Section 5-4 The number e and the function ex
 
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  Definition of the irrational number e  
 
Without getting into a discussion of limits right now, we can get an idea of what's happening by taking increasingly larger values of n.  We will talk about limits later on in the year.  Study the following table of values and use your calculator to double check the results:
 
n (1 + 1/n)n 
10 2.593742
100 2.704814
1000 2.716924
10,000 2.718146
100,000 2.718268
1,000,000 2.718280
 If you study this chart, you see that the number e approaches a value of 2.718 . . .  The function ex is called the natural exponential function.  The graph of  ex and e-x are graphed below:
Notice, they fit the pattern of the previous section.  The number e appears in many applications of physics and statistics.  We will take a close look at the number e and how it relates to compound interest.

Compound Interest Formula
 
Where:
A(t) = amount after time t.
Ao = Initial amount
r = rate in decimal
n = number of times compounded in a year.
t = time in years
Thus, if the interest was paid semiannually, n = 2.  Paid quarterly would make n = 4,  Paid monthly, n = 12, etc.

Sample Problems
 
1)  Find the value of a $1 if it is invested for 1 year at 10% interest compounded quarterly.
                 Solution:  Initial amount is $1 with r = .10, n = 4 and t = 1.
                         A(1) = 1(1 + .10/4)4 = 1.1038  This means that at the end of a year, each dollar invested in worth 1.1038 or slightly more than $1.10.  The effect of compounding adds another .0038 % to the interest rate.  Thus the effective annual yield is 10.38%.
 
2)  You invest $5000 in an account paying 6% compounded quarterly for three years.  How much will be in the account at the end of the time period?
 
             Solution:  Initial amount is $5000, with r = .06, n = 4 and t = 3
                         A(3) = 5000(1 + .06/4)12 = $5978.09.  This account pays $978.09 in interest over the three years.
 
3)  What is the effective annual yield on $1 invested for one year at 15% interest compounded monthly?
 
                 Solution:  Initial amount $1, with r = .15, n = 12 and t = 1
                          A(1) = 1(1 + .15/12)12 = 1.1608.  The effective annual yield is 16.08%.  This is a relatively big increase because of the number of times compounded in the year.


The above problems all had one thing in common.  The number of times compounded was a finite number.  We can also have continuous compounding.  That is, compounding basically every second on the second.  This would be rather cumbersome to calculate because the compounding is extremely large.  We can use a similar formula if the compounding is continuous.
P(t) = Poert
Notice the appearance of the number e.  If you look closely at the compound interest formula, you will see imbedded the definition of the number e.  Only use this formula if you are sure the compounding is continuous.

Problems
 
1)  $500 is invested in an account paying 8% interest compounded continuously.  They leave it in the account for 3 years.  How much interest is accumulated?
                 Solution:  Initial amount $500, with r = .08 and t = 3.
                       P(3) = 500(e.08(3)) = 635.62.  This means the interest is $135.62.
 
2)  A population of insects rapidly increases so that the population after t days from now is given by A(t) = 5000e.02t.  Answer the following questions:
     a)  What is the initial population?
     b)  How many will there be after a week?
     c)  How many will there be after a month? (30 days)
             Solutions:
             a)  The initial population is 5000 from the formula.
             b)  A(7) = 5000e.14 = 5751
             c)  A(30) = 5000e.6 = 9111

On to log functions: 
 
Back to the previous section: 
 

 
 
Current quizaroo #  5a
 
 
1)  (3-2 +3-3)-1
 
a)  36
b)  1/36
c)  27/4
d)  4/27
e)  35
 
 
 
2)  92 = 273x, find x

          a) 4/9

b) 9/4
c) 1/3
d) 2/3
e) 0
 

 
 
3)  The half-life of an isotope is 6 days.  If 4.5 kg are present now, how much will be present after 12 days?
 
a)  2.25 kg
b)  1.125 kg
c)  3 kg
d)  9 kg
e)  .565 kg
 
 
 
4)  Find the amount of interest earned if $1000 is invested for 3 years at 7% compounded quarterly.
 
a) $1231.44
b) $210
c) $1210
d) $100
e) $231.44
 
 
  5)  Find the amount of money you will have if you invest $2000 at 11% compounded continuously for 6 years.
 
a) $1869.58
b) $3869.58
c) $1320
d) $3320
e) $4000
 
 
 
 
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