Pre Calculus Version 1.5.2

Pre Calculus Version 1.5.2 - P RE C ALCULUS A guide This...

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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 x-value 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 x-term and y-term. 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 y-intercept 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 Y-intercept X-intercept 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 non-zero 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 Piece-wise 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 y-axis, 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 (+) Non-Linear 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 re-establish 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 X-Coordinate 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 5-1 Pre Calculus Page 32 4-1 3-1 2-1 1-1 0-1 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. +0-------0+++0------0+++++++0 -4 -3 -2 +++++++++0-----0+++++++++++ -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) x-intercepts* of correspond to extrema in x-intercepts* of correspond to extrema on as well as points of inflection on Tone of is the concavity of and the y-output of *as long as there's a sign change at X-intercept 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 ++++++0-----------0+++++++++++++++++++ 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 A-III B-I C-II 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 change-points I determine change-points 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) 0-----------0++++++++++++++++++++++++++ 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+++++++0--------------------------------0++++++++++++++++++++++ For SU &SD, I need to compare T 0---------------------------0+++++++++++++++++++++++++++++++ 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 12-14-10 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 non-stupid 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 x-axis! Sector: A part of a circle, like a pizza slice. Pre Calculus Page 93 Chapter 4-2 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 y-axis is effectively the "opposite" leg, and the x-axis 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 y-axis is effectively the "opposite" leg, and the x-axis is effectively the "adjacent" leg. Pre Calculus Page 100 Unit 9 • • • • • • Trigonometry for Non-Right 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 Non-Right Triangles Chapter 5.5 Friday, March 04, 2011 Determining the Area of Non-Right 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 non-right 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 'mini-function' 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 Mid-Line 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 Odd-Even 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 #1-25 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 ...
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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.

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