Math Problem Statement
For the given exponential function, find the average rate for x equals 1 and x equals 3. f left parenthesis x right parenthesis equals e Superscript x minus 1 Baseline minus 3
Solution
To find the average rate of change of the function between and , we use the formula for the average rate of change over an interval :
Here, , , and .
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Evaluate :
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Evaluate :
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Compute the average rate of change:
Simplifying further:
Thus, the average rate of change of from to is:
Would you like further details, or do you have any questions?
Here are some related questions to consider:
- How does the average rate of change differ for other intervals, such as to ?
- What is the derivative of , and how does it relate to the instantaneous rate of change?
- Can the average rate of change be negative for other intervals of this function?
- How does changing the constant term in affect the average rate of change?
- How would this calculation differ if were a polynomial instead of an exponential function?
Tip: The average rate of change for exponential functions often varies significantly across intervals due to the rapid growth rate of exponential terms.
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Math Problem Analysis
Mathematical Concepts
Exponential Functions
Average Rate of Change
Formulas
Average Rate of Change = (f(b) - f(a)) / (b - a)
Theorems
Definition of Average Rate of Change
Suitable Grade Level
Grades 11-12
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