MAT 166 Calculus for Bus/Soc Chapter 4 Notes Techniques for - - PowerPoint PPT Presentation

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MAT 166 – Calculus for Bus/Soc Chapter 4 Notes

Calculating the Derivative David J. Gisch

Techniques for Finding the Derivative

The Derivative The Power Rule

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Power Rule

Example: Use the power rule to calculate the derivative of the following functions, 3 3 2

Power Rule

Example: Find ′ when

• Power Rule

Example: If

• , find

.

Power Rule

Example: Calculate the derivative of

7

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The Rate of Change and Business

• In business and economics the rate of change of such

variables as cost, revenue, and profit are important.

• Economists use the word m arginal to refer to the rates
• f change.
• For example marginal cost refers to the change of cost.

▫ Or we can think of it as the cost to produce the next increment of production at any given production level.

Marginal Cost

Example: Suppose that the total cost (in hundreds of dollars) to produce thousand barrels of a beverage is given by 4 100 500. a) What is the marginal cost for 5? b) Explain what this means.

Marginal Cost

Example: Suppose that the total cost (in hundreds of dollars) to produce thousand barrels of a beverage is given by 4 100 500. a) What is the marginal cost for 20? b) Explain what this means.

Marginal Cost

Compare the marginal cost between 5 and 20 and explain what the difference means.

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The Derivative

• Recall that the derivative is the instantaneous rate of

change.

• What does it mean when 0 at ?

The Derivative

Example: Find all (exact) values of where the tangent line is horizontal if 5 6 3.

Derivatives of Products and Quotients

Product Rule

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Product Rule - Derivative

Example: Find the derivative of 74 .

Product Rule - Derivative

Example: Find the derivative of 3 2.

Product Rule - Derivative

Example: Find the derivative of 1 2.

Quotient Rule

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Quotient Rule - Derivative

Example: Find the derivative of

.

Quotient Rule - Derivative

Example: Find the derivative of

.

Quotient Rule - Derivative

Example: Find the derivative of

.

Derivative

Example: Find the equation of the line tangent to the graph

• f
• at the point 3.

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Marginal Average Cost

• The average cost per item can be found by taking the

total cost divided by the number of items. The rate of change of tis function is called the m arginal average cost.

• The same can be said for profit and revenue.

The Chain Rule

Composite Functions

Example: 2 1 Then 2 1 Example: 2 1

• Then
• The Chain Rule

Think “peel the onion.”

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Chain Rule - Derivative

Example: Find the derivative of 5 3.

Chain Rule - Derivative

Example: Find

if

• .

Chain Rule - Derivative

Example: Find the derivative of

.

Chain Rule - Derivative

Example: Find the derivative of 3 7 1.

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Application: Page 226 #58

Suppose the cost in dollars of manufacturing q items is given by 200 3500 And the demand equation is given by 15,000 1.5 a) Find an equation for the revenue.

Application: Page 226 #58

Suppose the cost in dollars of manufacturing q items is given by 200 3500 And the demand equation is given by 15,000 1.5 b) Find an expression for the profit.

Application: Page 226 #58

Suppose the cost in dollars of manufacturing q items is given by 200 3500 And the demand equation is given by 15,000 1.5 c) Find an expression for the marginal profit.

Application: Page 226 #58

Suppose the cost in dollars of manufacturing q items is given by 200 3500 And the demand equation is given by 15,000 1.5 d) Determine the value of the marginal profit when the price is $5000.

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Derivatives of Exponential Functions

Exponentials Exponentials (Chain Rule) The Derivative

Example: Find the derivative of 4.

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The Derivative

Example: Find

if .

The Derivative

Example: Find the derivative of 1.

Logistic Modeling

In a logistic m odel, the population after time obeys the equation

• 1

where , , and are constants with 0. The model is growth if 0 and decay if 0.

• The domain is all real numbers, and the range is 0, .
• There are horizontal asymptotes of 0 and ;

hence the range.

• We call the carrying capacity as it is the maximum

that a population can grow due to restraints.

The Derivative (Logistic Models)

A company sells 990 units of a new product in the first year and 3213 units in the fourth year. They expect that sales can be approximated by a logistic function, leveling off at around 100,000 in the long run. a) Find a formula for the sales as a function of time.

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Logistic Model

100,000 1 100.

The Derivative (Logistic Models)

b) Find the rate of change of sales after 4 years.

100,000 1 100.

Derivatives of Logarithmic Functions

Logarithms

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The Derivative

Example: Find the derivative of log .

Assumption of the Absolute Value

• ln 5

1 5 5 1

• ln5 1

5 5 1

• The Derivative

Example: Find

if ln2 3.

The Derivative

Example: Find the derivative of log5 3.

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The Derivative

Based on projections from Kelly Blue Book, the resale value

• f a 2010 Toyota Corolla 4-door sedan can be

approximated by the following function 15,450 13,915 log 1 where is the number years since 2010. Find and interpret 4 and ’4.