M140. Homeworks assigned. Dr. Leisinger, Spring 2013. Updated 5/03/2013, 12:05 am. Name of this file: http://math.umb.edu/~aleising/M140/Homeworks.txt ============================================================= Syllabus for course: To be provided by 1/31/2013 ============================================================= Please note that the numbers below in the first column are the homework # for each assignment ! HW#, Date Date Assignment Assigned Due 0:1/29 1/31 Read math.umb.edu/~aleising/General_Information.txt and follow the directions there 1:1/29 1/31 Read math.umb.edu/~aleising/M140/Homeworks.txt Read math.umb.edu/~aleising/M140/Textbook.txt Get a hardbound homework notebook! Only the sew-in, 9 3/4" x 7 1/4" 100-page is acceptable. Carefully number its pages as directed. Set up your contents on sides 1,2,3,4. ====> For good and bad examples of setting up your HW notebook: see: math.umb.edu/~aleising/HwNotebookSetup.pdf 2:1/29 1/31 Take the two diagnostic tests: (A) for algebra and (C) for functions. These are found in the preliminary section of the textbook. For those who do not have a text yet, the diagnostic tests are found in this directory at DiagnosticTests.pdf WORK IS TO BE DONE IN YOUR HW NOTEBOOK! 3:2/1 2/4 Section 1.1: Functions WORK IS TO BE DONE IN YOUR HW NOTEBOOK! Read and study the following examples on pp 11-17: Ex #1-4,6-10. [ note: " * " means written work! ] * pp 19-22: #1,2,7-10,12,18,28,31,38,42,43,47,68,71,72 For those still without a textbook, pages 11-22 will be found in this directory in text_pp_11-22.pdf 4:2/5 2/7 Section 1.2: Functions * pp 33-34: 1,2,3,4,8,9,10,19,22,28. 5:2/9 2/12 Section 1.3 * pp42-44 #1,2,9,10,13,14,15,18,28,31,32,33,41,42 Section 1.4: Read example 1 p. 45. * Section 1.4 pp 49. #5 6:2/9 2/12 Section 1.5: Limits Read examples pp 51-58 1-10 Read definitions 1-6 pp 50-57 (in boxes) * p59 #1,2,3,4 Section 1.7: Compare the definition of limit on p.73 with the one given in class. Note the in-class use of "For All" and "There Exists" quantifiers. Understand how these two definitions mean exactly the same thing. Memorize the in-class definition. (or, if you missed class, memorize the p.73 definition). 7: 2/12 2/14 Some limit problems. * Section 1.5. p59-61 #5,7,38,39. Section 1.6,Limit laws. Learn the first 5 limit laws. (see box p. 62) Don't forget, these laws are INCORRECT as stated in the text. In each case, the statements should read: "WHENEVER THE LIMITS ON BOTH SIDES EXIST". Read and study Examples 1-5 pp 63-66. * Exercises 1.6,p69; #1 (except part c);#3,4,5,10-14,20. 8: 2/17 2/19 More limit problems. Learn the definition of Infinite Limits (box, p56) * p70 ALL #21-26,47. Study examples #1,2,5 pp74-79. * p81 #19,21,22,41,42,43 Learn the definition of continuity in the box on p.82. Study p 82-83 examples 1,2 9: 2/19 2/26 Continuity exercises. Note: this is a rather long assignment. The "LEARNING" section of the assignment is to be done first. It should be completed by 2/21. The exercises should be started. Complete them if you can. However, they are not due until 2/26. [Review the epsilon-delta definitions of limit, continuity at a point; and infinite two-sided and one-sided limits.] Learn the definition of continuity on an interval.[Box,p.84] Learn: the following functions are continuous on their domains: any polynomial with real coefficients. any root function (positive integer roots) any rational function, that is, a quotient of two polynomials over the reals. any of the six trigonometric functions [see Theorem 7, p. 86] log(x) (to any base); b^x (for any base b) Learn: whenever f(x) and g(x) are continuous, then so are: fg(x) (function product); (f+g)(x); (f-g)(x); (f/g)(x); [See theorems 4 and 5, pp 84 and 85] function composition: f(g(x)) [see Theorem 8 p.87] on any interval on which they are defined. ==> Learn and understand the Intermediate Value Theorem, [Theorem 10, p. 89.] Remember that we have not proved it, because the textbook has not sufficiently defined the real numbers. Understand the example #9 on p. 89. This is a special case of the Fundamental Theorem of Algebra. It can be shown that every polynomial of odd degree over the rational numbers has at least one real root. That is proved almost exactly as in this example. Study the examples in the continuity section. * Section 1.8, pp 90-91. ALL: #1,3-8,11,13,15,23,24,28,29,33,35. For #33, do not use a calculator. Instead, ask yourself: for which values of x does sin(x) = -1? 10:2/23 2/26 The Derivative *p112 #27,28 *p124 #19-21,23,29 11:2/26 2/28 Derivative: using the sum, difference, power and product rules To reference these rules: sum and difference rules: text, p.129, two boxes. power rule: text, p.134 in the box at the bottom of the page. [we have proved this when the power n = an integer > -3. product rule: text, p.130 in the box at the bottom of the page. * section 2.3, p136, #1,3-8,11,15,16,23,26 [ALL] There will be a quiz on 2/28. Contents were announced in class. 12:2/28 3/05 Derivative: using: function composition rule (chain rule), general power rule, quotient rule, and derivatives of the six trig functions, ln(x) and e^x * Section 2.4 p146 ODD #1-15 * Section 2.5 p154 ODD #1-45 Please expect a quiz on this on 3/5. 13:3/5 3/7 Using the derivative. * p137 ODD #31-43. [just some more derivatives to calculate] * p137 ODD #51-58. Please note: the NORMAL to a curve at a point is a line that is PERPENDICULAR to the tangent line at that point. Example: the NORMAL to the circle of radius 1 centered at the origin at a point (cos(pi/4),sin(pi/4)) is the (extended) radius of the circle there, or the line y=x. 14:3/7 3/12 Second and higher derivatives $1.2: Read p120-122. * p125 #41,43,45,46,47.(in #45,46, don't bother with the graphs) 15:3/7 3/12 Implicit differentiation $2.6: Study examples p158-161 [also done in class] * $2.6 pp161-162; Ex.#1-4 (all); ODD 5-29 AND #24,28 16:3/12 3/14 Implicit Differentiation, Expect a quiz on 3/14, on finding the equation of a tangent line using implicit differentation. PROBLEMS of NORMAL DIFFICULTY: * p173 #1,3,5,9,13,15,21 DO NOT EXPECT TO GET A COMPLETE ANSWER TO THE NEXT PROBLEM. You may work on it over the vacation. * p164 #2. (Don't use a computer graphing system or graphing calculator.) This is a challenge problem. It is not due at any particular time. Work on it for a few hours, if you like. At least, evaluate dy/dx implicitly. Try the values c=0; c=-1; c=1; Draw the graphs on the same sheet of graph paper. Use values of dy/dx to help you visualize the curve. 17:3/13 3/26 Related Rates Read the examples in section 2.8, Related Rates. * $2.8, p180, #1,3,5,7,9,17. 18:3/26 3/28 Crit.Points; Linear Approx; Differentials; Taylor Polynomial * $2.8, p181#21,27 * $2.9, p187 ODD #1-7,11-17 * $3.1, p205 #15,17,19,20,25,27,31,33,35 19:3/30 4/2 Curve sketching. $3.5 (Summary of Curve Sketching) Read the explanation and examples in $3.5 pp 239-242 * #2.8 p242. #1,5,9,10,11,13,14 When doing these problems, use as many techniques from the summary pp239-242 as you need. Graphs should be done on graph paper. Use GraphPaper.pdf. Graph enough points to get a reasonable picture of each curve. You should use a separate piece of graph paper for each curve. Hold the graph paper vertically. Use the scale 1 box = 1 unit. Label each piece of graph paper with your name, hw#, problem #. 20:4/2 4/4 Prepare for a quiz on curve sketching on 4/4. Review all of our past lessons. Especially: limits. derivative. Intermediate Value Theorem Mean Value Theorem Fermat Lemma on f'(x) = 0 and local maxima/minima Critical points Second derivative test for max/min No other written homework. 21:4/4 4/9 Antiderivatives Read carefully: Section 3.9, pp 269-271 (for now, ignore examples 6 and 7) * p273 ODD #1-39. 22:4/9 4/11 Area by Antiderivatives Worksheet (do problems in your HW notebook; put answers on the worksheet if you wish.) Area_by_Antiderivatives.pdf (worksheet in this directory) 23:4/13 4/16 Area by lower and upper sums. * $4.1, p293 #1 (no calculations; just count rectangles) * $4.1, p293 #3 --> If you remember your trigonometry and half-angle formulas, you should be able to do this without using a calculator for cos(x). You'll need cos(pi/8),cos(pi/4),cos(3 pi/8) and cos(pi/2) --> After doing it that way, do the problem a second time using a calculator for the cosine. Compare your results. * $4.1, p293 #5 * $4.1, p293 #19,21,23. For problem #23, "i" is an integer, not sqrt(-1). 24:4/18 4/25 Using the Fundamental Theorem of Calculus. Read the proof of the Fundamental Theorem I, pp 312-313. On pp 313-314, read the examples #2 and #3. Read and be able to use the Fundamental Theorem Part 2 (box, p315). On pp 315-317, read the examples #5,6,7,8. * $4.3. p318 #5,6, and ODD #7-23. 25:4/23 4/25 Using the Fundamental Theorem of Calculus. * $4.3. p318 ODD #25-41. Note that #37,39,41 involve analysis of continuity. 26:4/23 4/30 Substitution rule for integration. Read p330-333. * $4.5 p335 ALL #1-4; ODD 5-17. 27:4/30 5/2 Areas between curves * $5.1: p349 #1,3,5,7,13,15,19 28:4/30 5/2 Volumes * $5.2: p360 #1,3,5,7,13,39 [and find the volume in #39, also] 29:5/2 5/7 Newton-Raphsen method Use the method shown in class to do the following: * (a) Calculate square root of 2 to 4 decimal places, using the starting value of 2. Remember, you are finding a root of f(x) = x^2 - 2 . Do calculations exactly, until you need a calculator. Make a table like the one shown in class. Terminate the iterations when your answer agrees with a calculator result for sqrt(2) to 4 significant digits. * (b) Calculate square root of 10 to 5 decimal places. use the starting value of 3. * (c) Calculate square root of 3 to 5 decimal places. use the starting value of 2. * (d) Calculate square root of 3 to 5 decimal places. use the starting value of 5 (not a very good guess!) * (e) Discuss the difference between your results in #c,#d. Write something intelligent. THERE WILL BE ANOTHER HW ASSIGNED, likely later.