We will have class as usual today and it’s a very important lesson, so see you then!

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A good introduction to the binomial theorem is on this page at Math is Fun. They also explain how to construct and use Pascal’s Triangle to get the binomial coefficients. At the end of the page are some practice problems, which are very good to work through.

The page which explains how to use the calculator in the binomial theorem is here at Mathbits.

Here is a video from Math Gives You Power which explains the binomial theorem a bit and then shows key-by-key how to use the TI-84 calculator to get all the binomial coefficients (in an easier way than the Mathbits uses) and then works some examples of finding specific terms in the binomial expansions. (Warning: It’s Youtube, so another video may auto play when this is done.)

There are two video examples from PatrickJMT:

After you’ve viewed those, work through the quiz from Mathopolis (Math is Fun). This will give you practice and help you to learn! (These are the same problems linked at the bottom of the Math is Fun page I mentioned above.) The answers are provided, once you’ve entered your answer, along with complete solutions. Make sure you work through these!

For more practice you can do the following from the textbook:

Exercises 25.4(a-d) Â and 25.5(a-d)

]]>An improved version of the notes I handed out in class can be found here:

MAT1375PolarFormComplexNumbers

Here are some videos from PatrickJMT that may be helpful:

Multiplying and dividing in polar form

Converting from polar form to standard a+bi form

**Homework:**

â€¢ Do the homework on vectors from last time, if you have not already done it.

â€¢ Study the examples I gave in the notes (and view the videos if you find them helpful).

â€¢ Do the following problems in the textbook (which include the problems I put on the board for you to do in class): Exercises 21.5(c, d); 21.6(a, b, c); and 21.7(a, b)

â€¢Â If you want to get a jump ahead on the new topics we will discuss next time, I intend to make use of this web resource on the binomial theoremÂ (from MathBits) which explains very succinctly how to do what we have to do, using your calculator. **Please make sure to bring your calculator with you on Wednesday in any case!**

â€¢ Please look at the Final Exam Review problems and choose some that you will want to solve for us, as time permits. Probably we will not have time to do them all, so I will concentrate on certain topics in class. Any of these problems that are not solved in class will be posted to Piazza as questions and you may post your solutions there.

My plan is to post video and other links for each type of problem on the Final Exam Review. Problems we will certainly discuss on Wednesday are #4, 8, and 9. Here is the information I posted last time:Â I will ask for volunteers for at least some parts of these.

Review links related to problem 4: please read or view these even if you think you don’t need to!Â Difference quotients from PatrickJMT: Example 1 Example 2

For #8, see the review self-tests for Test 4Â as well as the Test 4 solutions and Example 14.3 in the textbook – make sure that you are using the properties of logarithms!

For #9, see this post

Other links posted soon!

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

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**Topics:**

â€¢ Filling out the unit circle important points in the first quadrant (for use in solving trig equations today)

You can practice this using a copy of the handout:Â Â Unit circle for practice. You should be able to fill out all the important values along with their angles in less than 5 minutes. It is also useful to put the values of tangent for each of the important angles outside the circle (near the point on the circle).

â€¢ Solving the basic trig equations , , , (continued): see the summary on p. 277

â€¢ Solving more complicated trig equations by breaking them down into one or more of the basic trig equations.

We discussed Examples 20.11(a) and 20.12(a,b), as well as the solution to Exercise 20.4(h)

â€¢ Vectors: finding the magnitude and direction angle.

The handout I gave out in class is here:Â MAT1375VectorsMagnitudeDirectionAngle

We worked through all but the last two examples. You will finish those for homework. (They are also done in the textbook.)

Always make sure that you draw the vector in the coordinate plane before doing anything else. Sometimes it is obvious what the magnitude or the direction angle is without doing any computations! And you always need to know which quadrant your vector lies in (or along which axis).

**Homework:**

â€¢ Review the examples discussed in class and make sure that you understand how we are proceeding!

â€¢ Practice filling out the unit circle from memory and check your results. You need to be able to do this very quickly and accurately.

â€¢ There is no good WeBWorK for these topics, alas. So you will do the following problems from the textbook, and I would like to see some of them on the board next time:

Exercise 20.4(all parts, but especially f, g, i, and k)

Exercise 22.2(a-1): also finish the last two examples on the handout

â€¢ Also I would like to see some ofÂ Exercise 17.6(a-k) (previously assigned, sine and cosine graphs) on the board, with explanations of how you found the five important points. Please see my new post which includes an improved version of my notes and a video (slideshow with audio) giving two worked-out examples.

â€¢ Please take a look at the Final Exam Review sheet which is here

I am breaking the review into two parts. For Monday please prepare the following problems: #4, 8, 9. I will ask for volunteers for at least some parts of these.

Review links related to problem 4: please read or view these even if you think you don’t need to!

Difference quotients from PatrickJMT: Example 1 Example 2

For #8, see the review self-tests for Test 4 and Example 14.3 in the textbook – make sure that you are using the properties of logarithms!

For #9, see this post

â€¢ My plan for next class new topics is to do polar form of complex numbers. You may wish to view these videos from PatrickJMT to see a bit of where we are going:

What we are doing is basically looking at complex numbers as vectors in the plane!

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

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The five important points are the x-intercepts and the maximum and minimum points. You should know how to locate them and give their coordinates over one basic period of your function. (If you draw more than one period, you are telling us that you don’t know what “period” means!)

We look at functions whose equations have the form orÂ

The procedure for finding the x-coordinates is the same for all the graphs. The only thing that is different is what the y-coordinate will be: will it be 0 or a maximum or minimum? That depends on whether your function is a sine or a cosine, and also on whether the coefficient **a** is positive or negative. **So the first step is to figure out which basic graph shape your graph should have.**

The basic period of a sine function looks like this: The five important points have been marked off in purple dots. Notice that the basic period starts and ends on the x-axis.

Sine with a>0:

Sine with a<0:

The basic period of a cosine function looks like this: The five important points have been marked off in purple dots. Notice that the basic period starts and ends on either a maximum or a minimum, depending on the sign of a.

Cosine with a>0:

Cosine with a<0:

You just have to keep those general shapes in your mind as you proceed.

First order of business is to find the amplitude, period, and phase shift for your function.

Remember that the amplitude is – amplitude is always positive!

The period is

The phase shift is

To find the x-coordinates of the five important points:

â€¢ The first point has x-coordinate equal to the phase shift. Mark this x-coordinate lightly with your pen or pencil (don’t put a dot on it, not yet anyway)

â€¢ The last point on the basic period (which is the second x-coordinate we find) has x-coordinate equal to the phase shift plus the period. Mark off this x-coordinate lightly with your pen or pencil.

[It is a good thing to keep in mind that the total length you will end up using on the x-axis is going to the one period long! So the distance between the first point and the last point has to be one period. See the picture on p. 246 in the textbook, for example.]

â€¢ We will now find the middle x-coordinate. This is halfway between the two ends, so it will be the average of the two x-coordinates we already found. See how I do this in the examples! Mark this lightly – you can do this even before you find it as a number,of course.

â€¢ We now find the two remaining x-coordinates: each one is halfway in between two of the points we have marked off so far. Mark them lightly on the x-axis.

â€¢ Now we will find the actual points that go with those x-coordinates. That will depend on which of the above four basic shapes of graphs we are dealing with. For the points which are on the x-axis, the y-coordinate is 0. For the points which are maximum points, the y coordinate is the amplitude (positive). For the points which are minimum points, the y coordinate is the negative of the amplitude.

â€¢ Now connect the dots with a NICE SMOOTH CURVE (no corners!) to draw your graph. I like to extend the graph a little past the first and last points so that people know the actual graph of the function does not end there.

Here isÂ video slideshow with voiceover showing two examples: These are taken from the textbook, Examples 17.10(a) and 17.10(d). Sorry for the glitch in the second example, which I did not have time to edit.

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The topics are listed in the Self-Tests sheet, and also in the links below. But make sure that you practice, donâ€™t just read or watch videos or watch someone else work problems!

This test is shorter than the previous tests, so it will include a problem of solving a rational inequality (similar to problem 4 in Test 3). You may do that problem or you may skip it as you choose. If you do it and receive a higher score than you did on problem 4 if Test 3, the higher score will be credited to your Test 3 score. (See the practice for Test 3 and the Test 3 solutions posted here.)

Â ThinkingStrategicallyPreTestSurveyÂ – take a moment and complete this as usual!

MAT1375Test4ReviewFall2017Â Review self-tests and topics

MAT1375Test4ReviewAnswersFall2017

I have tried to make sure that there are no errors in these, but please let me know if you find anything!

Here are links to some sources to help you if you get stuck:

An exponential growth problemÂ (video) by PatrickJMT – it’s just one example, but he shows the relationship between the form of the exponential function we have used so far and the form where you are given a rate.

For exponential growth, the base is (1+rate), and for exponential decay the base is (1-rate). Be careful, and make sure that you are using this!

Here is another video with some useful examples.

Please make sure that you are using the method where the basic functions are like in those videos, and NOT with base e, as that other method will give you wrong answers with the kind of rates we are talking about!

The five-point method for drawing graphs of sine and cosine functions: (my old notes)Â Â The five-point method,

Make sure to look at my new post with notes and a link to video!

Look at the graphs in Example 17.10 in the book and pay attention to how they say that they are getting the x-coordinates of the important points. They are basically following this method, but not showing any computations as they find “halfway between” points.

For the logarithm problems, please see Example 14.3 in the textbook. I have not yet found any good sources online for this.

An exponential growth problemÂ (video) by PatrickJMT – it’s just one example, but he shows the relationship between the form of the exponential function we have used so far and the form where you are given a rate.

For exponential growth, the base is (1+rate), and for exponential decay the base is (1-rate).

I have extended the WeBWorK on inverse trig functions so that you can use it for practice. Here are some links to help with that:

Unit circle for practice

A trick to remember values on the unit circle (video)

How to remember all the important points in the unit circle (video)

Here is a youtube video on finding the exact values of the inverse trig functions that may help: he assumes that you know the important values that come from the two important right triangles and the points on the unit circle on the axes.

]]>**Topics:**

â€¢ Discussion of exponential functions in applications. The solutions to the quiz are posted here.

â€¢ How to find the coordinates of the five important points (the x-intercepts and the maximum and minimum points) in one period of the graph of sine or cosine functions, and sketching the graphs. The notes I used are here:Â Â The five-point method but I am working on making a better version.

â€¢ More inverse trig functions: inverse cosine

Please see the notes below for the informal word definitions of the inverse trig functions, which are very useful to say to yourself when you have to find a value of one of them (just as remembering that a logarithm is an exponent is useful).

â€¢ Solving a basic tangent equation to find all the solutions.

We will follow a similar procedure for each of the three basic types of trig equations: first we will do an example in some depth in order to invent a method, and then we will apply the method to solve more examples. It is a very good idea to study the first example of each type and make sure that you understand the method before you go on.

For tangent equations, the first example is Ex. 20.1. We get one solution by using the inverse tangent function, and then all the other solutions are found by repeatedly adding or subtracting the period from that first solution.

**Definitions of the inverse trig functions (in words):**

â€¢ To find an inverse to the tangent function, we had to restrict the domain of tangent to the interval , so that will be the range of the inverse tangent function.

Definition: or is the angle (or rotation) in the intervalÂ whose tangent is x.

Note:Â is an odd function:Â

â€¢ To find an inverse to the sine function,Â we had to restrict the domain of sine to the interval , so that will be the range of the inverse sine function.

Definition: or is the angle (or rotation) in the intervalÂ whose sine is x.

Note:Â is an odd function:Â

â€¢ To find an inverse to the cosine function,Â we had to restrict the domain of cosine to the interval , so that will be the range of the inverse cosine function.

Definition: or is the angle (or rotation) in the intervalÂ whose cosine is x.

Note:Â is neither even nor odd: but it does satisfy the identity

Homework:

â€¢ Review the definitions of the inverse trig functions and how we used them to find exact values. (Session 19.) Note that most of the time we will be working with exact values (not using a calculator) so you need to be able to do this. Review the unit circle if necessary – see the videos from PatrickJMT I linked last time.

There are also useful videos in Khan Academy: introduction to arcsine,Â introduction to arctangent.Â introduction to arccosine.

â€¢ Study Example 20.1 and make sure that you understand the method it creates for solving tangent equations.

â€¢ Do the WeBWorK on inverse trig functions. You are to find exact values in these, so do NOT use your calculator – you will be hurting yourself if you do. Just use your knowledge of the important right triangles and the unit circle picture.

â€¢ Find the coordinates of the five important points and sketch the graph for each of these: Exercise 17.6(a-k): I will request volunteers to put these on the board next time, for extra credit (as usual). Note that there is no WeBWorK for this and it will be on the Final Exam, so make sure you practice it!

â€¢ There will be a quiz next time: the topic will be finding exact values for the inverse trig functions. You will have to explain how you got your answers, either by drawing the unit circle with its important points or by using the important right triangles. Answers given without explanation will receive no credit!

â€¢ Don’t forget that Test 4 is scheduled for next Wednesday. The review self-tests will be in a separate post. This will be a shorter test than usual, so I will add to it a “makeup” problem similar to one that was commonly missed on a previous test. Make sure you look for it. This will give you an opportunity to improve a previous test score.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

]]>â€¢ Quick review of the three definitions of the trig functions – we need to use all three from time to time, but the Unit Circle definitions are the most important!

â€¢ Important facts that follow from the definitions: sine is an odd function, cosine is an even function, and they satisfy the Pythgorean identity &latex \sin^{2}(x) + \cos^{2}(x) = 1$.

â€¢ Thinking in radians:

The main thing when thinking in radians is not to keep translating into degrees (even in your head), but think about what the radian measure angles look like. One full rotation around the circle is ; half of that, a straight-line angle, is ; half of the straight-line angle is a right angle, which is .

To get the other important angles:

We can take half of a right angle, which will be . This is the acute angle in the isosceles right triangle, because the two acute angles in a right triangle must add up to a right angle. (All 3 angles must add up to a straight line: theorem of geometry!)

We can take a third of a straight line angle, and that will be the angle of the equilateral triangle, which is (Again, all three angles must add up to a straight line.) When we cut the equilateral triangle in half to make a right triangle, the smallest angle must be half of , namely it must be .

Visualize these angles as you go through this and think about their sizes relative to each other: you don’t have to think of degrees at all! The relative sizes are really very important also and associating them with the radian measure will help you become very fluent in radian language.

â€¢ The two important right triangles: in order to avoid thinking in degrees, we will call them by their more proper names: The equilateral right triangle, and the half-equilateral (or hemieq) right triangle. They are shown in this page

Here is a description of how to learn these triangles as I did in class (more or less), good although the drawings are a bit sloppy.

â€¢ To learn all the values of the trig functions at the important points on the unit circle: there is a pictureÂ here: You can find the blank unit circle to print out and use for practice here:Â Unit circle for practice

There are patterns in the coordinates of those points, and those may help you to remember them. Here is are some videos from Patrick’s Just Math Tutorials which show how this is done:

A trick to remember values on the unit circle

How to remember all the important points in the unit circle

â€¢ Basic graphs of sine, cosine, and tangent. (Was done last time with the sub)

When we consider these functions as functions and look at their graphs, we are always using radians as the input. It is best to think of the input as being a rotation on the unit circle, rather than as an angle. There is a geogebra resource I like to use for the basic graphs:Â Using the unit circle to generate the graphs.

The trig functions are periodic functions and that is one of the most important things about them. The **period** is the length along the horizontal axis it takes for the function to start repeating. For sine and cosine, the basic functions have period . The period for tangent is .

Sine and cosine functions have two other important parameters: the height from the midline of the wave to the top of the wave is called the **amplitude**, and the position of the beginning of the first period of the wave is called the **phase shift**. The phase shift is the amount of horizontal shifting of the basic wave, that is. All three of the parameters have physical significance: for example, for a sound wave, the period is related to the pitch of the sound: the reciprocal of the period is called the frequency. For lightwaves and radiowaves (same thing) the period is called the wavelength. The amplitude has to do with the amount of energy in the wave: for sound waves, large amplitude means a loud sound. The phase shift has to do with what happens when there is more than one wave interacting: when the phases are not the same, the waves can interfere with each other and result in “noise” – microphone feedback squealing is an example of this.

The period, amplitude, and phase can be changed by the same kind of transformations of the basic graphs we have already looked at:

Multiplying the output by a number changes the amplitude

Multiplying the input by a number changes the period

Adding a number to the input changes the phase

Drawing the graph with the important five points:Â The five-point method

We will do a bit more with this method next time. On the Final Exam you will be asked to sketch the trig graphs and label the five important points.

â€¢ New topic: Inverse trig functions. We defined the inverse tangent and the inverse sine functions, following the way they are defined in the textbook (Session 17). We will continue with the inverse cosine function next time, and then do some solving trig equations.

**Homework:**

â€¢ Learn and practice the two important right triangles and the important points on the unit circle – see the videos

A trick to remember values on the unit circle

How to remember all the important points in the unit circle

â€¢ Do the WeBWorK: the second Exponential Functions Applications has been extended to tomorrow, and also start working on the Trig Functions Short: you should do a bit every day or so, not wait until the last minute! (Distributed practice) Remember that when you are asked for the exact values of the trig functions, you should be using the two important triangles and the unit circle picture, NOT your calculator, and also make an effort not to translate into degrees – think in radians.

â€¢ Discussion of some homework problems (Exercise 15.1 parts 1, b, c)

Here is something that came up in (c):

* We need to know how to interpret what the calculator is telling us. In particular, when the result of a calculation is given in scientific notation, you should not just write down what you see on the screen, but interpret it as a number (in decimal form preferably, if possible, but otherwise in proper scientific notation).

â€¢ Applications of exponential functions: word problems

Please note, the methods and formulas we use follow one of several methods that you may find if you search the web (or talk to other people). Be careful to use our method and not some other method, because you may get wrong answers using the other methods. (The methods differ based on what type of information is given in the problem and what base is used for the exponential function.)

In particular: **do not** use any method which uses a “continuous rate of growth”, and **do not** use any method where the base of the exponential function is required to be e. To be safe, use your textbook or the links that I will provide as your resources.

We worked through Examples 15.2 and 15.3 in the textbook. Make sure that you understand how we got our exponential functions from the given information in the word problems, and how we solved the other parts once we had the exponential functions.

The next thing we will be working on (aiming at the kind of problems that will be on the Final Exam) is finding exponential function if we are given a rate of growth or decay per unit time, and using those to find out when a quantity will double or triple or become half tis original size, etc.

To make this easier, you can prepare by viewing the videos linked in the homework section. I have specifically chosen those videos because they use our method. Please do not go searching for other videos, they almost certainly will be using the wrong method for our problems!

**Homework:**

â€¢ Review the examples discussed in class. We did not finish Exercise 15.1(c), which was assigned last time, and you should certainly complete it!

â€¢ Do the (short) WeBWorK “Exponential Function Applications”.Â You do not need to do the other one at this time.

â€¢ Also, from the textbook, do Exercises 15.2 and 15.3. You may put one on the board next time, after the quiz, if you like.

â€¢ To prepare for next time, view the following videos which show how to find an exponential function if you are given a rate of growth or decay:

An exponential growth problem by PatrickJMT – it’s just one example, but he shows the relationship between the form of the exponential function we have used so far and the form where you are given a rate.

Exponential growth word problems by Kevin DoreyÂ (It’s YouTube, so be warned that the next video may autoplay right after this one. You’ll want to turn off autoplay.)

â€¢ There will be a quiz next time: the topic will be exponential functions, problems like Exercise 15.1(a-d)

â€¢ Don’t forget to fill out the Post-Test survey which I handed out in class. (not to hand in: this is just for yourself) Doing this kind of reflection has been shown to improve test grades!

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