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
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
. 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
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
to the point
. We use this derivative in marginal analysis
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
The marginal profit at
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
. An estimate of the profit from the 301st
item is -$20, meaning that production of the 301st
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 301st
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 301st 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 i
s very close to the exact value.
An estimate of the amount of profit from the 301st 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