Chapter 5.3 ~ Compound Interest

Equations:

Compound Interest (annually):
A=P(1+r)^t
A= accumulated amount after t years
P=Principal or starting amount to be invested
r= interest rate
t= term (years)
*Use this equation only when interest is being compounded annually.

Compound Interest (multiple conversions):
A=P(1+r/m)^(mt)
A= Accumulated amount after t years
P= Principal or starting amount to be invested
r= interest rate
t=term (years)
m=number of conversions or compundings in one year (quarterly, daily, etc.)
*Use this equation if interest is not being compounded annually

Effective Rate of Interest:
reff= (1+r/m)^m-1
reff= Effective rate of interest or true rate
r= nominal interest per year
m= number of conversions or compoundings in one year (quarterly, daily, etc.)
*Use this equation if interest is being compounded annually or for other conversion periods (quarterly, daily, etc.)

Present Value for Compound Interest:
P=A(1+r/m)^(-mt)
P= present value
A= future value
r= interest rate
m= number of conversions or compoundings in one year (quarterly, daily, etc.)
t= term (years)

Present Value for Continuous Compounding:
P=Ae^(-rt)
P= present value
A= future value
r= interest rate
t= term (years)

Continuous Compound Interest Formula
A=Pe^(rt)
A= Accumulated amount after t years
P= Principal or starting amount to be invested
r= interest rate
t= term (years)

Vocabulary Necessary for Understanding Compound Interest:

compound interest- interest is added to the principal and then earns itself that same rate which is then added to the new principle and so on and so forth
accumulated amount- sum of the principal and the interest after t- years
term- how long the money is gaining interest
nominal or stated rate- interest rate on the investment
conversion period- the time intervals between when interest is taken(such as quarterly, daily, etc.)
effective or true rate- the interest rate which produces the equal to the accumulated amount in one year that it would take the nominal rate compounded m- times a year
present value- amount which you have now
future value- amount which is gotten in the future (such as: at 9%, and at the end of nine years, the accumulated amount will be 20,000)
*present and future value are used when a question such as "how much money should I invest at a fixed rate, so that I will end up with x amount of money?"

Extra Notes:

Effective annual rate is used to compare to investment opportunities to see which would be the better deal. You plug in $1 as your principle, for one year to see which gives the most interest.
Present Value is needed when you need a certain amount of money for the future, so you need to find how much you need to start with at what interest rate to your future value.

Sample Question:

Find the accumulated amount after 12 years if $7000 is invested at 5.7% per year compounded annually and daily.

Plug in the quantities into the Annual Compound Interest Formula for one year A=P(1+r)^t, you get A= 7,000(1+.057)^12 = $13614.4 which is how much you would make after 12 years at that interest rate.
Plug in the quantities into the Multiple Conversion Compound Interest Formula for multiple conversions A=P(1+r/m)^(mt) to get A= 7000(1+(.057/365))^(12*365)= $13871.80

Internet Links for Further Information:
http://www.math.hawaii.edu/~ramsey/CompoundInterest.html
http://en.wikipedia.org/wiki/Compound_interest
http://mathworld.wolfram.com/CompoundInterest.html

Reminder to Danika for your post on Tuesday October 3rd due October 4th about Chapter 2.3 Functions and their Mathematical Models.

Personalization-
If you owe the bank $100, that's your problem. If you owe the bank $100 million, that's the bank's problem.
~ John Paul Getty

5.2: Logarithmic Functions

Formulas and Properties:

Logarithm of x to the Base b:
y = logb(x) if and only if x = b^y (x>0)

Logarithmic Notation:
log(x) = log10(x)
ln(x) = loge(x)

Laws of Logarithms:
logb(mn) = logb(m) + logb(n)
logb(m/n) = logb(m) - logb(n)
logb(m^n) = nlogb(m)
logb(1) = 0
logb(b) = 1

Logarithmic Function:
f(x) = logbx (b>0, b does not = 0)

Properties of the Logarithmic Function:
1. Its domain is (0, infiniti)
2. Its range is (-infiniti, infiniti)
3. Its graph passes through the point (1,0)
4. It is continuous on (0, infiniti)
5. It is increasing on (0, infiniti) if b>1 and decreasing on (0, infiniti) if b<1

Properties Relating e^x and lnx:
e^(lnx) = x (x>0)
ln(e^x) = x (for any real number x)

Graph of Logarithms:
A logarithmic function is the inverse of an exponential function. Thus, a logarithmic function is the reflection of an exponential function over the equation y = x:




















Practice Problem:

Q: Simplify the following logarithm:

5log7(x) + log7(7^1) - 2log7(x)






A:
5log7(x) + log7(7^1) - 2log7(x)

log7(x^5) + log7(7) - log7(x^2)

log7[(x^5)/(x^2)] + 1

log7(x^3) + 1



Link to a Logarithm Page

Brian, you're up next and you'll be covering Section 5.3: Compound Interest.

Logarithm Joke:
Question: How do we reduce an exponential growth of grass?
Answer: Using a "ln"-mower!

Thursday's Quiz Topics

Ok, here’s a list of topics for the quiz on Thursday:

Convert degrees to radians (positive and negative angles!)
Convert radians to degrees (positive and negative angles!)
Evaluate special angle trig functions (positive and negative angles!)
Solve a trigonometric equation (remember to find all values, not just the first quadrant solutions…)
Verify a trigonometric identity
Evaluate an exponential expression
Simplify an exponential expression
Solve an exponential expression

That’s it! I’ll be online after 9PM on Wednesday if you have any last minute questions. Good luck studying!

If you need a break, check out these “criminals.”

Test Chapter 1, 2.1-2 Topics

Here’s a list of topics that will be covered on this Thursday’s Chapter 1, 2.1-2 Test.  

• Absolute value


• Sketch a graph – piecewise function


• Determine values – piecewise function (2)


• Find the domain of a function


• Simplify an expression – radicals


• Simplify an expression – rational (fraction)


• Find roots of an equation


• Find points given distances and coordinates


• Find slope of a line (graph)


• Find slope and y-intercept (equation)


• Find composite function


• Find composite function


• Evaluate composite function


• Simplify a rational expression


• Find an equation for a line given points

Blog Postings

As a reminder, the requirements for each blog posting will consist of:

• A review of the main point of the class lecture/demonstration.  This summary should highlight any relevant formulas and/or graphs and communicate your interpretation of the concept covered in class. (15 pts)  For this part of the posting, I am looking for quality, not quantity.


• An example problem, including a statement of the problem, the answer, and the solution method.  (For your first post, using an example covered in class is acceptable, for additional postings, original examples will be required.) (10 pts)


• A link to an additional Internet resource supporting the Topic of The Day. (5 pts)


• A reminder to the next BlogMaster of their responsibility to post. (5 pts)


• A “personalization” of your posting.  This personalization can be a comment about the day’s class, an image, a quotation, a question posed for discussion, a joke, or something else that reflects you as a student.  These personalizations must be in good taste!  (5 pts)

In addition to your posting, you will be expected to comment on a minimum of two (2) of your classmates postings during each quarter.  These comments must either further enhance your classmates’ understanding of the posted topic or further a discussion question posed in the original posting.

Additional Notes:

• Postings will be due within 24 hours of class.  I will post a schedule of class scribes for the first quarter once everyone has joined the class blog.


• For help with posting equations and graphs, please feel free to come ask me for assistance.


• Initially, the blogs will be hosted on blogger.com.  As the year progresses, we hope to migrate to an internal website.


• While we are on blogger.com, there is some software available through the website that allows creation/editing of posts via Microsoft Word.

Blog Policies

There are some things I want you to remember about blogging. Many of things have been discussed by other teachers and classes, so I will paraphrase them here and try to give them proper credit:

First of all, our class will not be the only people to view our postings. The Internet is accessible almost everywhere these days, and even if a post is deleted, there’s no guarantee that the posting hasn’t been copied and propagated to other sites or linked to from those sites. This has a couple of implications:

First, privacy. We will only be using first names on the site. If I post pictures or video, no one will be identified, other than “Mr. French’s class”. Do not use pictures of yourself for your profile here. If you want a graphic image associated with your profile, use an “avatar” – a picture of something that represents you but is not you. Here’s a link to a fun image creator.

Second, etiquette, appearance and common sense. Bud the Teacher has these suggestions, among others:

1. Students using blogs are expected to treat blogspaces as classroom spaces. Speech that is inappropriate for class is not appropriate for our blog. While we encourage you to engage in debate and conversation with other bloggers, we also expect that you will conduct yourself in a manner reflective of a representative of this school.


2. Never EVER EVER give out or record personal information on our blog. Our blog exists as a public space on the Internet. Don’t share anything that you don’t want the world to know. For your safety, be careful what you say, too. Don’t give out your phone number or home address. This is particularly important to remember if you have a personal online journal or blog elsewhere.


3. Again, your blog is a public space. And if you put it on the Internet, odds are really good that it will stay on the Internet. Always. That means ten years from now when you are looking for a job, it might be possible for an employer to discover some really hateful and immature things you said when you were younger and more prone to foolish things. Be sure that anything you write you are proud of. It can come back to haunt you if you don’t.


4. Never link to something you haven’t read. While it isn’t your job to police the Internet, when you link to something, you should make sure it is something that you really want to be associated with. If a link contains material that might be creepy or make some people uncomfortable, you should probably try a different source.

Are there other considerations we should take into account? Use the comment feature to add any others or to clarify/expand on one of the above.

Welcome

Congratulations! You found our class blog! This is where we as a team will hopefully create a resource to help us conquer any issues that arise during our class this year. This is the place to talk about what’s happening in class; to ask a question you didn’t get to ask in class; to share your knowledge with fellow classmates and any other Internet users who choose to read our notes;…and most importantly it’s a place to reflect on what we’re learning.

A large part of retaining knowledge requires reviewing and discussing new information on a regular basis. This blog is intended to help each of you do just that. Between creating your own posts and commenting on your classmates’ posts, you will have the opportunity to explore each of the topics we cover this year in greater depth. I hope you will use this forum to help yourself and your classmates in whatever ways you can think of.

Blogging Prompt

Occasionally I will include a posting of my own, either to clarify a concept or to generate some further discussion. These postings will have a title similar to the one above this paragraph.

To get things rolling, here’s a question for you to think about and respond: Is God a mathematician? Why or why not?

Don’t forget to email me with the information I requested in class so I can include you on the team!