To calculate the average rate of change of the exponential function over the interval defined by the points \((-6, 13)\) and \( (5, 0)\), we follow these steps:
Step 1: Average Rate of Change of the Exponential Function
The average rate of change of a function \( f(x) \) over an interval \([a, b]\) is given by the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
For the given points, \( a = -6 \) and \( b = 5 \).
Step 2: Values of the Function
Assuming we need to calculate average rate of change for an exponential function \( f(x) = e^x \), we'll compute the values:
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Calculate \( f(-6) \): \[ f(-6) = e^{-6} \approx 0.002478752 \]
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Calculate \( f(5) \): \[ f(5) = e^5 \approx 148.41316 \]
Step 3: Average Rate of Change Calculation
Now substitute these values into the average rate of change formula:
\[ \text{Average Rate of Change} = \frac{f(5) - f(-6)}{5 - (-6)} = \frac{148.41316 - 0.002478752}{5 + 6} \]
\[ = \frac{148.41068}{11} \approx 13.5491 \]
Step 4: Estimate the Average Rate of Change of \( f(x) = (x - 0.1)^2 - 24.01 \)
Next, we find the average rate of change of the given quadratic function over the same interval.
First, compute \( f(-6) \) and \( f(5) \):
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Calculation of \( f(-6) \): \[ f(-6) = (-6 - 0.1)^2 - 24.01 = (-6.1)^2 - 24.01 = 37.21 - 24.01 = 13.20 \]
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Calculation of \( f(5) \): \[ f(5) = (5 - 0.1)^2 - 24.01 = (4.9)^2 - 24.01 = 24.01 - 24.01 = 0 \]
Step 5: Average Rate of Change Calculation for the Quadratic Function
Now substitute these values into the average rate of change formula:
\[ \text{Average Rate of Change} = \frac{f(5) - f(-6)}{5 - (-6)} = \frac{0 - 13.20}{5 + 6} \]
\[ = \frac{-13.20}{11} \approx -1.20 \]
Step 6: Comparison
- Average Rate of Change of the assumed exponential function: \(\approx 13.5491\)
- Average Rate of Change of the quadratic function: \(\approx -1.20\)
Conclusion
The average rate of change for the exponential function \( e^x \) over the interval is significantly greater than the average rate of change of the quadratic function \( (x - 0.1)^2 - 24.01 \). This reflects the rapid growth of the exponential function compared to the behavior of the quadratic function in the given interval.
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