(b) find a formula for the linearization of $f(x)$ at $x = 3$. describe in a sentence what the linearization…

Explanation:

Step1: Recall the linearization formula

The linearization $L(x)$ of a function $y = f(x)$ at $x = a$ is given by $L(x)=f(a)+f^{\prime}(a)(x - a)$. Here $a = 3$, so $L(x)=f(3)+f^{\prime}(3)(x - 3)$.

Step2: Explain what linearization represents

The linearization $L(x)$ of the function $f(x)$ at $x = 3$ is the linear - function that best approximates $f(x)$ near $x = 3$. It uses the value of the function and its derivative at $x = 3$ to create a straight - line approximation of the function in a small neighborhood around $x = 3$.

Answer:

Formula: $L(x)=f(3)+f^{\prime}(3)(x - 3)$. The linearization represents the linear function that best approximates $f(x)$ near $x = 3$.