Answer: Finding a Pattern Consider the function where is a constant. (a) Find the

Chapter 3, Problem 177

(choose chapter or problem)

Finding a Pattern Consider the function \(f(x)=\sin \beta x\), where \(\beta\) is a constant.

(a) Find the first-, second-, third-, and fourth-order derivatives of the function.

(b) Verify that the function and its second derivative satisfy the equation \(f^{\prime \prime}(x)+\beta^{2} f(x)=0\)

(c) Use the results of part (a) to write general rules for the even- and odd-order derivatives \(f^{(2 k)}(x)\) and \(f^{(2 k-1)}(x)[Hint: \((-1)^{k}\) is positive if k is even and negative if k is odd.]

Text Transcription:

f(x)=sin beta x

beta

f^prime prime(x)+beta^2f(x)=0

f^(2k)(x)

f^(2k-1)(x)

(-1)^k