Concept This module allows you to graph a function and its Taylor and Maclaurin polynomial approximations for several fu

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Concept This Module Allows You To Graph A Function And Its Taylor And Maclaurin Polynomial Approximations For Several Fu 1 (95.32 KiB) Viewed 26 times

Concept This module allows you to graph a function and its Taylor and Maclaurin polynomial approximations for several functions. You can select the degree of module also provides a graph of the error associated with each Taylor polynomial, allowing you to observe the error shrinking as the degree increases Instructions Select a function from the pull-down menu at the top. The graph of the function is shown along with the graph of a Taylor polynomial approximation. Yo 12, with the n slider at the top. Click the Animate button to show a progression of Taylor polynomial functions of increasing degrees. Selecting the che The table on the right gives the coefficients of the Taylor polynomial of degree n. In the lower diagram, a graph of the magnitude of the error R (X)| = |fx) - T(X)| associated with the Taylor polynomial approximation is shown. Then Simulation TAYLOR AND MACLAURIN SERIES CONVERGENCE f(x) = sin(x) CONCEPT 72 2 Animate Step - Function Taylor Series - * = 2 (slider) = 0 0 2 4 6 8 10 12 YOU WILL LEARN ABOUT: Taylor and Maclaurin polynomials. v Show all 3 - = 10 12 7.(x) = (x - ay -- = Mano pholow Shutnalo.com i 0 1 0.000000 1.NO ODOK000 X -21 -3.72 -12 0 -S A 3./2 27 Electromagnetic waves are frequently modeled using sums of sine and cosine functions. Such sums can be approximated using Maclaurin or Taylor polynomials. The Taylor series of the functionſ centered at a is a 0 Size of error R, (x) = f(x) – 1,( 1 0.3 860)=LATE f@ (a) (x-a)and ! indicaresic > 10 x WHEN WOULD I USE THIS - 0 -2. - -72 0 22 IT 37/2 21 IN STRUCTIONS EXERCISES

Click here to access the simulation in a new window. Exercise Select fix) = sin(x) from the drop-down menu and set a = 0. (a) Graph T1(x) and T2(x). What do you notice? T1(x) = T2(x) OT1(x) has a steeper slope than T2(X). T2(x) has a steeper slope than 7(x). T(x) is a reflection about the y-axis of T2(x). T1(x) is a reflection about the x-axis of T2(x) = What about T3(x) and T4(x)? 73(x) = T4(x). T3(x) has a steeper slope than T4(x). T4(x) has a steeper slope than 73(x). T3(x) is a reflection about the y-axis of T4(x). T3(x) is a reflection about the x-axis of T4(x). Explain why f(x) = cosx : "(x)=sinx....and will repeat sin(x) = 0+1x+0.x^2+-1/3!43+0.x4...etc. T3(x) = x - x^3/3!...14%) = x-x^3/3! Score: 0.11 out of 0.11 Comment: (b) on what interval would you say T3(x) is a close approximation to sin(x)? (Enter your answer using interval notation.)

(b) On what interval would you say T3(x) is a close approximation to sin(x)? (Enter your answer using interval notation.) On what interval would you say Tg(x) is a close approximation to sin(x)? (Enter your answer using interval notation.) On what interval would you say T7(x) is a close approximation to sin(x)? (Enter your answer using interval notation.) On what interval would you say To(x) is a close approximation to sin(x)? (Enter your answer using interval notation.) On what interval would you say T11(x) is a close approximation to sin(x)? (Enter your answer using interval notation) (c) For T11(x), estimate the interval on which the magnitude of the error is less than 0.1. (Round your answer to three decimal places. Enter your answer using interval notation.)