evaluate l(8.94) as the linear approximation of 1/8.94. (if necessary, round to six decimal places.) the…

Explanation:

Step1: Define the function

Let $f(x)=\frac{1}{x}$, and we want to find the linear - approximation at $a = 9$. The derivative of $f(x)$ using the power rule $(x^n)^\prime=nx^{n - 1}$ is $f^\prime(x)=-\frac{1}{x^{2}}$.

Step2: Calculate $f(a)$ and $f^\prime(a)$

When $a = 9$, $f(9)=\frac{1}{9}\approx0.111111$ and $f^\prime(9)=-\frac{1}{9^{2}}=-\frac{1}{81}\approx - 0.012346$.

Step3: Write the linear - approximation formula

The linear - approximation formula is $L(x)=f(a)+f^\prime(a)(x - a)$. Substituting $a = 9$, we get $L(x)=\frac{1}{9}-\frac{1}{81}(x - 9)$.

Step4: Evaluate $L(8.94)$

Substitute $x = 8.94$ into $L(x)$: [ \begin{align*} L(8.94)&=\frac{1}{9}-\frac{1}{81}(8.94 - 9)\ &=\frac{1}{9}-\frac{1}{81}\times(-0.06)\ &=\frac{1}{9}+\frac{0.06}{81}\ &=\frac{9}{81}+\frac{0.06}{81}\ &=\frac{9 + 0.06}{81}\ &=\frac{9.06}{81}\ &\approx0.111852 \end{align*} ]

Answer:

$0.111852$