We will use the derivative of profit, cost, and revenue functions to make estimates.

- To understand this process, we look first at an example function What is the rate of change of at ?

We calculate the derivative and then substitute 1 for *x*:

Graphs illustrating this process are shown below. The tangent

line at

is

We have

as an

equation of the tangent line. Examine the points on the

graph and on the tangent where

. They are not far apart. The tangent

line gives an estimate of the actual

point on the original function

The actual

function value on the

graph is 10. The tangent

line estimates this at 9. This estimate is better the closer we are to the

point of tangency, in this case the

point where

. This estimate is helpful for more complicated functions in arriving quickly at a value which is “close” to the next value of the function.

Notice that the

point on both the graphs of

and the tangent

line at

is (1, 4). The slope, or derivative of

, of the tangent

line is 5 at that

point as we calculated above. This means the

function increased about 5 units from the

point where

to the

point where

. We use this derivative in

marginal analysis.

**Let’s Practice:**

**
**

Marginal profit is the derivative of the profit

function (the same is true for cost and revenue). We use this marginal profit

function to estimate the amount of profit from the “next” item.

For example: using our profit

function from test #1, we have

Therefore,

The marginal profit at

is

P'(300) = 80 + 0.2(300) - (40 + 0.4(300)) = -$20

Using the ideas from the example above, we know that -20 represents an estimate of the change in profit from the points on the profit

graph at

and

. An estimate of the profit from the 301

^{st} item is -$20, meaning that production of the 301

^{st} item will decrease profit by $20.

It is important to be able to compare this to the exact change in profit as we move to the 301

^{st} item. Remember that the derivative gives us a

point on the tangent line, which is not exactly on the actual

graph of the profit function.

The exact profit from the 301^{st} item is given by

P(301) - P(300) = $979.90 - $1000 = -$20.10

We can see that this estimate of -$20 from the derivative of

is very close to the exact value.

**Summary:**

An estimate of the amount of profit from the 301^{st} item (the change in profit as we go from 300 items to 301 items) is given by

P'(300) = -20

The exact change in profit is given by

P(301) - P(300) = -$20.10