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Find the Taylor polynomial 73(x) for the function f centered at the number a. π f(x) = cos(x), a = - 2 T₂(x)= Graph f and T3 on the same screen. y 1.0 0.5 -0.5 -1.0 1.0 0.5 -0.5 -1.0 y 0.5 1.0 1.5 2.0 2.5 3.0 x 0.5 1.0 1.5 2.0 2.5 3.0 1.0 0.5 y A -0.5 -1.0 1 2 3 4 1.0 0.5 -0.5 -1.0 y x 0.5 1.0 1.5 2.0 2.5 3.0 (a) For what values of k does the function y = cos(kt) satisfy the differential equation 49y" = -81y? (Enter your answers as a comma-separated list.) k = (b) For those values of k, verify that every member of the family of functions y = A sin(kt) + B cos(kt) is also a solution. We begin by calculating the following. y = A sin(kt) + B cos(kt) y' = Ak cos(kt) - Bk sin(kt) = y"= Note that the given differential equation 49y" = -81y is equivalent to 49y" + 81y= Now, substituting the expressions for y and y" above and simplifying, we have LHS 49y" + 81y = 49 =-49 = (81 – 49k²)([ = 0 - +81(A sin(kt) + B cos(kt)) 49Bk2 cos(kt) + 81A sin(kt) + 81B cos(kt) +(81 - 49k2) B cos(kt) since for all value of k found above, k² =

Find the Taylor polynomial 73(x) for the function f centered at the number a. π f(x) = cos(x), a = - 2 T₂(x)= Graph f and T3 on the same screen. y 1.0 0.5 -0.5 -1.0 1.0 0.5 -0.5 -1.0 y 0.5 1.0 1.5 2.0 2.5 3.0 x 0.5 1.0 1.5 2.0 2.5 3.0 1.0 0.5 y A -0.5 -1.0 1 2 3 4 1.0 0.5 -0.5 -1.0 y x 0.5 1.0 1.5 2.0 2.5 3.0 (a) For what values of k does the function y = cos(kt) satisfy the differential equation 49y" = -81y? (Enter your answers as a comma-separated list.) k = (b) For those values of k, verify that every member of the family of functions y = A sin(kt) + B cos(kt) is also a solution. We begin by calculating the following. y = A sin(kt) + B cos(kt) y' = Ak cos(kt) - Bk sin(kt) = y"= Note that the given differential equation 49y" = -81y is equivalent to 49y" + 81y= Now, substituting the expressions for y and y" above and simplifying, we have LHS 49y" + 81y = 49 =-49 = (81 – 49k²)([ = 0 - +81(A sin(kt) + B cos(kt)) 49Bk2 cos(kt) + 81A sin(kt) + 81B cos(kt) +(81 - 49k2) B cos(kt) since for all value of k found above, k² =

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.5: Rational Functions
Problem 51E
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Find the Taylor polynomial 73(x) for the function f centered at the number a. π f(x) = cos(x), a = - 2 T₂(x)= Graph f and T3 on the same screen. y 1.0 0.5 -0.5 -1.0 1.0 0.5 -0.5 -1.0 y 0.5 1.0 1.5 2.0 2.5 3.0 x 0.5 1.0 1.5 2.0 2.5 3.0 1.0 0.5 y A -0.5 -1.0 1 2 3 4 1.0 0.5 -0.5 -1.0 y x 0.5 1.0 1.5 2.0 2.5 3.0
Transcribed Image Text:Find the Taylor polynomial 73(x) for the function f centered at the number a. π f(x) = cos(x), a = - 2 T₂(x)= Graph f and T3 on the same screen. y 1.0 0.5 -0.5 -1.0 1.0 0.5 -0.5 -1.0 y 0.5 1.0 1.5 2.0 2.5 3.0 x 0.5 1.0 1.5 2.0 2.5 3.0 1.0 0.5 y A -0.5 -1.0 1 2 3 4 1.0 0.5 -0.5 -1.0 y x 0.5 1.0 1.5 2.0 2.5 3.0
(a) For what values of k does the function y = cos(kt) satisfy the differential equation 49y" = -81y? (Enter your answers as a comma-separated list.) k = (b) For those values of k, verify that every member of the family of functions y = A sin(kt) + B cos(kt) is also a solution. We begin by calculating the following. y = A sin(kt) + B cos(kt) y' = Ak cos(kt) - Bk sin(kt) = y"= Note that the given differential equation 49y" = -81y is equivalent to 49y" + 81y= Now, substituting the expressions for y and y" above and simplifying, we have LHS 49y" + 81y = 49 =-49 = (81 – 49k²)([ = 0 - +81(A sin(kt) + B cos(kt)) 49Bk2 cos(kt) + 81A sin(kt) + 81B cos(kt) +(81 - 49k2) B cos(kt) since for all value of k found above, k² =
Transcribed Image Text:(a) For what values of k does the function y = cos(kt) satisfy the differential equation 49y" = -81y? (Enter your answers as a comma-separated list.) k = (b) For those values of k, verify that every member of the family of functions y = A sin(kt) + B cos(kt) is also a solution. We begin by calculating the following. y = A sin(kt) + B cos(kt) y' = Ak cos(kt) - Bk sin(kt) = y"= Note that the given differential equation 49y" = -81y is equivalent to 49y" + 81y= Now, substituting the expressions for y and y" above and simplifying, we have LHS 49y" + 81y = 49 =-49 = (81 – 49k²)([ = 0 - +81(A sin(kt) + B cos(kt)) 49Bk2 cos(kt) + 81A sin(kt) + 81B cos(kt) +(81 - 49k2) B cos(kt) since for all value of k found above, k² =
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