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11. Let on(x)=x" lim 84x for 0≤x≤ 1 and show that So, (0, 0≤x≤ 1, x = 1. 1. n(x) =

11. Let on(x)=x" lim 84x for 0≤x≤ 1 and show that So, (0, 0≤x≤ 1, x = 1. 1. n(x) =

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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a. Determine on (t) for an arbitrary value of n. G b. Plot n(t) for n = 1, ..., 4. Observe whether the iterates appear to be converging. nn c. Express lim n(t) = o(t) in terms of elementary 818 functions; that is, solve the given initial value problem. Gd. Plot (t) = n(t)| for n = 1, ..., 4. For each of 01(t),...,04(t), estimate the interval in which it is a reasonably good approximation to the actual solution. 3. y' = 2(y+1), y(0) = 0 4. y'=-y/2+t, y(0) = 0 In each of Problems 5 and 6, let po(t) = 0 and use the method of successive approximations to solve the given initial value problem. a. Determine on(t) for an arbitrary value of n. G b. Plot on(t) for n = 1, ..., 4. Observe whether the iterates appear to be converging. or no c. Show that the sequence {on(t)} converges. y'=ty + 1, y(0) = 0 y(0) = 0 (TS) In each of Problems 7 and 8, let po(t) = 0 and use the method of successive approximations to approximate the solution of the given initial value problem. of au-wolls of N Problems In each of Problems 1 and 2, transform the given initial value problem into an equivalent problem with the initial point at the origin. 1. dy/dt = 1² + y², y(1) = 2 2. dy/dt = 1- y³, y(-1) = 3 In each of Problems 3 through 4, let po(t) = 0 and define {on(t)} by the method of successive approximations. N 5. 6. y'=t²y-t, a. Calculate 1(t), ..., 3(t). mil odi G b. Plot 1(1), ..., 3(t). Observe whether the iterates appear to be converging. y' = 1² + y², y(0) = 0 y(0) = 0 In each of Problems 9 and 10, let po(t) = 0 and use the method of successive approximations to approximate the solution of the given initial value problem. 7. 8. y' = 1- y³, a. Calculate 1(t), ..., 4(t), or (if necessary) Taylor approximations to these iterates. Keep terms up to order six. Gb. Plot the functions you found in part a and observe whether they appear to be converging. 9. y'= -sin y + 1, y(0) = 0 31² +41 +2 2(y - 1) 10. y' = y(0) = 0 11. Let on(x) = x" for 0 ≤ x ≤ 1 and show that So, 0, 0≤x < 1, 1, x = 1. lim bn(x) = 81x This example shows that a sequence of continuous functions may converge to a limit function that is discontinuous. 2nxe-nx², 0≤x≤ 1. 12. Consider the sequence on(x) = 2nxe-nx² a. Show that lim n(x) = 0 for 0 ≤ x ≤ 1; hence alb sillar of bemolo them 81x b. Show that bus lansuper sito son lim on(x) dx = 0. S 3 81x hoitamps of mantel wol 2nxe-nx² dx = 1 - e¯n; hence [2 Thus, in this example, rb lim 818 lim 100 6, Ja S 13. a. Verify that (t) = k=0 on(x) dx = 1. Hence the 4919 even though lim n(x) exists and is continuous. 818 = dx # + Tº Σ k=1 k=0 t2k - k! k! piwo lim n(x) dx, a 818 equation (9). b. Verify that (t) is also a solution of the initial value problem (6). c. Use the fact that is a solution of the integral elementary functions. d. Solve initial value problem (6) as a separable equation. e. Solve initial value problem (6) as a first order linear equation. In Problems 14 through 17, we indicate how to prove that the sequence {on(t)), defined by equations (4) through (7), converges. a. Verify that o(t) t2k = e' to evaluate (t) in terms of Σ is a solution of the integral k! k=1 equation (9). b. Verify that (t) is also a solution of the initial value problem (6). c. Use the fact that = e' to evaluate (t) in terms of elementary functions. d. Solve initial value problem (6) as a separable equation. e. Solve initial value problem (6) as a first order linear equation.
Transcribed Image Text:a. Determine on (t) for an arbitrary value of n. G b. Plot n(t) for n = 1, ..., 4. Observe whether the iterates appear to be converging. nn c. Express lim n(t) = o(t) in terms of elementary 818 functions; that is, solve the given initial value problem. Gd. Plot (t) = n(t)| for n = 1, ..., 4. For each of 01(t),...,04(t), estimate the interval in which it is a reasonably good approximation to the actual solution. 3. y' = 2(y+1), y(0) = 0 4. y'=-y/2+t, y(0) = 0 In each of Problems 5 and 6, let po(t) = 0 and use the method of successive approximations to solve the given initial value problem. a. Determine on(t) for an arbitrary value of n. G b. Plot on(t) for n = 1, ..., 4. Observe whether the iterates appear to be converging. or no c. Show that the sequence {on(t)} converges. y'=ty + 1, y(0) = 0 y(0) = 0 (TS) In each of Problems 7 and 8, let po(t) = 0 and use the method of successive approximations to approximate the solution of the given initial value problem. of au-wolls of N Problems In each of Problems 1 and 2, transform the given initial value problem into an equivalent problem with the initial point at the origin. 1. dy/dt = 1² + y², y(1) = 2 2. dy/dt = 1- y³, y(-1) = 3 In each of Problems 3 through 4, let po(t) = 0 and define {on(t)} by the method of successive approximations. N 5. 6. y'=t²y-t, a. Calculate 1(t), ..., 3(t). mil odi G b. Plot 1(1), ..., 3(t). Observe whether the iterates appear to be converging. y' = 1² + y², y(0) = 0 y(0) = 0 In each of Problems 9 and 10, let po(t) = 0 and use the method of successive approximations to approximate the solution of the given initial value problem. 7. 8. y' = 1- y³, a. Calculate 1(t), ..., 4(t), or (if necessary) Taylor approximations to these iterates. Keep terms up to order six. Gb. Plot the functions you found in part a and observe whether they appear to be converging. 9. y'= -sin y + 1, y(0) = 0 31² +41 +2 2(y - 1) 10. y' = y(0) = 0 11. Let on(x) = x" for 0 ≤ x ≤ 1 and show that So, 0, 0≤x < 1, 1, x = 1. lim bn(x) = 81x This example shows that a sequence of continuous functions may converge to a limit function that is discontinuous. 2nxe-nx², 0≤x≤ 1. 12. Consider the sequence on(x) = 2nxe-nx² a. Show that lim n(x) = 0 for 0 ≤ x ≤ 1; hence alb sillar of bemolo them 81x b. Show that bus lansuper sito son lim on(x) dx = 0. S 3 81x hoitamps of mantel wol 2nxe-nx² dx = 1 - e¯n; hence [2 Thus, in this example, rb lim 818 lim 100 6, Ja S 13. a. Verify that (t) = k=0 on(x) dx = 1. Hence the 4919 even though lim n(x) exists and is continuous. 818 = dx # + Tº Σ k=1 k=0 t2k - k! k! piwo lim n(x) dx, a 818 equation (9). b. Verify that (t) is also a solution of the initial value problem (6). c. Use the fact that is a solution of the integral elementary functions. d. Solve initial value problem (6) as a separable equation. e. Solve initial value problem (6) as a first order linear equation. In Problems 14 through 17, we indicate how to prove that the sequence {on(t)), defined by equations (4) through (7), converges. a. Verify that o(t) t2k = e' to evaluate (t) in terms of Σ is a solution of the integral k! k=1 equation (9). b. Verify that (t) is also a solution of the initial value problem (6). c. Use the fact that = e' to evaluate (t) in terms of elementary functions. d. Solve initial value problem (6) as a separable equation. e. Solve initial value problem (6) as a first order linear equation.
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