Marginal cost Consider the following marginal cost...

Marginal cost Consider the following marginal cost functions. a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units. b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units. $$C^{\prime}(x)=3000-x-0.001 x^{2}$$

Marginal cost Consider the following marginal cost functions.
a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.
b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.
$$C^{\prime}(x)=3000-x-0.001 x^{2}$$

Key Concepts

Marginal Cost Function

The marginal cost function represents the instantaneous rate of change of the total cost with respect to the number of units produced. In other words, it provides the cost of producing one additional unit at any given production level, making it a crucial tool in analyzing cost behavior and production efficiency.

Definite Integral

The definite integral is used to accumulate the effect of a continuous rate of change over a specific interval. In the context of marginal cost, integrating the marginal cost function over a production interval yields the exact total additional cost incurred when production is increased between two levels.

Accumulation of Incremental Costs

This concept involves summing up the small incremental costs over a production interval to determine the total additional cost of production changes. It uses the marginal cost function and the definite integral to translate instantaneous cost changes into an aggregate cost over a finite range of production levels.

Transcript

00:03 Okay, so we have a function, it's a cost function, marginal cost function, and we're going to figure out the additional cost if we change x from 100 to 150 or from 500 to 550.

00:18 So we're going to do for above.

00:19 So to do that, we're going to use an integral.

00:22 So we're going to write out the integral because i think that's important.

00:25 So we have the integral, we have from 100 to 150, and we have 3 ,000 minus x minus 0 .001.

00:36 X squared, which is equal to, if we integrate this, we get 3 ,000 x minus x squared over 2, minus 0 .001, which in this case, i'm actually going to change 0 .001.

00:52 I'll write it up here.

00:53 I'm going to change 0 .001 to 1 over 3 ,000.

00:58 I think it'll make it a bit easier, with all the numbers and stuff.

01:01 So we're going to do that.

01:03 So here we get, sorry, 1 over 1 ,000.

01:07 There should be 1.

01:09 So we get x squared over 3 ,000.

01:14 X cubed over 3 ,000.

01:16 So we have x cubed over 3 ,000, and we're doing that from 100 to 150.

01:21 If we do that, we get 3 ,000 times 150 minus 150 minus 150 squared divided by 2 minus 150 cubed divided by 3 ,000, minus 350 cubed, divided by 3 ,000, minus 3 ,000.

01:45 Don't forget the parentheses because we're multiple or subtracting times a whole expression times 100 minus 100 squared over 2 minus 100 cubed over 3 ,000.

02:09 If we calculate all those out, we're going to get 450 ,000 minus 11 ,250 ,000, minus 11, 125 ,000, minus 11, 125 ,000, minus 11125, minus 325, minus 325, minus 3 ,000.

02:28 Plus 5 ,000 because remember a negative minus subtracting a negative creates a positive and then we have plus 3 ,333 .33 .3...

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