials
Idea: Approximate a function with a polynomial.
Suppose we want to approximate a function f (x) near x = 0. Assume that f’s derivative,
second derivative, third derivative, etc all exist at x = 0.
253
§11.1 - APPROXIMATING SERIES WITH POLYNOMIALS
Review. Let f (x) be a function whose derivatives all exist near x = 0. Suppose that
f (x) can be approximated by a degree 3 polynomial of the form
P3(x) = c0 + c1x + c2x2 + c3x3
in such a way that the function and the polynomial have the same value at x = 0 and
also have the same first through third derivatives at x = 0.
Write an expression for the polynomial coefficient c3 in terms of f (3)(0).
254
§11.1 - APPROXIMATING SERIES WITH POLYNOMIALS
Review. Let f (x) be a function whose derivatives all exist near x = 5. Suppose that
f (x) can be approximated by a degree 4 polynomial of the form
P4(x) = c0 + c1(x − 5) + c2(x − 5)2 + c3(x − 5)3 + c4(x − 5)4
in such a way that the function and the polynomial have the same value at x = 5 and
also have the same first through fourth derivatives at x = 5.
Suppose f (5) = 1, f 0(5) = 3, f 00(5) = 7, f (3)(5) = 13, and f (4)(5) = −11. What are the
coefficients of the polynomial?
255
§11.1 - APPROXIMATING SERIES WITH POLYNOMIALS
Note. For a function f (x) whose derivatives all exist near a, suppose we have a degree n
(n)
00
(n)
polynomial Pn(x) such that Pn(a) = f (a), P0n(a) = f 0(a), P00
n (a) = f (a), · · · Pn (a) = f (a).
If Pn(x) is written in the form c0 + c1(x − a) + c2(x − a)2 + c3(x − a)3 + · · · cn(x − a)n, what
are the coefficients c0, · · · cn in terms of f ?
Definition. For the function f (x) whose derivatives are all defined at x = a, the polynomial of the form
is called the nth degree Taylor polynomial for f , centered at x = a.
In summation notation, the Taylor polynomial can be written as:
256
§11.1 - APPROXIMATING SERIES WITH POLYNOMIALS
We use the conventions that:
• f (0)(a) means
• 0! =
• (x − a)0 =
257
§11.1 - APPROXIMATING SERIES WITH POLYNOMIALS
Example. For f (x) = ln(x),
(a) Find the 3rd degree Taylor polynomial centered at a = 2.
(b) Use it to approximate ln(2.1).
T9 (x)
5
T3 (x)
f (x)
2
-2
4
6
T6 (x)
-5
258
§11.1 - APPROXIMATING SERIES WITH POLYNOMIALS
Example. Find the 7th degree Taylor polynomials for f (x) = sin(x) and g(x) = cos(x),
centered at a = 0.
259
§11.1 - APPROXIMATING SERIES WITH POLYNOMIALS
Example. Find the 4th Taylor polynomial for f (x) = ex centered at a = 0. What is the
error when using it to approximate e0.15?
260
§11.1 - APPROXIMATING SERIES WITH POLYNOMIALS
Example. Use polynomials of order 1, 2, and 3 to approximate
261
√
8.
§11.1 - APPROXIMATING SERIES WITH POLYNOMIALS
Definition. For a function f (x) and its Taylor polynomial Pn(x), the remainder is written
Rn(x) =
Theorem. (Taylor’s Inequality) If | f (n+1)(c)| ≤ M for all c betwen a and x inclusive, then the
remainder Rn(x) of the Taylor series satisfies the inequality
|Rn(x)| ≤
262
§11.1 - APPROXIMATING SERIES WITH POLYNOMIALS
Example. Approximate f (x) = cos(x) by a Maclaurin polynomial of degree 4. Estimate
the accuracy of the approximation when x is in the interval [0, π/2].
263
§11.1 - APPROXIMATING SERIES WITH POLYNOMIALS
Example. Approximate f (x) = cos(x) by a Maclaurin polynomial of degree 4 (again).
For what values of x is the approximation accurate to within 3 decimal places?
Check out the approximation graphically.
264
§11.1 - APPROXIMATING SERIES WITH POLYNOMIALS
Example. How many terms of the Maclaurin series for ex should be used to estimate
e0.5 to within 0.0001?
265
§11.1 - APPROXIMATING SERIES WITH POLYNOMIALS
Extra Example. Approximate f (x) = ex/3 by a Taylor polynomial of degree 2 at a = 0.
Estimate the accuracy of the approximation when x is in the interval [−0.5, 0.5].
266
S11.2 PROPERTIES OF POWER SERIES
S11.2 Properties of Power Series
After completing this section, students should be able to:
• Determine if an expression is a power series.
• Determine the center, radius, and interval of convergence of a power series.
• Create new power series out of old ones by multiplying by a power of x or composing with an expression like 3x2.
• Differentiate and integrate power series.
267
S11.2 PROPERTIES OF POWER SERIES
Informally, a power series is a series with a variable in it (often ”x”), that looks like a
polynomial with infinitely many terms.
Example.
∞
X
(2n + 1)xn
n=0
3n−1
5x2 7x3 9x4 11x5
= 3 + 3x +
+
+
+
+ ···
3
9
27
81
is a power series.
Example.
∞
X
(5n)(x − 6)n
n=0
n!
52(x − 6)2 53(x − 6)3 54(x − 6)4 55(x − 6)5
= 1 + 5(x − 6) +
+
+
+
+ ···
2!
3!
4!
5!
is a power series centered at 6.
268
S11.2 PROPERTIES OF POWER SERIES
Definition. A power series centered at a is a series of the form
∞
X
cn(x − a)n =
n=0
where x is a variable, and the cn’s are constants called coefficients, and a is also a
constant called the center .
Definition. A power series centered at zero is a series of the form
∞
X
cn xn =
n=0
269
S11.2 PROPERTIES OF POWER SERIES
Example. For what values of x does the power series
∞
X
n=0
270
n! (x − 3)n converge?
S11.2 PROPERTIES OF POWER SERIES
Example. For what values of x does the power series
∞
X
(−2)n(x + 4)n
n=0
271
n!
converge?
S11.2 PROPERTIES OF POWER SERIES
Example. For what values of x does the power series
∞
X
(−5x + 2)n
n=1
END OF VIDEO
272
n
converge?
S11.2 PROPERTIES OF POWER SERIES
Review. Which of the following are power series?
1 (x + 1) (x + 1)2 (x + 1)3 (x + 1)4
A. +
+
+
+
+ ···
2
5
8
11
14
B.
1 1
2
3
4
+
+
1
+
x
+
x
+
x
+
x
+ ···
x2 x
C. 1 + 3 + 32 + 33 + 34 + · · ·
D. None of these.
Example. Find the center of any power series above.
273
S11.2 PROPERTIES OF POWER SERIES
Example. Find the center of the power series
∞
X
n=1
it converge?
1 n
Hint: lim 1 +
= e.
n→∞
n
274
nn(7 + 3x)n. For what values of x does
S11.2 PROPERTIES OF POWER SERIES
∞
X
(−5)n(2x − 3)n
Example. Find the center of the power series
. For what values of x
√
3n + 1
n=0
does it converge?
275
S11.2 PROPERTIES OF POWER SERIES
∞
X
x2n
Example. For what values of x does the power series
converge?
(3n)!
n=0
276
S11.2 PROPERTIES OF POWER SERIES
Theorem. For a given power series
gence:
∞
X
cn(x − a)n, there are only three possibilities for conver-
n=0
1.
2.
3.
Definition. The radius of convergence is
1.
2.
3.
Definition. The interval of convergence is the interval of all x-values for which the
power series converges.
1.
2.
3.
277
S11.2 PROPERTIES OF POWER SERIES
Question. If the interval of convergence of a power series has length 6, then the radius
of convergence of the power series is:
Question. Which of the following could NOT be the interval of convergence for a
power series?
A. (−∞, ∞)
B. (−4, 1]
C. (0, ∞)
D. [ 29 , 100
3 ]
P∞
Question. If the series n=1 cn5n converges, which of the following definitely converges? (The cn represent real numbers.)
P
n
A. ∞
n=1 cn (−3)
P∞
B. n=1 cn(−5)n
P
n
C. ∞
n=1 cn (−7)
D. None of these.
278
S11.2 PROPERTIES OF POWER SERIES
We can think of power series as functions.
∞
X
Example. Consider f (x) =
xn =
n=0
1. What is f ( 31 )?
2. What is the domain of f (x)?
3. What is a closed form expression for f (x)?
4. What is the domain for the closed form expression?
279
S11.2 PROPERTIES OF POWER SERIES
We can think of the partial sums of
with polynomials:
∞
X
xn as a way to approximate the function
n=0
s0 =
s1 =
s2 =
s3 =
sn =
280
1
1−x
S11.2 PROPERTIES OF POWER SERIES
Example. Express
2
as a power series and find the interval of convergence.
x−3
281
S11.2 PROPERTIES OF POWER SERIES
Example. Find a power series representation of
END OF VIDEO
282
x
1 + 5x2
S11.2 PROPERTIES OF POWER SERIES
Review.
1
1−x
Question.
can be represented by the power series:
1
1−x
is equal to its power series:
A. when x , 1
B. when x < 1
C. when −1 < x < 1
D. for all real numbers
E. It is never exactly equal to its power series, only approximately equal.
283
S11.2 PROPERTIES OF POWER SERIES
Example. Express each of the following functions with a power series.
1
1.
1 − x4
2.
1
1 + x4
x3
3.
1 + x4
284
S11.2 PROPERTIES OF POWER SERIES
Example. Find a power series representation of f (x) =
gence.
285
3
2+5x .
Find its radius of conver-
S11.2 PROPERTIES OF POWER SERIES
Summary: It is possible to make new power series out of old by:
•.
•.
•.
286
S11.2 PROPERTIES OF POWER SERIES
Differentiation and Integration
Recall how to differentiate and integrate polynomials:
d
[5 + 3(x − 2) + 4(x − 2)2 + 8(x − 2)3] =
dx
...
Z
5 + 3(x − 2) + 4(x − 2)2 + 8(x − 2)3 dx =
Power series are also very easy to differentiate and integrate!
Theorem. If the power series
f (x) = c0 + c1(x − a) + c2(x − a) + c3(x − a) + c4(x − a) · · · =
2
3
4
∞
X
cn(x − a)n
n=0
has a radius of convergence R > 0, then f (x) is differentiable on the interval (a − R, a + R) and
(i) f 0(x) =
Z
(ii)
f (x) dx =
The radius of convergence of the power series in (i) and (ii) are both R.
287
S11.2 PROPERTIES OF POWER SERIES
Question. Why does the summation sign for the derivative of a power series start at
n = 1 instead of n = 0?
Question. If a power series converges at the endpoints of its interval of convergence,
do the derivative and integral power series also converge at the endpoints?
288
S11.2 PROPERTIES OF POWER SERIES
Example. Find a power series representation for ln |x + 2| and find its radius of convergence.
289
S11.2 PROPERTIES OF POWER SERIES
2
1
Example. Find a power series representation for
.
4x − 1
290
S11.2 PROPERTIES OF POWER SERIES
Z
Example. Find a power series representation for
Z 1
x
dx, accurate to two decimal places.
3
8
+
x
0
291
x
dx and use it to approximate
8 + x3
S11.2 PROPERTIES OF POWER SERIES
Summary:
∞
X
1
1 + x + x2 + x3 + · · ·
with the power series
• We started by representing the function
1−x
n=0
∞
X
1
= 1 + x + x2 + x3 + · · · as a template to find the
1−x
n=0
power series for many other related functions, by:
• We used the equation
–
–
–
–
–
• These same techniques can be used with other templates to build new power series
out of old ones.
292
§11.3 - TAYLOR SERIES
§11.3 - Taylor Series
After completing this section, students should be able to:
• Use the definition of Taylor series to find a Taylor series for a function and write it
in summation notation.
• Determine the interval of convergence for a Taylor series.
• Build new Taylor series out of old by substituting a power of x, or multiplying by
a power of x, differentiating, or integrating.
• Use the binomial series to approximate square roots and other roots.
• Prove that the MacLaurin series for ex actually converges to ex, and likewise for the
Maclaurin series for sin(x) and cos(x) and closely related series like sin(2x).
293
§11.3 - TAYLOR SERIES
Definition. Suppose a function f (x) has derivatives f (k)(a) of all orders at the point a.
The power series
of f (x) centered at a.
is called the
We use the conventions that:
• f (0)(a) means
• 0! =
• (x − a)0 =
Definition. The power series
for f (x).
is called the
294
§11.3 - TAYLOR SERIES
Question. What is the difference between a Taylor series and a Maclaurin series?
Question. What is the difference between a Taylor series and a Taylor polynomial?
295
§11.3 - TAYLOR SERIES
Example. Find the Taylor series for f (x) =
1
centered at a = 5.
x
296
§11.3 - TAYLOR SERIES
Example. Find the Maclaurin series for f (x) = sin(x) and g(x) = cos(x). Find the radius
of convergence.
297
§11.3 - TAYLOR SERIES
Question. If a function has derivatives of all orders at x = a, then it is possible to
write down the Taylor series for f centered at a. But how do we know that it actually
converges to f ?
Note. The Taylor series for f centered at a converges to f on an interval I if and only if
...
Question. Does the power series of sin(x) actually converge to sin(x) on its radius of
convergence?
298
§11.3 - TAYLOR SERIES
Example. Find the Maclaurin series for f (x) = ex. What is the radius of convergence?
299
§11.3 - TAYLOR SERIES
Example. Use the Maclaurin series for f (x) = ex to find the Maclaurin series for g(x) =
2
x3e−x .
300
§11.3 - TAYLOR SERIES
Example. Find the Taylor series for f (x) = (1 + x)π centered at x = 0.
301
§11.3 - TAYLOR SERIES
Definition. The expression
pronounced
Note.
p(p − 1)(p − 2) . . . (p − n + 1)
is written as
n!
, and is also called a
,
.
p
0
Example. Write the Taylor series for f (x) = (1 + x)π using choose notation.
Definition. The binomial series is the Maclaurin series for (1 + x)p, where k is any real
number. That is, the binomial series is the series:
(1 + x)p =
This series converges when
.
302
§11.3 - TAYLOR SERIES
Example. Find the Maclaurin series for
√ 1
.
1+2x3
303
§11.3 - TAYLOR SERIES
Example. Find a Maclaurin series for f (x) =
304
1
1−x
§11.3 - TAYLOR SERIES
Question. Is it possible for a function to be represented by two different power series
∞
∞
X
X
with the same center? That is, if f (x) =
cn(x − a)n =
dn(x − a)n, does it necessarily
follow that cn = dn for all n?
n=1
n=1
305
§11.3 - TAYLOR SERIES
∞
X
5
5
5
Extra Example. If P(x) =
(x − 2)n = 5 + (x − 2) + (x − 2)2 + · · ·, find P000(2).
n!
1!
2!
n=0
A. 5
5
B.
2!
5
C.
3!
5 · 23
D.
3!
E. None of these.
Extra Example. Find a power series P(x) such that P(n)(5) = n for all n ≥ 0.
∞
X
n(x − 5)n
A.
B.
C.
n=1
∞
X
n=1
∞
X
n=1
(x − 5)n
(n − 1)!
(x − 5)n
n!
D. None of these
306
§11.3 - TAYLOR SERIES
S11.4 Working with Taylor Series
After completing this section, students should be able to
• Use Taylor series to find limits.
• Use Taylor series to compute approximate values of integrals.
• Use Taylor series to find the sum of a series.
• Use Taylor series to solve differential equations.
• List uses of Taylor series.
307
§11.3 - TAYLOR SERIES
Question. What are Taylor series good for?
•
•
•
•
•
•
308
§11.3 - TAYLOR SERIES
2
e−x − 1 + x2
Example. Use a Taylor series to evaluate lim
x→0
x4
309
§11.3 - TAYLOR SERIES
Example. Use Taylor series to prove L’Hospital’s Rule.
310
§11.3 - TAYLOR SERIES
Example. 1. Find a power series representation for e
2. Find a power series representation for
R
e
2
− x2
2
− x2
.
dx.
1
3. Use the first three terms of your power series to estimate √
2π
4. What does this number represent?
311
Z
1
2
− x2
e
−1
dx.
§11.3 - TAYLOR SERIES
Example. Use the MacLaurin series for arctan(x) to show that
π
1 − 1/3 + 1/5 − 1/7 + · · · =
4
312
§11.3 - TAYLOR SERIES
Example. Use a MacLaurin series from this table to find the sum of the Alternating
Harmonic Series.
313
§11.3 - TAYLOR SERIES
Example. Find a power series for the solution of the differential equation. Can you
guess what function this power series represents?
y0(t) = 6y + 9
y(0) = 2
314
§11.3 - TAYLOR SERIES
Example. Find the Maclaurin series for g(x) = eix, where i =
315
√
−1.
§11.3 - TAYLOR SERIES
Summary: What are Taylor Series good for?
316
§12.1 - PARAMETRIC EQUATIONS
§12.1 - Parametric Equations
Definition. A cartesian equation for a curve is an equation in terms of x and y only.
Definition. Parametric equations for a curve give both x and y as functions of a third
variable (usually t). The third variable is called the parameter.
Example. Graph x = 1 − 2t, y = t2 + 4
t
-2
-1
0
Find a Cartesian equation for this curve.
317
x
5
3
y
8
5
§12.1 - PARAMETRIC EQUATIONS
Example. Plot each curve and find a Cartesian equation:
1. x = cos(t), y = sin(t), for 0 ≤ t ≤ 2π
2. x = cos(−2t), y = sin(−2t), for 0 ≤ t ≤ 2π
3. x = cos2(t), y = cos(t)
318
§12.1 - PARAMETRIC EQUATIONS
Example. Write the following in parametric equations:
√
1. y = x2 − x for x ≤ 0 and x ≥ 1
2. 25x2 + 36y2 = 900
319
§12.1 - PARAMETRIC EQUATIONS
Example. Describe a circle with radius r and center (h, k):
a) with a Cartesian equation
b) with parametric equations
320
§12.1 - PARAMETRIC EQUATIONS
Review. Cartesian equations are ...
Parametric equations ...
Review. Which of the following graphs represents the graph of the parametric equations x = cos t, y = sin t. (The horizontal axis is the x-axis and the vertical axis is the
y-axis.)
A.
C.
B.
321
§12.1 - PARAMETRIC EQUATIONS
Example. Find a Cartesian equation for the curve.
√
1. x = 5 t, y = 3 + t2
Methods:
2. x = et, y = e−t
3. x = 5 cos(t) + 3, y = 2 sin(t) − 7
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§12.1 - PARAMETRIC EQUATIONS
Example. Find parametric equations for the curve.
1. x = −y2 − 6y − 9
Methods:
2. 4x2 + 25y2 = 100
3. 4(x − 2)2 + 25(y + 1)2 = 100
323
§12.1 - PARAMETRIC EQUATIONS
Example. What is the equation for a circle of radius 8 centered at the point (5, -2)
1. in Cartesian coordinates ?
2. in parametric equations?
324
§12.1 - PARAMETRIC EQUATIONS
Example. Find parametric equations for a line through the points (2, 5) and (6, 8).
1. any way you want.
2. so that the line is at (2, 5) when t = 0 and at (6, 8) when t = 1.
325
§12.1 - PARAMETRIC EQUATIONS
Example. Use the graphs of x = f (t) and y = g(t) to sketch a graph of y in terms of x.
326
§12.1 - PARAMETRIC EQUATIONS
Extra Example. The graphs of x = f (t) and y = g(t) are shown above. Select the graph
of the parametric curve described by these equations.
A.
B.
C.
D.
327
§12.1 - PARAMETRIC EQUATIONS
Example. A sailboat’s position at time t is given by the equations x = 3 − t, y = 2 − 4t.
A rowboat’s position is give by the equations x = 5 − 3t, y = −2 + t.
1. Do the boats collide?
2. Do the boats’ paths cross?
328
§12.1 - PARAMETRIC EQUATIONS
ARC LENGTH
Example. Find the length of this curve.
329
§12.1 - PARAMETRIC EQUATIONS
Note. In general, it is possible to approximate the length of a curve x = f (t), y = g(t)
between t = a and t = b by dividing it up into n small pieces and approximating each
curved piece with a line segment.
Arc length is given by the formula:
330
§12.1 - PARAMETRIC EQUATIONS
Set up an integral to express the arclength of the Lissajous figure
x = cos(t), y = sin(2t)
.
331
§12.1 - PARAMETRIC EQUATIONS
Review. The length of a parametric curve x = f (t), y = g(t) from t = a to t = b is given
by:
Example. Find the exact length of the curve x = cos(t) + t sin(t), y = sin(t) − t cos(t),
from the point (1, 0) to the point (−1, π).
332
§12.1 - PARAMETRIC EQUATIONS
Example. Write down an expression for the arc length of a curve given in Cartesian
coordinates: y = f (x).
2
Example. Find the arc length of the curve y = 12 ln(x) − x4 from x = 1 to x = 3.
333
§12.1 - PARAMETRIC EQUATIONS
The Arclength Function
Recall that the arclength of a curve x = f (t), y = g(t) from t = a to t = b is given by:
If we fix the t-value where the curve starts (t = a), but vary the t-value where the curve
ends (t = b), we can think of this as a function of b:
Often, this is written as a function of t instead of b by replacing b by t and using a
different variable (like s) in the integrand.
334
§12.1 - PARAMETRIC EQUATIONS
TANGENT LINES
The slope of the tangent line for a curve y = p(x) (given in Cartesian coordinates) is:
If the curve is given by parametric equations x = f (t), y = g(t), then the slope of its
tangent line is:
335
§12.1 - PARAMETRIC EQUATIONS
Example. For the Lissajous figure:
x = cos(t), y = sin(2t)
1. Find the slopes of the tangent lines at the center point (0, 0).
2. Find where the tangent line is horizontal.
336
§12.1 - PARAMETRIC EQUATIONS
Review. The slope of the tangent line for a parametric curve x = f (t), y = g(t) is given
by:
Example. The graph of the curve x(t) = 2 cos(t) + cos(2t), y(t) = sin(2t) for 0 ≤ t ≤ 2π is
drawn below.
1. Find the equations of the tangent lines at the point (−1, 0) on the curve.
2. Find the coordinates of all the points on the curve where the tangent line is vertical.
337
§12.2 POLAR COORDINATES
§12.2 Polar Coordinates
Cartesian coordinates: (x, y)
Polar coordinates: (r, θ), where r is:
.
and θ is:
Example. Plot the points, given in polar coordinates.
1. (8, − 2π
3 )
2. (5, 3π)
3. (−12, π4 )
Note. A negative angle means to go clockwise from the positive x-axis. A negative
radius means jump to the other side of the origin, that is, (−r, θ) means the same point
as (r, θ + π)
338
§12.2 POLAR COORDINATES
Note. To convert between polar and Cartesian coordinates, note that:
•x=
• y=
•r=
• tan θ =
Example. Convert (5, − π6 ) from polar to Cartesian coordinates.
Example. Convert (−1, −1) from Cartesian to polar coordinates.
339
§12.2 POLAR COORDINATES
Review. Points on the plane can be written in terms of Cartesian coordinates (x, y) or
in terms of polar coordinates (r, θ)
where r represents ...
and θ represents ...
The quantities x and y and r and θ are related by the equations ...
Review. Convert the point P = (4, −2π
3 ), which is in polar coordinates, to Cartesian
coordinates.
A.
B.
√
1
( 2 , 23 )
√
1
(− 2 , − 23 )
√
C. (−2, 2 3)
√
D. (−2, −2 3)
E. None of these.
340
§12.2 POLAR COORDINATES
√
Review. Convert the point P = (− 3, 3), which is in Cartesian coordinates, to polar
coordinates. (More than one answer may be correct.)
A. (1, π3 )
√ π
B. (2 3, 3 )
√
C. (2 3, − π3 )
√ 2π
D. (2 3, 3 )
√ −π
E. (−2 3, 3 )
341
§12.2 POLAR COORDINATES
Example. Plot the following curves and rewrite using Cartesian coordinates.
A. r = 7
B. θ = 1
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§12.2 POLAR COORDINATES
Example. Plot the following curves and rewrite the first one using Cartesian coordinates.
C. r = 12 cos(θ)
D. r = 6 + 6 cos(θ) (an example of a limacon)
343
§12.2 POLAR COORDINATES
Example. Describe the regions using polar coordinates.
344
§12.2 POLAR COORDINATES
Example. Convert the Cartesian equations to polar coordinates:
1. 4y2 = x
2. y = x
3. x2 + (y − 1)2 = 1
345
§12.2 POLAR COORDINATES
Example. Match the polar equations with the graphs.
1
2
3
4
5
6
(a) r = ln(θ)
(b) r = θ2
(c) r = cos(3θ)
(d) r = 2 + cos(3θ)
θ
(e) r = cos( )
2
3θ
(f) r = 2 + cos( )
2
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§10.4 AREA IN POLAR COORDINATES
§10.4 Area in polar coordinates
Goal: Find a formula for the area of a region whose boundary is given by a polar
equation r = f (θ).
Step 2: Find a formula for a sector of a circle.
347
§10.4 AREA IN POLAR COORDINATES
Step 2: Divide our polar region with boundary r = f (θ) into slivers ∆A that are
approximately sectors of circles.
Step 3: Approximate the total area with a Riemann sum.
Step 4: Take the limit of the Riemann sum to get an integral.
348
§10.4 AREA IN POLAR COORDINATES
Example. Find the area inside one leaf of the flower r = sin(2θ)
349
§10.4 AREA IN POLAR COORDINATES
Extra Example. Find the area of the region that lies inside both flowers: r = sin(2θ)
and r = cos(2θ)
350
§4.4 - L’HOSPITAL’S RULE
§4.4 - L’Hospital’s Rule
ln(x)
Example. lim √
x→∞ 3 x
Example. lim+ sin(x) ln(x)
x→0
351
§4.4 - L’HOSPITAL’S RULE
ex
Example. lim+
x→5 x − 5
Example. lim+ xx
x→0
352
§4.4 - L’HOSPITAL’S RULE
Example. lim ln(x2 − 1) − ln(x5 − 1)
x→∞
Tips for using L’Hopital’s Rule:
353
§4.4 - L’HOSPITAL’S RULE
Form
Example
What to do
354
§9.1, 9.2, 9.3 - DIFFERENTIAL EQUATIONS
§9.1, 9.2, 9.3 - Differential Equations
Differential equations are equations that involve functions and their derivatives. For
example,
√
dy
1. dx = x
2. y0 = 1 + y2
3.
d2 y
dx2
0
= −4y
4. y = x + y
Solving a differential equation means to find all functions y = f (x) that satisfy it.
Sometimes it is useful to find a particular solution, with a given initial condition, such
as y(2) = 5.
355
§9.1, 9.2, 9.3 - DIFFERENTIAL EQUATIONS
Example.
dy
dx
√
= x
1. Solve this differential equation.
2. How do you know you have found all solutions?
356
§9.1, 9.2, 9.3 - DIFFERENTIAL EQUATIONS
Example. y0 = 1 + y2
1. Verify that y = tan(x) is a solution to this equation.
2. Is y = tan(x) + 3 a solution?
3. Is y = 3 tan(x) a solution?
4. Is y = tan(x + 3) a solution?
5. Find a solution that satisfies the initial condition y(0) = 1.
357
§9.1, 9.2, 9.3 - DIFFERENTIAL EQUATIONS
Example. ”Separate” the differential equation by moving all y’s to the left side and all
x’s to the right side, to find all solutions to the equation
y0 = 1 + y2
358
§9.1, 9.2, 9.3 - DIFFERENTIAL EQUATIONS
Example. Find a solution the equation
dy
= xy2
dx
1. with the initial condition y(0) = 4.
2. with the initial condition y(1) = 0.
359
§9.1, 9.2, 9.3 - DIFFERENTIAL EQUATIONS
Definition. An equation of the form
dy
= g(x) f (y)
dx
is called a separable differential equation.
Equivalently, an equation of the form
dy g(x)
=
dx h(y)
is called a separable differential equation. Here, f (y) =
1
h(y)
Separable differentiable equations can be solved by moving expressions with y’s in
them to the left side of the equals sign and expressions with x’s in them on the right
and integrating both sides:
360
§9.1, 9.2, 9.3 - DIFFERENTIAL EQUATIONS
Example. Which of these equations are separable?
x
1. y0 = √
y
2. y0 = x + y
3. y0 = yex+y
4. y0 = ln(xy)
5. y0 = ln(x y)
xy + y
6. y0 =
2x − 3xy
7. y0 = xy − 2x + y − 2
361
§9.1, 9.2, 9.3 - DIFFERENTIAL EQUATIONS
Example.
d2 y
dx2
= −4y
1. Show that an equation of this form describes the motion of a spring.
2. Find as many solutions as possible for this equation.
362
§9.1, 9.2, 9.3 - DIFFERENTIAL EQUATIONS
Example. y0 = x + y
1. This equation is harder to solve or guess solutions for, but we can get approximate
solutions by plotting the “slope field”.
x
y
y0 (note: y0 = x + y)
363
§9.1, 9.2, 9.3 - DIFFERENTIAL EQUATIONS
Slope field for y0 = x + y
2. Sketch some curves whose tangent lines fall on this slope field.
3. Sketch an approximate solution to the differential equation that satisfies the initial
condition y(−1) = 1.
364
§9.1, 9.2, 9.3 - DIFFERENTIAL EQUATIONS
Example. For each situation, set up a differential equation. If you have extra time at
the end, you can solve the equations.
1. The rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. Write a differential equation to describe the
temperature of a cup of coffee that starts out at 90◦ C and is in a 20◦ room.
2. A population is growing at a rate proportional to the population size .
3. The logistic population model assumes that there is a maximum carrying capacity
of M and that the rate of change of the population is proportional to the product
of the population and the fraction of the carrying capacity that is left.
365