Thursday 27 March class

 

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!