Math Problem Statement
Use the given table of values for g(x) to do the following. x 2 2.3 3.1 4 4.3 5 g(x) 16.2 14.1 11.7 14.2 15.4 20.1 (a) Find the average rate of change of g(x) over [2, 5]. (Round your answer to three decimal places.) (b) Approximate g′(4) as accurately as possible.
Solution
Let's solve each part step by step.
Part (a): Average Rate of Change
The average rate of change of a function over an interval is calculated using the formula:
For the interval :
- ,
- ,
So, the average rate of change of over is:
Part (b): Approximate
To approximate the derivative , we can use the symmetric difference quotient:
Since we don't have from the table, we can instead approximate using points close to 4, such as:
- The points and
- The points and
Using and :
Using and :
Average Approximation:
Taking the average of the two approximations:
Thus, the approximation of is approximately:
Summary:
- (a) The average rate of change over is .
- (b) The approximation for is .
Would you like a more detailed explanation or have any questions about the process?
Further Questions:
- What does the average rate of change of a function represent in a real-world scenario?
- How does the choice of points affect the approximation of ?
- Why is using the symmetric difference quotient often more accurate for derivative approximations?
- Can you find the average rate of change over other intervals, such as ?
- What does a positive or negative average rate of change indicate about the function?
Tip:
When approximating derivatives, using points as close as possible to the point of interest gives a more accurate estimate.
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Math Problem Analysis
Mathematical Concepts
Average Rate of Change
Derivative Approximation
Symmetric Difference Quotient
Formulas
Average Rate of Change = (g(b) - g(a)) / (b - a)
Symmetric Difference Quotient: g'(x) ≈ (g(x+h) - g(x-h)) / (2h)
Theorems
-
Suitable Grade Level
Grades 10-12
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