Notes, Lesson 4.7
Applications to Economics
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The ability to use calculus to find minima and maxima is very useful in many areas of study. Economics is no exception. If we can maximize our profit and minimize our costs, our business goals can approach the optimum. Below is a chart of economic terms and formulas that will allow us to solve some economics problems and make use of our derivative skills:

Total Cost C(x)
Marginal Cost

C'(x)

Average Cost
Price Function p(x)
Revenue Function R(x) = x p(x)
Marginal Revenue R'(x)
Profit Function P(x) = R(x) - C(x)
Marginal Profit P'(x) = R'(x) - C'(x)

We see that whenever we find Marginal Cost or Marginal Revenue, or Marginal Profit, we are finding the instantaneous rate of change or derivatrive. As we often want to minimize average cost in business, it becomes important to business problem solving to recognize that:




Example Problem:

Given the cost function:
(a) Find the average cost and marginal cost functions.
(b) Use graphs of the functions in part (a) to estimate the production level that minimizes the average cost.
(c) Use calculus to find the minimum average cost.
(d) Find the minimum value of the marginal cost.

Given Problem, #8, Lesson 4.7
To find average cost, we know that we need to use the formula: .
To find the marginal cost, we use the formula: C'(x)

We now move on to part (b)

We plot the graph of the average cost (red), and the graph of the marginal cost (green). By observation, it appears that the value of x (number of items produced) where the two graphs intersect is about x = 140.
We now work on part (c)

Because of our principle that: "If the average cost is a minimum, then marginal cost = average cost," we set the average cost formula equal to the marginal cost formula and attempt to solve.
> solve(0=.0008*x^3-.09*x^2-339,x);

-11.52974818 - 54.70830486 I, -11.52974818 + 54.70830486 I, 135.5594964

Here we need a computer algebra system to find the solution to this cubic equation. We see there are two imaginary solutions and one real solution. The answer is approximately, This agrees with our visual estimate above.

 



Check Concepts
Check Concepts
Check Concepts


#1: True or False: To find the minimum value of the average cost, find the average cost function, differentiate, set equal to zero, and solve.
   
#2: True or False: The derivative of the cost function is the average cost function.
   
#3: To find the profit function, you subtract the cost function from the....
   
#4: The marginal profit function is the derivative of the...
   
#5 True or False: If the profit is a maximum, then the marginal revenue is equal to the marginal cost.

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