Chapter 9: Modeling Our World
Lecture notes
Math 1030 Section C
Section C.1: The Exponential Function
Exponential function
A exponential function grows or decay by the same relative amount per unit time. For any quantity Q
growing exponentially with a fractional growth rate r
Q = Q0 × (1 + r)t
where
Q = value of the exponentially growing quantity at time t
Q0 = initial value of the quantity (at t = 0)
r = fractional growth rate (which may be positive or negative) for the quantity
t = time
Negative values of r correspond to exponential decay. Note that the units of time used for t and r must
be the same. For example, if the fractional growth rate is 0.05 per month, then t must also be measured in
months.
Remark
You may notice that the exponential function is identical to the compound interest formula if we identify
Q as the accumulate balance A, Q0 as the starting principal P , r as the interest rate, and t as the number of
times interest is paid. In other words, compound interest is a form of exponential growth.
Ex.1 U.S. population growth.
The 2000 census found a U.S. population of about 281 million with an estimated growth rate of 0.7% per
year. Write an equation for the U.S. population that assumes exponential growth at this rate. Use the
equation to predict the U.S. population in 2100.
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Chapter 9: Modeling Our World
Lecture notes
Math 1030 Section C
Ex.2 Declining population.
China’s one-child policy was originally implemented with the goal of reducing China’s population to 700
million by 2050. China’s population was about 1.2 billion. Suppose China’s 2000 population declines at
a rate of 0.5% per year. Write an equation for the exponential decay of the population. Will this rate of
decline be sufficient to meet the original goal?
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Chapter 9: Modeling Our World
Lecture notes
Math 1030 Section C
Graphing Exponential Functions
How to graph exponential functions
The easiest way to graph an exponential function is to use points corresponding to several doubling times
(or half-lives in the case of decay), that is the points
(0, Q0 ), (Tdouble , 2Q0 ), (2Tdouble , 4Q0 ) · · ·
in the case of exponential growth, or
(0, Q0 ), (Thalf ,
Q0
Q0
), (2Thalf ,
)···
2
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in the case of exponential decay.
Ex.3
The growth rate of the U.S. population has varied substantially during the past century. It depends on
the immigration rate, as well as birth and death rates. Starting from a 2000 population of 281 million,
project the population in 2100 using growth rates that are just 0.2 percentage point lower and higher
than the 0.7% used in Example 1. Make a graph showing the population through 2100 for each growth
rate.
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Chapter 9: Modeling Our World
Lecture notes
Math 1030 Section C
Alternative Form of Exponential Functions
Forms of the exponential function
Let Q represent the quantity at any time t and Q0 represent the initial value.
• If given the growth or decay rate r, use the exponential function in the form
Q = Q0 × (1 + r)t
Remember that r is positive for growth and negative for decay.
• If given the doubling time Tdouble , use the exponential function in the form
t
Q = Q0 × 2 Tdouble
• If given the half-life Thalf , use the exponential function in the form
1T t
half
Q = Q0 ×
2
Section C.2: Selected Applications
Inflation
Inflation
Because the prices tend to change with time, price comparisons from one time to another are meaningful
only if the prices are adjusted for the effects of inflation. We can model the effects of inflation with an
exponential function in which r represents the rate of inflation.
Ex.4 Monthly and annual inflation rates.
The U.S. government reports the rate of inflation (as measured by the Consumer Price Index) both monthly
and annually. Suppose that, for a particular month, the monthly rate of inflation is reported as 0.8%. What
annual rate of inflation does this imply? Is the annual rate 12 times the monthly rate? Explain.
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Chapter 9: Modeling Our World
Lecture notes
Math 1030 Section C
Physiological Processes
Physiological processes are exponential
Many physiological processes are exponential. For example, the concentration of many drugs in the bloodstream decays exponentially. Alcohol is an exception since its concentration decays linearly.
Ex.5 Drug concentration.
Consider an antibiotic that has half-life in the bloodstream of 12 hours. A 10-milligram injection of the
antibiotic is given at 1:00 p.m. How much antibiotic remains in the blood at 9:00 p.m.? Draw a graph that
shows the amount of antibiotic remaning as the drug is eliminated by the body.
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