Transcript Pre-Calculus Quiz Review

Pre-Calculus Quiz
Review
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•
Overview
• Exponents
• Logarithms
• Unit Circle/Trig Functions
Exponents
Exponent
2  2  2  2  2 16
4
Base
Simple Exponent Rules to Remember!
b b b
x y
Remember these rules
only apply to LIKE BASES
x
x y
b
x y

b
y
b
x y
xy
(b )  b
Simple Exponent Rules to Remember!
(bc)  b c
x
x x
x
b x b
( )  x
c
c
Simple Exponent Rules to Remember!
b
1
x
y
b
x
 b
x
1
y x
 [(b) ]
Examples:
Expressas an intvalue:
64 3
2

Logarithms
bc = a
logba = c
Logarithmic functions are inverses of
exponential
Simple Logarithm Rules to Remember:
logb (UV)  logb (U)  logb (V )
logb (U /V )  logb (U)  logb (V )
logb (U )  plogb (U)
p
logb (1)  0
logb b  1
Pay attention to LIKE
bases!!!!
Couple of Side Notes
b
logb x
x
logb b  x
x
Couple of Side Notes
‘ln’ is “natural log” which is the SAME thing as logex.
‘e’ is a constant with a value 2.718. It has special meaning because
the slope of a tangent line of a function of ‘e’ is the function itself.
(Dont worry if you don’t understand this bit)
DON’T GET CONFUSED BY THE TERM ‘ln’. It just means log
base ‘e’. You don’t need to memorize the value of ‘e’. You can just
express your answer in terms of ‘e’!!
Example Problem:
5 4  2
x
x 2
Problems for you to try:
ln 34534
e
1
log8 2
4
x 1
1

2  4
x
64
Unit Circle
Unit Circle
• Circle of Radius 1
• Constructed from pythagoream’s theorem
•Angles correspond to points along the circle: these define our
values for cosine and sine
Trigonometric Functions
• From
the unit circle we can deduce:
• sin(x) = y/r
• cos(x) = x/r
• tan(x) = y/x
• …and of course we all know that
x 2 + y 2 = r2
Example Problems with Right Triangle:
Given that cos (θ) = − 3/ 4 and that sin (θ) > 0, find tan (θ) and express your
answer in simplified form (without trigonometric functions).
For some real number x, it is known that sin(x) = ¼ and cos(x) > 0. The value
of cos(x) can be written as √α / 4. What is α?
Common Trig Values to Memorize:
Note how sin is ascending
and cos is descending
If you have these down, then the
tan is simple sin/cos!
Add npi to go to different
quadrants on the coordinate axes
All
Students (sin is positive in 2nd)
Take (tan is positive in 3rd)
Calculus (cos is positive in 4th)
Inverse Trig Functions
The expression sin(x) = ½ reads “the sine of x is one half”.
The expression sin-1(1/2) = ? is the opposite of the above
expression. It can also be interpreted as: “The sine of some
angle is ½, what is the angle?”
Example Problem
Evaluate:
tan(sin-1(1/√2))
tan(cot-1(1/3))