Questions: given values. f(x)=x^2+1 at x=5 Show the relative rate of change

Transcript text: given values. \[ f(x)=x^{2}+1 \] at $x=5$ Show the relative rate of change

Solution

Solution Steps

Step 1: Define the Function

The function is given by \[ f(x) = x^2 + 1. \]

Step 2: Identify the Point of Interest

We are interested in evaluating the function at \[ x = 5. \]

Step 3: Calculate the Average Rate of Change

To find the average rate of change of $ f(x) $ around $ x = 5 $, we consider a small increment $ h $ such that $ x_2 = 5 + h $ where $ h = 0.0001 $. Thus, we have: \[ x_1 = 5 \quad \text{and} \quad x_2 = 5.0001. \]

Step 4: Apply the Average Rate of Change Formula

The average rate of change is calculated using the formula: \[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}. \]

Step 5: Evaluate the Function at the Points

Calculate $ f(5) $ and $ f(5.0001) $: \[ f(5) = 5^2 + 1 = 25 + 1 = 26, \] \[ f(5.0001) = (5.0001)^2 + 1 = 25.00010001 + 1 = 26.00010001. \]

Step 6: Substitute into the Formula

Substituting the values into the average rate of change formula gives: \[ \text{Average Rate of Change} = \frac{26.00010001 - 26}{5.0001 - 5} = \frac{0.00010001}{0.0001}. \]

Step 7: Simplify the Expression

This simplifies to: \[ \text{Average Rate of Change} \approx 10.0001. \]

Step 8: Conclusion

Thus, the relative rate of change of $ f(x) $ at $ x = 5 $ is approximately \[ 10.0001. \]