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by: Oswald Boyle


Oswald Boyle
GPA 3.94


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This 8 page Class Notes was uploaded by Oswald Boyle on Monday October 12, 2015. The Class Notes belongs to MATH 143 at Idaho State University taught by Staff in Fall. Since its upload, it has received 28 views. For similar materials see /class/222165/math-143-idaho-state-university in Mathematics (M) at Idaho State University.




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Date Created: 10/12/15
MATH 143 CHAPTER 12 TOPICS AND REVIEW PROBLEMS 121 Complex numbers7the complex plane real and imaginary parts the arithmetic of complex numbers complex conjugates 122 Long division of polynomials7the algorithm and what it tells you about polynomials and rational functions The relationship between the dividend divisor quotient and remainder Large scale behavior of rational functions slant asymptotes 123 Roots of polynomials and the factor and remainder theorems The multiplicity of a root 124 The fundamental theorem of algebra Recovering polynomials from their roots 126 The conjugate roots theorem 1 Find a real quadratic function that has a root at 2i 7 3 2 Determine a possible formula for a real polynomial with the following graph assuming that you also know that 71 H is a root of the polynomial 3 2i x 3 Let u23i v72i and w37i Compute a 2u7v buw cv3 d Uw euv 4 Compute the quotient and remainder when 2x3 3 is divided by 2 2x and determine all asymptotes horizontal vertical or slant for the rational function x 5 When a polynomial pz is divided by 2x2 3 the quotient is 3x 7 5 and the remainder is z 4 What is pm 6 MULTIPLE CHOICE The graph y 171 has which line as a slant asymptote a ym71 by27x cy5z dy87x ey9725m 7 MULTIPLE CHOICE Which polynomial has a graph most similar to the given graph a y 2mm722z3 b y 3m227mz3 c y 7mm22z73 d y 72 m 7 22 m 32 e y 72m27x 7173 8 One root of pz 6x3 7 2 7 21x 1 10 is z 72 Verify this and use this fact to help you factor pz completely 9 Let pz be a nonconstant polynomial with real coefficients What does the factor theorem tell us about pm What does the fundamental theorem of algebra tell us about pz To what extent can pz be factored if the factors are required to have real coefficients How is the conjugate roots theorem relevant 10 Design a polynomial with real coefficients that has 2 H as a root of multiplicity 2 11 TRUEFALSE Assuming that f is a polynomial and that the graph shows all relevant features which of the following statements are necessarily true possibly true or necessarily false bio a The de ree of is 4 g y b The degree of x 1s even c m 2 is a factor of I x d z3 is a factor of f l MATH 143 CHAPTER 12 Page 2 ANSWERS 1 If 902i73 then 954732 2i2 74 sothat z26z130 2 We require a polynomial with roots 71 i 2 and by the conjugate roots theorem 71 7 i Let pz a m 1 7 1 7 2 az2 2 z 2m 7 2 The y intercept gives a 71 and thus one possible solution is pz 71 3 m 1 3 a 65i b 97i c 7211i d 7170 e 774i i i i i 8 m 3 4 The quotient 1s 2 7 4 With a remainder of 8m 3 It follows that fz 2 7 4 w 1 1 so that y 2x 7 4 is a slant asymptote There are also vertical asymptotes at the roots of 2 1 2m ie the lines z 72 and z O 5 pz 2m2 33z7 5 71 4 6x3 710m2 10m711 6 By long division the quotient is 7m 8 with a remainder of 725 1 9 so that the slant asymptote is d y 8 7 m 7 We require simple roots at z 73 and z 0 and a double root at z 2 Both a and e have these roots However the large scale behavior shows that the degree must be even and the leading coefficient must be negative Thus the correct answer is e y 72 z 2 7 z2 m 3 8 p72 748 7 4 42 10 0 so 72 is a root and m 2 is a factor of By long division pz m 26 2 7 13x 1 5 By the quadratic formula the other two roots are and so that pz m 23z 7 52z 71 9 Let pz be a nonconstant polynomial with real coefficients The factor theorem tells us that r is a root if and only if 77 is a factor ie pr 0 if and only if pz 77 for some polynomial The fundamental theorem of algebra tells us that pz must have at least one root within the complex number system If r a bi is a nonreal root then by the conjugate roots theorem a 7 bi is also a root and thus m 7 a 7 7 a bi 2 7 2 am 1 a2 1 b2 must be a factor It follows that pz can be factored as a product of real linear factors and irreducible quadratic factors where an irreducible quadratic factor is a real quadratic with nonreal conjugate roots 10 If 2i is a root of multiplicity 2 ofa real polynomial then by the conjugate roots theorem 27i must also be a root of multiplicity 2 Thus one possibility is pz 7 2 i2 7 27i2 274x52 x478m3 26m2740m25 11 a The graph shows a simple root at z 2 and a 3 fold possibly 5 fold 7 fold root at z 0 We also cannot see complex roots if any So although the graph is consistent with the graph of a polynomial of degree 4 the degree could be greater than 4 ie this part is possibly true b The large scale behavior shows that the degree of the polynomial is even and that the leading coefficient is negative so this is necessarily true c This is de nitely false fz has a root at z 2 not at z 72 d This is necessarily true the behavior near z 0 indicates a root of odd multiplicity greater than or equal to 3 A triple root at z 0 corresponds to a factor of 3 MATH 143 CHAPTER 5 51 52 53 54 55 56 57 TOPICS AND REVIEW PROBLEMS Exponential functionsithe base of an exponential function domain range graph asymptote large scale behavior rules for exponents simplifying expressions The number 5 and the special role of the exponential to the base 5 Logarithms to the base I and the natural logarithmide nition domain range asymptote intercept graphs inverse relation The basic properties of the logarithm functionithe ve basic properties and the change of base formulas must be learned Solving equations and inequalities involving logarithm and exponential functions extraneous solutions Compound interest continuous compounding the doubling time Exponential growth and decay half life doubling time l to 03 U F 90 to H H H to H 03 MULTIPLE CHOICE The domain of z ln2 7 795 is b 7 00 c 700 0 d 7ooln27 MULTIPLE CHOICE If x ln2z 7 1 then the inverse function f 1m equals a 5 12 b gem d lnz 1 e 1ln2x 7 1 Sketch graphs of the following functions Be sure to determine all intercepts and asymptotes a yln27m b c y21 71 State the rules for logarithms and exponents a oo e something else 1 as c 25 342759 dylnlml71 Solve the following equations exactly a ln3m 1 2 lnm b log22m 1 log23m 1 1 c Solve the inequality log23 7 2x g 1 Simplify lnel 2 111 7 MEN Determine rational numbers a b c and 1 so that 7 1 m 2 7 z3Vm6 2x73 57202 C Vm3m3 c z d z 2 13 2 a a b b dlt gt3 Vm4Vm7 Which of the following isare valid identities assume a b gt 0 a ln 017 blna b log2a b log2a log2b An account earns 5 interest compounded monthly C 61112 eab alnb blna a If 2000 is deposited in the account how much will be in the account after 6 years assuming that there are no additional deposits or withdrawals b What initial deposit would be needed to have 3000 in the account after 6 years It is found that 3 grams of a 358 gram sample of a radioactive material have decayed after 5 years What is the half life of the material computed to the nearest year Abacterial population grows from 4000 to 5000 in 4 hours Assuming exponential growth when will the population reach 10000 A recent experiment at the National Superconducting Cyclotron Laboratory in East Lansing Michigan showed that the half life of titanium 44 is 592 years How much of a 37129 sample of titanium 44 decayed during the six month experiment In 3 hours the population of Tribles on the Enterprise has grown from 4 to 37 Estimate the population after an additional 7 hours assuming a linear growth b exponential growth MATH 143 CHAPTER 5 Page 2 ANSWERS 1 e Something else foo7 2 b ex1 3 a b I I 2 m2 39 39 u 1 1 2 W an2 1 I I l l c d 1 o m 76 Q 16 m 1 K 4 Exponents For all positive numbers a and b and all real numbers z and y a ay awry axay axiy awy ax39y iv 01 a I U ULb LEb a0 1 Logarithms For all positive numbers a7 67 z and y and real numbers u i 10am 10am 1060 ii 10gby 10am 71000 m 10m u 10005 00 legal 0 v 10m 1 1 31 2 5 a z e2 7 3 b x The solution x 71 is extraneous c z 6 E Note domain 7 7 1n2 8 a 7 b 1 c d 0 9 Only part d is a valid identity7 as can be veri ed by taking the natural logarithm of both sides Parts a7 b7 and c are all erroneous versions of standard identities 005 12396 10 a 2000 1 269804 0 05 126 b P0 1 300000 will give P0 222384 12 51n12 11 t 412 2 ln355358 years 12 t W 16425 hours after the population was 4000 13 37 17 55 51Xo 2160 pg 14 a Linear Pt 411t gives P10 114 b Exponential Pt 4 374 3 gives P10 6646 MATH 143 CHAPTER 4 41 42 TOPICS AND REVIEW PROBLEMS Linear functions7what they are and the connection between slopes and rates of change Economic examples marginal cost linear depreciation average velocity Linear interpolation ie two points vs a regression line Using linear models The algebra and geometry of quadratic functions Parabolas vertex and axis of symmetry extreme values intercepts Completing the square and the vertex form of a quadratic function SKIP 43 44 Sections 44 and 45 involve a variety of word problems Ideas to consider include the use of diagrams choosing variables expressing functions in terms of variables and the use of constraints to express functions in terms of a single variable Basic geometrical formulas that should be learned include the pythagorean theorem and distance in the coordinate plane areas and perimeters of squares circles rectangles and triangles and the volumes and surface areas of spheres cylinders and rectangular boxes Composite gures are also considered 45 The problems in section 45 combine ideas from 42 and 44 They are all word problems where the extreme values can be understood by determining the vertex of a quadratic 46 Polynomial functions and the degree of a polynomial The leading coefficient and large scale behavior Zeros intercepts factored form behavior near multiple roots sign determination Continuity smoothness and the fact that the number of turning points is less than the degree 47 Rational functions7de nition domain vertical asymptotes Large scale behavior horizontal and slant asymptotes Factored form intercepts sign determination graphing 1 Which of the following functions are polynomials For those that are give the degree for those that are not indicate why not a xgx3745 d x3x22x 377x4 b 2 x3 7 4x 122x 3 e 3x3 7 4x 7r22x c x3x22x7142x53 f x717 2 Give a possible formula for a polynomial with the given graph a b y 1990 39 g 6190 2 72 71 r x 0 3g x 2 X 3 For each of the following functions describe the behavior of the function as x 7 00 and as x 7 700 by one of the following x 7 00 x 7 700 or x 7 L where L is a particular real number whose value is expressed clearly 3 2 i 27 3 i x 723x372x a xix x c xi2im e xi m17mm4 2x23 3x72 x273x2 b d f m M47335 m n12 m Mowing 4 Find the vertex of y ix2 3x 7 7 by completing the square MATH 143 CHAPTER 4 H H Page 2 5 Give a piecewise de nition for the function x with the following graph y 37 5 5 4 17 3 3 I yf 721 o 39 1 27 2 I I I I I I m 72 71 0 1 2 3 6 Sketch graphs of the following functions Be sure to determine intercepts7 asymptotes7 sign7 and large scale behavior 7 13 90 4 2 2 7 2 2 QE a M x W gt c M wim m 2 7 m 6 b d 1 7 M m D2 M w z 3 7 2 m i i 7 Solve 1 g 1 Verify your result graphically 8 Find the point on the curve y that is closest to the point 37 0 9 A box with a square bottom is to contain 1000 mg The material for the top and the sides of the box costs SgtM2 while the bottom costs 7 in2 There are also labor costs of 200 per box Assume that there are no other costs Express the total cost of making a box as a function of the width of the base 0 A farmer uses 1000 ft of fence to create two separate pens7 one in the shape of a square7 the other in the shape of a rectangle that is twice as long in one direction as the other Determine the total area of the two pens as a function of the length of a side of the square pen 1 A ski area has xed costs of 12000 per day and variable costs averaging 4 per customer They estimate that there will be 2000 7 25p customers per day when the ticket price is 19 a Estimate the area s daily costs as a function of the ticket price b Estimate the area s daily revenues as a function of p c What is the daily pro t if the ticket price is 30 d What ticket price will maximize pro ts 2 The production of solid waste in the United States has been increasing7from 823 million tons in 1960 to 1391 million tons in 1980 Use a linear model to estimate the amount of waste produced in 1990 and 2000 What does the slope represent7 and what are its units 3 The table below shows the population of Idaho from 1950 to 1990 The equation of the regres sion line is y 11130m 7 211450007 where z is the year and y is the population NW 1950 1960 1970 1980 1990 yp0pulation 588637 667191 713015 944127 1006749 a Use the regression line to estimate the population of Idaho in 2010 b The 2000 Idaho census is 1293953 How accurate is the regression line c What7 speci cally7 does the number 11130 have to say concerning ldaho s population MATH 143 CHAPTER 4 Page 3 ANSWERS 1 a polynomial of degree 3 d not a polynomial because of x f b not a polynomial because of division e not a polynomial beacuse of 0 polynomial of degree 11 f polynomial of degree 07a constant polynomial 2 a 19x72x7x722 b qxax22xx73 with alt0 3 as xH I ltgto7 x H700 and as xaioo7 x Hoo as xH I ltgto7 x H7 and as xaioo7 fxgt7 as xH I ltgto7 f R VVVV H700 and as xaioo7 x H700 as gtOO7 f HOandasxaioo7 xa0 as xgtoo7 f H712 and as xaioo7 xai12 as xH I ltgto7 xaioo and as xaioo7 xaoo 4 y x I 62 716 so the vertex is 767 716 AAA W919 79 x g 72 lt x g 1 5 7x4 1ltxlt2 3x 7 4 2 lt x g 3 6 a b I quot 2 0 0 x 2 c y d y I JI I 3 I I l I quot r 4 Iquot 5 y LI 7 3 1 4 N 1 I2 m 3 m 3973 xquot I 39I39 I I I 3 7 I I39 I IIquot I II MATH 143 CHAPTER 4 H H 8 to H Page 4 The distance from a typical point z on the curve to the point 3 0 is d 732H02 x275z9 This will be a minimum at the vertex of the parabola y 2 7 5 z 9 ie at z 5 5 Let the length of a side of the base of the box be m and let the height of the box be y The volume 2 y must equal 1000 i713 It follows that the total cost of making the box is C 200 005 x2 4mg 007 m2 200 012352 E on Let x represent the length of a side of the square pen and let y represent the length of the shorter side of the rectangular pen The other side has length 2y so that the total area of the two pens is A 2 2 342 We are also told to use 1000 ft of fencing so that 1000 4x 1 6y Thus A m2 2 w a Cp 12000 4 2000 7 25p 20000 7 100p b Rp p 2000 7 25p c R30 7 C30 30 x 1250 717000 20500 d Since the pro t function Rp 7 C19 2100p7 25 p2 7 20000 is quadratic the maximum occurs at the vertex ie when p 210050 42 Let Wt represent the waste produced in year 25 measured in millions of tons A linear model gives Wt 823 284t 7 1960 The slope 284 13917 8231980 7 1960 represents the rate at which the waste is increasing ie 284 million tons per year The linear model gives W1990 1675 and W2000 1959 millions of tons a 1226300 b Regression line predicts 1115000 This is low by 178953 or 14 c As a rate of change the number 11130 is saying that on average the population of Idaho has been growing at a rate of 11130 people per year


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