Math 6 Practice for Exam 3 Generated December, 5 Name: SOLUTIONS Instructor: Section Number:. This exam has 7 questions. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck.. Do not separate the pages of the exam. If any pages do become separated, write your name on them and point them out to your instructor when you hand in the exam. 3. Please read the instructions for each individual exercise carefully. One of the skills being tested on this exam is your ability to interpret questions, so instructors will not answer questions about exam problems during the exam. 4. Show an appropriate amount of work (including appropriate explanation) for each exercise so that the graders can see not only the answer but also how you obtained it. Include units in your answers where appropriate. 5. You may use any calculator except a TI-9 (or other calculator with a full alphanumeric keypad). However, you must show work for any calculation which we have learned how to do in this course. You are also allowed two sides of a 3 5 note card. 6. If you use graphs or tables to obtain an answer, be certain to include an explanation and sketch of the graph, and to write out the entries of the table that you use. 7. You must use the methods learned in this course to solve all problems. Semester Exam Problem Name Points Score Winter 3 6 Winter 3 9 Fourier 9 Fall 4 3 chicken coop 8 Fall 3 5 circuit Fall 3 3 Fall 3 3 4 9 Fall 3 3 8 Total 7 Recommended time (based on points): 84 minutes
Math 6 / Final (April 3, ) page 8 6. [ points] Consider the function f(x) = ln(+x) and its Taylor series about x =. a. [4 points] Determine the first four non-zero terms of the Taylor series for f(x) = ln(+x) about x =. Be sure to show enough work to support your answer. ln(+x) = x x + 3 x3 4 x4 +... ) b. [4 points] Find the first three non-zero terms of the Taylor series for g(x) = ln( +x x about x =. Be sure to show enough work to support your answer. (Hint: You may find it helpful to utilize properties of logarithms.) Using our answer from part (a), we have ln( x) = ( x) ( x) + 3 ( x)3 4 ( x)4 +... = x x 3 x3 4 x4... Since g(x) = ln(+x) ln( x), we have g(x) = x+ 3 x3 + 5 x5 +... c. [ points] Find the exact value of the sum of the series ( ) 3 + 4 3 ( ) 3 3 + 4 5 ( ) 3 5 +.... 4 Using our answer for g(x) is part (b), we see this is the sum for the Taylor series evaluated at x = 3 4, so the sum of this series is g( 3 4) = ln(7). Winter, Math 6 Exam 3 Problem 6 Solution
Math 6 / Final (April ) page 9. [9 points] a. [ points] Find the Taylor series about x = of sin(x ). Your answer should include a formula for the general term in the series. sin(x ) = n= ( ) n x 4n+ (n+)! = x x6 3! + + ( )n x 4n+ + (n+)! b. [points] Letmbeapositiveinteger,findtheTaylorseriesaboutx = ofcos(mπx). Your answer should include a formula for the general term in the series. cos(mπx) = ( ) n (mπx) n n= (n)! = m π x! + + ( )n (mπx) n (n)! + c. [5 points] Use the second degree Taylor polynomials of sin(x ) and cos(mπx) to approximate the value of b m, where b m = sin(x )cos(mπx)dx. (The number b m is called a Fourier coefficient of the function sinx. These numbers play a key role in Fourier analysis, a subject with widespread applications in engineering and the sciences.) b m b m ( ) x m π x dx! x m π x4 dx b m x3 3 m π x5 b m 3 m π 5 Winter, Math 6 Exam 3 Problem 9 (Fourier) Solution
Math 6 / Final (December, 4) page 3. [8 points] Franklin, your friendly new neighbor, is building a large chicken sanctuary. You decide to help Franklin build a special chicken coop with volume (in cubic km) given by the integral x cos(x )dx. This integral is difficult to evaluate precisely, so you decide to use the methods you ve learned this semester to help out Franklin. Your friend and president-elect, Kazilla, stops by to give you a hand. She suggests finding the 4th degree Taylor polynomial, P 4 (x), for the function cos(x ) near x =. a. [4 points] Find P 4 (x). We can use the Taylor series expansion for cos(x ) so Therefore cos(y) = cos(x ) = n= ( ) n y n (n)! ( ) n (x ) n n= (n)! = y! + + ( )n y n + (n)! = (x )! P 4 = ( x4! ) = x4! + + ( )n (x ) n (n)! + b. [4points]SubstituteP 4 (x)for cos(x )intheintegralandcomputetheresultingintegral by hand, showing all of your work. x P 4 (x)dx = = = x4 x x 4 x 3 4 = 4 x= Fall, 4 Math 6 Exam 3 Problem (chicken coop) Solution
Math 6 / Final (December 5, ) page 8 5. [ points] When a voltagev in volts is applied to a series circuit consisting of a resistor with resistancerin ohms and an inductor with inductancel, the currenti(t) at timetis given by I(t) = V R ( e Rt L a. [ points] Show thati(t) satisfies ) di dt = V L wherev,r, and L are constants. ( RV I ). di dt = V ( e Rt/L R ) = V R L L e Rt/L = V ( ( e Rt/L ) ) L ( RV ) I = V L b. [6 points] Find a Taylor series for I(t) about t =. Write the first three nonzero terms and a general term of the Taylor series. Instead of taking derivatives of I(t) at t =, we can use the given expansion for e x = +x+ x + x3 +...+ xn +... about. Here,x = Rt.! 3! n! L I(t) = V ( ) e Rt L R [ ( )] +( Rt L )+ ( Rt L )! + ( Rt L )3 3! +...+ ( Rt L )n +... n! = V R = V [ ( R = V R RL t+ R!L t R3 3!L 3t3 +...+ ( )n R n ( R R t + R3...+ ( )n+ R n t n +... L!L t 3!L 3t3 n!l n = V VR t + VR...+ ( )n+ VR n t n +... L!L t 3!L 3t3 n!l n )] t n +... n!l n ) Fall, Math 6 Exam 3 Problem 5 (circuit) Solution
Math 6 / Final (December 5, ) page 9 c. [ points] Use the Taylor series to compute I(t) lim t t ( V = lim t t = lim t = V L, I(t) lim. t t ) t n +... n!l n ) VR t + VR...+ ( )n+ VR n L!L t 3!L 3t3 ( V L VR!L t + VR 3!L 3t...+ ( )n+ VR n n!l n t n +... since all of the terms involvingtevaluate to. Fall, Math 6 Exam 3 Problem 5 (circuit) Solution
Math 6 / Final (December 4, ) page 5 3. [ points] Let I = ( )5 + t dt a. [5 points] Approximate the value of I using Right() and Mid(). Write each term in your sums. Right() = ( (9 8 )5 + ( 3 ) )5 (.34+.755).494 ( (33 Mid() = )5 ( ) )5 4 + 3 3 (.8+.858).4697 b. [ points] Are your estimates of the value of I obtained using Right() and Mid() guaranteed to be overestimates, underestimates or neither? Right = Overestimate (increasing) Mid = Underestimate (concave up) c. [3points] FindthefirstthreenonzerotermsoftheTaylorseriesforg(t) = about t =. Using the binomial series: (+x) 5 = + 5 x+ ( 5 = + 5 5 x+ ( ) t ( )5 + t 5 = + )( ) 3 x! + 8 x + + 5 ( ) t + 8 = + 5 4 t + 5 3 t4 + ( )5 + t d. [ points] Use your answer from part (c) to estimate I. I + 5 4 t + 5 3 t4 dt = t+ 5 t3 + 3 3 t5 t= = + 5 + 3 3 = 45 96.547 Fall, Math 6 Exam 3 Problem 3 Solution
Math 6 / Final (December 7, 3) page 7 4. [9 points] Determine if each of the following sequences is increasing, decreasing or neither, and whether it converges or diverges. If the sequence converges, identify the limit. Circle all your answers. No justification is required. a. [3 points] a n = n 3 dx. (x +) 5 Converges to DIVERGES. INCREASING Decreasing Neither. (Not required) Since dx dx which diverges by p-test (with p = (x +) 5 x 5 ). Hence, 5 n 3 lim a n = lim dx = dx =. The sequence is increasing n n (x +) 5 (x +) 5 since >. (x +) 5 b. [3 points] b n = n k= ( ) k (k +)!. CONVERGES TO sin Diverges. Increasing Decreasing NEITHER. (Not required) ( ) k x k+ Since sinx =, then lim (k +)! b ( ) k () k+ n = = sin. Since the series n (k +)! k= k= is alternating, the sequence is neither increasing or decreasing. c. [3 points] c n = cos(a n ), where < a <. CONVERGES TO Diverges. (Not required) Since lim n an =, then lim n cosan = cos =. INCREASING Decreasing Neither. Fall, 3 Math 6 Exam 3 Problem 4 Solution
Math 6 / Final (December 7, 3) page 8. [ points] a. [4 points] Let a be a positive constant. Determine the first three nonzero terms of the Taylor series for f(x) = (+ax ) 4 centered at x =. Show all your work. Your answer may contain a. Using the binomial series (+u) p with u = ax and p = 4 p(p ) f(x) = (+ax +pu+ u = 4ax +a x 4 ) 4 b. [ points] What is the radius of convergence of the Taylor series for f(x)? Your answer may contain a. Since the interval of convergence of the binomial series is < u <, then the interval of convergence of the series for f(x) is < ax <, or a < x < a. Hence the radius of convergence is R = a. c. [3 points] Determine the first three nonzero terms of the Taylor series for g(t) = t (+ax ) 4dx, centered at t =. Show all your work. Your answer may contain a. g(t) = t (+ax ) 4dx t 4ax +a x 4 dx = x 4 3 ax3 +a x 5 t = t 4 3 at3 +a t 5 Fall, 3 Math 6 Exam 3 Problem 8 Solution
Math 6 / Final (December 7, 3) page d. [3 points] The degree- Taylor polynomial of the function h(x), centered at x =, is The following is a graph of h(x): P (x) = a+b(x )+c(x ). What can you say about the values of a,b,c? You may assume a,b,c are nonzero. Circle your answers. No justification is needed. a is: Positive NEGATIVE Not enough information b is: POSITIVE Negative Not enough information c is: Positive NEGATIVE Not enough information Fall, 3 Math 6 Exam 3 Problem 8 Solution