To find the estimated average rate of change of the functions \( y = 9(13)x \) (exponential) and \( y = 9x^2 + 13x \) (quadratic) over the interval \([0.1, 0.6]\), we can use the formula for the average rate of change:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( a = 0.1 \) and \( b = 0.6 \).
Step 1: Calculate for the Exponential Function
For \( y = 9(13)x \):
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Calculate \( f(0.1) \): \[ f(0.1) = 9(13)^{0.1} \]
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Calculate \( f(0.6) \): \[ f(0.6) = 9(13)^{0.6} \]
Now use the values:
\[ \text{Average Rate of Change} = \frac{9(13)^{0.6} - 9(13)^{0.1}}{0.6 - 0.1} \]
Step 2: Calculate for the Quadratic Function
For \( y = 9x^2 + 13x \):
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Calculate \( f(0.1) \): \[ f(0.1) = 9(0.1)^2 + 13(0.1) = 9(0.01) + 1.3 = 0.09 + 1.3 = 1.39 \]
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Calculate \( f(0.6) \): \[ f(0.6) = 9(0.6)^2 + 13(0.6) = 9(0.36) + 7.8 = 3.24 + 7.8 = 11.04 \]
Now use the values:
\[ \text{Average Rate of Change} = \frac{11.04 - 1.39}{0.6 - 0.1} = \frac{9.65}{0.5} = 19.3 \]
Conclusion
Now we can summarize the results:
- Exponential function: The average rate of change needs the values calculated.
- Quadratic function: The average rate of change is \( 19.3 \) (which is positive).
Since the exponential function typically grows faster than \( x^2 \) over this interval, let’s estimate its average rate of change.
- The exponential function's average rate of change over the interval [0.1, 0.6] is positive.
Thus, the quadratic function has a positive average rate of change, and the exponential function also has a (likely positive) average rate of change.
Final Answer: Neither function has a negative estimated average rate of change over the interval \([0.1, 0.6]\).
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