AP Calculus. Extreme Values: Graphically. Slide 1 / 163 Slide 2 / 163. Slide 4 / 163. Slide 3 / 163. Slide 5 / 163. Slide 6 / 163

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1 Slide 1 / 163 Slide 2 / 163 AP Calculus Analyzing Functions Using Derivatives Slide 3 / 163 Table of Contents click on the topic to go to that section Slide 4 / 163 Extreme Values - Graphically 1 st Derivative Test Concavity & 2 nd Derivative Test Connecting Graphs of f, f', and f'' Curve Sketching Rolle's Theorem Mean Value Theorem Newton's Method Optimization Extreme Values: Graphically Return to Table of Contents Slide 5 / 163 Slide 6 / 163 Horizontal Tangents Recall from the previous unit... we analyzed graphs and discovered the locations of horizontal tangent lines. Slopes Surrounding Point a Looking specifically at point a, we know at the peak, the slope is zero. What do you notice about the slope on either side of a? Looking at locations a, b, and c, while they all share the trait that they have horizontal tangents, what is different about each point? a b c a

2 Slide 7 / 163 Slopes Surrounding Point c Similarly, we have a change in slopes at point c, however the slope is changing from negative to positive at this point. Slide 8 / 163 Slopes Surrounding Point b Now, consider point b. We know the slope is zero at b; however, the function's slope does not change signs at this point. b c Slide 9 / 163 Local (Relative) Extrema Local Maximum: a high point on any interval relative to points around it. At this point, the slope changes from positive to negative, and the function changes from increasing to decreasing. Local Minimum: a low point on any interval relative to points around it. At this point the slope changes from negative to positive, and the function changes from decreasing to increasing. NOTE: Local max/mins CANNOT occur at endpoints! Slide 10 / 163 Absolute (Global) Extrema Absolute Maximum: occurs at c if f(c)>f(x) for all x in domain Absolute Minimum: occurs at c if f(c)<f(x) for all x in domain NOTE: Absolute max/mins can occur at endpoints! Slide 11 / 163 Slide 12 / 163 Extrema Identify/Label each of the following with Local or Absolute Maximum or Minimum. What do you notice about what is occuring at the star?

3 Slide 13 / 163 Critical Value A critical value (or critical point) is a point on the interior of the domain of a function at which the slope is zero or undefined. When asked to find local extrema, only critical values must be considered. If asked to find absolute extrema, critical values as well as endpoints are considered. Slide 14 / 163 Extrema & Endpoints An extrema can only occur at critical values or endpoints (absolute); however, the presence of a critical value does not guarantee an extrema at that value. What does this mean? Slide 15 / Using the given graph, which of the following are occurring at point b? Slide 16 / Using the given graph, which of the following are occurring at point d? A B C D E F local maximum local minimum absolute maximum absolute minimum slope is zero slope is undefined A B C D E F local maximum local minimum absolute maximum absolute minimum slope is zero slope is undefined Slide 17 / 163 Slide 18 / Using the given graph, which of the following are occurring at point c? 4 Using the given graph, which of the following are occurring at point a? A B C D E F local maximum local minimum absolute maximum absolute minimum slope is zero slope is undefined A local maximum B local minimum C absolute maximum D absolute minimum E slope is zero F slope is undefined

4 Slide 19 / On which interval(s) is the function increasing? Slide 20 / Using the given graph, which of the following are critical values? A B C (a,b) (b,c) (c,d) A B C E F G D H Slide 21 / Using the given graph, which of the following are occurring at? A B C D E F local maximum local minimum absolute maximum absolute minimum slope is zero slope is undefined Slide 22 / On which interval(s) is the function decreasing? A B C D E F G Slide 23 / 163 Slide 24 / 163

5 Slide 25 / 163 Slide 26 / On which interval(s) is the function increasing? Slide 27 / 163 Slide 28 / If a function has a critical value at x=3, then there must be a local or absolute extrema at that value. True False Slide 29 / 163 Slide 30 / 163 Calculating Extrema Algebraically 1 st Derivative Test We have discovered how to graphically interpret local and absolute extrema, and now we will extend our understanding to calculate extrema algebraically. Return to Table of Contents

6 Slide 31 / 163 Recall: A local maxima/minima occurs when the slope of the function changes from positive to negative, or negative to positive at a critical value. Another way of thinking is that the original function changes from increasing to decreasing, or decreasing to increasing. For a function to contain an absolute extrema, it must be the highest or lowest extrema on the interval, including endpoints. Slide 32 / 163 Algebraically, we can find critical values and test points on either side to determine the change in slope, if there is one. This is known as the 1 st Derivative Test. Helpful Steps: The 1 st Derivative Test 1. Find the derivative of the given function. 2. Find x-values where the derivative equals zero or is undefined. These are the critical values (locations of possible extrema). 3. Test x-values on either side of the critical values, substitute into the derivative and observe sign change for maximum or minimum. 4. Substitute x-values into original function to get corresponding y-values for the extrema. Slide 33 / 163 Slide 34 / 163 Critical Values First let's practice finding critical values. Remember, a critical value is the potential location for an extreme value. Find the location of any critical value(s): Slide 35 / 163 Example For the function, identify the intervals of increasing/decreasing: Slide 36 / 163 Example, Continued For the previous function, find any local extrema.

7 Slide 37 / 163 Example For the following function, identify the intervals of increasing/decreasing. Slide 38 / 163 Example, Continued For the previous function, find any local extrema: Find any local extrema for Slide 39 / 163 Example Find any local extrema for Slide 40 / 163 Example Slide 41 / 163 Slide 42 / 163 Grading on the AP Exam Drawing a sign chart is a great way to help visualize what is happening; however, a sign chart alone is not enough of an explanation on the AP Exam. Be sure to defend your answer, describing the sign change of the derivative. Absolute Extrema Now that we feel comfortable solving for local extrema, we can go further and determine where absolute extrema occur. Remember, for absolute extrema we must compare all extrema within the given interval, as well as the endpoints of that interval. For example: The function has a local maximum at (-3,7) because f '(x) changes from positive to negative at x=-3.

8 Slide 43 / 163 Extreme Value Theorem Given that is continuous on the interval then must attain an absolute maximum and minimum on that interval. Slide 44 / 163 Extreme Value Theorem Which of the following graph(s) meet the criteria for the Extreme Value Theorem? Circle your answer, and for those graph(s) that do not, explain why. Slide 45 / 163 Slide 46 / 163 Absolute Extrema Let's take a look back at an example we previously worked with: Example: Find the absolute extrema for on We found this function had a local minimum at and a local maximum at Now, to find the absolute extrema we need to compare these with the endpoints. Slide 47 / 163 Slide 48 / 163

9 Slide 49 / 163 Slide 50 / 163 Slide 51 / 163 Slide 52 / If a function has a local minimum at then True False Slide 53 / 163 Slide 54 / 163

10 Slide 55 / 163 Concavity & 2 nd Derivative Test Slide 56 / 163 Relationship Between Derivatives & Functions As we saw in the previous section, derivatives can tell us a lot of information about the original function. Next, we will discover what information the 2 nd derivative provides about the original function. Return to Table of Contents Slide 57 / 163 Concavity As we know, in mathematics, not all functions are linear. There are infinite curves created by functions and they take on unique shapes. Let's cover some new vocabulary to describe these curves. Slide 58 / 163 Concavity Recall that the sign of the 1 st derivative told us whether the function was increasing or decreasing. The sign of the second derivative will tell us where the function is concave up or concave down. Why is this helpful? Slide 59 / 163 Some functions remain concave up or down for all x in their domain, but many others can change concavity at multiple points. For example: Concavity Slide 60 / 163 Inflection Point A point x=c is an inflection point (or point of inflection) if a function is continuous at that point and the function changes concavity at that point. Label each section with the correct concavity. Note: Each star represents an inflection point for the function above.

11 Slide 61 / 163 Concavity & Points of Inflection Use 2 colors to shade the different regions (either concave up or concave down) or shade one with pencil and leave the other unshaded. Then identify any points of inflection. Slide 62 / Which accurately describes the shape at the point? d e f g h i j m n p q r s a b c k A concave up; decreasing B concave up; increasing C concave down; decreasing D concave down; increasing E point of inflection F none of the above Slide 63 / Which accurately describes the shape at the point? Slide 64 / Which accurately describes the shape at the point? a b c d e f g h i j k m n p q r s a b c d e f g h i j k m n p q r s A concave up; decreasing B concave up; increasing C concave down; decreasing D concave down; increasing E point of inflection F none of the above A concave up; decreasing B concave up; increasing C concave down; decreasing D concave down; increasing E point of inflection F none of the above Slide 65 / 163 Slide 66 / 163

12 Slide 67 / Which accurately describes the shape at the point? Slide 68 / Which accurately describes the shape at the point? a b c d e f g h i j k m n p q r s a b c d e f g h i j k m n p q r s A concave up; decreasing B concave up; increasing C concave down; decreasing D concave down; increasing E point of inflection F none of the above A concave up; decreasing B concave up; increasing C concave down; decreasing D concave down; increasing E point of inflection F none of the above Slide 69 / Which accurately describes the shape at the point? Slide 70 / Which accurately describes the shape at the point? a b c d e f g h i j k m n p q r s a b c d e f g h i j k m n p q r s A concave up; decreasing B concave up; increasing C concave down; decreasing D concave down; increasing E point of inflection F none of the above A concave up; decreasing B concave up; increasing C concave down; decreasing D concave down; increasing E point of inflection F none of the above Slide 71 / Which accurately describes the shape at the point? Slide 72 / Which accurately describes the shape at the point? a b c d e f g h i j k m n p q r s a b c d e f g h i j k m n p q r s A concave up; decreasing B concave up; increasing C concave down; decreasing D concave down; increasing E point of inflection F none of the above A concave up; decreasing B concave up; increasing C concave down; decreasing D concave down; increasing E point of inflection F none of the above

13 Slide 73 / 163 Algebraically Determining Concavity Since we are able to graphically interpret concavity, now we will discover how to algebraically determine concavity and points of inflection. Slide 74 / 163 Critical Values The first step in finding intervals of concavity algebraically is to find critical values for the 2 nd derivative. These critical values will occur when the 2 nd derivative equals zero or is undefined. Slide 75 / 163 Sign of 2 nd Derivative The sign of the 2 nd derivative determines the concavity of the original function. If for on an interval,, then is concave up on that interval. If for on an interval,, then is concave down on that interval. If has a critical value at and changes signs at, then is a point of inflection. Slide 76 / 163 Example Find the critical values for the 2 nd derivative using the following function: Slide 77 / 163 Example, Continued Find the intervals of concavity as well as any points of inflection: Slide 78 / 163 Example Find the intervals of concavity as well as any points of inflection:

14 Slide 79 / 163 Slide 80 / 163 Slide 81 / 163 Slide 82 / 163 Slide 83 / 163 Slide 84 / 163 The 2 nd Derivative Test This test allows us to identify local extrema for functions by combining information about the 1 st and 2 nd Derivative. Given that is a critical value of, and is continuous at. Then, if then is a local maximum and if then is a local minimum.

15 Slide 85 / nd Derivative Test Note: The 2 nd Derivative Test does not replace the 1 st Derivative Test. Both are useful tools in finding extrema of functions. The 2 nd Derivative Test simply provides an alternative method for classifying the extrema. Slide 86 / 163 Example Using the 2nd Derivative Test, determine if the function below has any relative extrema. That being said, there are situations where the 2 nd Derivative Test would fail to supply any information about extrema: does not exist but does not exist and In any of these scenarios, the 1 st Derivative Test would need to be used. Slide 87 / 163 Slide 88 / The 2 nd Derivative Test is used to find absolute extrema of functions. True False Slide 89 / 163 Slide 90 / 163

16 Slide 91 / 163 Slide 92 / 163 Connecting Graphs of f, f', and f'' Return to Table of Contents Slide 93 / 163 Analyzing Graphs In mathematics, we are often accustomed to being given a picture of a graph and being asked questions about that specific function which is graphed (much like we were practicing in the first section). However, as we venture further in to Calculus, we may be given the graph of the 1st or 2nd derivative and asked questions about the original function. We must become familiar with the information that each graph provides us in order to accurately answer the questions. Slide 94 / 163 Review So, let's review what we discovered in the first few sections: is increasing when is decreasing when has a local max when has a local min when is concave up when is concave down when has a point of inflection when Slide 95 / 163 Important Advice The #1 piece of advice when proceeding with questions involving graphs from now on: Ask yourself: Which graph am I looking at? the original function? the first derivative? the second derivative? At right is the graph of the derivative of a function, f, whose domain is the set of all real numbers and is continuous everywhere. Using the graph, determine the following: Slide 96 / 163 Example 1. Intervals on which f is increasing: 2. Intervals on which f is decreasing: 3. Relative extrema: 4. Intervals where f is concave up: 5. Intervals where f is concave down: 6. Point(s) of inflection:

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18 Slide 103 / 163 Slide 104 / 163 Slide 105 / 163 Slide 106 / 163 Curve Sketching Curve Sketching Return to Table of Contents Interpreting information from the first or second derivative can allow us to come up with a fairly accurate sketch of the original function without knowing the equation itself. Think of yourself as a criminal sketch artist - although you may have never seen the actual person, based on eye witness description and detail, you can create a fairly accurate picture. Slide 107 / 163 Slide 108 / 163 Example Below is the graph of the derivative of a function f, whose domain is the set of all real numbers and which is continuous everywhere. Use the provided information to create a sketch of the original function, if the only zeros of f(x) are x=-4 and x=0.

19 Slide 109 / 163 Slide 110 / 163 Curve Sketching Practice Questions Slide 111 / Below is the graph of the derivative of a function f, whose domain is the set of all real numbers and which is continuous everywhere. Use the provided information to create a sketch of the original function, if the only zeros Students type their answers here of f(x) are x = -3.5, x = 0, x = 6, and x = 9.5. When you are finished, type "done" with your SMART Responder. Slide 112 / Below is the graph of the derivative of a function f, whose domain is the set of all real numbers and which is continuous everywhere. Use the provided information to create a sketch of the original function, if the only zeros of f(x) are x = -7.5, x = -1.5, and x = 3. Students type their answers here When you are finished, type "done" with your SMART Responder. Slide 113 / 163 Slide 114 / Below is the graph of the derivative of a function f, whose domain is the set of all real numbers and which is continuous everywhere. Use the provided information to create a sketch of the original function, if the only zeros of f(x) are x = -3, x = 1, and x = 4. When you are finished, type "done" with your SMART Students type their answers here Responder.

20 Slide 115 / 163 Slide 116 / If the graph of is below, which of the graphs at right is the graph of? A B C D Slide 117 / 163 Slide 118 / If the graph of is below, which of the graphs at right is the graph of? 59 If the graph of is below, which of the graphs at right is the graph of? A B A B C D C D Slide 119 / 163 Slide 120 / 163 Michel Rolle ( ) Rolle's Theorem Return to Table of Contents Interesting fact about French mathematician Michel Rolle: He originally criticized the subject of calculus, thinking it did not use sound reasoning and resulted in errors. Later on, he published Rolle's theorem and thus approved the usefulness of calculus. He also contributed to the work on Gaussian elimination.

21 Slide 121 / 163 Activity On the graph below, connect the dots using any function that is continuous and differentiable. Identify any points on your graph with a horizontal tangent. Is it possible to connect the dots without creating a horizontal tangent, while still maintaining continuity and differentiability? Slide 122 / 163 Rolle's Theorem Suppose is a function that satisfies all of the following: is continuous on the closed interval [a,b] is differentiable on the open interval (a,b) Then there exists a number c, such that a<c<b, and Slide 123 / 163 A Visual Understanding of Rolle's Theorem: Slide 124 / 163 Example Find the value, c, that satisfies Rolle's Theorem for the function: Slide 125 / 163 Slide 126 / 163

22 Slide 127 / 163 Slide 128 / Find the value(s), c, that satisfies Rolle's Theorem for the following function on the given interval: Slide 129 / 163 Slide 130 / Find the value(s), c, that satisfies Rolle's Theorem for the following function on the given interval: Mean Value Theorem Return to Table of Contents Slide 131 / 163 Road Trip! Consider going on a road trip with your friends. You are driving along and at noon you decide to enter a toll road. You pick up a toll entrance ticket and notice you are at mile marker 1. You also notice the posted speed of 55mph and see no police. At 2:00pm you exit the toll road at mile marker 141. You exit and hand the attendant your toll ticket. After a moment, the attendant says, "That will be $5.00 for the toll and $ for the speeding violation." Slide 132 / 163 Mean Value Theorem Suppose is a function that satisfies the following: is continuous on the closed interval [a,b] is differentiable on the open interval (a,b) Then there exists a number c, such that a<c<b, and You sit in the car a few moments thinking; you didn't notice any police cars or speed cameras. Were you speeding on the toll road? And if so, how did the attendant know?

23 Slide 133 / 163 A Visual Understanding of the Mean Value Theorem Slide 134 / 163 Discussion: How are Rolle's Theorem and Mean Value Theorem related? How are they different? Slide 135 / 163 Slide 136 / 163 Example Find the value, c, which satisfies the Mean Value Theorem for the function: Slide 137 / 163 Slide 138 / Find the value(s), c, which satisfy the Mean Value Theorem for the function:

24 Slide 139 / 163 Slide 140 / Find the value(s), c, which satisfy the Mean Value Theorem for the function: Slide 141 / Find the value(s), c, which satisfy the Mean Value Theorem for the function: Slide 142 / Find the value(s), c, which satisfy the Mean Value Theorem for the function: Slide 143 / 163 Slide 144 / 163 Newton's Method Newton's Method Return to Table of Contents Newton's Method is a tool for approximating zeros (solutions) of functions. It has very useful applications, as this is a process we complete often in mathematics. The basic idea of Newton's Method is to start with an approximation for a zero and use the tangent line to narrow in on the actual solution. This process will become more clear as we work through some examples.

25 Slide 145 / 163 A Visual Representation of Newton's Method We are wanting to find the solution to a function f(x). We start by choosing an estimate, Notice: The solution to the tangent line at produces a closer approximation to the actual solution of f(x). Slide 146 / 163 Newton's Method If we continue this process, of using the tangent line, we can gain closer and closer approximations to the actual solution. Now, we need to define a way to calculate each of these subsequent x values. Slide 147 / 163 Newton's Method If is an approximate solution of and if then the next approximation is given by: Slide 148 / Frequently Asked Questions: How do we decide on our first "guess" for the root? This can be done a variety of ways: the "guess" is provided for you in the question graphically or knowledge of the function, itself Intermediate Value Theorem How many times should we apply Newton's Method? This depends on the accuracy you seek for the solution to the original function. Typically, the question will inform you of how many digits are needed. Note: If the question asks for 6 decimal places, this does not mean that you simply get an answer with at least 6 decimal places. You must get 2 consecutive answers to agree for at least 6 decimal places. Slide 149 / 163 Slide 150 / 163 Newton's Method With a Calculator Calculators are extremely useful during this section, due to the fact that the numbers/results contain so many digits, and need to be substituted time and time again. Next, we will cover how to effectively use your calculator on Newton's Method questions.

26 Slide 151 / 163 Slide 152 / 163 Newton's Method Calculator Instructions 1. Use your initial guess and substitute value into algorithm: 2. Using the ANS button, enter the same algorithm. 3. Keep pushing ENTER until you get the desired number of digits to match/agree. Slide 153 / 163 Slide 154 / Calculator OK Use Newton's Method to approximate the root to accurate to 6 decimal places. Choose your own initial estimate. Slide 155 / 163 Slide 156 / 163 Optimization Return to Table of Contents

27 Slide 157 / 163 Optimization in the Real World Often in businesses and real world situations, there is a need to maximize profit, minimize cost, minimize travel time, etc. We call these optimization problems. Now that we have a deeper understanding of calculus, we can apply it to these types of problems using the given constraints. Slide 158 / 163 Helpful Steps in Solving Optimization Problems 1. Assign variables to unknown quantities (drawing pictures is helpful). 2. Define the constraints for your variables, if there are any. 3. Decide on the equation that needs to be maximized or minimized. Write the equation in terms of only one variable. 4. Take the derivative of the equation from step 3 and set equal to zero, calculating max/min. 5. Check your answer against the constraints to make sure it is feasible. 6. Answer question fully. Note: Make sure to test endpoints if the interval is closed. Slide 159 / 163 Example What is the minimum distance between and the point? Slide 160 / 163 Example A farmer is creating a rectangular pen for animals and has 160 feet of fence. The side of a barn will be used for one side of the fence. What length and width would produce the largest area for the pen? What would the area be? Slide 161 / 163 Calculator OK 75 A can of tomatoes is being constructed from aluminum. The volume of the can is 180 cubic inches. What should the height of the can be in order to minimize the amount of aluminum used? Slide 162 / 163 Calculator OK 76 An open top box is created by cutting out squares of side length x from each corner, and bending up the sides. The cardboard is 18 by 24 inches. What is the maximum possible volume of the box?

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