Session 26: The relevant Khan academy videos can be found here. Bill Witte has a nice video as well, with lots of bells, whistles and graphics. We are going to use a somewhat lengthy video; please at least watch the first 6 minutes or so. As one of the comments suggest, you may want to increase the speed, especially if you are familiar with the material already. Keep in mind that the presenter uses function notation; in precalculus, this notation will become standard.

As an anticipation of precalculus, do you understand the function notation? As practice write down general formulas for both linear and exponential functions, as well as examples of each. In class we quickly graphed examples of both exponential growth and decay. Turning to the graphs, the same presenter gives 3 examples of exponential growth and then shows that decay can be thought of as a flip over the y-axis of an exponential growth function. Again the video is somewhat lengthy and you want to speed it up is are already familiar with the material.

The example we did in class was suppose that a 1 kg of a radioactive substance decays in 1 year, 1/3 of the substance is left. Use the presenter’s methods to graph both the decay as well as the mirror growth function.  Be sure to label the 2 graphs with their equations.

Session 27: We will first cover a special class of exponential equations. This Thinkwell video will be useful for the ExponentialEquations webwork assignment:

Logarithmic functions are the inverse of exponential functions. Here are the Khan academy videos. If you liked the presenter featured in the previous session, you should watch his video on logs. Instead, I will balance the gender of our presenters by switching to Nancy. The video is 19 minutes long. If you are short of time, please get through the first 9.5 minutes.

https://www.youtube.com/watch?v=ZIwmZ9m0byI

What do you think of her snail making? Does it help you convert a log equation into an exponential equation? In your webwork problem set, the first 5 problems are conversions. Write down these 5 expressions and draw snails to go with each one. Use the snails to help you in the conversion. If you like her videos, Nancy does a sequel where she makes use of the properties to solve log equations.

Session 28: Khan academy videos for Logarithmic properties can be found here. We are going to go back to Bill Witte and his bell and whistles. I chose it as your webwork tasks are largely paralleled by what he does starting around 4.5 minutes. He starts with expansion (your first 4 webwork problems) and switches to contraction around 6.5 minutes (last 2 webwork problems).

One idea that pops in mathematics a lot is inverse processes. Hopefully, after watching the video, you have good idea not only what the properties are, but also that you can use them for 2 opposite tasks, namely expansion and contraction. The last example presented is the most complicated. Write down the starting and finished expressions; try going backwards. In another words, try expanding the answer and see if you can get it to match what the starting expression. There are many steps. In some sense, this exercise is reminiscent of the trig identities. Do you see why?

A second topic for this session is compound interest. You might start with some Khan academy videos. As an alternative from study.com, try this unit. For a conceptual understanding, try Patrick’s derivation for the annual formula. In his sequel video, he derives the more general formula; keep in mind that APR (annual percentage rate) is a problematic term. It can either refer to the nominal rate, which is the way Patrick is using it, or the effective rate or yield. Read the wikipedia article. In your own words, explain what is the difference between nominal and effective and provide a concrete example to illustrate.

Session 29: Patrick’s video does two examples illustrating the 2 techniques used for webwork assignments (1st example: ExponentialEquations, which already have done; example 2: ExponentialEquations-Calc). As we transition to using logs to solve exponential equations, you may want to look at the Khan academy videos.

If you understand the compound interest formula, a difficult task is to go backwards. For example, suppose that you borrow $10,000 from your rich cousin with a 10% nominal rate compounded quarterly. How long will it be before you owe your cousin $20,000? In the next Thinkwell video, a similar problem is solved:

Once you have plugged all information into the compound interest formula, use the video to write down the steps needed to solve one of these backwards problems. Note in particular the places where log properties are used. In the ExponentialEquations-Calc webwork problem set, problem 5 part 3 could be easily rephrased as a backwards compound interest problem and problem 7 is a backwards interest problem, but using the continuous compounding formula.As your last writing exercise, rewrite problem 5 as in interest problem and write your 3 answers as sentences in this interest language.

If the role of e (and the natural logarithm) is of interest to you (no pun intended), you might take a peak at the Khan academy videos. This topic will be super important in precalculus.