p.304-5
Read through the discussion on page 304-5. Note that the amounts calculated (for the yield from compounding with a fixed nominal annual interest rate) get bigger as the compounding frequency goes up. The increases get less however, and eventually there is no visible effect on the final amount (ie it does go up, but by less than one cent so the rounded results do not change). No matter how frequently we compounded, the amount would never exceed $1094.18 and so there is a limiting value (somewhere between 1094.17 and 1094.18). This limiting case is called continuous compounding.
p.305-307
The special case r=1 (interest at 100%) may not seem like a typical interest rate, but the calculation on page 307 will show that the result for any interest rate can be computed in terms of that for rate r=1, so working out that one special limit will allow us to figure out the result for any other rate without going through the limit process again.
The table on page 306 suggests that the limiting value is about 2.7, and its exact value (which we cannot write down as a finite decimal) is given the name e.
p.307
The graph of can be quickly drawn by locating the points at x=0 (y=1) and x=1 (y=e ) which we can plot to whatever accuracy we require. (Just putting it somewhere between y=2.5 and y=3 is often good enough.) And of course, since e is between 2 and 3 the graph of is between and as shown.
The algebra on page 307 may look a little tricky, but the idea is just to use laws of exponents to rewrite the compound interest formula with the P, t and r on the “outside” and an expression in the middle whose limiting value as is independent of them (so that once we’ve done that one limit we can quickly calculate whatever we need).
The k in the middle part goes to infinity as n does, so, as , the expression also approaches the number e, and we get the simple formula .
(So in the end continuous compounding is easier to work with than any other kind!)
p.308
The phenomenon of a quantity growing (or decaying) continuously at a rate proportional to its size occurs in many contexts other than compound interest. Example#4 on page 308 deals with population growth and several other examples are used in the problems at the end of the section. When the authors say that a quantity “grows continuously” at rate of, say, 10% per year, they do not mean that in one year it grows to 10% more than its size at the start of the year, but rather that in a tiny fraction of a year it grows by the same tiny fraction of 10%, and that future growth is based on that new size so the growth “compounds” just like interest to yield an amount after t years of times what it started at. For one year this gives a growth factor of , so it actually grows by more than 10%. Similarly for any growth rate r, a quantity Q which grows continuously at rate r per year has a value after t years given by (where is the initial value).
Be careful about wording though. If a quantity grows continuously at a rate of 10% per year it grows by about 10.5% over a whole year and in general the actual growth over a full year is greater than the continuous growth rate. Conversely, if we are told the % growth over a whole year then the continuous growth rate would be less. So if a problem says a quantity “grows continuously to increase by 10% every year” and another says the quantity “grows continuously at a rate of 10% per year” they do not mean the same thing.
A continuous growth rate is sometimes referred to as an “instantaneous growth rate”, or in finance, as a “nominal interest rate” or “force of interest”; and the actual growth over a full year is often referred to in finance as the “effective annual interest rate”.
p.309-311
In some of the examples and applications on pages 309-311 the number e is being used in situations where any other number would be just as good. The point in these examples is to get more practice in working with exponential formulas and functions involving exponentials while at the same time getting comfortable with e as just another number.
When you have finished reading the section, go back to page 307 and look at the graphs. If you compare them as they cross the y-axis you will see that they all do so at the same point (y=1). You will also see that the slope of the tangent to at that point is a bit less than 1 (in fact the slope from (0,1) to (1,2) is exactly 1, but the graph curves up, so the slope at the left end is a bit less). On the other hand, if you measure it accurately you will see that the slope of is a tiny bit bigger than 1. But the slope of at x = 0 appears to be exactly equal to 1. In fact it is, and this property is really what makes the number e so important in calculus. If you want to check these slopes more carefully, you can zoom in on a graphing calculator or computer-based graphing program.
Definition of e
The number e can be defined as the limiting value approached by as
Its value is a bit less than 3 (about 2.7). A main reason for being interested in this limit is due to the fact that it is useful for the description of any quantity which grows continuously at a rate proportional to its size.
Applications
Money with interest “compounded continuously” is one example of a quantity which grows continuously at a rate proportional to its size. With interest at rate r per year, an initial sum of $P (often called the “Principal” of the loan) grows in t years to an amount of $A, where .
In general, when we say that a quantity grows continuously at rate of times its size per unit time we do not mean that over a full unit of time it grows by r times its size, but rather that over a small fraction of a time unit it grows by the corresponding fraction of r times its size, with future growth being based on that new size (just like in the case of frequently compounded interest). A quantity that grows continuously at rate of r times its size per unit time will actually grow over one full time unit by a factor of . So after t units of time it will be given by , where is the initial amount.
Population growth may be considered as continuous growth and modeled by a similar formula – i.e. , where is the initial population.
If the world population in year 2000 was 6 billion and grows continuously at a rate of 2% per year, what will it be in 2010? (See answer #1 below)
A continuous growth rate is sometimes referred to as an “instantaneous growth rate”, or in finance, as a “nominal interest rate” or “force of interest”; and the actual growth over a full year is often referred to in finance as the “effective annual interest rate”.
What is the effective annual interest rate for a nominal rate of 10% compounded continuously? (See answer #2 below)
Be careful about wording when reading growth problems. If a quantity grows continuously at a rate of 10% per year it grows by about 10.5% over a whole year and in general the actual growth over a full year is greater than the continuous growth rate. Conversely, if we are told the % growth over a whole year then the continuous growth rate would be less. So if a problem says a quantity “grows continuously to increase by 10% every year” and another says the quantity “grows continuously at a rate of 10% per year” they do not mean the same thing.
In some of the examples and applications on pages 308-311 of the text, the number e is being used in situations where any other number would work just as well. The point in these examples is to get more practice in working with exponential formulas and functions involving exponentials while at the same time getting comfortable with e as just another number.
Estimating and Graphing
Values and graphs of functions involving e often can be quickly approximated by using the fact that e is just a bit less than 3.
Draw a quick qualitative sketch of the graph of .
How does it compare with ? (See answer #3 below)
The Slope Property of
Before leaving this section, go back to page 307 of the text and look at the graphs. If you compare them as they cross the y-axis you will see that they all do so at the same point (y=1). You will also see that the slope of the tangent to at that point is a bit less than 1 (in fact the slope from (0,1) to (1,2) is exactly 1, but the graph curves up, so the slope at the left end is a bit less). On the other hand, if you measure it very carefully, you will see that the slope of at x=0is a tiny bit bigger than 1. But the slope of at x = 0 appears to be exactly equal to 1. If you want to check these slopes more carefully, you can do so by “zooming in” on a graphing calculator or computer-based graphing program.
In fact the slope of is indeed exactly one as it crosses the y-axis. This property is really what makes the number e so important in calculus, and this is why the exponential function with base e is often called the “natural exponential function”.
What is the slope of as it crosses the y-axis? (See answer #4 below)
Check your understanding and practice for speed by working through some of the Exercises on pp311-314.
Do enough of the odd numbered questions of each type to convince yourself that you can get the right answers. (Note that, as usual, the answers are in the back of the text and complete worked solutions are in the student study guide - but try to avoid looking at answers or solutions until you have made your own best effort).
Many of these problems are just to get used to working with e in the same way that you have worked with other numbers in previous sections.
Rather than using the solution guide, try to match the questions with text examples or problems you have solved for other bases in previous sections
Eg for #1-4 and 11-12 try comparing with questions from Section 4.2
for #13-16 try comparing with Example#6
for #5-10 try comparing with Examples#2&3
for #19-32 try comparing with Example#4.
#35-56 give you an opportunity to practice using your calculator or computer on similar examples involving numbers you might not want to crank out “by hand”.
As a bare minimum you should do ##1,5,11,15 and 21.
1. Here, if we start with t=0 in year 2000, =6 billion. The continuous growth rate of 2% corresponds to r=0.02 , and year 2010 corresponds to t=10. So the population in 2010 will be given in billions by
2. With continuous growth at rate r=0.1 per annum, the amount at the end of a year is given by so the actual growth over one year is by about 10.5%.
3. The facts that y is always between 0 and 1, with y(0)=1, and that , are true just as we discussed for in Section 4.2.
But at , , and the bell shape through these points appears similar to, but a bit narrower than, that for .
4. Since comes from by replacing x with kx, its graph is obtained by scaling horizontal distances by , and the slope at x=0 will be equal to k.