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(a) Write out the the general expression for the Taylor series expansion of f(x) about the point x = a. Include all terms up to the 4th order. (b) Write out the Taylor series expansion of the function f(x) = cos (x – n) about x = 7, up to the 2nd order. (c) Write out the Taylor series expansion of the function f(x) = V1+x about x = 2nd order. 0, up to the (i) Using a plotting program such as Matlab or Mathematica, plot both the original func- tion, f(x) on your plot to values between -1.0 and 1.0. = V1+ x , and the truncated Taylor series expansion. Restrict the x-range

(a) Write out the the general expression for the Taylor series expansion of f(x) about the point x = a. Include all terms up to the 4th order. (b) Write out the Taylor series expansion of the function f(x) = cos (x – n) about x = 7, up to the 2nd order. (c) Write out the Taylor series expansion of the function f(x) = V1+x about x = 2nd order. 0, up to the (i) Using a plotting program such as Matlab or Mathematica, plot both the original func- tion, f(x) on your plot to values between -1.0 and 1.0. = V1+ x , and the truncated Taylor series expansion. Restrict the x-range

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Many interesting problems in physics cannot be solved exactly, and we need to make (clever) ap- proximations in order to solve them. Physicists often employ the Taylor series expansion in order to make approximations, as it allows for the magnitude of the error introduced by the approxima- tion to be quantified. Many students misunderstand or misuse this important tool, so it is good to review a few examples. Note: 'up to the nth order' means you should include the nth derivative in the expansion. (a) Write out the the general expression for the Taylor series expansion of f(x) about the point x = a. Include all terms up to the 4th order. (b) Write out the Taylor series expansion of the function f(x) the 2nd order. = cos (x – T) about x = ™, up to (c) Write out the Taylor series expansion of the function f(x) = /1+x about x = 0, up to the 2nd order. (i) Using a plotting program such as Matlab or Mathematica, plot both the original func- tion, f(x) = v1+ x , and the truncated Taylor series expansion. Restrict the x-range on your plot to values between -1.0 and 1.0. (ii) Calculate the definite integral (to 6-decimal precision): r0.25 V1+x dx (1) -0.25 (iii) Calculate the definite integral (to 6-decimal precision) of the truncated Taylor series expansion, using the same limits of integration.
Transcribed Image Text:Many interesting problems in physics cannot be solved exactly, and we need to make (clever) ap- proximations in order to solve them. Physicists often employ the Taylor series expansion in order to make approximations, as it allows for the magnitude of the error introduced by the approxima- tion to be quantified. Many students misunderstand or misuse this important tool, so it is good to review a few examples. Note: 'up to the nth order' means you should include the nth derivative in the expansion. (a) Write out the the general expression for the Taylor series expansion of f(x) about the point x = a. Include all terms up to the 4th order. (b) Write out the Taylor series expansion of the function f(x) the 2nd order. = cos (x – T) about x = ™, up to (c) Write out the Taylor series expansion of the function f(x) = /1+x about x = 0, up to the 2nd order. (i) Using a plotting program such as Matlab or Mathematica, plot both the original func- tion, f(x) = v1+ x , and the truncated Taylor series expansion. Restrict the x-range on your plot to values between -1.0 and 1.0. (ii) Calculate the definite integral (to 6-decimal precision): r0.25 V1+x dx (1) -0.25 (iii) Calculate the definite integral (to 6-decimal precision) of the truncated Taylor series expansion, using the same limits of integration.
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