Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review:...

Math 112 Precalculus MathematicsSections 1 and 4

Instructor: Christopher Davis

Instructor: Christopher Davis Math 112 Precalculus Mathematics Sections 1 and 4 1 / 19

Who am I?

Christopher Davis,Office: Hibbard 533eMail: daviscw@uwec.eduOffice hours:11AM - 12PM Monday, Tuesday, Wednesday, Thursday(also by appointment, but please give me warning)

Syllabus, the course webpage and role:

Overview of this class

today:

A review: Real numbers, intervals, Absolute value.

Solving equations to get the information you want.

Functions and Graphs - Encoding an visualizing data.

Polynomials and their applications.I The easiest of functions to understand.I Applications to optimization (making the best possible choice).

Exponentials and LogarithmsI Population growth, radioactive decay and compound interest are all the

same thing.

Trigonometry: The mathematics of triangles and ballistics.

today:

same thing.

today:

same thing.

today:

same thing.

Topic 1: Real numbers.

Examples of Real numbers:

7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

A real number is any number that can be expressed in decimal form

7 = 7.000000 . . .−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

. . . greater numbers

−2−3

lesser numbers . . .

Examples of Real numbers:7, 2/3, −3/5, .7, 325.4435256343, π ∼ 3.141596, 5.23 = 5.232323 . . .

Definition

0 1−1 2 3

−2−3

Definition

A real number is

any number that can be expressed in decimal form

0 1−1 2 3

−2−3

Definition

0 1−1 2 3

−2−3

Definition

7 = 7.000000 . . .

−3/5 = − .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

−2−3

Definition

7 = 7.000000 . . .−3/5 =

− .6 = −0.66666You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

−2−3

Definition

7 = 7.000000 . . .−3/5 = − .6 = −0.66666

You should think of Real numbers as points on a line, The Number Line.Bigger numbers go to the right of lesser numbers.

0 1−1 2 3

−2−3

Definition

0 1−1 2 3

−2−3

Definition

1−1 2 3

−2−3

Definition

−1 2 3

−2−3

Definition

0 1−1

−2−3

Definition

0 1−1 2 3

−2−3

Definition

0 1−1 2 3

−2−3

Definition

0 1−1 2 3

−2−3

Definition

0 1−1 2 3

−2−3

Intervals

An interval is an uninterrupted piece of the real line.

Example: All the numbers between and −3 and 1.

0 1−3 4

Some Polls

Is 0 be in this interval?

Is 4 in this interval? no

Is −3 in this interval? It depends on what you mean by “between.”

As Mathematicians we abhor ambiguity.

We need a way to say if we mean the interval between −3 and 1 whichincludes or excludes the boundary points.

Intervals

An interval is an uninterrupted piece of the real line.Example: All the numbers between and −3 and 1.

0 1−3 4

Some Polls

Intervals

0 1−3 4

Some Polls

Intervals

0 1−3 4

Some Polls

Is 4 in this interval?

Is −3 in this interval?

It depends on what you mean by “between.”

Intervals

0 1−3 4

Some Polls

Is 0 be in this interval? yes

Is 4 in this interval?

Intervals

0 1−3 4

Some Polls

Intervals

0 1−3 4

Some Polls

Intervals

0 1−3 4

Some Polls

Intervals

0 1−3 4

Some Polls

Intervals

How many different things might we mean by the interval from −3 to 1?

We could include −3 and 1.

We could exclude −3 and 1.

We could include −3 and exclude 1.

We could exclude −3 and include 1.

In order to distinguish these, we will use a parenthesis “(” or “)” if anendpoint is excluded and a bracket “[” or “]” if it is included.Example:

Excludes −3 and 1.

Excludes −3 and includes 1.

Draw the interval which includes both endpoints.

Intervals

In order to distinguish these, we will use a parenthesis “(” or “)” if anendpoint is excluded and a bracket “[” or “]” if it is included.

Example:

Intervals

Making notation easier.

Drawing a picture like:

is hard.

Shorthand: (−3, 1] is the interval between −3 and 1 which includes 1 andexcludes −3.

Similarly we we will write (−3, 1), [−3, 1) and [−3, 1].

Comprehension check: True or false?

1 is in (−4, 8].

−4 is in (−4, 8].

−4.25 is in (−4, 8].

8 is in (−4, 8].

is hard.

1 is in (−4, 8].

−4 is in (−4, 8].

−4.25 is in (−4, 8].

8 is in (−4, 8].

is hard.

1 is in (−4, 8].

−4 is in (−4, 8].

−4.25 is in (−4, 8].

8 is in (−4, 8].

is hard.

1 is in (−4, 8].

−4 is in (−4, 8].

−4.25 is in (−4, 8].

8 is in (−4, 8].

is hard.

1 is in (−4, 8]. True

−4 is in (−4, 8].

−4.25 is in (−4, 8].

8 is in (−4, 8].

is hard.

1 is in (−4, 8]. True

−4 is in (−4, 8]. False

−4.25 is in (−4, 8].

8 is in (−4, 8].

is hard.

1 is in (−4, 8]. True

−4 is in (−4, 8]. False

−4.25 is in (−4, 8]. False

8 is in (−4, 8].

is hard.

1 is in (−4, 8]. True

−4 is in (−4, 8]. False

−4.25 is in (−4, 8]. False

8 is in (−4, 8]. True.

A Bat-Example

After unraveling a puzzle, Batman knows that the Riddler is in the interval[3, 6]. After investigating, Batman knows he is not to be found in [4, 5].

Where could the Riddler still be hiding? Draw a picture and write downthe names of the intervals where the Riddler might be.

A Bat-Example

After unraveling a puzzle, Batman knows that the Riddler is in the interval[3, 6]. After investigating, Batman knows he is not to be found in [4, 5].

Where could the Riddler still be hiding? Draw a picture and write downthe names of the intervals where the Riddler might be.

Inequalities define intervals

We said that 1 is in (−4, 8]. Why was that?

One answer: 1 is greater than −4 but less than or equal to 8.

The interval (−4, 8] is defined by all numbers x satisfying −4 < x ≤ 8.

Concept check:

What interval does the inequality 3 ≤ x < 5 define? Answer: [3, 5).

What inequalities define (4, 6)? Answer: 4 < x < 6 (or 6 > x > 4)

Concept check:

What interval does the inequality 3 ≤ x < 5 define?

Answer: [3, 5).

Concept check:

What inequalities define (4, 6)?

Answer: 4 < x < 6 (or 6 > x > 4)

Concept check:

Unbounded intervals.

On the number line, shade in the set of all numbers greater than or equalto -2.

This looks kind of like an interval. It certainly is an “uninterrupted set ofreal numbers.”

It is defined by the inequality −2 ≤ x .

In the spirit of the previous definitions we will use the notation [−2,∞) forthis interval. ∞ or “infinity” is not really a number. We are merely usingit as a symbol.

For you: What do you think that (−∞, 4) should mean? As a set definedby an inequality? As a picture? Is 4 in this set?

A handy table:notation inequality picture notation inequality picture

(a, b) a < x < b(

(a,∞)

[a, b] [a,∞)

[a, b) (−∞, b)

(a, b] (−∞, b]

(a, b) a < x < b(

(a,∞)

[a, b] a ≤ x ≤ b [a,∞)

[a, b) (−∞, b)

(a, b] (−∞, b]

(a, b) a < x < b(

(a,∞)

[a, b] a ≤ x ≤ b[

[a,∞)

[a, b) (−∞, b)

(a, b] (−∞, b]

(a, b) a < x < b(

(a,∞)

[a, b] a ≤ x ≤ b[

[a,∞)

[a, b) a ≤ x < b (−∞, b)

(a, b] (−∞, b]

(a, b) a < x < b(

(a,∞)

[a, b] a ≤ x ≤ b[

[a,∞)

[a, b) a ≤ x < b[

(−∞, b)

(a, b] (−∞, b]

(a, b) a < x < b(

(a,∞)

[a, b] a ≤ x ≤ b[

[a,∞)

[a, b) a ≤ x < b[

(−∞, b)

(a, b] a < x ≤ b (−∞, b]

(a, b) a < x < b(

(a,∞)

[a, b] a ≤ x ≤ b[

[a,∞)

[a, b) a ≤ x < b[

(−∞, b)

(a, b] a < x ≤ b(

(−∞, b]

(a, b) a < x < b(

(a,∞) a < x

[a, b] a ≤ x ≤ b[

[a,∞)

[a, b) a ≤ x < b[

(−∞, b)

(a, b] a < x ≤ b(

(−∞, b]

(a, b) a < x < b(

(a,∞) a < x(

[a, b] a ≤ x ≤ b[

[a,∞)

[a, b) a ≤ x < b[

(−∞, b)

(a, b] a < x ≤ b(

(−∞, b]

(a, b) a < x < b(

(a,∞) a < x(

[a, b] a ≤ x ≤ b[

[a,∞) a ≤ x

[a, b) a ≤ x < b[

(−∞, b)

(a, b] a < x ≤ b(

(−∞, b]

(a, b) a < x < b(

(a,∞) a < x(

[a, b] a ≤ x ≤ b[

[a,∞) a ≤ x[

[a, b) a ≤ x < b[

(−∞, b)

(a, b] a < x ≤ b(

(−∞, b]

(a, b) a < x < b(

(a,∞) a < x(

[a, b] a ≤ x ≤ b[

[a,∞) a ≤ x[

[a, b) a ≤ x < b[

(−∞, b) x < b

(a, b] a < x ≤ b(

(−∞, b]

(a, b) a < x < b(

(a,∞) a < x(

[a, b] a ≤ x ≤ b[

[a,∞) a ≤ x[

[a, b) a ≤ x < b[

(−∞, b) x < b)b

(a, b] a < x ≤ b(

(−∞, b]

(a, b) a < x < b(

(a,∞) a < x(

[a, b] a ≤ x ≤ b[

[a,∞) a ≤ x[

[a, b) a ≤ x < b[

(−∞, b) x < b)b

(a, b] a < x ≤ b(

(−∞, b] x ≤ b

(a, b) a < x < b(

(a,∞) a < x(

[a, b] a ≤ x ≤ b[

[a,∞) a ≤ x[

[a, b) a ≤ x < b[

(−∞, b) x < b)b

(a, b] a < x ≤ b(

(−∞, b] x ≤ b]b

Topic 2: Absolute value. The muchness of numbers

Definition

The Absolute value of a number, x , denoted |x | is the distance from x to0 on the number line.

Example: What is |5|?0 5

distance is 5

Example: What is | − 5|?

distance is 5

| − 5| =

Evaluate |4− 3|?

Definition

Example: What is |5|?

distance is 5

| − 5| =

Evaluate |4− 3|?

Definition

distance is 5

| − 5| =

Evaluate |4− 3|?

Definition

distance is 5

| − 5| =

Evaluate |4− 3|?

Definition

distance is 5

| − 5| =

Evaluate |4− 3|?

Definition

distance is 5

5Example: What is | − 5|?

distance is 5

| − 5| =

Evaluate |4− 3|?

Definition

distance is 5

|5| = 5

distance is 5

| − 5| =

Evaluate |4− 3|?

Definition

distance is 5

|5| = 5Example: What is | − 5|?

distance is 5

| − 5| =

Evaluate |4− 3|?

Definition

distance is 5

| − 5| =

Evaluate |4− 3|?

Definition

distance is 5

| − 5| =

Evaluate |4− 3|?

Definition

distance is 5

| − 5| =

5Evaluate |4− 3|?

Definition

distance is 5

| − 5| = 5Evaluate |4− 3|?

Symbolically working with the absolute value

Complete these sentences.

When x is positive, |x | =

When x is negative, |x | =

− x .

Compute:

5− |1− 7|

5− | − 6| = 5− 6 = −1||2− 3| − | − 4||

|| − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4|

− (π − 4) Since π − 4 < 0= 4− π

Given that x > 4 express |4− x | without any absolute value bars.

Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

When x is positive, |x | = x .

When x is negative, |x | =

− x .

Compute:

5− |1− 7|

5− | − 6| = 5− 6 = −1||2− 3| − | − 4||

|| − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4|

− (π − 4) Since π − 4 < 0= 4− π

When x is negative, |x | = − x .

Compute:

5− |1− 7|

5− | − 6| = 5− 6 = −1||2− 3| − | − 4||

|| − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4|

− (π − 4) Since π − 4 < 0= 4− π

|0| = 0

Compute:

5− |1− 7|

5− | − 6| = 5− 6 = −1||2− 3| − | − 4||

|| − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4|

− (π − 4) Since π − 4 < 0= 4− π

|0| = 0

Compute:5− |1− 7| =

5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| =

|| − 1| − | − 4|| = |1− 4| = | − 3| = 3

|π − 4| =

− (π − 4) Since π − 4 < 0= 4− π

|0| = 0

Compute:5− |1− 7| = 5− | − 6| =

5− 6 = −1

||2− 3| − | − 4|| =

|| − 1| − | − 4|| = |1− 4| = | − 3| = 3

|π − 4| =

− (π − 4) Since π − 4 < 0= 4− π

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| =

|| − 1| − | − 4|| = |1− 4| = | − 3| = 3

|π − 4| =

− (π − 4) Since π − 4 < 0= 4− π

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| =

|1− 4| = | − 3| = 3

|π − 4| =

− (π − 4) Since π − 4 < 0= 4− π

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| =

| − 3| = 3

|π − 4| =

− (π − 4) Since π − 4 < 0= 4− π

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| = | − 3| =

|π − 4| =

− (π − 4) Since π − 4 < 0= 4− π

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4| =

− (π − 4) Since π − 4 < 0= 4− π

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4| = − (π − 4) Since π − 4 < 0

4− π

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4| = − (π − 4) Since π − 4 < 0

= 4− π

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4| = − (π − 4) Since π − 4 < 0

= 4− π

Given that x > 4 express |4− x | without any absolute value bars.Since x > 4, 4− x < 4− 4 = 0.

4− x is negative.|4− x | = −(4− x) = x − 4.

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4| = − (π − 4) Since π − 4 < 0

= 4− π

Given that x > 4 express |4− x | without any absolute value bars.Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.

|4− x | = −(4− x) = x − 4.

|0| = 0

Compute:5− |1− 7| = 5− | − 6| = 5− 6 = −1

||2− 3| − | − 4|| = || − 1| − | − 4|| = |1− 4| = | − 3| = 3|π − 4| = − (π − 4) Since π − 4 < 0

= 4− π

Given that x > 4 express |4− x | without any absolute value bars.Since x > 4, 4− x < 4− 4 = 0. 4− x is negative.|4− x | = −(4− x) = x − 4.

A harder example

When x is negative, |x | = −x .

|0| = 0

Suppose that x is in the interval (4, 5). What is |x − 4|+ |x − 5|?

First we evaluate |x − 4|.

What does “x is in the interval (4, 5)” mean? Answer: 4 < x < 5.

Since 4 < x , x − 4 is positive. |x − 4| = x − 4.

Next we evaluate |x − 5|.

Since x < 5, x − 5 is negative. |x − 5| = −(x − 5).

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

For you: What is |x − 4|+ |x − 5| when x > 5?

A harder example

|0| = 0

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

A harder example

|0| = 0

What does “x is in the interval (4, 5)” mean?

Answer: 4 < x < 5.

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

A harder example

|0| = 0

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

A harder example

|0| = 0

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

A harder example

|0| = 0

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

A harder example

|0| = 0

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

A harder example

|0| = 0

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) =

x − 4− x + 5 = 5− 4 = 1

A harder example

|0| = 0

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 =

5− 4 = 1

A harder example

|0| = 0

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

A harder example

|0| = 0

|x − 4|+ |x − 5| = (x − 4) +−(x − 5) = x − 4− x + 5 = 5− 4 = 1

The distance between numbers

What is the distance from −3 to 2?

−3 2

distance is 5

What about from 2 to −3? Answer: 5.

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

The distance between two numbers a and b is given by |a− b| = |b − a|.

Translate the following statements into statements involving absolutevalue

The distance between 2 and 4 is equal to 2.

The distance between 0 and y is at least 5.

The distance between x and 2.5 is less than or equal to .5.

What is the distance from −3 to 2?−3 2

distance is 5

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

distance is 5

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

distance is 5

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

distance is 5

What about from 2 to −3?

Answer: 5.

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

distance is 5

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

distance is 5

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

distance is 5

Notice that | − 3− 2| = |2− (−3)| = 5.

Proposition

Intervals and absolute value

Find the interval containing all numbers for which the distance between 0and y is at most 2.

0 2−2

What interval is this? Answer: [−2, 2].

What numbers satisfy that |y | > 2?

−2 2

Answer: |y | > 2 if y is in (2,∞) or (−∞,−2).

0 2−2

−2 2

0 2−2

−2 2

0 2−2[ ]

What interval is this?

Answer: [−2, 2].

−2 2

0 2−2[ ]

−2 2

0 2−2[ ]

−2 2

0 2−2[ ]

−2 2

0 2−2[ ]

−2 2

0 2−2[ ]

0−2 2

0 2−2[ ]

0−2 2

As a table

Complete this tableinequality withabsolute value

interval(s)inequality without

absolute value

|y | < 2

y is in (−2, 2)

)( −2 < y < 2

|y | ≤ 2y is in [−2, 2]

−2 ≤ y ≤ 2

|y | > 2y is in (2,∞) or (−∞, 2)

y < −2 or 2 < y

|y | ≥ 2

y is in [2,∞) or (−∞, 2]

() y ≤ −2 or 2 ≤ y

What should this table look like if 2 is replaced with any positive numberd?

As a table

absolute value

|y | < 2

y is in (−2, 2)

)( −2 < y < 2

|y | ≤ 2y is in [−2, 2]

][ −2 ≤ y ≤ 2

|y | > 2y is in (2,∞) or (−∞, 2)

() y < −2 or 2 < y

|y | ≥ 2

y is in [2,∞) or (−∞, 2]

() y ≤ −2 or 2 ≤ y

As a table

absolute value

|y | < 2y is in (−2, 2)

)( −2 < y < 2

|y | ≤ 2y is in [−2, 2]

][ −2 ≤ y ≤ 2

|y | > 2y is in (2,∞) or (−∞, 2)

() y < −2 or 2 < y

|y | ≥ 2y is in [2,∞) or (−∞, 2]

() y ≤ −2 or 2 ≤ y

One last problem

What is the set of all x such that |x − 5| ≤ 4?Does it include any negative numbers?Your group must check your answer with me before leaving.

Homework: Appendix A.1: 39, 40, 49, 50, 57, 58; Appendix A.10: 1, 2,3, 4, 10, 11, 12, 13, , 14, 23, 24, 25, 26.

Instructor: Christopher Davispeople.uwec.edu/daviscw/oldClasses/math112Fall2014/... · A review:...

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Monday, September 24, 2012. Review: Perfect and major intervals Review: Minor intervals Introduce: Augmented & diminished intervals Aural Skills:

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Specific Intervals

CLASS XII APPLICATION OF DERIVATIVES …...Find the absolute maximum value and the absolute minimum value ofthe following functions in the given intervals: x — sinx+ COSX , x E [0,