MATH 10B Review Session

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MATH 10B Review Session

• Franklin Kenter
• March 15, 2009

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Instructions

• For this review session we will be doing actual
  problems so that you know what to study for, and
  moreover, so that you can identify your
  weaknesses and focus in those areas t
• Get into groups of 3 or 4. If possible, make a
  new friend or two.
• For each question, I will taking 3 or 4 groups to
  do the problems on the board. Each group that
  does so will be rewarded. Then we will go over
  the problems afterwards.

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Instructions continued

• While doing these problems, you reference your
  notes (even if your group is at the board).
  However, every time you reference your notes, YOU
  HAVE TO WRITE WHAT YOU REFERENCED ON YOUR NOTE
  SHEET!
• Lastly, this is a work in progress, please look
  out for and point out any mistakes.

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FINALS TIME AND PLACE

• TIMEWed. Mar. 18,   700pm-1000pm
• PLACE depends upon your lecture
• John Eggers (9am Lecture) MANDE AUD
• Michael Volpato (12pm Lecture) WLH 2005
• Shengli Kong (1pm Lecture)
• Last names A L PCYNH 109
• Last names M Z PCYNH 106
• Evelyn Lunasin (4pm Lecture)
• Last names A O LEDDN AUD
• Last names P Z HSS 1330

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Does anyone not know where his or her final is?

• John Eggers (9am Lecture) MANDE AUD
• Michael Volpato (12pm Lecture) WLH 2005
• Shengli Kong (1pm Lecture)
• Last names A L PCYNH 109
• Last names M Z PCYNH 106
• Evelyn Lunasin (4pm Lecture)
• Last names A O LEDDN AUD
• Last names P Z HSS 1330

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Additional Sources For Help This Week

• I will stay here for a short time after.
• All of this information (including this
  slideshow) can be found at math.ucsd.edu/fkenter/
  
• Dan Minsky has office hours Monday and Tuesday
  from 4-5pm in APM 6446.
• Jacob Hughes will be holding another review
  session Tuesday 8-930pm in CENTER 119

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What is the best way to study?

• DO PRACTICE PROBLEMS- as many as you can until
  you know it inside and out
• Speaking or which
• THERE ARE ALSO OTHER PRACTICE FINALS ON MY
  WEBSITE math.ucsd.edu/fkenter/

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Color code!

• ANSWERS will be in a green glowing box
• IMPORTANT CONCEPTS will be in an orange glowing
  box.
• Guess what my favorite colors are?

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Course Outline (not all inclusive)

• Chapters 5-6 The concept of an integral
• Chapters 5 7.5 Estimating integrals
• Chapter 7 Solving Integrals Algebraically- with
  the tools of substitution, by parts, and
• partial fractions and trig substitution
• Chapter 7.7-8 Improper Integrals
• Chapter 8 Applications of integrals
  specifically- Volumes of solids and
• Value of money over time
• Chapter 11 Differential Equations

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Question 1a

• Here is our first question. Its easy, but has a
  small little trick
• Estimate using
  LEFT(1)

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Solution 1a

• Estimate using
  LEFT(1)
• This is the number of subdivisions
  NOT the size of ?x.
• This means that there is 1 subdivision. ?x
  

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Solution 1a continued

• So our table (if that is really what you want to
  call it a table) looks like this
• To get the final answer
• We simply add up all of the rectangles

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Question 1b

• That might have been too easy, let us try this
• Estimate
• using LEFT(4), RIGHT(4), MID(4), and TRAP(4)

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Solution 1b (for LEFT(4) )

• ?x
• Again, to get the final answer
• We simply add up all of the rectangles

6.75
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Solution 1b (for RIGHT(4) )

• ?x
• Again, to get the final answer
• We simply add up all of the rectangles

10.75
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Solution 1b (for MID(4))

• ?x
• Again, to get the final answer
• We simply add up all of the rectangles

8.625
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Solution 1b (for TRAP(4))

• You could do a table for this one, but tell you
  what, it is much easier to take advantage of the
  fact that
• TRAP(n) ½ (LEFT(n) RIGHT(n))
• So TRAP(4) ½ (LEFT(4) RIGHT(4))
• ½ (6.75 10.25) 8.5

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Question 1c

• That might have been too easy, let us try this
• Referring to
• Which ones of LEFT(4), RIGHT(4), MID(4), and
  TRAP(4) are overestimates? Underestimates?
• Why?

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Solution 1c

• Note the following rules
• LEFT is an underestimate if f(x) is increasing,
  and an overestimate if f(x) is decreasing.
• Conversely, RIGHT is an overestimate if f(x) is
  increasing and an underestimate if f(x) is
  decreasing.
• Likewise, TRAP is an underestimate if f(x) is
  concave down, and an overestimate if f(x) is
  concave up.
• MID is an overestimate if f(x) is concave down,
  and an underestimate if f(x) is concave up.
• f(x) , is concave up and increasing on
  (1,3).
• Apply the above rules to determine the answer.

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Estimating Integrals

• A few notes the rules for Question 1c only work
  if the integral is strictly increasing (or
  decreasing, concave up, or concave down) on the
  interval.

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Before we move onThese are basic integrals you
should know!
Except for n-1
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More basic integrals you should know!
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The sine and cosine wheel
Clockwise to Differentiate
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The sine and cosine wheel
Counterclockwise to Integrate
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Question 2a,b,c,d

• Compute the following integrals

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Solution 2a

• What tool do we use? Why?

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Solution 2a

• What tool do we use? Why?
• By parts- namely because substitution will not
  work (i.e. no part is the derivative of the
  other)

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Solution 2a

• What tool do we use? Why?
• By parts- namely because substitution will not
  work (i.e. no part is the derivative of the
  other)
• The By Parts Formula

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Solution 2a continued

• But which one is u and which on is v'?

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Solution 2a continued

• But which one is u and which on is v'?
• We will make u ln(x). Why?

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Solution 2a continued

• Because is easier to integrate!
• A ranking of functions to choose for u is (in
  general) given by
• Log
• Inverse Trig
• Polynomials
• Sine and Cosine

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Solution 2a continued
Dont forget C
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Solution 2b

• What tool do we use? Why?

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Solution 2b

• What tool do we use? Why?
• Substitution- because is similar to the
• derivative of (except for a constant
  factor).

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Solution 2b

• What tool do we use? Why?
• Substitution- because is similar to the
• derivative of (except for a constant
  factor).
• Make a quick glance for substitution first-
  because substitution is easier!

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Solution 2b
Sometimes solving for dx in terms Of x and du can
be helpful
Cancel xs
Substitution
Resubstitute
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Solution 2b
When using substitution, There should be no xs
in your Substituted integral!!!
Sometimes solving for dx in terms Of x and du can
be helpful
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Short Solution to 2c

• This problem is an example where by parts wraps
  around. After doing by parts once, then again,
  you will end up with (Remember use LIPS with u
  cosx)

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Short Solution to 2c continued

• Now we can solve for

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Question 3a,b,c

• Let R be the area in the plane bounded by the
  curves yx, y-x, y0, and y1.
• Find the volume of the solid when R is revolved
  around the
• y-axis
• The line x1
• The line y2

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Notes on Question 3a,b,c

• Always, always draw a picture. It will make life
  easier.
• Remember that the formula for volumes of
  revolution is
• furthest function axis of rotation
• nearest function axis of rotation
• Determine the bounds of your integral by finding
  the extreme of x (for dx) or y (for dy)

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More Notes on Question 3a,b,c

• If the axis of rotation is vertical, then width
  of your donuts slices are going to be dy.
  Everything should be in terms of y and dy!
• If the extremes for x or y, as appropriate, are
  not given to you, try finding the intersections
  of curves (i.e. if the region is bounded by
  ycos(x) and ysin(x) and you are integrating dx,
  solve cos(x)sinx)

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Question 4

• Do the following integrals converge or diverge?
  Why?

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Question 5
a) If f(x) is even, find b) If f(x) is odd,
find c) If f(x) is even, find
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Question 6

• Is positive or negative? Why?
• Evaluate

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Question 7
Find the general solution
Find y(1) if t10 when y0
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Question 8

• As a lunar module takes off from the moon, its
  acceleration in is given by (t in seconds)
• Find the distance traveled in the 3 seconds.

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Solution 8

• Note the following relation
• Acceleration
• Velocity
• Position
• So given Acceleration, to find Position, all we
  have to do is integrate twice.

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Solution 8 continued

• Though this is slightly said than done. Every
  time you integrate you will left with a C, and
  you have to figure out what C is before you
  integrate again!
• Note that since the lunar module is taking off
  we assume that it is stationary (i.e., V0 when
  t0).

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Solution 8 continued
WATCH OUT! C is not always 0!

• Now we integrate again.

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Solution 8 continued

• Generally, the ground is where P0, since the
  lunar module starts on the ground, P0 when
  t0, so we can solve for C

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Question 9

• Frank decides to give his son, Franklin, a deal
• Franklin can pay Frank 35 to permanently raise
  his allowance 0.25 per month (paid
  continuously). If the bank rate is always
  constant 5 per year, and Frank will pay
  allowance for another 13 years, should Franklin
  take the deal? Why or why not?

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Solution 9
Annual Rate.. IN DECIMAL FORM

• Remember this formula? If not, put it on your
  notesheet.
• Present Value
• So the present value of receiving an extra
  quarter every month for 20 years is

Annual cash flow as a function of t
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Solution 9 continued

• So the present value of receiving an extra
  quarter every month for 20 years is
• Franklin should take the deal because the present
  value of the deal is more than 35.