Stewart algebra and trigonometry 4th edition c2016 txtbk

exponents and radicals x x m2n xn x 2n n x x n xn a b n y y x m x n 5 x m1n x m n  5 x mn n n n xy2  5 x y n x 1/n ! x n n n n ! xy ! x! y m n geometric formulas m n m Formulas for area A, perimeter P, circumference C, volume V: Rectangle Box A 5 l„ V 5 l„ h P 5 2l 1 2„ n x m/n !x m !x2 m n x ! x n Åy !y n mn " !x " ! x ! x special products x 1 y 2 5 x 2 1 2 x y 1 y x 2 y 2 5 x 2 2 2 x y 1 y h „ Triangle Pyramid A 5  12 bh V 5 13 x 1 y 3 5 x 3 1 3x y  1 3x y 2 1 y h x 2 y 3 5 x 3 2 3x y  1 3x y 2 2 y FACtORING formulas x 2 2 y 2 5  x 1 y x 2 y x 2 1 2xy 1 y 2 5  x 1 y 2 2 x  2 2xy 1 y  5  x 2 y 2 x 3 1 y 3 5  x 1 y x 2 2 xy 1 y 2 x 3 2 y 3 5  x 2 y x 2 1 xy 1 y 2 „ l l h a a b Circle Sphere V 5 43 pr A 5 pr C 5 2pr A 5 4pr r r QUADRATIC FORMULA If ax 2 1 bx 1 c 5 0, then x5 2b "b 2 4ac 2a inequalities and absolute value Cylinder Cone V 5 pr 2h V 5 13 pr 2h r h h r If a , b and b , c, then a , c If a , b, then a 1 c , b 1 c If a , b and c . 0, then ca , cb If a , b and c , 0, then ca . cb heron’s formula If a . 0, then x  5 a  means  x 5 a  or  x 5 2a x  , a  means  2a , x , a x  . a  means  x . a  or  x , 2a B Area 5 !s1s a2 1s b2 1s c2 a1b1c where s c A a b C Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it distance and midpoint formulas Graphs of Functions Distance between P1 x , y and P2 x , y 2 : Linear functions:   f1x2 mx b y d "1 x2 x1 2 1y2 y1 2 Midpoint of P1P2:   a lines x1 x2 y1 y2 b , 2 y b b x x Ï=b y2 y1 m5 x2 x1 Slope of line through P1 x , y and P2 x , y 2      Ï=mx+b Power functions:   f1x2 x n y 2 y 1 5 m x 2 x Point-slope equation of line through P1 x 1, y with slope m Slope-intercept equation of line with slope m and y-intercept b y 5 m x 1 b Two-intercept equation of line with x-intercept a and y-intercept b y x 51 a b y y x x Ï=≈      n Root functions:   f1x2 ! x logarithms y y y 5 log a x  means  a y 5 x Ï=x£ a log a x 5 x log a a x 5 x log a 1 5 0 log a a 5 1 x x log x 5 log 10 x ln x 5 log e x log a a}x}b 5 log a x  2 log a y y loga x log a x b 5 b log a x log b x 5  loga b Ï=œ∑ x log a x y 5 log a x 1 log a y      Ï=£œx ∑ Reciprocal functions:   f1x2 1/x n y y exponential and logarithmic functions y y y=a˛ a>1 y Ï= x y y=log a x a>1 x x y=a˛ 0 F If E and F are mutually exclusive, then P1E < F P1E P1F (c) A card is picked from a deck Let E, F, and G be the events “the card is an ace,” “the card is a spade,” and “the card is king,” respectively Are E and F mutually exclusive? E and G? Find P1E < F and P1E < G E and F are not mutually exclusive (the ace of spades is in both events): P1E < F 13 52 52 52 13 E and G are mutually exclusive (a card cannot be an ace and a king): P1E < G 4 52 52 13 (a) What is meant by the conditional probability of E given F? How is this probability calculated? This is the probability that E occurs if we know that F has occurred P1E F P1E > F P1F (b) What are independent events? Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other event (b) What is an event? An event is any subset of the sample space (continued) Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it chapter 14 Review: Concept Check Answers (continued) (c) If E and F are independent events, what is the probability of E and F occurring? What if E and F are not independent? P1E > F P1E 2P1F E If E and F are independent, then P1E > F P1E 2P1F (d) A jar contains white and black balls Let E and F be the event “the first ball drawn is black” and “the second ball drawn is black,” respectively (i) Find P1E > F if the balls are drawn with replacement In this case E and F are independent, so P1E > F P1E 2P1F 7 # 49 10 10 100 (ii) Find P1E > F if the balls are drawn without replacement In this case E and F are not independent, so P1E > F P1E 2P1F E # 57 10 15 (a) An experiment has two outcomes, “success” and “failure,” where the probability of “success” is p The experiment is performed n times What type of probability is associated with this experiment? Binomial probability (b) What is the probability that success occurs exactly r times? P1r successes in n trials C1n, r 2pr 11 p2 n2r (c) An archer has probability 0.6 of hitting the target Find the probability that she hits the target exactly times in attempts P13 hits in tries C15, 32 10.62 10.42 0.3456 (a) Suppose that a game gives payouts a1, a2, , an with probabilities p1, p2, , pn What is the expected value of this game? E a1 p1 a2 p2 an pn (b) You get $10 if you pick an ace from a deck, and you must pay $2 if you pick any other card What is your expected value? E 10 a 48 b 2 a b < 21.08 52 52 Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it sequences and series counting Arithmetic Fundamental counting principle Suppose that two events occur in order If the first can occur in m ways and the second can occur in n ways (after the first has occurred), then the two events can occur in order in m 3 n ways a, a 1 d, a 1 2d, a 1 3d, a 1 4d, an 5 a 1  n 2 d n a an n Sn a ak 32a 1n 12d4 n a b 2 k51 Permutations and combinations Geometric The number of permutations of n objects taken r at a time is a, ar, ar 2, ar 3, ar 4,  .       an  5 ar n21 P1n, r n rn Sn a ak a 12r k51 The number of combinations of n objects taken r at a time is If r , 1, then the sum of an infinite geometric series is a S5 12r n a b an n a b C1n, r n! r! 1n r 2! The number of subsets of a set with n elements is n The number of distinguishable permutations of n elements, with ni elements of the ith kind (where n1 1 n2 1   1 nk 5 n), is the binomial theorem a b2 n n! 1n r 2! an21b a n b n21 ab n21 n anb bn finance n! n 1! n 2! n k ! probability Compound interest r nt b n A Pa1 where A is the amount after t years, P is the principal, r is the interest rate, and the interest is compounded n times per year Amount of an annuity n Af R 11 i2 Present value of an annuity 11 i2 2n i where Ap is the present value, i is the interest rate per time period, and there are n pay­ments of size R Installment buying R5 If S is a sample space consisting of equally likely outcomes, and E is an event in S, then the probability of E is P1E n1E n1S2 number of elements in E number of elements in S P1E92 5 1 2 P1E2 Union of two events: P1E < F P1E P1F 2 P1E > F Conditional probability of E given F: P1E F Intersection of two events: n1E > F n1F P1E > F P1E 2P1F E Intersection of two independent events: iAp 11 i2 Complement of an event: i where Af is the final amount, i is the interest rate per time period, and there are n pay­ments of size R Ap R Probability of an event: 2n where R is the size of each payment, i is the interest rate per time period, Ap is the amount of the loan, and n is the number of pay­ments P1E > F P1E 2P1F Binomial Probability: If an experiment has the outcomes “success” and “failure” with probabilities p and q p respectively, then P1r successes in n trials2  5 C 1n, r2 p rq n2r If a game gives payoffs of a1, a2,  . . , an with probabilities p1, p2,  . . , pn , respectively, then the expected value is E 5 a p1 1 a p2 1 1 a n pn Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ... teaching comes in many forms and that there are many different approaches to teaching and learning the concepts and skills of algebra and trigonometry The organization and exposition of the topics... any time if subsequent rights restrictions require it FOURTH edition ALGEBRA AND TRIGONOMETRY James Stewart M c Master University and University of Toronto Lothar Redlin The Pennsylvania State... of algebra and trigonometry In all these changes and numerous others (small and large) we have retained the main features that have contributed to the success of this book New to the Fourth Edition