A Taylor series is something we often use and looks likedyy(xo+h) = y(xo) + (x) +dxh² d²y2! dx²(x0) +where (ro) is the first derivative of y(x) evaluated at îo. Similarly (ro) is the second derivative of y(x)evaluated at ro and so on. If h is small, we can truncate this series and get a good local approximation toy(x) in the vicinity of xo.Try the equation y(x) = 3x5 and set x = 3 and h = 0.05. Evaluate y at x = 3.05 by using the Taylor seriestruncated at the second (linear) and then the third (quadratic) term and compare with the actual answer.

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Answered: A Taylor series is something we often…

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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Question

A Taylor series is something we often use and looks like
dy
y(xo+h) = y(xo) + (x) +
dx
h² d²y
2! dx²
(x0) +
where (ro) is the first derivative of y(x) evaluated at îo. Similarly (ro) is the second derivative of y(x)
evaluated at ro and so on. If h is small, we can truncate this series and get a good local approximation to
y(x) in the vicinity of xo.
Try the equation y(x) = 3x5 and set x = 3 and h = 0.05. Evaluate y at x = 3.05 by using the Taylor series
truncated at the second (linear) and then the third (quadratic) term and compare with the actual answer.

Expert Solution

Step 1: Introduction of given problem

$y open parentheses x close parentheses equals 3 x to the power of 5 x subscript 0 equals 3 h equals 0.05 y open parentheses x subscript 0 plus h close parentheses equals y open parentheses x subscript 0 close parentheses plus h fraction numerator d y over denominator d x end fraction open parentheses x subscript 0 close parentheses plus fraction numerator h squared over denominator 2 factorial end fraction fraction numerator d squared y over denominator d x squared end fraction y open parentheses x subscript 0 close parentheses plus..$