review topics for math 1400 elements of calculus table of...

Math 1400 - Manyo Table of Contents - Review - page 1 of 2

Review Topics for MATH 1400 – Elements of Calculus

Table of Contents

MATH 1400 – Elements of Calculus is one of the Marquette Core Courses for Mathematical Reasoning.

Pre-Requisites: Three years of high school Mathematics.

This course has students from various backgrounds. All students have taken three years of high school

mathematics, one of which is Algebra. Some have taken four years of mathematics and the last course may

have been Pre-Calculus or Calculus. The course is taught so that students with various backgrounds should be

able to successfully complete the course, if they put in the effort to do so. This means that you must attend all

classes and do the assigned problems in a timely manner (i.e. keep up with the class; do not fall behind).

Some of you have not taken a math course in several years. For many of you, your Algebra skills are weak or

simply a memory of things past. While I will review Algebra as we work through the development of Calculus,

you will be best served with a review of basic Algebra skills before embarking on learning and understanding the

concepts of Calculus. What follows is seven Review topics. The first six should be review for all, the last review

on logarithms may be brand new material for many. The first six topics are encountered in the first chapter that

we cover, Chapter 10 of College Mathematics for Business, Economics, Life Sciences and Social Sciences, Barnett,

Ziegler, Byleen, 12th Edition, Pearson Prentice Hall.

Because we will be using the material in the first 4 Reviews when cover Section 10.1, I strongly suggest that you

review these topics before the semester begins. If you cannot, I cannot stress enough the importance of

understanding these topics before the first test of the semester. What follows is a table of contents for the

review exercises when they will be needed. The first chapter that we will cover is Chapter 10, followed by

Chapters 11, 12, 13 and parts of 14.

Review 1 – The Real Numbers – this should be completed before the first class. The concepts in this are

fundamental to understanding Calculus. We begin with section 10.1, which will take us three days to cover.

Review 2 – Functions– this should be completed before the first class. The concepts in this are fundamental

to understanding Calculus.

Review 3 – Multiplication of Polynomials – this needs to be understood before completing Review 4. The topics

in this review will be used during the entire semester. This should be completed before the 2nd day of class.

Review 4 – Factoring Polynomials – This should be completed before the 3rd day of class, (at the very latest).

Review 5 – Exponents – This should be completed before we cover section 10.2

Review 6 – Linear Equations – This should be completed before we get to section 10.4. The Derivative (the basis

for the rest of the semester.) After Section 10.4, we will be on a faster pace trying to cover approximately one

section per day.

Review 7 – Logarithms – This can wait until immediately after the first test. However, this is new material for

some of you. You may want to go through it twice to be sure that you are comfortable with it. I am more than

willing to help you with this outside of class.

Answers to Review Questions

Math 1400 - Manyo

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Math 1400 - Manyo Review 1

Review 1*- The Real Numbers

To be completed prior to the first class

*Review sheets contain material that should be a review for most students and are meant to be completed in less than 30 minutes in

preparation for class. Feel free to discuss and complete these with your classmates. However, simply copying someone's work defeats

the purpose and is a waste of your time.

Sets of numbers - see Textbook Appendix A-1 for further explanation You should be familiar with the following sets of numbers. The letter names given to them are their traditional

names in all of mathematics.

Integers: Z Integers include the counting numbers (1, 2, 3, …), their opposites and zero

Rational Numbers: Q

Rational numbers are numbers that can be expressed as the quotient of two integers and in decimal form are either terminating or repeating decimals.

Irrational Numbers: I

Irrational numbers are numbers that in decimal form are non-terminating and non-repeating.

Real Numbers: The Real Numbers is the union of the Rational and Irrational numbers.

Q1: The Venn diagram shows the relationships between these four sets of numbers. Complete the chart by checking each set to which the number belongs. Then place each number on the Real number line below.

0

Z

I I Q

The Universe is Indicate to which sets, each number belongs.

number Z Q I

Review 1 - Page 2 of 2

Using Interval Notation – see Texbook pages 6-7* for further explanation

graphically inequality interval

-3 -2 -1 0 1 2 3

On a number line, exclusion is shown with an empty (or open) circle and inclusion is shown with a solid

circle.

An inequality,uses < or > to show exclusion and or to show inclusion.

Interval notation uses a paren, ( or ), to show exclusion and a square bracket, [ or ], to show inclusion.

Q2: In the interval , the largest number is _______ and the smallest number is ______________. Q3: Complete the chart to describe each interval graphically, as an inequality and using interval notation.

How many Real Numbers are there? Q4: A set of numbers is dense if, for any two numbers and in the set, where , there is another

number in the set such that . (i.e between any two numbers in the set there is another number

belonging to the set).

A. Illustrate that "the set of Real Numbers is dense " by finding a real number between each and .

-0.6 -0.5 2 2.01

-3.9686 -3.9682 4.0039354 4.0039365

B. Given that the set of Real Numbers is dense, how many numbers are there between any two numbers? C. Is the set of integers dense? Explain.

graphically inequality interval

-3 -2 -1 0 1 2 3

This is also the way we indicate a point in the coordinate plane. e.g. P is the point (3,5). The correct interpretation comes from the context of what you are doing.

- and + (negative infinity and infinity) are

never included in an interval since they are not

numbers. They represent the idea that the

interval continues without bound, or infinitely.

The set of Real Numbers is (-, +)

We can indicate an interval of Real numbers graphically, with a compound inequality or using interval notation. The interval (-2, 1] does not include the endpoint -2 and does include the endpoint 1.

Math 1400- Manyo Review 2

Review 2 – Functions

A complete review of functions can be found in our textbook in Chapter 2.

Before we state a formal definition of a function let's discuss, informally the concept of a function.

Think of a function as a black box that takes as input a number, does something to the input number and

outputs a new number. For example, below we show a function that outputs 5, when the input number is 2.

2 5

Continuing with the same function, here are some additional inputs into the function box and outputs from the

function box.

3 10

4 17

5 26

It would be wrong to assume that only positive integers can be input into the function. This function box also

accepts negative numbers, fractions and, in fact, accepts any real number. Here are some more examples.

-2 5

-5 26

3

1.25

In general, we often think of the input number as and the output number as , and we give the function a

name, such as .

The statement says is a function of

The statement tells us that for the function when .

Were you able to determine what function is inside the black box?


Lets's look inside the box.

Definition: A function is a correspondence between two sets of numbers, the domain and the range, such that

for each element in the domain, there exists one and only one element in the range.

Comments: The input numbers, belong to the domain and the output numbers, belong to the range. The

definition says that for every value in the domain, there is one and only one, value in the range.

Therefore, the next two examples are not functions.

does not exist

So, what are the domain and range for the function

To determine the domain, we ask ourselves " Are there any real numbers that we cannot substitute for "

Since any real number can be squared and added to 1, the answer is no, since any number can be

substituted for . Therefore, the domain of the function is all real numbers, which can be represented as

or, using interval notation, . We call the independent variable, since we can let equal any

value in the domain.

To determine the range, we ask ourselves "What values can this function output?"

Since is always positive or zero, is always greater than or equal to 1. Therefore, using interval

notation, the range is . We call the dependent variable, since the value of depends upon the

value of

2

-2

This example shows This is not

possible if is a function since must have only one

value. This is why we define the square root function,

to mean the positive square root, i.e.

. If we want the negative square root, we

use the function , i.e.

A function must be defined at every element in its

domain. Therefore, if is not defined (ND) ,

is not in the domain of the function.


Graphing a Function

The expression, "a picture is worth a thousand words", is especially true in mathematics. If you look at the

inside back cover of your textbook, you will see the graphs of some elementary functions. These are the types

of functions we will be working with this semester in MATH 1400. You should be able to visualize these

functions by looking at the function, by graphing a few points on the function, or by using your graphing

calculator.

Let's graph the function we have been talking about, , by plotting some points on the coordinate

plane. The independent variable, in this example, is always plotted on the horizontal axis and the dependent

variable, in this example, is always plotted on the vertical axis.

First we list the coordinates of some points that satisfy the function. Then we plot those points, and then

connect the dots with a curve.

List points Plot points Sketch curve

-3

-2

-1

0

1

2

3

is an example of a second degree polynomial, or a quadratic function. Graphs of quadratic

functions are parabolas. This parabola has a vertex at the point (0,1) and is congruent to the graph of the

function , which has its vertex at (0,0).

Another way to say this is that is the graph of shifted vertically up one unit.

A Polynomial has the form

, where each is any real number

(including 0) and each is a positive integer or zero. The term with the highest power is called the leading or

dominant term and its value of is the degree of the polynomial. A constant is a polynomial with degree 0.

Examples of polynomial funtions:

Examples of functions that are not polynomials:

x

y

x

y


Exercises

1. For each of the functions, find

A. B. C.

2. Let A. Find the value of when _________________________________________________________ B. What is _______________________________________________________________________ C. What values of give a value of 11? _____________________________________________________ D. What values of give a value of 1? _____________________________________________________ E. The domain is __________________________ and the range is ______________________________ 3. For each function (continued on the next page)

Complete the list of points (some may be undefined, indicating the value is not in the domain)

Plot the points and sketch the curve.

List the domain and range using interval notation.

-3

-2

-1

0

1

2

3

x

y

x

y

The domain is

____________________

The range is

_____________________


-2

-1

0

1

2

-4

-1

0

1

4

9

Recall: The absolute value of a number is its positive distance from zero on the real number line. We know

x

y

The domain is

____________________

The range is

_____________________

x

y

The domain is

____________________

The range is

_____________________


-3

-2

-1

0

1

2

3

The functions in the next two examples have a numerator and denominator. Remember that any fraction with 0

in the denominator is not defined. e.g.

are undefined

-3

-2

-1

0

1

2

3

x

y

The domain is

____________________

The range is

_____________________

x

y

The domain is

____________________

The range is

_____________________


-4

-3

-2

-1

-1/2

-1/4

0

1/4

1/2

1

2

3

4

x

y

The domain is

____________________

The range is

_____________________

Math 1400 - Manyo

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Math 1400- Manyo Review 3

Review 3 – Multiplication of Polynomials

Basic properties of Addition and Multiplication on the Set of Real Numbers. are Real numbers

Properties of Addition Properties of Multiplication

Commutative: e.g.

Commutative: e.g.

Associative: e.g

Associative: e.g.

Additive Identity is 0: e.g.

Multiplicative Identity is 1:

e.g. and

Additive Inverse of is – : e.g.

Multiplicative Identity of is

if

Distributive Property of and Multiplication over Addition: e.g .

Multiplication of a Monomial (single term) and a Binomial (two terms) Multiply

rewrite subtraction as addition of the opposite

apply Distributive property

apply Commutative(re-order) and Associate(re-group) properties

multiply

rewrite as subtraction

Multiplication of Two Binomials Multiply

apply Distributive property

apply the Distributive property two times

multiply

add like terms

Some of you may have learned this using the FOIL method : First-Outer-Inner-Last. All of the above steps can be done in one step: First + (Outer + Inner) + Last = . As long as you do it correctly, any method is fine.

Squaring a Binomial Multiply

F + (O+I) + L

Review 3- Page 2 of 2 Q1: Multiply the following:

Math 1400 - Manyo Review 4 - page 1 of 2

Review 4 - Exponents

Definitions: Examples:

(1) If is a positive integer then,

- times

(2) If

(3) If and are integers,

(4) If is positive, then

Note:

is the reciprocal of and

is the reciprocal of

Q1: Rewrite using positive exponents only. Omit any root symbols.

Q2: Evaluate without a calculator. Check your answer with your calculator.

Q3: Rewrite so that the expressions have no visible denominator and no root symbols.


Properties of Exponents Examples

Q4: Rewrite the expressions. Multiply factors when possible. All exponents should be positive and all root symbols should be removed.

Math 1400 - Manyo Review 5: Factoring Polynomials - page 1 of 2

Review 5 – Factoring Polynomials

Type 1: , where and are integers

We want to factor into two factors of the form where and can be either

positive or negative. If we complete the multiplication, we get (and you should verify this)

Therefore, and

The first clue in factoring this type of expression is to determine if is positive or negative.

If is positive, then either both and are positive or both and are negative.

o The sign of and is determined by the sign of . If is negative, and are negative.

If is positive, and are positive. And, is the sum of and (i.e. )

If is negative, then and have opposite signs.

o The sign of is the same as the sign of .

Example 1: Factor into

In this example and . We need two factors of whose sum is -5. Since is positive we

know that and are either both negative or both positive. Since is negative, we know that and

are both negative. Again, we need two factors of whose sum is -5. The possible factors are shown in the

chart to the left:

Example 2: Factor into (Notice the similarity to Example 1)

In this example and . Since is negative we know that and have opposite signs. Since is negative, the factor with the larger magnitude is negative. We need two factors of whose sum is -5. The possible factors are shown in the chart to the left:

6 1

2 3

-6 -1

-2 -3

6 -1

2 -3

-6 1

-2 3

Since the last entry in the table satisfies

o o

We conclude

Since the third entry in the table satisfies

o o

We conclude


Type 2: : the difference of two squares

The difference of two squares always factors into the product of two conjugates Definition: conjugates are two binomials such that one is the sum of two terms and the other is the difference of the two terms.

Example 3: Factor

Example 4: Factor

Type 3: any polynomial expression Look for common factors in all terms of the polynomial, which include common constant factors and variable factors .

Example : Factor

o The three terms have constant factors of 6, 4 and 10, which have a common factor of 2

o The three terms have variable factors of , which have a common factor of o Therefore, we can pull a factor or from each of the three terms in the expression.

Math 1400 - Manyo Review 6- page 1 of 2

Review 6 – Linear Equations

Read pages 13 – 20 in the textbook to review. I will just do some examples here, which are sufficient if you

remember the basics. If you need more practice complete pgs 23-24: 1-4, 5 – 27 odd, 37, 41, 43, 45

1. Forms of a linear equation:

A. form

B. form, where is the slope and is the

C. point - slope form, where is the slope and is a point on the line.

2. Calculating slope: Given two points on the line

Example 1: What is the equation of the line pictured to the right? Pick any two points on the line. I chose and

The Therefore,

The equation of the line, is

Also, you can calculate the slope directly from the graph.

From the top point (-1,3) move vertically down 4 units and then horizontally to the right 2 units to

the point (1, -1). i.e.

Q1: Through observation, which line has positive slope? ____ negative slope? ___ no slope?____ 0 slope? ___

Then, find the equation of each line above.

A.

B.

C.

D.

x

y

x

y

x

y

x

y

x

y

Math 1400 - Manyo Review 6- page 2 of 2

Q2: Write the equation of a line with slope 3 and y-intercept -5. Q3: Write the equation of the line that passes through the points (-2,5) and (-3, 2)

Example 2: The third form of a linear equation: form of a line

If we know one point on a line and the slope, then all other points on the line must satisfy

What is the equation of the line that passes through the point (-6, 2) and has slope

?

All points on the line must satisfy

go directly to this statement if you memorize the point-slope form

Q4: Find the equations of the lines with given slope through the given point. This can be done if you start with the slope-intercept form of the line or the point-slope form of the line. Try these both ways.

Math 1400 - Manyo Review 7: page 1 of 4

Introduction / Review 7 – Logarithms

Compete before Sections 11.1 & 11.2

Some of you have previously studied logarithms in high school and some of you have not. There is a good

review of logarithms in the textbook, Section 2.3 pages 105 – 118. I am including a brief introduction/review

here that gives you exactly what you need to know to successfully complete the rest of this course.

What is a Logarithm? A logarithm is an exponent. There are two related functions, the exponential and logarithmic functions.

If we write the general exponential function as , then the related logarithmic function is

For both of these function, is called the base of the function with the restrictions that and

Example 1: (read as )

Two special cases are and (read as "natural log" where is the natural number)

if no is shown, the base is 10

ln is a logarithm with base

Q1: Write the related exponential statement for each logarithmic function in the first column and write the related logarithmic statement for each exponential function in the second column.

Q2: Calculate the following without a calculator.

______

Q3: Your TI83 or TI84 has two keys LOG and LN in the left-hand column. Find the following rounded to 4

significant digits to the right of the decimal point.

Note about For all possible values of the following are true:


Inverse Functions The function is the inverse function of if whenever is a point on the graph of the point

is a point on the graph of

Example 2: and are inverse functions.

Since , the point is on the graph of And, since , the point is on the graph of The graph of is the reflection of the graph of about the line

Q4: Complete this chart.

exponential function

logarithmic function

-3

-2

-1

0

1

2

3

x

yQ5. Plot the points you found in the charts

above. The first point in each chart is already

plotted. The grey line is , the line about

which the reflection occurs. The red line

connecting the points

and

is perpendicular to and bisected by the line

(i.e. the pts are reflections of each

other), Connect the points that belong to each

graph.

Q6. The domain and range of are

domain ____________ range _____________

and the domain and range of are

domain ____________ range _____________


Because exponential functions and their related logarithmic functions are inverses of each other, what one

function does to a number , the other function undoes it. This is a crude way of saying

Example 3:

Example 4: When do we use these properties?

A. When you are solving for a value that is an exponent, take the log or ln(either will work) of both sides.

Solve for :

take the logarithm of both sides: i.e.

B. When you are solving for a value that is in the argument of a logarithmic function, raise both sides as the

exponent of a power function with the same base as the log function.

Solve for :

raise both sides to a power of 10 (since 10 is the base of log)

This can also be solved by simply writing the related exponential

statement

Q6. Solve for the unknown


So far we have discussed the Properties of Logarithms listed in the first column (#1 - #4) . We will look at the

other three properties (#5 - #7), that come from related properties of exponents. Each property is written using

the general base and rewritten using the natural logarithm, which we will usually be using in Chapter 11.

Example 5: Since the , we know that

Q7: Given the following logarithms (rounded to 4 decimal places) apply Logarithm Properties #5 - #7 to find the answers to A – D. Do not use a calculator

Example 6: Proof of #5. Let and

Then the related exponential statements are

Then substitution

property of exponents

apply

substitute definitions of

Q8. Using Example 6 as your guide, prove property #6.

Properties of Logarithms If

OR OR

OR

OR

OR OR

OR

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