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Derivative ApplicationsMAC 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 ______

______________________.

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• 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

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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

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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 _____ _______________________________.

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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:

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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

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To find the relative extrema:1. Find the critical values of f2. Determine the sign of f ’ on each side of each

critical valuea. If f ’ changes from _____________, then f(c)

is a relative maximumb. If f ’ changes from _____________, then f(c)

is a relative minimumc. 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 _______________.

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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 prepresents 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:

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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

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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:

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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.

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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.

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Solution

12

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Example

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

ordering 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 = (_____________)*(________ ____________)

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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

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If f is differentiable on (a, b) then

1. f is concave up on (a, b) if _____________on

(a, b).

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

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• 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 of 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

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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.

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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)