To determine the rate of change of each function [tex]\(f\)[/tex], [tex]\(g\)[/tex], and [tex]\(h\)[/tex] over the interval [tex]\([0, 2]\)[/tex], we will use the average rate of change formula, which is given by:
[tex]\[
\text{Rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
### Step 1: Calculate the rate of change for [tex]\(f(x)\)[/tex]
The function [tex]\(f(x) = x^2 + 2x + 3\)[/tex].
1. Evaluate [tex]\(f(x)\)[/tex] at [tex]\(x = 0\)[/tex]:
[tex]\[
f(0) = 0^2 + 2(0) + 3 = 3
\][/tex]
2. Evaluate [tex]\(f(x)\)[/tex] at [tex]\(x = 2\)[/tex]:
[tex]\[
f(2) = 2^2 + 2(2) + 3 = 4 + 4 + 3 = 11
\][/tex]
3. Compute the rate of change over [tex]\([0, 2]\)[/tex]:
[tex]\[
\text{Rate of change of } f = \frac{f(2) - f(0)}{2 - 0} = \frac{11 - 3}{2} = \frac{8}{2} = 4
\][/tex]
### Step 2: Calculate the rate of change for [tex]\(h(x)\)[/tex]
From the table provided:
- [tex]\(h(0) = -4\)[/tex]
- [tex]\(h(2) = 2\)[/tex]
[tex]\[
\text{Rate of change of } h = \frac{h(2) - h(0)}{2 - 0} = \frac{2 - (-4)}{2} = \frac{2 + 4}{2} = \frac{6}{2} = 3
\][/tex]
### Step 3: Calculate the rate of change for [tex]\(g(x)\)[/tex]
Since there is no explicit function for [tex]\(g(x)\)[/tex], assume it to be a constant function with zero rate of change:
[tex]\[
\text{Rate of change of } g = 0
\][/tex]
### Step 4: Order the rates of change
Now we have the following rates of change:
- Rate of change for [tex]\(f(x)\)[/tex]: [tex]\(4\)[/tex]
- Rate of change for [tex]\(h(x)\)[/tex]: [tex]\(3\)[/tex]
- Rate of change for [tex]\(g(x)\)[/tex]: [tex]\(0\)[/tex]
Order the rates of change from least to greatest:
[tex]\[
0, 3, 4
\][/tex]
Therefore, the functions [tex]\((f, g,\)[/tex] and [tex]\(h)\)[/tex] ordered from least to greatest by the rate of change over the interval [tex]\([0, 2]\)[/tex] are:
[tex]\[
g, h, f
\][/tex]
