Math 143 Calculus III Final Exam Key Fall 2024 PDF

Math 143 - Calculus lll Name KEY Instructor: Michael Miller Date 3/18/24 No Calculator Section: Your Score:...

Math 143 - Calculus lll Name KEY Instructor: Michael Miller Date 3/18/24 No Calculator Section: Your Score: Final Exam out of 200 Clearly show all relevant work in order to receive fullcredit. Simplify all answers when possible and leave them in exact form, unless stated otherwise. Relax and good luck! 1. True or False (2 pointseach) Circle T if the statement is always true, otherwise circle F. Youdo not need to justify your answer. (a) T or(F If lim,oa, =0, then a,is convergent. (b)(T or F n! e n=0 (c)Tor F If f() =2x - x² +x converges for all x, then f"(0) = 2. f"(o) =) f(o)- 2 n 31 (d) T or(F If x=f() and y g(t) are twice differentiable, then d'y d'ylar? dx? d'x/d1? (e) T or(F The vector (3, -1, 2) is parallel to the plane 6x- 2y + 4z =1 It is perpendiculas to tu plane. 2. Evaluating Series (12 points) Show whether each series converges or diverges using an appropriate test. If the series converges, then find the value of the sum if p0ssible. -3|"-1 (a) (-3)=1 n=1 23n + 8t3 Converges n'+ 1 (b) n=1 In 2n' + 1 im In In (1) to TF) Diverges by 3. Absolute and Conditional Convergence (12 points) Determine if the series converges absolutely, converges conditionally, or diverges. Give a clear justificationwith the appropriate tests and check their necessary conditions. (a)(-1)-1 cheel In(n + 1) Converses n=1 Coadikonally la AST: In(nt) But | Since Jnlar)> lnCat) diergs Senes) lim n ln nt) So laa diverges DCT (-9)" (b) n=] nl0+1 Check lan n lot7 Ratio Test lan-l im n n+)t n’o Converges Absolutely 3 4. Repeating Decimal Representation (8points) Express the given number as a ratio of integers in simplest form by first representing it as a series, then evaluating the series. 21 2 1.321 = 1.32121212121... (O,021= = |,3t 0.02 =o 1333 330 330 429 +7 330 q90 2218 436 330 330 5. Interval of Convergence (12 points) Find and state the values of xfor which the series converges by giving the interval of convergence. Also state the radius of convergence. |5x-4| (5x - 4)" Ratio Test: (ne) 5x-4|" n=1 n' |5x-4| pee P>| |5x-4|