Math Problem Statement
Solution
Let's solve the problem step by step:
Problem Statement:
The cost function is given as:
where is the cost in dollars of producing units of a commodity.
Part (a): Average Rate of Change
We are asked to find the average rate of change of with respect to when changes from to two given values:
Formula:
The average rate of change of a function between two points and is given by:
(i) and
- First, we calculate and .
Then, we apply the formula for the average rate of change.
(ii) and
- Similarly, calculate and .
Again, apply the formula for the average rate of change.
Part (b): Instantaneous Rate of Change
The instantaneous rate of change (or marginal cost) is the derivative of the cost function with respect to , evaluated at .
Formula for Marginal Cost:
Evaluate this at :
Let me calculate all of these values for you.Here are the results:
(a) Average Rate of Change:
-
For :
-
For :
(b) Instantaneous Rate of Change (Marginal Cost) at :
Would you like any further explanation or details?
Here are 5 related questions to deepen your understanding:
- How does the average rate of change differ from the instantaneous rate of change?
- What does the marginal cost represent in practical terms?
- How would the rates change if the cost function had a cubic term?
- What if we wanted to estimate costs for ? How would you use the marginal cost?
- Can the marginal cost ever be negative? Under what conditions?
Tip: The marginal cost is often used to make production decisions, as it indicates the cost of producing one additional unit.
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Math Problem Analysis
Mathematical Concepts
Rates of Change
Derivatives
Quadratic Functions
Formulas
Average Rate of Change = (C(x2) - C(x1)) / (x2 - x1)
Marginal Cost (Instantaneous Rate of Change) = d(C(x)) / dx
Theorems
Definition of Derivative
Suitable Grade Level
Grade 12 or College-Level Calculus
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