16 multiple choice 1 point find the linear approximation of f(x) = tan x, for x close to 0…
16 multiple choice 1 point find the linear approximation of f(x) = tan x, for x close to 0. f₁(x)=f(1)+f(0)(x - 1)=x - 1 none of these. f₁(x)=f(-1)+f(0)(x + 1)=x + 1 f₁(x)=f(0)+f(0)(x - 0)=x
Answer
Explanation:
Step1: Recall linear - approximation formula
The linear approximation of a function $y = f(x)$ near $x = a$ is given by $L(x)=f(a)+f^{\prime}(a)(x - a)$.
Step2: Find $f(0)$ and $f^{\prime}(x)$ for $f(x)=\tan x$
We know that $f(x)=\tan x$, so $f(0)=\tan(0) = 0$. Also, $f^{\prime}(x)=\sec^{2}x$, and $f^{\prime}(0)=\sec^{2}(0)=1$.
Step3: Apply the linear - approximation formula
Using $a = 0$ in the linear - approximation formula $L(x)=f(a)+f^{\prime}(a)(x - a)$, we get $L(x)=f(0)+f^{\prime}(0)(x - 0)$. Substituting $f(0) = 0$ and $f^{\prime}(0)=1$, we have $L(x)=0 + 1\times(x-0)=x$.
Answer:
D. $f_{1}(x)=f(0)+f^{\prime}(0)(x - 0)=x$