Use the Maclaurin series for $f(x)=\sin (4 x)$ to find the Maclaurin series for $f(x)=\cos (4 x)$ .
A) $4 x-\frac{4^{3} x^{3}}{3 !}+\frac{4^{5} x^{5}}{5 !}-\cdots$
B) $1-\frac{4^{2} x^{2}}{2 !}+\frac{4^{4} x^{4}}{4 !}-\cdots$
C) $x-\frac{4^{2} x^{3}}{3 !}+\frac{4^{4} x^{5}}{5 !}-\cdots$
D) $4 x-\frac{4^{2} x^{2}}{2 !}+\frac{4^{3} x^{3}}{3 !}-\cdots$

Question

Quizplus · Accepted Answer

The Maclaurin series for $\sin(4x)$ is $4x - \frac{(4x)^3}{3!} + \frac{(4x)^5}{5!} - \cdots$. To find the series for $\cos(4x)$, differentiate $\sin(4x)$ to get $\cos(4x)$, which shifts the series to $1 - \frac{(4x)^2}{2!} + \frac{(4x)^4}{4!} - \cdots$, matching option B.

Use the Maclaurin Series For f(x)=sin⁡(4x)f(x)=\sin (4 x)f(x)=sin(4x) To Find the Maclaurin Series For

Use the Maclaurin Series For $f(x)=\sin (4 x)$ To Find the Maclaurin Series For