Find the linear approximating polynomial for the function centered at a.
-f(x) = tan x, a = 0
A) L(x) = 0
B) L(x) = 5x + 1
C) L(x) = x
D) L(x) = -x

Question

Quizplus · Accepted Answer

The linear approximating polynomial, also known as the first-order Taylor polynomial, for a function $f(x)$ centered at $a$ is given by $L(x) = f(a) + f'(a)(x - a)$. For $f(x) = \tan(x)$ and $a = 0$, we have $f(0) = \tan(0) = 0$ and $f'(x) = \sec^2(x)$, so $f'(0) = \sec^2(0) = 1$. Thus, $L(x) = 0 + 1(x - 0) = x$.

Find the Linear Approximating Polynomial for the Function Centered at A