The function $f(x)=\sin x+0.5$ can be approximated by the function $g(x)=0.5+x-\frac{x^{3}}{6}+\frac{x^{5}}{120}$. On what interval do the two graphs look similar?
A) $-\pi

Question

The function $f(x)=\sin x+0.5$ can be approximated by the function $g(x)=0.5+x-\frac{x^{3}}{6}+\frac{x^{5}}{120}$. On what interval do the two graphs look similar?
A) $-\pi<x<0$
B) $0<x<\pi$
C) $-\frac{\pi}{2}<x<\frac{\pi}{2}$

Quizplus · Accepted Answer

The function $g(x)$ is a Taylor series approximation of $\sin x + 0.5$ around $x=0$. Taylor series approximations are most accurate near the point of expansion, which is $x=0$ in this case. Therefore, the interval $-\frac{\pi}{2}<x<\frac{\pi}{2}$ is where the two graphs will look most similar.

The Function f(x)=sin⁡x+0.5f(x)=\sin x+0.5f(x)=sinx+0.5 Can Be Approximated by the Function g(x)=0.5+x−x36+x5120g(x)=0.5+x-\frac{x^{3}}{6}+\frac{x^{5}}{120}g(x)=0.5+x−6x3​+120x5​ on What Interval Do the Two Graphs Look Similar

The Function $f(x)=\sin x+0.5$ Can Be Approximated by the Function $g(x)=0.5+x-\frac{x^{3}}{6}+\frac{x^{5}}{120}$ on What Interval Do the Two Graphs Look Similar