MAT 166 Calculus for Bus/Soc Chapter 3 Notes Limits The - PowerPoint PPT Presentation

5/29/2013 MAT 166 – Calculus for Bus/Soc Chapter 3 Notes Limits The Deriviative David J. Gisch Limits Limits 1

5/29/2013 Limits Limits • Why? Don’t they always agree? � � � � � � 2� � � � 2 Limits Limits Example: What is the limit of � � � �� � � � 2 1 � � 2 as � → 2 ? 2

5/29/2013 Limits Limits Example: What is the limit, if it exists of � lim � �→� Rules for Limits Rules for Limits 3

5/29/2013 Algebra of Limits Algebra of Limits Example: Suppose that Example: Find the limit, if it exists, for � � � � � 12 �→� � � � 3 ��� lim lim �→� � � � 4 lim � � 3 �→� � � � ���� � ? �→�� What is lim Algebra of Limits Algebra of Limits Example: Find the limit, if it exists, for Example: Find the limit, if it exists, for � � � � � 1 � � 2 lim lim � � 4 �→� � � 1 �→� 4

5/29/2013 Limits at Infinity Limits at Infinity (Graphically) 12� � � 15� � 12 • Sometimes it is very useful to look at the “end” behavior lim � � � 1 of a graph. We do this by contemplating the limit at �→� infinity or negative infinity. Example: Suppose a small pond normally contains 12 units of dissolved oxygen in a fixed volume of water. Suppose also that at time t=0 a quantity of organic waste is introduced into the pond, with the oxygen concentration t weeks later given by � � � 12� � � 15� � 12 � � � 1 As time goes on, what will be the ultimate concentration of oxygen? Algebra of Limits Example: Find the limit, if it exists, for 2� � � � � 4 lim 6� � � 5� � 7 �→�� Solution: Here, the highest power of � (in the denominator) is � � , which is used to divide each term in the numerator and denominator. 5

5/29/2013 Continuity Continuity Continuity Continuity Example: Examine the graphs and decide if they are Example: Examine the graphs and decide if they are continuous at the indicated point. continuous at the indicated point. At � � 3 At � � 0 At � � 4 6

5/29/2013 Continuity Continuity Example: Examine the graphs and decide if they are Example: Examine the graphs and decide if they are continuous at the indicated point. continuous at the indicated point. At � � 1 At � � �2 Continuity on Closed Intervals Continuity of Functions 7

5/29/2013 Continuity of Functions S hifts in Relation to Continuity • Understand that the last charts refer to “parent” graphs. • For example • But what if I have the function � � � log � �� � 5� . Continuity Continuity Example: Find all values � � � where the function is Example: Find all values � � � where the function is discontinuous. discontinuous. � � � 4� � 3 � � � � ���� 2� � 7 8

5/29/2013 Continuity Continuity Example: Find all values � � � where the function is Example: Find all values � � � where the function is discontinuous. discontinuous. �2� � � 1 �� � � 1 � � � � � � 3� � 4 �7� � 7��2 � 8�� � � � � �� 1 � � � 3 5 � � � � 3 � � 1, � � a) � ������� b) � � 0, 1, � c) � d) � � 1, � � Continuity (Poultry Farming, #41) Continuity (Poultry Farming, #41) 48 � 3.64� � 0.6363� � � 0.00963� � , � � � � 1 � � � 28 Researchers at Iowa State University and the University o �1004 � 65.8�, 28 � � � 56 Arkansas have developed a piecewise function that can be used to estimate the body weight (in grams) of a male Is ���� a continuous function? broiler during the first 56 days of life according to 48 � 3.64� � 0.6363� � , � � � � 1 � � � 28 �1004 � 65.8�, 28 � � � 56 Where t is the age of the chicken (in days). What is the weight of a male broiler that is 25 days old? Use a graphing calculator to graph ���� on 1, 56 by 0, 3000 . 9

5/29/2013 Continuity (Poultry Farming, #41) 48 � 3.64� � 0.6363� � � 0.00963� � , � � � � 1 � � � 28 �1004 � 65.8�, 28 � � � 56 Why would researchers use two different types of functions to estimate the weight of a chicken at various ages? Rates of Change Average Rate of Change Rate of Change Example: Find the average rate of change for the function on the indicated intervals. � � � 2� � � 6� � 4 a) 0, 6 • Average rate of change looks a lit like slope and it is in some sense. If a function is linear they are the same thing. If a function is not linear then they are two different things. b) 4, 10 10

5/29/2013 Average Rate of Change Average Rate of Change Average Rate of Change Average Rate of Change • Understand that Average Rate of Change is exactly what it says, an average. • To see what we mean lets look at the next example. 11

5/29/2013 Average Rate of Change Instantaneous Rate of Change Example: Find the average rate of change for each function on the 0, 10 . � � � 1 � � � 5 5 � � � 2� � 5 Recall that � � � � � � � � ���� . Instantaneous ROC Instant ROC Example: Find the instantaneous rate of change for the • What is going on visually? function � � � 2� � � 6� � 4 at the point � � 4. 12

5/29/2013 Practical Use Book Example: P . 10 #37 • We can often create functions describing the position of an object. • How does you position change? ▫ Changes based on velocity (speed). Tangent Line Definition of Derivative 13

5/29/2013 Tangent Line (Instant ROC) Tangent Line Tangent Line Tangent Line Example: Consider the graph of the function � � � � � � 2. Example: Consider the graph of the function � � � � � � 2 . a) Find the slope and equation of the secant line through b) Find the slope and equation of the tangent line at the points where � � �1 and � � 2 . � � �1 . 14

5/29/2013 Graph of Example Genetics Example The graph below shows the risk of chromosomal abnormalities in a child increases with the age of the mother at the child's birth. Find the rate that the risk is rising when the mother is 40 years old. The Derivative The Derivative Example: Consider the function � � � � � . a) Find the derivative of ���� (i.e. find �′���� . b) Find �′�5� . 15

5/29/2013 The Derivative Example: Consider the function � � � 2� � � 4� � . a) Find the derivative of ���� (i.e. find �′���� . b) Find �′��3� . Existence Existence 16

5/29/2013 Existence Graphical Differentiation Graphical Differentiation Graphical Differentiation • It is helpful to be able to look at a graph and be able estimate what the graph of its derivative looks like. • For each of the following graphs sketch the graph of its derivative. 17

5/29/2013 18 Graphical Differentiation