Question

determine the linear approximation for (f(x)=5x - \frac{2}{x^{2}}) at (a = - 2). (l(x)=) submit answer next item attempt 1: 2 attempts remaining.

Explanation:

Step1: Recall linear - approximation formula

The linear approximation of a function $y = f(x)$ at $x = a$ is given by $L(x)=f(a)+f^{\prime}(a)(x - a)$.

Step2: Calculate $f(a)$

First, find $f(x)=5x-\frac{2}{x^{2}}=5x - 2x^{-2}$. When $a=-2$, $f(-2)=5\times(-2)-2\times(-2)^{-2}=-10 - 2\times\frac{1}{4}=-10-\frac{1}{2}=-\frac{20 + 1}{2}=-\frac{21}{2}$.

Step3: Calculate the derivative $f^{\prime}(x)$

Differentiate $f(x)$ with respect to $x$. Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we have $f^{\prime}(x)=5+4x^{-3}=5+\frac{4}{x^{3}}$.

Step4: Calculate $f^{\prime}(a)$

When $a = - 2$, $f^{\prime}(-2)=5+\frac{4}{(-2)^{3}}=5-\frac{4}{8}=5-\frac{1}{2}=\frac{10 - 1}{2}=\frac{9}{2}$.

Step5: Find the linear approximation $L(x)$

Substitute $f(-2)=-\frac{21}{2}$, $f^{\prime}(-2)=\frac{9}{2}$, and $a=-2$ into the linear - approximation formula $L(x)=f(a)+f^{\prime}(a)(x - a)$.
$L(x)=-\frac{21}{2}+\frac{9}{2}(x + 2)$.
Expand the right - hand side: $L(x)=-\frac{21}{2}+\frac{9}{2}x+9=\frac{9}{2}x-\frac{21}{2}+\frac{18}{2}=\frac{9}{2}x-\frac{3}{2}$.

Answer:

$L(x)=\frac{9}{2}x-\frac{3}{2}$