M140. Homeworks assigned. Dr. Leisinger, Fall 2015. Updated 12/5/2015, 7:41 pm. ADDED REVIEW MATERIALS (See the bottom of this document!) Name of this file: http://math.umb.edu/~aleising/M140/Homeworks.txt ================================================ FINAL EXAM: Friday, December 18, 2015, 6:30 pm - 9:30 pm. DO NOT MISS THIS EXAM !!!!!!!!!!! DO NOT plan to leave Boston before the scheduled time for this exam. ================================================ ============================================================= Syllabus for course: To be provided Tutoring for course: see "Tutoring.txt" file in this directory [Tutoring.txt was last updated on 9/17/2015 at 2:40 pm] ============================================================= Please note that the numbers below in the first column are the homework # for each assignment ! HW#, Date Date Assignment Assigned Due 0:9/5 9/9 Read http://www.math.umb.edu/~aleising/General_Information.txt and follow the directions there. You should also read the file: http://www.math.umb.edu/~aleising/Preparation_For_My_Classes.txt 1:9/5 9/9 Take the diagnostic tests (A,B,C,D) on this website, in the file "DiagnosticTests.pdf". Write your work in neat format to be turned in at the first class. 2:9/5 9/11 Number Systems, and the Field of Real Numbers THIS HOMEWORK IS TO BE DONE IN YOUR HW NOTEBOOK. MAKE SURE IT IS IN CORRECT FORMAT. Learn the contents of: (a) NumberSystems_Notes.pdf (in this directory) 3:9/5 9/11 HW Notebook Format You will receive credit for this assignment when your HW notebook is in correct format. 3A: 9/10 9/11 Real Numbers [Note: this HW number is "3A"] Read and understand the file "RealNumbers.pdf" which is found in this directory. * on a separate sheet of paper, answer these questions: 1. Give several examples of real numbers. 2. Explain in your own words how to represent the number 2.57999999... (repeating) as a terminating decimal. 3. Convert 3/7 into a repeating decimal 4. Convert .51515151... (repeating) into a fraction. 5. Explain why a non-repeating decimal cannot be represented as a fraction. 3B: 9/14 9/14 Set theory and logic review Read and understand: ElementarySetTheory.pdf 1.1:9/10 9/14 Section 1.1: Functions (25 problems) [Note: this HW number is "1.1"] [PLEASE NOTE DUE DATE] 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 1.2:9/12 9/16 Section 1.2: Functions (10 problems) [Note: this HW number is "1.2"] * pp 33-34: 1,2,3,4,8,9,10,19,22,28. 1.3:9/12 9/16 Section 1.3 New Functions From Old (17 problems) * pp42-44 #1,2,5,9,10,13,14,15,18,28,31,32,33,41,42,47 Section 1.4: Read example 1 p. 45. * Section 1.4 pp 49. #5 1.5A:9/16 9/18 Section 1.5: Limits (6 problems) Read examples pp 51-58 1-10 Read definitions 1-6 pp 50-57 (in boxes) * p59 #1,2,3,4,5,6 1.7A:9/16 9/18 Limit definition 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). 1.5B:9/16 9/21 Some limit problems. (14 problems) * Section 1.5. p59-61 #7,29,31,38,39. 1.6A:9/16 9/21 Limit laws. (14 problems) 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. 1.6B:9/23 9/23 More limit problems. (16 problems) Learn the definition of Infinite Limits (box, p56) * p70 ALL #21-26,47. Study examples #1,2,5 pp74-79. 1.7B:9/21 9/23 Continuity introduction. (6 problems) NOTE: HW # has been changed (old # was 1.7A) * 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 1.8:2/18 9/25 Continuity exercises. (16 problems) Note: this is a rather long assignment. The "LEARNING" section of the assignment is to be done first. It should be completed by 9/28 The exercises should be started. Complete them if you can. However, they are not due until 9/28. [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 legal 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), assuming g(x) is not = 0 [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.] 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? 2.2:9/23 9/28 The Derivative (7 problems) *p112 #27,28 *p124 #19-21,23,29 2.3:9/23 9/30 Derivative: (12 problems) 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] 2.4:9/30 10/02 Derivative: (8 problems) 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 #1,2,ODD #3-15 2.5:9/30 10/02 Derivative Chain Rule: (22 problems) using: Function composition rule (Chain rule), * Section 2.5 p154 ODD #1-45 2.3B:9/30 10/05 Using the derivative. (11 problems) * p137 ODD #31-43. [just some more derivatives to calculate] * p137 ODD #51,53,55,56,57. 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. 2.2B:10/5 10/07 Second and higher derivatives (5 problems) Read p120-122. * p125 #41,43,45,46,47.(in #45,46, don't bother with the graphs) 2.6A:10/5 10/09 Implicit differentiation (19 problems) $2.6: Study examples p158-161 [also done in class] * $2.6 pp161-162; Ex.#1-4 (all); ODD 5-29 AND #24,28 2.6B:10/8 10/12 Implicit Differentiation, (2 + 1 problems) PROBLEMS of NORMAL DIFFICULTY: [ Previously, problems listed here were part of 2.6A. Sorry. ] * p161 #1,3 DO NOT EXPECT TO GET A COMPLETE ANSWER TO THE NEXT PROBLEM. * 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. 2.8:10/12 10/14 Related Rates (8 problems) Read the examples in section 2.8, Related Rates. * $2.8, p180, #1,3,5,7,9,10,14,17. 2.8A:10/14 10/16 Related Rates (2 problems) * $2.8, p181#21,27 2.9:10/14 10/16 Linear Approx; Differentials (8 problems) * $2.9, p187 ODD #1-7,11-17 3.1:10/14 10/19 Critical Points; Extreme Value Theorem; Fermat's Theorem (12 problems) * $3.1, p205 ODD #15-35, and #20,36 3.5:10/14 10/21 Curve sketching. $3.5.Summary of Curve Sketching (7 problems) 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 #. xx:10/26 10/28 Prepare for a quiz on curve sketching. [ No written work. ] 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 3.9A:10/26 10/30 Area by Antiderivatives (10 problems) Worksheet (do problems in your HW notebook; put answers on the worksheet if you wish.) Area_by_Antiderivatives.pdf (worksheet in this directory) 3.9B:10/26 11/02 Antiderivatives (19 problems) Read carefully: Section 3.9, pp 269-271 (for now, ignore examples 6 and 7) * p273 ODD #1-39. 4.0:11/5 11/6 Calculus theorems Learn and be able to state and use the following theorems: (find them in the index of the textbook) * WRITE THESE THEOREMS NEATLY IN YOUR HW NOTEBOOK. a. Completeness property of the Real Numbers b. Extreme Value Theorem (EVTh) NOTE: A proof of the extreme value theorem has been posted. See: ExtremeValueTheorem_Proof.pdf c. Intermediate Value Theorem (IVTh) d. Fermat's Theorem e. Rolle's Theorem f. Mean Value Theorem (MVTh) g. Fundamental Theorem of Calculus Part I (FTC#1) h. Fundamental Theorem of Calculus Part II (FTC#2) 4.1:11/5 11/9 Area by lower and upper sums. (6 problems) * $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). ------------------------------------------------------ NOTE: I have shifted the due dates for the next assignments back one class meeting. ------------------------------------------------------ 25:11/5 11/13 Using the Fundamental Theorem of Calculus. (11 problems) 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. 26:11/5 11/16 Using the Fundamental Theorem of Calculus. (9 problems) * $4.3. p318 ODD #25-41. Note that #37,39,41 involve analysis of continuity. 27:11/5 11/18 Substitution rule for integration. (10 problems) Read p330-333. * $4.5 p335 ALL #1-4; ODD 5-17. 28:11/5 11/20 Areas between curves (8 problems) * $5.1: p349 #1,3,5,7,13,15,19,23 29:11/5 11/23 Volumes (6 problems) * $5.2: p360 #1,3,5,7,13,39 [and find the volume in #39, also] 30:11/5 11/25 Newton-Raphsen method (5 problems) Use the method shown in class to do the following: * (a) Calculate square root of 3 to 4 decimal places, using the starting value of 2. Remember, you are finding a root of f(x) = x^2 - 3 . Do calculations exactly, until you need a calculator. Fractions are easier than decimals. Make a table like the one shown in class. Terminate the iterations when your answer agrees with a calculator result for sqrt(3) 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 2 to 5 decimal places. use the starting value of 2. * (d) Calculate square root of 2 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. 31:12/4 12/9 Volumes by Shells (6 problems) NOTE: This HW is not required. If you complete it, you will receive ten extra points for your HW grade. Read text section 5.3. * $5.3: p365 : work example #2. * $5.3: p367 #1,3,5,7,8. ============================= REVIEW MATERIALS FOR FINAL EXAM =============== Review materials for the final exam are found in the directory: Final_Review_2015_Fall_Calculus_I_Math_140.pdf Further review materials, from Spring 2015, are found in this sub-directory: M140_ReviewMaterials/