# Rate of change and limits

• Jul 11th 2007, 12:11 PM
starswept
Rate of change and limits
Hi,
I'm having trouble with the following question regarding limits and rates of change:

A manufacturer's total weekly cost in producing x items can be written as C(x) = F + V(x), where F, a constant, represents fixed costs such as rent and utilities, and V(x) represents variable costs, which depend on production level x. Show that the rate of change of the cost is independent of fixed costs.

I know that obviously it will be, since f remains constant and is not dependent on the x variable. However, I'm not certain how to go about proving it. I thought of maybe setting values for F, V, and X, then using the limit for rate of change to show that it does not change when F does, but that didn't work out. Any help is greatly appreciated :) Thank you!
• Jul 11th 2007, 07:04 PM
Jhevon
Quote:

Originally Posted by starswept
Hi,
I'm having trouble with the following question regarding limits and rates of change:

A manufacturer's total weekly cost in producing x items can be written as C(x) = F + V(x), where F, a constant, represents fixed costs such as rent and utilities, and V(x) represents variable costs, which depend on production level x. Show that the rate of change of the cost is independent of fixed costs.

I know that obviously it will be, since f remains constant and is not dependent on the x variable. However, I'm not certain how to go about proving it. I thought of maybe setting values for F, V, and X, then using the limit for rate of change to show that it does not change when F does, but that didn't work out. Any help is greatly appreciated :) Thank you!

"rate of change" here, screams "derivative". to find the rate of change of cost, you find the derivative of C(x). since F is constant, it will vanish, so the rate of change of cost does not depend on F. that's all you need to prove it ...wait, have you done calculus? i don't remember what level of math you are at
• Jul 11th 2007, 07:07 PM
starswept
Quote:

Originally Posted by Jhevon
"rate of change" here, screams "derivative". to find the rate of change of cost, you find the derivative of C(x). since F is constant, it will vanish, so the rate of change of cost does not depend on F. that's all you need to prove it ...wait, have you done calculus? i don't remember what level of math you are at

Nope, I'm just starting calculus. All I've learned so far is the difference quotient, how to evaluate limits, and continuity. We start derivatives tomorrow (:eek:).
• Jul 11th 2007, 07:10 PM
Jhevon
Quote:

Originally Posted by starswept
Nope, I'm just starting calculus. All I've learned so far is the difference quotient, how to evaluate limits, and continuity. We start derivatives tomorrow (:eek:).

what do you mean by "the difference quotient"?
• Jul 11th 2007, 07:12 PM
starswept
Quote:

Originally Posted by Jhevon
what do you mean by "the difference quotient"?

The text just uses that to refer to the slope of a tangent:

Lim
h->0

f(a+h) - f(a)
__________

h
• Jul 11th 2007, 07:14 PM
Jhevon
Quote:

Originally Posted by starswept
The text just uses that to refer to the slope of a tangent:

Lim
h->0

f(a+h) - f(a)
__________

h

oh. you meant the limit definition of a derivative. i thought so but i had to make sure. i don't think i've ever heard it referred to as that. yes, that's fine. i think you can use that as well to show that F will vanish if you plug C(x) into that formula (i haven't tried it, but in theory it should work ;):D)

This is my 28:):)th post!!!!!!!!