Question

let f be a continuous and differentiable function. use this table of values that represents function f.

x	y
1	0
2	2
3	16
4	24

which of the following is the best estimate of f(2) based on this table of values?

○ 2
○ 8
○ 10
○ 14

Explanation:

Step1: Recall the average rate of change formula

To estimate the derivative \( f'(2) \), we use the average rate of change around \( x = 2 \). The average rate of change between \( x = a \) and \( x = b \) is \( \frac{f(b)-f(a)}{b - a} \). We can use the points before and after \( x = 2 \), i.e., \( x = 1 \), \( x = 2 \), \( x = 3 \).

Step2: Calculate the average rate of change before and after \( x = 2 \)

First, the average rate of change between \( x = 1 \) and \( x = 2 \):
\( \frac{f(2)-f(1)}{2 - 1}=\frac{2 - 0}{1}=2 \)
Then, the average rate of change between \( x = 2 \) and \( x = 3 \):
\( \frac{f(3)-f(2)}{3 - 2}=\frac{16 - 2}{1}=14 \)

Step3: Find the average of these two rates

To get a better estimate, we take the average of these two average rates of change:
\( \frac{2 + 14}{2}=\frac{16}{2}=8 \)
Alternatively, we can also consider the interval from \( x = 1 \) to \( x = 3 \) directly:
\( \frac{f(3)-f(1)}{3 - 1}=\frac{16 - 0}{2}=8 \)

Answer:

8 (corresponding to the option "8")