let f(x)=rootx/x+1 what is the average rate of change of f between x = 2 and x = 10?

  


The average rate of change of a function \( f(x) \) between two points \( x = a \) and \( x = b \) is given by the formula:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

For the function \( f(x) = \frac{\sqrt{x}}{x + 1} \), we need to find the average rate of change between \( x = 2 \) and \( x = 10 \).

### Step 1: Calculate \( f(2) \)
\[
f(2) = \frac{\sqrt{2}}{2 + 1} = \frac{\sqrt{2}}{3}
\]

### Step 2: Calculate \( f(10) \)
\[
f(10) = \frac{\sqrt{10}}{10 + 1} = \frac{\sqrt{10}}{11}
\]

### Step 3: Apply the average rate of change formula
\[
\text{Average Rate of Change} = \frac{f(10) - f(2)}{10 - 2} = \frac{\frac{\sqrt{10}}{11} - \frac{\sqrt{2}}{3}}{8}
\]

### Step 4: Simplify the expression
Let's compute this step-by-step:

\[
f(10) - f(2) = \frac{\sqrt{10}}{11} - \frac{\sqrt{2}}{3}
\]

Now find a common denominator:

\[
f(10) - f(2) = \frac{3\sqrt{10} - 11\sqrt{2}}{33}
\]

Finally, compute the average rate of change:

\[
\text{Average Rate of Change} = \frac{\frac{3\sqrt{10} - 11\sqrt{2}}{33}}{8} = \frac{3\sqrt{10} - 11\sqrt{2}}{264}
\]

So, the average rate of change of \( f(x) \) between \( x = 2 \) and \( x = 10 \) is:

\[
\boxed{\frac{3\sqrt{10} - 11\sqrt{2}}{264}}
\]

Would you like more details or have any questions?

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Here are 8 related questions you might find useful:

1. How do you find the derivative of the function \( f(x) = \frac{\sqrt{x}}{x+1} \)?
2. What is the average rate of change of \( f(x) = \frac{1}{x} \) between \( x = 2 \) and \( x = 10 \)?
3. How does the average rate of change compare to the instantaneous rate of change at \( x = 2 \) for \( f(x) = \frac{\sqrt{x}}{x+1} \)?
4. What is the significance of the average rate of change being positive or negative?
5. How do you calculate the average rate of change for \( f(x) = x^2 + x \) between \( x = 3 \) and \( x = 7 \)?
6. What is the average rate of change of a linear function over any interval?
7. Can you find the instantaneous rate of change at \( x = 5 \) for \( f(x) = \frac{\sqrt{x}}{x+1} \)?
8. How does the average rate of change for \( f(x) = \sqrt{x} \) compare to \( f(x) = \frac{1}{x+1} \)?

**Tip:** The average rate of change formula is very similar to the slope formula for a line, as it represents the slope of the secant line passing through two points on the graph of the function.

Discover how to find the average rate of change of the function f(x) = sqrt(x)/(x+1) between x = 2 and x = 10 with step-by-step calculations. Learn the application of the Average Rate of Change formula in calculus.

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