Lecture notes and Practice Problems
on Exponential functions
Exponential Continuous Growth
1. In 1960 the population was 10,000 and in 1980 it was 20,000. What will be the
population in the year 2010? Assume exponential growth. (This statement usually
means to use the equation y=aekt).
The first step is to align the data. For t = 0 the population is 10,000 and for t = 20 the
population is 20,000.
If you put in the values when t = 0 then 10,000 = ae0 and so a = 10,000. Now put in the
values when t = 20 and the equation becomes 20,000 = 10,000e20k. Solve for k by
dividing by 10000 and taking the log of both sides so k = (ln 2)/20. Now we can answer
our question. For the year 2010, t = 50 and y = 10,000e50*ln (2)/20. Use your calculator to
find the answer.
2. Suppose $1000 is invested at 8% compounded quarterly.
(a) Set up the general equation that would calculate S(t), the value of the investment
at any time t where t is the number of years after the initial investment.
(b) How much interest is earned if the money is invested for 5 years?
(c) Convert the equation that you got in part a) to the equivalent formula in base e.
LOGISTIC GROWTH: N = M/(1 + Ae-kt). This is a growth formula with a maximum
y-value or upper bound (or decay with a lower bound).
Example: An Exercise Club wishes to increase its membership. But its facilities will
support at most 800 members. One year ago they had 50 members. Now they have 300
members. How many members will they have in 3 years?
You will use the formula N = M/(1 + Ae-kt) to solve this problem. Align your data so that
for t = 0 there were 50 members and for t = 1 there were 300 members. Now find out
how many members there will be when t = 4. M will equal the upper bound of 800. For
t = 0, 50 = 800/(1 + Ae0) = 800/(1 + A). Now solve to find that A = 15. For t = 1, 300 =
800/(1 + 15e-k) and now solve for the quantity e-k and it will equal 1/9. So the formula
needed to answer the question is N = 800/(1 + 15(1/9) t ) now fill in the value t = 4 to get
the final answer.
Practice problems
1. A new store is opening up in a small town. They want to advertise by word or mouth
only. The total population is 10,500. Initially, 200 people in the community know of
the new store. After 1 day, the number of people who have heard the rumor has
grown to 250. Assume the spread of the rumor follows logistic growth.
(b) Find an equation, N(t), that will give the number of people who know about the
store after t days.
(c) How many people will know about the store in one week?
(d) How many days will it take for 1/2 of the population to have heard about the
store?
2. A new population of wolves was introduced into a national park in 1970. Original
there were six wolves placed in the park. After 10 years there were 20 wolves in the
park. At most the park can only support 100 wolves. Develop a logistic equation that
will model the growth of the wolves in the park.
3. Initially the Va Tech health center reported 10 cases of flu on campus. One week later
the center reported 30 cases of flu. There is a total of 25000 students on campus.
Develop a logistic equation to model this situation. How many students will have the flu
in 5 weeks.
4. The following equation models the number customers for a cable company at any
10000
time t. N =
1 + 49e −.02t
a) What is the limit to the number of customers that they can serve?
b) What number of customers did they start with initially?
c) How many customers did they have 2 years after starting their business?
NEWTON'S LAW OF COOLING:
This equation, N = A + Bekt, deals with the rate at which the temperature of an object
cools in relation to the surrounding temperature A, but it also has applications in the
business world as well. This equation is also bounded.
Example: A cake that is removed from the oven has a temperature of 300 degrees F.
Three minutes later its temperature is 200 degrees F. The room temperature is 70 degrees
F. What will the temperature of the cake be in ten minutes?
Align your data and let t = 0 when the temperature is 300. For t = 3 the temperature is
200. What is the temperature when t = 10? In the equation above A is the surrounding
temperature, so A = 70. When t = 0 then 300 = 70 + Be0. Therefore you have 300 = 70 +
B. Now solve to find that B = 230. For t = 3, 200 = 70 + 230e3k and you need to solve for
k, k = [ln (130/230)]/3. Use your calculator to simplify the calculation and answer the
question for t = 10. N = 70 + 230e10*(ln 130/230)/3. Is k a negative number?
Example:. The Hokie Book Store found that t weeks after the end of a sales promotion
the volume of sales was given by the function S = A + Bekt for 0< t < 4. A is always
the usual volume of sales. Therefore A = 50,000 which is the average weekly volume of
sales before the promotion. The sales volumes at the end of the first and third weeks were
$83,515 and $65,055, respectively. Assume that the sales volume is decreasing
exponentially. Find the sales volume at the end of the fourth week. To find the sales
volume, align the data so that for t = 0, S = 83,515 and for t = 2,
S = 65,055. Fill in the equation for t = 0 and you have 83,515 = 50,000 + Be0 = 50,000 +
B. Now solve for B. Next put in the values for t = 2 and solve for k. Finally answer the
question for t = 3 (the fourth week). Graph the equation for t
Practice Problems
1. A company is trying to expose a new product to as many people as possible
through television advertising. There are 125,000 possible viewers. No one is
aware of the product at the start of the campaign, and after 10 days 40% of the
possible viewers are aware of the product. Newton’s Law is assumed. How many
days will it take to expose 90% of the viewers? Round answer to the nearest day.
2. A company has just hired 50 new employees. None of these new employees have
used a computer. The company places them in their basic computer training
course which runs for 10 weeks, at which time those who have not mastered the
skill are moved to another department. At the end of the 5 week, 20 of the
employees have mastered the required skills and are moved to the production line.
Use Newton’s Law to find the training equation. How many employees are ready
for the production line by the 8th week? Round answer down to the nearest person.
3. An annuity is begun with an initial deposit of $1000 at earnings of 5%
compounded continuously. If money is deposited at the continuous rate of $1000
per year into the account, using Newton’s Law the function for the amount in the
annuity at t years is: A(t) = 21,000 e 0.05t - 20,000 dollars. How much is in the
account in 5 years? When to the nearest year, will you have $100,000 in the
account?
4. According to the weight charts, Sue should weigh 125 pounds. She currently
weights 158 pounds. Her personal trainer has developed a diet that will get her
close to the ideal weight. In t days her weight will be A(t) pounds. On the 10th day
of her diet, her weight is 150 pounds. Assuming Newton’s Law and that she sticks
to the diet, how many days to the nearest day before she weighs 130 pounds?