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Class Notes Precalculus I, 10/27/98
Exponential Functions; Inverse Functions and Logarithms
The quiz problem was to
- write the exponential function P(t) which models and investment of $3,000, growing at 7
percent annual interest compounded annually,
to use this function to determine the principal at the end of 10 years, and
to determine the average rate at which principal grows between t = 9.9 years and t =
10.1 years.
We solve the problem as follows:
- Since in t years the existing principal is increased by seven percent a total of t
times, we can obtain the principal after t years by multiplying the original principal by
1 + .07, or 1.07, t times.
- Thus, as seen in the preceding lecture, the principal after t years will be P(t) = $3000
(1.07) ^ t.
- After 10 years the principal will be P(10) = $3000 (1.07) ^ 10, which is easily
evaluated using a calculator.
- To find the average rate at which principal grows between two given times, we must
divide the change in principal `dP by the time interval `dt.
- To find the change in principal
between t = 9.9 and t = 10.1, we calculate `dP = P(10.1) - P(9.9).
- This is easily accomplished with a
calculator, and we obtain `dP = $80 (approximately).
- The time interval is clearly `dt
= (10.1 - 9.9) years = .2 years.
- We therefore conclude that the
average rate is `dP / `dt = $400 / year (approximately).
We were also to write the Q(t) function modeling, as a function of time, the
exponential decay of a sample whose initial activity was 200 decays/second, if the
activity decreased by four percent every hour.
- The four percent hourly decrease results in a growth factor of 1 + growth rate = 1 +
(-.04) = .96, so that the initial activity will in t hours change by factor .96 ^ t.
- Since the initial activity was 200 decays/second, we can therefore write Q(t) = 200
(.96) ^ t.
- Some class members felt it necessary to change the 200 decays/second to an hourly rate,
since the clock time t was measured in hours. This is acceptable but unnecessary. When the
factor .96 ^ t is calculated, t must of course be in hours. But this factor will then
multiply the initial activity, whether that activity is given in decays/second,
decays/minute, decays/hour or any other valid units.
- So we could say that Q(t) = 200
decays / minute * .96 ^ t, and also that Q(t) = 720,000 decays / hour * .96 ^t, since 200
decays/minute is identical to 720,000 decays/hour.
Video File #01
http://youtu.be/Y6JSNcPiiCE
Inverse Functions
We can form a partial table of an inverse function by first creating a table
representing the original function, then switching the columns of the table.
- For example, below we see a table
for the function y = 10^x.
- We note that the domain of the
function y = 10 ^ x consist of all real numbers, since any real number can be substituted
for x with a resulting real-number of value.
- The range of this function is all
positive numbers, since 10 ^ x can be made as small as desired by letting x be a
sufficiently large negative number (10 ^ -p = 1 / 10 ^ p, which gets small as the
magnitude of t gets large), and as large as desired by letting x be a sufficiently large
positive number.
We form the table of the function inverse to y = 10 ^ x by and switching the
columns of the y = 10^ x function, obtaining the table at right in the figure
below.
- The domain of this function is the
range of the previous function, and the range of this function is the domain of the
previous function.
Video File #02
http://youtu.be/1oo3JrasGQw
The table we obtain when
we reverse the columns of y = 10 ^ x represents the function y = log(x), which is the
function inverse to y = 10 ^ x. http://164.106.222.225/pc1spring99/lectures/pc_981013_981029/pc1_1027/class_
notes.htm
- If we graph the two functions, we see that the coordinates of the graph of y = 10^x are
reversed to obtain the coordinates of the graph y = log (x).
- For example, (1, -.1) of the graph of y = 10 ^ x is reversed to obtain the point (-.1,1)
of the graph of y = log (x).
- When the coordinates of that point are reversed, the x and y axes are effectively
reversed.
- The result is that the point is reflected across the line y = x, with the points lying
at equal distances from the line and connected by a line segment perpendicular to the y =
x line.
- A graph depicting the relationship between the original function y = 10 ^ x and its
inverse y = log(x) is shown below.
Video File #03
http://youtu.be/c89i--u8Cx4
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