let f be a function such that f(2)=5. the graph of the derivative of f is given below: graph of y = f(x)…

let f be a function such that f(2)=5. the graph of the derivative of f is given below: graph of y = f(x) with x - axis ranging from - 1 to 4 and y - axis ranging from - 3 to 2 using a linear approximation to estimate the value of f(2.1), we obtain the approximation 5.1 0.2 2 - 1.9 4.8

let f be a function such that f(2)=5. the graph of the derivative of f is given below: graph of y = f(x) with x - axis ranging from - 1 to 4 and y - axis ranging from - 3 to 2 using a linear approximation to estimate the value of f(2.1), we obtain the approximation 5.1 0.2 2 - 1.9 4.8

Answer

Explanation:

Step1: Recall linear - approximation formula

The linear - approximation formula is $L(x)=f(a)+f^{\prime}(a)(x - a)$, where $a = 2$, $x=2.1$.

Step2: Identify values from given information

We know that $f(2)=5$. To find $f^{\prime}(2)$, we look at the graph of $y = f^{\prime}(x)$. When $x = 2$, from the graph of $y=f^{\prime}(x)$, $f^{\prime}(2)=1$.

Step3: Apply the linear - approximation formula

Substitute $a = 2$, $x = 2.1$, $f(2)=5$, and $f^{\prime}(2)=1$ into the linear - approximation formula $L(x)=f(a)+f^{\prime}(a)(x - a)$. So $L(2.1)=f(2)+f^{\prime}(2)(2.1 - 2)$.

Step4: Calculate the approximation

$L(2.1)=5+1\times(2.1 - 2)=5 + 0.1=5.1$.

Answer:

A. 5.1