Oakton Community College
MAT 251    Calculus II
Section 001:   MW  11am - 12:50   Room 2145   DP Campus
Paul Boisvert     Professor of Mathematics 
FALL  Semester  2009
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Homework Assignments
Note:  Below, exponents are written with the wedge ^ notation.  So "x raised to the third power" would be "x^3". 
Fractions are written with slashes and parentheses:  A fraction with a + b in the top and c - d in the bottom is written as  ( a + b )  /  ( c - d ) .
Square Roots are written as "sqt(x)", and higher roots as fractional exponents, i.e.,  cube root of x would be  "x^(1/3)"
Date Page.  Problems Read Sections
Mon
8/24   		 p. 323 7-45 odd, but SKIP 31,37,41                                                                                                                                                                                                                           4.6
Wed
8/26		 p. 323  47-55 ALL, SKIP 48
Answers:   50)  Taking L = ln of the limit, LH gives you ... that L = 1/e.   So True Limit = e^L  =  e^(1/e)  , or  "the e'th root of e" .
52)  Trick question:  Taking L = ln of the limit, you immediately get L = 1.  Since this is a constant, taking a limit is pointless.  The original function is thus the constant e^L  = e^1  =  e,
       and so x^(1/ln x ) is simply a weird way of writing the constant number e--weird, because it equals the same result "e" for any x you choose, which is quite unusual for a function that
        doesn't LOOK like a constant when you start.  Of course, any limit of a constant function simply equals the constant, so the answer here is  e.  
         NOTE:  since it is a constant, its derivative d/dx (x^(1/ln x ) = 0.   This is a typical trick question in Calc 1 for logarithmic differentiation.  
54) Taking L = ln of the limit, LH gives you that L = 2.   So True Limit is  e^L  =  e^2. 		 
Mon
8/31 		   Review and memorize all Basic Integrals, which are  in table on page 333
            Or you could use table on  page 538:  If you use this later table, you need not know all of them, just #1 through 11, as well as 14 and 15.  
            Also memorize #18, 19, and 20, but assuming that "a" = 1.   The rules are given for integrals with any constant  "a"  in them--but you should rewrite them assuming a = 1 (and thus
           dropping out all the now unecessary "a"'s.)  For example, #19 thus becomes  Integral of  ( du / 1 + u^2 )  =  tan-inverse (u) + C , once all the "a"'s are set = 1.
 The other rules in that table on page 538 we will cover sooner or later in class.
ALSO, memorize the integral of cos^2 and sin^2, as given in class.  		 5.5, 5.6 
  p. 410  1-13 odd,  part a.) only for each one
15-45 [ending in 1, 5, and 9]   The ones ending in 3 and 7 are OPTIONAL, and should be done if you have significant trouble with the ones ending in 1, 5, and 9. 		  
  p. 412  57--for this problem only, do it two ways, one using dx, the other using dy.
For the rest of the AREA problems, you decide whether it is better done by a dy or a dx integral, but do NOT do it both ways (unless you have a lot of extra time on your hands.)
59, 62, 85, 87, 89,   93-103 odd  		  
  p. 416  21-33 odd, 81-101 odd, 105, 49, 51, 55, 56, 57, 61, 71, 79   
Wed
9/2		 p. 435  3-11 odd  Cross-sections (Slices) could be squares, rectangles, circles...
17-35 odd  Here, all are Volumes of Revolution, so all slices are DISKS, with A = pi (radius ) squared, where radius = y  if revolved about x-axis, and radius = x  if about the y-axis.  		 6.1 
Mon
9/7		    LABOR DAY   COLLEGE CLOSED  
Wed
9/9 		 p. 435  37 - 55 odd    WASHERS:  dV =  Area (dx) if about x-axis, where Area  =  pi [  (outer Radius)^2  -  (inner radius)^2 ].  In this case "radius" is a  y-value,
                         so "outer" means Top y-value and "inner" means Bottom y-value.   Then Integrate...
                       
                       If revolved about the y-axis, it becomes a dy integral, the "radius" is an x-value,  and "outer" means "Right-hand x-value" with "inner" being "Left-hand x-value."		 6.1 
  p. 443   1-11 odd
15-21 odd       CYLINDRICAL SHELLS:  
     
Here   dV  =  2 pi r h  dx,  where the shells come from "tin cans" with a vertical height in the y-direction (and a circular base formed by revolving a function about the y-axis .)
                          Note the "switch":   y-axis of revolution and "d_x_" differential.  So these are not like discs or washers.
                          In this case, for "Vertical Shells" "r" = radius = just the variable "x" (not a function, just plain old "x"),
                          and h = "height" of shell from  TOP y-value  to Bottom y-value.  That is, "h" will equal f(x) - g(x), the difference between top and bottom (with g(x) sometimes = 0.)
                           When you integrate the results, that is, integrate 2 pi r h, you do it from an "innermost" r-value closest to the axis of revolution, to an "outermost" r-value.
                            But of course in this case the "r"-values are actually "x-values", as expected for a "dx" integral.

                     For revolution about the x-axis, instead, everything switches roles:  use dy,   "r" = "y" (just plain old y),  and "height" (sideways) = Right x - Left x, where each x is a function:  so h = f(y) - g(y), where it's "Right" - "left" x-values.  		 6.2 
Mon
9/14		   REVIEW For Test 1       Bring your Book.
      BEFORE COMING TO THIS CLASS, you should have ALREADY Reviewed all old HW.  If you have not looked at and reviewed all old HW before coming in, you will WASTE a lot of time trying to do so during the Review Session.  If you HAVE already reviewed, you will be able to use the in-class review session to answer your (few, we hope) remaining questions, and fine-tune your knowledge well for the test.  		  
Wed
9/16 		    TEST 1    Covers all material on which HW was asssigned, up through Section 6.2.    Closed Book, No Notes.
     BRING SOME PAPER of your own.   This can be paper you will rip out of a notebook (as long as it has nothing written on it.)  Or any other kind of blank paper you wish.  Bring several sheets.
BRING your OWN Calculator.  NO SHARING of calculators during the test. 		  
Mon
9/21		    All HW assigned before Test 1 is DUE.  I will have a stapler there so you can staple it into one big packet.  Make sure your name is on at least the first page of it.     
p. 452 9-17 odd,  1-7 odd 6.3
Wed
9/23		 p. 474 11-21 odd 6.5
Mon
9/28		 p. 463 5-11 odd  (one-dimensional thin rods:  find Mo and x-bar.)

13, 15, 19, 23, 25  (2-dimensional, use vertical strips, find Mx, My, M(total mass), x-bar, and y-bar)		 6.4
Wed
9/30		 p. 482 1-6 ALL  (Spring problems) 6.6
Mon
10/5		 p. 482 7, 9 (Rope problems)
10)  Force = k/(x^2) is given, just integrate w.r.t. "dx"
15, 19, 21   Pumping liquids
p. 530 1-10 ALL
13-23 odd		 7.4
Wed
10/7		 p. 530 41-59 odd
25-35 odd
67-74 ALL (part a. only for each one)
81-83 ALL		 7.4
Mon
10/12		 REVIEW For Test 2       Bring your Book.
      BEFORE COMING TO THIS CLASS, you should have ALREADY Reviewed all old HW.  If you have not looked at and reviewed all old HW before coming in, you will WASTE a lot of time trying to do so during the Review Session.  If you HAVE already reviewed, you will be able to use the in-class review session to answer your (few, we hope) remaining questions, and fine-tune your knowledge well for the test.
Wed
10/14		 TEST 2    Covers all material on which HW was asssigned, up through Section 7.4    
   You may use ONE  8 1/2 x 11 sheet of notes--both sides.  No other notes, no book.
     
BRING SOME PAPER of your own.   This can be paper you will rip out of a notebook (as long as it has nothing written on it.)  Or any other kind of blank paper you wish.  Bring several sheets.
BRING your OWN Calculator.  NO SHARING of calculators during the test. 
Mon
10/19		 URGENT:  SORRY, I had a bad day Monday--the formula I gave you to integrate  Integral of  [ sqrt(1 + cos x) ] dx  was STILL WRONG, even after I "fixed" it after break.

CORRECT Process:  (Eliminate all earlier formulas, use the ones below)  
TRIG IDENTITY:    From Double-Angle formla,   cos(2w)  =  2 cos^2 (w) - 1 , we get:
      cos^2 (w)  =  (1/2) [ 1 + cos(2w) ]     OR, replacing w with x/2,    cos^2 (x/2)  =  (1/2) [ 1 + cos(x) ]    OR    2 cos^2 (x/2)  =  1 + cos(x).    Take sqrt. of both sides of last equation, get:

 Integral of [ sqrt(1 + cos x) ] dx   =   Integral of  [sqrt(2) sqrt [cos^2 (x / 2) ] ] 
                             =  sqrt(2) Integral of [cos (x/2) ].  But what I botched in class is that this integral now requires a little u-subst., u = x/2, which produces an extra factor of 2 in the numerator of the answer.
                                 So, FINAL ANSWER, after integrating, =  2 sqrt(2) sin (x/2) .

So, put this on notesheet:                 Integral of [ sqrt(1 + cos x) ] dx    =  sqrt(2) Integral of [cos (x/2) ] dx   =   2 sqrt(2) sin (x/2) + C

And, similarly, skipping details,       Integral of [ sqrt(1 - cos x) ] dx    =  sqrt(2) Integral of [sin (x/2) ] dx   =   -2 sqrt(2) cos (x/2) + C

I will hand out a sheet in class Wed with a better looking version of these formulas.
p. 543 37, 39, 47, 49, 53, 55, 63, 65
Board Problems:   A)  Integrate  x [fourth root of (5x - 2) ]  dx     Use the "switch" method discussed in class.  
                             B)  Integrate   x^9 (4x^5 - 3)^21  dx               Use same "switch" method.
Answer:  A)   (1/25) [(4/9) (5x - 2)^(9/4)  +  (8/5) (5x - 2)^5/4 ] + C
B)   (1/80) [(1/23) (4x^5 - 3)^23  +  (3/22) (4x^5 - 3)^22 ] + C		 8.1
Wed
10/21		 p. 552 1 - 21  [ending in 1, 5, 9]   Other odd ones ending in 3 and 7 highly recommended if you have any trouble with these. 8.2
Mon
10/26		 p. 569 29 - 32 ALL  ,  
1-13 odd
Note:  for powers of sec, you only need to know how to integrate sec and sec^2.  In HW, however, you may find yourself needing to integrate sec^3--use the results from class for that (and it's also in the book.)		 8.4
Wed
10/28		 p. 575 1, 7, 11-19 odd, 23-35 odd 8.5
Mon
11/2		 p. 563 3, 5, 7, 11, 13 8.3
Wed
11/4		 p. 615 1-31 [ending in 1,5, and 9]   ones ending in 3 and 7 are OPTIONAL 8.8
Mon
11/9		 REVIEW For Test 3       Bring your Book.
      BEFORE COMING TO THIS CLASS, you should have ALREADY Reviewed all old HW.  
Wed
11/11		 NO CLASS -- COLLEGE CLOSED for Veterans Day
Mon
11/16		 TEST 3    Covers all material on which HW was asssigned, up through Section 8.8   NOTE:  Section 9.1 will NOT be on this test--it will be on Test 4.  
 
You may use ONE  8 1/2 x 11 sheet of notes--both sides.  No other notes, no book.

     BRING SOME PAPER of your own.   This can be paper you will rip out of a notebook (as long as it has nothing written on it.)  Or any other kind of blank paper you wish.  Bring several sheets.
BRING your OWN Calculator.  NO SHARING of calculators during the test. 
Wed
11/18		 HW Due
p. 632 9-17 odd   This was assigned before Test 3, but you were not tested on it.  Make sure you have done it, and I will answer questions on it next time. 9.1
p. 702 1-21 odd,
25-61 [ending in 1,5, 9]  ones ending in 3 and 7 are optional		 10.5
p. 708 1-11 odd, 21, 23  Graphs--do by hand.  Show enough points in a table to figure out the trends in "r".  If you have trouble, show more points, but then think about how you could have shown less points once you see the trend.
You may of course then check the graph on your graphing calculator--make sure you know how to do that, too, using the polar mode.
31-37 odd   Intersections:  graph them first--for this you may use the calculator, though if you need more by-hand practice, these might be good to do by hand.
                    Then solve for r or r^2, set things equal, solve equation.  (Factor if possible, avoid QF unless needed.)  If you get "trig fcn (theta) = k", use your inverse trig-button of k on your calculator, but then THINK about what
                     other quadrant would also have a solution with the same reference angle (and the correct plus/minus sign.)  
                     LAST, see if other OBVIOUS intersections show up on the graphs that the by-hand solutions missed (because the curves may reach the intersection points at different angles theta.)		 10.6
Mon
11/20
       
       
       
 
   
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