8 multiple choice 1 point the function f is differentiable and has values as shown in the table above. both…

Answer:

B. 27.5

Explanation:

Step1: Recall the definition of the derivative

The derivative $f^{\prime}(a)$ can be approximated by the difference - quotient $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$.

Step2: Calculate the left - hand approximation

Using $x_1 = 2.8$ and $x_2=3.0$, we have $\frac{f(3.0)-f(2.8)}{3.0 - 2.8}=\frac{45 - 39.20}{0.2}=\frac{5.8}{0.2}=29$.

Step3: Calculate the right - hand approximation

Using $x_1 = 3.0$ and $x_2 = 3.1$, we have $\frac{f(3.1)-f(3.0)}{3.1 - 3.0}=\frac{48.05 - 45}{0.1}=\frac{3.05}{0.1}=30.5$.

Step4: Use the increasing property of $f^{\prime}$

Since $f^{\prime}$ is strictly increasing on the interval $0\leq x\leq5$, and the left - hand approximation of $f^{\prime}(3)$ is 29 and the right - hand approximation is 30.5, then $f^{\prime}(3)$ must be less than the right - hand approximation and greater than the left - hand approximation of values of $f^{\prime}$ near $x = 3$. So $f^{\prime}(3)$ could be 27.5.