Precal Blog

Today in class we presented the claims for exercise 2.1. So I’m going to explain the problems that were presented and later I’ll try to clarify the concept of e in mathematics and how to use it.

Exercise 2.1 Practice Problems

3. A bank account paying 8% annual interest compounded quarterly actually pays 2% interest each quarter. The annual yield is slightly higher than 8% due to the compounding.

a) If $1500 is deposited when the account is opened, how much interest is earned during the first year?

If you took notes on Thursday, this problem is very similar to the first problem in activity2.2. So I first set up the equation 1500(1+.08/4)^n, because the interest is being compounded quarterly. Then to solve the problem, you must plug in the number 4 (because there are four quarters in a year), and get the account total being $1623.64824….

But that’s not the final answer for letter a, in which you have to subtract the total value you got from the initial $1500, to get the interest earned. And the final answer is 123.64824….

b) What is the annual yield?

For this they are looking for a percentage, and to find that percent you take your interest earned in letter a and divide it by the initial amount, $1500. Like so,

123.64824….(interest)

1500(initial total)

And the final answer for b would be 8.24….%

c) If the money is invested for a 5-year period, what will the Balance be at the end of that interval?

This question is similar to letter a, and all you have to do is plug in twenty, (number of quarters in five years) for n, into the equation 1500(1+.08/4)^n.

The final answer would be $2228.92….

4. As previously noted, if you deposit $1 in a bank account paying interest at an annual rate of 100% compounded continuously, you would end up with e dollars after one year.

The Constant e

Well, to first understand this question, I may need to clarify the concept of e.

To better understand e you can compare it to pi, something we already know. It’s a constant number that many people try to memorize the consecutive digits, like pi.

The number e frequently occurs in mathematics and is an irrational constant (like ?). Its value is

e = 2.71828182845904523536028747135266249775724709369995…

The number e is used as a limit to how much some can be or how little it can be. It can also be represented in a graph with an asymptote because a value has only potential to reach that e value but can’t exceed it. Like in the following table, the values reach a point, but doesn’t exceed past that point.

Here’s also a website to further understand the number e. http://en.wikipedia.org/wiki/E_%28mathematical_constant%29

Go to the compound interest problem section,  it explains similar problems that we did in class.

a) With continuous compounding, how much would be in the bank after two years?

This would be represented with the expression e^2, because it’s continuously compounding which is e and it does so for 2 years which is the exponent. So the final answer is 7.389….

b) With continuous compounding, how much would be in the bank after five years? After t years?

Just as before, this would be represented with the expressions e^5 and e^t. The final answer for e^5 is 148.41….

c) Use your calculator to find 100*e^(.08). How does that answer compare to the work done in item 1 of activity 2.2?

The answer on my calculator was 108.32…. which was the same as the limit in the table, which can be referenced above.

d) Review your answer to item 3 of activity 2.2 and this exercise. Then generalize that work to write an expression for the balance after A dollars at 100r% compounded continuously for t years. Use numbers to check your expression for a specific case.

The generalized expression I wrote for this was b=A*te^%

Hopefully that helped clear up any questions you had about e and problems involving it, and if you still have some questions you can look on the website previously mentioned or you can read the section Base e in the packet on page 87. But if your more of an auditory or visual learner here’s a video on youtube that can maybe help. But as a warning: this video is boring, but informational. So if you’re not understanding the subject I recommend it.

http://www.youtube.com/watch?v=dzMvqJMLy9c

The next scribe will be Heather.