Calculus 1.3

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1.3 Exponential Functions
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Although some of todays lecture is from the
book, some of it is not. You must take notes to
be successful in calculus.
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We will be using the TI-89 calculator in this
class. You may use either the TI-89 Titanium or
the older TI-89.
TI-89 Titanium
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If 100 is invested for 4 years at 5.5 interest,
compounded annually, the ending amount is
On the TI-89
At the end of each year, interest is paid on the
amount in the account and added back into the
account, so the amount of increase gets larger
each year.
This is an example of an exponential function
exponent
base
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Graph for in a
-5,5 by -2,5 window
MODE
Graph.
FUNCTION
Display Digits
FLOAT 6
Angle.
RADIAN
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Graph for in a
-5,5 by -2,5 window
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Graph for in a
-5,5 by -2,5 window
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Graph for in a
-5,5 by -2,5 window
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Population growth can often be modeled with an
exponential function
Ratio
The world population in any year is about 1.018
times the previous year.
Nineteen years past 1991.
in 2010
About 7.6 billion people.
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Radioactive decay can also be modeled with an
exponential function
Suppose you start with 5 grams of a radioactive
substance that has a half-life of 20 days. When
will there be only one gram left?
After 20 days
40 days
t days
In Pre-Calc you solved this using logs. Today we
are going to solve it graphically for practice.
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46 days
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Graph y(11/x)x in a -10,10 by
-5,10 window.
Use trace to investigate the function.
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We can have the calculator construct a table to
investigate how this function behaves as x gets
much larger.
tblStart .1000
Move to the y1 column and scroll down to watch
the y value approach e.
p
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The TI-89 has the exponential growth and decay
model built in as an exponential regression
equation.
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0,1,2,3,4,5,6,7,8,9
2nd

2nd

alpha
L 1
(Upper case L used for clarity.)
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3
2
alpha
L 1
alpha
L 2
,
2nd
MATH
The calculator should return
ExpReg
Done
Statistics
Regressions
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6
3
2
alpha
L 1
alpha
L 2
,
2nd
MATH
The calculator should return
ExpReg
Done
Statistics
Regressions
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8
2nd
MATH
Statistics
ShowStat
The calculator gives you an equation and
constants
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We can use the calculator to plot the new curve
along with the original points
x
y1regeq(x)
)
regeq
2nd
VAR-LINK
Plot 1
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Plot 1
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What does this equation predict for the
population in 1990?
This lets us see values for the distinct points.
This lets us trace along the line.
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Enters an x-value of 11.
Moves to the line.
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What does this equation predict for the
population in 1990?
In 1990, the population was predicted to be 283.4
million.
This is an over estimate of 33 million, or 13.
Why might this be?
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Enters an x-value of 11.
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To find the annual rate of growth
Since we used 10 year intervals with b1.160626
or
p
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1.4 Parametric Equations
There are times when we need to describe motion
(or a curve) that is not a function.
These are called parametric equations.
t is the parameter. (It is also the
independent variable)
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Example 1
MODE
Graph.
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PARAMETRIC
2nd
T
)
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Hit zoom square to see the correct, undistorted
curve.
We can confirm this algebraically
parabolic function
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Unit Circle
If we let t the angle, then
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Since
We could identify the parametric equations as a
circle.
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Graph on your calculator
Use a -4,4 x -2,2 window.
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Ellipse
This is the equation of an ellipse.
p
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