ch 3 applications of differentiation notebook.notebook January 17, 2018 Extrema on an Interval

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1 Extrema on an Interval Extrema, or extreme values, are the minimum and maximum of a function. They are also called absolute minimum and absolute maximum (or global max and global min). Extrema that occur at the endpoints are endpoint extrema. 1

2 Extrema on an Interval Extreme Value Theorem 2

3 Extrema on an Interval Example 1: Find the derivative at each relative extrema shown. a) b) c) 3

4 Extrema on an Interval Definition of Critical Number: 4

5 Extrema on an Interval 5

6 Extrema on an Interval Example 2: Find the extrema of interval [ 1, 2] on the 6

7 Extrema on an Interval Example 3: Find the extrema of interval [ 1, 3] on the 7

8 Extrema on an Interval Example 4: Find the extrema of interval on the 8

9 Extrema on an Interval 9

10 3.2 Rolle's Theorem & MVT Wednesday December 5th 10

11 3.2 Rolle's Theorem & MVT Wednesday December 5th 11

12 3.2 Rolle's Theorem & MVT Rolle's Theorem 12

13 3.2 Rolle's Theorem & MVT Example 1: 13

14 3.2 Rolle's Theorem & MVT Example 2: 14

15 3.2 Rolle's Theorem & MVT Mean Value Theorem 15

16 3.2 Rolle's Theorem & MVT Determine the value of c that satisfies the Mean Value Theorem 16

17 3.3 Increasing, Decreasing, First Derivative Test 17

18 3.3 Increasing, Decreasing, First Derivative Test Test for Increasing/Decreasing Functions 18

19 3.3 Increasing, Decreasing, First Derivative Test Example 1: 19

20 3.3 Increasing, Decreasing, First Derivative Test 20

21 3.3 Increasing, Decreasing, First Derivative Test 21

22 3.3 Increasing, Decreasing, First Derivative Test Example 2: 22

23 3.3 Increasing, Decreasing, First Derivative Test Example 3: 23

24 3.4 Concavity & Second Derivative Test Tuesday Dec. 12th 24

25 3.4 Concavity & Second Derivative Test 25

26 3.4 Concavity & Second Derivative Test Test for Concavity 26

27 3.4 Concavity & Second Derivative Test Example 1: 27

28 3.4 Concavity & Second Derivative Test Example 2: 28

29 3.4 Concavity & Second Derivative Test Points of Inflection: 29

30 3.4 Concavity & Second Derivative Test Example 3: 30

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32 3.4 Concavity & Second Derivative Test Find all points of inflection and determine the intervals on which the graph is concave up or concave down. 32

33 3.5 Limits at Infinity 33

34 3.5 Limits at Infinity 34

35 3.5 Limits at Infinity Limits at Infinity 35

36 3.5 Limits at Infinity Example 1: 36

37 3.5 Limits at Infinity Example 1: 37

38 3.5 Limits at Infinity Example 2: 38

39 3.5 Limits at Infinity Example 3: 39

40 3.5 Limits at Infinity 40

41 3.5 Limits at Infinity Example 4: 41

42 3.5 Limits at Infinity Example 5: 42

43 3.5 Limits at Infinity Example 6: 43

44 3.5 Limits at Infinity 44

45 3.5 Limits at Infinity Example 7: 45

46 3.5 Limits at Infinity Example 8: 46

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50 3.6 Curve Sketching So far, we know how to analyze the following things about a graph... 50

51 3.6 Curve Sketching Guidelines for Analyzing the Graph of a Function 51

52 3.6 Curve Sketching Example 1: 52

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54 3.6 Curve Sketching 54

55 3.6 Curve Sketching 55

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60 1.Find any critical numbers of the function and determine the intervals where the function is increasing or decreasing. Find any relative extrema. 2. Find all points of inflection and discuss concavity for the following function. 60

61 3. Find each limit, if possible. 4. Sketch the graph of the equation using extrema, intercepts, asymptotes... 61