To find the average rate of change of the function [tex]\( f(x) = 2x^2 - 3 \)[/tex] between the points [tex]\((2, 1)\)[/tex] and [tex]\((2+h, f(2+h))\)[/tex], follow these steps:
1. Identify the coordinates:
- The first point is [tex]\((2, 1)\)[/tex].
- The second point is [tex]\((2+h, f(2+h))\)[/tex].
2. Compute [tex]\(f(2+h)\)[/tex]:
- Here, [tex]\(f(x) = 2x^2 - 3\)[/tex].
- Therefore, [tex]\(f(2+h) = 2(2+h)^2 - 3\)[/tex].
3. Expand and simplify [tex]\(f(2+h)\)[/tex]:
[tex]\[
f(2+h) = 2(2+h)^2 - 3
\][/tex]
[tex]\[
(2+h)^2 = 4 + 4h + h^2
\][/tex]
[tex]\[
f(2+h) = 2(4 + 4h + h^2) - 3 = 8 + 8h + 2h^2 - 3 = 5 + 8h + 2h^2
\][/tex]
4. Calculate the average rate of change:
- The formula for the average rate of change between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] of the function [tex]\(f\)[/tex] is:
[tex]\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
- In our case, this becomes:
[tex]\[
\frac{f(2+h) - f(2)}{(2+h) - 2}
\][/tex]
5. Substitute the values:
- We know [tex]\(f(2) = 2(2)^2 - 3 = 8 - 3 = 5\)[/tex].
- So:
[tex]\[
\frac{f(2+h) - f(2)}{(2+h) - 2} = \frac{(5 + 8h + 2h^2) - 5}{h} = \frac{8h + 2h^2}{h}
\][/tex]
6. Simplify the expression:
[tex]\[
\frac{8h + 2h^2}{h} = \frac{h(8 + 2h)}{h} = 8 + 2h
\][/tex]
Therefore, the expression in terms of [tex]\(h\)[/tex] that gives the average rate of change between [tex]\((2, 1)\)[/tex] and [tex]\((2+h, f(2+h))\)[/tex] is:
[tex]\[
8 + 2h
\][/tex]
