Derivative Applications MAC 2233 Instantaneous Rates of Change of a - - PDF document

3/16/2010 1

Derivative Applications

MAC 2233

Instantaneous Rates of Change of a Function

• The derivative is:

▫ The slope of the tangent line at a point ▫ The instantaneous rate of change of the function

Marginal Analysis

• The study of the amount of change that results in

a function (cost, revenue, profit) from ______ ______________________.

3/16/2010 2

• The marginal cost function, C ’(x), is the

approximate cost of ___________________ _________.

• The marginal revenue function, R ’(x), is the

approximate gain or loss in revenue by ______________________________.

• The marginal profit function, P ’(x), is the

approximate gain or loss in profit by _______ _______________________.

Marginal Functions

Find the marginal cost, marginal revenue, and marginal profit if and

Example

• The marginal cost is
• The marginal revenue is
• The marginal profit is

Solution

3/16/2010 3 Example

The price-demand function for a collectable doll is found to be where x represents the number of collectable dolls produced in hundreds and p is the price of the dolls in dollars. a) Determine the revenue function. b) Determine the marginal revenue function. c) Evaluate and interpret R’(15).

From Brief Calculus,2nd ed. By Armstrong & Davis, 2003, problem 78, p.272.

a) Determine the revenue function

• ______________!
• For the next part, when we differentiate, we will

need a ______________!

b) Determine the marginal revenue function

3/16/2010 4 c) Evaluate and interpret R’(15)

• Plug 15 into the derivative!
• Producing the ____________________ will

_______ revenue by approximately $_____.

Homework

• p. 163 problems 1, 3, 5, 11, 13
• p. 337 problem 77

Increasing/Decreasing

• A function is increasing on an interval if _____

_______________________________.

• A function is decreasing on an interval if _____

_______________________________.

3/16/2010 5 Graphically

The critical values of a function f are the values that make the derivative ______________. Note: These are the only places where ________ ________________________!

Critical Values Example

• Determine where the function is increasing and

decreasing:

3/16/2010 6 Solution

• Take ________ and find the ___________!
• __________:

Solution continued

• Plot the _________ on a number line and test

the signs around each.

• The function is increasing on
• The function is decreasing on

Relative Maximum—f has a relative (local) maximum at x = c if there exists an open interval (a, b) containing c such that f (x) ≤ f (c) for all x in (a, b). Relative Minimum—f has a relative (local) minimum at x = c if there exists an open interval (a, b) containing c such that f (x) ≥ f (c) for all x in (a, b).

Relative Extrema

3/16/2010 7

To find the relative extrema:

• 1. Find the critical values of f
• 2. Determine the sign of f ’ on each side of each

critical value

• a. If f ’ changes from _____________, then f(c)

is a relative maximum

• b. If f ’ changes from _____________, then f(c)

is a relative minimum c. If f ’ ________________, then f(c) is not a relative extremum

First Derivative Test Example

• Find the relative extrema of
• The derivative is
• The critical values are

Solution continued

• Plot the critical values on a number line and test

the signs around each.

• Because the sign changed from + to — across __,

there is _______________.

• Because the sign changed from — to + across __,

there is _______________.

3/16/2010 8 Solution continued

• To find what the max and min are, plug into ___

___________________!

• The maximum is
• The minimum is

Example

Linguini’s Pizza Palace is starting an all-you-can- eat pizza buffet from 5:00 to 9:00 p.m. on Friday

• evenings. A survey of local residents produced the

price-demand function where x represents the quantity demanded and p represents the price in dollars. Determine where the revenue is increasing and decreasing. Find and interpret the relative extremum.

From Brief Calculus,2nd ed. By Armstrong & Davis, 2003, problem 58, p.335.

Solution

• Find the revenue function:
• Take the derivative
• Find the critical values:

3/16/2010 9 Solution continued

• Plot the critical value on a number line and test

the signs:

• The revenue is _______ until we sell __ buffet
• tickets. After that, the revenue is _________.
• The pizza place will have a _______ revenue of

$______ when ____ buffets are sold.

Homework

• p. 202 problems 1-8, 11, 13, 15, 17, 23-29 odd, 45,

47, 53, 55, 59, 61, 63, 69, 71

• p. 238 problem 49
• P. 254 problem 19
• p. 337 problems 69, 71
• p. 352 problem 39

Absolute Maximum—f has an absolute (global) maximum at x = c if f (x) ≤ f (c) for all x in the domain of f. Absolute Minimum—f has a absolute (global) minimum at x = c if f (x) ≥ f (c) for all x in the domain of f.

Absolute Extrema of a function

3/16/2010 10

If a function f is continuous on [a, b], then f has an absolute maximum value and an absolute minimum value of f on [a, b].

Extreme Value Theorem

• 1. Verify _____________on [a, b].
• 2. Determine _______________ of f in (a, b).
• 3. Evaluate f (x) at ____________ and find

_______________.

• 4. The biggest number in step 3 is the absolute

maximum & the smallest is the absolute minimum.

To find Absolute Extrema on a Closed Interval

Example

• Find the absolute extrema of the function on the

interval:

3/16/2010 11 Homework

• p. 254 problems 1-15 odd
• p. 336 problems 39-45 odd

Optimization

• We can use calculus to find the maximum or

minimum of a quantity!

• Procedure:

▫ Form an equation to describe the situation (a picture often helps!) ▫ Take the derivative ▫ Find the critical values ▫ Locate the maximum or minimum by

 The first derivative test or  The Extreme Value Theorem

For a rectangle with area 100 ft2 to have the smallest perimeter, what dimensions should it have?

Example

From Applied Calculus,4th ed. By Waner & Costenoble, 2007, problem 10, p.370.

3/16/2010 12

My orchid garden abuts my house so that the house itself forms the northern boundary. The fencing for the southern boundary costs $4 per foot, and the fencing for the east and west sides costs $2 per foot. If I have a budget of $80 for the project, what is the largest area I can enclose?

Example

From Applied Calculus,4th ed. By Waner & Costenoble, 2007, problem 18, p.371.

Solution

• Let x be the length of

the southern fence. ▫ The fence here costs $4 per foot!

• Let y be the length of

the eastern & western fences. ▫ The fence here costs $2 per foot!

x y

Vanilla Box Company is going to make open- topped boxes out of 12” x 12” rectangles of cardboard by cutting squares out of the corners and folding up the sides. What is the largest volume box it can make this way?

Example

From Applied Calculus,4th ed. By Waner & Costenoble, 2007, problem 32, p.372.

3/16/2010 13 Solution

12 12

Example

A manufacturer of medical monitoring devices uses 36,000 cases of components per year. The

• rdering cost is $54 per shipment, and the annual

cost of storage is $1.20 per case. The components are used at a constant rate throughout the year, and each shipment arrives just as the preceding shipment is being used up. How many cases should be ordered in each shipment in order to minimize total cost?

From Calculus for Business, Economics, and the Social and Life Sciences,10th ed. by Hoffman & Bradley, 2007, problem 32, p.273.

Solution

• Let x = ___________________
• Total Cost = ________+ ___________
• Storage Cost:

▫ If we reorder x every time we hit 0, __________ ______________________. ▫ Storage Cost = (_____________)*(________ ____________)

3/16/2010 14 Solution

• Let x = ____________________
• Total Cost = ________+ _________
• Reorder Cost:

▫ Reorder Cost = (____________)*(________ _________)

Solution

• Let x = ___________________
• Total Cost = _________+ ____________
• So, the total cost is:

Homework

• p. 255 problems 31, 33
• p. 270 problems 5, 7, 9, 11, 17, 21, 23, 27, 31, 33,

39

3/16/2010 15

If f is differentiable on (a, b) then

• 1. f is concave up on (a, b) if _____________
• n

(a, b).

• 2. f is concave down on (a, b) if ___________
• n (a, b).

Concavity

1. If __________ for each x in (a, b) then f is concave up on (a, b).

• 2. If __________ for each x in (a, b) then f is

concave down on (a, b).

• Procedure to find intervals of concavity:

▫ Find all values for which _________________ ____________________ ▫ Determine the sign of _____on the interval between each value in (1) by testing points.

• The point where concavity changes is called an

inflection point.

Theorem

Determine where the function is concave up and where it is concave down:

Examples

3/16/2010 16

• Where the rate of change of sales starts to

decrease

• This corresponds to _____________!

Point of Diminishing Returns Example

The Sucre Cola Company estimates that total sales

• f its new cola, TS(x), when spending x million

dollars on advertising, can be modeled by Locate the point of diminishing returns for TS(x) and interpret its meaning.

From Brief Calculus,2nd ed. by Armstrong & Davis, 2003, problem 60, p.350.

Homework

• p. 220 problems 1-4, 5, 7, 11, 39, 41, 45, 47, 53,

55, 61

• p. 237 problems 39, 41

3/16/2010 17

Elasticity of demand measures the impact that a change in price has on the demand for a product. Elastic—a small change in price results in a _______________________. Inelastic—a small change in price ___________ _______________________.

Definitions

• Elastic if E > 1

▫ Changing the price results in ___________________ ▫ _________the price to increase revenue

• Inelastic if E < 1

▫ Changing the price results in ___________________ ▫ _________the price to increase revenue

• Unit elasticity if E = 1

▫ Changing the price results in ___________________ ▫ Revenue is ______________!

Price Elasticity of Demand Example

The PackIt Company determines that the demand function for their lightweight daypack is given by where p is the unit price (dollars) of a daypack and q is the demand.

• a. Determine E(10) and state whether the

manufacturer should lower or raise the price to increase revenue.

• b. Set E(p)=1 and solve for p to determine at what

price the revenue is the greatest.

3/16/2010 18

• a. Determine E(10) and state whether

the manufacturer should lower or raise the price to increase revenue.

• The demand is

________. The price should be _____ to increase revenue.

• b. Set E(p)=1 and solve for p to

determine at what price the revenue is the greatest.

• The revenue will

be _______ when the price is set at $______ per unit.

Homework

• p. 254 problems 23, 25, 27, 39
• p. 337 problems 65 (a and c), 67 (a and c)