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Find the elasticity of demand at the indicated price for each of the following.(a) $p+4 x=80, p=40 $(b) $10 p+x=500, p=46$(c) $4 p+x=100, p=0 $ (d) $p x=8, p=4$(e) $p x^{2}=12, p=3 $(f) $p=\sqrt{9-x}, p=2$

(a) -1(b) $-23 / 2$(c) 0(d) -1(e) $-1 / 2$(f) $-8 / 5$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 5

Applications II - Business and Economic Optimization Problems

Derivatives

Baylor University

University of Nottingham

Idaho State University

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.

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all of these have similar steps. So starting with the first one to save you and I'm not going to write down the actual equations. I'm going to ultimately do the first step in salt for experts, but you want to solve for X in each of these will do one by one. So we'll start with a soapy plus four, X equals eighties. The equation were given to solve for X. We get 80 minus P all over four. And then the next step will be were given that P equals 40. So if we plug this into this P we get that X equals 10. The next step we want to do is find the derivative of dx we're D P. So that's the derivative of this equation right here. And so we can View it as 80/4 -4 P over for And so the derivative of pio reform native P over for do you P equals negative 1/4. So finally we can have the elasticity Equals 2 -1 4 times 40 over 10, like in this equation right here. And that will give us -1. Let's do that with the 2nd 1 might run out of room here. So for B we have to solve the original equation for X. And we get so there will be 10 P plus X equals 500 to solve for X. We get 500 minus 10 P. Were given that P equals 46. When you plug that into here, we get that x equals 40 and then we want to find dx over D P. So to find the derivative of this equation. And that one is fairly simple. Yeah. Sorry this is actually supposed to -10 p. There's no speak So are derivative is -10. And then we find the elasticity by multiplying the derivative with ratio of price over quantity 36. And that will give us -11.5. Yeah. Next we have four P plus X equals 100. So we solve for X. We get 100 -4 p. Yeah. And they were given that P equals zero. So when you plug that p into here we get x equals 100 solving for dx or dP. And you might notice when P equals zero. The L. S. E C will be a slightly different. It won't be a full number. So dx over D. P. We get -4. And then if you multiply and then for the elasticity you multiply by the derivative With price which is zero over X. And we get zero for us to see. So this is what I meant when it's not a full number is actually zero. Mhm. Okay so next one. I raised some stuff so we have room we have part D. Just P. X. It was eight. And so solving for X we get a to repeat and then we have P equals for When you plug that into here we get x equals two. And then finding the derivative it sounds a bit trickier when you have a fraction like this. I like to write it as P to the negative ones just so I can use the exponents rule. So that will give us negative eight P. To the negative too. And then if you multiply so here we're going to take this P. Value and actually plug it in twice because we have times P over X. So we plug in this P. Value here and here to get paid. So we will have -8 times 4 to the negative tooth Over 4/2. Now we'll give us native one. Okay next we have part E. So when you solve that for X we get X equals 12 or repeat square root and then we have P equals three. Pull that into here and we get x equals two DXDP. You have to be real careful with this one But I like to write as 12 over P. to the 1/2. Then using general we got native 1/2 12 overpay. I'm sorry not negative it's positive To the negative 1/2 times -12 Because you're taking the derivative of this inside. Pay to the negative too. And then from you can just plug your value of P. And directly and did this expression Which I simplified as negative 1 4th times negative 12 Times Native 1 9th. And then you take your P over or your P over X. Which is three halves and multiply that here too. And that will give you negative one half and then last one. I'll just see if I can put it down here. We have when solving for X. We get nine minus P squared kazaks P. We have two And then plug that in here to get x equals five dx over D. P will be, we're just looking at this because nines include the negative in there. So we got negative too. P to the first power times P over X, which will give us elasticity. So we get native two times two multiplied by 2/5, which is 8 -8/5 which will be

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