Warmup 11/30/16 Objective Tonight’s Homework If you aren’t Christian: Do you think there are any wise words or advice in the Bible? If you are Christian: Do you think some of the philosophers and founders of other religions have some wise things to say? Objective Tonight’s Homework To learn how to identify inflection points pp 263: 2, 3, 4, 5, 6, 7

Homework Help Let’s spend the first 10 minutes of class going over any problems with which you need help. 2

Notes on Concavity and Inflection Points Let’s look at the graph below:

Notes on Concavity and Inflection Points Let’s look at the graph below: There are a few things we can say about this function. - In these ranges, the graph opens upwards. We call this upward concavity. 4

Notes on Concavity and Inflection Points Let’s look at the graph below: There are a few things we can say about this function. - In these ranges, the graph opens upwards. We call this upward concavity. - In these ranges, the graph opens downward. We call this downward concavity. 5

Notes on Concavity and Inflection Points Let’s look at the graph below: There are a few things we can say about this function. - In these ranges, the graph opens upwards. We call this upward concavity. - In these ranges, the graph opens downward. We call this downward concavity. - Here, the graph changes between upward and downward. These are inflection points. 6

Notes on Concavity and Inflection Points With visual inspection, we can see this kind of thing easily. But we want to model it mathematically. 7

Notes on Concavity and Inflection Points With visual inspection, we can see this kind of thing easily. But we want to model it mathematically. How can we do this? By looking at how slope changes. If we look at the graph to the right, we can see that slope here goes from being negative to positive, always getting bigger. 8

Notes on Concavity and Inflection Points With visual inspection, we can see this kind of thing easily. But we want to model it mathematically. How can we do this? By looking at how slope changes. If we look at the graph to the right, we can see that slope here goes from being negative to positive, always getting bigger. We said this is change in slope and that it’s getting bigger. This means our second derivative is greater than zero. 9

Notes on Concavity and Inflection Points So there’s our mathematical relationship: We can determine the concavity of a function by taking the second derivative. 10

Notes on Concavity and Inflection Points So there’s our mathematical relationship: We can determine the concavity of a function by taking the second derivative. If f’’(x) > 0 concave upward If f’’(x) < 0 concave downward If f’’(x) = 0 inflection point 11

Notes on Concavity and Inflection Points We can now run two tests for any function. First Derivative Test If f’(c) = 0… We can look at the points around f’(c) to determine whether the region is a local maximum or local minimum - If it is positive left and negative right, it is a local maximum. - If it is negative left and positive right, it is a local minimum. - If it is the same sign on both sides, it is an inflection point. 12

Notes on Concavity and Inflection Points Second Derivative Test If f’(c) = 0… We can look at the second derivative. - If it is negative at “c”, then f(c) is a local maximum. - If it is positive at “c”, then f(c) is a local minimum. - If it is zero at “c”, then f(c) is an inflection point. 13

Group Practice Look at the example problems on pages 258 through 262. Make sure the examples make sense. Work through them with a friend. Then look at the homework tonight and see if there are any problems you think will be hard. Now is the time to ask a friend or the teacher for help! pp 263: 2, 3, 4, 5, 6, 7 14

Exit Question Which of the below would define "inflection point"? a) The top of a downward concavity point b) The bottom of an upward concavity point c) The point where a graph changes concavity d) The point where a graph equals zero e) The point where the derivative equals zero f) None of the above