Calculus 1.5


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1.5 Functions and Logarithms
Golden Gate Bridge San Francisco CA
Greg Kelly Hanford High School Richland
Washington
Photo by Vickie Kelly 2004
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A relation is a function if for each x there is
one and only one y.
A relation is a one-to-one if also for each y
there is one and only one x.
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To be one-to-one a function must pass the
horizontal line test as well as the vertical line
test.
one-to-one
not one-to-one
not a function
(also not one-to-one)
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Inverse functions
Given an x value we can find a y value.
Solve for x
Inverse functions are reflections about y x.
Switch x and y
(eff inverse of x)
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example 3
Graph
for
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example 3
Graph
for
Switch x y
Change the graphing mode to function.
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Consider
This is a one-to-one function therefore it has
an inverse.
The inverse is called a logarithm function.
Two raised to what power is 16
Example
The most commonly used bases for logs are 10
and e
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In calculus we will use natural logs exclusively.
We have to use natural logs
Common logs will not work.
is called the natural log function.
is called the common log function.
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Even though we will be using natural logs in
calculus you may still need to find logs with
other bases occasionally.
(base 10)
(base 2)
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And while we are on the topic of TI-89 Titanium
keyboard shortcuts
(square root)
(fifth root)
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Properties of Logarithms
Since logs and exponentiation are inverse
functions they un-do each other.
Product rule
Quotient rule
Power rule
Change of base formula
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Example 6
1000 is invested at 5.25 interest compounded
annually. How long will it take to reach 2500
We use logs when we have an unknown exponent.
17.9 years
In real life you would have to wait 18 years.
p
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Indonesian Oil Production (million barrels per
year)
Example 7
Use the natural logarithm regression equation to
estimate oil production in 1982 and 2000.
How do we know that a logarithmic equation is
appropriate
In real life we would need more points or past
experience.
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Indonesian Oil Production
20.56 million 42.10 70.10
60 70 90
607090
2nd

2nd

alpha
L 1
6
3
5
alpha
L 1
alpha
L 2

2nd
MATH
LnReg
The calculator should return
Statistics
Regressions
Done
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6
3
5
alpha
L 1
alpha
L 2

2nd
MATH
LnReg
The calculator should return
Statistics
Regressions
Done
6
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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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(No Transcript)
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What does this equation predict for oil
production in 1982 and 2000
This lets us see values for the distinct points.
This lets us trace along the line.
82
Enters an x-value of 82.
Moves to the line.
In 1982 production was 59 million barrels.
100
Enters an x-value of 100.
p
In 2000 production was 84 million barrels.