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Given a linear demand equation $x=m p+b$. Let $p$ change from $p_{0}$ to $p_{0}+h .$ Show that the ratio of the relative change in $x$ to the relative change in $p$ is exactly equal to the elasticity of demand at $p_{0}$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 5

Applications II - Business and Economic Optimization Problems

Derivatives

Oregon State University

University of Michigan - Ann Arbor

University of Nottingham

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.

08:48

Find the demand function i…

02:28

Find the elasticity of dem…

And this problem you could find a definition of elasticity of the men. To be fortunate GFP will be equal to the P. F. P over F. F. P. And there will be equation one. So let's fft me equal to Y. And as prime of people equal to wipe out than E. F. T. The equal to p times. White crime over what. And that's the equation too. Now the elasticity of demand is a linear function of the price given by E. F. P. Being equal to 80. Because the if you put E. F. P equals a B plus B. eight people have been the equation too. You get ap crusty times. Why will be equal to negative P. Tom's wife? Few expanders by using algebra. You got a few times wide con clause 80 plus B Times while be equal to zero sovereign for why? Yeah py pond equal to negative 80 plus B Tom's wife. So py prom because B Y. D. P. Now we can integrate both sides. If you integrate the the U. I. Was E. T. Back garden negative. You're integrating. Yes. So the integration of the O. I. D. P will be Eleanor block which is ego through negative integration of integration of A. D. P becomes negative 80 plus B times are enough key. You have elements why go to negative ap minus B times. Telling me The goal is to isolate for one. So you could just basic algebra. Uh huh. You have done enough. Why? Why B M mm Equal to negative 80 man. You can take the exponent the exponent in to both sides, taking E. To the Ellen of YP B. You get PB is equal to E. to the -80. Now you can solve the lie. So why gives you one over T. V. Times even a negative 18? So after F F P becomes this one, this right here, one over PB times E. To the negative A. P. Which is your demand function, which is F f t.

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