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The marginal revenue for $x$ items in dollars is given by $R^{\prime}(x)=-4 x+8$ Determine the (a) revenue function and (b) the demand function.

(a) $-2 x^{2}+8 x$(b) $-2 x+8$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Oregon State University

Harvey Mudd College

University of Nottingham

Boston College

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

01:36

Find the revenue and deman…

02:08

Demand Find the demand fun…

03:16

Demand function, $p=D(x),$…

02:55

Find the marginal revenue …

02:03

01:55

02:09

So we have those E Yeah, 310 you know, minus four X. Suppose we have this. So this is gonna be D. R A calls 310 D X. Okay. Okay. And we're just gonna integrate both sides and you know that this one is going to give you are because you're integrating one with respect to uh this are right here and you're integrating this with respect to X right here. So this is gonna be 310 minus X squared? Right? Two X squared actually. And then plus an arbitrary cost. And see, okay, now that has put extra quarter zero. Put X equals zero. Put are equal to zero, What do you have? So X equals zero, R equals zero. And then you can see that C is also going to be zero because this is zero, this is zero, this is zero. So C is going to be zero. So finally our is going to be 310 X mhm minus two X squared. So this is your function that satisfies this differential equation and satisfies the initial conditions, right?

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