Math 195 Topic Summaries
1.1 Real Numbers:
This section is review. Make sure you understand Examples 1, 2, 3, and 5 well. It is not necessary to know the names of the properties. You should spend only a little time studying open and closed interval notation. Interval notation is important. You will have many opportunities to practice it throughout the semester. Make sure you review PEMDAS and understand adding and subtracting fractions by finding a LCD. Surprisingly it is easier to multiply and divide fractions than it is to add and subtract fractions, since you do not need a common denominator to multiply or divide fractions.
Textbook Examples: 1-7
Textbook Exercises: 29-38, 47-76
Intervals and Interval Notation Video
1.2 Exponents and Radicals:
Read through the section and make sure you understand each of the Laws of Exponents on p.14. You may skip "Scientific Notation". We will not use calculators in this course.
Textbook Examples: 1-5, 8-13
Textbook Exercises: 9-80
Integer Exponents Basic Exercises
Powers of Products and Quotients Video
Powers of Products and Quotients Exercises
Intro Rational Exponents Video
Fractional Exponents Exercises
Mixed Radical and Exponential Expressions Video
Properties of exponents Exercises
Evaluating Fractional Exponents Video
Evaluating fractional Exponents: fractional base Video
Evaluating Mixed Radicals and Exponents Video
1.3 Algebraic Expressions:
You will study factoring and expanding polynomials. Make sure you understand the difference. One of the goals of middle school math is to factor integers. One of the goals of high school math is to factor polynomials. In both cases you factor a complicated thing into simpler parts. Scientists do the same thing but they do not call it factoring. Biologist factor organisms into cells because the cells are easier to understand than the whole organism. Physicists factor atoms into protons, neutrons, and electrons.... Factoring can be hard work. Therefore you do not often want to expand a factored object unless you have a good reason to do so. Make sure you understand the Examples in this section but you may skip the Examples on "factoring the sum and differences of cubes", "factoring with fractional exponents" (Example 12) and "factoring with rational expressions" (Example 14). You need not memorize the "Difference of cubes" or "Sum of cubes" special factoring formula on p.30. You should spend a lot of time practicing the "Trial and Error" method to factor.
Textbook Examples: 1-11, 13, 15.
Recommended Textbook Exercises: 1-94, 100-112
Factoring Difference of Squares: leading coefficient not 1 Video
Difference of Squares Exercises
Factoring Quadratics as (x+a)(x+b) Video
More Examples of factoring quadratics Video
1.4 Rational Expressions
A rational expression is a fraction of polynomials. Luckily there is not much new here. All you do is mix together the basic algebraic techniques for fractions from middle school (and from section 1.1) with the factoring and reducing techniques for polynomials from section 1.3. Just like in middle school you need to be able to reduce, add (find the LCD), subtract (find the LCD), multiply, and divide rational expressions. One of the keys to many exam problems is to factor and reduce as much as possible before adding, subtracting, multiplying and dividing. You should work through the Examples in this section. There is a nice table on "Avoiding Common Errors" at the end of this section. You should read this table of common errors multiple times and try not to make any of these errors. You can skip rationalizing the numerator and denominator.
Textbook Examples: 1-7
Textbook Exercises: 1-69
Reducing Rational Expressions to Lowest Terms Video
Reduce Rational Expression to Lowest Terms Exercises
Multiplying Rational Expressions Video
Multiply and Divide Rational Expression Exercises
Adding Rational Expression: Unlike Denominators Video
Subtracting Rational Expressions: factored denominators Video
Add and Subtract Rational Expressions Exercises
1.5 Equations.
You should already know the quadratic formula, the zero product property, and cross multiplying for solving rational equations. You may not yet be familiar with completing the square and solving equations involving a radical. When learning and reviewing the equation solving techniques in this section do not get lost in the techniques. The adding, subtracting, multiplying, dividing, reducing, and power techniques learned in previous sections are all helpful here but remember: you are searching for a number that you can plug in for x so that the equation is true. If you can guess a solution go ahead. You don't always need to follow the fancy techniques. You can check any solution by substituting the solution for x in the original equation.
Do not however get carried away in substituting your solution for x to check your answer. Many times students have gotten the correct solution to an equation and then made an error while checking the solution and convinced themselves that the correct answer was incorrect. Whether or not you decide to check your solutions to equations is a personal matter. You need to practice solving many equations to decide when it is a good idea to check your answer. Sometimes I do check sometimes I don't.
You may skip Example 12 "Fourth-Degree of Quadratic Type" and Example 13 "Fractional Powers". You should be comfortable using the quadratic formula.
Textbook Examples: 1-11, 14.
Textbook Exercises: 2-90, 99-109
Equations with Variable in the Denominator Video
Solving Quadratics by Taking Square Roots Video
Solve Quadratics by Taking Square Roots Exercises
Worked Example: Quadratic Formula Video
Using the Quadratic Formula: Number of Solutions Video
Worked Example: Rewriting and Solving Equations by Completing the Square Video
Rational Equations Intro Video
Rational Equations Intro Exercises
Equations with Rational Expressions Video
Square-Root Equations Intro Video
Square-Root Equations Intro Exercises
Square-Root Equations: One Solution Video
1.8 Inequalities
Here you will solve inequalities. You can use many of the same techniques to solve inequalities that you used to solve equalities in section 1.5. One technique that you absolutely cannot use is you cannot multiply or divide both sides of an equality be a negative number. If you do multiply both sides of an inequality by a negative number you must remember to flip the inequality symbol. You can however add and subtract both sides by the same number without changing anything. Most inequalities have an infinite number of solutions so you cannot write them all in order. You must use interval notation or graph notation to write your answers. This is why we learned interval notation in section 1.1.
Just like in equations you may test your solution interval to an inequality by substituting several numbers from your interval(s) into the original inequality. This is only a partial check and, like always, you may not want to check your answers on an exam lest you make an error in checking and then erase a correct answer. I find it helpful to test my answer when solving absolute value inequalities.
The section on nonlinear inequalities (quadratic inequalities, inequalities with repeated factors, and quotient inequalities) may be new to you. You should spend some time studying these problems, especially the technique of making a sign table. Learning how to make a sign table is an important skill. You will make sign tables again in calculus.
You may skip the part of this section on modeling.
Textbook Examples: 1-7
Textbook Exercises: 7-98
Testing Solutions to Inequalities Video
Inequalities with Variables on Both Sides Video
Intro to Absolute Value Inequalities Video
1.9 Coordinate Geometry
This should be a review of graphing from high school. You should memorize the distance and midpoint formulas. You should understand that the distance formula is derived from the Pythagorean Theorem. You should spend a lot of time graphing equations by making tables and plotting points. Later in the course you will learn faster ways to trace some graphs. The trouble with these faster methods is that they do not always work. Sketching by making a table always works! You should learn it well. Much later, after you learn some calculus, you will sketch by using a mixture of these faster techniques, some calculus techniques, as well as making a table of points and plotting points. You will always be making tables. Making tables is boring, but it is useful. It is important that you are able to quickly make a rough sketch of many simple graphs with labeling points before taking calculus.
A common question on CCNY final exams asks students to find the center and radius of a circle given by an equation. In order to complete such a question you must know the standard form of a circle (and how the formula is derived from the distance formula) as well as how to complete the square.
You may skip the part of this section about symmetry.
Textbook Examples: 1-10
Textbook Exercises: 25-45, 55-96
Features of a Circle From Its Graph Video
Features of a Circle from its Expanded Equation Video
Features of a Circle from its Expanded Equation Exercises
Graph of a Circle from its Expanded Equation Exercises
1.10 Lines
This is truly a review section. Every student has studied lines in high school and middle school. Nevertheless, you should spend some time doing many problems in this section. You need to understand lines well. One of the key ideas of calculus is to approximate curves by lines. It makes sense to do this because lines are easy and curves can be very complicated. However you will never be able to approximate complicated curves by lines if you do not first understand lines well.
You may skip Example 10 "Graphing a Family of Lines".
Textbook Examples: 1-9, 11
Textbook Exercises: 1-52, 57-92
Intro to Point-Slope Form Video
Point-Slope and Slope-Intercept Equations Video
Graphing a Linear Equation Video
Graph From Linear Standard Form Exercises
Writing Linear Equations in All Forms Video
Parallel Lines From Equation Video
Writing Equations of Perpendicular Lines
2.1 Functions
In middle school you used numbers to describe the world around us. You used numbers to measure distances (D) and weigh (W) things. In calculus the weight of some substance may change with time giving a function W(t). In middle school you always needed new numbers (first fractions and then irrationals) to describe the world because the world is a complicated place. Using functions to describe the world is no different. We always need new functions because the world is a complicated place. One of the main goals of precalculus is to become comfortable with as many functions as possible before starting a calculus course. Calculus studies functions in a more sophisticated way. If you are not comfortable with the basic function techniques introduced in precalculus you will struggle in calculus.
So where does one begin to study functions when there are so many of them? One answer is that we will divide functions into families: the polynomials (Chapter 3), the rational functions (Chapter 3), the algebraic functions (Chapter 2), the exponential functions (Chapter 4), and the trigonometric functions (Chapter 5). We will study the formulas of functions from each family in this course. By the end of the course you should be able to recognize the family from the formula.
Unfortunately these families are still not enough. Most phenomena studied with functions in science and engineering are more complicated and do not fit into any of the families. What do you do in that case? Calculus has a clever answer to this problem. In calculus you will learn how to approximate the complicated functions you will study in advanced science and engineering with the simple families we study here. In any case we want to be comfortable with as many families of functions as possible.
One great way to expand the amount of functions that you are comfortable working with is to learn inverse functions (section 2.8). In some sense learning inverse functions doubles the amount of functions you know. Thats all there is to it. Learn about inverse functions and then you "double" the functions you know to describe the world we live in. Most inverse functions are just called "inverse functions". Yet some inverse functions are so special that they get their own name: inverse exponential functions are called "logarithmic functions". Inverse trig functions are called "inverse trig functions" or "arc..." functions. It can get confusing.
And I am getting too far ahead. Please start reading section 2.1. Do not fuss too much about the definition of a function. You already know this definition, more or less. Instead start doing the webassign problems on domains, piecewise functions, evaluating functions, and making tables. This is most important. I will never ask you to state the definition of a function on a final exam.
Textbook Examples: 1-9
Recommended Textbook Exercises: 19-76
Determining Whether Values are in Domain of Function Video
Identifying Values in Domain Exercises
Examples Finding the Domain of Functions Video
Determining the Domain of Functions Exercises
What is the Range of a Function
2.2 Graphs of Functions
Every precalculus student must know how to graph functions. A very common final exam question asks a student to sketch a graph of function given by a formula. There are probably five or more questions like this on each final exam? You need to know how to do it. The most important and most basic way to graph a function is to first make a table of values like Example 1 "Graphing Functions by Plotting Points". You should know this example well. Unfortunately humans are not so good at making tables; computers are better. Humans make simple errors and are slow. Later (in section 2.6) we will learn a faster way to graph functions for humans.
Yet the faster way is not necessarily a better way. The faster way relies on recognizing the formula from one of our families of functions. Although one can often recognize the family and shape of the basic problems on precalculus exams, it is rarely possible to do this when graphing in the real world. In those cases we must make a table and plot points. Though it is slow and error prone, making a table and then plotting points always works. Eventually when you get to a science or engineering lab you will have a computer sketch graphs for you by plotting points. This may ultimately be the best method? Nevertheless it is a useless method until you have sketched many graphs by plotting points yourself, without a computer. If you do not put in the work and sketch many graphs on your own, you will never know what the computer is doing when it makes a sketch for you.
You should skip the formal definition of a graph with set builder notation on page 159. Instead work through EX 1 on page 160. You should skip the subsection "Graphing Functions with a Graphing Calculator" and Examples 2 and 3. You should spend some time to understand Examples 4 and 5 on p. 162. These are important graphs that you should understand.
The best definition of a function in precalculus is this: Functions have graphs that pass the vertical line test. You can better understand the vertical line test by doing Examples 8 and 9. Finally you should memorize the shapes (except fo the Greatest Integer Function) on p. 166. In order to graph many precalculus functions fast (without making a big table) you must memorize basic shapes.
Skip pages 166-67 on relations.
You should memorize the formula with shapes table on p. 168.
Textbook Examples:1, 2, 4, 5, 7-9
Textbook Exercises: 2-32, 37-46, 49-54
Worked Example: Graphing Piecewise Functions Video
2.3 Getting Information From the Graph
This is an easy section. There is a lot of basic information you can get from studying the graph of a function like the domain and range of the graph, where the function is increasing and decreasing, maximum and minimum values of a function. You should be able to figure out most of these ideas on your own. They follow from common sense.
In calculus you will learn how to find these same values without looking at the graph of the function. This is harder. Calculus is harder. You will have to wait until you take calculus before you understand how to find when a function is increasing and decreasing without looking at its graph. You may skip the subsection "Comparing Function Values: Solving Equations and Inequalities Graphically" and Examples 3 and 4.
Textbook Examples: 1, 2, 5-9
Textbook Exercises: 1-22, 37-40, 51-54
Worked Example: Domain and Range from Graph Video
Domain and Range from Graph Exercises
Intro to Minimum and Maximum Points Video
Worked Example: Absolute and Relative Extrema Video
Absolute Maxima and Minima Exercises
Worked Example: Domain and Range From Graph Video
Domain and Range from Graph Exercises
2.4 Average Rate of Change
The two fundamental ideas in calculus, the derivative and the integral, are respectively related to the average rate of change and the total change. We will not study the derivative and integral in this class but you should know how to compute (1) the average rate of change and (2) the total change of a function. If you already know these concepts, learning calculus will be easier.
Textbook Examples: 1-4
Textbook Exercises: 1-26
Introduction to Average Rate of Change Video
Worked Example: Average Rate of Change from Graph Video
Worked Example: Average Rate of Change from Table Video
Average Rate of Change: Graphs and Tables Exercises
2.6 Transformation of Functions
Here finally we meet a faster way to sketch some functions. Rather than making a table, plotting points, and then connecting the points, we will use geometric transformations to graph functions whenever possibles. When first learning the transformation method to sketch graphs it seems wildly improbable that the method is useful. Yet it is helpful in many problems in this course and calculus. Many basic graphs that show up on precalculus exams can be sketched much faster using the transformation method than by making a table. Indeed if you continue to sketch graphs by making tables, you will most likely run out of time while working on an exam. Moreover, in order to understand the theory in future courses where basic functions are used to approximate more complicated functions, understanding transformation methods is necessary.
The transformation method to sketch graphs boils down to this: Memorize some basic graph shapes (like the shapes on p.166) and then learn how to shift, stretch, and reflect these graphs into new graphs. This is a clever way to quickly expand your ability to sketch basic shapes. All you need to do is memorize a few basic shapes and then learn how to transform them. Since humans have limited brain capacity and limited computational abilities but have strong geometrical intuition, the transformation method is a useful. It only requires brain space to memorize a few shapes and then learn how to transform these shapes. This method is a little like the way you learned to multiply numbers in 4th grade. You first memorized how to multiply numbers up to twelve and then learned the multiplication algorithm to multiply any two numbers.
The trouble with the transformation method of graphing, unlike the multiplication algorithm, is that the transformation method often does not work (however it does often work on precalculus and calculus exams). The graphs that show up in many science and engineering problems are usually too complicated for the transformation method to be of much use. In these cases we have a computer make a table and plot points. Unlike humans, computers can make tables quickly and then flawlessly plot the points. Yet even after you have a computer sketch for you, you will still use the transformation method. The transformation method is helpful theoretically and also to visualize basic waves and vibrations that arise when you study differential equations.
You may skip "Even and Odd" functions
Textbook Examples: 1-7
Textbook Exercises: 1-72
Shifting Functions Examples Video
Graphing Shifted Functions Video
Scaling Functions Introduction Video
Scaling Functions Vertically: Examples Video
Scale Functions Vertically Exercises
2.7 Composing Functions
You should be able to go through this section quickly. Do not spend much time on "Sums, Differences, Products, and Quotients" of functions. Instead spend some time doing some problems related to composition of functions like Examples 3 and 4, which is review material from high school math. You may skip Example 5 "A Composition of Three Functions", Example 6 "Recognizing Composition of Functions", and Example 7 "An Application".
Textbook Examples: 3-6
Textbook Exercises: 27-72
Intro to Composing Functions Video
Evaluating Composite Functions Video
Evaluate Composite Functions Exercises
Evaluating Composite Functions: Using Tables Video
Evaluating Composite Functions: Using Graphs Video
Evaluate Composite Functions: Graphs and Tables Exercises
2.8 Inverse Functions
This is an important section. You will have to spend a lot of time on it. Learning about inverse functions is a great way to expand you library of functions. After this section you will "almost" double the number of functions that you know. For each function f(x) you know you will get an inverse function, more or less. Since one of the goals of precalculus is to learn as many functions as possible, learning about inverse functions is helpful. In fact, even if you do not want to double your library of functions, you still need to learn about inverse functions. Inverse functions arise naturally in applications, even students who do not know much math will ask about inverse functions without knowing it.
For example, the simple interest equation is I(t) = Prt when I(t) is the interest you receive for investing a principal amount P with rate r for time t. So if you invest $1000 at a 5% rate for 3 years, the interest you receive is I(3) = (1000)(.05)(3) = $150. Great you make $150 after three years. This is how we have though of functions. Simply plug in the domain value t=3 and out comes the range value I(3) = $150. But what if this is not the way you want to think of the relationship between I and t. What if you want I to be the domain and t to be the range? Suppose you do not ask "how much interest will I make after 3 years?" but ask instead "after how much time will I make $95 in interest?" Both questions are totally natural, even to someone who knows nothing about inverse functions. But the second question is an inverse question to the function I(t). In the second question I = $95 is the domain and t is the range. Inverse questions arise naturally in science and engineering all the time. You must learn about inverses.
In this course we have four ways to think about functions: 1. Verbally, 2. as a Formula, 3. as a Table, and 4. as a Graph. You should be able to think of inverse functions in these four ways too. I just explained how to think of inverse functions verbally in the last paragraph. It is usually impossible to find a formula for an inverse function but there is a method you will learn in this section that sometimes works---and it shows up on final exams. Finding a table for an inverse is easy: Just switch the x (domain) column of your table with the y (range) column. That is all there is to it. The graph of an inverse function is the mirror reflection of the function over the line y = x.
There is one slight problem with all this. We know that the graph of a function passes the vertical line test. This is our definition of a function. ALL functions must pass the vertical line test. It sometimes happens that after mirror reflecting a graph about the line y = x the new shape does not pass the vertical line test. Then the function does not have an inverse. All functions must pass the vertical line test. So we want to know when an inverse will pass the vertical line test and become an inverse function. The answer is that only functions that are "one to one" have an inverse function. A "one to one" function is a function that passes the horizontal line test. Only one to one functions have inverse functions.
This is almost the end of the inverse story. It may happen that the function y = f(x) you are studying does not pass the horizontal line test (it is not "one to one") so it does not have an inverse but you still need the inverse. In that case you must restrict the domain of y = f(x) so that it passes the horizontal line test. Then it will have an inverse. Restricting the domain is a messy part of the subject but you must learn it.
Textbook Examples: 1-10
Textbook Exercises: 7-74
Finding Inverses of Rational Functions Video
Finding Inverses of Rational Functions Exercises
Restrict Domains of Functions to Make Them Invertible Exercises
Reading Inverse Values from a Graph Video
Reading Inverse Values from a Table Video
Inverse Functions: Graphs and Tables Exercises
Using Specific Values to Test for Inverses Video
Verifying Inverses by Composition Video
Verify Inverse Functions by Composition: Not Inverse Video
Verify Inverse Functions Exercises
3.1 Quadratic Functions
All students in this course should be able to recognize a quadratic function y = f(x) and how to graph a quadratic function. The graph of a quadratic is a parabola. Some parabolas point up; some point down. Some parabolas have two x-intercepts; some have one x-intercept, some have zero x-intercepts. You can find the quadratics x-intercepts by solving the quadratic equation f(x) = 0 by factoring or using the quadratic formula. One way to sketch the graph of a quadratic function is to use completing the square to put the quadratic into standard form. If you hate completing the square you can just memorize the vertex formula x = -b/(2a) and then make a small table by choosing one x-value greater than and one x-value less than the vertex.
You may skip the subsection on "Modeling with Quadratic Functions" and Examples 5 and 6.
Textbook Examples: 1-8
Textbook Exercises: 1-44, 51-44
Vertex Form Introduction Video
Graphing Quadratics: Vertex Form Video
Graph Quadratics in Vertex Form Exercises
Finding the Vertex of a Parabola in Standard Form Video
Graphing Quadratics: Standard Form Video
Graphing Quadratics in Standard Form Exercises
Comparing Maximum Points of Quadratic Functions Video
3.2 Polynomial Functions and Their Graphs
The graphs of a degree one polynomial is a line. The graph of a degree two polynomial (a quadratic function) is a parabola. You learned to graph lines in section 1.10 and parabolas in section 3.1. In this section you will learn to graph some polynomials of higher degree. The method we will learn is to factor the polynomial is to factor the polynomial to find the x-intercepts and then to choose test values to make a sign table. After you have the sign table and x-intercepts you can make a rough sketch of the graph. You should know that this method rarely works because higher degree polynomials are mostly impossible to factor. Yet it does work on many final exam questions and you should learn the method. Besides mastering this technique is a good way to appreciate the intimate relationship between algebra (factoring) and geometry (graphing intercepts). Finally you should understand the end behavior of a higher degree polynomial and how it relates to its degree being even or odd.
You should understand Examples 4, 5, 6, and 8. You may skip the other EXAMPLES in the section. You may skip the "Intermediate Value Theorem for Polynomials" and subsection "Local Maxima and Local Minima of Polynomials".
Textbook Examples: 1-8
Textbook Exercises: 1-44, 51-44
Graphing Quadratics in Factored Form Video
Graphing Quadratics in Factored Form Exercises
Zeros of Polynomials: Plotting Zeros Video
Zero of Polynomials: Matching Equation to Zeros Video
Zeros of Polynomials: Matching Equation to Graph Video
Zeros of Polynomials Factored Form Exercises
Zeros of Polynomials (with Factoring): Common Factor Video
Positive and Negative Intervals of Polynomials Video
Positive and Negative Intervals of Polynomials Exercises
Multiplicity of Zeros of Polynomials Video
Zeros of Polynomials (Multiplicity) Video
Zeros of Polynomials (Multiplicity) Exercises
Intro to End Behavior of Polynomials Video
End Behavior of Polynomials Exercises
4.1 Exponential Functions
We have now studied polynomial, algebraic, and rational families of functions. In Chapter 4 you will learn about a new and important family of functions: the exponential functions and their inverse functions, the log functions. You absolutely need to understand this family well. These functions arise frequently as solutions to basic science and engineering problems. Since you do not know how to graph a basic exponential function, you should make a table to graph your first couple exponential functions. Making a table of a basic exponential graph, y = a^x, will illustrate the concept of a horizontal asymptote.
A table is the foundation of any graph. Once you graph one exponential function like f(x) = 2^x and then g(x) = (1/3)^x, you do not want to continue to make tables for every exponential function that you need to graph. You should then transform the basic exponential graphs as you transformed basic graphs in section 2.6 so you do not have to make many tables. You will be able to graph faster this way.
The most important fact about the graph of a basic exponential function is that it has the x-axis as a horizontal asymptote. The x-axis is not part of the graph of y = 2^x. It serves as a backbone of the graph. All exponential graphs will have a horizontal asymptote. You cannot graph an exponential functions without finding its horizontal asymptote. The graph of each basic exponential function y = a^x passes through the point (0,1) and has the x-axis as a horizontal asymptote. Do not forget this. It then becomes easy to sketch the graph of transformed exponential functions like y = 2^x + 5. The first thing you do is move the horizontal asymptote, the x-axis, up 5 units. So the horizontal asymptote of y = 2^x + 5 is then the horizontal line y = 5.
You may skip the subsection on "Compound Interest" as well as Examples 5, 6, and 7. You will not be tested on this material.
Textbook Examples: 1-4
Textbook Exercises: 1-44
Transforming Exponential Graphs Video
Transforming Exponential Graphs (example 2) Video
Graphing Exponential Functions Video
Graphs of Exponential Functions Exercises
4.2 The Natural Exponential Function
This is a quick section. Do not spend much time on it, 20 minutes tops. If you understand sections 4.1, you will understand this section. There is one exponential function that is more important than all the others in calculus. It is called "the natural exponential function" and is written f(x) = e^x or as f(x) = exp(x). The base e is ugly; e is irrational. It is approximately 2.71. However using the natural exponential function has many advantages. For instance if you insist on using the exponential function g(x) = 2^x instead of the natural exponential function, you will have ugly constants in many important calculus formulas. Calculus will then be more difficult to learn. It is better to learn and use the natural exponential function.
You only need to remember one basic exponential function f(x) = e^x and its graph in future courses. You can get all the other exponential graphs from it by the method of transformations in section 2.6. Learn the natural exponential function and its graph. Your future self studying calculus will be grateful.
You may skip reading this section and doing any of its included EXAMPLES (too much calculator problems). Do the assigned webassign problems instead. You should be able to do them quickly using your knowledge from section 4.1.
Textbook Examples: 2 only.
Textbook Exercises: 3-16
4.3 Log Functions
In this section you will utilize the full power of the inverse method you studied in section 2.8. Until this section you have not gained much by learning the inverse method. Each inverse function that you found so far has been a function that we already knew, an known algebraic function or a rational function. One of the goals of precalculus is to learn as many functions as possible. The inverse method is most powerful when it gives you new functions, functions that you did not already know. We then expand our library of functions. The inverse method explained in section 2.8 only works to your advantage if you memorize little to nothing about the inverse function. All the information you get from the inverse function should derive from section 2.8. The brain space you will save by not memorizing specific facts about an inverse function can be used to learn something else, maybe more math...
Ask yourself this: Why do we memorize a multiplication table in grade school but do not memorize a division table? The answer is that division is the inverse of multiplication. It is thus unnecessary to memorize a division table. Any division problem like 35/7 and converted to a multiplication problem 7x = 35. Do not waste your brain cells memorizing a division table.
Exponential functions are one-to-one (they pass the horizontal line test) so each exponential function has an inverse. The inverse exponential function is so important that it is given a special name, the log function. If you want to graph or make a table of a log function, first sketch or make a table of the corresponding exp function. If you need to compute the value of a log function, first convert to exponential form (this is how you first learned to divide---you converted division into multiplication).
When graphing a log function you should know that each log function has one vertical asymptote and no horizontal asymptote. You should find the vertical asymptote and trace it (with a different color pen) before tracing the log graph. In order to find the vertical asymptote of a log graph use the transformation method of section 2.6 and the fact that all basic log functions have the y-axis as the vertical asymptote.
Textbook Examples: 1-6, 9, 10.
Textbook Exercises: 1-78
Evaluating Natural Logarithm with Calculator Video
Evaluating Logarithms (Advanced) Video
Evaluating Logarithms (Advanced) Exercises
Relationship between Exponentials and Logarithms Video
Graphical Relationships between 2^x and Log_2(x) Video
Graphing Logarithmic Functions (Example 2) Video
Graphing Logarithmic Functions (Example 1) Video
Graphs of Logarithmic Functions Exercises
4.4 Laws of Logs
I know that I said last section that, in order to save brain space, you should not learn anything about log functions but instead convert all things log into their corresponding exponential things. I lied. You should memorize the laws of logs. You should know that the laws of logs are nothing more than the laws of exponents converted into logs (see the proof in our textbook). However in this case you should not convert the laws of exponents yourself on an exam. Memorize the laws of logs instead. It is safer that way.
This is a quick section. Do not spend much time on it. You will see the power of the laws of logs in the next section when you will solve exponential and log equations. ONLY go over Examples 1, 2, 3. Skip the others.
Textbook Examples: 1-3
Textbook Exercises: 1-5, 7-58
Intro to Logarithm Properties Video
Intro to Logarithm Properties (2 of 2) Video
Use the Properties of Logarithms Exercises
4.5 Exp and Log Equations
In this section we add two new tools to our equation solving toolkit. We can now 1. take log (using the same base) of both sides of an equation and 2. take exp (using the same base) of both sides of an equation. This is no different from middle school when you learned how to solve equations by adding the same number to both sides of an equation or by dividing both sides of an equation by the same number. The key is that you MUST do the same thing to both sides of an equation.
This all seems too easy---its just middle school all over again? This is not true. There are two tricky details you must practice: 1. Every time you learn a new equation solving technique you must mix the technique in with all the other techniques in the correct order (PEMDAS...) and 2. There is a domain issues with log (you may end up with false roots). In order to become comfortable with doing things in the correct order you should solve many equations in this section. In order to not have false roots, you should test your solutions to log equations to make sure they are in the domain.
You should skip Examples 4, 5, 6, 10 (graphical part), 11, 12, 14. You do not need to know how to solve geometrically or how to use a calculator to approximate an exact solution.
Textbook Examples: 1-3, 7-9, 10 (algebraic solution), 13
Textbook Exercises: 1-36, 47-66, 87-90
Solving Exponential Equations Using Exponent Properties Video
Solving Exponential Equations Using Exponent Properties Exercises
Solving Exponential Equations Using Exponents Properties (Advanced) Video
Solving Exponential Equations Using Exponent Properties (Advanced) Exercises
Solving Exponential Equations Using Logarithms base 10 Video
Solving Exponential Equations Using Logarithms base-10 and base-e Exercises
Solving Exponential Equations Using Logarithms base-2 Video
4.6 Modeling With Exponential Functions
In section 2.1 when you first learned function you learned to think of functions in four ways 1. verbally, 2. as a table, 3. graphically, and 4. as a formula. Each way to view a function has its advantages and disadvantages. When you apply the math you've learned in your science and engineering classes you will often start with a verbal description of the function. In this class we have focused mostly on tables, graphs and formulas. In this section however we will focus on the verbal description of functions. It is challenging. You must read and then translate what you've read into a formula. This is what scientists and engineers do.
You should ONLY read the subsection "Exponential Growth (Relative Growth Rate)" and ONLY work out EX3. You ONLY need to memorize (and understand how to use) the exponential growth (relative growth rate) model in the blue box on p.373.
Textbook Examples: 1-3, 7
Textbook Exercises: 11-16, 23-25
Constructing Exponential Models: Half Life Video
Exponential Model Word Problem: Medication Dissolve Video
Exponential Model Word Problem: Bacteria Growth
CHAPTER 6: Trig Functions: A Right Triangle Approach
In this chapter you begin your study of your final family of functions, the trig functions that arise in studying periodic (repetitive) behavior. Of course this is all routine by now (we do the same things every time we learn a new family of functions): 1. make a table to graph a couple (sin, cos) basic trig graphs, 2. then use transformations from section 2.6 to realize many more trig graphs, 3. finally study inverse trig functions. This would be fast if only it was easy to make a table for basic trig functions. Unfortunately it is not. In order to compute a table of trig values, you need to do geometry (luckily only basic high school geometry of triangles and circles is needed). So far in this class when you've made a function table you've substituted some number for x in f(x) and then made some computations. In trig tables you substitute some number for the Greek letter "theta" (our text uses "theta" for the independent variable in chapter 6) and then draw a triangle. It takes some getting used to.
And you should know that we will only make a table this semester. We are running out of time. You will graph trig functions and study their inverse functions next semester in math 195.
6.1 Angle Measure
In this section you will learn how to measure an angle using two measurements: 1. degrees and 2. radians. Even though you probably have already studied degree measure in high school, it turns out that radian measure is more useful. All your trigonometric formulas in calculus will be easier if you measure angles in radians. If you insist on measuring angles with degrees in calculus, your formulas trig formulas will all have ugly constants in them. All calculus students measure in radians rather than degrees. It is easier that way. Your future self studying calculus will appreciate it if you now learn to use radians. One way to begin to appreciate the power of measuring in radians is to look at the formulas in the blue boxes for the "Length of a Circular Arc" and "Area of a Circular Sector". These formulas are easy to memorize if you measure in radians (you must memorize them). They are not so easy to memorize using degrees.
You should spend some time learning about the "Standard Position" of an angle and the complications that arise with "Coterminal Angles". You should understand how to convert from degrees to radians and from radians to degrees. You should study Examples 1, 2, 3, 4. You should skip subsection "Circular Motion" and Examples 5, 6, 7.
Textbook Examples: 1-4
Textbook Exercises: 1-2, 4-66
Radian Angles and Quadrants Video
Arc Length from Subtended Angle Video
Subtended Angle from Arc Length Video
Arc Length from Subtended Angle: Radians Video
Radians and Arc Length Exercises
6.2 Trig of Right Triangles
You should memorize ONLY the trig ratios for sin t = opp/hyp, cos t = adj/hyp, and tan t = opp/adj. You can get the other three trig ratios taking the reciprocals of these three.
You should practice the basic high school geometry of triangles used in getting the "Special Ratios" from the "Special Triangles" at home. You should not mess around with these triangles on an exam. Just memorize the table of special values of sine and cosine (you can get the other four trig "special ratios" from these two). Some students like to write this table on the back of their exam once the get it so they do not forget anything.
One more thing: It is rare that we can find numbers we already know as sin(t) and cos(t). The special ratios are really special. Most theta-values that we plug in give new irrational numbers that we do not already know. For instance if someone asks you to compute the sine and cosine determined by theta = 2. The best answer is the exact answer the point cos(2) and sin(2. You could plug these into your calculator but this will only give an approximate answer. The best answer is the exact answer cos(2) and sin(2).
All included Examples should be understood without a calculator.
Textbook Examples: 1, 2, 4-6.
Textbook Exercises: 5-12, 17-46, 55-60.
Use a Trig Equation to Determine the Length of a Side of a Right Triangle Video
Find the Reach of a Ladder Video
Triangle Similarity and the Trigonometric Ratios Video
Trigonometric Ratios in Right Triangles Video
Trigonometric Ratios in Right Triangles Exercises
Solving for a Side in Right Triangles with Trigonometry Video
Solve for a Side in Right Triangles Exercises
CHAPTER 5 Trig Functions: The Unit Circle Approach
In this section you begin your study of your final family of functions, the trig functions that arise in studying periodic (repetitive) behavior. Of course this is all routine by now (we do the same things every time we learn a new family of functions): 1. make a table to graph a couple (sin, cos) basic trig graphs, 2. then use transformations from section 2.6 to realize many more trig graphs, 3. finally study inverse trig functions. This would be fast if only it was easy to make a table for basic trig functions. Unfortunately it is not. In order to compute a table of trig values, you need to do geometry (luckily only basic high school geometry of triangles and circles is needed). So far in this class when you've made a function table you've substituted some number for x in f(x) and then made some computations. In trig tables you substitute some number for t (our text uses "t" for the independent variable in chapter 5) and then draw a picture. It takes some getting used to.
One more thing. There are two ways to introduce trigonometry: 1. Unit Circle Trig (Chapter 5), and 2. Right Triangle Trig (Chapter 6). They are the same. Anything you learn in Chapter 5 you can also learn in Chapter 6 and vise versa. So you may say, why bother learning the same thing two ways? The answer is that the unit circle and the right triangle approach trig from different viewpoints. It is helpful to know both viewpoints, especially the unit circle approach using radian. In fact you always want to use radians from here on out; radians are a more natural measure. Your future self studying calculus will be thankful. The calculus-trig formulas using radians are simpler. The calculus-trig formulas using degrees contain ugly constants.
Many of you have already studied right triangle trig in high school. We will not cover Chapter 6 in this course (besides a quick review of degrees). Knowing right triangle trig and degree measure will be helpful to you as you learn Chapter 5 but make sure you train yourself to think in terms of the unit circle and radians. Calculus will be much easier to understand that way.
5.1 The Unit Circle
In this section you need to learn the vocabulary of finding a terminal point on the unit circle determined by the real number t using "The Reference Number" and "Table 1" from page 404. In essence making trig tables are challenging. It is better to just memorize Table 1 and then use reference numbers to expand this table. That is what everyone does. However you should know that the table is not so complicated. It can be found using the basics of right triangles. You should try to find the table for yourself sometime. Just don't try to find the table during an exam. It is easy to get stuck. You should memorize the table for exams.
Textbook Examples: All
Textbook Exercises: 1-60
Proof of the Pythagorean theorem Video
Reference Angle for an Angle (Using Degrees)
Reference Angle for an Angle (Using Radians)
5.2 Trigonometric Functions of Real Numbers
This section is just a repetition of the last section with some new vocabulary. In section 5.1 you learned how to turn a real number into a Terminal Point, (x,y). In this section you use the components Terminal Point (x, y) in order to learn two new functions (one function for the x-component, one for the y-component) like so, cos(t) = x and sin(t) = y. To recap the cosine function works by 1. plugging in t, 2. doing geometry to find the Terminal Point determined by t, and 3. take the first component of the Terminal Point to find cos(t). The same steps work for sin(t) but you take the second component, the y-component.
I know. I know. You do not like these unit circle formulas for sine and cosine. You prefer and are used to seeing sin(t) = opp/hyp and cos(t) = adj/hyp. That is right triangle trig and it is taught in Chapter 6. We are learning unit circle trig here; it will turn out to be more useful to you in the future. In any case, if you truly miss right triangle trig, study p. 411. You will see that unit circle trig and right triangle trig are the same. It will be easier for you to use the unit circle approach as much as possible.
One more thing: It is rare that we can find numbers we already know as Terminal Points. The same goes for sine and cosine. Most t-values that we plug in give new irrational numbers. For instance if someone asks you to compute the Terminal Point determined by t = 2. The best answer is the exact answer the point (cos(2), sin(2)). You could plug these into your calculator but this will only give an approximate answer. The best answer is the exact answer (cos(2), sin(2)).
Do not waste brain space memorizing tables for tan, cot, csc, and sec. Just memorize tables for sine and cosine ONLY. You can find all the other tables from those. You should learn the even/odd properties for sine and cosine ONLY. You can get all other even/odd properties from those two. You should memorize all the fundamental identities from the blue box on p. 415. You should understand ALL the EXAMPLES from this section except EX3 using a calculator.
Textbook Examples: 1, 2, 4, 5.
Recommended Textbook Exercises: 5-38, 47-54, 67-74
Evaluating Trig Functions Using Reference Angles (Degrees) Video
Evaluating Trig Functions Using Reference Angles (Radians) Video
Cosine, Sine, and Tangent of pi/6 and pi/3 Video
6.3 Trig Functions and Angles
Very similar to section 5.2 and should be studied simultaneously with that section.
In this section you will learn how to leverage the table of "special" trig angles you found using basic geometry and then memorized in section 6.2. This method allows you to determine many many more trig values then the trig values for the ''special'' angles. It is best to follow our textbook's method in this case:
Find the reference angle associated to the given angle.
Hope that the reference angle matches with one of the "special angles". If the reference angle does indeed match with one of the ''special'' angles then the answer will be the almost the same ratio as the matching ''special'' angle.
Determine if you will use + or - the answer from 2. by finding the quadrant in which the original angle lies. You need not memorize the formula for the area of a triangle.
Skip "Areas of Triangles" and Example 8.
Textbook Exercises: 1-54
Trig Values of Special Angles Exercises
5.3 Trig Graphs
After all the hard geometric work to make tables for sine and cosine, this section now follows easily from section 2.6 "Transformations": 1. Use the sine and cosine tables to sketch the basic graphs y = sin(x) and y = cos(x), 2. Use transformations to get many trig graphs related to sine and cosine. Indeed it is even easier than that. You should not even use all the values from your sine and cosine tables from section 5.2. Use only the multiples of pi/2 (90 degrees), i.e 0, pi/2, -pi/2, pi, -pi, 3pi/2, -3pi/2, .... (or 0, 90 degrees, -90 degrees, 180 degrees, -180 degrees, 270 degrees, -270 degrees,....). You do not need to use the challenging t-values pi/6, pi/4, pi/3, ... (30 degrees, 45 degrees, 60 degrees, ....). You could plot these points but they would only "muck-up" your graph. Just plot all multiples of pi/2 (90 degrees). You will then get all the local (in this case also absolute) maximums, minimums, and intercepts of the basic graphs y = sin(t) and y = cos(t). Your basic graphs will look best this way.
As you work through this section you will understand the periodicity (repetition of a pattern) in the trig graphs. You need not plot all multiples of pi/2 (90 degrees), just enough to recognize the pattern. One of the goals of graphing trig graphs on a 195 final exam is to recognize the pattern as quickly as possible.
In this section you will have to come to grips with the "Horizontal Stretch/Compress" transformation like y = sin(2x). Up until now there were always methods to avoid grappling with this challenging transformation. Now there is no hiding. You need to learn it. What is worse is this new horizontal transformation can interfere with the left/right transformation you already know. If you do them in the wrong order, you will mess up the problem. You must follow PEMDAS. Probably the easiest way to deal with this tricky transformation is to always use the form in the blue box on p.424 and then study Examples 4 and 5.
Examples 1, 2, 3, 4, and 5 are all good. You should study them carefully. You may skip subsection "Using Graphing Devices.." and all its EXAMPLES 6, 7, 8, 9, 10.
By the way, you may have noticed that we really did not need to memorize the shape of both sine and cosine. All we need to memorize is one of them and then use a horizontal shift to get the other. If you follow the guide of section 2.6 on "Transformations", this should be done. In this section we learned to memorize as few shapes as possible and then use transformations to get other shapes. Then use the brain space you saved memorizing basic shapes to learn more math. Don't follow this advice in this case. Everyone memorizes both the shape of y = sin(x) and y = cos(x). You should too.
Textbook Examples: 1-5
Textbook Exercises:4-44
Animation: Graphing the Sine Function
Animation: Graphing the Cosine Function
Amplitude and Period of Sine and Cosine
Horizontal and Vertical Translations of Sine and Cosine
Intersection Points of y = sin(x) and y = cos(x) Video
Example: Graphing y = 3sin(1/2x) - 2 Video
Example: Graphing y = -cos(pi x) + 1.5 Video
Amplitude and Period of Sinusoidal Functions from Equations Video
Transforming Sinusoidal Graphs: Vertical Stretch + Horizontal Reflection Video
Transforming Sinusoidal Graphs: Vertical and Horizontal Stretches Video
Amplitude of Sinusoidal Functions from Equation Exercises
Period of Sinusoidal Functions from Equations Exercises
5.5 Inverse Trig Functions
Like always after we graph some basic graphs in a new family of functions we first transform these basic graphs into more interesting graphs of the same family of functions and then second study the inverse functions of the basic graphs. As you can see the graphs of y = sin(x) and y = cos(x) fail the horizontal line test miserably (they are not one-to-one). Yet the inverse sine and inverse cosine functions are incredibly important in mathematics and its applications to science and engineering. You must learn to restrict the domain of both y = sin(x) and y = cos(x) so that they become one-to-one and have an inverse. The one rub in all this is that, unlike in section 2.8 "Inverse Functions" where you were allowed to restrict the domain as you saw fit, when it comes to inverse sine and cosine functions, everyone around the world restricts the domain the same way. For y = sin(x) you should restrict the domain to [pi/2, pi/2] and for y = cos(x) you should restrict the domain to [0,pi]. You should also study inverse tangent. But that is all. You will only be tested on arcsin, arccos, and arctan.
Textbook Examples: 1, 3, 4, 5.
Textbook Exercises: 3-10, 23-48.
Restricting Domains of Functions to Make Them Invertible
Inverse Trig Functions, Part 1 (Basic Introduction) Video
Inverse Trigonometric Functions Part 2 (Evaluating Inverse Trig Functions) Video
7.1 Trig Identities
Often your textbook asks you to memorize too many formulas. This section is no different. There are too many formulas. You should not memorize them all.
The only identities that you really need to know for this section are "The Reciprocal Identities" (blue box page 538) and the single "Pythagorean Identity" sin^2(x) + cos^2(x) = 1 (blue box page 538). You can derive all the other identities in the blue box on page 538 from these. For practice try to derive second "Pythagorean Identity" tan^2(x) + 1 = sec^2(x) from the identity sin^2(x) + cos^2(x) = 1 and by adding fractions. It should be easy. Math majors spend a log of time deriving things. Their arguments in the deriving is called a proof. All math is ultimately based on proof. In this section you will learn to prove some trig identities.
Textbook EXAMPLES 1-7.
Textbook Exercises: 3-26, 33-67
Using Trigonometric Identities Video
Using the Pythagorean Trig Identity Video
Use the Pythagorean Identity Exercises
7.2 Addition and Subtraction Formulas
Like the last section, there are too many formulas to memorize here. Only memorize the formulas 1. sin(-t) = -sin(t), 2. cos(-t) = cos(t), 3. sin(a + b) = sin(a)cos(b) + sin(b)cos(a), and 4. cos(a + b) = cos(a)cos(b) - sin(a)sin(b). You can derive the other formulas from these.
You then should look over Examples 1, 2, and 3. If you are interested in becoming a math major, you should study the proof of the addition formula for cosine (it is not hard). You should skip all examples and content in this section following Example 3.
Textbook Examples: 1, 2, 3, 7
Textbook Exercises: 1-40, 55, 56.
Trig Angle Addition Identities Video
Using the Cosine Angle Addition Identity Video
Using the Cosine Double-Angle Identity Video
Using the Trig Angle Addition Identities Exercises
Finding Trig Values using Angle Addition Identities Video
7.3 Double-Angle, 1/2-Angle, Product-Sum Formulas
You should not memorize the double angle formulas. They are easily derived from the angle sum formulas for sine and cosine which you should have memorized from section 7.2. Don't waste brain space on memorizing things that are so easy to derive yourself. Save the brain space to learn something else.
The 1/2-angle formulas for sine and cosine are not as easily to derive (though not too hard) as the double angle formulas. Most students memorize the 1/2-angle formulas for sign and cosine (you can find them in the blue box on the bottom of page 556). Do not bother memorizing the 1/2-angle formula for tan.
You should skip the subsections "Evaluating Expressions Involving Inverse Trig Functions" and "Product-Sum Formulas" in their entirety. You should only look at Example 1 and Example 5 inside the section.
Textbook Examples: 1, 2, 5
Textbook Exercises: 3-10, 17-29, 37, 38.
7.4 Basic Trig Equations
You should spend some time thinking about how many trig equations have an infinite number of solutions. In order to do this look at EX1, EX2, and EX5. You should skip the other EXAMPLES. You should skip the subsection "Solving Trig Equations by Factoring".
Textbook Examples: 1-5.
Textbook Exercises: 1-36
