MAT1375 Precalculus Spring2014 MW

Topics:

• Identifying the properties of logarithms that are being used: see the handout for their names

On Test 3 you will be asked to identify the property of logs you are using at each step. Please try to use only one property at a time.

• Solving logarithmic equations: see Example 14.5(e-h): we discussed (e) and (g)

To solve a logarithmic equation, first (if necessary) rewrite each of the sides of the equation as a single logarithm to the same base.

Then use Observation 14.4: if the logarithms are equal, then their inputs (also called the arguments) must be equal.

Then solve the resulting simpler equation.

Finally, check all the answers you got solving the simple equation to see whether or not they are solutions of the original logarithmic equation (before you made any changes to it). Remember when checking, do not move any term or number across the equals sign: this can invalidate your check. Just simplify each side separately to see if they are equal.

Then state your solution or solutions to the original equation: or state that there are no solutions, if that is what happened.

• Application: Exponential growth and exponential decay:

see Examples 15.1 for general procedures for finding the form of the function, and 15.2, 15.3 for applications

For the application to exponential growth or decay, we are using functions of the form

with .

[Please be aware that there is an alternate way to analyze these where the base is always taken to be , so the function will have the form . We do not use that form in this course! Please only use the form given above or the formula which we will use next time (when the rate of growth is given).]

Homework:

• Review all of the examples discussed in class, including the homework problems where we identified the properties of logarithms.

• Do the assigned parts of Exercises 14.4, 15.1, 15.2, and 15.3

• Do the WeBWorK: due by Sunday 11 PM. Start early!

• No Warm-Up this time

• Don’t forget that Test 3 is scheduled for the first 50 minutes or so of class on Wednesday 9 April. There will be a separate post about that. Please use Piazza to discuss the problems on the Review Self-Tests.

Topics:

• Transformations of the basic graph for b>1, and finding the domain (Example 13.13)

• Using the properties of logarithms to simplify expressions containing logarithms (Example 14.2)

• Using the properties of logarithms to expand a logarithm (Example 14.3)

• Solving exponential equations when both sides can be written as a power of the same base (Example 14.5(a-d))

• Solving exponential equations using logarithms (We generally use the natural logarithm for this: see Example 14.6)

Be very careful, when working with logarithms, that you use the properties in the list I handed out. A common error is to think that a quotient of logarithms can be simplified: it cannot. So, for example, there is no way to simplify log(20)/log(4), and it certainly is not equal to log(5).

Homework:

• Study the definition and the properties of logarithms: review the Examples, to see how the properties are being used. You should have the list of properties in front of you as you work the problems.

• Do the assigned parts of Exercise 13.6, and the assigned problems in Session 14, except SKIP Exercise 14.4 for now.

• Do the WeBWorK: start early! Due by tomorrow 11 PM. (Fixed link!)(Fixed a second time – it keeps changing from what I put in it, I don’t know why. The direct link is

http://mathww.citytech.cuny.edu/webwork2/MAT1375-Shaver-MW/

• Do the Warm-Up – also due by tomorrow 11 PM.

There are a couple of questions posted on the Piazza discussion board that are seeking student solutions, as well as other goodies that you may find useful. Remember that you can also help by editing an answer to improve it! This is another way to get homework points if you have not been putting problems on the board.

Topics:

• Basic exponential functions and their graphs (for b>1 and for 0<b<1): See Example 13.2 and Observation 13.3, also Example 13.5

Note that all of these basic graphs have the y-intercept at (0,1) and have a horizontal asymptote y=0 (the x-axis). The graph is asymptotic to the x-axis on only one side, though. The domain is  and the range is the interval . (The graph always lies above the x-axis.)

• The Euler number , also known as the base of the natural logarithm. See Definition 13.4.

• Transformations of the basic exponential graphs: see Example 13.7 (I changed (c) though)

• Definition of the logarithm as the inverse of a basic exponential function: For any base b>o, b not equal to 1, we have

• Special cases:

When the base is 10, we call it the common logarithm, and we write it

When the base is , we call it the natural logarithm, and we write it .

(However, be aware that many mathematicians use  to mean the natural logarithm, because the natural logarithm is by far the most commonly used logarithm for us!)

• Graphs of basic logarithmic functions with b>1. See Example 13.12.

Note that these graphs have x-intercept at (1,0) and have a vertical asymptote x=0 (the y-axis). The domain is the interval and the range is . [Remember that the inverse of a function interchanges the roles of the inputs and outputs: compare these to what we had for the basic exponential graphs!]

• Computing some values of logarithms using the definition to rewrite into exponential form: see Example 13.10.

A few other things that came up during class today (while discussing the homework): I am putting them below the fold, at the bottom of this post.

Homework:

• Review all the definitions and examples worked in class, also the additional examples listed above in the textbook.

• Do the assigned problems from Session 13, except SKIP #13.2(d) and #13.6 (for now)

• Do the WeBWorK: start early, and make sure that you have an email address in WeBWorK (look under “password/email” in the left sidebar in WeBWorK). This WeBWorK is due by 11 PM Sunday.

• Do the Warm-Up for Properties of Logarithms: also due by 11 PM Sunday!

A few other things that came up during class today below the fold…

Continue reading →

Topics:

• Quick review of the first method of solving absolute value inequalities, expanded to also look at the graph. See also Example 12.4(d) in the textbook.

• Solving linear inequalities: this is just like solving linear equations, except that when you divide or multiply both sides by a negative number, you must reverse the direction of the inequality. This may result in a double inequality being put in the wrong order, and then you have to fix that up: see Example 12.1(c).

• Solving polynomial inequalities of degree greater than 1: the method is basically the same as the first method we used for absolute value inequalities, except that we want to make one side 0 first, in order to make it easier to solve the related equation. See Examples 12.2

•  Solving rational inequalities: the method is basically the same as the method for solving polynomial inequalities, except that we also need to consider the zeroes of the denominator along with the solutions of the related equation when we cut the real line up into intervals. (However, zeroes of the denominator will never be part of the solution set!) See the example on the handout: also Examples 12.4(a-c)

The methods for solving polynomial inequalities (of degree >1) and rational inequalities are outlined below the fold, at the bottom of this post.

Homework:

• Reread and review the examples worked in class. It is also very useful to study the other parts of Examples 12.1, 12.2, and 12.4.

• Do the assigned problems from Session 12

• Do (if you have not already done it) Exercise 11.4, especially 11.4(d). I will call someone at random to put this on the board.

• Do the WeBWorK: start early! Due by Tuesday (tomorrow!) 11 PM

• No Warm-Up this time.

• Make sure you check the Checklist!

Methods for solving polynomial and rational inequalities outlined below the fold…

Continue reading →

Please go through the checklist and make sure that you have everything set up. Especially make sure you have an email address in WeBWorK, as that will be used to send out midterm grades later on.

(after Test 2 on Wednesday)

Topics: Rational Functions and their graphs

• Domain: the domain of a rational function will be all real numbers except the x-values which make the denominator 0.

• Vertical asymptotes: these occur where the denominator is 0 and the numerator is not 0 (and see also the second box on the handout).

• Holes in the graph (also called removable singularities): these occur where both the numerator and denominator are 0 to the same multiplicity (and see also the second box on the handout).

• Horizontal asymptotes: these represent the “end behavior” of the graph, so they depend only on the leading terms of the numerator and denominator. See the discussion of Example 11.2(a-d) for details.

• y-intercept: if 0 is in the domain, the y-intercept is f(0)

• x-intercept(s): these are where f(x) = 0, which means that the numerator of the rational function is 0 (and the denominator is not 0).

Before looking at the graph on your calculator, determine all of the above algebraically, and then consider what viewing window will be appropriate. Be careful in interpreting what you see on your calculator display! The graphing calculator will sometimes connect parts of the graph which are actually separated by vertical asymptotes. This is another reason that you need to know what you expect the graph to look like before you ever put it into your graphing calculator!

Note: there is a corrected and improved version of the handout on Graphing Rational Functions posted over on Piazza.

Homework:

• In Monday’s class I told you if you still need to put an email address in WeBWorK. Please take care of this today if you have not done it yet!

• Review the discussion of Example 11.2(a-d) as it pertains to the vertical and horizontal asymptotes.  You will probably also want to study the Examples 11.5.

• Do the assigned problems from Session 11

• Do the WeBWorK: due by Sunday evening 11 PM. Start early!

• Do the Warm-Up: also due by Sunday 11 PM.

Spring 2014 PLTL Schedule (link to pdf of the schedule, or click on the image below to embiggen it)

Test 2 is scheduled for the first 50 minutes of class on Wednesday 19 March.

The review Self-Tests were handed out in class and are also available on the Piazza discussion board, where you can ask questions if you get stuck!

Below the fold are references to the sections of the textbook that you should study for each problem if you are having trouble with it.

Continue reading →

Topics:

• Finding all roots of a polynomial and using them to find a complete factorization of the polynomial over the complex numbers.

• For a polynomial with real coefficients, if it has non-real complex roots they occur in conjugate pairs: if a+bi is a root, then so is a-bi.

• Using knowledge of the roots and other information to find the formula for a polynomial.

Homework:

• Review the examples discussed in class, and also study Example 10.9 in the textbook.

• Do (if you have not already done it) Exercise 9.4(a-d). In each case, start with the standard viewing window. Write down each change you make to the standard viewing window and why you made it – what was it about the graph that you wanted to see that you could not see.

• Do the assigned problems from Session 10

• Do the WeBWorK – due by Sunday 11 PM

• Do the Warm-Up for Rational Functions – also due by Sunday 11 PM

• Don’t forget that Test 2 is scheduled for Wednesday 19 March. There will be a separate post with more information.