Section 7.5 Average Rate of Change: Velocity and Marginals

Vergara Math 71 Lecture Notes

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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

function. For

( ) over the interval [ ], the average rate of change of over [ ] is:

() ()

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

. The instantaneous velocity is the derivative of the

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 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 , ,

and also depend on . We call the demand function

( ). A formula for the revenue

can be found if we know the price per unit; this formula is

.

Average & Instantaneous Rate of Change Example

Find the average rate of change of ( )

over the interval [

rate of change of at the endpoints of the interval.

] and find the instantaneous

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

seconds) is given by ( )

. Find the average velocity over (a) [ ] and (b) [ ] as

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 feet high. His initial velocity is feet per

second, and his position at time is ( )

.

(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 .

2. Find remember that this notation is equivalent to the notation ( )

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

to find a formula for the profit .

3. Find 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 ( ) Page 4 of 4

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