Suppose that the cost, in dollars, for a company to produce...

Suppose that the cost, in dollars, for a company to produce x pairs of a new line of jeans is C(x) = 1800 + 2x + 0.02x^2 + 0.0005x^3. (a) Find the marginal cost function. (b) Find C'(100). C'(100) = What does this predict? The exact cost of the 101st pair of jeans. The approximate cost of the 100th pair of jeans. The exact cost of the 99th pair of jeans. The exact cost of the 100th pair of jeans. The approximate cost of the 101st pair of jeans. (c) Find the difference between C'(100) and the actual cost of manufacturing the 101st pair of jeans. $

Transcript

00:01 Let's determine the marginal cost function, given that the cost in dollars to produce x pairs of new line of genes is given by this cost function, c of x, equals 200 plus 3x plus 0 .05x squared plus 0 .002 x cube.

00:18 So to determine the marginal cost function, we basically find the derivative of the cost function, that is c prime of x, gives the marginal cost function.

00:28 So that means we have to find the derivative of this c of x function with respect to x.

00:35 So let's find the derivative of the right side expression.

00:38 The derivative of 200, which is a constant, is 0.

00:42 And then the derivative of 3x is 3 times the derivative of x.

00:47 And that is 1.

00:48 Next we have 0 .05 and its derivative is 0 .05 times the derivative of x squared using power rule it is 2x, plus the last term that we have is 0 .002 x cube.

01:03 So first i put this constant point 0 .02.

01:08 Multiply this with the derivative of x cubed and according to the power rule, it is 3x squared.

01:13 So let's simplify this.

01:16 We have c prime of x and this equals 3 times 1 is 3 plus 0 .05 times 2x and that equals 0 .0 .1.

01:27 X and then we have a point triple zero two times three x squared so that will be 0 .0.

01:38 0 .0 6 x squared.

01:42 So this is the expression for the marginal cost function.

01:53 Let's do part b.

01:54 Here we have to find c prime of 100.

01:59 So this equals we have to replace x in the marginal cost function by 100.

02:05 So when you do that, we get 3 plus 0 .1 times of 100 plus 0 .06 times 100 and this is square.

02:22 And evaluating this is in the calculator, we get this exactly equals 19.

02:30 So we see that c prime of 100 equals 19.

02:39 Okay, now let's predict this.

02:42 This one is the marginal cost at x equals 100, which means the marginal cost of producing the extra one pair of chains at x equals 100.

02:56 That means the marginal cost of producing the 101st item.

03:01 And that one is always an approximate and not the exact value.

03:05 So in this answer choices, we strike out this first three answer choices because they say exact cost.

03:15 And if you look at the fourth one, it says like approximate cost of the hundredth pair of genes...

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