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Math 1431

Section 12485 MWF 12‐1pm 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 and course material!

If you e-mail 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.

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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 1x , 2x in I ,

1 2x x implies that 1 2f x f x .

We say that f is decreasing on I if for every two numbers 1x , 2x in I ,

1 2x x implies that 1 2f x f x .

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.

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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?

2 1f x x

4 35 2f x x x x or 2f x x cos x ?

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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 0f ' x for all x in I , then f increases on I .

If 0f ' x for all x in I , then f decreases on I .

Remark: Conversely, we can say the following: If f increases on I , then 0f ' x for all x in the interior of I .

If f decreases on I , then 0f ' x for all x in the interior of I .

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Example: Given 2 10 2f x x x , when is this function increasing? When is

it decreasing?

Example: Given 5 36 40 10f x x x , when is this function increasing? When

is it decreasing?

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Example: Given 2 32 1 2 4f ' x x x x (be careful; derivative is

given!) , when is the function f x increasing? When is it decreasing?

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Example: Given 2

4

xf x

x

, when is this function increasing? When is it

decreasing?

2

8

4

x xf ' x

x

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Example: Given 22

1

xf x

x

, when is this function increasing? When is it

decreasing?

2

2 2

2 1

1 1

xf ' x

x x

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Example: Given 2 34 /f x x , when is this function increasing? When is it

decreasing?

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In some cases, you may want to sketch the graph to answer questions about increasing/decreasing.

Example: Given 5 2 1f x x , when is this function increasing? When is it

decreasing?

Exercise: Given

2 1 0

2 0 5

5

x , x

f x x, x

x, x

, when is this function increasing?

When is it decreasing?

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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 c if

f c 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 c , we say that f c is a

local extreme value of f .

When the graph of a function is given, we can easily find the local extreme values by inspection.

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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

0f ' c 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

0f ' c or f ' c does not exist

are called critical points for f .

Example: Find critical points of the function: 2 4f x x x, .

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POPPER#

Question# 2 4f x x x,

When is the function increasing?

A) (4,infinity) B) (-infinity,0) C) (0,infinity) D) (2,infinity) E) None

Question# 3 5f x x x,

When is the function increasing?

A) (-infinity,infinity) B) (-infinity,0) C) (0,infinity) D) (1,infinity) E) None

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