Mohammed Yaaseen Gomdah WeBWorK assignment number Assignment 11 W14 is due : 04/06/2016 at 03:00am EDT. The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information. MATH 203 Winter 2016 F This le is /conf/snippets/setHeader.pg you can use it as a model for creating les which introduce each problem set. The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind of mistake. Usually you can attempt a problem as many times as you want before the due date. However, if you are having trouble guring out your error, you should consult the book, or ask a fellow student, one of the TA's or your professor for help. Don't spend a lot of time guessing - it's not very efcient or effective. Give 4 or 5 signicant digits for (oating point) numerical answers. For most problems when entering numerical answers, you can if you wish enter elementary expressions such as 2 ^ 3 instead of 8, sin(3 pi/2)instead of -1, e ^ (ln(2)) instead of 2, (2 + tan(3)) (4 sin(5)) ^ 6 7/8 instead of 27620.3413, etc. Here's the list of the functions which WeBWorK understands. You can use the Feedback button on each problem page to send e-mail to the professors. (G) Use interval notation to indicate where f (x) is concave down. Concave down: (H) List all horizontal asymptotes of f . If there are no horizontal asymptotes, enter 'NONE'. Horizontal asymptotes y = (I) List all vertical asymptotes of f . If there are no vertical asymptotes, enter 'NONE'. vertical asymptotes x = (J) Use all of the preceding information to sketch a graph of f . When you're nished, enter a \"1\" in the box below. Graph Complete: 1. (1 pt) Consider the function f (x) = x2 e4x . f (x) has two inection points at x = C and x = D with C D where C is and D is Finally for each of the following intervals, tell whether f (x) is concave up (type in CU) or concave down (type in CD). ( ,C]: [C, D]: [D, ) 2. (1 pt) Suppose that f (x) = x2 3. (1 pt) Let f (x) = x4 8x3 + 3x + 4. Find the open intervals on which f is concave up (down). Then determine the x-coordinates of all inection points of f . 1. f is concave up on the intervals 3x . 16 (A) List all critical numbers of f . If there are no critical numbers, enter 'NONE'. 2. 3. Critical numbers = (B) Use interval notation to indicate where f (x) is decreasing. Note: Use 'INF' for , '-INF' for , and use 'U' for the union symbol. Decreasing: (C)List the x-values of all local maxima of f . If there are no local maxima, enter 'NONE'. x values of local maxima = (D) List the x-values of all local minima of f . If there are no local minima, enter 'NONE'. x values of local minima = (E) List the x values of all inection points of f . If there are no inection points, enter 'NONE'. Inection points = (F) Use interval notation to indicate where f (x) is concave up. Concave up: f is concave down on the intervals The inection points occur at x = Notes: In the rst two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word \"none\". In the last one, your answer should be a comma separated list of x values or the word \"none\". 4. (1 pt) A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 11 x2 . What are the dimensions of such a rectangle with the greatest possible area? Width = Height = 5. (1 pt) A cylinder is inscribed in a right circular cone of height 4 and radius (at the base) equal to 6.5. What are the dimensions of such a cylinder which has maximum volume? Radius = Height = 6. (1 pt) If 1500 square centimeters of material is available to make a box with a square base and an open top, nd the largest possible volume of the box. Volume = cubic centimeters. 1 7. (1 pt) A rancher wants to fence in an area of 1000000 square feet in a rectangular eld and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use? 10. (1 pt) Let f (x) = x3 8x2 + 4x 1. Find the open intervals on which f is concave up (down). Then determine the x-coordinates of all inection points of f . 1. f is concave up on the intervals 8. (1 pt) Find the point on the line 5x + 3y + 4 = 0 which is closest to the point (3, 4). ( , ) 2. 3. 9. (1 pt) A fence 5 feet tall runs parallel to a tall building at a distance of 6 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? Length of ladder = feet. f is concave down on the intervals The inection points occur at x = Notes: In the rst two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word \"none\". In the last one, your answer should be a comma separated list of x values or the word \"none\". Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 2 Mohammed Yaaseen Gomdah E WeBWorK assignment due : 04/06/2016 at 03:00am EDT. c1 = 6x 1. (1 pt) Represent the function as a power series 7+x c2 = n=0 c3 = f (x) = cn x n c0 = c4 = c1 = Find the radius of convergence R of the power series. . R= c2 = c3 = 6. (1 pt) The function f (x) = power series: c4 = Find the radius of convergence R = . f (x) = n2 n=1 The above series converges for cn x n Find the rst few coefcients in the power series. 2)n (x is represented as a n=0 2. (1 pt) Find the convergence set of the given power series: 6 1+16x2 c0 = x . c1 = 3. (1 pt) Find all the values of x such that the given series would converge. c2 = c3 = (4x)n n6 n=1 c4 = Answer: Find the radius of convergence R of the series. R= . Note: Give your answer in interval notation. 4. (1 pt) Find all the values of x such that the given series would converge. 7. (1 pt) Let F(x) = Z x 0 sin(3t 2 ) dt. Find the MacLaurin polynomial of degree 7 for F(x). n=1 (x 4)n 4n Use this polynomial to estimate the value of Answer: Z 0.7 0 sin(3x2 ) dx. 8. (1 pt) Find Taylor series of function f (x) = ln(x) at a = 3. Note: Give your answer in interval notation ( f (x) = 5. (1 pt) Suppose that cn (x n=0 6 = cn xn (13 + x) n=0 c0 = c1 = Find the following coefcients of the power series. c0 = c2 = 1 3)n ) c3 = A. B. C. D. c4 = Find the interval of convergence. 10. (1 pt) Find the Maclaurin series of the function f (x) = (7x2 )e 10x . The series is convergent: from x = , left end included (Y,N): to x = , right end included (Y,N): f (x) = cn x n n=0 9. (1 pt) Match each of the Maclaurin series with correct function. Determine the following coefcients: c1 = 2n xn n=0 n! ( 1)n 2x2n+1 2. 2n + 1 n=0 2x2n+1 3. ( 1)n (2n + 1)! n=0 ( 1)n 22n x2n 4. (2n)! n=0 1. 2 arctan(x) 2 sin(x) cos(2x) e2x c2 = c3 = c4 = c5 = Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 2