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Jim Lamb ers Math 1B Fall Quarter 2004- Le ture 2 Notes
These notes orresp ond to Se tion 4.1 in the text.
Exp onential Fun tions
In many appli ations, the relationship b etween two quantities x and y is des rib ed by an equation of the form y = f (x), where f is a fun tion, and it is known that the rate of hange of y with resp e t to x is prop ortional to the value of y. When this is the ase, the fun tion f (x) is an exponential fun tion. In this le ture, we b egin our dis ussion of exp onential fun tions and learn ab out some appli ations in whi h they are useful. We b egin with a pre ise de nition of an exp onential fun tion.
De nition 1 (Exponential fun tion) Let b be a positive real number that is not equal to 1. The exp onential fun tion with base b is the fun tion f (x) that is de ned by
f (x) = bx^ (1)
where x is any real number. The number b is al led the base of the fun tion f (x), and the number x is al led the exp onent.
Certainly, the most familiar exp onential fun tion is the exp onential fun tion with base 10, f (x) = 10 x^. This fun tion is used in s ienti notation, whi h is used to express numb ers in su h a way that its magnitude and its signi ant digits are separated and therefore more easily read. For example, the numb er 123,456.789 is written as 1 : 23456789 105 , while 0 : 000001234 is written as 1 : 234 10 ^6. Another well-known exp onential fun tion is the one with base 2, f (x) = 2 x^. Various p owers of 2 are often used in omputer s ien e. Before ontinuing, we re all ertain basi prop erties of exp onents. b^0 = 1 for any real numb er b (in luding zero!) If x is a p ositive integer, then bx^ is the pro du t of b b b, where b app ears as a fa tor x times. For example, b^1 = b, b^2 = b b, and so on. If x is a negative integer, then bx^ = (1=b) x^.
If x is a rational numb er, then x = p=q , where p and q are integers, and bx^ is equal to the q th ro ot of bp^.
If x is an irrational numb er, su h as
p 2, we an de ne bx^ as a limit of the sequen e bx^1 , bx^2 , and so on, where x 1 ; x 2 ; : : : is a sequen e of numb ers that onverges to x. We will provide a more pra ti al de nition in an up oming le ture.
Basi Exp onential Graphs
The graphs of two exp onential fun tions, with bases 2 and 1 =3, are shown in Figure 1. From these
−5 −5 0 5
0
5
10
15
20
25
30
35
x
y
y=2x
−5 −3 −2 −1 0 1 2 3
0
5
10
15
20
25
30
x
y
y=(1/3)x
Figure 1: Graphs of y = 2 x^ (left plot) and y = (1=3)x^ (right plot). In ea h plot, the p oint (0; 1) is indi ated by a ir le.
two graphs, we make the following observations, whi h apply to exp onential fun tions in general.
The graph of any exp onential fun tion ontains the p oint (0; 1). This is due to the fa t that any numb er raised to the zeroth p ower is equal to 1. The graph is a ontinuous urve that do es not have any breaks or sharp orners in it. The graph of y = bx^ has a horizontal asymptote at y = 0. Sp e i ally, the graph approa hes the horizontal line y = 0 as x approa hes 1 if b > 1, and it approa hes this same line as x approa hes 1 if 0 < b < 1. If b > 1, then bx^ is in reasing. This means that if x > y , then bx^ > by^. On the other hand, if 0 < b < 1, then bx^ is de reasing, meaning that x > y implies that bx^ < by^.
It follows that p opulations tend to grow exp onentially, so a graph of p opulation as a fun tion of time lo oks somewhat like the graph of an exp onential fun tion. As we will see, this fa t allows us to use urrent and past p opulation data to estimate future p opulation growth. One way of mo deling p opulation growth using an exp onential fun tion is to use the doubling time growth model to measure the p opulation, denoted by P , at a given time t. This mo del onsists of the equation P = P 0 2 t=d^ ; (9)
where P 0 is the p opulation at t = 0, and d is the doubling time, whi h is the amount of time that is needed for the p opulation to double. Any unit of time, su h as hours or years, may b e used for t, but d must use the same unit as t. If the doubling time and the initial p opulation is known, then the p opulation at any time t an b e estimated using equation (9). Later in this ourse, we will learn how to solve more diÆ ult problems involving p opulation growth.
Radioa tive De ay
A radioa tive substan e de ays at a rate that is prop ortional to the amount that urrently exists. It follows that radioa tive de ay over time, like p opulation growth over time, an b e mo deled using an exp onential fun tion. The half-life de ay model an b e used to measure the amount A of a radioa tive substan e that will exist at time t. The mo del onsists of the equation
A = A 0
t=h ; (10)
where A 0 is the amount of the substan e that exists at t = 0, and h is the half-life, whi h is the amount of time that is needed for half of the substan e to de ay. If the half-life and the initial amount is known, then the amount at any time t an b e measured using equation (10). Later in this ourse, we will learn how to solve more diÆ ult problems involving radioa tive de ay.
Comp ounding of Interest
When an amount of money, alled the prin ipal, is dep osited into a savings a ount, or is loaned, it earns interest at a given interest rate, whi h is a p er entage of the prin ipal. The interest is added to the prin ipal after a given p erio d of time, at whi h p oint the ombined amount of the prin ipal and interest may b e reinvested, and therefore earn additional interest. The interest that is paid on reinvested interest is alled ompound interest. Sin e the interest is prop ortional to the prin ipal, the growth of the ombined amount of prin ipal and omp ound interest over time an b e mo deled by an exp onential fun tion that has the form
A = P
r n
nt ; (11)
where P is the prin ipal that is originally invested, r is the annual interest rate, n is the numb er of times p er year that interest is omp ounded (at regular intervals), t is the amount of time, measured in years, that the prin ipal is invested, and A is the amount of money, in luding prin ipal and omp ound interest, that exists after P has b een invested for t years. Later in this ourse, we will learn how to use this mo del to determine how mu h interest an b e earned when interest is omp ounded ontinuously, instead of at regular intervals.
