find the linear approximation to the following function at the given point a. f(x)=e^{-3x + 2}; a = -2 l(x)= (type an exact answer.)

Question

Accepted Answer

# Explanation:
## Step1: Find $f(a)$
Substitute $x = a=-2$ into $f(x)=e^{-3x + 2}$.
$$
\begin{align*}
f(-2)&=e^{-3\times(-2)+2}\
&=e^{6 + 2}\
&=e^{8}
\end{align*}
$$
## Step2: Find $f^{\prime}(x)$
Use the chain - rule. If $y = e^{u}$ and $u=-3x + 2$, then $y^{\prime}=e^{u}\cdot u^{\prime}$.
Since $u^{\prime}=-3$, so $f^{\prime}(x)=e^{-3x + 2}\times(-3)=-3e^{-3x + 2}$.
## Step3: Find $f^{\prime}(a)$
Substitute $x = a = - 2$ into $f^{\prime}(x)$.
$$
\begin{align*}
f^{\prime}(-2)&=-3e^{-3\times(-2)+2}\
&=-3e^{8}
\end{align*}
$$
## Step4: Use the linear - approximation formula $L(x)=f(a)+f^{\prime}(a)(x - a)$
Here $a=-2$, $f(a)=e^{8}$, $f^{\prime}(a)=-3e^{8}$.
$$
\begin{align*}
L(x)&=e^{8}+(-3e^{8})(x-(-2))\
&=e^{8}-3e^{8}(x + 2)\
&=e^{8}-3e^{8}x-6e^{8}\
&=-3e^{8}x - 5e^{8}
\end{align*}
$$

# Answer:
$L(x)=-3e^{8}x - 5e^{8}$

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QUESTION IMAGE

Question

Explanation:

Step1: Find \(f(a)\)

Step2: Find \(f^{\prime}(x)\)

Step3: Find \(f^{\prime}(a)\)

Step4: Use the linear - approximation formula \(L(x)=f(a)+f^{\prime}(a)(x - a)\)

Answer: