MAT 142 Calculus For Business For Marginal And Average Cost Function

 

A company lists its profit function as

P(x) = 8x – 0.02x2 – 500

Where x is the number of napkins made (in millions) and P is the profit in hundreds of dollars.

a. Graph the function.

b. Find the instantaneous rate of change in profit at x = 100.
(You should not use the definition of the derivative.)

c. Find the instantaneous rate of change in profit at x = 200.
(You should not use the definition of the derivative.)

d. Find the profit at x = 200.

e. Write a sentence interpreting question b. Use the correct units.

f. Interpret parts b, c, and d on the graph of part a.

 

Answer:

From the given situation C(x) is the total cost of producing x units of a product hence the marginal cost is defined as the rate of change of c(x) with respect to x

Marginal Cost, MC= and

Average Cost function is c(x) =C/x

1. The marginal cost function is found by differentiating both sides of the equation with respect to x

C(x) = 0.08x² + 2

Marginal Cost, MC=

=

=

=0.08(2x) +0=0.16x

Marginal Cost, MC=0.16x

1. Average Cost function is given by c(x)=C(x)/x

=

=

Differentiating both sides of the equation with respect to x

=

1. Marginal Cost for x=5

Marginal Cost=0.16x=01.6(5)=0.8

Interpretation: When the level of production is 5 units, the total cost function is increasing at the rate of 0.8

2. =
3. f) Graph MC(x), , and 

Observations: As can be noticed from the graph, the curve for marginal cost goes through the minimum point of the average cost curve. As a result of this when,

MC<AVC, AVC is said to be falling

MC>AVC, AVC is said to be on the rise

1. a) Graph of P(x) =8x-0.02x²-500 

1. b) Given

Profit P(x) =8x-0.02x²-500, then the instantaneous range of range in the profit would be determined by 

=8-0.02(2x)

=8-0.04x

Thus 

8-0.04(100) =8-4

=4 

1. c) At x=200 

=8-0.04x

=8-0.04(200)

8-8=0

1. d) P(x) at x=200=

=300

1. e) Interpretation of part b: When the company makes 100 pumpkins it makes profit that is 4 times the cost of production of the pumpkins
2. f) At x=100, p(x) =100 and the slope at this point is positive.  

References

Bisch, G. I. (2009). Nonlinear Oligopolies: Stability and Bifurcations. New Delhi: Springer Science & Business Media.

Costenoble, S. (2017). Finite Mathematics and Applied Calculus. Manchester: Cengage Learning.

Naito, T. (2015). Sustainable Growth and Development in a Regional Economy. Tokyo: Springe.

Schotter, A. (2010). Microeconomics: A Modern Approach. London: Cengage Learning.

Waner, S. (2017). Finite Mathematics and Applied Calculus, Loose-leaf Version. London: Cengage Learning.

Wheeler, R. (2016). Brief Calculus: A Graphing Calculator Approach. London: Wiley.