Calculus 1.5

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1997 BC Exam
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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 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
3
menu
3
enter
1
menu
4
Zoom to -1,7 x -1,3
menu
4
Zoom Square
B
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example 3
Graph
for
Switch x y
Clear the previous graph.
Graph the curves as functions.
ctrl
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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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A Few Historical Notes
Logarithms were "invented" by a Scottish nobleman
named John Napier (1550-1617).
Logarithm is the combination of two Greek roots,
Logos (reason or ratio) artihmus (number).
The word logarithm was introduced in Napiers
1614 work, Mirifici Logarithmorum canonis
descriptio, (description of the wonderful canon
of logarithms), originally published in Latin.
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Astronomer Johannes Kepler read Napiers work in
1616, and used logarithms in developing his Third
Law of Planetary Motion.
Keplers Third Law states The square of the
orbital period of a planet is directly
proportional to the cube of the semi-major axis
of its orbit.
Kepler published the third law in 1620 in a book
titled Euphemerides, dedicated to Napier. He
later published his own work on logarithms.
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The term natural logarithm (actually
logarithm Ephemerides us naturalis) was first
used by an Italian mathematician, Pietro Mengoli
(1626-1686).
Surprisingly, the notation ln for natural logs
was not first used until 1893 by an American,
Washington Irving Stringham (1847-1909).
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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.
If you leave the base blank, it assumes you want
a common log. For example
Or you can specify the base
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Here are shortcuts for accessing the various
symbol palettes on the TI-inspire
ctrl
Characters/Symbols
Expression Templates
Trig Symbols
Equalities Inequalities
ctrl
Punctuation Mark
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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