James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

242 Chapter 3 Differentiation Rules
You can see that the interest paid increases as the number of compounding periods snd
increases. If we let n l `, then we will be compounding the interest continuously and
the value of the investment will be
Astd − lim
A0S1 1 r nt
n l ` nD
− lim
n l ` A0FS1
− A 0F lim
− A 0F lim
1
nD
r nyr
1 r nyr
n l `S1
nD
1 1 m
ml `S1
mD
G
G
G
rt
rt
rt
(where m − nyr)
But the limit in this expression is equal to the number e (see Equation 3.6.6). So with
continuous compounding of interest at interest rate r, the amount after t years is
If we differentiate this equation, we get
Astd − A 0 e rt
dA
dt
− rA 0 e rt − rAstd
which says that, with continuous compounding of interest, the rate of increase of an
investment is proportional to its size.
Returning to the example of $1000 invested for 3 years at 6% interest, we see that
with continuous compounding of interest the value of the investment will be
As3d − $1000e s0.06d3 − $1197.22
Notice how close this is to the amount we calculated for daily compounding, $1197.20.
But the amount is easier to compute if we use continuous compounding.
■
3.8
Exercises
1. A population of protozoa develops with a constant relative
growth rate of 0.7944 per member per day. On day zero the
population consists of two members. Find the population size
after six days.
2. A common inhabitant of human intestines is the bacterium
Escherichia coli, named after the German pediatrician Theodor
Escherich, who identified it in 1885. A cell of this bacterium
in a nutrient-broth medium divides into two cells every
20 minutes. The initial population of a culture is 50 cells.
(a) Find the relative growth rate.
(b) Find an expression for the number of cells after t hours.
(c) Find the number of cells after 6 hours.
(d) Find the rate of growth after 6 hours.
(e) When will the population reach a million cells?
3. A bacteria culture initially contains 100 cells and grows at a
rate proportional to its size. After an hour the population has
increased to 420.
(a) Find an expression for the number of bacteria after t hours.
(b) Find the number of bacteria after 3 hours.
(c) Find the rate of growth after 3 hours.
(d) When will the population reach 10,000?
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.