Monday 26 November class

(after Test 3)

Topic: graphing one period of sine and cosine functions:

Notes and a video of my slideshow with voiceover are in this post

Homework:

• Review the examples discussed in class, and start reviewing the unit circle.

• Do the WeBWorK assignments on the Unit Circle and on graphing trig functions (4 assignments, broken down). The recommended order to do them is as follows:

  1. Trigonometry – Unit Circle
  2. Trigonometry – Graphing Amplitude
  3. Trigonometry – Graphing Period
  4. Trigonometry – Graphing Phase Shift
  5. Trigonometry – Graphing Comprehensive

• Also make sure to do the following problems from the textbook:

p. 251 Exercise 17.6 a, b, c, d, f, g, (h was done in class), i, k

You may put one of these on the board at the start of class next time.

• Here are some useful videos from PatrickJMT to help with the review  of the unit circle from today and next time:

Deriving points on the unit circle (using the two special right triangles and other things)

Remembering the important points on the unit circle: part one, part two.

(Remember that the points on the unit circle give you the cosine and sine of an angle, in that order.)

 

Remember that if you get stuck on any of the problems or have a question about any of the material, you can post a question to the Piazza discussion board.

 

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Graphing Trigonometric Functions

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 $latex f(x) = a\sin(bx+c)$ or $latex f(x) = a\cos(bx+c)$

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 $latex |a|$ – amplitude is always positive!

The period is $latex \frac{2\pi}{b}$

The phase shift is $latex -\frac{c}{b}$

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

• 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)

• 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.]

• 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.

• 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.

• 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.

• 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 is  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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Math Help schedule

MathTutoringFall2018

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Inverse Trig Functions for Wednesday 26 November

Definitions of the inverse trig functions (in words):

• To find an inverse to the tangent function, we had\ve to restrict the domain of tangent to the interval $latex \left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ on which tangent is one-to-one and all values of tangent are covered in this restricted domain, so that will be the range of the inverse tangent function.

Definition: $latex \tan^{-1}(x)$ or $latex \arctan(x)$ is the angle (or rotation) in the interval $latex \left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ whose tangent is x.

Note: $latex \tan^{-1}(x)$ is an odd function: $latex \tan^{-1}(-x) = -\tan^{-1}(x)$

• To find an inverse to the sine function, we have to restrict the domain of sine to the interval $latex \left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ on which sine is one-to-one and all values of sine are covered in this restricted domain, so that will be the range of the inverse sine function.

Definition: $latex \sin^{-1}(x)$ or $latex \arcsin(x)$ is the angle (or rotation) in the interval $latex \left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ whose sine is x.

Note: $latex \sin^{-1}(x)$ is an odd function: $latex \sin^{-1}(-x) = -\sin^{-1}(x)$

• To find an inverse to the cosine function, we have to restrict the domain of cosine to the interval $latex \left[0,\pi\right]$ on which cosine is one-to-one and all values of cosine are covered in this restricted domain, so that will be the range of the inverse cosine function.

Definition: $latex \cos^{-1}(x)$ or $latex \arccos(x)$ is the angle (or rotation) in the interval $latex \left[0,\pi\right]$ whose cosine is x.

Note: $latex \cos{-1}(x)$ is neither even nor odd: but it does satisfy the identity $latex \cos^{-1}(-x) = \pi -\cos^{-1}(x)$

 

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Test 3 review Self-tests and answers, now with sources

Test 3 is scheduled for the first hour of class on Monday 26 November

Here are the review self-tests:

MAT1375Test3ReviewFall2018

Here are the answers:

MAT1375Test3ReviewAnswersFall2018

Sources for extra help with some of these problems: you should also look at the relevant WeBWorK assignments, which will show you solutions to their problems.

For problem 1, here are my notes:

MAT1375GraphRationalFnsNew

For problems 2 and 3, see the video sources in this post

 

Remember that if you get stuck on any of the problems or have a question about any of the material, you can post a question to the Piazza discussion board. I have posted several of the review problems so that you could post your solutions for a homework point if you wish. f you post by taking a photo (or better, a scan) of your work, please make sure that it is clearly legible.

Also, you may post a solution to any other problem from the review by posting the problem itself as a Question on Piazza and then posting your solution as the student solution to that question.

Genius scan is a very nice app for taking scans with a smartphone.

 

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Test 3 review Self-tests and answers (now with sources)

Test 3 is scheduled for the first hour of class on Monday 26 November

Here are the review self-tests:

MAT1375Test3ReviewFall2018

Here are the answers:

MAT1375Test3ReviewAnswersFall2018

Sources for extra help with some of these problems: you should also look at the relevant WeBWorK assignments, which will show you solutions to their problems.

For problem 1, here are my notes:

MAT1375GraphRationalFnsNew

For problems 2 and 3, see the video sources in this post

 

Remember that if you get stuck on any of the problems or have a question about any of the material, you can post a question to the Piazza discussion board. I have posted several of the review problems so that you could post your solutions for a homework point if you wish. f you post by taking a photo (or better, a scan) of your work, please make sure that it is clearly legible.

Also, you may post a solution to any other problem from the review by posting the problem itself as a Question on Piazza and then posting your solution as the student solution to that question.

Genius scan is a very nice app for taking scans with a smartphone.

 

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Wednesday 19 November class

Topics:

• Applications of exponential functions: see Examples 15.3, 15.6, 15.7. I added a question to Ex. 15.3: in what year will the population size have doubled from its original size (in 2000)?

• Quick review of Trigonometry basics we need for this course (to be continued a bit next time):

The right-triangle definitions of the trig ratios

Two special right triangles – you need to know these very well!

Finding sine, cosine, and tangent for the angles in the special triangles

The coordinate plane definition of the trig functions

The unit circle definition of the trig functions (MOST IMPORTANT!)

The signs of sine, cosine, and tangent in the four quadrants

Radian measure for angles – we will almost always use radian measure, so you should learn to think in radians as much as possible and, especially, be familiar with the radian measure of the important angles.

It is also useful to read and work through Section 17.1 in the textbook, which contains this and more.

 

Homework:

• Review the examples discussed in class, and start reviewing trig basics.

• Do the WeBWorK Exponential functions – growth and decay. Problem #3 is required (as it is most similar to what will probably be on the Final Exam). If you do any one or more of the others by the due date (no extensions for this!) it will count as extra credit.

• Also make sure to do the following problems from the textbook:

Exercises 15.3 – 15.8

You may put one of these on the board at the start of class next time.

• Here are some useful videos from PatrickJMT to help with the review fro today and next time:

Deriving points on the unit circle (using the two special right triangles and other things)

Remembering the important points on the unit circle: part one, part two.

(Remember that the points on the unit circle give you the cosine and sine of an angle.)

 

Remember that if you get stuck on any of the problems or have a question about any of the material, you can post a question to the Piazza discussion board.

 

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Monday 12 November and Wednesday 14 November class

Topics:

• Logarithms and their properties
• Exponential functions
• Graphs of exponential functions
• Graphs of logarithmic functions
• Using properties of logarithms to simplify or expand expressions
• Solving exponential equations using the natural logarithm (For more information about the base e and the natural logarithm and what is “natural” about it, see here.)
• Applications of exponential equations (introduction)

 

Here is my sheet containing the properties of logarithms: MAT1375RulesAboutLogs

 

Homework:

• There are several WeBWorK assignments. It is recommended that you do them in the following order:

(1) Exponential Functions – Graphs
(2) Logarithmic Functions – Graphs
(3) Logarithmic Functions – Properties
(4) Exponential Functions – Equations

• Here are some problems from the textbook that you may choose to put on the board at the start of class:
Exercise 14.2
Exercise 14.5(a-d)

• I also answered a student question about inequalities on Piazza and I have re-opened the
WeBWorK assignment on rational inequalities so you can go back and work on it some more if you like.

 

Remember that if you get stuck on any of the problems or have a question about any of the material, you can post a question to the Piazza discussion board.

 

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Homework from the textbook due Monday 12 November

These are problems on sketching complete graphs. You may put one on the board at the start of class, but make sure that you show your work in computing the intercepts and asymptotes, etc. Do not just put up the graph!

Session 11, p. 168 Exercise 11.4

 

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Polynomial and rational inequalities

Topics:

Here are some videos from PartickJMT which show how to use test points:

Example 1 (polynomial inequality)

Example 2 (polynomial inequality)

Example 1 (rational inequality)

Example 2 (rational inequality)

Example 3 (rational inequality)

Here is a video which shows how to use the graphs: you can also look at the textbook’s examples, which use the graphs as well. (Examples 12.4)

Solving a rational inequality using its graph (Sorry, this is from YouTube so another video may autoplay afterward: I don’t know how to stop this.)

Homework:

• Review the examples discussed in class. You may also want to view the videos I have linked above.

• Do the following exercises from the textbook (there is no WeBWorK for this): Exercises 13.1(a-f) and 13.2(a-e)

 

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