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The total cost $C$ (in dollars) of production can be interpreted as an area. If the cost per unit $C^{\prime}$ (in dollars per unit) of producing $x$ units is given by $100 /(0.01 x+1)^{2}$, find the total cost of producing 100 units by finding the area under the curve of $C^{\prime}$ vs $x$.

Calculus 2 / BC

Chapter 26

Applications of Integration

Section 2

Areas by Integration

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given this cost function like to compare the marginal cost to the average cost. So the marginal cost That's the first derivative. See prime of X. That's going to be 0.2 x less 40 and the average cost is the cost to make X items divided by X. So if we take our function CFX and divide by X, we're going to get 0.1 x squared plus 40 x less 3600 over X. And if we simplify that a bit will have 0.1 x less 40 plus 3600 over X. So this is our average cost. And let's compare at 100 100 items, the marginal cost would be see prime of 100. That'll be 0.2 times 100 plus 40 which equals 42. The marginal cost is 42 at X equals 100. The average cost at X equals 100. It's going to equal the 0.1 times 100 plus 40 plus 3600 divided by 100 which equals 77. So we can see the marginal cost of ah 100 which equals 42 is less than the average cost to make 100 items, which is 77. If we were to make another item, the average cost for 101 items would be. This same function only will substitute in 101. I was 3600 divided by 100 and what? And that turns out to be 76.6, which is less than 77 so we can see that the average price goes down as the number increases.

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