M119 Notes, Lecture 2 - DOC by Osf3YB8

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									1.5 Exponential Functions and Compounding
                        x
                  1
Graph y  2 ; y    What is the y-intercept? What happens as x   , x   ?
              x

                  2




An exponential function has a constant ratio, while linear functions has a constant
difference.

t                  1        2         3         4         5          6
y                  4        7        10        13        16         19




t                  1        2         3         4         5          6
y                  2        6        18        54       162        486




An exponential function has an equation:
y  P0 a t . What is an equation for the exponential function listed above?




              y2
a x2  x1 
              y1
Exponential models are very useful for many models, including how money grows.
Let’s take $1000 and let it grow by 6% per year under different forms of compounding
for 40 years.

Simple Interest: no compounding




A  Pr t  P  mt  b

Annual Compounding
You get interest once per year and calculate the interest off the balance from the prior
period.




 A  P(1  r )t
Monthly Compounding
You get interest once per month (but only 1/12th of it) and calculate the interest off the
balance from the prior period.




                    12 t
          r 
 A  P 1  
        12 
                   nt
             r
 A  P 1     
             n

Continuous Compounding
What happens if we let n = number of statements per year go to infinity?

Ex. P=1, t=1, r=1 and let just n change




 A  Pert
Back to original example: Let’s take $1000 and let it grow by 6% per year under
different forms of compounding for 40 years.
1.6 The Natural Log
Ex. 10 x  1000 . This can be solved with any base of logarithm, each of which makes the
answer look different, but are mathematically equivalent.




This gives us now all 4 log rules:
1) ln(AB) = ln(A) + ln(B)
2) ln(A/B) = ln (A) – ln (B)
3) ln A B )  B ln( A)
                 ln B log B log x B
4) log a ( B )                   (change of basis)
                 ln a log a log x a

Solve the following using natural logs:
1000  8(5) t




20  7  e 3 x




3e x  10  4 x

								
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