Answered: (A) Find the radius of convergence of… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section: Chapter Questions

Problem 18T

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Question

(A) Find the radius of convergence of the power series
x3
x6
x⁹
+
3.2 (6.5) (3-2) (9.8) (6-5)-(3-2)
y = 1
+
Remark: The absolute value of the ratio of terms has a very simple and
obvious expression and the ratio test indicator can be easily computed
from that.
(B) Show that the function so defined satisfies the differential equation
y" + xy = 0.

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6.2 Determine the highest real rool of Rx)%= 2x-11.Zx + 17.7x- 5 (a) Graphically (b) Fixed-point iteration method (three iterations, , = 3). Note: Make certain that you develop a solution that converges on the root. (c) Newton-Raphson method (three iterations, x, = 3).
A 8-year projection of population trends suggests that t years from now, the population of a certain community will be P(t) = t³ + 15t² + 21t+ 60 thousand people. (a) At what time during the 8-year period will the population be growing most rapidly? 6 (b) At what time during the 8-year period will the population be growing least rapidly? (c) At what time during the 8-year period will the rate of population growth be growing most rapidly?
3. (a) Find the roots of the function, f(x) = 2 - - 1, analytically up to 3 decimal places. (b) and determine if they will converge to any roots or not. (Make sure that one of the g(r) that you construct converges to a root (s). Construct 2 fixed point functions g(r), such that f(r) = 0. Also find their converging rate, A, Using ro = 60, and the fixed point funetion g(r) that converges to the root(s), find the root of the (c) above function aganin using fixed point iterations accurate up to 3 decimal places.
(vi) Suppose that capital expenditure varies with time in such a way that t months from now there will be K(t) thousand dollars of capital expenditure, whereK(t) = 2t4 + 3t + 149 t + 2 (a) What will be the capital expenditure 3 months from now? How many units will be produced at this time? (b) At what rate will production be changing with respect to time 5 months from now? Will production be increasing or decreasing at this time?
(4 -3) Please show all working -2 '5 (1) Given that P = 2 9. ) and Q = (: %3| -3 4 -3. Calculate: (i) P + Q (ii) 3Q (iii) ½P
(2) The annual net profit for a company, from 1995 to 2005, can be approximated by the model an 10e0.2n, n = 1, 2, 3, ., N, where an is the annual net profit in millions of dollars, and n represents the year, with n =1 corresponding to 1995. Estimate the total net profit during this period. (3) A deposit of 15 dollars is made at the beginning of each month, for a period of 5 years, in an account that pays 0.9 % interest, compounded monthly. Find the balance in the account at the end of the 5 years. (4) A deposit of 20 dollars is made every 3 months, for a period of 10 years, in an account that pays 1% interest, compounded quarterly. Find the balance in the account at the end of the 10 years. (5) A deposit of $500 can be made with two options: earn 2% interest, compounded quarterly for 3 years, or earn 1.5%, compounded monthly for 3 years. Find the balance after 3 years of both options.
2- An efficiency study of the morning shift at a certain factory indicates that an average worker who arrives on the job at 8:00 A.M. will have produced Q(t) = -1³ + 6t² + 24t units t hours later. a. Compute the worker's rate of production at 11:00 A.M. b. At what rate is the worker's rate of production changing with respect to time at 11:00 A.M. ?
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13) Match the graph with the correct function. (b) S) = -1 2r + 1 (a) S(1) %3D %3D 2r +1 ? + 2r + 2 2r - 1 (e) None of these x' + 2r? + x- 2 2x +1 (c) S(x) = (d) S(x) 14)
4. Consider the function, f(x) = x³ = x² - 9x +9. Answer the following: State the exact roots of f(x). (a) (b) Construct three different fixed point functions g(x) such that f(x) = 0. (Make sure that one of the g(x)'s that you constructed converges to at least a root). (c) (d) Find the approximate root, x, of the above function using fixed point iterations up to 4 significant figures within the error bound of 1 × 10-³ using xo = 0 and any fixed point function g(x) from part(b) that converges to the root(s). Find the convergence rate/ratio for g(x) constructed in previous part and also find which root it is converging to?
! It is known that the reliability function for a critical solid state power unit for use 10 ,t20 and t measured in years. 10+t in a communication satellite is R(t) (a) How many units must be placed in parallel in order to achieve a reliability of 0.98 for 5 year operation? (b) If there is an additional common mode with constant failure rate of 0.002 as a result of environmental factors, how many units should be placed in parallel?
Most functions can be rearranged in several ways to give x = g(x) with which to begin the fixed-point method. For f(x) = e² – 2x² = 0, one g(x) is ex x = ±. 2 (a) Show that this converges to the root near 1.5 if the positive value is used and to the root near -0.5 if the negative is used. (b) There is a third root near 2.6. Show that we do not converge to this root even though values near to the root such as p. o = 2.5 or p, = 2.7 are used to begin the iterations. (c) Find another rearrangement that does converge correctly to the third root.

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