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College Algebra: Continuously Compounded Interest


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About this Lesson

  • Type: Video Tutorial
  • Length: 5:06
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 54 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
Algebra: Exponential and Logarithmic Functions (36 lessons, $49.50)
College Algebra: Exponential Growth & Decay (4 lessons, $5.94)

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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So when we think about interest rates and making investments, of course the more that we have compounding, the better and happier we are because in fact the more compounding, the sooner we get little teeny injections of interest and income, and those little teeny injections, even that are small, start to accrue income in and of themselves. So, in fact, the more you have, the earlier you have, the better you will end up doing. So in the best of all possible worlds what you want would be actually an account that would compound continuously, not just annually, not just semi-annually, not just quarterly, not just monthly, not just daily, but, in fact, continuously - every single moment you are getting microscopic injections.
Well, if you will remember, the formula for figuring out interest or the future value or the amount you have for some period, given the initial rate, which is p, is to take the rate of interest, divide it by the compounding per year, and then raise that to the power of the number of years times the number of compounding. It turns out that if you want to do this continuously, then what you actually see is the exponential function arise, and that's because if you take and raise it to the power, and you put in larger and larger values of n, you head towards the number e. You can try that. In fact, when we talked about e a while back, we actually reminded or described the fact that if you put higher and higher powers into here, you'll get closer to 2.718 and so on.
So, in fact, if you look at this formula, you can sort of see hidden in there is sort of to the m power. So if you let this m get really, really big, somehow e should be creeping into this formula. In fact, if you want to compound continuously, then the formula becomes a = p(e)^rt. So this converts. This is the one where it's compounding non-continuously in sort of a finite period; but, in fact, if you compound continuously, the amount you have at the end of some time is just your principal - the amount you started with - times e raised to the power - the rate - times the number of years. That's it.
So, for example, a great question that people like to ask you is, "Suppose I had a whole bunch of money like $300, and suppose I want to invest it in an account that compounds continuously, and it pays a rate of 5 percent." Well, the question is: "How long would I have to wait in order for me to double my money?" Of course, if I wait long enough, this is going to keep making interest and interest, but then at the end of some period I want to have actually twice as much money. How long do I have to wait? Well, first of all will that depend on the initial amount? For example, if I put in a little teeny bit of money, like if I put in like $50, will it double faster than if I put in like a $1,000? That's an interesting question. Maybe if you put in $1,000 you'd have to wait longer because it's so much money. Or maybe if you put in like 25¢, it will be really fast - the next day.
Well, it turns out that the answer is that it doesn't depend at all on how much you start with. You just want to double the amount you have, so it's all relative to the amount that you start with. If you think about it, you can see that immediately here because if you want to find the doubling time for money, what should a be? What should the amount be at the end of that period? It should be two times what you started with, so 2p. And notice that once I have a 2p here, those p's will cancel, so this has nothing to do with how much we actually start with. So if you put in a quarter and you want to get 50¢, you have to wait just as long if you put in $1 billion and you wanted to wait for $2 billion. Sorry.
Anyway, let's actually figure that out for this particular account. So I want to double my money, so I want to have 2p at the end of the period. I start with some p amount, and e - the rate - I told you was 5 percent, so that's .0525t. Well, notice the p's can cancel, and so I'm just left with 2 = e^.0525t. Now it's an exponential thing. I have the unknown in the exponent - don't want that - so I'll take the natural logs of both sides. By the way, you could take any log you want of both sides, but the natural log will feed off of the e very nicely, and so it'll simplify. That's why I take natural logs. Now the whole point of this is that using logs of exponents inside of algorithms, that becomes a coefficient, and so now my unknown now becomes on the ground level. I see the natural log of 2 = .0525t times the natural log of e, and that's the whole greatness about using the natural log versus any other log, the natural log of e - that's log base of e is one. So it goes away. Now I can solve for t. t must just equal the natural log of two, and I divide by .0525. What's that? If you plug it into a calculator, and it produces 13.202 years. So just over thirteen years, and you should double your money no matter how much you put in. A good investment? Well, a lot of investors say that you should be doubling your money every seven years. So, in fact, you should be doing better than this. Good luck, and careful investing.
Exponential and Logarithmic Functions
Word Problems Involving Exponential Growth and Decay
Continuously Compounded Interest Page [2 of 2]

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