MATH 1300 Lecture Notes Tuesday, August 27, 2013
1. Section 1.2 of HH - Exponential Functions
(a) Exponential Functions: An exponential function has the form y=abx, or using
function notation, f(x) = abx. Note that the variable, x, is in the exponent.
•ais the initial value, the value of the function when x= 0, because b0= 1.
•bmust be a positive real number. If b > 1, then the relationship is called
exponential growth. If 0 < b < 1, then the relationship is called exponential
decay.
•Just as linear functions have a constant rate of change, exponential functions
have a constant percent change. For example, if a population increases by 5%
every year, then the growth factor bis computed to be 1 + .05 = 1.05. If the
population decreases by 5% every year, then the decay factor bis computed
to be 1 −.05 = .95.
(b) The Family of Exponential Functions: Changing the values of aand bin an
exponential function changes the graph of that function. The value of ajust
determines the y-intercept of the graph. The value of bdetermines whether the
graph increases or deacreases, and how quickly it increases or decreases. Refer to
Figures 1.19 and 1.20 in section 1.2 to review these patterns.
(c) Continuous Growth/Decay: Every exponential function f(x) = abxcan be written
in the form f(x) = aekx (in this form, b=eksince f(x) = aekx =a(ek)x).
•If kis a positive number, then ek>1 so this is continuous growth.
•If kis a negative number, then 0 < ek<1, so this is continuous decay.
•The positive number |k|is the continuous rate of growth/decay. It is impor-
tant to be able to convert exponential functions between these two forms. For
example, if you were asked for the growth rate of the exponential function
whose continuous growth rate is 50%, then f(x) = ae.5x=a(e.5)x=a·1.65x
shows that the growth rate is 1.65 (so there is a constant growth of 65%).
(d) Half - life/Doubling - time:
•The half-life of a substance is the amount of time it takes for the initial amount
to be cut in half. If the half-life of a substance is t0and the initial amount
is a, then the exponential function modeling this situation is f(t) = a(1
2)
t
t0
(note that f(t0) = 1
2a, which is the definition of half-life). For example, if
the half-life of a radioactive substance was 2 years, then the amount of this
substance left after time twould be f(t) = a(1
2)t
2=a(.71)t. So the decay
factor is .71 (meaning the substance is decreasing by 29% each year).
•The doubling time of a substance is the amount of time it takes for the initial
amount to be doubled. If the doubling time of a substance is t0and the
initial amount is a, then the exponential function modeling this situation is
f(t) = a(2)
t
t0(note that f(t0) = 2a, which is the definition of doubling time).
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