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    theWorldScholars Cup

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    DemiDec, The World Scholars Cup, Power Guide, and Cram Kit are registered trademarks of the DemiDec Corporation.

    Academic Decathlon and USAD are registered trademarks of the United States Academic Decathlon Association.

    DemiDec is not affiliated with the United States Academic Decathlon.

    MATHCRAM KIT

    I. WHAT IS A CRAM KIT?................................................................. 2II. CRAMMING FOR SUCCESS 2III. GENERAL MATH................................. 3IV. ALGEBRA........................................................ 5V. GEOMETRY...........................................................15VI. TRIGONOMETRY...................................................22VII. CALCULUS......................................................................................26VIII. CRUNCH KIT..................................................................... 33IX. ABOUT THE AUTHOR.35

    BYSTEVEN ZHUHARVARD UNIVERSITY

    FRISCO HIGH SCHOOL

    EDITED BY

    DEAN SCHAFFERSTANFORD UNIVERSITY

    TAFT HIGH SCHOOLSOPHY LEEHARVARD UNIVERSITY

    PEARLAND HIGH SCHOOL

    DEDICATED TO PYTHAGORAS,

    FOR BEING SUCH A HOMIE.

    2009 DEMIDEC

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    WHAT IS A CRAM KIT?A Word from the Editor

    COMPETITION IS NEARING STRUCTURE OF A CRAM KIT

    The handful of days before competition can be the mostoverwhelming. You dont have enough time to revieweverything, so a strategic allocation of your resources is

    crucial. Cram Kits are designed with one goal in mind-----to provide you with the most testable and most easilyforgotten facts.

    Math. The very word strikes fear into the hearts ofmany. But dont be discouraged-----math, like any otherevent, can be mastered through studying, and perhapsmore than any other event, through test-taking.

    Sounds simple, right? Unfortunately, Decathlon math isso broad that no guide could possibly hope to cover allof nooks and crannies. This Cram Kit, then, is meant asa quick review tool to cover last-minute formulas and to

    correct minor misconceptions that may cost you pointsin competition. I advise you to go through this guidewith textbooks nearby. Doing example problems is thebest way to reinforce the concepts that you learn.

    The main body of the Cram Kit is filled with charts anddiagrams for efficient studying. Youll also find helpfulquizzes to reinforce the information as you review.

    The Crunch Kit presents the most important formulasthat you need to know for the math test. Realize,however, that knowing when to apply each formula ishalf the battle. Plugging in the numbers is often theeasiest step.

    Last, but not least, remember to relax. In the finalmoments before you open your test booklet, confidenceis your most important asset.

    Good luck and happy cramming!

    Sophy Lee

    CRAMMING FOR SUCCESSA Word from the Author

    PIECES OF THE MATH PIE

    TIME IS TICKING!

    If you have one day left, read the whole guide.

    *

    If you have one hour left, read the Crunch Kit.

    *

    If you have one minute left, scan the List of Lists

    *

    If you have one second left, good luck.

    10%

    30%

    30%

    20%

    10% General Math

    Algebra

    Geometry

    Trigonometry

    DifferentialCalculus

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    GENERAL MATHThe Deceptively Simple and the Utterly Confusing

    INTEGERS, FRACTIONS, DECIMALS,

    AND PERCENTSBASIC COUNTING TECHNIQUES

    FRACTIONS

    Fractions must have a common denominatorbefore we can add or subtract them

    When multiplying fractions, try to cancel outcommon factors

    When dividing fractions, flip over the secondfraction and multiply it by the first one

    Step 1. Turn the second fraction upside-down(the reciprocal):

    1 4

    4 1

    Step 2. Multiply the first fraction by the reciprocal of thesecond:

    1 4 = 14 = 4

    2 1 21 2Step 3. Simplify the fraction: 2

    PERCENTAGES

    1% represents one in 100 Divide a percentage by 100 to convert it to a

    decimal

    Multiply a decimal by 100 to convert it t

    25 20$40.00 (1- ) (1- ) $24

    100 100x x o a percentage

    Formula for sale prices: x(1- )(originalprice)100

    Successive discounts do NOT have the sameeffect as a cumulative discount

    If more than one discount applies to an item, keepmultiplying the right side of the above formula by

    discount(1- )

    100

    A shirt originally priced $40.00 is markeddown by 25%. Joe uses a 20%-off coupon topurchase the shirt. How much does he haveto pay for the shirt before tax?

    MULTIPLICATION PRINCIPLE

    Helps us find the total number of possibilities whenwe are choosing one item from each of severalgroups

    Multiply the number of choices from each group If Sally can choose an outfit from 4 pairs of jeans,

    5 shirts, and 3 pairs of shoes, she has 4 x 5 x 3 =60 outfit choices

    FACTORIALS, PERMUTATIONS, AND COMBINATIONS

    FACTORIALS

    ! denotes a factorial (50! 50 49 48... 2 1) x x x x

    PERMUTATIONS

    Arrangements of a set of objects in which ordermatters

    When arranging r objects out of a set of n totalobjects, the number of permutations is n r

    n!P

    (n-r)!

    A club of 12 people wants to elect a president, avice-president, and a treasurer. How manydifferent results can this election have?

    The three positions are different, so order matters 12 3 12! 12!P 12 x 11x 10 1320

    (n-3)! 9!

    COMBINATIONS

    Arrangements of a set of objects in which order doesNOT matter

    When arranging r objects out of a set of n totalobjects, the number of combinations is

    n rn!

    C(r!)(n-r)!

    A club of 12 people wants to elect three people toa committee. How many different results can thiselection have?

    The three seats on the committee are the same,so order does not matter

    12 3 12! 12! 12 x 11x 10C 2203!(12-3)! 3!9! 3 x 2 x 1

    TRY THIS MNEMONIC!

    Permutations = Prizes (order matters) Combinations = Committees (order doesnt matter)

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    GENERAL MATHMore Counting; Vegas

    BASIC COUNTING TECHNIQUES (PT. 2) PROBABILITY OF EQUALLY LIKELY EVENTS

    ARRANGEMENT RULES

    ARRANGEMENT PRINCIPLE

    When a set has two or more identical objects, weneed to take away the redundant arrangementscaused by the identical objects

    To arrange the letters in CALIFORNIA, weneed to find the number of permutations anddivide by the factorials of the identical letters

    CALIFORNIA has two As and two Is possible arrangements

    ARRANGING OBJECTS IN CIRCLES

    When we arrange objects in circles, we need tomake sure that each arrangement represents adistinct ordering of objects, not a mere rotation ofanother arrangement

    Number of possible circular arrangements = hk

    We have to keep one object in place to mark thebeginning of the arrangements

    How many different ways can four people sitaround a circular table?

    Keep one person in place and rearrange theother three

    53

    When arranging keys on a keychain, we mustdivide the result by 2 since we can flip thekeychain over, which makes arrangements thatare mirror images of each other identical

    In how many different ways can 4 keys bearranged on a keychain?

    53

    PROBABILITY

    The chance that an event will happen

    RULES

    The probability that event A happens is P(A) csc 6x 2 8 The probability that independent, unrelated events A

    and B will occur is P(A+B) = P(A) x P(B)

    If events A and B are not mutually exclusive, theprobability of one or the other occurring isP(A or B) = P(A) + P(B) --- P(A+B)

    USEFUL FACTS

    A standard poker deck has 52 cards Such a deck has 4 suits (2 red and 2 black) of 13

    cards each

    A decks face cards are the Jack, Queen, and King ofeach suit (12 face cards total in a standard pokerdeck)

    A standard die has 6 facesEXAMPLES

    What is the probability of rolling a sum of 9 with twodice?

    We have 4 outcomes with a sum of 9 (3-6, 6-3,4-5, 5-4) The total possible number of outcomes is 6 x 6 =

    36

    The probability of rolling a sum of 9 is Asin(kx h) b or Acos(kx h) b

    What is the probability of drawing a red Queen froma standard deck of cards?

    A 52-card deck has four Queens, two of whichare red

    r2 1

    P(Q ) 52 26

    What is the probability of a coin landing heads fourtosses in a row?

    For each toss, the chance of landing heads is 12

    The tosses are independent events, since eachtoss does not affect the result of any other toss

    41 1

    P(4H) P(H) P(H) P(H) P(H)2 16

    CALCULATOR USE

    When dealing with permutations and combinations,use the built-in functions on your scientific or

    graphing calculator to avoid typing in the formulas.Master these (and other calculator techniques)

    beforethe test!

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    ALGEBRA

    Separate but Equal

    SOLVING POLYNOMIAL EQUATIONS

    (THE BASICS)

    SOLVING POLYNOMIAL EQUATIONS (LINEAR)

    EQUATION

    A mathematical statement that two expressions areequal

    Examples 3 + 7 = 14 --- 4 4x + 5 = 2y

    POLYNOMIAL

    An expression containing variables 4 25x x 23

    9

    The variables cannot be contained in fractiondenominators

    The variables also cannot be contained inexponents

    Polynomials with only one term are calledmonomials

    212x y is an example of a monomial Even though the expression has two variables,

    x and y, the variables are contained in oneterm

    The degree or order of a polynomial is the sameas the degree of the term with the highest sum ofexponents

    Consider 4xyz + 3x4y2 --- 81z 4xyz has a degree of 1 + 1 + 1 = 3 3x4y2 has a degree of 4 + 2 = 6 -81z has a degree of 1 Thus, 4xyz + 3x4y2 --- 81zis a 6th order

    polynomial

    The leading coefficient of a polynomial is thecoefficient of the term with the highest degree

    The leading coefficient of 7x --- 9x3 + 15x2 - 64is -9

    LINEAR POLYNOMIALS

    Equations that have a degree of 1 and straight-line graphs

    SLOPE-INTERCEPT FORM

    y = mx + b m is slope

    2 1

    2 1

    y ym

    x x, given points (x1, y1) and (x2, y2)

    b is the y-intercept b is the value of y when the line crosses the y-

    axis, when x = 0

    POINT-SLOPE FORM

    1 1y y m(x x ) m is slope (x1, y1) is a given point

    STANDARD FORM

    Ax By C

    A

    B is slope

    C(0, )B

    is the y-intercept

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    ALGEBRAThe Root of the Problem

    SOLVING POLYNOMIAL EQUATIONS

    (QUADRATIC AND HIGHER POWERS)

    SOLVING POLYNOMIAL EQUATIONS

    (SPECIAL THEOREMS)

    QUADRATIC EQUATIONS

    Equations that have a degree of 2

    The roots of a quadratic equation are the valuesof x for which y = 0 (where the graph intersectsthe x-axis)

    Roots are also called zeroes or x-intercepts If the equation is in the form y = Ax2 + Bx + C, we

    can use the quadratic formula to find the roots

    Quadratic formula: 2B B 4ACx2A

    The part of the quadratic formula under theradical sign, B2 --- 4AC, is called the discriminant If the discriminant is positive, then the

    equation has two real roots (graph crossesthe x-axis twice)

    If the discriminant is 0, then the equation hasone real root (graph touches the x-axis once)

    If the discriminant is negative, then theequation has no real roots (graph does notintersect the x-axis)

    Sometimes we can solve quadratic polynomialsby factoring

    Think of factoring as reverse distribution 24x 4x 3 0 (2x 3)(2x 1) 0 If either factor equals 0, the whole expression

    equals 0

    Thus, we will set both factors equal to 0 tofind the roots

    32x 3 0 x2

    1

    2x 1 0 x 2

    HIGHER ORDER EQUATIONS

    Equations that have a degree higher than 2

    Some cubic polynomials are factorable Sum of cubes formula:

    x3

    + y3

    = (x + y)(x2

    --- xy + y2)

    Difference of cubes formula:x

    3--- y

    3= (x --- y)(x

    2+ xy + y

    2)

    REMAINDER AND FACTOR THEOREMS

    Remainder Theorem: To find the remainder when apolynomial is divided by (x --- c), plug c into thepolynomial

    What is the remainder when x4 --- 5x + 27 isdivided by x + 3?

    In this example, c = ---3, as x + 3 = x --- (---3) The remainder is (---3)4 --- 5(---3) + 27 = 123

    Factor Theorem: If dividing a polynomial by (x --- c)yields a remainder of 0, then (x --- c) is a factor of thepolynomial

    The remainder when x3 --- 5x2 --- x + 5 is divided by(x --- 5) is (5)

    3

    --- 5(5)2

    + 5 = 0 Thus, (x --- 5) is a factor of x3 --- 5x2 --- x + 5

    ROOT THEOREMS

    Rational Roots Theorem: To find all of the possiblerational roots of a polynomial, divide all the factors ofthe constant by all the factors of the leadingcoefficient

    Find all possible rational roots of 3x2 --- 6 + 5x3 +2x

    The constant is ---6, and the leading coefficient is5 because the third term has the highest degree

    Now we list all the positive and negative factorsof -6 over all of the positive and negative factorsof 5

    1 2 3 6 1 2 3 6, , , , , , ,1 1 1 1 5 5 5 5

    The list includes all possiblerational roots, butnone of them has to be a root of the polynomial

    Given a polynomial in the form Ax2 + Bx + C, twoformulas exist for finding the sum and the product ofthe roots

    1. Sum of roots formula: BA

    2. Product of roots formula:

    Cfor odd numberedpolynomials

    A

    Cand for even numberedpolynomials

    A

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    ALGEBRAMore or Less

    SOLVING INEQUALITIES

    INEQUALITY

    A mathematical statement that two expressions

    are not equal

    As with solving an equation, solve an inequalityby isolating the variable

    When multiplying or dividing by a negative term,flip the sign of the inequality

    LINEAR INEQUALITY

    An inequality with a degree of 1 18 < ---5x --- 7 25 < ---5x 5 > x

    QUADRATIC INEQUALITY

    An inequality with a degree of 2 22x 3x 8 43 22x 3x 35 0 (2x 7)(x 5) 0 At this point, we will plot the roots on a

    number line, dividing it into three regions

    We will pick a value in each of the threeregions to test the inequality in each region

    We will use -6, 0, and 4 Plugging -6 and 4 into the polynomial satisfy

    the inequality, so we will place checks inthose regions

    Plugging 0 into the polynomial makes theinequality false, so we will place an x in thatregion

    The inequality is true when x < ---5or x > 72

    ABSOLUTE VALUE INEQUALITIES

    A numbers absolute value is its distance from 0 on anumber line

    Absolute value is always non-negative (by definition) When an inequality contains an absolute value, we

    have to solve two inequalities based on the original

    Consider 2x --- 3 < 5 The first inequality is the same as the original, but

    without the absolute value signs

    2x 3 5 2x 8 x 4

    For the second inequality, we multiply the rightside by -1 and flip the sign of the inequality

    2x 3 5 2x 2 x 1

    Thus, 2x --- 3 < 5 holds true when x < 4 and x > ---1

    5

    72

    -5

    7

    2

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    ALGEBRAPutting the Fun in Function!

    FUNCTIONS (BASICS) FUNCTIONS (COMPOSITE AND INVERSE)

    WHAT IS A FUNCTION?

    A relationship between an independent variable x and

    a dependent variable y

    f(x) denotes a function Functions can only have one value of y for each

    value of x

    Vertical-line test: If you can place a vertical line atevery x-value of an equations graph, and the linecrosses the graph at no more than one point, thenthe equation is a function

    The following graph is not a function becausea vertical line would cross the graph at twopoints whenever x > 0

    DOMAIN AND RANGE

    The domain of a function consists of all the x-values that have corresponding y-values Find the domain of 1f(x)

    x

    At x = 0, the function is undefined (nocorresponding y-value), so the domain is allreal numbers except 0

    The range of a function consists of all its possibley-values

    The following graph has a range of -1 to 1

    TYPES OF FUNCTIONS (PART 1)

    COMPOSITE FUNCTION

    Combines two or more functions together For two functions f(x) and g(x), a possible composite

    function is f(g(x)) or, written in another form,(f g)(x)

    In function (f g)(x) , plug x into g(x) and plug thatresult into f(x)

    Find a(b(x)) if a(x) = 3x2, b(x) = 5x + 7, and x = 2 b(2) = 5(2) + 7 = 17 a(b(2)) = a(17) = 3(17)2 = 867

    INVERSE FUNCTIONS

    To find the inverse function f-1(x) of a function f(x),replace f(x) with y and switch the positions of x and y

    The inverse of y = 3x +2 is x = 3y + 2 Because we switch the xs and the ys, the graphs of

    inverse functions are mirror images of the originalgraphs across the line y = x

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    ALGEBRAFunctions: The Logarithm Strikes Back (With Rational Exponential Force)

    FUNCTIONS

    (RATIONAL, EXPONENTIAL, LOGARITHMIC)

    FUNCTIONS

    (OPERATIONS ON LOGARITHMIC FUNCTIONS)

    TYPES OF FUNCTIONS (PART 2)

    RATIONAL FUNCTIONS

    Functions in which variables are in thedenominators of fractions

    Fractions are ratios, hence, rationalfunctions 4 35x 412x is a rational function

    EXPONENTIAL FUNCTIONS

    Functions in which the independent variable x isin an exponent

    3x is an exponential function A common exponential function is ex e is a constant like and can be found on a

    scientific or graphing calculator

    e = 2.71828LOGARITHMIC FUNCTIONS

    Functions in which the independent variable x isin the argument of a logarithm

    Logarithms are the reverse of exponents Logarithms follow the form logbase (argument) =

    exponent, such that baseexponent

    = argument

    Log7(49) = 2 because 49 is 7 to the 2ndpower

    When the logarithm does not have a base written,assume that the base is 10

    Log(1000) = 3, since 103 = 1000 Logs with a base of e are called natural logarithms Natural logarithms are denoted ln(x) Logarithms and exponential expressions canceleach other out to yield the exponent when the

    bases are the same

    13Ln(e ) 13 x4Log (4 ) x

    WORKING WITH LOGS

    ADDITION

    When adding two logarithms of the same base, wecan combine them into one logarithm with thearguments multiplied together

    12 12 12log (x 1) log (x 3) log ((x 1)(x 3))

    SUBTRACTION

    When subtracting two logarithms of the same base,we can combine them into one logarithm with thefirst argument divided by the second

    12 12 12x 1log (x 1) log (x 3) logx 3

    OTHER CASES

    When the entire argument of a logarithm has anexponent, we can turn the exponent into a coefficientof the logarithm

    2 3 2log((5x 9) ) 3log(5x 9) We can pull the 3 out because it applies to the

    whole argument

    We cannot pull the 2 out because it only appliesto one term in the argument

    REVERSAL

    These three rules can also be used in reverse A logarithm whose argument is a product can be split

    into the sum of two logarithms whose arguments arethat products factors

    Log12((x --- 5)(x + 9)) = Log12(x --- 5) + Log12(x + 9) A logarithm with one argument divided by another

    can be split into the difference of two logarithms,such that the divisor becomes the argument of thesubtracted logarithm

    12 12x 5

    log log (x 5) log (x 9)x 9

    A coefficient of a logarithm can become the exponentof the logarithms entire argument

    3(Log(5x2 + 9)) = Log((5x2 + 9)3)

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    ALGEBRAUse Your Imagination; Walk the Line

    COMPLEX NUMBERS READING GRAPHS OF FUNCTIONS (LINEAR)

    WHAT IS A COMPLEX NUMBER?

    Any number in the form a + bi

    a and b are real numbers i is an imaginary number such that i 1

    OPERATIONS WITH COMPLEX NUMBERS

    We can simplify higher powers of i Find i47 We know that i2 = ---1 i47 is the same as (i46)(i) (i46)(i) = (i2)23(i) Thus, i47 = (---1)23(i) i47 = ---1

    COMPLEX CONJUGATES

    Pairs of complex numbers in forms a + bi and a ---bi

    A fraction with an imaginary number in thedenominator is simplified by multiplying itsnumerator and denominator by the complexconjugate of the denominator

    Simplify

    1 i

    2 3i

    1 i 2 3i 2 3i 2i 3 1 5i

    2 3i 2 3i 4 6i 6i 9 13

    Notice that multiplying by the complexconjugate removes i from the denominator

    COMPLEX QUADRATIC ROOTS

    In a quadratic equation whose discriminant (b2 ---4ac) is negative, the roots are complex numbers

    If the roots are complex numbers, they will becomplex conjugates

    A polynomial with the root 35 + 9i must alsohave the root 35 --- 9i

    LINEAR FUNCTIONS

    Linear functions are always straight lines First, we find the y-intercept of the function

    The line above crosses the y-axis at y = 3 In slope-intercept form, which is y = mx + b, the y-

    intercept is b, so b = 3

    To find m, the slope, we need two points from thegraph

    We already know that the y-intercept is (0,3) We can also read the x-intercept from the graph,

    which is (---6,0)

    Using the formula for slope,

    2 1

    2 1

    y ym

    x x, we find

    that the slope is

    0 3 3 1

    m6 0 6 2

    Therefore, the graph above represents 1y x 32

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    ALGEBRA

    Read Between the Curves

    READING GRAPHS OF FUNCTIONS

    (QUADRATIC)

    READING GRAPHS OF FUNCTIONS (HIGHER

    ORDER)

    QUADRATIC FUNCTIONS

    Quadratic functions are always U-shaped or n-shaped

    The graphs of quadratic equations are calledparabolas

    The standard form for the equation of a parabolaisy = A(x --- h)

    2+ k

    The point (h,k) is the vertex-----the turning point ofthe curve

    In the graph above, the vertex is (---2,1) We can plug points into the standard form for the

    equation of a parabola to obtain the equation ofthe graph

    We can plug the vertex of the graph above togety = A(x --- (---2))

    2+ 1, which becomes

    y = A(x + 2))2

    + 1

    We still need to find A by plugging in a pointfor (x,y)

    We can read from the graph the point (0,---1) 21 A(0 2) 1 2 A

    4

    1A2

    Thus, the equation of the graph above is 2

    1y (x 2) 1

    2

    HIGHER ORDER EQUATIONS

    If the degree of the equation is even, the graph willstart and end on the same side of the y-axis The following graph represents 6 31y x x

    4,

    which starts and ends on the positive side of they-axis

    If the degree of the equation is odd, the graph willstart and end on opposite sides of the y-axis

    The following graph represents y = ---x7 + x4,which starts on the positive side of the y-axis andends on the negative side

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    ALGEBRAFlipped Functions and Arithmetic Arrangements

    READING GRAPHS OF FUNCTIONS

    (EXPONENTIAL AND LOGARITHMIC)

    SEQUENCES, SERIES, AND MEANS

    (ARITHMETIC)

    EXPONENTIAL FUNCTIONS

    Exponential functions create graphs withhorizontal asymptotes Asymptotes are lines at which the x or y value of

    a function approaches infinity or negative infinity(but never reaches it)

    The following graph represents y = ex, which has ahorizontal asymptote at y = 0

    As x approaches negative infinity, y willapproach 0 but will never reach it

    LOGARITHMIC FUNCTIONS

    Logarithmic functions create graphs with verticalasymptotes

    The following graph represents y = ln(x), whichhas a vertical asymptote at x = 0

    As x approaches 0, y approaches negativeinfinity

    ARITHMETIC SEQUENCE

    Pattern of numbers that has a common differenced 1, 8, 15, 22, 29 Common difference is 7 because each term is 7

    more than the previous one

    Formula to find the nth term of an arithmeticsequence: nth term = first term + d(n --- 1)

    Find the 9th term of the sequence: 68, 64, 60,56

    n = 9 and d = ---4 (---4 = 64 --- 68 = 60 --- 64, and soon)

    9th term = 68 + (---4)(9 --- 1) = 68 --- 32 = 36ARITHMETIC SERIES

    The sum of an arithmetic sequence

    Formula to find the sum of the first n terms:

    (firstterm last term)

    n2

    Formula to find n, the number of terms in the series:

    (lastterm first term)

    n 1d

    Find the sum of the arithmetic progression: 17,20, 2344, 47, 50

    d = 3, the last term is 50, and the first term is 17 (50 17)n 1 12

    3

    Now we can find the (17 50)sum 12 4022

    Summation problems may use sigma ()notation

    5

    k 1

    k = the sum of the numbers 1 through 5

    The index k starts at 1, the lower bound, andincreases by 1 for each term until it reaches 5

    The expression on the right side of the sigma sign(here, k) represents an element of the series

    The expression above is the same as 1+2+3+4+5ARITHMETIC MEAN

    The average of two or more numbers

    The arithmetic mean of 1, 4, 7, 10, and 13 is

    1 4 7 10 13

    75

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    ALGEBRARational Commonists

    SEQUENCES, SERIES, AND MEANS

    (GEOMETRIC)

    SEQUENCES, SERIES, AND MEANS

    (GEOMETRIC AND INFINITE)

    GEOMETRIC SEQUENCE

    Pattern of numbers with a common ratior 2, 6, 18, 54 The common ratio is 3 because each term is 3

    times the previous one

    Formula to find the nth term of a geometricsequence: n 1nth term (firstterm)r

    What is the 8th term of the sequence thatbegins: 625, 125, 25, 5?

    The common ratio is 15

    71 1 1

    8th term (625) (625)5 78125 125

    GEOMETRIC SERIES

    The sum of a geometric sequence

    Formula to find the sum of the first n terms of ageometric sequence:

    n(first term)(1 r )

    1 r

    Find

    k 110

    k 1

    3(4)

    2

    We plug in k 1 to find the first term:

    1 13

    (4) 42

    Were trying to find the sum of the termsfromk =1 to k = 10, so n = 10

    The ratio that we multiply to find eachconsecutive term is

    3

    2 , so r =3

    2

    Thus, the sum is

    103

    (4) 12

    453.323

    12

    GEOMETRIC MEAN

    The square root of the product of two terms

    Find

    k 110

    k 1

    3(4)

    2

    What is the geometric mean of 4 and 64? 4 64 16 4, 16, and 64 form a geometric series with a

    common ratio of 4

    INFINITE SERIES

    The sum of a sequence with an infinite number of terms

    For an infinite series to be solvable, r has to be lessthan 1

    The infinite series of the sequence that beginswith

    1 1 12,1, , , ...

    2 4 8will have a value because

    each term is 1

    2times the previous one

    The terms will eventually be so close to 0 thatadding them to the series does not change thesum

    These types of series are said to converge, or reach adefinite sum

    If r is 1 or higher, the sequence will keep generatinglarger numbers, and the series will have an indefinitevalue

    The series of the sequence that begins with ---2, 4,---8, 16, ---32 does not have a value because everyterm is -2 times the previous one

    The terms will keep increasing, and the sum willnever stay at a definite number

    These types of series are said to diverge, or not reacha definite sum

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    ALGEBRA

    Can You See the Pattern?

    SEQUENCES, SERIES, AND MEANS (GRAPHING) SEQUENCES, SERIES, AND MEANS (GRAPHING)

    In arithmetic sequences, the terms have equalvertical distances between them because the

    common difference d never changes

    In the above geometric sequence, each term is twiceas large the previous one

    In an arithmetic series, the sums do not haveequal vertical distances between them becauseeach term added is larger than the previous term

    In the above geometric series, the sum approaches 4as n extends to infinity, meaning that the seriesconverges

    In a diverging series, the sum would approach infinity

    0

    5

    10

    15

    20

    0 5 10

    Arithmetic Sequence

    0

    50

    100

    150

    200

    250

    300

    0 2 4 6 8 10

    Geometric Sequence

    0

    10

    20

    30

    40

    50

    60

    7080

    0 5 10

    Arithmetic Series

    0

    1

    2

    3

    4

    5

    0 50 100 150

    Geometric Series

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    GEOMETRY

    Triangles with Little Squares in the Corner

    RIGHT TRIANGLES SPECIAL RIGHT TRIANGLES

    PYTHAGOREAN THEOREM

    A right triangle contains a right angle (90) The two sides adjacent to the right angle are

    called legs

    In the above diagram, a and b are legs The hypotenuse is the side opposite the right

    angle

    In the above diagram, c is the hypotenuse The Pythagorean theorem states a relationship

    between the three sides

    2 2 2a b c The theorem can also give us information about

    other types of triangles in which c is the longestside

    If 2 2 2a b c , then the triangle is acute (allangles are less than 90)

    If 2 2 2a b c , then the triangle is obtuse (oneangle is greater than 90)

    A Pythagorean triple is a set of three integers thatfit the theorem

    3, 4, 5 5, 12, 13 7, 24, 25 8, 15, 17 9, 40, 41

    Any multiple of a Pythagorean triple will also be aPythagorean triple 6, 8, 10 10, 24, 26

    45-45-90 TRIANGLES

    45-45-90 triangles are right triangles with legs ofequal length

    They are also called right isosceles triangles The hypotenuse is equal to 2 times a side

    30-60-90 TRIANGLES

    The shorter leg is opposite the 30 angle The hypotenuse is twice the length of the shorter leg The longer leg, which is opposite the 60 angle, is

    3 times the shorter leg

    2ss

    s 3 30

    60

    s

    s

    s 2

    45

    45

    a

    b

    c

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    GEOMETRYPoint-Line Coordination

    COORDINATE GEOMETRY (POINTS) COORDINATE GEOMETRY (LINES)

    MIDPOINT

    The point that is exactly in the middle of two other

    points

    Given two points (x1, y1) and (x2, y2), theirmidpoint is the average of their coordinates:

    1 2 1 2x x y y,2 2

    Find the midpoint of (-2, 3) and (5, -6)

    2 5 3 ( 6) 3 3, ,

    2 2 2 2

    SLOPE

    The rate of change of a line

    In other words, slope is a ratio of how fast the lineis changing vertically over how fast the line ischanging horizontally

    Given two points (x1,y1) and (x2,y2) that lie on thesame line, the slope of the line is

    2 1

    2 1

    y ym

    x x

    Note that slope is change in y (vertical) overchange in x (horizontal) Thus, slope can be remembered as rise over

    run

    In equations, slope is usually denoted as m

    DISTANCE FORMULA

    The distance between two points (x1, y1) and(x2, y2) is:

    2 21 2 1 2(x x ) (y y )

    PARALLEL AND PERPENDICULAR LINES

    Two lines are parallel if they have the same slope A line that crosses two parallel lines is called atransversal

    Two angles that add up to 180 degrees are calledsupplementary angles (1 & 2, 4 & 3, 1 & 4, etc.)

    Two angles that add up to 90 degrees are calledcomplementary angles

    All of the larger angles (1, 3, 5, 7) are equal to eachother

    All of the smaller angles (2, 4, 6, 8) are equal to eachother

    The sum of any larger angle and any smaller angle is180

    Two lines are perpendicular if they intersect and formright angles

    The slopes of perpendicular lines are negativereciprocals of each other (the product of their slopesis -1)

    Find the slope of a line perpendicular to the line

    4y x 3

    7

    The slope of the given line is 47

    , so the slope of

    the perpendicular line is 7

    4

    1

    23

    4

    5

    67

    8

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    GEOMETRYFour-sided Shapes That Are Almost, but Not Entirely, Unlike Triangles

    COORDINATE GEOMETRY (QUADRILATERALS)

    QUADRILATERAL

    A four-sided polygon

    TRAPEZOID

    A quadrilateral with one pair of parallel sides

    The parallel sides are called bases The non-parallel sides are called legs The height is the distance from one base to the

    other

    Area = 12

    (base1 + base2)(height)

    In a coordinate system, the two parallel baseshave the same slope, and the two legs havedifferent slopes

    PARALLELOGRAM

    A quadrilateral with two pairs of parallel sides

    Opposite sides are congruent (equal inmagnitude)

    Opposite angles are congruent Consecutive angles are supplementary (add up to

    180)

    Area = (base)(height) In the above diagram, the base is the side on the

    bottom, and the height is the vertical dotted line

    In a coordinate system, opposite sides have thesame slope and length

    RECTANGLE

    A parallelogram with four right angles

    Area = (base)(height) In a coordinate system, opposite sides have the same

    slope and length, and adjacent sides must beperpendicular

    RHOMBUS

    A parallelogram with four congruent sides

    The diagonals form right angles The diagonals bisect each other and bisect theangles, forming four congruent right triangles Area = 1

    2(diagonal1)(diagonal2)

    In a coordinate system, the diagonals areperpendicular, and the side lengths are all equal

    SQUARE

    A quadrilateral with four congruent sides and fourright angles, making it both a type of rectangle andrhombus

    Area = (side)2 In a coordinate system, all sides have the same

    length, and adjacent sides are perpendicular

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    GEOMETRYMovin On Up, Dimensionally

    PLANE AND SOLID FIGURES (AREA) PLANE AND SOLID FIGURES (VOLUME)

    AREA OF A TRIANGLE

    1Area (base)(height)2

    Works best for right triangles and triangleswhose base and height are known

    Herons Formula: Area (s)(s a)(s b)(s c) a, b, and c are the sides of the triangle, and

    a b cs

    2

    When using this formula, find s first and storeit as a variable in your calculator

    Be careful to calculate the formula correctly This formula works for any triangle, but you

    need to know the lengths of all three sides

    Area = 12

    ab(sinC)

    a and b are two sides, and C is the anglebetween them

    SURFACE AREA OF SOLID FIGURES

    Prism: SA = Area of 2 bases + area of lateral faces Pyramid: SA = Area of the base + area of lateral

    triangles

    Cylinder: SA = 2 r2 + 2 rh r is the radius of the base, and h is the height

    of the cylinder

    Sphere: SA = 4 r2 r is the radius of the sphere

    Cone: SA = r2 + r 2 2r h r is the radius of the base, and h is the height

    of the cone

    2 2r h is the lateral height, the distancefrom the edge of the base to the apex of thecone

    If the lateral height is given, substitute it for2 2r h

    VOLUME OF SOLID FIGURES

    Prism: V = (area of base)(height) Pyramid: 1V (areaofthebase)(height)

    3

    Cylinder: V = r2h r is the radius of the base h is the height of the cylinder

    Sphere: V = 43

    r3

    r is the radius of the sphere Cone: V = 1

    3 r2h

    r is the radius of the base h is the height of the cone

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    GEOMETRYCircle Time

    PLANE AND SOLID FIGURES (CIRCLES)

    MEASURING CIRCLES

    CIRCUMFERENCE OF A CIRCLE

    Circumference = 2 r Circumference is the perimeter of a circle

    AREA OF A CIRCLE

    Area = r2 r is the radius of the circle

    LOOKING INSIDE

    ANGLES IN A CIRCLE A circle has 360 or 2 radians

    180 = radians A central angle has the same measure as its

    intercepted arc

    An inscribed angle has half the measure of itsintercepted arc

    LINES AND CIRCLES (PART 1)

    Tangents are lines that intersect a circle at one point A tangent will be perpendicular to the radius of

    the circle at the point where it touches the circle

    Secants are lines that intersect a circle at two points

    Chords are line segments that have endpoints on therim of a circle

    The longest chord is the diameter If two chords are the same distance from the

    center of a circle, they have the same length andintercept the same-sized arc

    90

    45

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    GEOMETRYCircle Time: Part Deux

    PLANE AND SOLID FIGURES (CIRCLES) (CONTD)

    LINES AND CIRCLES (PART 2)

    TWO CHORDS

    In the above diagram, two chords intersect at apoint E

    AB DCAEB CED2

    and

    AD BCAEB BEC

    2

    AE EC BE ED

    LINES AND CIRCLES (PART 3)

    A TANGENT AND A SECANT

    In the above diagram, AB is a tangent and AC is asecant that intersects the circle at point D

    BC BDA2

    2(AB) AD AC

    TWO TANGENTS

    In the above diagram, two tangents have acommon endpoint at A and intersect circle O at Band C

    The lengths of the two tangents are the same The two radii OB and CO are perpendicular to

    their respective tangents

    major arc BC minor arc BCA

    2

    TWO SECANTS

    In the above diagram, two secants originating frompoint A intersect a circle at points D and E

    BC DEA2

    AD AB AE AC

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    GEOMETRYA Striking Resemblance

    CONGRUENCE SIMILARITY

    PROPERTIES OF CONGRUENT FIGURES

    Two figures are congruent if their correspondingsides have the same length and the sides form thesame angles

    The figures may be flipped or rotated The following figures are all congruent

    CONGRUENT TRIANGLES

    SSS (Side-Side-Side): If the corresponding sidesof two triangles are congruent, the triangles arecongruent

    A triangle with side lengths 3, 4, and 5 iscongruent to a triangle with side lengths 3, 4,and 5

    SAS (Side-Angle-Side): If two triangles have thesame angle, and the corresponding sides adjacentto the angle are congruent, then the triangles arecongruent

    A triangle with side lengths of 2 and 6separated by an angle of 54 degrees iscongruent to another triangle with sidelengths of 2 and 6 separated by 54 degrees

    ASA (Angle-Side-Angle): If two triangles havetwo matching angles, and the sides between bothangles are congruent, then the triangles arecongruent

    A triangle with angles of 34 and 89 degreesseparated by a side of length 7 is congruent toanother triangle with angles of 34 and 89degrees separated by a side of length 7

    PROPERTIES OF SIMILAR FIGURES

    Two figures are similar if corresponding sides formequal ratios and the sides form the same angles

    The figures may be flipped or rotated The following figures are all similar

    SIMILAR TRIANGLES

    SSS: If the corresponding sides of two triangles formequal ratios, then the triangles are similar

    A triangle with side lengths 4, 7, and 9 is similarto a triangle with side lengths 8, 14, and 18

    SAS: If two triangles have the same angle, and thecorresponding sides adjacent to the angle form equalratios, then the triangles are similar

    A triangle with side lengths of 3 and 5 separatedby an angle of 80 degrees is similar to a trianglewith side lengths of 12 and 20 separated by 80degrees

    AA (Angle-Angle): Triangles with two correspondingangles are similar

    Since a triangle only has three angles, the thirdone can be found if two of them are known

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    TRIGONOMETRYSine Here

    RIGHT TRIANGLE RELATIONSHIPS TRIGONOMETRIC FUNCTIONS

    SIDES AND ANGLES

    To remember what the trig functions mean, usethe mnemonic SOHCAHTOA (soak-a-toe-a)

    OppositeSine(angle)Hypotenuse

    AdjacentCosine(angle)Hypotenuse

    OppositeTangent(angle)Adjacent

    asinA cosBc

    bsinB cos Ac

    atan A cotBb

    btanB cot Aa

    csec A cscBb

    csecB csc Aa

    csc (cosecant) is the reciprocal of sin (sine) sec (secant) is the reciprocal of cos (cosine) cot (cotangent) is the reciprocal of tan (tangent)

    TRIG FUNCTIONS AND QUADRANTS

    Each trig function is only positive in certain quadrants(mnemonic: All Students Take Classes)

    All of the trig functions have positive values inQuadrant I

    Sine is positive in Quadrant II Tangent is positive in Quadrant III Cosine is positive in Quadrant IV Each reciprocal function-----cosecant, secant, and

    cotangent-----has the same sign as its correspondingfunction

    REFERENCE ANGLES

    When drawing angles, we place the initial side at thepositive x-axis and go counter-clockwise, ending witha terminal side

    A reference angle is the angle between the terminalside and the x-axis

    The sine, cosine, and tangent of an angle isnumerically equivalent to its corresponding referenceangle, but the sign may need to be adjusteddepending on the quadrant in which the terminal sideis located

    The above angle is 225, and it lies in Quadrant III Its reference angle is 225 --- 180 = 45 sin(225) is numerically equivalent to sin(45),

    but sine values are negative in Quadrant III

    sin(225) = ---sin(45) = ---0.707

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    TRIGONOMETRYThe Arc Side and Graphic Descriptions

    INVERSE TRIG FUNCTIONS PROPERTIES OF TRIG GRAPHS

    THE BASICS

    Inverse trig functions reverse the effects of trigfunctions

    If sinA = B, then arcsinB = A 1sin(30 )

    2, and

    1arcsin 30

    2

    The inverse trig functions are arcsin, arccos,arctan, arccsc, arcsec, and arccot

    The inverse trig functions can also be notated: 1 1 1 1 1 1sin ,cos ,tan ,csc ,sec ,cot

    Unlike sin2x, which means (sinx)2, 1sin x does notmean

    1

    (sinx)

    DOMAIN AND RANGE

    Inverse trig functions do not pass the vertical linetest unless we limit their domains and ranges

    The following limits allow us to work with inversetrig functions as true functions

    Function Domain Range

    arcsin [ 1, 1]

    -[ , ]2 2

    arccos [ 1, 1] [0, ]

    arctan ( , )

    ( , )2 2

    arccsc ( ,1][1, )

    [ ,0) (0, ]2 2

    arcsec ( ,1][1, )

    [0, ) ( , ]2 2

    arccot ( , ) (0 , )

    PERIOD

    The smallest interval taken for function values to repeat

    All trig functions are periodic (they repeat) The period of a function is always positive Sine, cosine, and their reciprocal functions (cosecant

    and secant) have a period of2

    k, where k is the

    coefficient of x in the argument

    The function sin(6x) has a period of 26 3

    Tangent and cotangent have periods of k

    , where k is

    the coefficient of x

    The function cot(---7x) has a period of 7

    AMPLITUDE

    Half of the distance between the maximum and

    minimum values of the function

    Sin and cos have amplitudes determined by thecoefficient of the function

    The function 3cos(5x) has an amplitude of 3

    HORIZONTAL (PHASE) SHIFT

    A constant term inside the function shifts the graphhorizontally

    A function with argument (kx --- h) is shifted hk

    units

    from x = 0

    What is the phase shift of the functiontan(3x + 5)?

    First, we need to put the argument into the form(kx --- h)

    tan(3x + 5) = tan(3x --- (---5)) We know k = 3 and h = ---5, so the function is

    shifted h 5

    k 3units from x = 0 (in the negative

    direction, or to the left)

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    TRIGONOMETRYOoh, Pretty Wave; Identity Quandary

    MORE PROPERTIES OF TRIG GRAPHS IDENTITIES

    VERTICAL SHIFT

    A constant term outside the function shifts thegraph vertically What is the vertical shift of csc(6x + 2) --- 8? The constant term outside the function is ---8,

    so the graph is shifted 8 units in the negativedirection (down)

    CONSOLIDATION (SINE/COSINE)

    Asin(kx h) b or A cos(kx h) b Amplitude = A Period = 2

    k

    Horizontal shift = hk

    Vertical shift = b Note that for tangent and cotangent functions,

    period is equal tok

    , and amplitude is largely

    irrelevant in graphs

    ALL TOGETHER NOW

    The following graph represents 5sin(4x --- 8) + 2

    Amplitude (marked by the green line from themiddle to the trough of the wave) is 5

    Period (marked by the bracket that covers onecomplete cycle) is

    2

    4 2

    Horizontal shift is h 8 2k 4

    units from x = 0 (to

    the right)

    5sin(4x 8) 2 5sin(4x 8) 2 Vertical shift is 2 units up because the constant

    term outside the function is 2

    WHY DO WE USE IDENTITIES?

    To convert between different trigonometric

    functions to solve a problem

    RECIPROCAL IDENTITIES

    1 1sinx ; cscxcscx sinx

    1 1cosx ; secxsecx cosx

    1 1tanx ; cotxcotx tanx

    QUOTIENT IDENTITIES

    sinxtanxcosx

    cosxcotxsinx

    PYTHAGOREAN IDENTITIES

    2 2sin x cos x 1 2 2tan x 1 sec x 2 21 cot x csc x

    OTHER IMPORTANT IDENTITIES

    sin(x y) (sinx)(cosy) (cosx)(siny) cos(x y) (cosx)(cosy) (sinx)(siny)

    tanx tany

    tan(x y)1 (tanx)(tany)

    sin(x y) (sinx)(cosy) (cosx)(siny) cos(x y) (cosx)(cosy) (sinx)(siny) tanx tanytan(x y) 1 (tanx)(tany) sin(2x) 2sinxcosx

    2 2

    2

    2

    cos(2x) cos x sin x

    1 2sin x

    2cos x 1

    22tanx

    tan(2x)1 tan x

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    TRIGONOMETRYTriangular Relationships; Finding a Good Angle

    LAW OF SINES AND COSINES ALGEBRAIC EQUATIONS INVOLVING TRIG

    FUNCTIONS

    LAW OF SINES

    In a triangle, the ratio of the sine of an angle to itsopposite side is the same for all three angles

    sinA sinB sinCa b c

    LAW OF COSINES

    With a slight modification, the Pythagoreantheorem can work for any triangle, producing theLaw of Cosines

    Given two sides and the angle between them, wecan find the length of the third side

    2 2 2c a b 2ab(cosC) 2 2 2a b c 2bc(cosA) 2 2 2b a c 2ac(cosB)

    SOLUTIONS

    Unless domain and range are limited, trig functionscan have an infinite number of solutions The answers to these functions will repeat every

    360 or 2 radians

    The same reference angle in different quadrants canproduce the same result in a trig function

    SOLVING

    We usually want to turn all the different types of trigfunctions into just one type by substituting identitiesor by canceling out common terms

    Then, we can isolate the trig expression and solve forthe angle 1 --- cos2x + sin2x = 0 1 --- (1 --- sin2x) + sin2x = 0 sin2x + sin2x = 0 2sin2x = 0 sin2x = 0 sinx = 0 x = 0, 180, 360

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    CALCULUSTake It to the Limit

    BASIC LIMITS CONTINUITY AND LHOPITALS RULE

    LIMIT

    The y-value of a function as it gets infinitely close

    to an x-value

    Limits are notatedx c

    limf(x) , where c is the value

    that x approaches

    When evaluating a limit, plug in c for x

    2 2

    x 3lim (x 5) 3 5 4

    A limit can be evaluated at infinity or negativeinfinity

    2

    2x

    x 5xlim

    3x 1

    At infinity, the terms containing x2 will be solarge in value that the other terms will havelittle effect on the result of the limit

    Thus, the limit becomes essentially

    2

    2x

    xlim

    3x

    Canceling out the x2 gives us 13

    If the denominator of a rational expressionhas a higher degree than the numerator, thelimit as x approaches infinity is 0, since the

    denominator will become much larger thanthe numerator

    If the numerator of a rational expression has ahigher degree than the denominator, the limitas x approaches infinity is infinity, since thenumerator will become much larger than thedenominator

    Limits can be specifically left-handed or right-handed

    A left-hand limit approaches the x-value from theleft side of a graph

    x clim f(x) is a left-hand limit where x

    approaches c from the left (negative) side

    A right-hand limit approaches the x-value fromthe right (positive) side of a graph

    x c

    lim f(x) is a right-hand limit where x

    approaches c from the right side

    A function has a limit at c only when the left-handlimit and the right-hand limit at c are equal

    In other words, a function has a limit at cwhen f(x) approaches the same y-value onboth sides of c

    CONTINUITY

    A function exhibits continuity when its graph has nogaps

    A function is continuous at an x-value c if its limit at cequals its y-value at c

    x climf(x) f(c) for the function to be continuous at

    c

    If the limit exists at c, but it does not equal f(c) , thenwe say that a removable discontinuity exists at c

    If the limit does not exist at c, then we say that a non-removable discontinuity exists at c

    LHOPITALS RULE

    This topic requires knowledge of derivatives, so skipahead if you need to refresh (or perhaps learn howthey work)

    After plugging c into a limit, if the limit isindeterminate, we can use LHopitals rule to try toconvert the limit into a determinate one

    Indeterminate limits come in the form of

    0and

    0

    LHopitals rule takes the derivative of the numeratorand denominator of a limit separately

    After the derivatives, plug in c again to see if thelimit has become determinate

    2

    x 4

    x 2x 8lim

    x 4

    If we plug in ---4, we get 00

    , an indeterminate

    form

    After taking the derivative of the numerator andthe denominator separately, we have

    x 4

    2x 2lim

    1

    Plugging in ---4 again, we find that the limit hasbecome determinate and equals ---6

    If the limit does not become determinate after thefirst application of the rule, you can keep using therule until you reach a definite answer

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    CALCULUSSpin-Offs

    DERIVATIVES

    WHAT IS A DERIVATIVE?

    The rate of change of a function at any point

    FIRST DERIVATIVES

    Notated dydx

    , y', or f'(x)

    The first derivative of a curve or line is essentiallyan instantaneous slope

    The derivative of f(x) is the slope of a linetangent to the function at x

    In the above graph, the tangent line touchesthe curve where x = 1

    The tangent line has a slope of 2, so the curveatx = 1 has a derivative of 2

    If a function represents displacement, then itsfirst derivative represents velocity

    Displacement tells us wheresomething is Velocity us tells how fastsomething is

    moving

    Thus, velocity is the derivative, or the rate ofchange, of displacement

    The sign of the first derivative indicates how theoriginal function is changing

    If the first derivative is negative, then theoriginal function is decreasing in value

    If the first derivative is positive, then theoriginal function is increasing in value

    If the first derivative is 0, then the originalfunction is not changing in value

    Taking (finding) a derivative is calleddifferentiation

    ELEMENTARY POWER RULE

    The derivative of a term axn is naxn-1 In other words, we take the exponent and multiply it

    by the coefficient; then we subtract 1 from theexponent

    The derivative of a constant is 0 Constant a can be written as a(1) = ax0; taking

    the derivative will yield 0

    If f(x) = 4x3 --- 5x + 27, find f'(2) Each term either only has one variable or is a

    constant, so we can apply the power rule to eachterm

    ' 3 1 1 1

    f (x) 3(4x ) 1(5x ) 0 ' 2f (x) 12x 5 ' 2f (2) 12(2) 5 43

    SECOND DERIVATIVES

    The rate of change of a functions first derivative

    The second derivative of a displacement function isacceleration

    The second derivative is the changing rate of velocity Second derivatives reveal the concavity of a function

    and possible inflection points

    If the second derivative is positive, then the originalfunction is concave up

    If the second derivative is negative, then the originalfunction is concave down

    If the second derivative is 0, then the original functionmay have a point of inflection

    A point of inflection is where concavity changesfrom up to down or from down to up

    Concave Up Concave Down

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    CALCULUSDerive Safely

    DIFFERENTIATION RULES

    PRODUCT RULE

    When taking the derivative of a term that is theproduct of two expressions, we need to use theproduct rule

    If f(x) = uv, then f'(x) = (u')(v) + (v')(u) 2f(x) (x 5)(3x 2) 2u x 5 u' 2x v 3x 2 v' 3 2f '(x) (2x)(3x 2) (3)(x 5) 2 2f '(x) 6x 4x 3x 15 2f '(x) 9x 4x 15

    QUOTIENT RULE

    When taking the derivative of a term that is thequotient of two expressions, we need to use thequotient rule

    If uf(x)v

    , then

    2

    (v)(u') (u)(v')f'(x)

    v

    32

    x 7xf(x)

    9x

    3u x 7x 2u' 3x 7 2v 9x v' 18x 2 2 3

    2 2

    (9x )(3x 7) (x 7x)(18x)f'(x)

    (9x )

    4 2 4 24

    27x 63x 18x 126xf'(x)81x

    4 24

    9x 63xf'(x)

    81x

    22

    x 7f'(x)

    9x

    CHAIN RULE

    When taking the derivative of a function within afunction (a composite function), we need to use thechain rule

    If f(x) = g(h(x)), then f '(x) g'(h(x)) h'(x) 2 4f(x) 6(8x 13) 2h(x) 8x 13 h'(x) 16x 4g(x) 6x 3g '(x) 24x

    2 3

    f '(x) 24(8x 13) (16x) 2f '(x) 384x(8x 13)

    We can think of the chain rule as multiplying thenormal derivative, 24(8x

    2+ 13)

    3, by the derivative

    of the inside of the parentheses, 16x

    In other words, multiply the derivative of theoutside piece and by the derivative of the insidepiece

    OTHER COMMON DERIVATIVES

    d sin(u) (cos(u))(u')dx

    d cos(u) (sin(u))(u')dx

    2d tan(u) (sec (u))(u')dx

    2d cot(u) (csc (u))(u')dx

    d

    sec(u) (sec(u))(tan(u))(u')dx

    d csc(u) (csc(u))(cot(u))(u')dx

    u ud e e (u')dx

    d 1ln(u) (u')

    dx u

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    CALCULUSAint It Great?

    INDEFINITE INTEGRALS DEFINITE INTEGRALS

    ANTIDERIVATIVE

    A possible function that has a known derivative; also

    known as an integral

    ANTIDIFFERENTIATION

    The process of finding antiderivatives is calledantidifferentiation or integration

    The integration symbol is

    HOW TO FIND AN INDEFINITE INTEGRAL

    We basically reverse the steps of differentiation

    2(6x 3x 5)dx

    The dx at the end of the expression tells usthat the argument in the integration is aderivative

    Each term either only has one variable or is aconstant, so we can reverse the power rule foreach term

    To reverse the power rule, we will add one tothe exponent and divide the coefficient by thenew exponent

    The first term has a power of 2, which meansits integral must have a power of 3

    The first term of the integration becomes3 3

    6x 2x

    3

    The second term becomes 23 x2

    The third term becomes ---5x Put together, the integral is 3 232x x 5x C

    2

    The C at the end is an unknown constant Because all constants differentiate to 0,we have to account for the possibility of a

    constant when we integrate

    The C is what makes this integral indefinite: Ccould be any value

    INDEFINITE VS. DEFINITE

    Definite integrals produce a value because they have

    bounds. Indefinite integrals do not produce a valuebecause they include an unknown constant C.

    HOW TO FIND A DEFINITE INTEGRAL

    We begin the integration the same way we do forindefinite integrals

    4 5(15x 3x 2)dx Notice that the integral now has bounds at 0 and

    1

    After integrating, we get 5 6 113x x 2x2 0 Because this is a definite integral, we do not need

    a C at the end of the polynomial

    Next, we plug in each bound 5 61 13(1) (1) 2(1) 4

    2 2

    5 613(0) (0) 2(0) 02

    Finally, we find the difference between the resultof the upper bound and the result of the lowerbound

    1 14 0 42 2

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    CALCULUSDivination

    GRAPHS OF DERIVATIVES GRAPHS OF DERIVATIVES

    1.

    2.

    3.

    READING INFORMATION FROM GRAPHS

    Graph 1 is the original function, y Graph 2 is the first derivative, y'

    From --- < x < 0, the y' is negative, which meansthat y is decreasing

    At x = 0, y' is 0, which means that y is neitherincreasing nor decreasing

    From 0 < x < , y' is positive, which means that yis increasing

    The change from a negative y' to a positive y' at x = 0means that y has a minimum (explained later) at x =0

    Graph 3 is the second derivative, y'' Y'' stays positive over its entire domain Thus, y is concave up over its entire domain

    If we were only given graph 2 or graph 3, we would beable to find the shape, but not the vertical alignment,of their integrals

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    CALCULUSThe Magic Touch; Speed

    TANGENT LINES RATES OF CHANGE

    WHAT IS A TANGENT LINE?

    A line which touches a graph at a point and has the

    same slope as the graph at that point

    HOW TO FIND A TANGENT LINE

    Find the tangent line of f(x) = x3 --- 5x2 --- 8 at x = 2 The derivative of any function is its slope, so the

    derivative at a point is also the slope of thetangent line at that point

    2f '(x) 3x 10x f '(2) 8 We know that the slope of the tangent line is -

    8

    Since the tangent line meets the function at x= 2, it will have the same y-value as thefunction at that point

    Thus, one of the points on the tangent line is(2,-20), as f(2) = 8 --- 5(2)

    2--- 8 = ---20

    We have the slope and one point, so we canuse point-slope form, which is y---y1=m(x---x1),to find the equation of the tangent line

    y ( 20) 8(x 2) y 20 8x 16 The tangent line is y = ---8x --- 4

    SINGLE VARIABLE PROBLEMS

    Single variable rate of change problems usuallyinvolve displacement, velocity, and acceleration

    If the displacement of a car is represented by thefunction s(t) = 15t

    2+ 5t + 10, what is the acceleration

    of the car when t =4?

    To find the acceleration equation, we need totake two derivatives of the displacementequation

    s'(t) = v(t) = 30t + 5 s''(t) = v'(t) = a(t) = 30 In this case, acceleration is a constant, so it will

    equal 30 when t = 4

    RELATED RATE PROBLEMS Related rate problems involve at least two variables When more than two variables exist in a problem, we

    will try to reduce them to two variables

    Implicit differentiation is often used because youoften have to take the derivative of a physicaldimension (volume, radius, etc.) with respect to time

    An inverted cone with a height of 10 ft and a baseradius of 5ft is being filled with sand at a rate of

    3ft2

    min. How fast is the height of the sand changing

    when it is 6 ft high? The sand is essentially forming a cone that

    increases in volume

    The formula for the volume of a cone is 2

    1V r h

    3, (r is the radius of the base, h is the

    cones height)

    We know that the ratio of the height to the radiusis

    10

    5or

    2

    1, so we can substitute

    1r h

    2into the

    volume equation to get

    2

    31 1 1V h h h3 2 12

    Take a derivative to get 2dV 1 dhhdt 4 dt

    Plugging in h = 6 and dV 2dt

    , we get

    21 dh

    2 (6)4 dt

    dh 2 ftmindt 9

    , so the height of the sand is

    changing at a rate of about ft0.07min

    when the

    height is 6 ft

    IMPLICIT DIFFERENTIATION

    Usually, you take the derivative of a term with respectto the same variable as the one in the term. Forexample, taking the derivative of 2x

    3with respect to x

    yields6x2. When taking the derivative with respect to

    a variable other than the one in the term, useimplicitdifferentiation. Taking the derivative of 2x

    3with

    respect to t yields 2dx

    6xdt

    .

    This problem combines the power and productrules

    Derive x5 + 3xy2 = 15y --- 18with respect to x 5 1 2 dy dy5x 3y (3x)(2y) 15 0

    dx dx

    4 2 dy dy5x 3y 6xy 15dx dx

    4 25x 3y dy

    15 6xy dx

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    CALCULUSHighs and Lows

    MAXIMA AND MINIMA

    DEFINITIONS

    The absolute/global maximum of a function is thehighest y-value that it reaches

    The absolute/global minimum of a function is thelowest y-value that it reaches

    A relative/local maximum is a point whose y-value is higher than its surrounding points

    The peaks of hills in a graph A relative/local minimum is a point whose y-

    value is lower than it surrounding points

    The bottoms of valleys in a graph Maxima and minima are collectively known as

    extrema

    FINDING LOCAL EXTREMA

    Take the first derivative of a function Solve for the zeroes of the first derivative These zeroes are called critical points Place the critical points on a number line, dividing

    the number line into regions

    Find the sign of the first derivative in theseregions

    Going from left to right, if the sign changes fromnegative to positive around a critical point, then alocal minimum exists at that point

    The original function goes from decreasing toincreasing, forming a valley

    If the sign changes from positive to negativearound a critical point, then a local maximumexists at that point

    The original function goes from increasing todecreasing, forming a hill

    FINDING ABSOLUTE EXTREMA

    Take the first derivative of a function and Find all the critical points Plug all the critical points into the original equation to

    find their y-values

    If the graph has endpoints, plug the endpoints intothe original equation to find their y-values

    Compare the y-values to find the highest (maximum)and the lowest (minimum)

    Not all functions have absolute extrema since their y-values may go to infinity or negative infinity

    MAX/MIN WORD PROBLEMS

    Problems may ask you to optimize a constructionunder a constraint

    A farmer wants to build a fence that encloses thelargest possible area. He only has 30 yards offence. What should the rectangles dimensionsbe?

    The area of a rectangle is A = LW, where L islength and W is width

    The perimeter of a rectangle is P = 2L + 2W We know that the perimeter is 30, so 2L + 2W =

    30

    L + W = 15 L = 15 --- W Substituting into the area formula, we get

    A = (15 --- W)(W)

    A = 15W --- W2 Next, we take the derivative and find the critical

    point(s)

    A' = 15 --- 2W 0 = 15 --- 2W 2W = 15 W = 7.5 L = 15 --- W = 15 --- 7.5 = 7.5 The dimensions should be 7.5 yards by 7.5 yards Note that a square maximizes area for rectangles

    A square only maximizes area if all four sidesare limited by the perimeter

    If fewer than four sides are limited (say thefence is being built against a barn), then asquare will not maximize area

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    CRUNCH KITFormula Frenzy (Page 1)

    GENERAL MATH

    Permutations: n r n!P(n-r)!

    Combinations: n r

    n!

    C (r!)(n-r)!

    Circular arrangements: (n --- 1)! Probability that two independent events will occur:

    P(A+B) = P(A) x P(B)

    Probability that one of two mutually exclusive eventswill occur: P(A or B) = P(A) + P(B) --- P(A+B)

    ALGEBRA

    Slope:

    2 1

    2 1

    y ym

    x x

    Point-slope form: 1 1

    y y m(x x )

    Slope-intercept form: Standard form of a linear function: Ax By C Quadratic formula: 2B B 4ACx

    2A

    Sum of cubes: x3 + y3 = (x + y)(x2 --- xy + y2) Difference of cubes: x3 --- y3 = (x --- y)(x2 + xy + y2) Sum of roots: B

    A

    Product of roots:

    C

    for odd numberedpolynomialsA

    Cand foreven numberedpolynomials

    A

    nth term of an arithmetic sequence:nth term = first term + d(n --- 1)

    Number of terms in an arithmetic series:

    (lastterm first term)

    n 1d

    Sum of first n terms of an arithmetic series:

    (firstterm lastterm)

    n

    2

    nth term of a geometric sequence:nth term --- (first term)r

    n---1

    Sum of first n terms in a geometric series:

    n(first term)(1 r )

    1 r

    Geometric mean: xy

    GEOMETRY

    Pythagorean theorem: 2 2 2a b c Midpoint formula:

    1 2 1 2x x y y

    ,

    2 2

    Distance formula: 2 21 2 1 2

    (x x ) (y y )

    Area of a trapezoid: Area = 12

    (base1 + base2)(height)

    Area of a parallelogram: Area = (base)(height) Area of a rectangle: Area = (base)(height) Area of a rhombus: Area = 1

    2(diagonal1)(diagonal2)

    Area of a square: Area = (side)2

    1

    2Area (diagonal1)

    2

    Area of a triangle: 1Area (base)(height)2

    Area (s)(s a)(s b)(s c) , where

    a b c

    s2

    Area =1

    2ab(sinC)

    Surface area of prism:SA = Area of 2 bases + area of lateral faces

    Surface area of pyramid:SA = Area of the base + area of lateral triangles

    Surface area of cylinder: SA = 2r2 + 2rh Surface area of sphere: SA = 4r2 Surface area of cone: SA = r2 + r 2 2r h Volume of prism: V = (area of base)(height) Volume of pyramid: 1V (areaof thebase)(height)

    3

    Volume of cylinder: V = r2h Volume of sphere: V = 4

    3r

    3

    Volume of cone: V = 13r

    2h

    Circumference of circle: 2r Area of circle: r2 180 = radians Central angle = intercepted arc Inscribed angle = 1

    2intercepted arc

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    CRUNCH KITFormula Frenzy (Page 2)

    TRIGONOMETRY

    OppositeSine(angle)Hypotenuse

    Adjacent

    Cosine(angle) Hypotenuse

    OppositeTangent(angle)Adjacent

    1 1sinx ; cscxcscx sinx

    1 1cosx ; secxsecx cosx

    1 1tanx ; cotxcotx tanx

    sinxtanxcosx

    cosxcotxsinx

    2 2sin x cos x 1 2 2tan x 1 sec x 2 21 cot x csc x sin(x y) (sinx)(cosy) (cosx)(siny) cos(x y) (cosx)(cosy) (sinx)(siny) sin(x y) (sinx)(cosy) (cosx)(siny) cos(x y) (cosx)(cosy) (sinx)(siny) sin(2x) 2sinxcosx

    2 2

    2

    2

    cos(2x) cos x sin x

    1 2sin x

    2cos x 1

    Law of sines: sinA sinB sinCa b c

    Law of cosines: 2 2 2c a b 2ab(cosC)

    CALCULUS

    Power rule: b b 1d (ax ) abxdx

    Product rule: If f(x) = uv, then f'(x) = (u')(v) + (v')(u) Quotient rule: If uf(x)

    v, then

    2

    (v)(u') (u)(v')f'(x)

    v

    Chain rule: If f(x) = g(h(x)), then f '(x) g'(h(x)) h'(x) d sin(u) (cos(u))(u')

    dx

    d cos(u) (sin(u))(u')dx

    2d tan(u) (sec (u))(u')dx

    2d cot(u) (csc (u))(u')dx

    d sec(u) (sec(u))(tan(u))(u')dx

    d csc(u) (csc(u))(cot(u))(u')dx

    u ud e e (u')dx

    d 1ln(u) (u')

    dx u

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    FINAL TIPS AND ABOUT THE AUTHOR

    FINAL TIPS ABOUT THE AUTHOR

    Do the easy problems first; all the questions areworth the same points, and the easy problemsmay be at the end of the test

    Use a timer in practice and at competition Use all 30 minutes to work-----dont give up! When you have 5 minutes left, guess on all

    remaining unanswered questions before returningto your current problem

    Be familiar with your calculator If you dont know how to do a problem, try

    plugging in the answers, since theyre given to you

    Make sure your calculator is in degree modewhen working with degrees and in radian modewhen working with radians

    Before you begin the test, pick your favoriteguessing letter, and use it every time you cannoteliminate any answer choices

    They say Steven Zhu shot aman down in Reno, but thatwas just a lie. Keb Moreferences aside, this much

    is known about Steven: he isan economics major atHarvard University, hecompeted with the FriscoHigh School decathlonteam, and he once won astate championship in someplace called Texas. After a stint at the Federal ReserveBank of Dallas this summer, Steven hopped aroundvarious cities in China, land of Mao and slow internets. Hewould like to maximize happiness instead of utilitysomeday, but in the meantime, he will settle for a nap.

    ABOUT THE EDITOR

    SOPHY LEE

    After 19 years of planning and pondering, Sophy Lee has decided that the best things in life emergefrom coincidence. She discovered her favorite book, Zen and the Art of Motorcycle Maintenance,tucked away in the corner of an lawyers bookshelf. At the age of 11, she learned about Alexander theGreat in middle school and asked her parents to name her little brother Alex. In her junior year, shejoined Academic Decathlon and watched the program change the lives of her entire team. A year later,she led the Pearland High School Acadec team to its first state championship. These days, you canfind Sophy looking for coincidences and braving the cold at Harvard University. She welcomes yourthoughts on Zen and motorcycles at [email protected].

    ABOUT THE EDITOR/POWER ALPACA

    DEAN SCHAFFER

    Dean Schaffer believes that in his former life, he was either an owl (wise and nocturnal), alolcat (prone to nonsensical utterances), or a Microsoft Word spellchecker (compulsive butvulnerable to glitches). In this life, he attends Stanford University, majors in AmericanStudies, minors in Classics, and doesnt really know what he wants to do when he grows

    up-----something he constantly hopes hell never have to do.

    Since joining DemiDec to write the Renaissance Music Power Guide, Dean has taken turnsmaking the Power Guide more powerful, the flashcard a lot flashier, and the Cram Kit abitcrammier? This season marks Deans fifth with DemiDec, and his lengthy tenure has,thus far, given him a glimpse of the ineffable quirks of the English language and, morenotably, of the ineffable cuteness of the three puppies which inhabit DemiDec HQ (and areprobably the single biggest productivity drain on DemiDec Dan).