MATH 2300 review problems for Exam 3 ANSWERS

Transcription

1 MATH 300 review problems for Eam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justif our answer b either computing the sum or b b showing which convergence test ou are using, wh and how it applies (depending on the case). (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) n (n diverges integral or limit comparison test + ) ( ) n n (n converges alternating series test + ) n (n converges integral, comparison, or limit comparison test + ) n + n n diverges nth term test (that is, the n the term does not go to zero) p n 4 + 5n diverges limit comparison test or nth term test sin n (hint: consider n converges ratio test (n)! (n + 3)! (n + )! diverges ratio test ) converges limit comparison test n converges comparison or limit comparison test test converges ratio test nn. Find the values of a for which the series converges/diverges: (a) (b) n= n[ln(n)] a a > () a a > 0 3. Consider the series ln n. Are the following statements true or false? Full justif our answer. n (a) The series converges b limit comparison with the series (b) The series converges b the ratio test. False n. False

2 (c) The series converges b the integral test. False 4. Consider the series answer. ( ) n ln n. Are the following statements true or false? Full justif our n (a) The series converges b limit comparison with the series (b) The series converges b the ratio test. False (c) The series converges b the integral test. False (d) The series converges b the alternating series test. True n. False 5. The series P a n is absolutel convergent. Are the following true or false? Eplain. (a) P a n is convergent. True (b) The sequence a n is convergent. True (c) P ( ) n a n is convergent. True (d) The sequence a n converges to. False (e) P a n is conditionall convergent. False (f) X a n n converges. True 6. Does the following series converge or diverge? 3 n n You must justif our answer to receive credit. Yes, because, if a n denotes the nth term of this series, then the limit of a n+ /a n as n! is zero. 7. Let f() =. (a) Find an upper bound M for f (n+) () on the interval ( /, /). n+ (n + )! (b) Use this result to show that the Talor series for converges to on the interval ( /, /). B part (a) and b the Lagrange Error Bound Formula, We have E n () apple n+ n+ = ( ) n+ on ( /,, ). But the fact that < / on this interval tells us that lim E n () apple lim ( ) n+ = 0 on this interval. But remember that E n () = f() P n (), where P n () is the nth degree Talor polnomial for f(). So P n ()! f() as!, and we re done. 8. If P b n ( ) n converges at = 0 but diverges at = 7, what is the largest possible interval of convergence of this series? What s the smallest possible? Largest: [ 3, 7). Smallest: [0, 4).

3 9. (a) Which of the slope fields (i) (iv) below could be the slope field for the logistic di erential equation d d Please eplain how ou got our answer. = (5 )? (i) (ii) (iii) (iv) Slope field (ii), because this is the onl one that has zero slope at = 0 and = 5. (b) On the slope field ou chose for part (a) of this problem, sketch in the solution curve for the above logistic di erential equation that has initial condition (0) =. 0. (a) Write down the second degree Talor polnomial P () approimating f() = ln + ( ) 3 near = 0. P () =. (b) Use our result from part (a) to approimate ln(.09). Hint: P (0.) = = ln(.09) = f(0.) (c) What does Lagrange s error bound sa about the error in the approimation ou found in part (b)? You should find it useful to note that f 000 () = ( ) + 4 ( ) 3, and that f 000 () is a decreasing function on the interval (0, /0). E () apple f 000 (0) ! = ( )(4) ! =

4 . Show below are the slope fields of three di erential equations, A, B, and C. For each slope field, the aes intersect at the origin. A B C For each of the following functions, indicate which, if an, of the di erential equations, A, B, and C it could be the solution of. Note that an of the functions could be a solution to zero, one, or more than one of the di erential equations. If a function is a solution to none of the di erential equations clearl write None as our anwer. (a) = 0 C (b) = A,B,C (c) = + ke A, B,C. Solve (make sure to write our final answer in the form = a function of ): (a) d d =, () = = 3p 7 3 / (b) 0 = ( + ), () = = p 5e (c) 0 = cos(), (0) = 3 = 3e sin() (d) ( 3 + ) d 3 = 0, () = ln = ln( + 3 ) d d q (e) e 3 + d 3 = 0, () = e = 3 ln + e e Consider a continuous function f() with f(0) = and f() =. Consider the solution of the di erential equation f() d f 0 () = 0, which satisfies the initial condition (0) =. What is the d value of this solution at =? ln + 4. Let f() = Find the intervals of convergence of f and f 0. For f: [ 5, 3]. For f 0 : [ 5, 3). 5. Consider the function = f() sketched below. ( + 4) n n

5 Suppose f() has Talor series about = 4. f() = a 0 + a ( 4) + a ( 4) + a 3 ( 4) (a) Is a 0 positive or negative? Please eplain. a 0 > 0, because the function is positive at = 4. (b) Is a positive or negative? Please eplain. a > 0, because the function is increasing at = 4. (c) Is a positive or negative? Please eplain. a < 0, because the function is concave down at = How man terms of the Talor series for ln(+) centered at = 0 do ou need to estimate the value of ln(.4) to three decimal places? We will use the error bound. The error bound corresponding to P n (0.4) is given b M(0.4)n+, where M is the maimum of f n+ (u) on the interval [0, 0.4]. (n + )! For n, the derivatives of f() = ln( + ) are given b the following formula: f n () = ( )n (n )! ( + ) n Clearl, f n+ (u) = (0.4)n+ M = = The error bound is then ( + 0) n+ (n + )! error bound is smaller than is n = 6. is decreasing on the interval [0, 0.4], so ( + u) n+ = (0.4)n+. The first n for which the n + 7. A car is moving with speed 0 m/s and acceleration m/s at a given instant. Using a second degree Talor polnomial, estimate how far the car moves in the net second. m 8. Find the integral and epress the answer as an infinite series. Z e d = n n + C

6 9. Using series, evaluate the limit lim!0 sin 3 = Use the Lagrange Error Bound for P n () to find a reasonable bound for the error in approimating the quantit e 0.60 with a third degree Talor polnomial for e centered at = 0. For n = 3, the Lagrange error bound is given b M4, where M is the maimum of f 4 (u) = e u 4! on the interval between 0 and. For = 0.6, M is the maimum of e (an increasing function) on the interval [0, 0.6], so M = e 0.6. The bound is: e0.6 (0.6) 4. 4!. Consider the error in using the approimation sin 3 /3! on the interval [, ]. Where is the approimation an overestimate? Where is it an underestimate? For 0 apple apple, the estimate is an underestimate (the alternating Talor series for sin is truncated after a negative term). For apple apple 0, the estimate is an overestimate (the alternating Talor series is truncated after a positive term).. Find the Talor series around = 0 for cosh = e + e. (Your answer should involve onl even powers of.) n=0 n (n)!