Key C oncepts

CHAPTER 0 Preparing for Precalculus

Powers of / (p. P6 ) To find the value of let R be the remainder when n is divided by 4.n < 0 n > 0

R = - 3 -> i n = / R = 1 —> n _ /'/? = —2 —> / ” = —1 R = 2 —> n = - 1

R = ~ 1 _ ♦ / " = _ / R = 3 —►n _ —i/? = 0 -> / " = 1 R = 0 _ ►n _

1

Graph of a Quadratic Function (p. P9) Consider the graph of y = ax2 + bx + c, where a ± 0 .• The y-intercept is a(0) 2 + b(0) + co r c.• The equation of the axis of symmetry is x = —

The x-coordinate of the vertex is ■2 a

Maximum and M inimum Values (p. P10) The graph of f(x) = ax2 + b x + c, where a=f=0,• opens up and has a minimum value when a > 0 , and• opens down and has a maximum value when a < 0 .

Zero Product Property (p. P11) For any real numbers a and b, if ab = 0, then either a = 0, b = 0, or both a and b equal zero.

Completing the Square (p. P12) To complete the square for any quadratic expression of the form x 2 + bx, follow the steps below.Step 1 Find one half of b, the coefficient of x.Step 2 Square the result in Step 1.Step 3 Add the result of Step 2 to x2 + bx.

■■♦-(ir=MrQuadratic Formula (p. P12) The solutions of a quadratic equation of the form ax2 + bx + c = 0, where a =/= 0,

are given by the following formula.- b ± \ ! b l - 4 acx = 2 a

nth Root of a Number (p. P14) Let a and b be real numbers and let n be any positive integer greater than 1.• If a = bn, then b is an nth root of a.• If a has an nth root, the principal nth root of a is the root having the same sign as a.The principal nth root of a is denoted by the radical expression \ [a , where n is the index of the radical and a is the radicand.

Basic Properties of Radicals (p. P15) Let a and b be real numbers, variables, or algebraic expressions, and m and n be positive integers greater than 1 , where all of the roots are real numbers and all of the denominators are greater than 0. Then the following properties are true.

< f a - \ f b = \ fa b_ nfa

<Tb V b

Product Property

Quotient Property

R1

Rational Exponents (p. P16) If b is a real number, variable, or algebraic expression and m and n are positive integers greater than 1 , then

• b" = \ fb , if the principal nth root of b exists, and• b~n = ( \ [ b ) m or if y is in reduced form.

Substitution Method (p. P18)Step 1 Solve one equation for one of the variables in terms of the other.Step 2 Substitute the expression found in Step 1 into the other equation to obtain an equation in one variable.

Then solve the equation.Step 3 Substitute the value found in Step 2 into the expression obtained in Step 1 to find the value of the

remaining variable.

Elim ination Method (p. P19)Step 1 Multiply one or both equations by a number to result in two equations that contain opposite terms. Step 2 Add the equations, eliminating one variable. Then solve the equation.Step 3 Substitute to solve for the other variable.

Solving Systems of Inequalities (p. P21)Step 1 Graph each inequality, shading the correct area. Use a solid line to graph inequalities that contain > or

< . Use a dashed line to graph inequalities that contain > or < .Step 2 Identify the region that is shaded for all of the inequalities. This is the solution of the system.Step 3 Check the solution using a test point within the solution region.

Adding and Subtracting Matrices (p. P24) To add or subtract two matrices with the same dimensions, add or subtract their corresponding elements.

A + B = A + B

' a b' e f ' a + e b + f+ —

. c d , . 9 h . c + 9 d + h .

A — B = A - Ba b ' e a - e b - f'

. c d. . 9 h . . c - 9 d - h .

Properties of M atrix Operations (p. P25)For any matrices A, B, and Cfor which the matrix sum and product are defined and any scalar k, the following properties are true.Commutative Property of Addition A + B = B + AAssociative Property of Addition (A + B) + C = A + (B + C)Left Scalar Distributive Property k(A + B) = kA + kBRight Scalar Distributive Property (A + B)k = kA + kB

Fundamental Counting Principle (p. P28) Let A and B be two events. If event A has n, possible outcomes and is followed by event B that has n2 possible outcomes, then event A followed by event B has n: • n2 possible outcomes.

Permutations (p. P29)The number of permutations of n objects taken n at a time is

nPn = w!.

The number of permutations of n objects taken ra t a time is

n n\(n - r ) \ '

Combinations (p. P30) The number of combinations of n objects taken ra t a time isnl

n^r — (n - r)\ r ! '

Measures of Central Tendency (p. P32)Mean The sum of the numbers in a set of data divided by the number of items.

Population Sample

n — — X = —P n nThe middle number in a set of data when the data are arranged in numerical order. The number or numbers that appear most often in a set of data.

MedianMode

R2 | Key Concepts

Measures of Spread (p. P32)

RangeVariance

StandardDeviation

The difference between the greatest and least values in a set of data.The mean of the squares of the deviations from the mean.

Population Variance Sample Variance

a "s 2 = _ £ ( x - x ) :

n n - 1

The average amount by which individual items deviate from the mean of all the data.

Population Standard Deviation Sample Standard Deviation

< - x ) 2

Real Numbers (p. 4)

Real Numbers (

Functions from a Calculus Perspective

Letter Set Examples

Q rationals 0.125,-1,1 = 0.666...O 0I irrationals V3 = 1.73205...z integers -5,17,-23,8w wholes 0,1,2,3...N naturals 1,2,3,4...

Function (p. 5) A function f from set A to set B is a relation that assigns to each element x in set A exactly one element y in set B.

Vertical Line Test p. 6 ) A set of points in the coordinate plane is the graph of a function if each possible vertical line intersects the graph in at most one point.

Tests fo r Symmetry (p. 16)

Graphical Test Model Algebraic TestThe graph of a relation is symmetric with respect to the x-axis if and only if for every point (x, y ) on the graph, the point (x, - y ) is also on the graph. 1

. y

ii1 x

, U -y)

Replacing y with - y produces an equivalent equation.

The graph of a relation is symmetric with respect to the y-axis if and only if for every point (x, y) on the graph, the point (—x, y) is also on the graph.

(-* , y) j -

i /

- \ u , y )

Replacing xwith —x produces an equivalent equation.

1 ° , \ *

The graph of a relation is symmetric with respect to the origin if and only if for every point (x, y) on the graph, the point (—x, - y ) is also on the graph.

1

\ J r .X

Replacing xwith -x a n d ywith - y produces an equivalent equation.

rj=

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Key Concepts

Even and Odd Functions (p. 18)

Type of Function Algebraic TestFunctions that are symmetric with respect to the y-axis are called even functions.

For every x in the domain of f, f(-x ) = f(x).

Functions that are symmetric with respect to the origin are called odd functions.

For every x in the domain of /, /(—x) = -f(x).

Lim its (p. 24} If the value of f(x) approaches a unique value L as x approaches cfrom each side, then the limit of f(x) as x approaches c is L.

Types of D iscontinuity (p. 24)

A function has an infinite discontinuity at x = c if the absolute value of the function increases or decreases indefinitely as x-values approach cfrom the left and right.

A function has a jump discontinuity at x = c if the limits of the function as x approaches cfrom the left and right exist but have two distinct values.

Example

A function has a removable discontinuity if the function is continuous everywhere except for a hole at x = c.

Example

Continuity Test (p. 25 A function f(x) is continuous at x = c if it satisfies the following conditions.• f(x) is defined at c. That is, f(c) exists.• f(x) approaches the same value from either side of c. That is, jimc f(x) exists.• The value that f(x) approaches from each side of c is f(c). That is, Nm, f(x) = f(c).

Intermediate Value Theorem (p. 27) If f(x) is a continuous function and a < b and there is a value n such that nis between f(a) and f(b), then there is a number c, such that a < c < b and f(c) = n.

Increasing, Decreasing, and Constant Functions (p. 34)

• A function f\s increasing on an interval I if and only if for any two points in /, a positive change in x results in a positive change in f(x).

• A function f is decreasing on an interval I if and only if for any two points in /, a positive change in x results in a negative change in f(x).

• A function f is constant on an interval / if and only if for any two points in /, a positive change in x results in a zero change in f(x).

Relative and Absolute Extrema (p. 36)

• A relative maximum of a function / is the greatest value f(x) can attain on some interval of the domain. If a relative maximum is the greatest value a function f can attain over its entire domain, then it is the absolute maximum.

• A relative minimum of a function /is the least value f[x) can attain on some interval of the domain. If a relative minimum is the least value a function f can attain over its entire domain, then it is the absolute minimum.

Average Rate of Change (p. 38) The average rate of change between any two points on the graph of f is theslope of the line through those points. The line through two points on a curve is called a secant line. The slopeof the secant line is denoted msec. The average rate of change on the interval [xv x2] is

f(x2) - f(x: )Msec = — V----X2 — X|

R4 Key Concepts

J

Square Root and Reciprocal Parent Functions p. 45)

Absolute Value and Greatest Integer Functions (p. 46)

The absolute value function, denoted f(x)= |x |, is a V-shaped function defined as

f(x) = ' ~x if * < 0

The greatest integer function, denoted f(x) = [x ], is defined as the greatest integer less than or equal to x.

y

- f ix = h nJJ

°LX

•

Vertical and Horizontal Translations (p. 47)

• Vertical Translations: The graph of g(x) = f(x) + k is the graph of f(x) translated /(units up when k > 0, and k units down when k < 0 .

• Horizontal Translations: The graph of g[x) = f(x - h) is the graph of f(x) translated h units right when h > 0 , and h units left when h < 0 .

Reflections in the Coordinate Axes (p. 48)

• Reflection in x-axis: g(x) = - f ( x ) is the graph of f(x) reflected in the x-axis.• Reflection in y-axis: g(x) = f ( - x ) is the graph of f(x) reflected in the y-axis.

R5

Key Concepts

• Vertical Dilations: If a is a positive real number, then g(x) = a • f(x), is the graph of f(x) expanded vertically,if a > 1 ; the graph of f(x) compressed vertically, if 0 < a < 1 .

• Horizontal Dilations: If a is a positive real number, then g(x) = f(ax), is the graph of f(x) compressedhorizontally, if a > 1 ; the graph of f(x) expanded horizontally, if 0 < a < 1 .

Vertica l and Horizonta l D ila tions (p. 49)

Transformations with Absolute Value (p. 51)

gix)

g (x )= |f(x)|This transformation reflects any portion of the graph of f(x) that is below the x-axis so that it is above the x-axis.

g(x) = f(|x|)This transformation results in the portion of the graph of f(x) that is to the left of the y-axis being replaced by a reflection of the portion to the right of the y-axis.

Operations with Functions (p. 57) Let f and g be two functions with intersecting domains. Then for all x-values in the intersection, the sum, product, difference, and quotient of f and g are new functions defined as follows.Sum ( f + 0 )(x) = f(x) + g(x)

Difference ( f - g)(x) = f(x) - g(x)

Product (f-g )(x ) = f(x) • g(x)

Quotient ( | j ( x ) = ^ | , g(x) + 0

Composition of Functions (p. 58) The composition of function fw ith function g is defined by [ fo g](x) = f[g (x )]. The domain of fo g includes all x-values in the domain of grthat map to fif(x)-values in the domain of f.

Horizontal Line Test (p. 65) A function f has an inverse function M if and only if each horizontal line intersects the graph of the function in at most one point.

Finding an Inverse Function (p. 6 6 )Step 1 Determine whether the function has an inverse by checking to see if it is one-to-one using the

horizontal line test.Step 2 In the equation for f(x), replace f(x) with yand then interchange xand y.Step 3 Solve for yand then replace y with M (x) in the new equation.Step 4 State any restrictions on the domain of H . Then show that the domain of f is equal to the range of

H and the range of f is equal the domain of M .

Compositions of Inverse Functions (p. 6 8 ) Two functions, f and g, are inverse functions if and only if

• f i m = x fo r every x in the domain of g(x), and• g [ f(x)] = x fo r every x in the domain of f(x).

R6 Key Concepts

— r rPower, Polynomial, and Rational Functions

Power Function (p. 8 6 ) A power function is any function of the form f(x) = ax" where a and n are nonzero constant real numbers.

Monomial Functions (p. 8 6 ) Let f be the power function f(x) = axn, where n is a positive integer.

n Even, a Positive

Domain: ( - 00, 00) Range: ( - 00,0]

x- and y-intercept: 0

Continuity: continuous for x e R

Symmetry: y-axis Minimum: (0,0)

Decreasing: ( - 00,0) Increasing: (0 ,00)

End behavior: lim fix) = - 0 0 and lim fix) = - 0 0x -t-oo r x—too r '

n Odd, a Positive

Domain and Range: (— 0 0 , 0 0 )

x- and y-intercept: 0

Continuity: continuous on (—00, 00)

Symmetry: originExtrema: none Increasing: (—00, 00)

End Behavior: lim fix) = - 0 0 and lim fix) = 00X—*—00 ' ' X—>00 ' '

Domain: ( - 00, 00) Range: ( - 00,0]

x- and y-intercept: 0

Continuity: continuous forx e R

Symmetry: y-axis Maximum: (0,0)Decreasing: (0 ,00) Increasing: (—00,0)

End behavior: lim fix) = - 0 0 and lim fix) = — 00X-*—oc v x~*oo v '

Domain and Range: ( - 00, 00)

x- and y-intercept: 0

Continuity: continuous forx e R

Symmetry: origin

Extrema: none Decreasing: ( - 00, 00)

End Behavior: lim fix) = 00 and lim f(x) = — 00X—*—00 X-+00 x '

Radical Functions (p. 90) Let fbe the radical function f(x) = < fx where n is a positive integer.

nEven

Domain and Range: [0 ,00)

x- and y-intercept: 0

Continuity: continuous on [0 ,00)

Symmetry: none Increasing: (0, 00)

Extrema: absolute minimum at (0,0)

End Behavior: lim f(x) = 00

nOdd

Domain and Range: (—00, 00)

x- and y-intercept: 0

Continuity: continuous on ( - 00, 00)

Symmetry: origin Increasing: ( - 00, 00)

Extrema: none

End Behavior: lim f(x) = — 00 and lim f(x) = c

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Leading Term Test for Polynomial End Behavior (p. 98)The end behavior of any polynomial function f[x) = anx n + . . . + a^x + a0 can be described in one of the following four ways.

n odd, a „ positive

lim f ix ) = -o o and lim f ix ) = <X->-oo ' ' X—>00 ' '

n even, a„ positive

lim f(x ) = 00 and lim f(x ) = 00X—►—00 ' ' X—>oo

lim f(x) = < lim f(x) = <

n odd, a„ negative

lim f ix ) = 00 and lim f(x ) = - 0 0X—►—00 X—»CX>

lim f{x) = — 00X—too

n even, an negative

lim f ix ) = -0 0 and lim f(x ) = - 0 0X-y- 00 v X—XX)

lim f(x) = -cx) lim f ix ) = ■X—»—00 X—>00

Zeros and Turning Points of Polynomial Functions (p. 100) A polynomial function f of degree n > 1 has at most n distinct real zeros and at most n - 1 turning points.

Quadratic Form (p. 100; A polynomial expression in x is in quadratic form if it is written as au2 + b u + c for any numbers a, b, and c, a =£ 0 , where u is some expression in x.

Repeated Zeros of Polynomial Functions (p. 102) If (x - c)m is the highest power of (x - c) that is a factor of polynomial function f, then c is a zero of multiplicity m of f, where m is a natural number.• If a zero c has odd multiplicity, then the graph of f crosses the x-axis at x = c and the value of f(x) changes

signs at x = c.

• If a zero c has even multiplicity, then the graph of f is tangent to the x-axis at x = c and the value of f(x) does not change signs at x = c.

Polynomial Division (p. 110) Let f(x) and d(x) be polynomials such that the degree of d(x) is less than or equal tofix)

the degree of f(x) and d(x) ± 0. Then there exist unique polynomials q(x) and r(x) such that = q(x) + rix) or f(x) = d(x) • q(x) + r(x), where r(x) = 0 or the degree of r(x) is less than the degree of d(x). If0 {X )

r(x) = 0 , then d(x) divides evenly into f(x).

Synthetic Division A lgorithm (p. 111) To divide a polynomial by the factor x - c, complete each step.Step 1 Write the coefficients of the dividend in standard form. Write

the related zero c of the divisor x - c in the box. Bring down the first coefficient.

Step 2 Multiply the first coefficient by c. Write the product under the second coefficient.

Step 3 Add the product and the second coefficient.

Step 4 Repeat Steps 2 and 3 until you reach a sum in the last column. Thenumbers along the bottom row are the coefficients of the quotient. The power of the first term is one less than the degree of the dividend. The final number is the remainder.

Example

Divide 6 x 3 - 25x2 + 18x + 9 by x - 3.

j J

coefficients remainder of quotient

j = Add terms. X = Multiply by c.

R8 Key Concepts

Remainder Theorem (p. 112, If a polynomial f(x) is divided b y x - c, the remainder is r = f(c).

Factor Theorem (p. 113) A polynomial f(x) has a factor (x - c) if and only if f(c) = 0.

Rational Zero Theorem (p. 119) If f is a polynomial function of the form f(x) = anxn + a n _ 1x " 1 + ... + a2 x2 + a^x + a0, with degree n > 1 , integer coefficients, and a0 =/= 0 , then every rational zero of f has the form where

• p and q have no common factors other than + 1 ,• p is an integer factor of the constant term a0, and• q is an integer factor of the leading coefficient an.

Upper and Lower Bound Tests (p. 121) Let f be a polynomial function of degree n > 1 , real coefficients, and a positive leading coefficient. Suppose f(x) is divided by x - c using synthetic division.• If c < 0 and every number in the last line of the division is alternately nonnegative and nonpositive, then c is

a lower bound for the real zeros of f.• If c > 0 and every number in the last line of the division is nonnegative, then c is an upper bound for the real

zeros of f.

Descartes’ Rule of Signs (p. 123) If f(x) = anxn + an_ 1x n_ 1 + ... + a^x+ a0 is a polynomial function with real coefficients, then• the number of positive real zeros of f is equal to the number of variations in sign of f(x) or less than that

number by some even number, and• the number of negative real zeros of f is the same as the number of variations in sign of f( -x ) or less than

that number by some even number.

Fundamental Theorem of Algebra (p. 123) A polynomial function of degree n, where n > 0, has at least one zero (real or imaginary) in the complex number system.

Linear Factorization Theorem (p. 124) If f(x) is a polynomial function of degree n > 0, then fhas exactly n linear factors and

f[x) = an( x - c , ) ( x - c2) ■■■ ( x - cn)

where a „ is some nonzero real number and cv c 2, ..., cn are the complex zeros (including repeated zeros) of f.

Factoring Polynomial Functions Over the Reals (p. 124) Every polynomial function of degree n > 0 with real coefficients can be written as the product of linear factors and irreducible quadratic factors, each with real coefficients.

Vertical and Horizontal Asymptotes (p. 131)The line x = c is a vertical asymptote of the graph of f if lim f(x) = ±oo or lim f{x) = ± o o .

x—>c+The line y = c is a horizontal asymptote of the graph of f if lim ^ f(x) = c or Jim_ t(x) = c.

Graphs of Rational Functions (p. 132) If f is the rational function given byfix) = JW = a°xn + an. - i* " -1 + - + a 1x+a„

b{x) bmxm + + . . .+ b^x+ b0'

where b(x) =/= 0 and a(x) and b(x) have no common factors other than + 1 , then the graph of f has the following characteristics.Vertical Asymptotes Vertical asymptotes may occur at the real zeros of b(x).Horizontal Asymptote The graph has either one or no horizontal asymptotes as determined by comparing the degree n of a(x) to the degree m of b(x).

• If n < m, the horizontal asymptote is y = 0.• If n = m, the horizontal asymptote is y = ^ - .

Dm• If n > m, there is no horizontal asymptote.Intercepts The x-intercepts, if any, occur at the real zeros of a(x). The y-intercept, if it exists, is the value of f when x = 0 .

Oblique Asymptotes (p. 134) If f is the rational function given by= aM a„xn + a „ _ 1x n - 1 + . . . + a1x + a „

W bmxm + 6 m_ i x m ~ 1 + . . . + 6 , x + b0'

where b(x) has a degree greater than 0 and a(x) and b(x) have no common factors other than 1 , then the graph of fhas an oblique asymptote if n = m + 1. The function for the oblique asymptote is the quotient polynomial q(x) resulting from the division of a(x) by b(x).

Key

Conc

epts

Exponential and Logarithmic Functions

Exponential Function (p. 158) An exponential function with base b has the form f(x) = abx, where x is any real number and a and b are real number constants such that a ± 0 , b is positive, and b =/= 1.

Properties of Exponential Functions (p. 159)

Exponential Growth

Domain: (-oo, oo)

y-lntercept: 1

Extrema: none

Range: (0, oo) x-lntercept: none

Asymptote: x-axis

End Behavior: lim fix) = 0 and lim fix ) = ooX—*—oo x—>ooContinuity: continuous on (-oo, oo)

Exponential Decay

y-lntercept: 1

Extrema: none

x-lntercept: none

Asymptote: x-axis

End Behavior: lim f(x) = oo and lim fix ) = 0X—*—oo ' ' x—toc ' '

Continuity: continuous on (-oo, oo)

Compound Interest Formula (p. 162) If a principal P is invested at an annual interest rate r(in decimal form) compounded n times a year, then the balance A in the account after t years is given by

r \n tA - P ( 1 + 3 “

Continuous Compound Interest Formula (p. 163) If a principal Pis invested at an annual interest rate r(in decimal form) compounded continuously, then the balance A in the account after t years is given by

A = Pert.

Exponential Growth or Decay Formulas (p. 164)If an initial quantity N0 grows or decays at an exponential rate ror k(as a decimal), then the final amount N after a time t is given by the following formulas.

Exponential Growth or Decay

W=W0(1 + r ) 1 If r is a growth rate, then r > 0.If r is a decay rate, then r < 0.

Continuous Exponential Growth or DecayN = N0e kt

If k is a continuous growth rate, then k > 0. If k is a continuous decay rate, then k < 0.

Relating Logarithm ic and Exponential Forms (p. 172) If b > 0, b ± 1, and x > 0, then

Logarithmic Form Exponential Form

loghx = y if and only if

IIbase

h x = yf tby = x.

exponent base exponent

Basic Properties of Logarithms (p. 173) If b > 0, b ± 1, and x is a real number, then the following statements are true.

• log6 jb* = xlogf i 1 = 0

logf t6 = 1£log„x _ x x > g Inverse Properties

Basic Properties of Common Logarithms (p. 173) If x is a real number, then the following statements are true.

log 1 = 0

log 1 0 = 1

log 1 0 * = xlog I 0 log* = x, x > 0

Inverse Properties

R 1 0 Key Concepts

Basic Properties of Natural Logarithms (p. 174) If x is a real number, then the following statements are true.

• In 1 = 0• In e = 1

Properties of Logarithm ic Functions (p. 176)

Logarithmic Growth

• In ex = xJn x = X, X > 0

Inverse Properties

Domain: (0, oo) Range: (-oo, oo)y-lntercept: none x-lntercept: 1Extrema: none Asymptote: y-axisEnd Behavior: lim f(x) = - o o and lim f(x) = oo

Continuity: continuous on (0, oo)

Logarithmic Decay

Domain: (0, oo) y-lntercept: none Extrema: none

Range: (—oo, oo) x-lntercept: 1 Asymptote: y-axis

End Behavior: lim fix ) = oo and lim f(x) = -ooX—►0+ X—>ooContinuity: continuous on (0, oo)

Properties of Logarithms (p. 181) If b, x, and yare positive real numbers, b =/= 1, and p is a real number, then the following statements are true.Product Property Quotient Property

log6 x y = log6 x + logf lyX

log * y = iog *J f— log b y Power Property log b xp = p log b x

Change of Base Formula (p. 183) For any positive real numbers a, b, and x, a =/= 1, b =/= 1,log a x

0gftx = - ^ _ . b log a b

One-to-One Property o f Exponential Functions (p. 190) For b > 0 and b f 1 , bx = by if and only if x = y.

One-to-One Property o f Logarithm ic Functions (p. 191 ) For b > 0 and b =/= 1, log b x = log b y if and only if x = y.

Logistic Growth Function (p. 202) A logistic growth function has the form f(t) =

number, a, b, and c are positive constants, and c is the limit to growth.

Transformations fo r Linearizing Data (p. 204) To linearize data modeled by:• a quadratic function y = ax2 + bx + c, graph (x, \J y ) .• an exponential function y = abx , graph (x, In y).• a logarithmic function y = a In x + b, graph (In x, y).• a power function y = axb, graph (In x, In y).

1 + ae~btwhere fis any real

Trigonometric Functions

Trigonometric Functions (p. 220)Let 9 be an acute angle in a right triangle and the abbreviations opp, adj, and hyp refer to the length of the side opposite 6, the length of the side adjacent to 9, and the length of the hypotenuse, respectively. Then the six trigonometric functions of 9 are defined as follows.

sine (9) = sin 9 = hyp

cosine (9) = cos 9 = ^ - hyp

tangent (9) = tan 9 = 3QJ

cosecant (9) = csc 0 =hypopp

secant (9) = sec 9 = ^ adj

cotangent (9) = cot 0 = - | |

Key Concepts

Key

Conc

epts

Trigonometric Values of Special Angles (p. 222)

30°-60°-90° Triangle

V3x

45°-45°-90° Triangle

oOCO 45° 60°

sin 0 V l V2 V 32 2

cos o E i 9 V 2 1I H 2 2

tan 0 1 V 3

CSC 0 H T 9 V 2 2\/33

B P P f f i 2V 3 : i l f l 3 V 2 2

cot 0 L 1 V 33

Inverse Trigonometric Functions (p. 223)

Inverse Sine If 9 is an acute angle and the sine of 0 is x, then the inverse sine of x is the measure ofangle 9. That is, if sin 0 = x, then sin~ 1 x = 9.

Inverse Cosine If 9 is an acute angle and the cosine of 9 is x, then the inverse cosine of x is themeasure of angle 9. That is, if cos 9 = x, then cos - 1 x - 9.

Inverse Tangent If 9 is an acute angle and the tangent of 9 is x, then the inverse tangent of x is themeasure of angle 9. That is, if tan 9 = x, then tan -1 x = i

Radian Measure (p. 232) The measure 9 in radians of a central angle of a circle is equal to the ratio of the length of the intercepted arc s to the radius a of the circle. In symbols, 0 = j , where 9 is measured in radians.

Degree/Radian Conversion Rules (p. 233)

• To convert a degree measure to radians, multiply by

• To convert a radian measure to degrees, multiply by

t t radians 180° ' 180°

t t radians'

Coterminal Angles (p. 234)

Degrees If a is the degree measure of an angle, then all angles measuring a + 360n°, where n is an integer, are coterminal with a.

Radians If a is the radian measure of an angle, then all angles measuring a + 2mr, where n is an integer, are coterminal with a.

Arc Length (p. 235i If 9 is a central angle in a circle of radius r, then the length of the intercepted arc s is given by s = r9, where 9 is measured in radians.

Linear and Angular Speed (p. 236) Suppose an object moves at a constant speed along a circular path of radius r.• If s is the arc length traveled by the object during time f, then

the object’s linear speed ins given by v -

• If 9 is the angle of rotation (in radians) through which the object moves during time t, then the angular speed of the object is given by us = 4 .

R12 | Key Concepts

Area of a Sector (p. 237) The area A of a sector of a circle with radius rand central angle 0 is /1 = ^ r 29, where 9 is measured in radians.

Trigonometric Functions of Any Angle (p. 242) Let 9 be any angle in standard position and point P(x, y) be a point on the terminal side of 9. Let r represent the nonzero distance from Pto the origin. That is, letr = \ f y 2x L + y 2 ± 0. Then the trigonometric functions of 9 are as follows.

CSC Isin 9 = -r

cos 9 = j

tan 9

' = Lr y * osec 9 = x ± 0

c o t 0 = | , y # O

Common Quadrantal Angles p. 243)

\y(0 , r)

9

y y y

9

O ooOII

( / V * 1 0

0 radians 9 = 90° or

* V r , 0) 0

^radians 6= 180°o

*x [ O

t t radians 0 = 270° or

X

(0, - r )

• y - radians

Reference Angle Rules (p. 244)If 9 is an angle in standard position, its reference angle 9 ’ is the acute angle formed by the terminal side of 9 and the x-axis. The reference angle 9 ' for any angle 9 ,0° < 6 < 360° or 0 < 9 < 2n, is defined as follows.

Evaluating Trigonometric Functions of Any Angle (p. 245)

S tep l Find the reference angle 0'.Step 2 Find the value of the trigonometric function for 9 ’.Step 3 Using the quadrant in which the terminal side of 9 lies, determine the sign of the trigonometric

function value of 9.

Trigonometric Functions on the Unit Circle (p. 248) Let f beany real number on a number line and let P(x, y ) be the point on fwhen the number line is wrapped onto the unit circle. Then the trigonometric functions of fare as follows.

sin f =

csc f = ■>y+ 0

cos t = x

sec f = j , x 0

tan f = j , x £ 0

cot f = -p, y # 0

Therefore, the coordinates of P corresponding to the angle f can be written as P(cos f, sin t).

Periodic Functions (p. 250) A function y = f(f) is periodic if there exists a positive real number c such that f ( f + c) = f(t) for all values of fin the domain of f.The smallest number c for which f is periodic is called the period of f.

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Key Concepts

Properties of the Sine and Cosine Functions (p. 256)

Sine Function Cosine FunctionDomain: (—00 , 00) Range: [—1,1] Domain: (—00 , 00) Range: [—1,1]y-intercept: 0 y-intercept: 1

x-intercepts: m , n e Z x-intercepts: f n , n e Z

Continuity: continuous for all real numbers Continuity: continuous for all real numbersSymmetry: origin symmetry (odd function) Symmetry: y-axis symmetry (even function)Extrema: max. of 1 at x = ^ + 2mr, n e Z Extrema: max.of 1 a tx = 2 m , n e z

Q/rrmin. of —1 at x = + 2 m t .n s Z min of - 1 at x = tt + 2mr, n e Z

End Behavior:2

lim sin x and lim sinxX—» -o o X—>00

End Behavior: lim cos x and lim cosxX—* -0 0 X—►oo

do not exist. do not exist.Oscillation: between - 1 and 1 Oscillation: between - 1 and 1

Amplitudes of Sine and Cosine Functions (p. 257) The amplitude of a sinusoidal function is half the distance between the maximum and minimum values of the function or half the height of the wave. For y = a sin (bx + c) + cfand y = a cos {bx + c) + d, amplitude = \a\.

Periods of Sine and Cosine Functions (p. 259) The period of a sinusoidal function is the distance between anyO /rr

two sets of repeating points on the graph of the function. So, period = — .

Frequency of Sine and Cosine Functions (p. 260) The frequency of a sinusoidal function is the number of cycles1*1the function completes in a one unit interval. The frequency is the reciprocal of the period. So, frequency = — .Z'K

Phase Shift of Sine and Cosine Functions (p. 261) The phase shift of a sinusoidal function is the difference between the horizontal position of the function and that of an otherwise similar sinusoidal function. So, phase

S|* = “ W '

Properties of the Tangent Function (p. 269)

Domain: x e R , x # y + mr, n e RRange: (—oo, oo)

x-intercepts: mr, n e By-intercept: 0Continuity: infinite discontinuity at x = j + mz,

n e RAsymptotes: x = y + m r, n e RSymmetry: origin symmetry (odd function)Extrema: noneEnd Behavior: lim tanxand lim tanxX—»—oo X—*oc

do not exist. The function oscillates between -oo and oo. period: ir

Period of the Tangent Function (p. 270) The period of a tangent function is the distance between any two consecutive vertical asymptotes. So, period =

R 14 Key Concepts

P roperties o f the C otangent Function (p. 272)

Domain: x e R ,x= £ nix, n e R

Range: (—00 , 00)/-intercepts: y + rnv, n e R

y-intercept: noneContinuity: infinite discontinuity at x = h-k, n e R

Asymptotes: x = n it, n e RSymmetry: origin (odd function)

Extrema: noneEnd Behavior: lim cot x and lim cot x do not exist.X—»—00 X—*00

The function oscillates between - 0 0 and 00 .

Properties of the Cosecant and Secant Functions (p. 273)

Domain:

Range:/-intercepts:y-intercept:Continuity:Asymptotes:Symmetry:End Behavior:

Cosecant Functionx e R , nix, n e R (—00 , —1] and [1 ,00)

none noneinfinite discontinuity at x = n-rc, n e R

x = n-K, n e t

origin (odd function) lim csc x and lim csc x do not exist. TheX—* - 0 0 X—too

function oscillates between - 0 0 and 00 .

period: 2iv

Domain:

Range:x-intercepts:^intercept:

Continuity:Asymptotes:Symmetry:Behavior:

Secant Functiont, x £ i + im, n e :

—1] and [1,oo)x e R( - 00 ,

none1infinite discontinuity at x = y + niv, n e R

x = y + m x,n e R y-axis (even function)

lim sec x and lim sec xdo not exist. The

period: 2 ix

Damped Harmonic Motion (p. 276) An object is in damped harmonic motion when the amplitude is determined by the function a(t) = ke~ct.

Domain of Compositions of Trigonometric Functions (p. 286)

f ( f~1 ( * ) ) = X f~ H f(x)) = x

If - 1 < x < 1, then sin (sin-1 x) = x. If —y < x < y , then sin-1 (sin x) = x.

If - 1 < x < 1, then cos (cos"1 x) = x. If 0 < x < it, then cos-1 (cos x) = x.

If —00 < x < 00 , then tan (tan-1 x) = x. If - y < x < y , then tan-1 (tan x) = x.

Law of Sines (p. 291! If AABC has side lengths a, b, and c representing the lengths of the sides opposite the angles with measures A, B, and

sin A _ sin B _ sin Ca ~ b ~ c

C, then •

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Key Concepts

The Ambiguous Case (SSA) (p. 293)

Law of Cosines (p. 295' In AABC, if sides with lengths a, b, and c are opposite angles with measures A, B, and C, respectively, then the following are true.• a 2 = b 2 + c 2 - 26c cos• b2 = a2 + c2 - 2accos B• c2 = a2 + b2 - la b cos C

Heron’s Formula (p. 296) If the measures of the sides of AABC are a, b, and c, then the area of the triangle is given by Area = \ J s ( s - a ) ( s - b ) ( s - c), where s = l ( a + b + c).

Area of a Triangle Given SAS (p. 297; The area of a triangle is one half the product of the lengths of two sides and the sine of their included angle.

P T F R Trigonometric Identities and Equations

Reciprocal and Quotient Identities (p. 312)

Reciprocal Identitiessin 9 =

csc 9 =

1csc 9

1sin 9

COS 9 =

sec 8 =

1sec e

1cos 9

tan i

cot (

1cot I

1tan f

Quotient Identitiessin 9tan

c o te =

cos 9 COS 6sin 9

Pythagorean Identities (p. 313)sin2 9 + cos2 6 = 1 tan2 9 + 1 = sec2 9 cot2 9 + 1 = csc2 9

Cofunction Identities (p. 314)sin 9 = cos f e -

cos 9 = sin( f - « )

tan 9

cot 9

- c o t ( f - » )

= tan ( f - 0 )

sec

csc

i = csc ( y - 9)

> = sec ( y - 0 )

Odd-Even Identities (p. 314)sin ( - 9) = - s in 9 csc ( - 9 ) = -c s c 9

cos [ - 9 ) = cos 9 sec ( - 9 ) = sec 9

tan ( - 9 ) = - ta n 9 cot (—0) = - c o t 9

Sum and Difference Identities (p. 337)

Sum Identitiescos (a + (3) = cos a cos (3 - sin a sin (3sin (a + (3) = sin a cos [3 + cos a sin j3. . _. tan a + tan /3tan a + /3 = - — -------— ^

1 - tan a tan (1

Difference Identitiescos ( a - (3) = cos a cos [3 + sin a sin (3 sin ( a - 1 3) = sin a cos f3 - cos a sin (3

tan a - tan f3tan ( a - f3) = 1 + tan a tan /3

R 16 Key Concepts

Double-Angle Identities (p. 346)sin 0 ci 2 tan 9

sin 20 = 2 sin 0 cos 0 cos 20 = cos2 0 - sin2

tan 20 ---------- —1 - tan2 9

Power-Reducing Identities (p. 347)sin2 0 = 1 ~ c°s 261

Half-Angle Identities (p. 348)

cos 20 = 2 cos2 0 - 1

cos 20 = 1 - 2 sin2 0

C0S2 0 = i±CO S20 tan2 0 ■ l - cos 26 l + cos 28

sin | = ±

cos | = +

- cos 9 tan — V i— cos 6+ cos (

t a n f = _ 3 ! “ i L 2 1 + cos (

+ cos ( tan I = I j^ c o s j 2 sin 8

Product-to-Sum Identities (p. 350)sin a sin /3 = ^ [cos ( a - ( 3 ) - cos (a + p)]

cos a cos (3 = 1 [cos (a - (3) + cos (a + 0)]

Sum-to-Product Identities (p. 350). _ n . l a + /3\ l a — f3\sin a + sin /3 = 2 sin I— ^ -1 cos I— -— I

• n r, la t + P \ . ( a - f isin a - sin 0 = 2 cos I— ~ J sin I— -

sin a cos (3 = -|[s in (a + /3) + sin (a - /?)]

cos a sin f3 = -|[s in (a + (3) - sin (a - (3)]

_ „ l a + (3\ ( a - /3\cos a + cos (3 = 2 cos I— cos I— ^ 1

_ I a + p \ . I a — /3\cos a - cos (3 = - 2 sin I— 1 sin I —-- - I

C H A P T E R 6 Systems of Equations and Matrices

Operations tha t Produce Equivalent Systems (p. 364) Each of the following operations produces an equivalent system of linear equations.• Interchange any two equations.• Multiply one of the equations by a nonzero real number.• Add a multiple of one equation to another equation.

Elementary Row Operations (p. 366) Each of the following row operations produces an equivalent augmented matrix.• Interchange any two rows.• Multiply one row by a nonzero real number.• Add a multiple of one row to another row.

Row-Echelon Form (p. 367) A matrix is in row-echelon form if the following conditions are met.• Rows consisting entirely of zeros (if any) appear at the bottom of the matrix.• The first nonzero entry in a row is 1, called a leading 1.• For two successive rows with nonzero entries, the leading 1 in the higher row is farther to the left than the

leading 1 in the lower row.

M atrix M ultip lication (p. 375) If A is an m x r matrix and B is an r x n matrix, then the product AB is an m x n matrix obtained by adding the products of the entries of a row in A to the corresponding entries of a column in B.

Properties of M atrix M ultip lication (p. 377 For any matrices A 6 , and Cfor which the matrix product is defined and any scalar k, the following properties are true.Associative Property of Matrix Multiplication (AB)C = A(BC)

Associative Property of Scalar Multiplication k(AB) = (kA)B = A(kB)

Left Distributive Property C(A + B) = CA + CB

Right Distributive Property (-4 + B)C = A C + BC

Identity Matrix (p. 378) The identity matrix of order n, denoted /„, is an n x n matrix consisting of all 1s on its main diagonal, from upper left to lower right, and Os for all other elements.

Inverse of a Square Matrix (p. 379) Let A be an n x n matrix. If there exists a matrix B such that A B = BA = /„, then B is called the inverse of A and is written as A~1. So, AA~ 1 = A- 1 A = /„.

Inverse and Determinant of a 2 x 2 Matrix (p. 3 8 1 ) Let A =

If A is invertible, then / T 1 = 1a d - cb

matrix and is denoted det(/4) = |/4| =

d - b - c a a b c d

a b. A is invertible if and only if ad - cb=f= 0.

c d

The number ad - cb is called the determinant of the 2 x 2

= ad - cb.

Invertible Square Linear Systems (p. 388) Let A be the coefficient matrix of a system of n linear equations in n variables given by A X = B, where X is the matrix of variables and B is the matrix of constants. If A is invertible, then the system of equations has a unique solution given by X = A ~ 1 B.

Cramer’s Rule (p. 390) Let A be the coefficient matrix of a system of n linear equations in n variables given by AX = B. If det(/4) ± 0, then the unique solution of the system is given by

x - K l'X° ~ W

where A{ is obtained by replacing the /'th column of A with the column of constant terms B. If det(/4) = 0, then AX = B has either no solution or infinitely many solutions.

Vertex Theorem for Optimization (p. 405) If a linear programming problem can be optimized, an optimal value will occur at one of the vertices of the region representing the set of feasible solutions.

Linear Programming (p. 406) To solve a linear programming problem, follow these steps. Step 1 Graph the region corresponding to solution of the system of constraints.Step 2 Find the coordinates of the vertices of the region formed.Step 3 Evaluate the objective function at each vertex to determine

which x- and y-values, if any, maximize or minimize the function.

Partial Fraction Decomposition of f (x ) ld (x ) (p. 401)

1. If the degree of f(x) > the degree of d(x), use polynomial long division and the division algorithm to

write = q(x) + Then apply partial fraction decomposition to

fix)2. If ^ is proper, factor d(x) into a product of linear and/or prime quadratic factors.

3. For each factor of the form (ax + b)n in the denominator, the partial fraction decomposition must include the sum of n fractions

+ -a x + b (gx+ b)2 (ax + b)

An(ax+ b)n

where A , , A,, A3 An are real numbers.

4. For each prime quadratic factor that occurs n times in the denominator, the partial fraction decomposition must include the sum of n fractions

B x + C]■ + ■

BjX + Co■ + - •+ ••

BgX + Cn

(ax2 + bx+ c)naxl + b x + c (ax2 + bx+d)i (ax2 + b x + c )J

where Bv B2, B3, .. . , Bn and Cv C2, C3 C„are real numbers.

5. The partial fraction decomposition of the original function is the sum of q(x) from part 1 and the fractions in parts 3 and 4.

R 18 Key Concepts

Conic Sections and Parametric Equations

Standard Form of Equations for Parabolas (p. 423)

(x — A) 2 = 4p(y — *) ( y - k ) 2 = 4 p ( x - A)

yl\ •* / - J

y i1 rf"y y i • /

\ i / : / t > \

i---

i

\ / ° / ; \ x a a

\ ! / - ! \ 1*0 \ i f / x /

r \1 --

/ 0 0 ! x1 1

1

t1

t

p > 0 p < 0 p > 0 p < 0

Orientation: opens vertically Orientation: opens horizontally

Vertex: (A, k) Focus: (A, k + p) Directrix d :y = k - p Vertex: (A, k) Focus: (A + p,k) Directrix d\x = h — p

Axis of Symmetry a:x = h Axis of Symmetry a \y = k

Standard Form of Equations for Ellipses (p. 433)

(x -h )2 (y -k )+ = 1

Center: (A, k) Foci: (A ± c, k) Vertices: (A ± a, k)Co-vertices: (A, k ± b) Major axis: y = k Minor axis: x = ha, b, c relationship: c2 = a2 - b2 or c = V o 2 - b2

(x - h ) 2 (y - k ) 2b2 a2

Orientation: vertical major axisCenter: (h, k) Foci: (A, k ± c ) Vertices: (A, k ± a)Co-vertices: (A ± b, k) Major axis: x = A Minor axis: y = k a, A, c relationship: c2 = a2 - b2 or c = V o 2 - b2

Eccentricity (p. 435) For any ellipse,

the eccentricity e = 4-a

( x - h ) 2 , ( y - k ) 2 . ( x - h ) 2 ( y - k ) 2= 1 or -A2

= 1 , where c 2 = a2 - b 2,

Standard Form of Equations for Circles (p. 437) The standard form of an equation for a circle with center (ft, k) and radius r is (x - ft) 2 + (y - k)2 = r 2.

Standard Forms of Equations for Hyperbolas (p. 443)

(x -h )2 (y -k )2b2

= 1

Orientation: horizontal transverse axis

Center: (A, k) Vertices: (A ± o, k)Foci: (A ± c, k)' Transverse axis: y = kConjugate axis: x = A Asymptotes: y - k = ±% (x - h)a, A, crelationship: c2 = o2 + A2 or c = \ fa 2~+~b2

(y -k )2 (x -h )2 = 1

Orientation: vertical transverse axis Center: (A, k) Vertices: (A, k ± a)Foci: (A, k ± c) Transverse axis: x = hConjugate axis: y = ka, A, c relationship: c2 = o2 + A2 or c = \Ja2 + b

Asymptotes: y — k = ± ^ - ( x - h )D

&

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Key Concepts

Key

Conc

epts

BxyClassify Conics Using the Discriminant (p. 447) The graph of a second degree equation of the form Ax2 + Cy2 + D x + E y + F = 0 is• a circle if B2 - 4AC < 0; B = 0 and A = C.• an ellipse if B2 - 4AC < 0; either B ± 0 or A ± C.• a parabola if B2 - 4A C = 0 .• a hyperbola if B2 - 4AC > 0.

Rotation of Axes of Conics (p. 454) An equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 in the xy-plane can be rewritten as A(x ')2 + C (y ')2 + Dx’ + Ey' + F = 0 in the x'y'-p lane by rotating the coordinate axes through an angle 0. The equation in the x'y'-plane can be found using x = x ' cos 0 - y ' sin 0, and y = x ' sin 0 + y ' cos 0.

Angle of Rotation Used to Eliminate xy-Term (p. 455) An angle of rotation 0 such that cot 20 ■■ A - CB ; B ± 0,

0 < 0 < y , will eliminate the xy-term from the equation of the conic section in the rotated x'y'-coordinate

system.

Rotation of Axes of Conics (p. 457) When an equation of a conic section is rewritten in the x 'y '-p lane by rotating the coordinate axes through 0, the equation in the xy-plane can be found using x ' = xcos 0 + y sin 0, and y ' = ycos 0 - xs in 0.

Parametric Equations (p. 464) If f and g are continuous functions of t on the interval /, then the set of ordered pairs (f(t), g(t)) represent a parametric curve. The equations x = f( t ) and y = g(t) are parametric equations for this curve, f is the parameter, and / is the parameter interval.

Projectile Motion (p. 467) For an object launched at an angle 0 with the horizontal at an initial velocity v0, where g is the gravitational constant, t is time, and ft0 is the initial height:

Horizontal Distance X = tv0 COS I Vertical Position y = tv0 sin 0 - \gt

C H A P T E R 8 VectorsFinding Resultants (p. 484)

Triangle Method (Tip-to-Tail) Parallelogram Method (Tail-to-Tail)

To find the resultant of o + b, follow these steps.

Step 1 Translate b so that the tail of b touches the tip of a.

Step 2 The resultant is the vector from the tail of a to the tip of b.

To find the resultant of a + b, follow these steps.

Step 1 Translate b so that the tail of b touches the tail of a.

Step 2 Complete the parallelogram that has a and b as two of its sides.

Step 3 The resultant is the vector that forms the indicated diagonal of the parallelogram.

Multiplying Vectors by a Scalar (p. 485) If a vector v is multiplied by a real number scalar k, the scalar multiple k \i has a magnitude of \k\ |v|. If k > 0, k \i has the same direction as v. If k < 0, k \i has the opposite direction as v.

Component Form of a Vector (p. 492) The component form of a vector ~AB with initial point A(x^, y ^ and terminal point fi(x2, y 2) is given by (x 2 - x 1, y 2 - y ^ .

Magnitude of a Vector in the Coordinate Plane (p. 493) If v is a vector with initial point (x ,, y ,) and terminal

point (x2, y 2), then the magnitude of v is given by |v| = \j(x2 - x,)2 + (y2 - y t ) 2 . If v has a component form

of (a, ft), then |v| = \Ja2 + ft2 .

R 20 | Key Concepts

Vector Operations (p. 493) If a = ( av a2) and b = {b v b2) are vectors and k is a scalar, then the following are true.Vector Addition a + b = (a, + b : , a2 + b2)Vector Subtraction a - b = (a 1 - , a2 - b2)Scalar Multiplication ka = {kav ka2)

Dot Product of Vectors in a Plane (p. 500) The dot product of a = ( a1 , a2) and b = ( ^ , b2) is defined as a • b = a1 6 1 + a2b2.

Orthogonal Vectors (p. 500 The vectors a and b are orthogonal if and only if a • b = 0.

Properties of the Dot Product (p. 501) If u, v, and w are vectors and k is a scalar, then the following properties hold.Commutative Property u • v = v • uDistributive Property u - ( v + w) = u - v + u - wScalar Multiplication Property k(u • v) = ku • v = u • k\iZero Vector Dot Product Property 0 • u = 0Dot Product and Vector Magnitude Relationship u • u = |u|2

Angle Between Two Vectors (p. 502) If 9 is the angle between nonzero vectors a and b, thena • bCOS I|a||

Projection of u onto v (p. 503) Let u and v be nonzero vectors, and let w , and w 2 be vector components of u such that w 1 is parallel to v as shown. Then vector w 1 is called the vector projection of u onto v, denoted projvu,and projvu =

Distance Formula in Space (p. 511) The distance between points A(xv y v z ^ and B(x2,y 2,z 2) is given by

A B = \ j ( x 2 - x ,)2 + (y2 - y / + (z2 - z , ) 2 .

M idpoint Formula in Space (p. 511) The midpoint M of AB is given by M x! + *2 y^+ y2

Vector Operations in Space (p. 513) If a = (a v a2, a3), b = (b v b2, b3), and any scalar k, thenVector Addition a + b = (a1 + fr,, a2 + b2, a3 + b3)Vector Subtraction a - b = a + (—b) = (a , - bv a2 - b2, a3 - b3)Scalar Multiplication ka = ( kav ka2, ka3)

Dot Product and Orthogonal Vectors in Space (p. 518 The dot product of a = {av a2, a3) and b = (b v b2, b3) is defined as a • b = a ^ + a2b2 + a3b3. The vectors a and b are orthogonal if and only if a • b = 0 .

Cross Product of Vectors in Space (p. 519) If a = a^\ + a2j + a3k and b = b{\ + b2j + b3k, the cross product of a and b is the vector a x b = (a2b3 - a3b2)i - (a: b3 - a3b j i + (a1 £>2 - a2b i)k.

Triple Scalar Product (p. 521) If t = ^ i + f2j + f 3 k, u = u {i + u2\ + u3k, v = ^ i + v2\ + i/3 k, the

triple scalar product is given by t • (u x v) =

Polar Coordinates and Complex Numbers

Polar Distance Formula p. 537) If , 0 J and P2(r2, 92) are two points in the polar plane, then the distance P2 is given by

Vr 12 + r22 - 2r-ir2 cos (02 - 9-\).

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Key Concepts

S ym m etry o f Polar Graphs (p. 543)

Quick Tests fo r S ym m etry in Polar Graphs (p. 544) The graph of a polar equation is symmetric with respect to

the polar axis if it is a function of cos 9, and the line 9 = ~ if it is a function of sin 9.

Special Types of Polar Graphs (p. 546)

Circles r = a cos 9 or r = a sin 6

Limagons r = a + b cos 9 or r = a ± b sin 9, where a and b are both positive

Roses r = a cos n9 or r = a sin n9, where n > 2 is an integerThe rose has n petals if n is odd and 2n petals if n is even.

Lemniscates r 2 = a2 cos 29 or r 2 = a2 sin 29

Spirals of Archimedes r = a9 + b

Convert Polar to R ectangular Coordinates (p. 551) If a point P has polar coordinates (r, 9), then the rectangular coordinates (x, y) of P are given by x = rcos 9 a n d y = /-sin fl.That is, (x, y) = (rcos 9, rs in 9).

Convert R ectangular to Polar Coordinates (p. 553) If a point P has rectangular coordinates (x, y) then the polar coordinates (r, 9) of P are given by

r = \ J x 2 + y 2 and 9 = tan -1 w h e n x > 0

9 = tan -1 | + 7r or 9 = tan -1 ^ + 180°, when x < 0.

Polar Equations of Conics (p. 562) The conic section with eccentricity e > 0 , d > 0, and focus at the pole has the polar equation:

P f l• r = - — —— - if the directrix is the vertical line x = d,1 + e cos 6

pfl• r = -— —— - if the directrix is the vertical line x = - d ,1 - e cos 9

pci• r = - — . - if the directrix is the horizontal line y = d, and1 + esin 9

pci• r = - — if the directrix is the horizontal line y = - d.1 - esin 9

Absolute Value of a Com plex Num ber (p. 569) The absolute value of the complex number

z = a + b i is \z\ = \a + b i\ = V a2 + b 2 .

Polar Form of a Com plex N um ber (p. 570) The polar or trigonometric form of the complex number z = a + b i isz = r(cos 9 + /'sin 9), where r = |z| = \l a2 + b 2 , a = r cos 6 ,b = rs in 9, and 9 = ta n ~ 1 | f o r a > 0 or

9 = tan” 1 | + i t for a < 0.a

Product and Quotient of Complex Num bers in Polar Form (p. 572) Given the complex numbers zA = (cos 0 1 + / sin 9^) and z 2 = r 2(cos 92 + / sin 92):Product Formula z^z2 = a / 2 [cos (91 + 92) + i sin (0 , + 02)]

Q uotient Form ula T~= T tcos ^ 1 ~ ^ + ' sin ^ 1 “ where z 2 and r 2 ^ 0

R 22 Key Concepts

De Moivre’s Theorem (p. 574) If the polar form of a complex number is z = r(cos 9 + /'sin 9), then for positive integers n, z n = [r(cos 9 + i sin 9 )]n or r n{cos n9 + /s in n9).

Distinct Roots (p. 575) For a positive integer p, the complex number r (cos 9 + /'sin 9) has p distinct pth roots. They are found by

ri>[cr " | cos g + p mr + /'sin 9 + p2 — where n = 0 , 1 , 2 p - 1 .

Sequences and Series

Sigma Notation (p. 594) For any sequence av a2, a3, a4 the sum of the first /(terms is denotedk

X ] 3n = a + a2 + a3 + a4 + + ak, where n is the index of summation, /(is the upper bound ofn= 1summation, and 1 is the lower bound of summation.

The n th Term of an A rithm etic Sequence (p. 599) The nth term of an arithmetic sequence with first term a1

and common difference d is given by a „ = a1 + (/? - 1 )d.

Sum of a Finite A rithm etic Series (p. 603) The sum of a finite arithmetic series with n terms or the nth partial sum of an arithmetic series can be found using one of two related formulas.Formula 1 Sn = -^ (a , + a„) Formula 2 S„ = | [ 2 a 1 + ( n - 1 ) d ]

The nth Term of a Geometric Sequence (p. 609) The nth term of a geometric sequence with first term a1 andcommon ratio r is given by a„ = a1r n ~ 1.

Sum of a Finite Geometric Series (p. 612) The sum of a finite geometric series with n terms or the nth partialsum of a geometric series can be found using one of two related formulas.

a, - a„rFormula 1 Formula 2 Sn--

The Sum of an Infin ite Geometric Series (p. 613) The sum S of an infinite geometric series for which |r | < 1 isa,

given by S = — J—.

The Principle of Mathematical Induction (p. 621) Let Pn be a statement about a positive integer n. Then Pn is true for all positive integers n if and only if P is true, and for every positive integer k, if Pk is true, then Pk+1 is true.

Formula fo r the Binomial Coefficients of (a + b)n (p. 630) The binomial coefficient of the an rb r term in theexpansion of (a + b)n is given by nCr - n\

(n - r)! r ! '

Binomial Theorem (p. 632) For any positive integer n, the expansion of (a + b)n is given by (a + b)n = nC0an b° + „C i an ~ 1 i ) 1 + nC2 an~ 2b 2 + ■■■ + nCr an ~ rb r + ■■■ + nCn a°bn, where r = 0 , 1 ,2 ..........n.

OOPower Series (p. 636) An infinite series of the form ^ anx n = a0 + a: x + a2x 2 + a3x 3 + ■ ■ •, where x and a

n=0can take on any values and n = 0 ,1 , 2 is called a power series in x.

Exponential Series (p. 638) The power series representing ex is called the exponential series and is given by

ex = V ' ^ T = 1 + x + 4r- + 4 T- + 4 r + 4 T - + ' " . wtlictl is convergent for all x.Hi 2 ! 3 ! 4 ! 5 !

R 23

Key Concepts

Key

Conc

epts

Power Series fo r Cosine and Sine (p. 639) The power series representations for cos xand sin xare given by

c o s x - V (~ 1)n x2 n _ 1 X2 x< / /C 0S X - n (2/7)! 2! + 4! 6! 8! ■, and

°° / -j \ /7 2n + 1 3 5 7 9s in x = = ; | , ~ 4 r + 4 r ~ 4 r + 4 r ~ which are convergent for all x.n=o + M o' i- yi

Euler’s Formula (p. 640) For any real number 9, eie = cos 9 + / sin 9.

Exponential Form of a Complex Number (p. 640) The exponential form of a complex numbera + b i\s given by a + b i = reie, where r = V a 2 + b 2 and 9 = tan - 1 | for a > 0 and 9 = for a < 0 .

tan - 1 4 + iv

■ ISymmetric and Skewed D istributions (p. 654)

Inferential Statistics

Negative or Left-Skewed Distribution

/ N mean median

In a negatively skewed distribution, the mean is less than the median, the majority of the data is on the right, and the tail extends to the left.

Symmetrical Distribution

mean, median

In a symmetrical distribution, the data are evenly distributed on both sides of the mean. The mean and median are approximately equal.

Positive or Right-Skewed Distribution

median mean

In a positively skewed distribution, the mean is greater than the median, the majority of the data is on the left, and the tail extends to the right.

Choosing Summary S tatistics (p. 654) When choosing measures of center and spread to describe a distribution, first examine the shape of the distribution.• If the distribution is reasonably symmetrical and free of outliers, use the mean and standard deviation.• If the distribution is skewed or has strong outliers, the five-number summary (minimum, quartile 1, median,

quartile 3, maximum) usually provides a better summary of the overall pattern in the data.

Symmetric and Skewed Box Plots (p. 657)

Discrete and Continuous Random Variables (p. 664)• A discrete random variable can take on a finite or countable number of possible values.• A continuous random variable can take on an infinite number of possible values within a specified interval.

Probability D istribution (p. 665) A probability distribution of a random variable X is a table, equation, or graph that links each possible value of X with its probability of occurring. These probabilities are determined theoretically or by observation. A probability distribution must satisfy the following conditions.• The probability of each value of X must be between 0 and 1. That is, 0 < P(X) < 1.• The sum of all the probabilities for all the values of X must equal 1. That is, £ P(X) = 1.

Mean of a Probability D istribution (p. 6 6 6 ) To find the mean of a probability distribution of X, multiply each value of A'by its probability and find the sum of the products.

R 24 | Key Concepts

Variance and Standard Deviation of a Probability D istribution (p. 667) To find the variance of a probability distribution, subtract the mean of the probability distribution from each value and square the difference. Then multiply each difference by its corresponding probability and find the sum of the products. The standard deviation is the square root of the variance.

Binomial Experiment (p. 6 6 8 ) A binomial experiment is a probability experiment that satisfies the following conditions.• The experiment is repeated for a fixed number of independent trials n.• Each trial has only two possible outcomes, success S or failure F.• The probability of success P(S) or p is the same in every trial. The probability of failure P(F) or q is 1 - p.• The random variable X represents the number of successes in n trials.

Binomial Probability Formula (p. 669) The probability of X successes in n independent trials of a binomialnxnn - x njexperiment is P(X) = nCx p xqn ■ ( / j_ ^

probability of failure for an individual trial.

p xqn x, where p is the probability of success and q is the

Mean and Standard Deviation of a Binomial D istribution (p. 670) The mean, variance, and standard deviation of a random variable X that has a binomial distribution are given by the following formulas, mean: variance:

<t = V c r 2 or yfnpq

p = n p <t2 = npq

standard deviation:

Characteristics o f the Normal D istribution (p. 674)• The graph of the curve is bell-shaped and symmetric with respect to the mean.• The mean, median, and mode are equal and located at the center.• The curve is continuous.• The curve approaches, but never touches, the x-axis.• The total area under the curve is equal to 1 or 100%.

The Empirical Rule (p. 675) In a normal distribution with mean p and standard deviation u \

/ 1 34% ! 34% ; \2.35% / 1 X 2.35%

f l3 .5 % [p — 3 a p — 2(7 p — a p p + a p + 2 cr p + 3cr

X - p, where X is the

I 68% -----1--------------- 95% ------------------------------ 99.7% ---------------

• approximately 6 8 % of the data values fall between p - cr and p + a.• approximately 95% of the data values fall between p - 2 u and p + 2a.• approximately 99.7% of the data values fall between p - 3cr and p + 3cr.

Formula fo r z-Values (p. 676) The z-value for a data value in a set of data is given by z = data value, p is the mean, and a is the standard deviation.

Characteristics o f the Standard Normal D istribution (p. 677)• The total area under the curve is equal to 1 or 100%.• Almost all of the area is between z = - 3 and z = 3.• The distribution is symmetric.• The mean is 0, and the standard deviation is 1.• The curve approaches, but never touches, the x-axis.

Central L im it Theorem (p. 6 8 6 ) As the sampling size n increases:• the shape of the distribution of the sample means of a population with mean p and standard deviation a will

approach a normal distribution, and• the distribution will have a mean p and standard deviation <r7 = =.

x Vn

z-Value of a Sample Mean (p. 6 8 6 ) The z-value for a sample mean in a population is given byX — LL _

z = , where x is the sample mean, /x is the mean of the population, and <r? is the standard error.

Approximation Rule fo r Binomial D istributions (p. 689) The normal distribution can be used to approximate a binomial distribution when n p> 5 and nq > 5.

Normal Approximation of a Binomial D istribution (p. 690) The procedure for the normal approximation of a binomial distribution is as follows.Step 1 Find the mean p, and standard deviation cr.Step 2 Write the problem in probability notation using X.Step 3 Find the continuity correction factor, and rewrite the problem to show the corresponding area under

the normal distribution.Step 4 Find any corresponding z-values for X.Step 5 Use a graphing calculator to find the corresponding area.

Maximum Error of Estimate (p. 697) The maximum error of estimate E for a population mean p, is given byE = z • o v or z • -%=, where z is a critical value that corresponds to a particular confidence level, and

* V n *or ~^= is the standard deviation of the sample means. When n > 30, s, the sample standard deviation, may be

V nsubstituted for a.

Confidence Interval for the Mean (p. 697} A confidence interval Cl for a population mean p is given byC l= x + E or x ± z ~ ~ , where x is the sample mean and E is the maximum error of estimate.

V n

Characteristics of the /-D istribu tion (p. 699)• The distribution is bell-shaped and symmetric about the mean.• The mean, median, and mode equal 0 and are at the center of the distribution.• The curve never touches the x-axis.• The standard deviation is greater than 1.• The distribution is a family of curves based on the sample size n.• As n increases, the distribution approaches the standard normal distribution.

Confidence Interval Using f-D istribu tion (p. 699) A confidence interval Cl for the f-distribution is given by

Cl = x ± t • -4= , where x is the sample mean, f is a critical value with n - 1 degrees of freedom, s is theV n

sample standard deviation, and n is the sample size.

M inimum Sample Size Formula (p. 701) The minimum sample size needed when finding a confidence interval for the mean is given by n = | - y j 2, where n is the sample size and E is the maximum error of estimate.

Steps fo r Hypothesis Testing (p. 707)Step 1 State the hypotheses, and identify the claim.Step 2 Determine the critical value(s) and region.Step 3 Calculate the test statistic.Step 4 Reject or fail to reject the null hypothesis.

Correlation Coefficient (p. 713) For n pairs of sample data for the variables x and y, the correlation coefficient r

between xand y is given by r = 7 7 ^ S ( “ !s7 “ ) ( ^ s ^ ) where x (-and y, represent the values for the /th pair

of data, x and y represent the means of the two variables, and sx and sy represent the standard deviations of the two variables.

Formula for the f-Test for the Correlation Coefficient (p. 715) For a t-test of the correlation between two variables, the test statistic for p is the sample correlation coefficient rand the standardized test statistic t is

given by t = where n - 2 is the degrees of freedom.

R 26 | Key Concepts

Equation of the Least-Squares Regression Line (p. 716) The equation of the least-squares regression line for an explanatory variable x and response variable y is y = ax + b. The slope a and y-intercept b in this equation

are given by a = r - f and b -■°x: y - ax, where /represents the correlation coefficient between the two

variables, x and y represent their means, and sx and sy represent their standard deviations.

Limits and Derivatives

Independence of L im it from Function Value at a Point (p. 737) The limit of a function f(x) as x approaches c does not depend on the value of the function a t point c.

One-Sided Lim its (p. 738)

Left-Hand Limit If the value of f(x) approaches a unique number L as x approaches cfrom the left,the n jim _ f(x) = Lv

Right-Hand Limit If the value of f(x) approaches a unique number L2 as x approaches c from the right,then lim f(x) = L2.

X— >c+

Existence of a L im it a t a Point (p. 738) The limit of a function f(x) as /approaches c exists if and only if both one-sided limits exist and are equal. That is, if

lim f(x) = lim f(x) = L, then lim f(x) = L.X - ^ C x —»c+ x ~ * c

Lim its a t In fin ity (p. 740)• If the value of f(x) approaches a unique n u m b e r^ as x increases, then lim f(x) = Lv which is read the

lim it o f f(x) as x approaches infinity is Lv

• If the value of f(x) approaches a unique number L2 as x decreases, then lim f (x )= L2, which is read the lim it o f f(x) as x approaches negative infinity is L2.

Lim its of Constant Functions (p. 746) The lim it of a constant function at any point is the constant value of the function or lim k = k.

x — >c

Lim its of the Identity Function (p. 746) The lim it of the identity function at any point c is c or Jim, x = c.

Properties of L im its (p. 746) If /cand care real numbers, n is a positive integer, and lim f(x)and lim g(x) both exist, then the following are true.Sum Property

Difference Property

Scalar Multiple Property

Product Property

Quotient Property

Power Property

nth Root Property

Jim [f(x) + g(x)] = Jim f(x) + Jim g(x)

Jim [f(x) - g(x)\ = Jimc f(x ) - Jim g(x)

Jim [/tf(*)] = /( Jim f(x)

Jim [f(x) • g(x)] = Jim f(x) • Jim g(x)

lim f(x)

Jimc [fWn] = Lriimc fW ]n

lim \ [ f(X ) = V/lim f(x), if lim f(x) > 0 when n i:X c v x c x y c

is even.

Lim its of Polynomial Functions (p. 748) If p(x) is a polynomial function and c is a real number, thenJim p(x) = p(c).

P{x)Lim its of Rational Functions (p. 748 If r{x) = ^ is a rational function and c is a real number, thenq(x)

Jim r ( x ) = r(c) = if q(c) ± 0.

L im its of Power Functions at In fin ity p. 750) For any positive integer n,• lim x n = o o .

X— >co

• lim x n = o o if n is even.

lim x n = — o o if n is odd.

R 2 7

Key Concepts

Limits of Polynomial Functions at Infinity (p. 751) Let p be a polynomial functionplx) = anx n + . . . + a ,x + an.Then lim p(x) = lim anx n and lim p (x )= lim anx n.rv ' " 1 u X—>00 X— >oo X- -00 X—►—00

Lim its of Reciprocal Function at In fin ity (p. 751) The lim it of a reciprocal function at positive or negative infinity

isO or lim 1 = lim 1 = 0 -x—>00 x x—>—00 x

Instantaneous Rate of Change (p. 758) The instantaneous rate of change of the graph of f(x) at the point (x, f(x))

is the slope m of the tangent line given by m = Hrn f x + h ^— — , provided the limit exists.

Average Velocity (p. 760) If position is given as a function of time f(t), for any two points in time a and b, thechange in distance f{b) - f(a)average velocity v is given by vm = change in time

Instantaneous Velocity (p. 760) If the distance an object travels is given as a function of time f(t), then the

instantaneous velocity v(t) at a time / is given by v(t) = jim ^ + ^ — — , provided the limit exists.

Power Rule for Derivatives (p. 767) The power of x in the derivative is one less than the power of x in the original function, and the coefficient of the power of x in the derivative is the same as the power of x in the original function.

Rules for Derivatives (p. 767)Constant The derivative of a constant function is zero. If f(x) = c, then f'(x ) = 0.Constant Multiple of a Power If f(x) = cx11, where c is a constant and n is a real number, then f '(x ) = cnxnSum and Difference If f(x) = g(x) ± h(x), then f \ x ) = cf(x) ± H(x).

Extreme Value Theorem (p. 769) If a function f \s continuous on a closed interval [a, b], then f(x) attains both an absolute maximum and an absolute minimum on [a, b].

Product Rule for Derivatives (p. 770) If f and g are differentiable at x, then

fy f ( x ) g ( x j\ = f'(x )g (x) + f(x)g'(x).

Quotient Rule for Derivatives (p. 771 If f and g are differentiable at x and g(x) =f= 0, then

- 1

dxf(x)g(x)

f'(x)g(x) - f(x)g'(x)[g(x)]2

Definite Integral (p. 777) The area of a region under the graph of a function isrb n

f(x )d x = lim YfJa

where a and b are the lower limits and upper limits respectively, A x = and x, = a + /A x . This method is referred to as the right Riemann sum.

Rules for Antiderivatives (p. 785)yll+ 1

Power Rule If fix) = x " , where n is a rational number other than - 1 , F(x) = - — - + C.' ’ n+ 1Constant Multiple of a Power If fix) = kxn, where n is a rational number other than - 1 and k is a constant,

kvn+ 1then F(x) = r - + C.' ' n+ 1

Sum and Difference If the antiderivatives of /(x) and g(x) are F(x) and G(x) respectively, then theantiderivative of fix) ± g(x) is F(x) ± G(x).

, f nIndefinite Integral (p. 785) The indefinite integral of /(x) is defined by / f(x)dx = F(x) + C, where F(x) is an antiderivative of f{x) and C is any constant.

Fundamental Theorem of Calculus (p. 786) If / is continuous on [a, b] and F(x) is any antiderivative of fix), thenbrb

JJaf(x )d x = F{b) - F(a). The difference F(b) - F(a) is commonly denoted by F(x)

R 28 j Key Concepts