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The cost, in dollars, of producing $x$ bicycles is given by $C(x)=60+$ $10 x+1000 / x(x \geq 10) .$ (a) What does it cost to produce: (i) 99; (ii) 1 00 bicycles? (b) What is the cost of producing the 100 th bicycle? (i) Find it exactly; (ii) Use the derivative to find it approximately. (c) What is the error in using the derivative to approximate the marginal cost of the 100 th bicycle?(d) What is the average cost per bicycle when producing the 100 bicycles?

$(a)$(i) 1060.10(ii) 1070(b)(i) 9.90(ii) 9.90(c) 0.002d) 10.70

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 7

Marginal Functions and Rates of Change

Derivatives

Harvey Mudd College

University of Michigan - Ann Arbor

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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so the number for the cost of producing both the 99 100 bicycles for part A. What we can do is write the cost function and just plug and chug. So we have C f X equals 60 last 10, Alex Husband 1000 over X and this just is the X here is given that X is greater than or equal to 10. So for 99 and 100 you just plug that in for X and solve with those you get 1060 and 10 cents for 99 bicycles and 1070 for 100 bicycles moving on to park the for the first part. What you do is you just take the difference between these two to get the cost of producing the 100 bicycle. And when you subtract 1060 from or 1060 and 10 from 1070 you get 9 90. And if we were rounding, you would actually if we were running this to the next place, it would actually be a 0.102 so we can put appointment zero to their for the second part. Use the derivative. So we find the derivative of this equation. The derivative of the first term would be zero, because 60 is a constant and then the second term would be 10, because ex the one minus one would be 10 and then the derivative of 1000 over X. I like to write it as for something to t t as 1000 x to the negative one because it makes it easier to be able to see that the negative one goes here and then you add one or how you subtract one. And that would make that term negative 1000 X to the negative two or negative 1000 over X squared. That's the derivative of that. So the derivative of the cost function is yeah, 10 minus 1000 over X squared and then using that to find the cost of the 100 bicycle you would use X equals 100. Let's move this up here actually was 100 so the derivative, or the marginal cost of the 100 bicycle equals 9 90. And then for part C, it's asking, what's the error? So you take 9 90 minus 9. 90 to 9.902 and you find that the air is 0.2 And the last part. I don't know if I'll have room here, but, um, to find the average cost of each bicycle for 100 bicycles, you just take 100 or the cost for 100 bicycles, which is 1070 and divided by the number of bicycles, and that would give you 10 70 per bicycle.

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