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For the demand equation $p=(a-b x)^{1 / 2},$ with $a>0$ and $b>0,$ show that the price at which the total revenue is maximized is independent of $b$.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

03:11

for the given demand funct…

01:34

If total demand $(Q)$ is s…

06:39

$\begin{array}{l}{\text { …

01:02

The demand equation for a …

04:17

(a) Show that if the profi…

18:04

for the demand equation P equals a minus B x one half. Yeah. For a is greater than zero and B is greater than zero. We want to show that the price at which total revenue is maximized is independent of B. And we know this to be the case because if we take a P. Prime, which is the next step of maximizing or function, um we end up getting one half A -7. X. It Uh to the -1 House. Okay. Mhm. And a negative thing. We're gonna set this equal to zero though, and that's going to mean that since this right here is going to get put in the denominator, we're gonna have one over to route this and this is just going to get factored out. Um So we see it to be independent of B.

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