Math 232 Calculus 2 - Fall 2018 · x6.1 - 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

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 afunction 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 ofthat quantity over time.

• Find an equation for velocity and position from an equation for acceleration plusinitial conditions.

• Find an equation for the amount of a quantity from an equation for its rate ofchange plus an initial condition.

2


Example. A squirrel is running up and down a tree. The height of the squirrel fromthe ground over time is given by the function s(t) graphed below, where t is in secondsand 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


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 ismy displacement and what is my distance traveled?

4


Example. A swimmer is swimming left and right in a long narrow pool. Her velocityover time is given by the following graph, where velocity v(t) is in meters per secondand time t is in seconds.

Here, distance is measured from the left end of the pool, so a positive velocity meansand a negative velocity means .

A. Describe the swim. Was the swimmer swimming at a constant speed? When wasthe swimmer swimming left vs. right? At what time(s) did the swimmer turnaround?

5


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


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


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 attime t.

8


Example. Suppose f (t) represents the rate of change of a quantity over time (e.g. therate of water flowing out of a resevoir). Then

•∫ b

af (t) dt represents ...

• If F(0) is the amount of the quantity at time 0, then F(0) +∫ b

af (t) dt represents ...

•∫ b

a| f (t)| dt represents ...

9


Example. The population of bacteria is changing at a rate of f (t) = e−t − 1/e. What isthe net change in population between time t = 0 and time t = 2?

10


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) = 2and s(0) = −1. Find the velocity and position functions. What is its displacement inthe first 2 seconds? How much total distance did it travel in the first 2 seconds.

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§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.

12


Recall: to compute the area below a curve y = f (x), between x = a and x = b, we candivide 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


To compute the area between the curves y = f (x) and y = g(x), between x = a andx = b, we can divide up the region into rectangles.


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


Example. Find the area between the curves y = x2 + x and y = 3 − x2

15


Review. The area between two curves y = f (x) and y = g(x) between x = a and x = bis given by:

16


Review. The area between the curves y = 2x + 1 and y = 5 − 2x2 is given by:

A.∫ 1−2

2x + 1 − 5 + 2x2 dx

B.∫ 1−2

5 − 2x2 − 2x + 1 dx

C.∫ 1−2

5 − 2x2 − 2x − 1 dx

D.∫ 5−3

5 − 2x2 + 2x + 1 dx

E. None of these.

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Example. The shaded area between the curves y = cos(5x), y = sin(5x), x = 0, andx = π4 is given by:

A.∫ π/4

0sin(5x) − cos(5x) dx

B.∫ π/4

0cos(5x) − sin(5x) dx

C. Both of these answers are correct.

D. Neither of these answers are correct.

18


Extra Example. Set up the integral to find the shaded area bounded by the three curvesin the figure shown.

• f (x) = x2 − x − 6• g(x) = x − 3• h(x) = −x2 + 4

19


Note. The area between two curves x = f (y) and x = g(y) between y = c and y = d isgiven by:

This formula works as long as f (y) g(y).

20


To compute the area between the curves x = f (y) and x = g(y), between y = c andy = d, we can again divide up the region into rectangles.


The approximate area between the two curves is

The exact area between the two curves

21


Example. Find the area between the curves f (y) = sin(y)+5, g(y) =y2

√36 + y3

6, y = −2,

and y = 2.

22


Example. The area between the curves y = x2 and y = 3x2, and y = 4 is given by:

A.∫ 2

0x2 − 3x2 dx

B.∫ 2

03x2 − x2 dx

C.∫ 2

0

√y −

√y3

dy

D.∫ 4

0

√y −

√y3

dy

E.∫ 4

0

√y3− √y dy

23


Extra Example. In the year 2000, the US income distribution was: (data from WorldBank, see http://wdi.worldbank.org/table/2.9)

Income Category Fraction of Fraction of Cumulative CumulativePopulation Total Income Fraction of Fraction of

Population IncomeBottom 20% 0.20 0.05 0.20 0.05

2nd 20% 0.20 0.11 0.40 0.163th 20% 0.20 0.16 0.60 0.324th 20% 0.20 0.22 0.80 0.54

Next 10% 0.10 0.16 0.90 0.70Highest 10% 10 0.30 1.00 1.00

The Lorenz curve plots the cumulative fraction of population on the x-axis and thecumulative fraction of income received on the y-axis.

The Gini index is the area between the Lorenz curve and the line y = x, multiplied by2.Estimate the Gini index for the US in the year 2000 using the midpoint rule.

24

http://wdi.worldbank.org/table/2.9


Extra Example. Find the area between the curves 2x = y2 − 4 and y = −3x + 2 that liesabove the line y = −1

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§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.

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§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 =

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§6.2 - VOLUMES

Example. Find the volume of the solid whose base is the ellipsex2

4+

y2

9= 1 and whose

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 or theshape of a .

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 weintegrate with respect to .When the region is rotated around the y-axis, or any other vertical line, then weintegrate with respect to .

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§6.2 - VOLUMES

Example. Consider the region bounded by the curve y = 3√

x, the x-axis, and the linex = 8. What is the volume of the solid of revolution formed by rotating this regionaround the x-axis?

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§6.2 - VOLUMES

Example. Consider the region in the first quadrant bounded by the curves y = 3√

x andy = 14x. What is the volume of the solid of revolution formed by rotating this regionaround the x-axis? The y-axis?

END OF VIDEO

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§6.2 - VOLUMES

Review. Suppose a 3-dimensional solid can be sliced perpendicular to the x-axis andthe slice at position x has area given by the function A(x). Then the volume is givenby:

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

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§6.2 - VOLUMES

Example. The region between the curves y = ex, x = 0, and y = e3 is rotated aroundthe x-axis, to make a solid of revolution. When computing the volume, what are thecross-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.

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§6.2 - VOLUMES

Example. The region between the curve y = ex, x = 0, and y = e3 is rotated aroundthe y-axis, to make a solid of revolution. When computing the volume, what are thecross-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 = 5instead of the y-axis.

35

§6.2 - VOLUMES

Extra Example. Consider the region bounded by y = 6x2 , x = 1, x = 2, and the x-axis.

Set up an integral to compute the volume of the solid obtained by rotating this regionabout the line x = 12.

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§6.2 - VOLUMES

Example. Find the volume of the solid whose base is the region between y =√

x, thex-axis, and the lines x = 1 and x = 5, and whose cross sections perpendicular to thex-axis are equilateral triangles.

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§6.2 - VOLUMES

Example. Find the volume of the solid whose base is the region between y =√

x, thex-axis, and the lines x = 1 and x = 5, and whose cross sections perpendicular to they-axis are equilateral triangles.

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§6.2 - VOLUMES

Extra Example. Find the volume of a pyramid with a square base of side length a andheight h.

39

§6.2 - VOLUMES

Extra Example. Find the volume of a cone with a circular base of radius a and heighth.

40

§6.2 - VOLUMES

Extra Example. Set up an integral to find the volume of a bagel, given the dimensionsbelow.

41

§6.5 - ARCLENGTH

§6.5 - Arclength


• Explain the relationship between the formula for arc length and the distance for-mula.

• 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) betweenx = a and x = b by dividing it up into n small pieces and approximate each curvedpiece 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 isgiven by the formula:

Example. Set up an integral to calculate the arc length of the curve y =√

x betweenx = 0 and x = 3.

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§6.5 - ARCLENGTH

Example. Find a function a(t) that gives the length of the curve y = ex+e−x

2 between x = 0and x = t.

47

§6.5 - ARCLENGTH

Note. Although arc length integrals are usually straightforward to set up, the squareroot sign makes them notoriously difficult to evaluate, and sometimes impossible toevaluate.

48

§6.6 - SURFACE AREA

§6.6 - Surface Area


• Identify the components of the formula for the area of a surface of revolution thatcorrespond to circumference and slant height.

• Compute the area of a surface of revolution.

49


How could you calculate the surface area 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


To find the surface area of a surface of revolution, imagine approximating it with piecesof cones.

We will need a formula for the area of a piece of a cone.

51


The area of this piece of a cone isA = 2πr`

where r =r1 + r2

2is the average radius and ` is the length along the slant. (See textbook

for derivation.)

52


Use the formula for the area of a piece of cone A = 2πr` to derive a formula for surfacearea.

53


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 formulasbe?

54


Example. Find the surface area of the surface of revolution formed by rotating aboutthe x-axis the curve y =

√x between x = 0 and x = 2.

55


Example. Find the surface area when the curve y =√

x between x = 0 and x = 2 isrotated around the y=axis.

56


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


• Use integration to calculate the work done when a varying force, given by afunction, 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


Definition. If if a constant force F is applied to move an object a distance d, then thework done to move the object is defined to be

Question. What are the units of force? What are the units of work?

59


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 5feet high?

Example. How much work is done to lift a 0.9 kg book off the floor onto a shelf that is1.5 meters high?

60


Example. A particle moves along the x axis from x = a to x = b, according to a forcef (x). How much work is done in moving the particle? (Note: the force is not constant!)

61


Example. How much work is required to lift a 1000-kg satellite from the earth’s surfaceto an altitude of 2 · 106 m above the earth’s surface?The gravitational force is F =

GMmr2

, M is the mass of the earth, m is the mass of thesatellite, and r is the distance between the satellite and the center of the earth, and G isthe 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


Review. In the expression W =∫ b

aF(x) dx what do W, F(x), and dx represent?

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 (inmetric 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 (inEnglish units), you need to multiply by 32 f t/s2.

C) Both.

D) Neither.

63


Example. In the alternative universe of the Golden Compass, the souls of humans andtheir animal companions, called daemons, are closely tied. Suppose that the forceneeded to separate a human and its daemon is given by f (x) = 10xe−x

2/1000 pounds,where x represents the distance between the human and the daemon in feet. Lyra andher daemon are currently 5 feet apart. How much work will it take to separate theman additional 5 feet?

64


Example. A 200-kg cable is 300 m long and hangs vertically from the top of a tallbuilding. 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


Example. An aquarium has a square base of side length 4 meters and a height of 3meters. The tank is filled to a depth of 2 m How much work will it take to pump thewater out of the top of the tank through a pipe that rises 0.5 meters above the top ofthe tank?

66


Extra Example. A bowl is shaped like a hemisphere with radius 2 feet, and is full ofwater. How much work will it take to pump the water out of the top of the bowl? Usethe fact that water weights 62.5 pounds per cubic foot.

67


Springs follow Hooke’s law: the force required to stretch them a distance x past theirequilibrium position is given by f (x) = kx, where k is a constant that depends on thespring.

Example. A spring with natural length 15 cm exerts a force of 45 N when stretched toa 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 cmbeyond its natural length.

3. Set up the integral/s needed to find the work done in stretching the spring from alength of 20 cm to a length of 25 cm.

68


Other Applications - Mass from DensityExample. Find the mass of a wire that lies along the x-axis if the density of the wire at

position x is given by ρ(x) =1

4 − x for 0 ≤ x ≤ 3.

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Other Applications - Force on a Dam

If you are standing under water, the pressure from a column of water above your headis:

This pressure is the same in all directions, so the pressure on a vertical wall of theswimming pool is:

The force of water on a strip of a vertical dam is given by:

70


The force of water on a vertical dam is give by:

71


Example. Find the total force on the face of this vertical dam, assuming that the waterlevel 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 divideby n:

For a continuous function f (x) on an interval [a, b], we could estimate the averagevalue 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 = limn→∞

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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 theinterval b − a is analogous to dividing by .Example. Find the average value of the function g(x) =

11 − 5x on the interval [2, 5].

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


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

[a, b], there is a number c with a ≤ c ≤ b such that f (c) =∫ b

a f (x)dx

b − a .

Proof:

75


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


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 tothe area under the curve from 0 to π.

77


Example. Suppose g(x) is a continuous function and∫ 5

2g(x) dx = 12. Which of the

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.

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Extra Example. The temperature on a July day starts at 60◦ at 8 A.M., and rises (withoutever 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.?

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§8.1 - INTEGRATION REVIEW

§8.1 - Integration Review


1. Compute an integral of a function, like∫

sec2(x) dx, by recalling the antiderivativeof 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.

4. Use u-substitution for integrals like∫

x√

1 + x dx in which x must be rewritten interms of u.

80


Example.∫

cos4(θ) sin(θ) dθ

Example.∫ 3

2

dy5 − 3y

81


Example.∫

1x−1 + 1

dx

Example.∫

1 + x4 + x2

dx

82

§8.2 - INTEGRATION BY PARTS

§8.2 - Integration by Parts


• Use integration 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.• Use integration by parts to compute integrals like

∫arctan(x) dx, by using 1dx as

du

• Use integration by parts to compute integrals like∫

sin(x)ex dx, in which the in-tegrands cycle around, and it possible to solve for the integral without ever fullycomputing it.

83


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 termsof something that is hopefully easier to integrate. Integrating using this method iscalled:

84


Example. Find∫

xex dx.

85


Review. ∫u dv =

Example. Integrate∫

t sec2(2t) dt using integration by parts. What is a good choice foru and what is a good choice for dv?

86


Example. Find∫

x(ln x)2dx

87


Example. Integrate∫ 2

1arctan(x)dx.

88


Example. Find∫

e2x cos(x)dx.

89


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? (Morethan one answer is correct.)

A.∫

x3 dx

B.∫

ln(x) dx

C.∫

x2ex dx

D.∫

xex2 dx

E.∫

ln y√

ydy

90

§8.3 - INTEGRATING TRIG FUNCTIONS

§8.3 - Integrating Trig Functions


• Compute integrals of powers of sine and cosine that include at least one odd powerby converting sines to cosines or vice versa and using u-substitution.

• Compute integrals of even powers of sine and cosine using the trig identitiescos( θ) =

12

+12

cos(2θ) and sin( θ) =12− 1

2cos(2θ)

• Compute some powers of sec and tan by converting them to sine and cosine, or byapplying u-substitution.

• Compute∫

sec(x) dx and∫

csc(x) dx.

• Compute∫

sec2(x) dx and∫

tan2(x) dx

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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


Example. Find∫

sin4(x) cos(x) dx

Example. Find∫

sin4(x) cos3(x) dx

93


Example. Find∫

sin5(x) cos2(x) dx

94


Example. Consider ∫cos2(x)dx

.

According to a TI-89 calculator∫cos2(x) dx =

sin(x) cos(x)2

+x2

.

According to the table in the back of the book,∫cos2(x) dx =

12

x +14

sin 2x

.

Are these answers the same?

95


Compute∫

cos2(x) dx by hand. Hint: cos2(x) =1 + cos(2x)

2

Example. Compute∫

sin2(x) dx by hand.

96


Example. Compute∫

sin6(x) dx by hand.

97


Review. What tricks can be used to calculate∫

cos7(5x) sin4(5x) dx?

98


Which of these integrals can be attacked in the same way, using the identitysin2(x) + cos2(x) = 1 and u-substitution?

A.∫

sin3(x) cos4(x) dx

B.∫

cos3(√

x)√x

dx

C.∫

cos2(x) sin4(x) dx

D.∫

sin3(2x)√

cos(2x) dx

E.∫

tan3(x) dx

F.∫

sin2(x) dx

99


Even powers of sine and cosine.

Review. What trig identities are most useful in evaluating∫

cos2(x) sin4(x) dx?

Example. Compute∫

cos2(x) sin4(x) dx by hand.

100


Conclusions:

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


Note. Often the answers that you get when you integrate by hand do not look identicalto the answers you will see if you use your calculator, Wolfram Alpha, or the integraltable in the back of the book. Of course, the answers should be equivalent. Why doyou think the answers look so different?

102


These integrals have their own special tricks.

Example.∫

tan2(x) dx

Example.∫

sec(x) dx

103

§8.4 - TRIG SUBSTITUTIONS

§8.4 - Trig Substitutions


• 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


The following three trig identities are useful for doing trig substitutions to solve somekinds of integrals with square roots in them.

sin2(x) + cos2(x) = 1 tan2(x) + 1 = sec2(x) cot2(x) + 1 = csc2(x)

105


Example. According to Wolfram Alpha,∫x2√

49 − x2dx =

12

(49 sin−1

(x7

)− x√

49 − x2)

Let’s see where that answer comes from using a trig substitution.

END OF VIDEO

106


Review. To compute∫

x2√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


Example. Find∫

1√x2 + a2

dx. (Assume a is positive.)

108


Example. Compute the integral∫ 2/3

1/3

√9x2 − 1

xdx

109


Which trig substitutions for which problems?

110


What trig substitutions would be most useful for these integrals?

1.∫

2√4 + x2

dx

2.∫

(100x2 − 1)3/2 dx

3.∫

x

√4 − x

2

9dx

4.∫

(25 − x2)2 dx

5.∫ √

−x2 − 6x + 7 dx

111


Extra Example. Use calculus to find the volume of a torus with dimensions R and r asshown.

112

§8.5 - INTEGRALS OF RATIONAL FUNCTIONS

§8.5 - Integrals of Rational Functions


1. Recognize whether an integral is a good candidate for the method of partial frac-tions.

2. Rewrite a rational expression as a sum of appropriate partial fractions, performinglong division first if the numerator has degree greater or equal to the denominator.

3. Compute an integral using the method of partial fractions.

113


Example. According to Wolfram Alpha,∫3x + 2

x2 + 2x − 3 dx =54

ln |1 − x| + 74

ln |x + 3| + C

Let’s see where this answer came from.

114


Review. True or False:∫

12x2 − 7x − 4 dx = ln |2x

2 − 7x − 4| + C

115


Example. Find∫

2x2 + 7x + 19x2 − 5x + 6 dx

116


Example. How would you set up partial fractions to integrate this?∫5x + 7

(x − 2)(x + 5)(x) dx

117


Example. How would you set up partial fractions to integrate this?∫

4x2 + 3x + 7x3 − 4x2 + 4xdx

A.4x2 + 3x + 7

x(x − 2)2 =Ax

+B

x − 2

B.4x2 + 3x + 7

x(x − 2)2 =Ax

+B

(x − 2)2

C.4x2 + 3x + 7

x(x − 2)2 =Ax

+B

x − 2 +C

(x − 2)2

D.4x2 + 3x + 7

x(x − 2)2 =Ax

+Bx + C(x − 2)2

118

§8.6 INTEGRATION STRATEGIES

§8.6 Integration Strategies


• 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 thefirst step. You do not need to finish any of the problems.

1.∫

x3 ln x dx

2.∫

cos2(x) dx

3.∫

dxx ln(x)

4.∫

arcsin(x) dx

5.∫

x2 + 1√x

dx

6.∫

sin(x)3 + sin2(x)

dx

7.∫

x3

25 − x2 dx

8.∫

x3√25 − x2

dx

9.∫

e√

x dx

120

PHILOSOPHY ABOUT INTEGRATION

Philosophy about Integration

Definition. (Informal Definition) An elementary function is a function that can be builtup 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


Question. Is it always true that the derivative of an elementary function is an elemen-tary function?

Question. Is it always true that the integral of an elementary function is an elementaryfunction?

122


Techniques of integration ... and their limitations.

123

§8.9 -IMPROPER INTEGRALS

§8.9 -Improper Integrals


• 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 computeit, 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 Com-

parison Theorem.

• Use the Comparison Theorem to determine if integrals converge or diverge withoutactually integrating.

• Give an example to show how failing to notice that an integral is improper andcomputing it as if it were proper can lead to nonsense.

124


Here are two examples of improper integrals:∫ ∞1

1x2

dx

and

∫ π2

0tan(x) dx

Question. What is so improper about them?

Definition. An integral is called improper if either

(Type I)

or,

(Type II)

or both.

125


Type 1 Improper Integrals

To integrate over an infinite interval, we take the limit of the integrals over expandingfinite intervals

Example. Find∫ ∞

1

1x2

dx

Definition. The improper integral∫ ∞

af (x) dx is defined as ...

We say that∫ ∞

af (x) dx converges if ...

and diverges if ...

126


Definition. Similarly, we define∫ b−∞

f (x) dx as ...

and say that∫ b−∞

f (x) dx converges if ...

and diverges ...

Example. Evaluate∫ −1−∞

1x

dx and determine if it converges or diverges.

END OF VIDEO

127


Review. Which of the following are NOT improper integrals?

A.∫ ∞

1e−x dx

B.∫ 3

0

1x2

dx

C.∫ 5−5

ln |x| dx

D.∫ 0−∞

4x + 4

dx

E. They are all improper integrals.

Example. Evaluate∫ ∞

1

1√x

dx and determine if it converges or diverges.

128


Question. For what values of p > 0 does∫ ∞

1

1xp

dx converge?

129


Example. Find the area under the curve y = e3x−2 to the left of x = 2.

130


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)→ −∞ asx→ b−, then∫ b

af (x) dx =

Definition. If f (x)→∞ or f (x)→ −∞ asx→ a+, then∫ b

af (x) dx =

131


Example. Find the area under the curve y =x√

x2 − 1between the lines x = 1 and x = 2.

1 2 3 4 5

1.11.21.31.41.5

END OF VIDEO

132


Review. True or False: If f (x) is continuous on (1, 2] and f (x) → ∞ as x → 1+, then∫ 21

f (x) dx diverges. (Hint: remember the pre-class video on Type 2 integrals.)

Example. Find∫ 10

1

4(x − 3)2 dx .

Note. Since4

(x − 3)2 blows up at x = 3, this integral must be computed as the sum oftwo indefinite integrals.

If you compute it without breaking it up YOU WILL GET THE WRONG ANSWER!

133


Question. For what values of p > 0 does∫ ∞

1

1xp

dx converge?

Question. For what values of p > 0 does∫ 1

0

1xp

dx converge?

134


Theorem. Comparison Theorem for Integrals: Suppose 0 ≤ g(x) ≤ f (x) on (a, b) (where a orb could be −∞ or∞).

(a) If∫ b

af (x) dx , then

∫ ba

g(x) dx also.

(b) If∫ b

ag(x) dx , then

∫ ba

f (x) dx also.

135


Example. Does∫ ∞

2

2 + sin(x)√x

dx converge or diverge?

136


Review. If 0 ≤ f (x) ≤ g(x) on the interval [a,∞), then which of the following are true?A. If

∫ ∞a f (x) dx converges, then

∫ ∞a g(x) dx converges.

B. If∫ ∞

a f (x) dx converges, then∫ ∞

a g(x) dx diverges.

C. If∫ ∞

a f (x) dx diverges, then∫ ∞

a g(x) dx converges.

D. If∫ ∞

a f (x) dx diverges, then∫ ∞

a g(x) dx diverges.

E. None of these are true.

137



1

cos(x) + 74x3 + 5x − 2 dx converge or diverge?

138



7

3x2 + 2x√x6 − 1

dx converge or diverge?

139


Extra Example. Does∫ ∞

2

√x2 − 1

x3 + 3x + 2dx converge or diverge?

140



0e−x

2dx converge or diverge?

141


Question. What are some useful functions to compare to when using the comparisontest?

Question. True or False: Since −1x≤ 1

x2for 1 < x < ∞, and

∫ ∞1

1x2

dx converges, the

Comparison Theorem guarantees that∫ ∞

1−1

xdx also converges.

142


Comparison Test Practice Problems

Decide what function to compare to and whether the integral converges or diverges.

1.∫ ∞

1

1e5t + 2

dt

2.∫ ∞

2

√x2 − 1

x3 + 3x + 2dx

3.∫ ∞

1

x2

x2 + 4dx

4.∫ 2

0

√t + 2t2

dt

5.∫ ∞

5

6√t − 5

dt

6.∫ 5−1

cos(t) + 4√t + 1

dt Hint: do a u-

substitution.

7.∫ ∞

1

5ez + 2z

dz

8.∫ ∞

7

4 sin(x) + 5√x3 + x

dx

9.∫ ∞

7

x + 3√x4 − x

dx

10.∫ ∞

0

5√xex + 1

dx

143


Example. Find∫ ∞−∞

x cos(x2 + 1) dx

144


Question. True or False:∫ ∞−∞

f (x) dx = limt→∞

∫ t−t

f (x) dx

145

§10.1 - SEQUENCES AND SERIES INTRO

§10.1 - Sequences and Series Intro


• 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


Definition. A sequence is an ordered list of numbers.Example. 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 9, . . .

A sequence is often denoted {a1, a2, a3, . . .}, or {an}∞n=1, or {an}.

Example. For each sequence, write out the first three terms:

1.{

3n + 1(n + 2)!

}∞n=1

2.{

(−1)kk + 33k

}∞k=2

147


Definition. Sometimes, a sequence is defined with a recursive formula (a formula thatdescribes how to get the nth term from previous terms), such as

a1 = 2, an = 4 −1

an−1Example. Write out the first three terms of this recursive sequence.

Note. Sometimes it is possible to describe a sequence with either a recurvsive formulaor a ”closed-form”, non-recursive formula.

148


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 havethe same common difference.

If a is the first term and d is the common difference, then the arithmetic sequence hasthe form:

(starting with n = 0)

An arithmetic sequence can also be written:

(with the index starting at n = 1.)

149


Example. For each sequence, write a formula for the general term an (start with n = 1or with n = 0).

B. {3, 0.3, 0.03, 0.003, 0.0003, · · · }

C.{

152 ,

754 ,

3758 ,

187516 , · · ·

}D. {3,−2, 43,−89, . . .}

Definition. A geometric sequence is a sequence for which consecutive terms have thesame 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


Example. For each sequence, write a formula for the general term an, starting withn = 1.

E. {−29, 416,− 825, 1636, . . .}

F. {−6, 5,−1, 4, 3, 7, 10, 17, . . .}

END OF VIDEO

151


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


Example. Consider the sequence{−1

2,

310,− 9

50,

27250

, · · ·}

1. What are the next three two terms in this sequence?


3. What is a explicit (closed form) formula for this sequence?

153


Example. Consider the sequence{(−1)n2 · 5

n!

}∞n=0

1. What are the first three terms in this sequence?


154


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, · · · }

B.{−1

2,

310,− 9

50,

27250

, · · ·}

C.{

12,

23,

34,

45, · · ·

}

155


Definition. For any sequence {an}∞n=1, the sum of its terms a1 + a2 + a3 + · · · is a series.Often this series is written as

∞∑n=1

an

Example. Consider the sequence{

12n

}∞n=1

. If we add together all the terms, we get theseries:

∞∑n=1

12n

=

What does it mean to add up infinitely many numbers?

156


Definition. The partial sums of a series∞∑

n=1

an are defined as the sequence {sn}∞n=1, where

s1 =

s2 =

s3 =

sn =

Definition. The series∞∑

n=1

an is said to converge if :

Otherwise, the series diverges.

Note. Associated with any series∞∑

n=1

an, there are actually two sequences of interest:

1.

2.

157


Example. For the series∞∑

n=1

1n2 + n

, write out the first 4 terms and the first 4 partial

sums. Does the series appear to converge?

158


Review. What is the difference between the following two things?

• the sequence{ 1

4k

}∞k=1

• the series∞∑

k=1

14k

Question. What does it mean for the sequence{ 1

4k

}∞k=1

to converge vs. diverge?

Question. What does it mean for the series∞∑

k=1

14k

to converge vs. diverge?

Question. Does the series∞∑

k=1

14k

converge or diverge?

159


Example. Using your calculator, Excel, or any other methods, compute several partialsums for each of the following series and make conjectures about which series convergeand which diverge.

A. 4 + 0.2 + 0.02 + 0.002 + · · ·

B.∞∑j=1

(−1) j

C.∞∑

k=1

kk + 1

160

10.2 SEQUENCES

10.2 Sequences


• 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 afunction 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 asequence 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, 43,−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, 43,−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? Decreasingand 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

Example. Is the sequence{n − 5

n2

}∞n=1

monotonic? Bounded?

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)n4n5n+2

}∞n=0

•{5 · 0.5n

3n−1

}∞2

•{4/3, 2, 3,

92,274. . .

}

• {2,−4, 8,−16, 32,−64, . . .}

167

10.2 SEQUENCES

Question. For which values of a and r does {a · rn}∞n=0 converge?

168

10.2 SEQUENCES

The following are some techniques for proving that a sequence converges:

Example. Does{

(−1)tet−13t+2

}∞t=0


Trick 1: Recognize geometric sequences

169

10.2 SEQUENCES

Example. Does{

ln(1 + 2en)n

}∞n=1


Trick 2: Suppose an = f (n) where n = 1, 2, 3, . . ., for some function f defined on allpositive real numbers. If lim

x→∞f (x) = L then ...

So ... replace an with f (x) and use l’Hospital’s Rule or other tricks from Calculus 1 toshow that lim

x→∞f (x) exists.

170

10.2 SEQUENCES

Example. Does{

cos(n) + sin(n)n2/3

}∞n=5


Trick 3: Use the Squeeze Theorem: trap the sequence between two simpler sequencesthat 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 asfor functions. (See textbook.)

For example, if limn→∞

an = L and limn→∞

bn = M, then

limn→∞

(an + bn) =

limn→∞

(anbn) =

limn→∞

(can) = (c is a constant)

Example. Does{

k2

2k2 − k +4 · πk

6k

}∞k=3


174

10.2 SEQUENCES

Question. Do the limit laws help establish the convergence of this sequence?{n +

3 − 2n2

}∞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{

(−1)nn2

}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 limx→∞

f (x) = L thenlimn→∞

an = L.

2. Suppose an = f (n) for some function f . If limn→∞

an = L, then limx→∞

f (x) = L.

178

10.2 SEQUENCES

Additional problems if additional time:

Do the following sequences converge or diverge? Justify your answer.

1.{

cos( j)ln( j + 1)

}∞j=1

2.{

(−1)t4t−132t

}∞t=3

3. 3√kln(k)

∞k=2

4.{3n

n!

}∞n=1

5.{n!

3n

}∞n=1

179

§10.3 - SERIES

§10.3 - Series


• 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 theconvergence 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∞∑

i=2

5(−2)i32i−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}∞n=0 converges to 0 when , converges towhen and diverges when .

Question. For what values of r does the geometric series∞∑

n=0

arn converge?

Stragegy:

1. Find a formula for the Nth partial sum sumNk=0a · rk.2. Take the limit of the partial sums.

182

§10.3 - SERIES

Conclusion: The geometric series∞∑

n=0

arn converges to when .

The geometric series∞∑

n=0

arn diverges when .

Example. Does∞∑

i=2

5(−2)i32i−3


END OF VIDEO

183

§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∞∑

n=2

3xn−1

2nconverge? What does it

converge to (in terms of x)?

184

§10.3 - SERIES

Trick 2: Recognize telescoping series.

Example.∞∑

k=2

ln(

kk + 1

)

185

§10.3 - SERIES

Example.∞∑

n=2

3n2 − 1

186

§10.3 - SERIES

Trick 3: Use Limit Laws.

Fact. If∞∑

n=1

an = A and∞∑

n=1

bn = B, then

∞∑n=1

an + bn =

∞∑n=1

an − bn =

∞∑n=1

c · an =

where c is a constant.

187

§10.3 - SERIES

Example. Does the series converge or diverge? If it converges, to what?

∞∑n=1

4 · 5n − 5 · 4n6n

188

§10.3 - SERIES

Question. True or False: If∞∑

n=1

an diverges and∞∑

n=1

bn converges, then∞∑

n=1

(an + bn)

diverges.


n=1

an diverges and∞∑

n=1

bn diverges, then∞∑

n=1

(an + bn) di-

verges.

189

§10.3 - SERIES


n=1

an converges, then so does∞∑

n=5

an.


n=5

an converges, then so does∞∑

n=1

an.

190

§10.3 - SERIES


n=1

an = A and∞∑

n=1

bn = B, then∞∑

n=1

an · bn = A · B


n=1

an = A and∞∑

n=1

bn = B, then∞∑

n=1

anbn

=AB

.

191

§10.4 - THE DIVERGENCE TEST AND THE INTEGRAL TEST

§10.4 - The Divergence Test and the Integral Test


• 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 valueof the sum based on partial sums and integrals.

• Use an integral, when appropriate, to determine how many terms are needed toapproximate the sum of a series to within a specified level of accuracy.

192


Example. Does this series converge or or diverge?

∞∑n=1

1n2

193


The series∞∑

n=1

1n2

is closely related to the improper integral∫ ∞

1

1x2

dx .

194


Example. Does this series converge or or diverge?

∞∑n=1

1√x

195


Theorem. (The Integral Test) Suppose f is a continuous, positive, decreasing functionon [1,∞) and an = f (n). Then

1. If∫ ∞

1f (x) dx converges, then

∞∑n=1

an converges.

2. If∫ ∞

1f (x) dx diverges, then

∞∑n=1

an diverges.

196


Example. Does∞∑

n=1

ln nn


END OF VIDEO

197


Example. Does∞∑

k=1

kk + 1

converge or diverges?

Note. If the sequence of terms an do not converge to 0, then the series∑

an ...

Theorem. (The Divergence Test) If

then the series∞∑

n=1

an diverges.

198


Example.∞∑

t=1

t sin(1/t)

Example.∞∑

t=1

(−1)n

Note. If the sequence of terms an do converge to 0, then the series∑

an.

199


Review. We know that∫ ∞

1

1x2

dx converges to 1. Which of the following are true?

A.∞∑

n=1

1n2

converges.

B.∞∑

n=1

1n2

= 1.

C. Both of the above.

D. None of the above.

200


Example. Does∞∑

n=1

nen


201


Example. Does the following series converge or diverge?

15

+18

+1

11+

114

+ · · ·

202


Question. For what values of p does the p-series∞∑

n=1

1np

converge?

203


Definition. The Harmonic Series is the series:

Question. Does the Harmonic Series converge or diverge?

204


Bounding the Error

Definition. If∞∑

n=1

an converges, and sn is the nth partial sum, then for large enough n, sn

is a good approximation to the sum s∞ =∞∑

k=1

ak. Define Rn be the error, or remainder:

Rn =

Use the pictures above to compare R2 to∫ ∞

2f (x) dx and

∫ ∞2

f (x) dx where f (x) is thepositive, decreasing function drawn with an = f (n).

205


Use the pictures above to compare Rn to∫ ∞

n f (x) dx and∫ ∞

n+1 f (x) dx where f (x) is thepositive, decreasing function drawn with an = f (n).

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


Example. (a) Put a bound on the remainder when you use the first three terms to

approximate∞∑

n=1

6n2

.

(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


Question. Which of the following are always true?

1. Suppose f is a continuous, positive, decreasing function on [1,∞) and for n ≥ 1,

an = f (n). Then∞∑

n=1

an converges, if and only if∫ ∞

1f (x) dx converges.

2. Suppose f is a continuous, positive, decreasing function on [5,∞) and for n ≥ 5,

an = f (n). Then∞∑

n=1

an converges if and only if∫ ∞

5f (x) dx.

3. Suppose f is a continuous, positive function on [1,∞) and for n ≥ 1, an = f (n).

Then∞∑

n=1

an converges if and only if∫ ∞

1f (x) dx converges.

208

§10.5 - COMPARISON TESTS FOR SERIES

§10.5 - Comparison Tests for Series


• For the (ordinary) comparison test, give conditions that will guarantee convergenceof a series and conditions that will guarantee divergence of a series, and justifywhy these conditions make sense.

• For the limit comparison test, state what values of the limit of the ratio of termsallows you to determine that a series converges or diverges, and what values areinconclusive.

• Determine what series to compare another series to, when using the comparisonor limit comparison test.

• Identify situations that make it preferable to use the ordinary comparison testinstead of the limit comparison test and vice versa.

209


Theorem. (The Comparison Test for Series) Suppose that∑∞

n=1 an and∑∞

n=1 bn are series and0 ≤ an ≤ bn for all n.

1. If converges, then converges.

2. If diverges, then diverges.

Note. The following series are especially handy to compare to when using the com-parison test.

1. which converges when

2. which converges when

210


Example. Does∞∑

n=1

3n

5n + n2converge or diverge?

211


Theorem. (The Limit Comparison Test) Suppose∑

an and∑

bn are series with positive terms.If

limn→∞

anbn

= L

where L is a finite number and L > 0, then either both series converge or both diverge.

Example. Does∞∑

n=1

3n

5n − n2 converge or diverge?

212


Review. The (Ordinary) Comparison Test for Series: Suppose that∑∞

n=1 an and∑∞

n=1 bnare series with positive terms and 0 ≤ an ≤ bn for all n.

1. If converges, then converges.

2. If diverges, then diverges.

213


Review. Suppose∑∞ an and ∑∞ bn are series with positive terms. Which of the follow-

ing will allow us to conclude that∑∞ bn converges? (More than one answer may be

correct.)

A. limn→∞

an = limn→∞

bn and∞∑

an converges.

B. limn→∞

anbn

= 0 and∑∞ an converges.

C. limn→∞

anbn

=13

and∑∞ an converges.

D. limn→∞

anbn

= 5 and∑∞ an converges.

Review. The Limit Comparison Test: Suppose∑

an and∑

bn are series with positiveterms. If

limn→∞

anbn

= L

where L ,then:

214


Advice on the Comparison Theorems:Question. What series are especially handy to compare to when using the comparisontest?

Question. How to decide whether to use the Ordinary Comparison Test or the LimitComparison Test?

215


Example. Decide if the series converges or diverges.

∞∑n=1

3n − 5√n3 + 2n

216


Example. Decide if∞∑

n=3

n sin2(n)n3 + 7n

converges or diverges.

217


Example. Decide if∞∑

n=3

n sin2(n)n3 − 7n converges or diverges.

218


Question. True or False: For an, bn > 0, if limn→∞

anbn

= 0, then the series∑

an and∑

bn have

the same convergence status.

Can anything be concluded if limn→∞

anbn

= 0?

219


Question. Find the error: Consider the two series∞∑

n=1

an = (−1) + (−2) + (−3) + (−4) + (−5) + (−6) . . .

and ∞∑n=1

bn = 2 + (−1) + (1/2) + (−1/4) + (1/8) + (−1/16) + . . .

Note that∑∞

n=1 bn is a geometric series with ratio r = −1/2.Since an ≤ bn for all n, and

∑bn converges,

∑an also converges, by the Ordinary

Comparison Test.

220


Note. Orders of magnitude:

221


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


• 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 alter-

nating series.

• Determine how many terms are needed to approximate an alternating series withina specified level of accuracy.

• Explain the relationship between convergent, absolutely convergent, and condi-tionally convergent.

• Prove that a series∞∑

n=1

an converges by showing that∞∑

n=1

|an| converges and using

the fact that absolutely convergent implies convergent.

223


Definition. An alternating series is a series whose terms are alternately positive andnegative. It is often written as

∞∑k=1

(−1)k−1bk

where the bk are positive numbers.Example. (The Alternating Harmonic Series)

224


Does the Alternating Harmonic Series converge? Hint: look at ”even” partial sumsand ”odd” partial sums separately.

225


Theorem. (Alternating Series Test) If the series∞∑

n=1

(−1)n−1bn = b1 − b2 + b3 − b4 . . .

satisfies:

1.

2.

3.

then the series is convergent.

226


Example. Which of these series are guaranteed to converge by the Alternating SeriesTest?

A. 5√2− 5√

3+ 5√

4− 5√

5+ 5√

6− 5√

7+ · · ·

B. 22 − 12 + 23 − 13 + 24 − 14 + 25 − 15 + · · ·

C. 18 − 14 + 127 − 19 + 164 − 116 + 1125 − 125 + · · ·

D. 2.1 − 2.01 + 2.001 − 2.0001 + 2.00001 · · ·

227


Question. Why is the condition limn→∞

bn = 0 necessary?

Question. Why is the condition bn+1 ≤ bn for all large n necessary?

228


Example. Does the series converge or diverge?

∞∑n=1

(−1)n n2

n3 − 2

229



∞∑k=1

(−1)k(1 + k)1/k

230


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∞ andthe nth partial sum sn.

1. Write an equation for the nth remainder Rn.

2. Find an upper bound on |Rn|:|Rn| ≤

231


Example. Consider the series −14 + 19 − 116 + 125 − · · ·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


Example. How many terms of the series

−14

+19− 1

16+

125− · · ·

do we need to add up to approximate the limit to within 0.01?

233


Definition. A series∑

an is called absolutely convergent if

Example. Which of these series are convergent? Which are absolutely convergent ?

1.∞∑

m=0

(−0.8)m convergent abs. convergent

2.∞∑

k=1

1√k

convergent abs. convergent

3.∞∑j=5

(−1) j1j

convergent abs. convergent

234


Question. Is it possible to have a series that is convergent but not absolutely conver-gent?

Definition. A series∑

an is called conditionally convergent if

Question. Is it possible to have a series that is absolutely convergent but not conver-gent?

235


Review. Which of the following statements are true about a series∞∑

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 betweenconvergence, absolute convergence, and conditional convergence?

236


Example. Does this series converge or diverge? If it converges, does it convergeabsolutely or conditionally?

∞∑n=1

cos(nπ/3)n2

237



∞∑n=2

cos(n) + sin(n)n3

238

§10.7 - RATIO AND ROOT TESTS

§10.7 - Ratio and Root Tests


• 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


Recall: for a geometric series∑

arn

Theorem. (The Ratio Test) For a series∑

an :

1. If limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = L < 1, then ∞∑

n=1

an is .

2. If limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = L > 1 or limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = ∞, then ∞∑

n=1

an is .

3. If limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = 1, then ∞∑

n=1

an .

240


Example. Apply the ratio test to∞∑

n=1

n2(−10)nn!

241


Review. In which of these situations can we conclude that the series∞∑

an converges?

A. limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = 0

B. limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = 0.3

C. limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = 1

D. limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = 17

E. limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = ∞

Review. (The Ratio Test) For a series∑

an :

1. If limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = L < 1, then ∞∑

n=1

an is .

2. If limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = L > 1 or limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = ∞, then ∞∑

n=1

an is .

3. If limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = 1 or DNE , then .

242


Example. Apply the ratio test to∞∑

n=1

(1.1)n

(2n)!

243


Example. Apply the ratio test to the series∞∑

n=2

3n2 − n

244


Extra Example. Apply the ratio test to the series

a1 = 1, an =sin n

nan−1

245


Theorem. (The Root Test)

1. If limn→∞

n√|an| = L > 1 or lim

n→∞n√|an| = ∞, then

∞∑n=1

an .

2. If limn→∞

n√|an| = L < 1, then

∞∑n=1

an .

3. If limn→∞

n√|an| = 1, then

∞∑n=1

an .

246


Example. Determine the convergence of∞∑

n=1

5n

nn

247


RearrangementsDefinition. A rearrangement of a series

∑an is a series obtained by rearranging its

terms.Fact. If

∑an is absolutely convergent with sum s, then any rearrangement of

∑an also

has sum s.

But if∑

an is any conditionally convergent series, then it can be rearranged to give adifferent sum.Example. Find a way to rearrange the Alternating Harmonic Series so that the rear-rangement diverges.

Example. Find a way to rearrange the Alternating Harmonic Series so that the rear-rangement sums to 2.

248

§10.8 - STRATEGY FOR CONVERGENCE TESTS FOR SERIES

§10.8 - Strategy for Convergence Tests for Series


• 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


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


Question. The limit comparison test and the ratio test both involve ratios. How arethey different?

251


Example. Which convergence test would you use for each of these examples? Carryout the convergence test if you have time.

1.∞∑

n=1

2n

n3

2.∞∑

n=1

(−1)n ln nn + 3

3.∞∑

n=1

13√

n2 + 6n

4.∞∑

n=1

1n!− 1

2n

5.∞∑

n=1

n2

en2

6.∞∑

n=1

3n 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


Review. Let f (x) be a function whose derivatives all exist near x = 0. Suppose thatf (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 andalso 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


Review. Let f (x) be a function whose derivatives all exist near x = 5. Suppose thatf (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 andalso have the same first through fourth derivatives at x = 5.

Suppose f (5) = 1, f ′(5) = 3, f ′′(5) = 7, f (3)(5) = 13, and f (4)(5) = −11. What are thecoefficients of the polynomial?

255


Note. For a function f (x) whose derivatives all exist near a, suppose we have a degree npolynomial Pn(x) such that Pn(a) = f (a), P′n(a) = f ′(a), P′′n (a) = f ′′(a), · · · P(n)n (a) = f (n)(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, whatare the coefficients c0, · · · cn in terms of f ?

Definition. For the function f (x) whose derivatives are all defined at x = a, the poly-nomial 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


We use the conventions that:

• f (0)(a) means• 0! =• (x − a)0 =

257


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).

f (x)T3(x)

T9(x)

T6(x)-2 2 4 6

-5

5

258


Example. Find the 7th degree Taylor polynomials for f (x) = sin(x) and g(x) = cos(x),centered at a = 0.

259


Example. Find the 4th Taylor polynomial for f (x) = ex centered at a = 0. What is theerror when using it to approximate e0.15?

260


Example. Use polynomials of order 1, 2, and 3 to approximate√

8.

261


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 theremainder Rn(x) of the Taylor series satisfies the inequality

|Rn(x)| ≤

262


Example. Approximate f (x) = cos(x) by a Maclaurin polynomial of degree 4. Estimatethe accuracy of the approximation when x is in the interval [0, π/2].

263


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


Example. How many terms of the Maclaurin series for ex should be used to estimatee0.5 to within 0.0001?

265


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


• 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 com-

posing with an expression like 3x2.

• Differentiate and integrate power series.

267


Informally, a power series is a series with a variable in it (often ”x”), that looks like apolynomial with infinitely many terms.

Example.∞∑

n=0

(2n + 1)xn

3n−1= 3 + 3x +

5x2

3+

7x3

9+

9x4

27+

11x5

81+ · · ·

is a power series.

Example.∞∑

n=0

(5n)(x − 6)nn!

= 1 + 5(x − 6) + 52(x − 6)2

2!+

53(x − 6)33!

+54(x − 6)4

4!+

55(x − 6)55!

+ · · ·

is a power series centered at 6.

268


Definition. A power series centered at a is a series of the form∞∑

n=0

cn(x − a)n =

where x is a variable, and the cn’s are constants called coefficients, and a is also aconstant called the center .

Definition. A power series centered at zero is a series of the form∞∑

n=0

cnxn =

269


Example. For what values of x does the power series∞∑

n=0

n! (x − 3)n converge?

270



n=0

(−2)n(x + 4)nn!

converge?

271



n=1

(−5x + 2)nn

converge?

END OF VIDEO

272


Review. Which of the following are power series?

A.12

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