Math Problem Statement
Find the average rate of change of
g
(
x
)
=
–
19
x
over the interval
[
19
,
20
]
.
Solution
To find the average rate of change of the function over the interval , we use the formula:
Where and .
- First, calculate :
- Then, calculate :
- Now, apply the average rate of change formula:
- Simplify the expression:
- Find a common denominator to combine the fractions:
- So, the average rate of change is:
Thus, the average rate of change of over the interval is .
Would you like further details or have any questions?
Here are some related questions you might find interesting:
- What would be the average rate of change over the interval ?
- How does the value of the average rate of change compare with the derivative of at ?
- What does the small value of indicate about the change in the function over this interval?
- How would the average rate of change differ if the interval was much larger, say ?
- How does the average rate of change help in approximating the function's behavior between the two points?
Tip: The closer the two points are on the interval, the more the average rate of change approximates the instantaneous rate of change (derivative) at a point in between.
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Math Problem Analysis
Mathematical Concepts
Average Rate of Change
Functions
Formulas
Average Rate of Change formula: \( \frac{g(b) - g(a)}{b - a} \)
Theorems
-
Suitable Grade Level
Grades 11-12
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