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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).