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

let f be a function such that f(2) = 5. the graph of the derivative of f is given below: graph description y = f(x) with x - axis from - 1 to 4 and y - axis 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 description y = f(x) with x - axis from - 1 to 4 and y - axis 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 $a$ is the known point and $x$ is the point we want to approximate. Here, $a = 2$ and $x=2.1$.

Step2: Find $f(a)$ and $f^{\prime}(a)$

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 $y=f^{\prime}(x)$, $f^{\prime}(2)=- 2$.

Step3: Apply the linear - approximation formula

Substitute $a = 2$, $x = 2.1$, $f(2)=5$ and $f^{\prime}(2)=-2$ into the formula $L(x)=f(a)+f^{\prime}(a)(x - a)$. Then $L(2.1)=f(2)+f^{\prime}(2)(2.1 - 2)$. [ \begin{align*} L(2.1)&=5+( - 2)\times(2.1 - 2)\ &=5+( - 2)\times0.1\ &=5-0.2\ &=4.8 \end{align*} ]

Answer:

$4.8$