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A manufacturer estimates that his daily cost, in dollars, of producing $x$ television sets is $C(x)=2000+500 x+8000 / x .$ He must manufacture at least2 sets, and due to storage limitations, he cannot produce more than 250 sets. Prove, by using Exercise 75 that there is some number of sets which will minimize his cost.

$C$ is continuous on $2 \quad x \quad 75$ so $\mathrm{Ex} 75$ is applicable

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Campbell University

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Idaho State University

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.

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The cost $C$, in dollars, …

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A manufacturer of office s…

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A company manufactures and…

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A manufacturer finds that …

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A manufacturer has been se…

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So they give us this cost equation, which is four X squared, minus 2000 X plus six. I gotta count. 12345 zeroes. 2345 zeros. He's got double check. Yeah, so without having calculus, all we know is that this is a parabola goes down and then comes back up. Um, So when they asked for the minimum of something, all we can do at this point is understand There are asking for the Vertex. Well, you can find the Vertex by doing that. X equals negative B over two, a formula and the B values the negative 2000. So the negative of negative 2000 ft positive 2000 over to a and A is falling This problem. So if you look at that of 2000 divided by eight always give me an answer of, uh, 250 and whatever the units are looks like it's t v sets, t v sets. So that's circle that circle that and green. That's your answer for letter it so and let her be, uh they just want you to figure out what that minimum cost to be, which is plugging in 250. And for all of those exits now, I would probably go to a calculator just to make sure I'm doing this correctly. Um, you could probably do some of this in your head. Most people know that. 25 square to 625 with two zeros in there. I'll see two times 25 is 50 with 1234 zeros. Plus this number. Yeah, but even then, I'd probably go to the calculator to get the correct answer of $350,000. Yeah, circle that. That's your answer. A letter B. So then and let her see, They're just saying, What's the cost per set? Um, well, that's going to be equal to the minimum cost you had per 250. So divide those numbers, so cost per cent is divide them to get the correct answer of $1400 per set. Uh, so a is 250 b is 350,000 and see as 1400

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