This preview shows page 1. Sign up to view the full content.
Unformatted text preview: P RE C ALCULUS
A guide
This guide was compiled for one purpose, to help people who are having
difficulty understanding the way Pre Calculus is taught. I think visually , so
I wrote this guide visually. I wish you good luck, and please enjoy this
resource. Austin Schafer
2011 PR EFAC E
Why m ak e di g i tal no te s
Many people have asked me why I would go through the effort of making a digital
guide to a mathematics course. I have many reasons, at first I really just wanted to
use my computer for something new; I thought I could make better notes using a
computer. I later learned that my digital notes, which would become this guide, were
not better because they were more extensive or better illustrated, they were better
because they are distributable. A guide of a class is not helpful when there is only
one copy of it. The power of these notes lies in their ability to be reproduced. I would
be proud to have my work spread to new people, please though, don’t take credit for
what you didn’t write. A g ui de b as e d o n w hat?
I have used many sources in creating this guide. The person who influenced it most
was my course teacher, Mr. Torre. I also used the course textbook for reference; the
textbook is titled, Precalculus: Graphical, Numerical, Algebraic: Seventh Edition
published by Pearson. My final resource was the internet, I would like to give some
extra credit to Wikipedia, which is an excellent free resource for even the most
complicated mathematics. Unit 1 •
•
•
•
• Lines
Systems of Equations
Matrixes
Determinants
Cramer's Rule of Expansion by Minors Pre Calculus Page 1 Lines and Stuff
Monday, August 30, 2010 Point: An infinitely small spot. That can be defined as a location
at the intercept of two or more axis values. An example is the
point xvalue 4, y value 3. This would be written (4,3), and is
located at 4 "over(right)" and 3 "up", from the origin, which is
an arbitrary point labeled (0,0). Linear Equation
Typically a linear has a first degree xterm and yterm.
A solution is (5,2), but there is infinitely more. I can represent all the points as a picture on a
graph. This linear equation forms a line that is infinitely long, and made of infinite solution points.
A line is a representation of all of a linear equations solutions.
If What is a Slope? then the line is horizontal It is a rational number that describes the 'tilt' of a line.
Commonly we think
Specifically, slope is a rate of change of y as x changes, and can be determined by any two points on a
line. (Slopes do exist in Quadratic Functions too, see "A New Way To Slope") then the line is vertical. Parallel Lines : Lines are parallel if their slopes are
equivalent .
Perpendicular Lines: Lines that intersect at a 90⁰ angle.
This means that the line's slopes have inverse reciprocals
of each other.
Intersecting Lines: This type of lines comprises all other
lines. Since all lines not that are not parallel must
intersect, all lines that are not parallel or perpendicular
must intersect at some other angles. These types of lines
are called intersecting. Writing the equation of the line..
Slope Intercept Form Standard Form Point Slope Form A and B must be
whole integers
slope yintercept Vocab Note: Singular Functions have no Inverse When a function is in standard form, the following equations can be used
to determine the characteristics of the line and quickly graph it.
Slope Yintercept Xintercept Practice Problem:
Suppose & line passes through (7,1), what is the equation of the line ? What is the standard form of the line? Practice Problem Answers:
The Equation is Pre Calculus Page 3 Systems of Equations
Wednesday, September 01, 2010 A system of Equations is a set of equations (duh). A system usually has a solution, if it does not then it is inconsistent.
When working with an system, there is some basic manipulation rules you can use.
1. Multiply an entire equation by a nonzero constant
2. Interchange any two equations
3. Replace an equation with a linear combination of addition An Example of Replacing an Equation with a Linear Combination of Addiction Pre Calculus Page 4 Graphing Vocabulary
Friday, September 03, 2010 Consistent & Independent Inconsistent Constant & Dependent One Solution Zero Solutions Infinite Solutions  If you don't want to substitute the 'y' you just
found (i.e. It would be messy), you can use the
same elimination process that you just used to
find the 'y', to find 'x'. Pre Calculus Page 5 Matrix Stuff
The Matrix Saturday, October 23, 2010 The Matrix is a simulated reality perceived by most humans created by sentient
machines. This is chronicled in The Matrix, a 1999 scientific documentary. The coding of
the matrix is show right. Alternatively in math, a Matrix is simply a rectangular arrangement of elements, which are numbers. This is a 2x2 matrix For Example, [1,0] is row 1 of Matrix A. Rows are the horizontal Columns are vertical These are elements Matrix A can be written as [A] The dimension are called order, as in the order of these matrixes are the same
and have the same order. Matrix Arithmetic
(A)
(B) (C) E.M.O.  Study Chapter 7.2 pg 579 Addition  Must be of Equal Order
Subtraction  Must be of Equal Order
Scalar Multiplication  In matrix land, there are nebular numbers called scalars or Rock Climbers
Matrix Addition, it works for Subtraction using the same principle. Augmented Matrixes
An augmented matrix is a square matrix which has an extra column.
This last column is not manipulated with the matrix. Augmented
Matrixes are used in matrixes for solving systems of equations. Pre Calculus Page 1 Identity and Augmented Matrixes
Tuesday, January 25, 2011 The Identity Matrix  the scalar matrix with the main diagonal all 1's Matrix Algebra Equation
Classic Linear System
This system of equations is equal to:
This Matrix can be "Augmented" to look like this:
this augmented matrix represents the standard form of the system of equations Pre Calculus Page 7 Solving with Matrixes
Sunday, February 06, 2011 Solving Systems of Equations with Matrixes
Since Matrixes are just representations of system of equations, they can be
manipulated just like systems of equations to solve the system.
We can convert a system of equations to a matrix in the following way:
Augment Line This Becomes We solve it like this:
Multiple Row 1 by
, add
result to Row 2, and replace
Row 2 with this result
Multiple Row 1 by
, add
result to Row 3, and replace
Row 3 with this result
Multiple Row 2 by
, add
result to Row 3, and replace
Row 3 with this result
Multiple Row 2 by , add result
to Row 1, and replace Row 1
with this result
Multiple Row 3 by , add result
to Row 2, and replace Row 2
with this result
Pre Calculus Page 1 Multiple Row 3 by , add result
to Row 2, and replace Row 2
with this result
Multiple Row 3 by , add result
to Row 1, and replace Row 1
with this result Next we convert
the matrix back
into a system of
equations This matrix form is called reduced row echelon form and is the easiest way to solve
systems of equations using Matrixes. A reduced row echelon matrix will be an
identify matrix with an augmented section. This allows for the simple conversion
to find the values of the variables. Pre Calculus Page 2 Matrix Multiplication
Saturday, November 06, 2010 Consider the matrix products [A] x [B]. This product only exists if the
number of rows in [B] is equal to the number of columns in [A]
Matrix multiplication works like this: You can determine the size of the resultant matrix of [A] x [B]
[A] x [B]
[B] x [A] Consider the Matrixes Obviously it makes a big difference what order the Matrixes are in. Pre Calculus Page 9 Matrix Determinants and Inverses
Monday, October 18, 2010 The Determinant represents the
numerical (ordinal) value of a matrix. Note: Determinants are defined only for square
matrixes. Determinants
There is a simple test that determines if a 2x2 matrix has an inverse: If ad  bc ≠ 0, then The Value of a Determinant
The value of can be written and solved like this: Matrix Inverses Pre Calculus Page 10 Unit 2 •
•
•
•
•
•
•
•
•
•
•
•
• Interval Notation
F(x) Notation
Extrema and Inflection Points
Piecewise functions
Pasta Sauce
Vertical, Horizontal and Slant Asymptotes
Types of Symmetry
Concavity and Tone
Implicit and Explicit Concavity
Function Characteristics
Continuity
Types of SPOD
Partial Fraction Decomposition Pre Calculus Page 11 Determining Domain and Range Graphically
Thursday, January 20, 2011 Domain: Not a function, does ; Pre Calculus Page 13 F(x) Notation
Wednesday, January 19, 2011 F(x) Notation Take, for example,
becomes using F(x) notation, use F(x) in lieu of y.
"F" is the Function Name, it could be any letter, but X is the
most common. "X" Indicates the domain variable Examples: Variable
Unspecified Constants Arithmetic Combinations of Functions:
Domain: All real
Domain: All Real
Domain: All Real
Domain: All Real
Domain: All Real
Domain: Denominator cannot equal 0, because you cannot divide by zero. Function Compositions: Compounding 2 Functions "F of G of x" Take these two functions as a given. Input g(x) into the f(x) equation
for f(g(x)) Pre Calculus Page 14 Input g(x) into the f(x) equation
for f(g(x)) Input f(x) into the g(x) equation
for g(f(x)) Pre Calculus Page 15 Extrema, Inflection, Decreasing, and Concave relations
Tuesday, October 05, 2010 Pre Calculus Page 16 Mr. Torres Pasta Sauce
Monday, October 11, 2010 Tomatoes 1 can + 3/4 can of water Onions 2 Garlic 4 cloves Salt, Pepper, Sugar 1, 1/2, 2 tsp
Olive Oil 3 TBSP Basil 1 bunch Fresh Oregano 1 Sprig Tomato Paste 1 can + 2 cans of water Simmer 90 minutes →Purée →cook 1/2 hour more Pre Calculus Page 17 Asymptotes
Wednesday, December 01, 2010 An asymptote is most commonly encountered are of curves of
the form y = ƒ(x). They are simply a line where as x approaches
a value, f(x) gets increasingly "wild".
A textbook example is when
, as x approaches zero,
like when it is
gets increasingly large. This is
demonstrated by the graph at right.
In this case, asymptotes, it is a vertical asymptote. As you may notice, at
, there is no value for
, all
vertical asymptotes have an undefined
or y value. This is
because they are caused by dividing , or more nearly dividing
by, zero. As the function approaches anything/0, the output
gets really strange.
Take for example , the function works well until the x value approaches x=3, because at x=3, the value of
is undefined. The graph looks like this. Horizontal Asymptotes
Friday, December 03, 2010
4:25 PM A horizontal asymptotes exists whenever a
value approaches a constant as x approaches
. Simply put, it is
when a function approaches a constant value of y as its x value gets REALLY big. A graph of a function with two
horizontal asymptotes is show at right. The dotted lines are the horizontal asymptotes, while the solid line is the
function. For example, the function ƒ(x) = 1/(x 2+1) has a horizontal asymptote at y = 0 when x tends both to
and
because, respectively, This function's asymptote would be described like this: Pre Calculus Page 18 Slant Asymptotes
Wednesday, December 01, 2010 When a linear asymptote is not parallel to the x  or yaxis, it is called an oblique asymptote or slant asymptote. This equation has the following slant asymptote;
BLUE is the asymptote
BLACK is the function This function also has a vertical asymptote at Pre Calculus Page 20 Sketching an Asymptotic Functions and Some Practice Problems
Friday, October 15, 2010 1)
2)
3)
4) Check for Spods
Vertical Asymptotes
Slant Asymptotes
Interesting features (aka single point discontinuity) Practice Problems F(x) = x2 + 8x  11
y= x2 + 8x 11
X=y2 + 8y  11
X+27= y2 + 8y + 16
X + 27 = (y + 4)2
Y = 4 Pre Calculus Page 21 N=Numerator
D=Denominator Partial Fractions Decomposition
Friday, October 15, 2010 Proper or Improper
Proper Numerator≥Denominator
Think 'Degree' now!!!
Step 1: Factor the denominator. If the denominator is not factorable, Stop!!! And put a sad face, it does
not exist.
Step 2:
Find A and B Expansion Expression
Step 3:
Step 4: Throw algebra at it until it dies Step 5: Extract the System
A+B=5
A  4B = 10
Step 6: Solve it A=2
B=3 Step 7: is the decomposition of Pre Calculus Page 22 Algebra Slugfest Problem
Tuesday, October 19, 2010 A+D=3
2D+E=4
6A+B+3D+2E=16
2B+C+6D+3E=20
9A+2C+6E=9 1 0 0 1 0 3 0 0 0 2 1 4 6 1 0 3 2 16 0 2 1 6 3 20 9 0 2 0 6 9 Pre Calculus Page 23 RREF A=1
B=4
C=0
D=2
E=0 Improper RPF
Tuesday, October 19, 2010 Use Polynomial Long Division
Top is larger degree, so it is improper How to use, See "Algebra Slugfest Problem" Pre Calculus Page 24 Unit 3 •
•
•
•
•
• New Sloping Method
Brutal Algebra
Slope of a Point
Tangent Line Equation
Exponent Rule Shortcut for Finding Derivatives
Horizontal Tangent Lines Pre Calculus Page 25 A New Way to Slope
Tuesday, October 19, 2010 Speed Formula Slope indicates the
inclination of a line Miles for every Hour
Speed is a rate of change over distance as time changes Distance
225 Miles Average Rate of Change of Distance with Respect to Time 4.5 hours Linear Functions Are Lines, and thus have a
Defined slope for the entire length of the line. Slope
Is
(+) NonLinear Slope
is BAD UNLESS
you look at
"Local Slope" Quadratic Functions have
no defined slope over their
entire length. This Function looks line ish if you zoom in
enough, that's called "local Linearity" Local linear at X=A
Defined at X=A This has a limit,
but will never
have a constant
slope until we
have only two
points Locally
Linear Continuous at X=A Continuous Defined This determines the
neighborhood, the
local slope, & we're
trying to find the exact
slope of f(x) at x=2
I'm gonna reestablish the neighborhood! What is the local
linearity slope of
f(x)=
at (2.9) M=11.41 (1.99,8.880599 M=11.9401 (2,9) (1.999,8.988005999) M=11.994001 (2,9) (2,9) (1.9,7.859) (2,9) (1.9999,8.9888000599) M=11.99990001 Connecting two points
locally is called the
secant line.
M=11.stuff
The closet the points are
together, the closer the
slope of the secant line is
to the actual local slope. Pre Calculus Page 27 (2,9)
Tangent
The tangent line is the
slope of the line at x=2,
or at x= anything on the
function.
In this case the tangent
line is the instantanious
slope Answer to Practice Problem Left Practice Problems
2nd Point x G(x) 257.542 208.47132 247.132402 3.001 206.24611 246.113024 3.0001 At 231.7542 3.01 At x=3
Find the slope of G(x) at x=3 Secant Line Slope 3.1 206.02460 246.0113002 So the slope of G(x) AT x=3 is 246
:) Mtan = 246 at x=3 Now, what is the equation of the tangent line to G(x) at x=3 point of tangency
246
G(3) so 206 Pre Calculus Page 28 Finding a Slope at ANY Point
Friday, October 22, 2010 Find the slope Point of Tangency H is not equal to
zero yet, this
slope is still a
secant line This is the
tangent Line
Equation
Specific to Lim h→0 Lim h→0 Lim h→0 Lim h→0 Lim h→0
2x is the TAN slope function!!! Pre Calculus Page 29 Lim h→0 Brutal Algebra to Find Tangent Line Slope Formula
Monday, October 25, 2010 Find Find the Tangent Line Slope Formula
For
In general your goal is to start with 'h' being the
difference
between the
two points Now find , Use and let h=0 The Secant Line
formula for G(x)
at any two points
(x,F(x)) and
another point H
away. Is For any point on G(x) is
ith h=0, so with h=0 Tangent Line Equation
Monday, October 25, 2010
12:10 PM Stay Variables XCoordinate of your Point of Tangnecy usually given
I.e. "At x=1" Pre Calculus Page 30 Exponent Rule Shortcut for
Wednesday, November 03, 2010 "F prime of X"
"The derivative of X"
"The tangent line slope function at any X"
Multiply coefficient
by exponent Find If
Subtract one from each Exponent 51 Pre Calculus Page 32 41 31 21 11 01 Horizontal Tangent Line
Friday, November 05, 2010 The local slope around an extrema is 0 The slope of the actual function at ay X
Where does F(x) have a horizontal Tangent Line The slope of the tangent Line at any X
Slope while f(x)=0 Set Setting F(x) = 0 solving finds the x coordinates for
"points of horizontal tangency" Wait, this doesn't work!!
What else has to be true about in order to prove Extremaness? There must be a sign change in the neighborhood!! +   + Maxima
Minima The organized way to explore the sign of
with a sign chart
0++++++++0 4 3 2 Pre Calculus Page 33 0 2 3 4 is More Horizontal Tangent Lines
Monday, November 08, 2010 Set equal to zero Find Points of Horizontal Tangency must change signs of locally for there to be an extremun.
+00+++00+++++++0 4 3 2 +++++++++00+++++++++++ 4 3 2 Pre Calculus Page 34 0 2 3 4 0 2 3 4 Unit 4 •
•
•
• Sign Charts
Prime Functions
Tone and Concavity
Rectilinear Motion Pre Calculus Page 35 Superimpose Sign Charts
Monday, November 08, 2010 Describes the y coordinate
describes the tone of F(x)
Describes the concavity of F(x)
Describes the tone of Study Sheet on Primes
Tuesday, November 09, 2010
11:23 AM For each of the following functions, characterize the entire function in terms of tone
and concavity, utilizing a sign chart and then sketch an accurate, though not
necessarily artistic, rendition of ƒ(x). Compare to your calculator’s graph.
1) ƒ(x) = x3 – 48x
2) ƒ(x) = x3 + 3x2 – 45x – 50
3) ƒ(x) = 2x3 + 27x2 + 84x
4) ƒ(x) = x3 – 12x2 – 66x – 92
5) ƒ(x) =
6) ƒ(x) = x3 + 9x2 + 15x + 2 Pre Calculus Page 37 Study Sheet on Primes Answers
Tuesday, November 09, 2010 For each of the following functions, characterize the entire function in terms of tone
and concavity, utilizing a sign chart and then sketch an accurate, though not
necessarily artistic, rendition of ƒ(x). Compare to your calculator’s graph.
1) ƒ(x) = x3 – 48x 2) ƒ(x) = x3 + 3x2 – 45x – 50 3) ƒ(x) = 2x3 + 27x2 + 84x 4) ƒ(x) = x3 – 12x2 – 66x – 92 5) ƒ(x) = 6) ƒ(x) = x3 + 9x2 + 15x + 2 Pre Calculus Page 39 Study Sheet Number 4
Wednesday, November 10, 2010 No Zeros, means that there is no horizontal tangent lines in the original equation
 4) ƒ(x) = x3 – 12x2 – 66x – 92 4
++++++++++0 Before x=4, from the negative side, F(x) has negative tone, and positive concavity
After x=4, from the negative side, F(x) has negative tone, and negative concavity Pre Calculus Page 40 Tone and Concavity of the Derivative
Wednesday, November 10, 2010 Consider the above graph. Y5 is the prime of Y4 Correspond in the following ways:
a)
b) c) xintercepts* of
correspond to extrema in
xintercepts* of
correspond to extrema on
as
well as points of inflection on
Tone of
is the concavity of
and the youtput of *as long as there's a sign change at Xintercept Pre Calculus Page 41 Rectilinear Motions
Monday, November 15, 2010 Rules and Conventions
Motion back and forth along a number line(axis) 1. The Placement/Orientation of the number axis is arbitrary
2. Unless otherwise stated, number axis is horizontal; right being (+) and left
being ().
3. The velocity and acceleration are and not necessarily constant!
4. We will have 3 functions to work with:
a. S(T) "Position"
b. V(T) "Velocity" T is the universal independent variable
c. a(T) "Acceleration"
5. "T" is in seconds unless otherwise stated In motion Example:
A particle in rectilinear motion (call him Fred), has a
position function of At T=2 Fred is at 55 on number line, you don't know
anything else except that at 2 seconds he is at 55
When T=3, then
number line. , Fred is now at 48 on the "Velocity is Speed and Direction" This is the Velocity Equation!! Velocity MPH , so 3 is the initial point, or = Example: Does Fred Turn Around at Where is Fred Stopped? Justify
0 2 5 Plug the times into the position formula
T=2,5
++++++00+++++++++++++++++++ 5.5
This is what it would look like it time was made a vertical dimension, stretching out
the overlapping movements. 5
4
2
3 28 40 55 0.5 Pre Calculus Page 42 Exam 4 Study Sheet With Answers
Tuesday, November 16, 2010 Do a sign chart to determine if
there is a sign change. Pre Calculus Page 43 AIII
BI
CII Pre Calculus Page 44 Cannot use exponent Rule Find the zeros in the prime
x=0 When Sally has stopped T=12, y Pre Calculus Page 45 Unit 5 •
•
•
•
•
• Velocity Function
Acceleration
Free Fall
Rounding Conventions
Rectilinear Motion
Rectilinear Displacement Bonus: My Responses to Exam 5 Pre Calculus Page 46 Velocity Function
Monday, November 29, 2010 I can stop without changing directions.
TRUE In 1 dimensional world I can change directions without stopping.
FALSE Why do I care about stopping points?
Because they represent possible direction changepoints I determine changepoints for certain by seeing a sign change in
0 0++++++++++++++++ Speed& direction "right" moving
"Left" moving Extrapolation Fred starts out left moving, and changes direction at 4 sec Extrapolation is inaccurate Data This is silly, or so I hope Copyright XKCD Acceleration How do we interpret a(T)? By itself, a(T) being positive, negative, or zero doesn't mean much. However
combined with velocity…. Monday, November 29, 2010
11:47 AM Speeding up: positive velocity and positive velocity
negative velocity and negative acceleration m
m/s Slowing Down: positive velocity and negative acceleration
negative velocity and positive velocity m/s2 Acceleration is the tending force on velocity
If Fred is said to be uniformly accelerated 0 0+++++++++++++ V(T) 00++++++++++++++++++++++++++ a(T)
During what intervals of time is Fred Slowing
Down? Speeding Up?
(0,2) SU
(2,4) SD
(4, + ) SU It always switches at 0 velocity, so there is
no sign to compare, so there is always
parenthesis SU SD SU T=4
S(T) T=2
32
Pre Calculus Page 48 0 In Class Problem
Tuesday, November 30, 2010 Plug in to S(T) to
find position when
not moving. S(T) Solve for 0 16 20 0+++++++00++++++++++++++++++++++ For SU &SD, I need to
compare T 00+++++++++++++++++++++++++++++++ 0 2 3 4 (0,2) SD
(2,3) SU
(3,4) SD
(4, + SU
Values of T Pre Calculus Page 50 Practice Problem (LONG)
Tuesday, November 30, 2010 Suppose that an object is moving up and down on an axis that
Let Feet, assume the object start out falling. a) What is V(6)?
b) What is S(6)?
c) What is the average velocity for the time interval [0,6]? a) Using by reverse engineering get b)
c)
V(T) Output is V(T)
TONE is a(T) SU Output +
Tone  SD Pre Calculus Page 51 T The Mathematical Model for Free Fall
Tuesday, November 30, 2010 Your number axis is vertical. The ground is zero as a position. Moving upward is positive velocity, therefore falling is
negative velocity. Acceleration is a constant, 32 ft/sec or 9.81 m/s2 No air resistance Height (h) a(T)=32 ft/sec You can choose any starting point, because all the values are
relative to each other. If the starting point of a 200 ft
building is the top, the street below is just 200 ft. Earth Not drawn to Scale Pre Calculus Page 52 Rounding
Monday, December 06, 2010 At
1.
2.
3.
4. For word questions S(T) , Fred is….
Since
so Fred is left moving.
Since S(T) is concave up at
,
SD BecauseC
&
have opposite signs.
Fred is in "Positivenessland" , so Fred has a positive acceleration. T V(T)
At T2 , Bob is….
1. Stopped because
2.
because at
,
decreasing
3. Neither! Since SU & SD result from the comparison of signs of
Output, since
it doesn't have a sign. T T2 "Displacement" is the NET difference between Si and Sf Displacement is not total distance traveled necessarily.
Example, start at 5, go to 1, then back to 4, displacement is 1. Pre Calculus Page 53 , V(T) +
+ + + Area =9
+ T, which is This tells you the Meters traveled, using this
You can determine the displacement and the
distance traveled using the velocity data and
initial position, or any position for that matter. Area = 2
  For V(T), your "Tools" output & the tone 2 units of distance in the negative direction Displacement is the sum of the signed area regions. Total distance
traveled is the sum of the absolute value of the signed area of all the
regions. An object is thrown downward from a height of 112
ft and reaches the ground in 2 sec. What is Vi
0= Try It
A car is traveling on a straight road & goes from 55 mi/hr
to 25 mi/hr in 30 sec. If the acceleration is constant, what
is it? Pre Calculus Page 54 Pre Calculus Page 56 Calculus Rectilinear Motion Problem
Monday, December 13, 2010 20
10
0
10 (2,20) (7,20)
(16,10)(18,10) A=140 A=25 2 4 6 8 10 12 14 16 18 20 22 24
A= 50
(10,10) (14,10) A squirrel starts at building A at time t=0 and travels along a straight horizontal wire connected to
building B. For
The squirrels velocity is model by the piecewise function shown above.
(a) At what times in
At t=9 and t=15
(b) At what time interval in does the squirrel change direction. Give a reason.
is the squirrel farthest from the building. Pre Calculus Page 57 121410 In Class Problem
Wednesday, December 15, 2010 Pre Calculus Page 58 Unit 6 •
•
•
•
• Polynomials
Graphing Polynomials
Descartes' Rule of Signs
Really Big Polynomials
Estimating Irrational Roots Pre Calculus Page 59 Polynomials
Monday, January 03, 2011 Variable This will be on Exam 6 Base Exponent The sum/difference of monomials
"ONE" "THING"
Coeff.
Constant Descending Order N=1 N=3 This is a Fourth Degree Polynomial N=1 N=3 Pre Calculus Page 61 N=2 N=4 N=2 N=4 N=5 N=5 N=6 These are all only possibilities, and do in no way represent all possible functions of the described type. In Summary…
1) If
then the right end goes up
then the right end goes down
2) If
3) If N is ODD, the ends are "Opposite"
4) If N is EVEN, the ends are the "Same" Pre Calculus Page 62 N=6 Graphing a Polynomial
Monday, January 03, 2011 Explanation of Derivation of the Formula Every derivative of a polynomial is itself a polynomial.
Let
Assume Let
Let
I want
( ) (
( have roots of 2,1,4 Throw Algebra at it, UNTIL IT DIES ) ) Finding the Roots: Assuming you don't randomly pick a root out of thin air….. Roots of F(x) There is an organized way to find roots…
If real roots exist, then they are either irrational or rational
If there are rational, they must be in the form where P is a factor of
& Q is a factor of Rational Root Theorem Pre Calculus Page 63 RRT Process
Factor This, to Get These Check These Roots: 3,
These are the roots of Using Pre Calculus Page 64 Descartes' Rule of Signs
Thursday, January 06, 2011 Descartes' Rule of Signs  Quantity of changes in sign in indicates the number of (+) real roots….potentially Descartes' Rule of Signs  Quantity of changes in sign in indicates the number of () real roots….potentially It has to have some combination of roots, either (), (+), or imaginary. Because roots must have pairs,
the number of each type must be even. We can chart all the possibilities, and will (below). Descartes' Table
(+)
()
Im 2
2
0
0 2
0
2
0 It will be one of these root combinations 0
2
2
4 Hint: it's this one. :) Find the Roots Pre Calculus Page 65 Practice Problem
Thursday, January 06, 2011 Step 1 Step 2 Since it's positive, rule out... Step 3 Step 4 Pre Calculus Page 67 More Complicated Practice Problems
Thursday, January 06, 2011 Pre Calculus Page 68 Really Big Polynomial Factoring
Monday, January 10, 2011 an TV sun Note: All roots are rational and have an absolute value less than 10.
27 in Plugin, Find Roots Make a Decartes' Table 1, 3, 6 all work Now divide the roots out Synthetic Division Rinse, Wash, Repeat. Pre Calculus Page 69 (+)
5
3
1
1
5
3 ()
2
2
2
0
0
0 Im
0
2
4
6
2
4 ext ext poi Pre Calculus Page 70 Estimating Irrational Roots
Tuesday, January 11, 2011 Intermediate value theorem
If is continuous on and if K is in , then there exists a "c" in Here's an example
I'm picking I know that Input Interval and the "c" will be 1) Between what 2 consecutive integers is a root? Why? I'm to lazy to write this in the computer, so here's a picture. Pre Calculus Page 71 such that Exam 6 Study Guide Answers
Thursday, January 13, 2011 Pre Calculus Page 73 Pre Calculus Page 74 Pre Calculus Page 75 Pre Calculus Page 76 Unit 7 • Logarithms
○
○
(ln)
• Exponential Growth and Decay
• Natural Log
• Log Rules Pre Calculus Page 77 Introduction to Logs, Exponents, & Stuff Note: Those Algebra Rules include: Monday, January 24, 2011 Exponent Rules
Please recall all exponent rules from Algebra 2 In short, Logarithms are exponents!!!
Vocabulary: Pronounced "The Log Base 10 of 10,000" If , then Decreasing LOG Graphing
Rule: LOGS are not defined for negative bases or negative
powers. Graph Pre Calculus Page 79 Never equals 12 Logarithm Manipulation
Monday, January 24, 2011 Formulae for Manipulating LOG Expressions
1) M & N are any factors of MN. 2)
3)
4) 5) "Logarithmic Property of Inequality"
If
Change of Base Formula a. For C, pick any nonstupid choice of a new base. Good choices are 10 and e. 6) Pre Calculus Page 80 Euler and His Girlfriend, Ellen Pronounced "Oiler" Ellen=ln haha….ha Friday, January 28, 2011 'e' Constant Value (Like ) that is Grossly equal to 3
Common log Natural Log Exponent Equations
This Problem Doesn't Require LOGs The only way to solve this problem is with LOGs Practice Problems
Solve these in terms of Ellen
1.
2.
3.
4. 5. Pre Calculus Page 81 One Problem for Calculator Exponential Growth and Decay Model
Monday, January 31, 2011 Exponential Growth and Decay Model Note: K is the Growth/Decay Constant. is the function whose input is "T" and whose
output is the quantity at that "T" value.
Notice something about C? < Example Problem
1990, world population was 6 billion and growing at an
exponential rate of 2% annually.
, T is years since 1990.
According to that model, when will the Earth's population reach
7,000,000,000? According to that model, what is the doubling time? Pre Calculus Page 82 More Practice
A city's population doubles every 23 years. Find K, OK? Pre Calculus Page 83 Miscellaneous Other Log Notes
Wednesday, February 02, 2011 If
If
If
If , then
, then
, then
, then Pre Calculus Page 84 Unit 8 •
•
•
•
•
•
• Radians
Arc Length
Negative Angles
Trigonometric Functions and Ratios
Unit Circle
Quadrantal Angles
Breaking the Triangle Model Pre Calculus Page 85 Angles and Stuff
Thursday, February 03, 2011 Vertex Angle of opening is measured in degrees. Internal and External Area Types of Angles "internal" inside the angle Acute Angles (0⁰ through 90⁰) Obtuse Angles (90⁰ through 180⁰) "external" outside the angle "Right Angle" 90⁰ Central Angles and Arcs
An angle vertexed at the center of a circle.
Arc subtended by A. It's length is notated Pre Calculus Page 87 Arc subtended by A. It's length is notated A
62⁰ L R=3
B Radian Angle Measure
Is the Real Number equivalent to any number of degrees. How many radians is 360⁰ equivalent to?
radians L=K This is 1 Radian A⁰
R=K One Radian is a unit. Definitions: The length of the arc on the unit circle
formed by the corresponding degree Angle Measure
Pre Calculus Page 88 OK
Not OK,
Don't get lazy Pre Calculus Page 89 2 , RAD, and Conversions
Wednesday, February 09, 2011 Conversion: Unit Circle Circumference simplified Conversion Shortcut: This is the number of radians Pre Calculus Page 90 Common Conversions: Negative Angles / Standard Position and Arc Length
Wednesday, February 09, 2011 120 I Between 0 and 1/2 III Between 1 and 3/2 II Between 1/2 and 1 IV Between 3/2 and 3 Positive Angle Measure Negative Angle Measure! Determining the Arc Length with the Radian Number and the Radius , assuming A is the central angle in radians Pre Calculus Page 91 Vocab: Coterminal angles are angles
that have the same terminal and initial
angles, but different measured, this is
most commonly caused by one angle
containing an extra
, or by one
angle being negative and the other not.
See below. Pre Calculus Page 92 Quadrantal Angles
Monday, February 14, 2011 Quadrantal Angles are: Angles in standard position whose
terminal sides coincide with one of the axes.
Let's Introduce Theta, the angle variable! For example: initial Every in standard position has its own Reference Angle is the acute* angled formed between the terminal
side of and the nearest xaxis! Sector: A part of a circle, like a pizza slice. Pre Calculus Page 93 Chapter 42 in text book Trigonometric Ratios
Monday, February 14, 2011 Trig + = Trigonometric Ratios Sine
Cosine
Tangent These ratios are angle dependent The Sine Ratios of Angle Sin
Cos
Tan
Cos
12 13 Sin
5 Tan
x Tan
Note: Do not (Yet!) consider 90 Pre Calculus Page 94 5 5 The 3 Secondary Trigonometric Ratios
Tuesday, February 22, 2011 Calculator Play
CoTangent Secant CoSecant Evaluate COT
:
Never, never, never do this Geometry Flashback:
Triangles Reciprocals
2S N
M S Reciprocals S Reciprocals N/2 Reciprocals Geometry Flashback:
Triangles S
S S Pre Calculus Page 95 This also means that, for example, Pre Calculus Page 96 Breaking the Triangle
Tuesday, February 22, 2011 What are
is a point on 's terminal side. for in S.P. such that (3,7) This allows you to determine the distance to any point from the origin Let Given the reference triangle, you can let
all ratios of with 7 3 !! Every can be drawn in S.P. (Standard Position)
In SP, every has a
(Reference angle, such as )
combined with terminal side creates a reference
triangle for every imaginable! Pre Calculus Page 97 Circular Function Re Definition and Ghost Triangles
Wednesday, February 23, 2011 There is new definitions for the Trig Ratios Where is any point on 's terminal side and Ghost Triangles Important: Remember, the yaxis is effectively the "opposite"
leg, and the xaxis is effectively the "adjacent" leg. Pre Calculus Page 98 Some Practice Problems
Wednesday, March 02, 2011 14 Pre Calculus Page 99 Trigonometric Functions Review
Wednesday, March 02, 2011 Important: Remember, the yaxis is effectively the "opposite"
leg, and the xaxis is effectively the "adjacent" leg. Pre Calculus Page 100 Unit 9 •
•
•
•
•
• Trigonometry for NonRight Triangles
Law of Sines
Area of Any Triangle Trig Formula
Graphing Trig Functions
Trig Functions Domain, Period, and Range Cycle
Trig Function and Angle Variable Coefficients Pre Calculus Page 101 Trig for NonRight Triangles Chapter 5.5 Friday, March 04, 2011 Determining the Area of NonRight Triangles
You may have noticed that normal trigonometric means of determining the
area, angle and leg length measures of a triangle only work with right triangles.
Since all triangles are not right (most aren't), we need a method to determine
these things using trigonometry for other nonright triangles. To do this, we will
divide the triangle into two right triangles and then and then calculate the
measures of those triangles. Vocab: Axillary Lines are lines that
are not part of an original
geometric construction, but
which can be proven to exist. Split the triangle in half with an axillary line. (Shown as
red.) Determine the length of Use standard methods to determine the areas of both triangles, and
take the sum for the area of the whole triangle.
The example triangle has an area of Triangle Labeling Convention c Partially Constructed Triangles
This triangle started with only the following information:
Pre Calculus Page 103 This triangle started with only the following information: a=8
See if you can construct it, here's a hint, it looks like this: Example Problem
Solve the triangle h Step 1: Dar what you know. 6 c Step 2: Draw the height.
Step 3: Dar the 'a' leg
Step 4: Calculate the height But wait, there is two possible triangles here 6 h c This triangle works anywhere that the constructed triangle with radius
Pre Calculus Page 104 This triangle works anywhere that the constructed triangle with radius
6 center point d (at the end of leg b) intersects leg c. basically, if we
draw a circle around the defined point at angle AB with the length of
leg a, a triangle can exist anywhere that circle intersects the third leg. Point d 6 h 6 c Triangle can exist Here and Here Math Behind It Note: Your calculator will show you the
acute one, but the obtuse one may still
exist. Find angle Answers Pre Calculus Page 105 Definition of a Sinusoid, Law of Sine, Law of Cosine
Thursday, March 10, 2011 Sinusoid
A sinusoid is a function that can be written in the form: Where a, b, c, and d are constants and neither a nor b are 0.
Law of Sines
The Law of Sines states that the ratio of the sine of an angle to the
length of its opposite side is the same for all three angles of any
triangle.
This can also be said as follows: In any
with angles A, B, and C
opposite sides a, b, c, respectively, the following is true. Law of Cosine
This can be used when:
1. Two sides of a triangle and their enclosed angle are known
2. All three sides are known. Pre Calculus Page 106 Area of Any Triangle
Friday, March 11, 2011 Area of Any Triangle Vocab: Semi Perimeter is half of the perimeter. Area =
to b and pass through Formula Semi Perimeter All Sides In this triangle,
Now look at the triangle on the right.
We can see here that therefore If we substitute this new expression for height, we have a formula
You can choose which equation to use based on what variables you have. This equation
could also be written:
or even Pre Calculus Page 107 Graphing Trig Functions
Monday, March 14, 2011 Quick notes:
, where is an angle measure. We will use 'Zoom  7"
Because that will give us the following integers when we use RAD mode, which we
will. Which makes it easier to look at sine waves, because degrees are pi
approximations for the integers we will get. Zoom 7 graphed in radians
Domain: All Real
Range: [1, 1] Pre Calculus Page 108 graphed in radians Domain: All Real
Range: [1, 1] graphed in radians Domain: All Real except multiples of
Range: All Real Pre Calculus Page 109 and together graphed in radians Quick tip. For approximations,
and
Are identical at very small values. They are
identical at the origin and multiples of 2π, and
become increasingly differentiated with
absolute value distance from the origin or
multiples of 2π. The Period of Sine
The period of Sine is the minimum repeatable domain interval. The Period of Sine
Interval of domain necessary to contain all possible domain values.
(Extrema to Extrema) Pre Calculus Page 110 Pre Calculus Page 111 Domain, Period, and Range Cycle
Tuesday, March 15, 2011 The Period: The minimum width of domain
needed to capture a repeatable pattern. The
period is simply the width value. Period Range Cycle Range Cycle: A domain interval that
creates a 'minifunction' that has the
same range as the entire function.
The enclosed domain has the same range
as the whole function and passes the
horizontal line test. Range Cycle for Tan COT COT has asymptotes where TAN has intercepts Pre Calculus Page 112 COT
COT has asymptotes where TAN has intercepts Range Cycle is SEC
Blue is the cosine, green is the
secant, red is the asymptotes.
Notice how the secant and the
cosine share extrema.
: All Real except multiples of Or
and Range Cycle Relation to ARCfunction
A range cycle is really a domain interval, that will eventually serve as
the range for ARCfunction
Range Cycle of SEC :
and Pre Calculus Page 113 : and Pre Calculus Page 114 Function and Angle Variable Coefficient
Sunday, March 20, 2011 Coefficients Graphing Coefficients and Amplitudes Function MidLine
3
1 2 Amplitude and Coefficients
For Sine and CoSine, the amplitude is 1 when the coefficient is 1.
This is useful, because it means that: Remember that Tangent and CoTangent do NOT have defined amplitudes
Pre Calculus Page 115 Remember that Tangent and CoTangent do NOT have defined amplitudes
because they have asymptotes. Angle Variable Coefficient Angle Variable Coefficient We know that
We also know that
Now, with
When
When Pre Calculus Page 116 Heron's Principle
Wednesday, March 30, 2011 Where S = of the perimeter
a,b, and c are the triangle sides
A is area
Pre Calculus Page 117 Amplitude and Period
Sunday, April 03, 2011 Amplitude
The Amplitude of the Sinusoid
is
Graphically the amplitude is half the height of the wave. (from y = 0 to
the extrema.)
Period
The Period of the Sinusoid
Similarly, is
is Graphically the period is the length of one full cycle of the wave The Graph above shows CoSine with increasing period (in integers starting at 1)
This next graph shows CoSine with increasing amplitude (integers starting at 1) Pre Calculus Page 118 Here are some other graphs… Increasing amplitude and Decreasing Period Pre Calculus Page 119 Increasing Amplitude and Increasing Period Pre Calculus Page 120 Analytical and Perspective Sketch
Sunday, April 03, 2011 Take
We graph this, because the amplitude is , and the Period is This graph is drawn to look like a standard CoSine graph, but this is not
the way the graph looks like on a 1x to 1y scale, this is just convenient.
This is called analytical graphing. This is what the graph looks like when compared to a standard CoSine
(y =cos x). Pre Calculus Page 121 The second graphing method is called perspective graphing. Pre Calculus Page 122 Unit 10 • More Trig Function Work Pre Calculus Page 123 Trigonometric Identities
Monday, April 18, 2011 An example ID such that , so Trig IDs are equations that are true for all values such that they are
defined.
Here are some basic IDs
Pythagorean IDs CoFunction IDs OddEven Ideas Pre Calculus Page 125 Pre Calculus Page 126 Sum and Diff IDs
Friday, April 15, 2011 Sum and Difference for Sine Why?
Find the exact value of Sum and Difference for Cosine Why?
Find the exact value of Pre Calculus Page 127 Trigonometric Functions Visualized
Monday, April 18, 2011 View all trig functions relative to a circle This isn't really necessary for most people, but can be really helpful
Need to Know Angles
Remember these you will need them and it can save time: Pre Calculus Page 128 Pre Calculus Page 129 Trigonometric Functions Graphed
Tuesday, April 19, 2011 Pre Calculus Page 130 Pre Calculus Page 131 Unit 11  Bits and Pieces •
•
•
•
•
• Binomial Expansion
Factorials
Pascal's Triangle...Again
Sigma Notation
Binomial Theorem
Converging Series Pre Calculus Page 132 Factorials
Friday, April 29, 2011 Note:
Definition
"!" is defined only for whole numbers!
Most Calculators overflow at
Mine overflows at
Which is equivalent to
,
which is over
times the number of atoms in the universe!! Lotteries
Choose 6 numbers from 1 to 52 inclusive,
and get them in the correct order.
How many combinations of 52 items chosen
6 at a time exist? Your odds of winning are or 0.000004911949% Pre Calculus Page 134 Pascal's Triangle
Friday, April 29, 2011 Derived like this:
So it ends up looking like this: Pre Calculus Page 135 Book Assignment Page 715 #125 Binomial Expansion
Friday, April 29, 2011 Using Pascal's Triangle
Expand: Expansion Power Pascal Coefficients
1, 5, 10, 10, 5,1 Interior Coefficients Pre Calculus Page 136 Binomial Theorem
Friday, April 29, 2011 The Binomial Theorem
Gives the Kth term of Example of Usage K What is the 47th term of 96 46
C= 96 46 46
Example Problem Find the 23rd term.
 Solve in terms of X and Y Pre Calculus Page 137 Sigma
Monday, May 02, 2011 Upper Index Red is optional Expression Lower Index of Summation Evaluation
Formula How it works: The Last Term Included This is the Formula that the terms are based off of. K is the integer
value of the term; for example, term 2 has
. is just a variable,
and is treated like any constant. The First Term Included Pre Calculus Page 138 When
Tuesday, May 03, 2011 Dealing with this You change the lower limit.
Use j, Pre Calculus Page 139 Converging (or not) Series
Tuesday, May 03, 2011 Pre Calculus Page 140 ...
View
Full
Document
This note was uploaded on 12/09/2011 for the course MATH 1113 taught by Professor Sills during the Summer '11 term at Montgomery College.
 Summer '11
 Sills
 Calculus, PreCalculus, Sine, Cosine, Tangent

Click to edit the document details