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Find (a) the elasticity of demand and (b) the range of prices for which the demand is elastic $(E<-1)$ $f(p)=60 p(10-p)$

\frac{20}{3}<p<10

Calculus 1 / AB

Chapter 3

Applications of Differentiation

Section 9

Rates of Change in Economics and The Sciences

Derivatives

Differentiation

Applications of the Derivative

Oregon State University

Harvey Mudd College

Baylor University

Lectures

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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.

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In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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Find (a) the elasticity of…

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Use the price-demand equat…

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were given a demand function. Athletes P is equal to 60 times p times a quantity 10 minus p. Let's make a quick observation here. The price is going to be positive. And if we want to have a positive demand, that means 10 minus p also has to be positive. Negative demand rose. Zero demands would not be a very good business. So we want to determine the elasticity and that's equal to the price times a relative change in demand based on that price. What means we're gonna need to determine the first derivative. So let's go ahead and multiply this out. 60 p. Times 10 minus b is 600 p minus 60. He squared. That's what our demand function is. So our first derivative is going to equal to 600 minus 120. He now we confined our elasticity functions. He is equal to p times, actually. Why don't we go ahead and factor this 60 out? Well, we're at it. Back to out of 60. We get 60 times 10 minus two. He over here We have our first derivative 60 times 10 minus two p. That's all over the demand function which was 60 p times 10 minus one p. So the peace cancel in the sixties cancel and we get that are elasticity. Function is 10 minus two p over 10. Linus he That's our elasticity of demand. We'd also like to know the values of P for which the demand is elastic. That means we're looking for be less than negative one. That means 10 minus two p over 10 minus p is less than negative one. I would have to multiply both sides by 10 minus P. Let's take a look up here. We already said 10 minus Pia's positive. So when we multiply both sides by 10 minus p, we leave the inequality as it is. We don't need to flip the sign. Now we can distribute 10 minus two p is less than negative, 10 less p. So 20 is less than three p, and that means he is going to be greater than 20/3. 20/3 is six and 2/3 so that will be 6.6. Repeating. That means our demand will be elastic if the price is greater than six and 2/3 and we also had constrained our price to be less than 10. So this is the range or which our demand will be elastic

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