Wednesday 1 November class

Posted on November 1, 2017 by Sybil Shaver

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 f(x) = c\cdot b^{x}, where the coefficient c is not 0, and the base  b> 0, b \neq 1. It should be pretty obvious why we need c\neq 0 and b\neq 1. 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 f(x) = b^{x}, 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 (0, \infty). 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 f(x) = 2^{x}: 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 f(x) = \left(\frac{1}{2}\right)^{x}: 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 f(x) = e^{-x} which we rewrote as f(x) = e^{-x} = \left(e^{-1}\right)^{x} = \left(\frac{1}{e}\right)^{x}

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!

 

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