Math 1431 DAY 16. Dr. Melahat Almus. OFFICE HOURS: MWF 11-11:30am, MW 1-2:15pm at 621 PGH

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2 Math 1431 DAY 16 Dr. Melahat Almus OFFICE HOURS: MWF 11-11:30am, MW 1-:15pm at 61 PGH If you me, please mention your course (1431) in the subject line. Check your CASA account for quiz due dates; don t miss any quizzes. BUBBLE IN PS ID VERY CAREFULLY! If you make a bubbling mistake, your scantron will not be saved in the system and you will not get credit for it even if you turned it in. Bubble in Popper Number. Be considerate of others in class. Respect your friends and do not distract anyone during the lecture. Section 3.3 Intervals of Increase and Decrease and Extreme Values Definition: Let f be a function whose domain includes an interval I. We say that f is increasing on I if for every two numbers x 1, x in I, x x implies that f x f x. 1 1 We say that f is decreasing on I if for every two numbers x 1, x in I, x x implies that f x f x

Example: The graph of f x is given below: The function is increasing over the intervals: f

3 If the graph of a function is given, it is very easy to find the intervals of increase and decrease. Simply observe whether the y values are going up or down. Example: The graph of f x is given below: The function is increasing over the intervals: f is decreasing over the intervals: What if the function is given by a formula? f x x or f x x x x f x x cosx?

4 Observe that over the intervals where the function is increasing, the tangent lines have positive slope. On the other hand, over the intervals of decrease, the tangent lines have negative slope. Theorem: Suppose that f is differentiable on the interior of an interval I and continuous on all of I. If f' x 0 for all x in I, then f increases on I. If f' x 0 for all x in I, then f decreases on I. Remark: Conversely, we can say the following: If f increases on I, then f' x 0 for all x in the interior of I. If f decreases on I, then f' x 0 for all x in the interior of I. 3

5 Example: Given it decreasing? f x x 10x, when is this function increasing? When is Example: Given 5 3 is it decreasing? f x 6x 40x 10, when is this function increasing? When 4

6 3 Example: Given f' x x 1 x x 4 given!), when is the function f (be careful; derivative is x increasing? When is it decreasing? 5

7 Example: Given decreasing? f x x, when is this function increasing? When is it x 4 6

8 Example: Given decreasing? f x x, when is this function increasing? When is it x 1 7

9 Example: Given f x x 4 / 3 decreasing?, when is this function increasing? When is it 8

10 In some cases, you may want to sketch the graph to answer questions about increasing/decreasing. Example: Given f x 5 x 1 decreasing?, when is this function increasing? When is it x 1, x 0 Exercise: Given f x x, 0 x 5 x, 5 x When is it decreasing?, when is this function increasing? 9

11 Next: Section 3.4 Extreme Values Local Extreme Values Definition: Suppose that f is a function defined on open interval I and c is an interior point of I. The function f has a local minimum at x c if f c f x for all x in I (that is, for all x sufficiently close to c ). The function f has a local maximum at x f c c if f x for all x in I (that is, for all x sufficiently close to c ). In general, if f has a local minimum or maximum at x local extreme value of f. c, we say that f c is a When the graph of a function is given, we can easily find the local extreme values by inspection. 10

This graph suggests that local maxima or minima occur at the points where the tangent line is horizontal or where the function is not differentiable and this is true.

12 This graph suggests that local maxima or minima occur at the points where the tangent line is horizontal or where the function is not differentiable and this is true. Fact: Suppose that the function f is defined on an open interval containing the number c. If f has a local minimum or maximum at x c, then f' c 0 or f ' c does not exist. This fact gives us one of the tools for finding local extreme values of a function defined by a formula. Definition: The interior points c of the domain of a function f for which f' c 0 or f ' c does not exist are called critical points for f. Example: Find critical points of the function: f x x 4x,. 11

13 POPPER# Question# f x x 4x, When is the function increasing? A) (4,infinity) B) (-infinity,) C) (0,infinity) D) (,infinity) E) None Question# 3 f x x 5x, When is the function increasing? A) (-infinity,infinity) B) (-infinity,0) C) (0,infinity) D) (1,infinity) E) None 1

Section MWF 12 1pm SR 117

Section MWF 12 1pm SR 117 Math 1431 Section 1485 MWF 1 1pm SR 117 Dr. Melahat Almus almus@math.uh.edu http://www.math.uh.edu/~almus COURSE WEBSITE: http://www.math.uh.edu/~almus/1431_sp16.html Visit my website regularly for announcements