Find the average rate of change of the function over the given interval.
y equals StartRoot 2 x EndRoot​;
between xequals2 and xequals8
Question content area bottom
Part 1
What expression can be used to find the average rate of​ change?
A.StartFraction f left parenthesis 8 right parenthesis plus f left parenthesis 2 right parenthesis Over 8 plus 2 EndFraction
StartFraction f left parenthesis 8 right parenthesis plus f left parenthesis 2 right parenthesis Over 8 plus 2 EndFraction
B.ModifyingBelow lim With h right arrow 0 StartFraction f left parenthesis 8 plus h right parenthesis minus f left parenthesis 8 right parenthesis Over h EndFraction
ModifyingBelow lim With h right arrow 0StartFraction f left parenthesis 8 plus h right parenthesis minus f left parenthesis 8 right parenthesis Over h EndFraction
C.ModifyingBelow lim With b right arrow a StartFraction f left parenthesis b right parenthesis minus f left parenthesis 2 right parenthesis Over b minus 2 EndFraction
ModifyingBelow lim With b right arrow aStartFraction f left parenthesis b right parenthesis minus f left parenthesis 2 right parenthesis Over b minus 2 EndFraction
D.StartFraction f left parenthesis 8 right parenthesis minus f left parenthesis 2 right parenthesis Over 8 minus 2 EndFraction
StartFraction f left parenthesis 8 right parenthesis minus f left parenthesis 2 right parenthesis Over 8 minus 2 EndFraction
Part 2
The average rate of change of y between xequals2 and xequals8 is 
  
enter your response here.
​(Simplify your​ answer.)

### Part 1: Expression for the average rate of change
The average rate of change of a function over an interval \([a, b]\) is given by:

\[
\frac{f(b) - f(a)}{b - a}
\]

In this case, the interval is \([2, 8]\) and the function is \(f(x) = \sqrt{2x}\). The correct expression to find the average rate of change is:

\[
\frac{f(8) - f(2)}{8 - 2}
\]

This corresponds to **Option D**.

### Part 2: Calculating the average rate of change
Now, let's calculate the average rate of change for \(f(x) = \sqrt{2x}\) between \(x = 2\) and \(x = 8\).

1. First, find \(f(8)\):
\[
f(8) = \sqrt{2 \times 8} = \sqrt{16} = 4
\]

2. Next, find \(f(2)\):
\[
f(2) = \sqrt{2 \times 2} = \sqrt{4} = 2
\]

3. Now, compute the average rate of change:
\[
\text{Average rate of change} = \frac{f(8) - f(2)}{8 - 2} = \frac{4 - 2}{8 - 2} = \frac{2}{6} = \frac{1}{3}
\]

So, the average rate of change is \( \frac{1}{3} \).

### Summary
The expression to find the average rate of change is:
\[
\frac{f(8) - f(2)}{8 - 2}
\]
And the average rate of change of \(y = \sqrt{2x}\) between \(x = 2\) and \(x = 8\) is:
\[
\frac{1}{3}
\]

Would you like more details or have any questions?

---

Here are some related questions:
1. How does the average rate of change differ from the instantaneous rate of change?
2. How would the average rate of change change if the interval were from \(x = 1\) to \(x = 9\)?
3. What is the geometric interpretation of the average rate of change for this function?
4. How would you compute the instantaneous rate of change of \(f(x) = \sqrt{2x}\) at \(x = 2\)?
5. How does the form of the function \(f(x) = \sqrt{2x}\) affect the average rate of change as \(x\) increases?

**Tip:** The average rate of change is essentially the slope of the secant line connecting two points on a curve.

Find the average rate of change of the function over the given interval. y equals StartRoot 2 x EndRoot; between x equals 2 and x equals 8. Part 1: What expression can be used to find the average rate of change? A. StartFraction f left parenthesis 8 right parenthesis plus f left parenthesis 2 right parenthesis Over 8 plus 2 EndFraction B. ModifyingBelow lim With h right arrow 0 StartFraction f left parenthesis 8 plus h right parenthesis minus f left parenthesis 8 right parenthesis Over h EndFraction C. ModifyingBelow lim With b right arrow a StartFraction f left parenthesis b right parenthesis minus f left parenthesis 2 right parenthesis Over b minus 2 EndFraction D. StartFraction f left parenthesis 8 right parenthesis minus f left parenthesis 2 right parenthesis Over 8 minus 2 EndFraction Part 2: The average rate of change of y between x equals 2 and x equals 8 is.

Learn how to find the average rate of change of the function y = √(2x) between x = 2 and x = 8. This step-by-step solution covers key algebraic concepts, including using the formula for the average rate of change and simplifying square root functions. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation