Vergara Math 71 Lecture Notes
Date: ________________
Section 7.5 – Average Rate of Change: Velocity and Marginals


The average rate of change of a function is the slope of a line created by joining two points on a
], the average rate of change of over [
] is:
( ) over the interval [
function. For
( )
( )
The instantaneous rate of change of a function is defined as the derivative of the function; the
instantaneous rate of change tells you how fast the function is change at any single value of .
The rate of change of position (distance) with respect to time is called velocity. The average
velocity is defined as

position function. If the position function is ( ), then the instantaneous velocity is
( )
( ).
Profit ( ), cost ( ) and revenue ( ) are related by the formula
. We can find the
marginal profit

. The instantaneous velocity is the derivative of the
or
( ), the marginal cost
or
( ), and the marginal revenue
or
( ) by taking the derivatives of the profit, revenue and cost functions. Here is the number
of units produced.
We can find a formula for the revenue in many situations. We let be the price per unit
produced, where is again the production level (the number of units produced). is
sometimes referred to as the “demand.” The value depends on the value of just as , ,
( ). A formula for the revenue
and also depend on . We call the demand function
can be found if we know the price per unit; this formula is
.
Average & Instantaneous Rate of Change Example
over the interval [
Find the average rate of change of ( )
rate of change of
] and find the instantaneous
at the endpoints of the interval.
1. Average rate of change of
over [
2. Instantaneous rate of change of
]:
at
and at
:
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Velocity Example 1
A falling object is dropped from a height of 100 feet. The height (in feet) of the object at time (in
] as
seconds) is given by ( )
. Find the average velocity over (a) [ ] and (b) [
well as the (c) instantaneous velocity at time
.
(a) [
(b) [
]
]
(c) Instantaneous velocity at
Velocity Example 2
At time
, a diver jumps from a diving board that is
second, and his position at time is ( )
feet high. His initial velocity is
.
feet per
(a) When does the diver hit the water?
(b) What is the diver’s velocity at impact?
(c) At what time is the diver’s velocity equal to zero?
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Marginal Profit Example
The profit derived from selling
units of an alarm clock is given by ( )
.
(a) Find the marginal profit for a production level of 50 units.
(b) Compare (a) with the actual gain in profit obtained by increasing the production level from 50 to
51 units.
Demand and Revenue Example
A business sells 2,000 items per month at a price of $10 each. It is estimated that monthly sales will
increase 250 units for each $0.25 reduction in price. Use this information to find the demand function
and the total revenue function.
Demand Function:
Revenue Function:
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Marginal Revenue Example
A fast-food restaurant has determined that the monthly demand for its hamburgers is given by
Find the increase in revenue per hamburger for monthly sales of 20,000 hamburgers. In other words,
find the marginal revenue when
.
1. Find .
 remember that this notation is equivalent to the notation
2. Find
3. Find
(
( )
).
Marginal Profit Example
Suppose that the cost of producing hamburgers in the previous example is ( )
,
where at most 50,000 hamburgers can be produced (that is,
). Find the profit and the
marginal profit for each production level: (a)
, (b)
, (c)
.
1. Recall that ( )
2. Use the formula
3. Find
to find a formula for the profit .
 remember that this notation is equivalent to the notation
( )
4. Find the values for (a), (b) and (c):
Production Value
(a)
(b)
(c)
Profit ( )
Marginal Profit
( )
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