Clicker Question 1
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At what point do the graphs of
f (x ) = 2x and g (x ) = 10x cross?
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A. (1, 0)
B. (0, 1)
C. (0, 0)
D. (2, 10)
E. (10, 2)
Clicker Question 2
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What is the value of P (t ) = 100(5t )
when t = -2 ?
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A. 4
B. -2
C. -4
D. 2500
E. 100
About that number e ….
(9/30/11)
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What is this mysterious number e
anyway? Well, it is the natural limit of
compounding, in the following sense:
If you invest $1 for 1 year at 100%
interest compounded annually, you
make $1(1+1)1 = $2.00
If you compound twice, you make
$1(1 + ½)2 = $2.25
The story of e, continued
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Compound quarterly: $1(1+1/4)4 = $2.44
Compound monthly: $1(1+1/12)12 = $2.61
Compound daily: $1(1+1/365)365 = $2.71
Definition: e = limit (1+1/n)n as n gets big.
So, compound continuously, make $e !!!
We will see that e is “natural” in various
ways as this course goes on.
New Functions from Old:
Inverse Functions
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Inverse functions: If a function can be
reversed (i.e., the output becomes the
input and the input becomes the
output), the reverse function is called
the inverse of the original.
For example: the inverse of f (x) = x 3
is the cube root function.
A function is (directly) invertible only if
its graph hits every horizontal line at
most once (it is “one-to-one”).
Inverses - Examples
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f (x ) = x 2 is not directly invertible.
(Why?) But if we restrict its domain so
that it is one-to-one, then we can invert
it on that domain. (To what?)
In general to invert a given formula:
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Write in the form y = f (x )
Solve for x .
Exchange x and y in your formula
Examples….
Clicker Question 3
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Find a formula (y as a function of x ) for
the inverse of y = (2x +1)3.
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A. y = (2x +1)1/3
B. y = (2x +1) -3
C. y = (x 1/3 – 2) + 1
D. y = (x 1/3 – 1) / 2
E. y = 2x 1/3 - 1
Inverse Trig Functions
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The sin and cos functions cannot be
directly inverted. (Why?) But restricting
the domains appropriately (How? Look
at the graphs!), we obtain the inverse,
or arc, trig functions, arcsin, arccos,
arctan, etc.
These functions take in a number and
put out an angle (in radians!). So, e.g.,
arcsin(x ) means
“the angle whose sin is x .”
Clicker Question 4
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What is the arcsin(-2 / 2)?
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A. 45 °
B. -  / 4
C.  / 4
D. - 45 °
E. 0
Assignment for Monday
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Read Section 1.6 pages 58-62
and 67-69.
In Section 1.6 do Exercises 1, 2, 3-13
odd, 21, 63-71 odd. (Suggestion on 69:
draw a right triangle and label its sides
so that one of the acute angles has sin
equal to x. Using Pythagoras, what is
the cos of that angle? Same technique
on 71.)