To determine the rate of change for the interval between [tex]\( x = 0 \)[/tex] and [tex]\( x = 2 \)[/tex] for the given function [tex]\( f(x) = 2x^2 + x - 3 \)[/tex], we need to calculate the change in the function's values over that interval and divide it by the change in [tex]\( x \)[/tex].
Here are the steps for calculating the rate of change:
1. Identify the values of [tex]\( f(x) \)[/tex] at the endpoints of the interval:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) \)[/tex] is [tex]\( -3 \)[/tex] (as given by the table).
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) \)[/tex] is [tex]\( 7 \)[/tex] (as given by the table).
2. Calculate the difference in the function's values:
[tex]\[
f(2) - f(0) = 7 - (-3) = 7 + 3 = 10
\][/tex]
3. Calculate the difference in [tex]\( x \)[/tex]-values:
[tex]\[
2 - 0 = 2
\][/tex]
4. Compute the rate of change using the formula:
[tex]\[
\text{Rate of change} = \frac{ \Delta f(x) }{ \Delta x } = \frac{ f(2) - f(0) }{ 2 - 0 } = \frac{ 10 }{ 2 } = 5
\][/tex]
Hence, the rate of change for the interval between [tex]\( x = 0 \)[/tex] and [tex]\( x = 2 \)[/tex] is [tex]\( 5.0 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{5} \][/tex]
