M130. Homeworks assigned. Dr. Leisinger, Spring 2013. Updated 5/07/2013, 2:38 pm. Name of this file: http://math.umb.edu/~aleising/M130/Homeworks.txt ============================================================= NOTE: Special Math130 Tutoring !!! ======================== EXTRA M130 Sessions with Tutor: ==== We have an excellent tutor for this semester! ===> SEE: Tutoring.txt ============================================================= ============================================================= Syllabus for course: To be provided by 2/11/2013 ============================================================= Please note that the numbers below in the first column are the homework # for each assignment ! HW#, Date Date Assignment Assigned Due 0:1/28 1/30 Read math.umb.edu/~aleising/General_Information.txt and follow the directions there 1:1/28 1/30 Graph lines (handout) WORK IS TO BE DONE ON THE GRAPH PAPER HANDED OUT! 2:1/28 1/30 Read math.umb.edu/~aleising/M130/Homeworks.txt Read math.umb.edu/~aleising/M130/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 3:1/30 2/1 Field Axioms. The Rational Numbers (Q), the Real Numbers (R), and the Complex Numbers (C) are FIELDS. Learn VERY WELL the FIELD AXIOMS (box, pp 4,5). WORK IS TO BE DONE IN YOUR HW NOTEBOOK Textbook p. 7-8, problems #1-10,ALL. SEE Foerster_pp_4-8_15.pdf in this directory. 4:1/30 2/4 Order of Operations WORK IS TO BE DONE IN YOUR HW NOTEBOOK p15, Odd problems, #1-9,25-35 SEE Foerster_pp_4-8_15.pdf in this directory. 5:2/1 2/4 Number systems. WORK IS TO BE DONE IN YOUR HW NOTEBOOK See Algebra_In_Number_Systems.pdf in this directory. * Do Exercises #1,2,3,4,5,6. [NOTE : "*" means "written work". ] Exercises #1,2,3 should be relatively easy. Draw a 2-clock, a 3-clock, and a 4-clock to help you. Watch out. The 3-clock has the numbers 0,1,2 (but not 3). The 4-clock has the numbers 0,1,2,3 (but not 4). To multiply 2 by 3 in Z/(4), for example, you can do either: (a) 2x3=6; Then start at zero and count six around the clock. or (b) Start at 0 on the clock. Count 2 places, and do that 3 times. Where do you end up? Exercises 4,5, and 6 are somewhat more difficult. 6:2/4 2/6 Factoring. See: FactoringAnything.pdf READ IT VERY CAREFULLY !! Work problems #1-10 IN YOUR HOMEWORK NOTEBOOK. Leave space for the rest of the problems, which you will continue after the next class. 7:2/4 2/6 Polynomial rings. a) All rings are fields. What property do fields have that rings do not? << NOTE CORRECTION HERE 11/35 pm 2/4/2013 b) Can you verify that the polynomial ring Z[X} of polynomials in one variable over the integers Z, is a ring? c) Can you verify that the polynomial ring Z/(2)[X} of polynomials in one variable over the integers modulo 2 Z/(2) is a ring? These are polynomials with coefficients in the 2-clock. 8:2/4 2/8 Look at Algebra_In_Number_Systems.pdf again. I added exercise 7. It is a bit more difficult than the earlier exercises. Please don't discuss this with anyone, and don't look online. Try to work out this problem by yourself. If you get an answer, don't share it with anyone. 9:2/7 2/8 More factoring exercises Text section 7-3. START this HW assignment now. Try to do as much as you can. DUE DATE: 2/11 Carefully read pp 328-332. LEARN box p329; box p331. * Carefully work examples 1,2,3,4 on pp 330-331 in your HW book. Follow the book's method. * p333 ODD PROBLEMS 3-31, 43-61, 63-81. DO NOT COPY ANSWERS FROM THE BACK OF THE BOOK. WORK THE PROBLEMS; THEN CHECK YOUR ANSWERS. 10:2/7 2/13 FactoringAnything.pdf "How to Factor (almost) anything)" * Fill out the answer sheet completely. Do the work in your HW notebook. Turn in the Answer Sheet on 2/13. 11:2/11 2/13 Distance Formula; Circles READ: Text pp462-464. (Disregard the example for now.) LEARN VERY WELL the three forms of the distance formula, p463. [These forms have circled numbers 1,2,3 on the right] UNDERSTAND:the distance formula IS the Pythagorean Theorem. Write the equation of each of these circles: a. Radius 5, center (0,0) b. Radius 5, center (1,3) c. Radius 7, center (-1,-3) d. Radius 1, center (-1,0) e. Radius = (square root of 2), center (5,7) f. Radius = (square root of 7), center (1,-6) g. Radius = (square root of 11), center (2,4) h. Radius = 11, center (2,4) CONTINUE TO STUDY FACTORING !!! 12:2/13 2/15 More factoring. Note that HW#10 WILL BE COLLECTED on Friday, 2/15. [Don't worry about #31,32 on the Factoring Anything sheet]. * p334 83,85. Learn BOX p. 338 (the Discriminant) * p340 ODD #5-35 13:2/17 2/19 Circles: Completing the Square. Read Example and discussion, pp464-466. * pp466-467. ALL problems #1-12 14:2/21 2/22 Circles: Finding the equation of a circle, if you know the center and a point on the circle. This is a two-step problem. Suppose the circle has center at C=(c,d), and it contains the point A(a,b). Step 1. use the distance formula to find the distance from C to A. d^2 = (c-a)^2 + (d-b)^2. That distance is the radius of the circle. Step 2. Now write the equation of the circle using the method you already know. * p 467 ALL #13-18,20. STUDY ALL PAST WORK, ALSO. 15:2/23 2/25 Quadratic functions (Section 5.1) Read carefully Section 5.1 pp 174-175. * Do exercise 5.1 p 175 ALL #1-7. Use a piece of graph paper for the graph. You can use "GraphPaper.pdf" in this directory as graph paper. STUDY ALL PAST WORK, ALSO. 16:2/26 2/27 Handout: Graphing Quadratics * (put answers on sheet) * (no graphing yet; don't calculate roots.) 17:2/27 3/01 Handout: Graphing Quadratics * (put answers on sheet) * calculate roots * graph #1-12 on graph paper handout page. DO NOT CHANGE THE SCALE ON THE GRAPH! 1 box = 1 unit. 18:3/1 3/4 Graphing Quadratics, again * p179-180 ODD #1-29 Please sketch any graphs in your HW notebook. 19:3/4 3/6 Examples on graphing quadratics Text, pp 181-186: study the examples. For the following equations, FIND THE ROOTS. If you can factor the polynomial, do so. Otherwise, use the Quadratic Formula to find the roots. * p186-187 #1-19 ODD and #16. Follow directions in the book for these problems: * p187 #21-29 ODD and #28. 20:3/7 3/8 Complete the Square to transform a quadratic to vertex form. Carefully work through example 1 on p. 178 of the text. Copy it into your HW notebook. The vertex form of a quadratic is: y - k = a (x-h)^2. The vertex is at the point (h,k). Just as with circles, the reference point [vertex] (h,k) is found as "y - (the y-center)" and "x - (the x-center}" The number "a" is the steepness. *THEN: for each equation below, transform to vertex form. *After that, find the roots using the Quadratic Equation. *What do you notice? *(a) y = 3x^2 + 4x - 5 *(b) y = 3x^2 + 4x + 5 3/8 NO NEW HOMEWORK 21:3/11 3/13 Transform Quadratic to vertex form Use the method given in class. Or see: ConvertQuadraticToVertexForm.docx If you were not in class, you may use the method on p.178. * p179 #1-5,7,13,18; ODD #19-29; #33. 22:3/15 3/25 Exponentiation * P234 #1-10 (no calculator) * P234 #11-14 (use a calculator) LEARN Properties of exponentiation (blue box, p.237) * P239-240 ODD #1-31 (you may use a calculator for #21,23,25) 23:3/25 3/27 Graphing simple exponential functions * Handout given in class. (do work on sheet) online copy: GraphingExponentialFunctions.pdf 24:3/27 3/29 Function Transformations * Try to take the practice quiz which is found online: * Functions_Shift_Mirror_Domain.pdf Problem 7 did not print correctly. It is: f(x) = |x| + 1/( sqrt(x+1) ) Do not use a calculator for this assignment. 25:3/30 4/1 Graphing logarithmic functions * .pdf file on line: GraphLogs.pdf Do your work on the worksheet. 26:4/1 4/3 Logarithms. Learn the following properties and definitions of logarithms. Blue boxes pp 264; 265; 269; 270. Be aware that because Foerster is an older book, it was written before calculators were generally available. The answers in the sections on logarithms, exponentials, and trigonometric functions were calculated using tables for logs and trig functions. These answers are correct only to 4 or 5 decimal places. Many problems do not ask for or require a calculator. But whenever you use your calculator for a problem in these sections, please give AS MANY DIGITS AS POSSIBLE in your answer. DO NOT ROUND OFF YOUR ANSWERS. THIS IS TRUE THROUGHOUT THIS COURSE. Read and study p276 examples #1-3. * p278-279. ODD #1-49. Calculator is used ONLY for #1-6. 27:4/4 4/8 Logarithms. Change of base formula. Read carefully: text pp 283-284. * p285 ODD #1-19 Also, READ and perhaps prove the following properties: Look at them until they don't seem strange. Try a few examples for yourself. * pp 286-287 ALL #27-34. 28:4/6 4/8 Logs & Exponentials as mathematical Models Read Section 6.14 p300-302. * Fully work the example "Bacteria Problem" p301-2 in your HW notebook. Try to work each step; then check your step with the text. * p303 #1 (Population problem.) Set up this problem with time t (years) so that 1970 is t=0. In part (c), use a web search to find the US population now. Compare it to your result from your formula. In part (f): in what year was the Declaration signed? What is the value of your variable t in that year? 29:4/8 4/10 Introduction to Trigonometry Note: our textbook uses these words for sin(), cos() etc: "circular function" : when the input is in radians "trigonometric function" : when the input is in degrees I will always call them trigonometric functions. An input argument in DEGREES MUST HAVE a degree mark. An input argument in RADIANS MUST NOT HAVE a degree mark. Read carefully: pp 710-716. Especially take note of notes #1,2 on p.712. Understand: COTERMINAL angles. REFERENCE angle. Read examples 1,2,3 pp 714-715. * p717 ODD #1-9 * p717 ALL, #13-21 30:4/10 4/12 Unit Circle (handout) Follow directions given in class 31:4/10 4/12 Graphing the Cosine function (handout) Follow directions given in class 32:4/12 4/17 Graphing the Cosine function (handout) On the same sheet as HW 31: * use your calculator (in radian mode!) to find cos(0), cos(.1), cos(.2), ..., cos(6.2), cos(6.3) Enter this data in a table in your HW notebook. Then, plot all of those points on the graph for HW#31. 33:4/17 4/19 Graphing sinusoidal functions a. Read text pp 742-746. [Thanks, Emeka!] b. Read "Al's_Box_Method_2.docx" on the website. c. Read text section 13.7 p 750-751. * p752 #Q1-Q10. Notes: for Q3: "sinusoidal axis location" is our "C", or "center line height". Notes: for Q4: "phase displacement" is our "D", the x-shift. * p754 #9,10,11,12. For each sinusoid, write the equation. Use the equation y-C = A cos (B (x-D) ). Notes: find the period (distance from crest to crest). Then find B = (2 pi)/period . 34:4/22 4/24 Graphing sinusoidal functions In EVERY problem, find A,B,C,D; calculate p. Write these values in a neat table! Then use the box method for graphing. * p749 #5,6,7,8,9,10. (Use GraphPaper.pdf, held |=======| this way, for graphing.) . . . . For the next group, find A,C,D,p from the graph; calculate B. Then write the proper equation. * p752 #1,3,5,6.[Keep everything in degrees. B=(360 degrees)/p] 35:4/24 4/26 Review of sine and cosine definitions On p713, read the notes to figure 13.2B and figure 13.2C. Understand: For any angle,the "reference angle" is the positive acute angle between the terminal side and the u-axis. Re-learn the definitions of the sine and cosine (box p718) Re-learn the definitions of the six trig. functions: (see box p.800) tan = sin/cos; cot = 1/tan; sec = 1/cos; csc = 1/sin * p717 ODD #1-23. (redo; this was part of HW #29) * p726 ODD #1-35. 36:4/24 4/26 [no written work; but prepare for class on 4/26 as follows:] What is an EVEN function? what is an ODD function? ( See box p815.) Which trig functions are EVEN? which are ODD? ( See box p816.) In the 4/26 class, we'll learn the COSINE OF A SUM formula: cos(A+B) = cos(A) cos(B) - sin(A) sin(B) The proof is on the website. I'll hand out copies of this proof in class. 36:4/26 4/29 Using cos(x+y) formula Use: sin(x+y) = sin(x)cos(y) + cos(x)sin(y) Use: cos(x+y) = cos(x)cos(y) - sin(x)sin(y) * p822-24 #1,3;#7,8,9,20, 27,29,31,33,34,35. (Note: 15 = 45 + -30; 75 = 45 + 30) 37:4/29 5/1 Trig: cos(x+y); law of cosines * p822-24 #38,39,41,44. Learn: law of cosines (box p874). Learn: law of sines (first box p884). * For examples #1-4 pp 876-878: Work each example as best you can. Then read the book's answer. Finally, make sure you can do each problem. * p879 #1,3,7,9. Use a calculator, give angles in degrees. =====>>>> NOTE: on p879, angles are given in degrees and minutes. For example, p879#1: angle C is given as 71 degrees 40 minutes. That means 71+(40/60) degrees, or 71 2/3 degrees, or 71.666666666666 degrees. Write ALL DECIMAL PLACES given by your calculator. Check your answers against the book's answers. Realize that the book's answers are based on trig. tables accurate only to a few decimal places. This is because the book was written before calculators were generally available. THERE IS NO NEW HW due on 5/3/2013. See note to HW#37, above, about degrees and minutes. 38:5/6 5/8 THE LAST HOMEWORK! Trig: Triangle Area Formula. cos(x/2) formula. astronomy. === Trigonometric equations: $14.9: read examples 1-6, although these are harder than the ones I am assigning. * p856 #1,3,5,7,8. === Triangle Area Formula $15.3: learn box p881. Read examples #1,2. * p883 #1,3,5. === Cos half angle formula (see box p847, #8, second entry). * find the following cosines in exact form (no calculator) A). cos(15 degrees) B). cos(7.5 degrees) C). cos(3.75 degrees) <<< CORRECTED FROM EARLIER VERSION (Thanks, Nathaniel!) After you have the exact form, use your calculator to compare your worked-out answer with your calculator answer for these cosines. The two answers should agree. === Astronomy. Light travels 3x10^8 meters / second. Light travels from the Sun to the Earth in 8 minutes. * D). How far is the sun from the earth? A parsec is the length of the long side of an isosceles triangle whose third side is the diameter of the Earth's orbit, having its smallest angle equal to one second of arc (= 1/3600 degree). Find the size of a parsec. * E). A light-year is the distance light travels in one year. Find the length of a light-year. * F). How many light-years is one parsec?