MAT1375 Precalculus FA2017

(After Test 3)

Techniques we will need for applications of exponential functions:

Solving for an unknown exponent

Example: find x if

Solution: take the logarithm of each side. Any base will do. We will use the natural logarithm.

by the “log of a power” property

divide both sides by

This is the exact value. We will approximate it using a calculator.

Solving for an unknown base (integer exponent)

Example: Find b if

Solution: we solve by taking the 15th root, which is the power:

We discussed Example 15.1 parts a, b, and c. These show how we will go about translating word problems into math language in the applications of exponential functions we are about to do.

Homework:

• Review the methods we used in these examples. We will need to use these (and other things we already know who to do) in working the application problems.

• There is no very good WeBWorK on this. You should do Exercise 15.1(a-d): I will ask for volunteers to put these on the board next time.

• Make sure that you finish the WeBWorK that was previously assigned!

• No Quiz next time, but do make sure you are prepared and know how to work exercises like Exercise 15.1, so we can go ahead and work the application problems!

• The Test 3 solutions are in this separate post, along with a Post-test Thinking Strategically survey. It will be very worth your while to fill out the Post-test survey even if you did not fill out the pre-test survey!

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!

Here is the Thinking Strategically Post-Test Survey: it will be useful for you to complete this even if you did not complete the pre-test survey (as it will help with your future test preparation)

ThinkingStrategicallyPostTestSurvey

Here are the solutions to the problems on Test 3. Please let me know if you find any typos or other errors in these!

Note: there was an error in the domain for problem 5. It is corrected in the version below.

MAT1375Test3-solutions

Topics:

• Review of finding x- and y-intercepts for various kinds of functions (the basic principle being the same)

To find the x-intercepts (which are the roots of the function) you set the function = 0 (namely, y=0).

To find the y-intercept, you set x=0.

This comes merely from looking at the graph. The x-intercepts are the points on the graph where y is 0, and the y-intercept is the point where x=0.

• Using the properties of logarithms to expand and to simplify logarithmic expressions.

Here are Paul’s Online Notes on properties of logarithms, which I referred to in class.

Here is a better version of the list of properties of logarithms:

MAT1375RulesAboutLogsFall2017

Homework:

• Review the examples we did in class. Always have the list of properties of logarithms in front of you when you work on these types of problems and make sure you see how the properties are being used! (It is very easy to invent fake properties that are actually false. We have already seen how this can happen.) See the Paul’s notes linked above, and also Examples 14.2 and 14.3 in the textbook.

• Do the following problems from the textbook: Exercises 14.1(a-e) and 14.2(a-f). I may request that certain problems from these be put on the board next time or on Monday next week, or I may post some to Piazza for your solutions.

• Finish the WeBWorK

• Don’t forget that Test 3 is scheduled for Wednesday. The review materials and links are on this separate post.

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!

Test 3 is scheduled for Wednesday 15 November, the first hour or so of class.

Here is the survey for you to use to plan how to make best use of the resources available to you: this is for your private use

ThinkingStrategicallyPreTestSurvey

Here are the review self-tests and answers. Links/references will be posted as soon as they are ready.

MAT1375Test3ReviewFall2017

MAT1375Test3ReviewAnswersFall2017 (corrected)

The answers to Self-test A have extensive hints and descriptions of how to do the computations.

There was an error in the answer to Self-Test B problem 4: the x-intercept should have been (103,0). I have corrected this version.

Here are some links to other resources:

My notes on how to find the asymptotes and intercepts for a rational function and sketch its graph (with an example):

MAT1375GraphRationalFnsNew

Topics:

• Graphs of logarithmic functions: transforming the basic log function graphs (see Session 5, section 5.2 to review the transformations)

• Finding the equation of the asymptote, the domain, the x-intercept, and the-y-intercept (if it exists) for the transformed log functions.

To find the equation of the vertical asymptote, set the argument (the input) of the function equal to 0. (You can also find the equation simply by looking at how the transformations have acted on the position of the vertical asymptote, and you can also find the position of the x-intercept this way some of the time.)

For example,

We added 5 to the input, so this moves the graph of 5 units to the right. So the asymptote moves to , and the x-intercept at (1,0) moves to (-4,0).

Alternatively, you could find the equation of the vertical asymptote by setting  the argument x+5=0 and solving: x=-5 is the equation of the vertical asymptote.

To find the domain, either consider how the transformations act on the graph, or else you can look for the x-values where the argument is greater than 0.

In the example, we can see from the fact that the graph was shifted 5 units to the left, that the domain is now x>-5, in other words the domain is the interval

Alternatively, we can find the domain by setting x+5 >0 and solving: x>-5.

To find the x-intercept: sometimes it is easy to see how the x-intercept has been moved, but we can always find the x-intercept (as for ANY function) by setting the function =0 and solving for x.

In the example, we set and solve by rewriting in exponential form:

, so the x-intercept is (-4,0).

To find the y-intercept, if there is one: there will only be a y-intercept if x=0 is in the domain of the function. In that case, set x=0 in the formula for the function to find the y-intercept (as you would do for ANY function).

In the example, since 0 is in the domain, we set x=0 to get .

So the y-intercept is at . It is not possible to compute $latex \log_{2}(5)$ exactly, but we can estimate it using a calculator – we will do this later. It is possible to see that $latex \log_{2}(5)$ is between 2 and 3 by translating to exponential form – try it!

When you sketch a graph, make sure that you show all of these features clearly.

Homework:

• Review the examples worked in class: one of them is in the notes above. Make sure that you understand how to find the asymptote, intercept or intercepts, and domain for the transformed logarithmic functions.

• Do the WeBWorK except you can omit problem #12 and 13 for now. Please do all the rest and do not wait to the last minute!

• Also do the following problems from the textbook, for extra practice: Exercise 13.6(a-h), find the equation of the vertical asymptote, the domain, the x-intercept, and the y-intercept if there is one, and sketch the graph. I will post answers to these.

• Don’t forget that Test 3 is scheduled for next Wednesday. The review materials will be on a separate post.

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!

Here are the solutions to Test 2, according to the version you have (look at the lower left on the cover sheet)

MAT1375Test2a-solutions    version

MAT1375Test2b-solutions   version

If you did not receive full credit for problems 3, 5 and/or 7, you have the opportunity to work similar problems to increase your score. In order for your score to increase, these must be done totally correctly and showing all work (except what you can easily do in your head: if in doubt, WRITE IT DOWN!)

Note: if you use a vertical line test or a horizontal line test, you should sketch a few vertical or horizontal lines to show what is happening. This is especially important to do when the graph fails one or the other of those tests: show a line where it fails!

These problems are due by the start of class on Wednesday 8 November.

MAT1375Test2-redoproblems

Topics:

• Discussion of using the graph to investigate the intervals in solving a rational inequality. Please see the links in the previous post for videos which show solving polynomial and rational inequalities: there is only one that I could find that used the graphs, unfortunately.

• Definition of logarithms, using the definition to translate from exponential form to logarithmic form and vice-versa, and some properties of logarithms that come from the definitions.

• Graphs of the basic logarithm functions. In this course we will only need to look at logarithms whose base is greater than 1.

The handout which I gave you, which contains all of the definitions and properties of logarithms that we need, is also available here:

MAT1375logpracticeNew

Homework:

• Review how to sketch the graph of a rational function, in particular, how to find the equations of its vertical and horizontal asymptotes. See examples 11.5 in the textbook, and the notes I used in class are also available here:

MAT1375GraphRationalFnsNew

• Review the problems we did using the definition of logarithms (on the handout). Make sure that you can easily translate back and forth from exponential form to logarithmic form and vice-versa.

• In the WeBWorK, do problems 4-11 for now. You can try the others as well if you like, but #4-11 are required for Wednesday’s class
• Also do from the textbook, Exercises 13.3 and 13.4. Please note that Exercise 13.4 is to be done without using a calculator, just as we did in class (and also see Example 13.10)

Here are some videos on logarthims:

from PatrickJMT: Logarithms properties 1

from Khan Academy (a sequence of 9 connected videos and lessons

• There will be a Quiz on Wednesday. The topics will be logarithms, translating from exponential form into logarithmic form and evaluating without a calculator (so calculators will be forbidden for this quiz only), just like Exercises 13.3 and 13.4.

Also don’t forget that if you did not receive full scores on Problems 3, 5, and/or 7 on Test 2, you have the opportunity to do a similar problem for half of the missing points. These problems must be written up completely correctly and are due at the start of class on Wednesday. See this post for more information and for the problems themselves. They are absolutely no excuses due on Wednesday, so don’t delay!

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!

Topics:

• Solving rational inequalities – was done last time with the sub. We discussed two homework problems in some detail. Make sure you understand how to use test points as well as how to use the graphs, when investigating the intervals.

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.)

• New topic: Exponential functions

An exponential function is a function which has the variable in the exponent. We will mostly be looking at exponential functions which have the form , where the coefficient c is not 0, and the base  , . It should be pretty obvious why we need and . Next time we will learn why we need the base to be positive. (Why the base cannot be negative, in particular.)

We saw that for the very basic functions of the form , their graphs have one of two basic shapes depending on whether b is greater than 1 or less than 1:

For all of these functions, the domain is the set of all real numbers, and the range is the interval . Also their y-intercepts are all at the point (0,1).

If b is greater than one, the graph looks much like the graph of : It grows exponentially to the right, and to the left it approaches the x-axis, which is its horizontal asymptote. This kind of shape is called exponential growth.

If b is between 0 and 1, the graph looks much like the graph of : It goes down very rapidly from the left, and as we go to the right it approaches the x-axis, which is its horizontal asymptote. This kind of shape is called exponential decay.

Different values for b will affect how rapidly the graph grows or decays. You should experiment with different bases to see this. It is good to use Desmos for this, since it lets you plot several graphs at the same time in different colors. For example, here are some that I showed in class: remember that we were looking at the graph of which we rewrote as

Also, we introduced that number e which appears above. It is called Euler’s number, and what you need to know for now is that it is an irrational number and is approximately equal to 2.7. If you need more digits, you can use your calculator.

Homework:

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

• Finish the WeBWorK on polynomial inequalities

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

• Start working on the Test 2 redo problems if you choose to do them. Please pay careful attention to the instructions. In particular, the due date/time is firm.

• There will be a quiz next time: the topic will be polynomial and rational inequalities.

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!

(Includes after the test on Wednesday)

Topics:

• Sketching a complete graph of a polynomial
• Rational Functions and their graphs
• Solving polynomial inequalities using the same method we used to solve absolute value inequalities – see Section 1.4

A complete graph of a polynomial must show all of the important features of the graph:

• The x-intercepts (real roots) if any, and the y-intercept. Note that the y-intercept of a polynomial is the constant term.
• The turning points (local maxima or minima) of the graph, if there are any.
• The end behavior.

Here is the summary of graphing rational functions that I showed in class: it includes a worked example

MAT1375GraphRationalFnsNew

Solving polynomial (and rational) inequalities:

We use the same method we have previously used to solve absolute value inequalities, that is:

First solve the corresponding equation

Then investigate the intervals to see if they solve the inequality or not

Then look at the endpoints

The only difference is that now we will make use of our knowledge of the graphs of polynomials (and later, rational functions) to investigate the intervals. This is equivalent to the “test point” method.

Homework:

• Review the examples discussed in class. Make especially sure that you understand how we are using the graphs to solve the polynomial inequalities in Ex. 12.2(a,b)

• Do the WeBWorK: the assignment on Polynomial Inequalities is not due until Tuesday, because it includes rational inequalities, which we have not yet discussed, but you can get started on the first few problems in the meantime.

• Also do the following problems from the textbook: Exercises 10.4(a-c and f-h), 11.4(all parts), and 12.2(a, c, d)

• No quiz next time.

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!

Test 2 is scheduled for the first 50 minutes or so of class on Wednesday 25 October.

The review self-tests and answers (with some hints or partial solutions) are on this page.

Of course you can also study the solutions to the quizzes.

More resources will be given in this post later on. Please check back!