Calculus 1.5

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1.5 Logarithms
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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
2nd
60,70,90

2nd

L 1
2nd
(on a Ti-89)
,
LnReg
The calculator should return
Statistics
Regressions
Done
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(on a Ti-84)
LnReg
Y-VARS
FUNCTION
Y1
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
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What does this equation predict for oil
production in 1982 and 2000?
TRACE
This lets us see values for the distinct points.
Enters an x-value of 82.
In 1982, production was 59 million barrels.
Moves to the line.
Enters an x-value of 100.
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In 2000, production was 84 million barrels.
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1.6 Trig Functions
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Trigonometric functions are used extensively in
calculus.
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Even and Odd Trig Functions
Even functions behave like polynomials with
even exponents, in that when you change the sign
of x, the y value doesnt change.
Secant is also an even function, because it is
the reciprocal of cosine.
Even functions are symmetric about the y - axis.
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Even and Odd Trig Functions
Odd functions behave like polynomials with odd
exponents, in that when you change the sign of x,
the sign of the y value also changes.
Cosecant, tangent and cotangent are also odd,
because their formulas contain the sine function.
Odd functions have origin symmetry.
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The rules for shifting, stretching, shrinking,
and reflecting the graph of a function apply to
trigonometric functions.
Vertical stretch or shrink reflection about
x-axis
Vertical shift
Positive d moves up.
Horizontal shift
Horizontal stretch or shrink reflection about
y-axis
Positive c moves left.
The horizontal changes happen in the opposite
direction to what you might expect.
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When we apply these rules to sine and cosine, we
use some different terms.
Vertical shift
Horizontal shift
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Trig functions are not one-to-one.
However, the domain can be restricted for trig
functions to make them one-to-one.
These restricted trig functions have inverses.
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