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Abstract algebra

bccalcclassact6avectors

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Name:________________________ Date:________________________ AP Calculus BC Class Activity 6a: Vectors Given the following vectors: and : Find the vector: Find the vector: Find the direction (in radians) and magnitude of the following vectors: Given the following direction and magnitude, find a vector that represents it: , 12 2 rad, 6 Find a unit vector in the direction of . Find the angle between the vectors and . Find the magnitude of the vector between the vectors and .
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Vectores Scalars Review

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Name _____________________ Date _____ Mr. Carcich CP Physics Vectors/Scalars Review Sheet You need to know: What is the difference between scalar and vector quantities? What are four examples of vector quantities? What are four examples of scalar quantities? How do we draw vectors? What does the length of the line represent? What does the direction of the arrow represent? How do we measure the angle of the vector? (From the resultant to the nearest x-axis, less than 90 degrees) When drawing vectors, why do we need to use a scale? What is a resultant vector?

ALGEBRA GUIDE

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Formula Sheet for College Algebra Final Exam Properties of Exponents p p mp npp m n mpnppmn nmmn mn m n mnmn bb b a b a baba aa aa a aaa 1.6 .5 )(.4 )(.3 .2 .1 = =??? ? ??? ? = = = = ? ? + Quadratic Formula a acbbx 2 42 ???= Circle rradius khcenter rkyhx = = =?+? ),( )()( 222 Vertex of Parabola ),( )()( 2 , 2 2 khatVertex khxaxf a bfa b +?= ??? ???? ? ?? ??? ? ?? Standard Form of Equation of Parabola phxdirectrix kphfocus khvertex hxpky pkydirectrix pkhfocus khvertex kyphx ?= += = ?=? ?= += = ?=? : ),( ),( )(4)( : ),( ),( )(4)( 2 2 Properties of Logarithms pb pb b xbiffxy p b p b b y b b = = = = == log.5 01log.4 log.3 1log.2 log.1 nm n m nm mn mpm bb b bb b b p b loglog

AHSME 1992

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USA AIME 1992 1 Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms. 2 A positive integer is called ascending if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there? 3 A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly .500. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than .503. What?s the largest number of matches she could?ve won before the weekend began?

AHSME 1991

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USA AIME 1991 1 Find x2 + y2 if x and y are positive integers such that xy + x+ y = 71x2y + xy2= 880. (0) Rectangle ABCD has sides AB of length 4 and CB of length 3. Divide AB into 168 congruent segments with points A = P0, P1, . . . , P168 = B, and divide CB into 168 congruent segments with points C = Q0, Q1, . . . , Q168 = B. For 1 ? k ? 167, draw the segments PkQk. Repeat this construction on the sides AD and CD, and then draw the diagonal AC. Find the sum of the lengths of the 335 parallel segments drawn. Expanding (1 + 0.2)1000 by the binomial theorem and doing no further manipulation gives ( 1000 0 ) (0.2)0 + ( 1000 1 ) (0.2)1 + ( 1000 2 ) (0.2)2 + ? ? ?+ ( 1000 1000 ) (0.2)1000 = A0 +A1 +A2 + ? ? ?+A1000, (0) where Ak = (1000 k )

AHSME 1989

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USA AIME 1989 1 Compute ? (31)(30)(29)(28) + 1. 2 Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices? 3 Suppose n is a positive integer and d is a single digit in base 10. Find n if n 810 = 0.d25d25d25 . . . 4 If a

Algebra 1

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Curriculum Map Staff Name: Mr. Kurth Course: Algebra 1 Month/Unit: Sept 6 ? Sep 17 UNIT 1 ? Tools of Algebra C O N T E N T Using Variables Exponents and Order of Operations Addition and Subtraction of Real Numbers Multiplication and Division of Real Numbers Distributive Property Various Properties of Real Numbers Graphing Data on a Coordinate Plane S K I L L S Add, Subtract, Multiply and Divide Variables Solve for variables Correct processing real numbers through the order of operations Use the distributive property Compare integers and real numbers

Rational Functions

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Extending function arithmetic to division results in the family of "rational" functions (Quotients of polynomials) with the property that Let be the degree of the numerator and be the degree of the denominator A rational function is said to be: Strictly proper if Proper if Improper if Any rational function which is not strictly proper can be expressed as the sum of a polynomial and a strictly proper rational function Given with Let be the result of the quotient and be the remainder of the same quotient, then EXAMPLE: Any strictly proper rational function can be expressed as the sum of simpler rational functions whose denominators are quadratic or linear polynomials Display Mode:

Inverse Functions

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Inverse Functions Given a function , if there is a function such that which equals the identity function The function is said to be "invertible" "undoes" what "does", and vice versa Such a function is called the inverse, denoted as (the inverse of ) The notation is not to be confused with an exponent In some cases the inverse of a function can be found through algebraic methods CONSIDER: Given to determine we must find a functions that must undo But, recall the set of outputs from , to undo we take Thus Observe that: Not every function has an inverse CONSIDER: But is not a function. An inverse only exists when different inputs in the domain always yield different outputs in the range Such functions are called one-to-one
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