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The cost and demand equations are given. (a) Determine the profit function. (b) What is the profit when the level of production is $100 ?$(c) Find the marginal profit function. (d) What is the price per item when the marginal profit is zero? (e) Sketch the profit function. (f) What is the level of production when the marginal profit is zero? What does it represent?$$C(x)=20 x+500,20 p+x=1000$$

(a) $P(x)=-\frac{1}{20} x^{2}+30 x-500$(b) 2000(c) $P^{\prime}=-1 / 10 x+30$(d) 35(e) 300

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 7

Marginal Functions and Rates of Change

Derivatives

Missouri State University

University of Michigan - Ann Arbor

Idaho State University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

01:10

Suppose it has been determ…

01:12

Profit The demand function…

04:04

Profit An analyst has foun…

01:09

Suppose the cost of produc…

03:30

18:04

(a) Show that if the profi…

Suppose the cost, in dolla…

04:58

Marginal revenue, cost, an…

So let's suppose it's been determined that the demand in thousands of dollars of a certain for a certain time Is given by the equation 20 overruled x. p. equals 20 overrode x. I'm in the cost of producing X00 items would be given by CFX is equal to hi that Carson. So then we want to determine the revenue function, the profit function. Um So the revenue function is going to be okay X times the profit function or the demand function, that's gonna be 20 x over DX. And then the profit function is going to equal the revenue function minus the cost function. So that's why I end up giving us all the functions we need to solve the problem.

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