MA1375 Precalculus Spring2014 TTh

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) and the start of (h)

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. (We will finish parts (b) and (c) of 15.3 next time, but they are done similarly to Example 15.2)

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 Monday 11 PM. Start early! There are two very short assignments • No Warm-Up this time

• Don’t forget that Test 3 is scheduled for the first 50 minutes or so of class on Thursday 910April. 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 graphs and   for b>1, and finding the domain for the logarithmic function (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 14.6a) – more next time.

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 Exercises 13.4 and 13.6, and do the assigned problems in Session 14, except SKIP Exercise 14.4 for now. Also, in Exercise 14.5 do (a-c) only for now.

• Do the WeBWorK: start early! Due by tomorrow 11 PM.

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

Topics:

• Using the tools of Session 12 to find the domain of a radical function (Exercise 12.3, which was not assigned, but a student did some of it for us.)

• 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.

• 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!)

• Rewriting exponential equations into logarithmic form: see Example 13.9.

Some things we did not have time to discuss in class: I will be putting up notes or questions on Piazza related to the following:

• Factoring by grouping (which can sometimes be used to factor a third-degree polynomial)

• A way to factor the numerator in Exercise 11.4(c) by elementary techniques (without looking at the graph!)

• Figuring out the graph for 11.4(c) – [The graphing calculator program on the computer that I’ve been using seems to be buggy. I’m going to avoid using it until I can get it checked out, so we will have to reply on our actual calculators!]

Please also look on Piazza for Wilson’s list of the steps to find all the roots of a polynomial, or to find its complete factorization (Session 10): and other questions whose answers you can help edit!

Homework:

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

• Make sure that you have done Exercise 12.4 (the assigned parts) and that you are following the method described in the handout I gave you! (Which is the method of the textbook for solving rational inequalities also.)

• Do the assigned parts of Exercises 13.1 and 13.3 (for now)

• Remind yourself of the definitions of negative exponents and fractional exponents: What does mean? What does mean?

• Read Session 13

• 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 Monday.

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

Topics:

• Quick review of the first method of solving absolute value inequalities, expanded to also look at the graph. See 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.

Update: a couple of other things that came up during today’s class:

• The difference of squares pattern (for factoring or multiplying) – very important!

• What does “undefined” mean when we say that is undefined? It means: the operation “Division by 0” is undefined (the operation does not have a way to be defined).

The normal definition of division is as an inverse of multiplication:

if and only if 

and this is also how we check a division. So suppose you could divide 6 by 0:

What multiplication would you use to check that? What goes wrong? Now try  and see if you can tell why it is even worse!

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(b, c, and especially d). I may call someone at random to put this on the board.

• Do the WeBWorK: start early! Due by Wednesday (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 Thursday)

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 Tuesday’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 Monday evening 11 PM. Start early!

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

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

Test 2 is scheduled for the first 50 minutes of class on Thursday 20 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:

• More on complete graphs of polynomials and recognizing when we do not have a complete graph.

Facts about polynomial graphs:

• The domain of a polynomial is the whole real line

• The y-intercept is the constant term

• For a polynomial of degree n, there are at most n real roots (which are the same as the x-intercepts of its graph) and there are at most n-1 local maxima or minima (turning points).

• The graph of a polynomial is continuous – no breaks or jumps

• The graph of a polynomial does not have any corners

• The graph of a non-constant polynomial does not contain any horizontal line segment

• The end behavior of the graph is determined by the leading term of the polynomial.

• Review of complex numbers

• The fundamental theorem of algebra and its consequences.

The important thing to remember is that the Fundamental Theorem implies that every polynomial of degree n>0 has exactly n roots, if you allow complex numbers as roots and you count each root with its multiplicity. (The multiplicity of a root is the number of times its factor appears in the factorization of f(x), in other words, it is the exponent that goes on that factor.)

• 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.

Homework:

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

• Optional, but recommended: Do 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 parts from Session 10 Exercises 10.3 and 10.4

• Do the WeBWorK – due by Monday 11 PM

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

• Don’t forget that Test 2 is scheduled for Thursday 20 March. There will be a separate post with more information.

Topics:

• Synthetic division (a short way to do long division when dividing by x-c)

• Roots, factors, and remainders: the Remainder Theorem and the Factor Theorem

SUMMARY: For any polynomial f(x), he following three things are equivalent:

f(c) = 0 (c is a root of f(x)

(x-c) is a factor of f(x)

The remainder is 0 when you divide f(x) by x-c

• Polynomial graphs: see below

Facts about polynomial graphs:

• The domain of a polynomial is the whole real line

• The graph of a polynomial is continuous – no breaks or jumps

• The graph of a polynomial does not have any corners

• For a polynomial of degree n, there are at most n roots (which are the same as the x-intercepts of its graph) and there are at most n-1 local maxima or minima. (We will discuss this next time.)

• The end behavior of the graph is determined by the leading term of the polynomial.

Homework:

• Reread and review the Observations and the examples discussed in class.

• Finish the assigned problems from Session 8

• In Session 9, do the assigned problems from 9.1, 9.2, and 9.3

• Do the Warm-Up which is in WeBWorK this time- due tomorrow by 11 PM! Start early.

• Do theWeBWorK – a short assignment – due tomorrow by 11 PM!