The cost function for production of a commodity is C ( x ) = 352 + 22 x ? 0.06 x

Question

The cost function for production of a commodity is

C(x) = 352 + 22x ? 0.06x2 + 0.0007x3.

(a) Find C'(100).


Interpret

C'(100).

This is the amount of time, in minutes, it takes to produce 100 items.This is the cost of making 100 items.    This is the rate at which the production level is decreasing with respect to the cost when x = 100.This is the number of items that must be produced before the costs reach 100.This is the rate at which costs are increasing with respect to the production level when x = 100.

(b) Find the actual cost of producing the 101st item. (Round your answer to the nearest cent.)
$

Explanation / Answer

Given cost function is:

C(x)=352+22x+0.06x2+0.0007x3

C'(x)=0+22(1)+0.06(2x)+0.0007(3x2) (Since, if f(x)=an then f'(x)=n.an-1)

Now for C'(100) we put x=100 in the above equation.

C'(100)=22+0.06(2*100)+0.0007(3*10000)

C'(100)=22+12+21=55

C is the cost function, when it is differentiated with respect to x, it gives the change in cost with respect to point x.

So for C'(100):

This is the rate at which costs are increasing with respect to the production level when x = 100.

-------------------------

(b)

Actual cost is given by C.

So for the cost of 101st item, we put x=101 in C.

C(101)=352+22(101)+0.06(101)2+0.0007(101)3

   =352+2222+612.06+721.2107

=3907.2707

So the actual cost is 3907.2707

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