quiz 6 - requires respondus lockdown browser + webcam\nstarted: sep 27 at 12:25pm\nquiz instructions\naccess…

Explanation:

Step1: Recall linearization formula

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

Step2: Find $f(a)$

Given $f(x)=\sqrt{x}=x^{\frac{1}{2}}$ and $a = 9$. Then $f(a)=f(9)=\sqrt{9}=3$.

Step3: Find the derivative of $f(x)$

Using the power - rule $(x^n)^\prime=nx^{n - 1}$, for $f(x)=x^{\frac{1}{2}}$, we have $f^{\prime}(x)=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}$.

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

Substitute $x = a = 9$ into $f^{\prime}(x)$. So $f^{\prime}(9)=\frac{1}{2\sqrt{9}}=\frac{1}{6}$.

Step5: Calculate $L(x)$

$L(x)=f(9)+f^{\prime}(9)(x - 9)$. Substitute $f(9) = 3$ and $f^{\prime}(9)=\frac{1}{6}$ into the formula: [ \begin{align*} L(x)&=3+\frac{1}{6}(x - 9)\ &=3+\frac{1}{6}x-\frac{9}{6}\ &=\frac{1}{6}x+\frac{18 - 9}{6}\ &=\frac{1}{6}x+\frac{3}{2} \end{align*} ]

Answer:

A. $\frac{x}{6}+\frac{3}{2}$