To find the average rate of change of the function [tex]\(f(x) = 2x^2 - x + 1\)[/tex] from [tex]\(x = 1\)[/tex] to [tex]\(x = 3\)[/tex], follow these steps:
### Step 1: Evaluate the function at [tex]\(x = 1\)[/tex]:
First, we need to find the value of the function at [tex]\(x = 1\)[/tex].
[tex]\[ f(1) = 2(1)^2 - 1 + 1 \][/tex]
[tex]\[ f(1) = 2 - 1 + 1 \][/tex]
[tex]\[ f(1) = 2 \][/tex]
### Step 2: Evaluate the function at [tex]\(x = 3\)[/tex]:
Next, we need to find the value of the function at [tex]\(x = 3\)[/tex].
[tex]\[ f(3) = 2(3)^2 - 3 + 1 \][/tex]
[tex]\[ f(3) = 2(9) - 3 + 1 \][/tex]
[tex]\[ f(3) = 18 - 3 + 1 \][/tex]
[tex]\[ f(3) = 16 \][/tex]
### Step 3: Calculate the average rate of change:
The average rate of change of the function over the interval from [tex]\(x = 1\)[/tex] to [tex]\(x = 3\)[/tex] is given by the formula:
[tex]\[ \frac{f(x_2) - f(x_1)}{x_2 - x_1} \][/tex]
Substitute [tex]\(x_1 = 1\)[/tex], [tex]\(x_2 = 3\)[/tex], [tex]\(f(x_1) = 2\)[/tex], and [tex]\(f(x_2) = 16\)[/tex]:
[tex]\[ \frac{f(3) - f(1)}{3 - 1} = \frac{16 - 2}{3 - 1} \][/tex]
[tex]\[ \frac{16 - 2}{3 - 1} = \frac{14}{2} \][/tex]
[tex]\[ = 7 \][/tex]
So, the average rate of change of the function [tex]\(f(x) = 2x^2 - x + 1\)[/tex] from [tex]\(x = 1\)[/tex] to [tex]\(x = 3\)[/tex] is [tex]\(7\)[/tex].
