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

6/10/2013 MAT 166 – Calculus for Bus/Soc Chapter 4 Notes Techniques for Finding the Derivative Calculating the Derivative David J. Gisch The Derivative The Power Rule 1

6/10/2013 Power Rule Power Rule Example: Find �′��� when � � � � � Example: Use the power rule to calculate the derivative of the following functions, � � � 3� � � 3� � 2 � Power Rule Power Rule Example: Calculate the derivative of � � � �� � � � Example: If � � � ������� � , find �� � � � 7 �� . ��� 2

6/10/2013 The Rate of Change and Business Marginal Cost Example: Suppose that the total cost (in hundreds of • In business and economics the rate of change of such dollars) to produce � thousand barrels of a beverage is variables as cost, revenue, and profit are important. given by � � � 4� � � 100� � 500 . • Economists use the word m arginal to refer to the rates of change. a) What is the marginal cost for � � 5 ? • 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. b) Explain what this means. Marginal Cost Marginal Cost Example: Suppose that the total cost (in hundreds of Compare the marginal cost between � � 5 and � � 20 and dollars) to produce � thousand barrels of a beverage is explain what the difference means. given by � � � 4� � � 100� � 500 . a) What is the marginal cost for � � 20? b) Explain what this means. 3

6/10/2013 The Derivative The Derivative Example: Find all (exact) values of � where the tangent line • Recall that the derivative is the instantaneous rate of is horizontal if � � � � � � 5� � � 6� � 3 . change. • What does it mean when � � � � 0 at � � � ? Product Rule Derivatives of Products and Quotients 4

6/10/2013 Product Rule - Derivative Product Rule - Derivative Example: Find the derivative of � � � �� � � 7��4 � � � � . Example: Find the derivative of � � � �3� � 2� � . Product Rule - Derivative Quotient Rule Example: Find the derivative of � � � �� � 1�� � � 2� . 5

6/10/2013 Quotient Rule - Derivative Quotient Rule - Derivative Example: Find the derivative of � � � ���� Example: Find the derivative of � � � � � ����� ���� . � � �� . Quotient Rule - Derivative Derivative Example: Find the derivative of � � � ���� Example: Find the equation of the line tangent to the graph � . � of � � � ��� at the point � � 3 . 6

6/10/2013 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 Chain Rule • The same can be said for profit and revenue. Composite Functions The Chain Rule Example: � � � � � � � � 2� � 1 Think “peel the onion.” � �2� � 1� � Then � � � Example: � � � � �� � 2��� � 1� � � � � � Then � � � � ���������� 7

6/10/2013 Chain Rule - Derivative Chain Rule - Derivative Example: Find the derivative of � � � �5� � 3� � . Example: Find �� � �� if � � � ����� � . Chain Rule - Derivative Chain Rule - Derivative Example: Find the derivative of � � � �3 7� � � 1 . Example: Find the derivative of � � � ����� ����� � . 8

6/10/2013 Application: Page 226 #58 Application: Page 226 #58 Suppose the cost in dollars of manufacturing q items is Suppose the cost in dollars of manufacturing q items is given by given by � � � 200� � 3500 � � � 200� � 3500 And the demand equation is given by And the demand equation is given by � � � � � 15,000 � 1.5� � � � � � 15,000 � 1.5� a) Find an equation for the revenue. b) Find an expression for the profit. Application: Page 226 #58 Application: Page 226 #58 Suppose the cost in dollars of manufacturing q items is Suppose the cost in dollars of manufacturing q items is given by given by � � � 200� � 3500 � � � 200� � 3500 And the demand equation is given by And the demand equation is given by � � � � � 15,000 � 1.5� � � � � � 15,000 � 1.5� c) Find an expression for the marginal profit. d) Determine the value of the marginal profit when the price is $5000. 9

6/10/2013 Exponentials Derivatives of Exponential Functions Exponentials (Chain Rule) The Derivative Example: Find the derivative of � � � 4 �� . 10

6/10/2013 The Derivative The Derivative Example: Find the derivative of � � � �� � � 1� � � �� . Example: Find �� �� if � � � �� � �� . Logistic Modeling The Derivative (Logistic Models) A company sells 990 units of a new product in the first year In a logistic m odel , the population � after time � obeys and 3213 units in the fourth year. They expect that sales the equation can be approximated by a logistic function, leveling off at � � � � around 100,000 in the long run. 1 � �� ��� where � , � , and � are constants with � � 0 . The model is growth if � � 0 and decay if � � 0 . a) Find a formula ���� for the sales as a function of time. • 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. 11

6/10/2013 Logistic Model The Derivative (Logistic Models) b) Find the rate of change of sales after 4 years. 100,000 � � � 1 � 100� ��.�� 100,000 � � � 1 � 100� ��.�� Logarithms Derivatives of Logarithmic Functions 12

6/10/2013 The Derivative Assumption of the Absolute Value Example: Find the derivative of � � � log � � . �� �5� �5 � 1 1 �� ln�5�� � 1 �� 5� 5 � 1 �� ln �5� � � � The Derivative The Derivative �� if � � ln�2� � � 3� . Example: Find the derivative of � � � log � �5� � 3� � � . Example: Find �� 13

6/10/2013 The Derivative Based on projections from Kelly Blue Book, the resale value of 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� . 14