Math 242 Fall 2009 Please before you start the practice problems, go over the illustrated examples in each section first. Cover up the solutions and try to work out the answers on your own. Practice Problems Go over the illustrated examples in each section. 4.8 Newton s Method #, 4, 23, 29, 30, 3, 33. Problems 29, 30 are good ones. 4.4 L Hôpital s Rules #49, 5, 56, 72, 75, 78, 8, 82. 6. Area # 5, 9, 25, 3, 49. 6.2 Volumes # 5, 7,, 35, 4, 5. 6.3 Volumes #, 3, 5,, 9, 37. 6.5 Average Values #, 9, 3, 8. (Answer to #8 (b) = 76.3 o C). 7. IP not covered in Quiz # 3 to 2, 2, 23, 33, 38, 44, 45, 5. 7.7 Numerical Integration Methods Please go over the illustrated examples. You may skip the mid-point rules her other than #47, 48. Most of the problems require a calculator. #3 (Do the Trapeziodal and Simpson s here), 7, 9, 27, 28, 29, 47, 48. The last two #47 and 48 are simple accounting 0. I ll do #46 in class. For the integral π sin(x)dx, use Simpson s rule with n = 4i to do an approximation. 0 What is an approximation of π obtained here? Note: the answer involves a 2.
7. Integration by parts There is not much here except you just have to do as many problems as possible.. All the illustrated examples are important. They ll appear in future math courses that use calculus. Try #23, 24, 27, 28, 3, 33, 38(good one), 43, 47, 48, 5, 66(a). 7.2 Trigonometric Integrals There is no need to use the table on p.465. Try # 5,, 3, 7, 2, 24, 29, 33, 57, 65, 70. 7.3 Trigonometric Substitutions Please go over all the illustrated examples first. # 4 to 7, 25, 27, 35(use #39 which was done in class) 7.4 PF Again go over examples to 9. Obviously, Example 8 is too long for a quiz or for a test. #7 to 2, 33, 33(let u = x 2 ), 35, 39, 4. 7.8 p.55, Improper Integrals Try # 9, 2, 25, 39(must do!), 59. 0. Parametric Curves # 5, 7, 9, 5, 25, 33, 4, 46. 0.2 Calculus on Parametric Curves # 3, 5, 7, 7, 25, 29, 30,, 4, 43, 45, 54, 57, 59, 6(straight forward problem, do your algebra carefully here), 73. For #73, find the position of T first. Express the line TP in terms of the angle θ and then use simple trigonometry as done in cycloid case, Quiz II Monday 0/26 on 7.4, 0. and 0.2 The quiz will run for 30 minutes. As a warm-up, allow yourself 45 minutes to work on the following straight forward problems: A. Evaluate Answer p.48 #5. x 4 x 4 dx. B. Give a reasonable plot of x = cos(t), y = sec(t) for 0 t < π/2. Find the equation of the tangent line at t = π/4. 2
C. Find the total length of the astroid defined by x = cos(t) 3, y = sin(t) 3. D. Set up a definite integral giving the surface area of a horn-like object obtained by rotating x = t, y = /t for t in [,5] about the x axis. The integral can actually be evaluated by changing it to the form 24 on the reference page in the back of text. A picture is given at the bottom of p. 537. E. p. 638, #73. 0.3 Polar Coordinates Here make sure you do not mix up graphs in rectangular coordinates to those in polar. #3, 5, 7, 9, 3, 33, 37, 43, 47, 49, 57, 59, 63, p.670: 7 and 8. 0.4 Area in Polar Coordinates For the problmes here, please review the basic graphs of some of the well-known polar curves: circle, cardiod, limacon, lemniscate, n-leaved rose. #7, 9, 3, 9, 2 23, 25, 27, 29, 35. 0.5 Conic Sections # 5, 7, 5, 6, 9, 23, 27, 29, 3, 37, 43, 53, 56, 57, 58. Review questions for this Chapter. Allow yourself two hours to do the following problems: pp.670-67 # 3, 35, 37, 4, 47, 53, 57.. Sequences Please go over Example 3 again. # 3, 7, 9,, 3, 9, 2, 23, 27, 35, 45, 54, 58, 6, 62, 65, 67 to 7, 79, 80. Please do 69 and 70 by induction. Solution to #70 by induction. Please review Example 3 first. We do two separate inductions on the boundedness and on monotonicity. Note that all the inequalities are derived by some scrap work first before they are presented here. A. a leq2. Assume a n leq2, then 2 a n. Hence 3 a n. Both quantities are positive. By switching sides, we get 3 a n. The left side is none other than a n+, so in particular, a n+ 3. B. a = 2, a 2 =. Therefore a a 2. 3
Assuming a n+ a n, then a n a n+. In particular 3 a n 3 a n+, which is equivalent to saying. 3 a n+ 3 a n Note that we used the fact that both quantities are positive. The last inquality is equivalent to saying a n+2 a n+. Now both assumptions in the Monotonic Convergence Theorem have been verified, the sequence has a limit L. To find L, we solve for the equation, just like in Example 3, that L = /(3 L). Solving for L, we pick L = 3 5. 2 What is wrong with the other solution? Induction Principle Let S(n) be a mathematical statement on an integer n. The induction principle allows you to prove the statement for all positive integers n if you can verify the following two steps: (a) S() holds. and (b) S(n + ) holds under the assumption that S(n) holds. Further exercise on Induction. () Prove that n k 2 = n(n + )(2n + )/6. k= (2) Prove that n k= k(k + ) = n +. (3) Prove that 3 n 2 n+ for n 2. In this case, step one starts from n = 2..2 Infinite Series A series is the limit of the n th partial sum of a sequence a n. #7, 9,, 3, 7, 2 to 3(odd ones), 35, 37, 4, 43, 47, 49, 55, 64, Further review problems for test two. Allow yourself exactly 70 minutes of time. p.58 # 6. Answer to #6. ln y 6 ln y + 2 + C. 8 8 pp. 670-67 #35, 4, 47, 57. Solution to #57 - longbow curve 4
The problem is to find the positions of Q and R respectively as a dunction of θ. Let T be the point on the top of the circle with coordinates (0, 2a). By trigonometry, OT/TR = tan(θ), i.e. TR = 2a cot(θ). Hence for the position of R, x = 2a cot(θ), y = 2a. For Q, the polar coordinates are r = 2a sin(θ). This is similar to Example 6. Therefore, the point Q on the circle has coordinates x = r cos(θ) = 2a sin(θ) cos(θ), y = r sin(θ) = 2a sin(θ) 2. P is the mid-point obtained by taking the average of the x and y coordinates of Q and R respectively. p.759 #6, 9, 27. Section.7 Review on convergence of Series Please go over the odd ones in # to 27 on p.722 as a review on various convergence tests. Difference between limit comparison test and ratio test Some of you who had high school calculus in series get these two tests mixed up. The Limit comparison test works on two different positive series a n and b n. The statement is that the convergence of one series implies the convergence of the other if a n lim = L > 0. n b n The big requirement is that the value L has to be strictly positive. It can be 4.5, 0.4,.... The ratio test also involves a limit but it is on two consecutive terms a n and a n+. The statement is lim a n+ = ρ. n a n The conclusion is that the limit ρ has to be smaller than, even zero limit is allowed, for the series a n to converge. So do not mix up these two tests..8 Power Series Try # to 9 odd ones. You need to do the end-point check..9 Geometric Series Please go over the examples first. Try #3, 5, 9 (do simple partial fractions first), 5, 2, 25, 27, 32 (For #32, differentiate the series twice and show that the 2nd derivative is -f(x) ). 5
.0 Taylor Series Go over the illustrated examples and the ones you saw in Maple. Make sure you can derive those seen in Table on p.743. Try # 2(good one!),, 3, 5, 7, 9, 3, 9, 29, 3, 4, 47, 49, 5, 53, 55, 63, 67.. Further Examples on Taylor Go over Examples and 2. Try # 5 to 7, (there is no need to plot), 3 to 2 odd ones in this group, 27 to 29. Review problems for this chapter. p.759-760 # 3, 7, 5, 7, 2, 27, 29, 3, 33 (hint: find the sum of the series for e x and e x ), 34 (ans: when ln(x) < which is /e < x < e), 4, 43, 45, 47, 5, 55, 56 (Use the Binomial series, first 3 non-zero terms), 59. 6