Question

x 11 13 14 15 17
f(x) -52 -63 -63 -63 -68
what is the best estimate for f(16) we can make based on this table?
choose 1 answer:
a -0.4
b -2.5
c -65.5
d -5

Explanation:

Step1: Recall derivative approximation formula

The derivative $f^{\prime}(a)$ can be approximated by the average rate of change $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$ for $x_1$ and $x_2$ close to $a$. For $a = 16$, we use the points $(x_1,f(x_1))=(15,f(15))$ and $(x_2,f(x_2))=(17,f(17))$ since 15 and 17 are closest to 16 among the given $x$-values.

Step2: Calculate the average rate of change

We know that $x_1 = 15$, $f(x_1)=-63$, $x_2 = 17$, and $f(x_2)=-68$. The average - rate of change formula is $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$. Substitute the values: $\frac{-68-(-63)}{17 - 15}=\frac{-68 + 63}{2}=\frac{-5}{2}=-2.5$.

Answer:

B. $-2.5$