Day 4 ~ Increasing/Decreasing, and Extrema. A Graphical Approach

Transcription

1 Day 4 ~ Increasing/Decreasing, and Extrema A Graphical Approach

2 Warm Up ~ Day 4 1) Find the a) domain b) x & y intercepts c) range d) discontinuities e) end behavior using limit notation ) g( x) 3x 7x 6 Find the a) domain b) x & y intercepts c) range

3 Warm Up ~ Day 4 ANSWERS 1) Find the a) domain D : (,3) (3,4) (4, ) b) x & y intercepts x int : none y int : (0, 1 ) 3 c) range R : (, 0) (0, 1) (1, ) d) discontinuities Hole (Removable Disc.) at (4,1) V.A. (NonRemovable Disc.) at x=3 e) end behavior using limit notation lim f ( x) 0 lim f ( x) 0 x x

4 Warm Up ~ Day 4 ANSWERS ) g( x) 3x 7x 6 Find the a) domain b) x & y intercepts c) range Domain : (, 3] [, ) 3 Range x int : ( 3,0) and (,0) 3 y int : none : [0, )

5 Warm Up ~ Day 4 1) Find the a) domain b) x & y intercepts c) range d) discontinuities e) end behavior using limit notation

6 Warm Up ~ Day 4 ) g( x) 3x 7x 6 Find the a) domain b) x & y intercepts c) range

7 Let s look at the Homework (Quiz 1 Review Sheet & Rational Summary and Practice Sheet) Quiz moved to Friday! 7

8 Tonight s Homework 1) Extra Practice Unit 4 ICM handout ALL (just the side with graphs) ) More Rational Functions Practice handout - Odds Quiz moved to Friday! 8

9 Part 1 Increasing, Decreasing, Max and Min 9

10 Local/Relative Extrema Local Maximum the maximum y-value of a function on some small interval Local Minimum the minimum y-value of a function on some small interval Local Extrema are also called relative extrema Local Maxima Local Minimum

11 Absolute Extrema Absolute extrema are the maximum and minimum of all range values. Report all extrema (local or absolute) as: *Max/Min y-value occurs at x-value* Absolute Maximum (4, 3) Occurs when y-values change from increasing to decreasing Absolute Max Example: Maximum 3 occurs at x = 4

12 Local & Absolute Extrema What are the local extrema of: Remember, report all extrema (local or absolute) as: Max/Min is #, occurs at # Max/Min is y-value, Occurs at x-value Remember, if given a function, you can use your calculator to find max s and min s! Max: 4.08 occurs at x = Min: -.08 occurs at x = 1.15

13 Increasing and Decreasing Increasing Decreasing

14 Constant and Mixed Intervals Constant Mixed

15 Increase, Decrease, and Constant Always use x-values to report increasing/decreasing/constant intervals! Mixed (0, 3) Increasing? Decreasing? (-5, -)

16 Increase and Decrease ANSWERS Always use x-values to report increasing/decreasing intervals! Mixed (0, 3) (-5, -) Increasing? [ 5,0] Decreasing? (, 5] [0, )

17 Increasing/Decreasing (Algebraic Models) Ex: f (x) = (x + ) What is the interval when this function is: Increasing Decreasing (-,0) 1 st Get any Max and/or Min Values in Calculator OR with Algebra! Always use x values!

18 Increasing/Decreasing (Algebraic Models) ANSWERS Ex: f (x) = (x + ) What is the interval when this function is: Increasing [, ) (-,0) Decreasing (, ] Always use x values!

19 Back to this example We found the extrema: (-1.15, 4.08) When is the function increasing? When is the function decreasing? (1.15, -.08)

20 Back to this example ANSWERS We found the extrema: Max: 4.08 occurs at x = Min: -.08 occurs at x = 1.15 (-1.15, 4.08) When is the function increasing? (, 1.15] [1.15, ) (1.15, -.08) When is the function decreasing? [ 1.15, 1.15]

21 Student Practice Find the intervals increasing/decreasing. State the domain and range. A) gx ( ) x x 1 Increasing: Decreasing: Domain: B) h( x) x 4 5 Range: Increasing: Decreasing: Domain: Range:

22 Student Practice ANSWERS Find the intervals increasing/decreasing. State the domain and range. A) gx ( ) x x 1 Increasing: Decreasing: Domain: (, 1) ( 1, 0] [0, 1) (1, ) (-, -1)È(-1, 1)È(1, ) B) h( x) x 4 5 Range: Increasing: Decreasing: Domain: Range: (, 0] (1, ) [4, ) (, 4] (-, ) [5, )

23 More Practice: Increasing/Decreasing 30. f ( x) x 1 x h( x) ( x ) f ( x) x x x Textbook p. 98 #30, 3, 34

24 More Practice ANSWERS: Increasing/Decreasing 30. f ( x) x 1 x 1 3 Inc : [1, ) Dec : (, 1] Inc h( x) ( x ) f ( x) x x x Inc : [, ) Dec : (, ] : (,.55] [1., ) Dec : [.55,1.] Textbook p. 98 #30, 3, 34

25 Practice Start HW & ask questions 5

26 Next slides Have #1 and #3 leftover problems to use later! 6

27 Warm Up ~ Day 5 Find the a) domain b) x & y intercepts c) range d) discontinuities e) end behavior using limit notation 1) ) 3) hx ( ) gx ( ) 7x 15 x 4x 3 5x 9 4) Find domain and range: 3x 7x 6

28 Warm Up ~ Find the a) domain b) x & y intercepts c) range d) discontinuities e) end behavior using limit notation 1) hx ( ) 7x 15 x D: (, 0) (0, ) R:[ 0.4, ) 15 x int :(,0) 7 y int : none lim f ( x) 0 lim f ( x) 0 x x ) D : (,3) (3,4) (4, ) R : (, 0) (0, 1) (1, ) Hole at (4,1) x int : none 1 y int : (0, ) 3 lim f ( x) 0 lim f ( x) 0 x x

29 Warm Up ~ Find the a) domain b) x & y intercepts c) range d) discontinuities e) end behavior using limit notation 3) gx ( ) 4x 3 5x 9 D: (, ) R:(, ) x int :(0,0) y int : (0,0) lim g( x) lim g( x) x x 4) Find domain and range: y 3x 7x 6 Domain : (, 3] [, ) 3 Range : [0, )

30 Warm Up ~ Find the a) domain b) x & y intercepts c) range d) discontinuities e) end behavior using limit notation 1) hx ( ) 7x 15 x )

31 Warm Up ~ Find the a) domain b) x & y intercepts c) range d) discontinuities e) end behavior using limit notation 3) gx ( ) 4x 3 5x 9 4) Find domain and range: y 3x 7x 6