Solution Found!
Derivative trick Here is an alternative way to evaluate
Chapter 8, Problem 65E(choose chapter or problem)
Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Explain why \(f^{(k)}(a)=k !\) multiplied by the coefficient of \((x-a)^{k}\).
Use this idea to evaluate \(f^{(3)}(0)\) and \(f^{(4)}(0)\) for the following functions. Use known series and do not evaluate derivatives.
\(f(x)=e^{\cos x}\)
Questions & Answers
QUESTION:
Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Explain why \(f^{(k)}(a)=k !\) multiplied by the coefficient of \((x-a)^{k}\).
Use this idea to evaluate \(f^{(3)}(0)\) and \(f^{(4)}(0)\) for the following functions. Use known series and do not evaluate derivatives.
\(f(x)=e^{\cos x}\)
ANSWER:Solution 65E
Step 1:
Given that
f(x) = ecos x