Expert: Paul Klarreich - 10/11/2008

Question
I am working through Morris Kline's Calculus: An Intuitive And Physical Approach and came across an answer of his that I did not agree with. I am hopeful you can help me understand.

In section 2-9 (The Differentiation of Simple Polynomials), Exercise 8 states:

"The total cost C of producing x units of some item is a function of x. (Of course physically x takes on only positive integral values but it is convenient to think of it as taking on all real values in some domain.) Economists use the term Marginal Cost for the rate of change of C with respect to x. Suppose that C = 5x^2 + 15x + 200, what is the marginal cost when x = 15? Would this marginal cost be the cost of the 16th unit?"

The rate of change   : C' = 10x + 15
Second derivative   : C'' = 10

I got the first part correct: C'(15) = 165 (ie, the marginal cost at the 15th unit is 165.)
For the second question, I thought No and the official solution guide says Yes (but without any explanation). I don't understand why it is Yes.

I calculated several values for C and C':

x       C       C'
----    ----    ----
0       200     15
1       220     25
2       250     35
3       290     45
...      ...    ...
15      1550    165
16      1720    175

To my way of thinking, the cost of the 16th unit is the cost of producing 16 units minus the cost of producing 15 units: 1720 - 1550 = 170 which does not equal 165.

In more general terms, I'm finding that the instantaneous rate of change in cost with respect to the number of units at unit x does not equal the cost of unit x+1. This makes sense to me since the rate of change is not constant, but is increasing by 10 for each additional unit (based on the second derivative).

Would you please help me understand the correct answer?

Thanks,  

Answer
Questioner:   Paul
Category:  Calculus
Private:  No
 
Subject:  Question in Morris Kline's Book
Question:  I am working through Morris Kline's Calculus: An Intuitive And Physical Approach and came across an answer of his that I did not agree with. I am hopeful you can help me understand.

In section 2-9 (The Differentiation of Simple Polynomials), Exercise 8 states:

"The total cost C of producing x units of some item is a function of x. (Of course physically x takes on only positive integral values but it is convenient to think of it as taking on all real values in some domain.) Economists use the term Marginal Cost for the rate of change of C with respect to x. Suppose that C = 5x^2 + 15x + 200, what is the marginal cost when x = 15? Would this marginal cost be the cost of the 16th unit?"

The rate of change : C' = 10x + 15

Second derivative : C'' = 10  << not relevant to this.

I got the first part correct: C'(15) = 165 (ie, the marginal cost at the 15th unit is 165.)
For the second question, I thought No and the official solution guide says Yes (but without any explanation). I don't understand why it is Yes.

I calculated several values for C and C':

x       C       C'
----    ----    ----
0       200     15
1       220     25
2       250     35
3       290     45
...      ...    ...
15      1550    165
16      1720    175

To my way of thinking, the cost of the 16th unit is the cost of producing 16 units minus the cost of producing 15 units: 1720 - 1550 = 170 which does not equal 165.

In more general terms, I'm finding that the instantaneous rate of change in cost with respect to the number of units at unit x does not equal the cost of unit x+1. This makes sense to me since the rate of change is not constant, but is increasing by 10 for each additional unit (based on the second derivative).

Would you please help me understand the correct answer?

Thanks,
....................................
Hi, Paul,

As that guy on 'Laugh-in' used to say, 'VERRRRY INTERESTING'.   I have to think you are correct here.  

If  C(x) is

WOW! ANOTHER HOME RUN!
(Sorry -- I've been watching the game.)

the cost of x units, then there is a disconnect between  x  as a continuous variable and  x as a discrete variable.  

So I have to agree with you, that based on these conditions, the cost of the 16-th unit should be C(16) - C(15), and not  C'(15).  

If the product in question is, say, Maple Syrup, then  C'(15) = 165 dollars/gallon, and this would be an accurate rate for producing the next DROP, but not the next gallon, because there should really be not much difference between  C'(x) and C'(x + 1), but there is.

So I agree with you.  Alas, Morris Kline has passed on and is no longer accepting feedback on his analyses.

You are really talking here about something called the Differential Approximation.  When you get there, write me again.