Precalculus I MA 107
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9 01 9 51 gt gt gt gt gt Hgtw Nl O Write as a string of simpler log expressions loga MA107 Precalculus Algebra Exam 3 Review 22 March 2008 Given the functions f and 9 find each of the following X X21 9X12 a f o g x b g o f c f o f d g o g x If x find gx such that f o gx xx2 4x 4 The volume of a right circular cylinder of height h and radius r is V 7TT2l39L If the height is 4 units more than the radius express the volume V as a function of T Is this function onetoone f l1495162 Is the function fx onetoone Hint draw a graph and use the horizontal line test Let x a Determine f 1 x b Determine the domain of f c Determine the range of f The graph of f lies in quadrant H In what quadrant does the graph of f 1 lie Consider the exponential function fx 6 a Use transformations to show a graph of gx ex1 l b What is the domain of g c What is the range of g If 3 15 then what does 3 X1 equal Change into a log expression 4 29 Change into an exponential expression log9x2 21 Write as a single log expression logt1 B Zloga C log1 D 7 lilogt1 E Xz l ljj X75 39 Show how to find the exact value of 210g1020 7 log10 4 without using a calculator gt gt gt gt 00OU1 NNNNNNNH mmewNHop Use the changeofbase formula to write the expression 1095 22 as a log expression with base 6 Find the domain of the function fx lnx 7 4 Find the domain of the function fx lnbc2 4x 4 3x777 Solve for x 3 7 27 636264 Solveforx 4x 1 e Solve for x lnbc2 7 3 0 Solve for x 4X1 12 Solve for x 58 20 Solve for x logzbc 7 l 7 logzbc 6 logzbc 7 2 7 logzbc 3 Solve for x 36 7 6 1 75 Compute the inverse function for x 264x The funds available to a certain account can be given as a function of time according to the formula At 500063913t where t is in years After how many years will be size of the account double U N DWNWQ mwaHo S MAl O7 Precalculus Algebra Exam 2 Review 17 February 2008 The following demand equation models the number of units sold x of a product as a function of price p x 74p 200 a Please write a model expressing the revenue R as a function of x b What price should the company charge to maximize revenue c What is the maximum revenue d What quantity x corresponds to the maximum revenue 9 How do you know there is a maximum and not a minimum point on the graph of this function Write a polynomial x which has the roots 72 2 4 and such that the graph of x touches the x axis at 72 and 2 and crosses the x axis at 4 and has a degree of 9 Write a polynomial x which has the roots 1 71 and such that the graph of x touches the x axis at both 71 and l and has a degree of 6 What are the roots of the polynomial 2x 7 3 x3 7x2 What is the domain of the rational function fx For the rational function above please state the vertical asymptotes For the rational function above please state the horizontal asymptotes For the rational function above please state the zeroes Let px x3 x2 7 6x What is the degree of px What are the roots of px What is the multiplicity of each root For each root state if the graph of px crosses or touches the xaxis at the intercept Please state the power function that the graph of x resembles for large values of lxl What is the remainder when p x is divided by x 7 5 Consider q x 7x5 29x4 7 22x2 5x 21 List all the possible rational roots Use the Intermediate Value Theorem to find an open interval which contains at least one real root of qlxl 74x22 3 Solve the rational inequality X X7 J gt 0 17 Sketch the graph offlx x 7 2 13xx 71 18 lwish to enclose a rectangular yard with a fence 1 have 400 feet of fencing available 21 Express the area A of the yard as a function of w the width of the rectangle 17 Express the area A of the yard as a function of l the length of the rectangle chmdii r m 1 t M 01 1 1 tit 19 A cy indrica box is to the constructed such that the sum of the height and radius is 100 inches Con struct a function which states the volume of the box as a function of its radius 20 Which polynomial below is thatwhose graph is depicted to the right ChoiceA i ljxllxi lx lf Choice B fajxflxi lxiw Choice c igxltwswxial ChoiceD iwglxiilllximl Choice E igj txislluia 21 Which polynomial below is thatwhose graph is depicted to the right Choice A r 3 Choice B r x711x2 x1lllx2 ChoiceC frilllfxrl ChoiceD frillffxrl Choice E ifxrl x72 3 4 3 MA107 Precalculus Algebra Exam 4 Review Solutions 12 April 2008 1 Find the amount that results from each of the following investments a 1000 dollars invested at 5 compounded semiannually after a period of 10 years The compound interest formula is A 131 j With P 1000n 21 05t 101get A 10001 191001 163862 b 1000 dollars invested at 45 compounded monthy after a period of 10 years WithP 100011 12 045t 101get A 10000 1021001 156699 c 1000 dollars invested at 4 compounded continously after a period of 10 years The continuous interest formula is A P6 With P 10001 04t 10 I get A mood041001 149182 2 Find the present value of 5000 dollars invested at 10 interest compounded daily The present value formula is P AU 1 For n 365A 5000 r 10Iget P 50001 13653 3 At what interest rate would an initial investment of 3000 dollars double in value over 5 years Com pounded continuously We have A P6 with P 3000 t 5 We want to solve for T when A 6000 6000 300065T 4 The population of a colony of bacteria obeys the law of uninhibited growth Initially there is a popu lation of 2 million After a period of 40 days there is a population of 3 million a What is the groth constant I use the exponential growth model Pt Po 6 where k is the growth constant P0 is the initial population and t is time days To find the growth constant I need to plug known values into the model and solve for k I know Po 2000000 and when t 40 Pt 3000000 3000000 2000000640k 40k ln 7 lne4 k ln 40k k b What is the size of the colony after 9 days 1mg We now have the model Pt 20000006 40 t Plugging in 9 for t see 1 Pt 2000000630 9 e 2191041 C When will the population double mg We now have the model Pt 20000006 40 t I want to know t when Pt 4000000 twice the initial population mg 4000000 ZOOOOOOe Act he t 1112 4 t ln2 he 7 t 40 t m 6038 5 A copy of this exam review has been sealed in a gold temple awaiting to enlighten the mankind of the future After the temple has been opened and the mummy s curse broken archaeologists set to work determining the age of the paper They find it contains 03 of its original amount of Carbon14 Assuming a halflife of 5 600 years determine the age of the paper The model for exponential decay is At A0 6quot First we need to find k The halflife tells me that when t 5600 then At leo So I plug these values into the model and solve for k lZAO A0675600k 1 675600k lnl lne 56 0k lnl2 600k ML Now that l have the model At A06 5650 t I can work on determining the age of the paper I want to know t when At 003A0 So I plug this information into the model and solve for t 003A0 Ace 5500 t 003 65600 t 5600 H003 7 t My 7 5600 t m 4693260 6 Convert 90 degrees to radians of 90 gtlt g 7 Convert 71 radians to degrees m105 l 8 Suppose sin6 The identities to find the exact value of each of the five remaining trigonometric functions cos6tan6csc6sec6cot6 Consider a right triangle The sine of an angle is the ratio of the lengths of the opposite leg to the hypotenuse So we can say the opposite leg has length 3 and the hypotenuse has length 5 To find the length of the adjacent leg use the Pythagorean Theorem 32X2 52 9x2 25 x2 16 x 4 So the length of the adjacent leg is 4 sin6 fse cos6 tame h 3i 212 0509 sinle 3 sec6 wslw cot6 1W g 9 Suppose tan6 The identities to find the exact value of each of the five remaining trigonometric functions sin6 cos6 csc6 sec6 cot6 Again consider a right triangle Tangent is the ratio of the opposite leg to the adjacent leg We consider the opposite leg to be 3 and adjacent leg to be 5 To get the hypotenuse use the Pythagorean Theorem 32 52 X2 9 25 x2 34 x2 m x2 7 opposite 7 i 51116 7 hypotenuse 7 34 7 adjacent 7 5 cosle 7 hypotenuse 7 W 7 adjacent 7 sin9 7 7 3 tame 7 hypotenuse 7 6059 7 7 5 csc6 m secle Tisha cot6 mule 7 g Without a calculator determine the value of sec283 7 tan283 Recall thatl tan26 sec26 So sec283 7 tan283 l tan283 7 tan283 1 tan283 7 tan283 1 11 Without a calculator determine the value of csc213 7 cot213 gtlt cos13 gtlt sec13 Recall that 1 cot26 csc26 Also recall that sec6 519 So csc213 7 cot213 gtlt cos13 gtlt sec13 1cot2113ojj7cotz 3 jj gtlt cos13 gtlt sec13 1 X cos13 gtlt sec13 1gtlt cos13 X W 1 X1 1 12 Without a calculator determine the value of SSEL 3 J in 57 39 Because 33 57 90 and cos and sin are cofunctions then cos33 sin57 cos33 7 sin57 71 sin57 rave 39 So I write 13 Without a calculator determine the value of cot10 7 cosq 00 Sam 00 ltgt 3383 7 comm so cot10 7 333 o 14 Given cos30 l determine the value of a cos60 In class we derived b 51112800 We know sin230 cosz30 1 cos30 so cosz30 jz i 4 Then sin230 1 sin230 sin30 15 C sec30 3 594300 cosl gt0 A d csc30 By Complementary Angle Theorem sin30 cos60 Then csc30 m 2 15 A ship offshore from a vertical diff known to be 900 feet inheight takes a sighting of the top of the diff The angle of elevationis found to be 30 How far offshore is the ship 0 We want to find X the distance from the diff Iknow tan30 gg so 577 xe Solve for X to get x m 155979 16 To measure the height of the Watson Towers two sightings are taken at a distance of 120 feet apart If the first angle of elevation is 66 and the second is 60 what is the height of the tower X 120 W The trick here is to look for right triangles There are two Both share the leg formed by the tower vertical red line But the base of one is the horizontal red line with a length of x The base of the other is the blue line with a length of 120 x The key is the tangent function tan6 I can get two relations tan66 2246 h x tan60 1732 mg l have two equations and two variables This gives me a system of two equations 2246 eqn 1 1732 Tzloler eqn2 Solve equation 1 for x Now plug this solution into equation 2 7 11 1732 712mm 1732120 2 Q46 11 20784 77lh 11 20784 22911 11 9076 17 A point in the terminal side of an angle 6 is 2 3 Assume the vertex of the angle rests on the origin Find each of the following a sin6 The origin the point 2 0 and the point 2 3 form a right triangle The line along the x axis has a length of 2 Call this line a The line from 2 0 to 2 3 has a length of 3 Call this line b Thencz22324913socm sin6 g b cos6 cos6 C tan6 tan6 d csc6 csc6 m ms e sec6 a sec6 051 m2 L n cot6 mode 23 g G in degrees and radians tan6 b 3 so 6 tan 1 g m 5631quot 3 18 The hypotenuse of a right triangle is 10 inches One leg is 4 inches Find the degree measure of each a le 0 We just need one o the angles Call it 0c Because the sum of the angles of a triangle is 180 and the triangle is a right triangle the other angle will be 90 7 0c Lets find the angle 0c that is formed by the hypotenuse and the known leg The known leg is adjacent to the angle so well use the cosine relationship ada ent C C051 m 5 cosoc 7 Then 0c cos 1 1 m 6642 Then the other angle is 90 7 664Z 2358 Milton is sitting on a wall 4 feet above the ground He is watching a squirrel in a tree 15 feet away If the angle of elevation is 55 how tall is the tree u 5 15 0 Look at the diagram We can use trigonornetry to find h Iknowtan55 so adacent 1 428 Solve for hto get 11 15 X 1428 2142 But the tree is an additional 4 feet tall So the height is 2542 20 Solve the right triangle shown for each of the given parameters ET X 4b5 l To get c use the Pythagorean Theorem 4252 02 1625 02 41 c2 m c opposite To get 5 note that sinl W W Then 5 sin l m 896055385radians 51342 To get a note that the sum of the angles of a triangle is 180 So a must be 180790751340 3866 15 0 5 l To get 5 note that the sum of the angles of a triangle is 180 So 5 must be 1807907150 75 7 adjacent Now note that coslfl 7 hypotenuse So cos75 Then c W350 m 1932 7 opposite Now note that sinl 7 ihypotenuse So sin75 T b 3239 Then 19 1932 sin75 m 1866 C oc40 B 500 There are multiple solutions to this problem Here is how to derive one First note that sin50o 7 hopp l so ypotenuse 766 b Q There are many choices for a and b Hence why there are multiple solutions to the problem The easiest is to letb 766 and a 1 Use the Pythagorean Theorem to get C 7662 12 CZ 586756 1 CZ 1586756 CZ 1260mc 21 Sketch a gteph of the tegton bounded byx m 3th x2e 2x 3 Where do the Curves intersect 0 To nd the mtersecuon solve for x x1 2x 3 x1 2x3 Sol have the cootdmetes 39 and 71 my 22 Sketch a graph of the region bounded by y xyy x2 Where do the curves intersect note the scale on the graph is 1 line for every unit 2 To nd the intersection solve for x x x x x x733 0 mix 0 So I have the coordinates 00 and 11 MAl O7 Precalculus Algebra Final Exam Review Questions 3rd Update Solutions 27 April 2008 DISCLAIMER This review does not cover every nook and cranny of the final exam Its purpose is to help remind you of important concepts we ve covered this semester You should review old exams old homeworks and old practice exams as well Anything we ve covered and anything from old exams quizzes WebAssigns book homework etc is fair game for the exam New material we ve covered Sec 64 is fair game as well With the exception of Exercise 1 I ve arranged the questions a through c in order from simplest to most involved 1 N o Find all your old tests a Without looking at the solution guide correct each problem which you missed points on b Check your answer against the solution guide c Find a similar problem in the WebAssigns or the book Work the problem and ensure you can get the correct answer Which of the following are functions If they are functions are they onetoone If they are oneto one state the inverse function a 2556146514121 Is a function Not onetoone Find the difference quotient Your goal is to simplify the result to the point that you can let h 0 without dividing by 0 a x x l l 1 b gx 36231 2 2 2 2 2 1117 J First the difference quotient is given by w Let s find gx 11 first To find 9 x h I will replace every instance of x in the formula 23 with an x 11 So I get Z w h x22hxhz1 xh71 Since 9 x h W then I can substitute W for gx h in the differ ence quotient Diff Quotient ma a 7 h Replace x with 36231 because they are equal by definition 2 2 2 x izhxih it xil x it xih t xh7 x7177 x7 x h 2 x22hxhztxit x txhrt W WWW x22hxh2txit7 x2txh7t i 2x it i it h x3ixz2hx272hxxhzihzxil x3hxzixzxhil 7 u u i u u t 7 h ihxzizhxzixhzizhxihzih x hx72x7ht h 2 2 7X Jilinjjszif l 71 factor out an 11 xzhx72xihil x2hx72xihl x2h72x7h71 x2h72x7h1 C hx J W 4 Find the domain a x 17 X 1 94an 1quot WWW XX023 z 7 C hbd 2i Interval 70072 711 300 O N Let x x2x1 a Find the average rate of change from x 1 to x 5 31flbjifla 514m 313 7 1 bid 4 4 b Find the slope of the secant line from x 1 to x c fcg11 1 3172 13711 02 1 c Find the equation of the secant line from x 1 to x c A line has the formy mx b As shown before the slope m is c 2 So I plug 0 2 in for m to see thaty c 21x 1 b Now what is b I know that when x 1 that y 3 So I plug that into the equa tion 3 c 1 211111 b and solve for b Hencey c2x17 c The functions 81p and D p model supply and demand of quantities of a particular brand of soup 5v1x2 8 D1 p 6074x a Find the quantity demanded when the price is 2 1 1 52 1 b Is the supply curve increasing or 139 Qquot How do you know The vertex is a minimum To the right of the vertex the curve is increasing To the left it is decreasing c Find the equilibrium point That is the price for which supply equals demand 54833 380668 1 Consider the piecewisedefined function below 0 x4 x34 a What is 75 1o 1 ifxlt74 if74 x 1 ifxgt1 x b What isf0 1 1 4 1 o c Whatis m has State the equation of each of the following after applying the given transformations Be able to show a graph of each function af x x2 1 shiftingl to the right 3 up re ecting aboutx axis lwxiuw l b 900 ex rl shifting 2 to the left Zdown reflecting about the x axis a7ex2zt 2 c hx lnx 7 l shifting l to the right 2 down re ecting about the line y x l a exZ 2 The graph of x is given to the right a What are the x andy intercepts l x 413 y 3 b L tn 4 t l Maxima 03 Minima 273 the relative minima and maxima c On whatintervalsis x increasing decreasing l Increasing 7150132 Decreasing 0 2 The following demand equation models the number of units sold x of a product as a function of price p x 77p 683 a Write a model expressing the revenue R as a function of x Revenue is the product of price and number of units sold Hence R px But I know x 77p 683 So I can substitute 77p 683 for x Rm M7710 683 77102 683p b How do you know there is a maximum and not a minimum point on the graph of this function Recall the formula is a quadratic which has a single vertex Either the vertex is the highest point or the lowest point The p2 coefficient is negative This means the parabola opens down so the vertex must be the highest point c What price should the company charge to maximize revenue Find the vertex Formula for the p coordinate is 7 lget p 4879 Let x 32 x a Find the zeroes of x Solve the numerator for zero to find the zeroes But I must throw out any solutions which are also roots of the denominator Why Because at these x values the denominator is zero So the value of x isn t defined at that point Hence those values of x aren t in the domain So they can t be roots of x x74 b What are the vertical asymptotes Solve the denominator for zero and throw out any solutions which are also roots of the numerator Or Write x in reduced form then look for the roots of the denominator x75 c What are the horizontal asymptotes If the degree of the numerator is less than the degree of the denominator the horizontal asymptote is y 0 If the degree of the numerator is greater than the degree of the denominator there is no horizontal asymptote If the degress are equal divide the leading coefficients The coefficients of the highest power term The degree of the numerator and denominator are both 2 The horizontal asymptote isy l 12 UPDATE Let x 7 2x4 715x3 719x2 714x 7 54 a Determine the remainder when x is divided by x l Recall if a polynomial x is divided by x 7 c then the remainder is Ho 0 in this case is 71 f 71 742 Remark if He is zero then there is no remainder In this case x 7 0 would be a factor of x b Use the Intermediate Value Theorem to determine if x has a real root on the interval 72 3 H72 50 H3 7510 There is a real root on this interval Because the signs alternate across the endpoints by IVT there is a real root c Use the Intermediate Value Theorem to find another interval which contains at least one real root 723 works 13 Write a polynomial which a Has real roots atx 71 andx1 fx1x11x711 b Has real roots at x 71 and x 1 and whose graph touches the x axis at these points fx1x112x7112 l c Whose graph touches the x axis at x 72 and x 3 crosses the x axis at x 0 and has degree 7 ltgtfx1x3x212x7312 l 14 Let x x2 1 gx 6x5hx a Determine f o gxj fgxj f6x5 6x52 1 36x2 60x 26 b Determine g o hx 0 gthtxn g1615 X l c Determine h o fx of mm htxz 11 fai i X 2 1 15 a Change into a log expression 69 32 log6l321 y l b Change into an exponential expression loga p3 5 9 a5 D3 c Write as a string of simple log expressions loga VZZL2 37QI logax 3logax1 7 1jlogabc2 1 flogabci 6 16 Determine the domain and the range of each of the following functions a x lnx 4 The argument of a logarithm cannot be negative or zero So I require x 4 gt 0 Domain x l x gt 74 To find the range I can compute the inverse function and determine its domain Or I can recall that the exponential function is the inverse of the logarithm function and we established in class that its range is infinite Range 700 00 b 9li c h x 6x71 The denominator cannot be zero So I require 6x 7 1 y 0 Domain x1 x y lg To get the range I have to compute the inverse function and determine its domain Range x1 x y lg x 2e As stated the domain of the exponential function is infinite Domain 700 00 1 1 f The inverse function is f 1 x lnl x To find its domain I require cause the argument of a logarithm cannot be negative x gt 0 be Range x l x gt 0 17 Solve for x a 3x22x lo 1 Note l 3 2 Then 333 3 Recall that if Au Aquot then u v So I know that x2 2x 72 or x2 2x 2 0 But there is no real solution To see this I can use the quadratic formula or look at a graph and see the parabola does not cross the xaxis g5x27112 log5x2 711 2 510g5x2711 52 x2 7 11 25 The exponential and logarithm functions are inverses of each other So they cancel each other out x2 7 36 0 x i6 c 64 7 8 1 7 18 Solve for x 64X 7 8X1 7 712 821 7 88 7 712 8X12788X 7712 LetA78X A2 7 8A 7 712 A2 7 8A 12 7 0 A76A72j 7o A 7 6 A 7 2 8X 7 6 8X 7 2 1mm 71 6 mam 71112 x1n8 1116 xln8j 1112 x 7 iz g x 13 a 6 7 24 2 6W 1 724 4X2 74 4X2 7 4 x2 1 x2 71 0 b lnbc2 7 8 0 lnbc2 7 8 0 elnxz78 ea x2 7 8 1 x2 7 9 0 X 31X 3 0 x i3 c lo2x7 3 7 log22x 6 log2x7 l 7 log2x4 logzbc 7 3 7 log22x 6 logz 2136 x73 2x6 X73JX4J x2x712 0 log2x7 l 7 logzlx4l log By Properties of Logarithms Recall If logu logv thenuv x712x6 2x24x76 x23x6 No real solution 19 Find the values of each of the following without a calculator a Elog525 7 3log5l M13 b cos22 7 sin68 w o c tan22 7 csc288 W 71 20 a Write as a single log expression log4A Zlog4B 7 Zlog4C 1 w l 10g4l J b Write as a series of simplier log expressions log W log4A l log4B 71 flog4x2 x 1 As mentioned in class on Thur be careful with addition inside logarithm arguments c If 4 13 then what does 4364 equal Recall all the properties of exponents 4quot quot 1 x271 4 14 21 9000 dollars is invested at 9 interest compounded monthly a How much money do I have after a period of 12 years A9OOO1144 b After how many years will my balance double in value Wt m2 121nl 1 c Find a new interest rate so that my balance will have doubled after 3 years a 129 71 e233 22 A skeleton of a porcupine has been recovered Archaeologists determine it contains 44 of its original amount of Carbon14 Assuming a halflife of 5 600 years determine the age of the porcupine 0 e 663278 m 23 A point in the terminal side of an angle 0 is 3 4 Assume the vertex of the angle rests on the origin Find each of the following a cos0 m ET tan0 0 g l csc0 0 3 l P d sec0 0 3 l cot0 0 g l 5 24 Milton is sitting on a chair It rises 2 feet above the ground He is watching a squirrel in a tree that is 19 feet away If the angle of elevation is 42 how tall is the tree 19tan42 21911 l 25 A right triangle has legs of length 2 3 What is the length of the hypotenuse What are the measures of the angles Hypotenuse m Angles 90 tan 1 g m E63l tan 1 g m 33690 26 A right triangle has one angle of measure 19 What are the measures of the other angles and legs Angles 90 l9o 71 There are multiple solutions for the legs Assuming the hypotenuse has length 1 then Leg 1 sinl9 Leg 2 sin7l m 326 m 946 Notice that 3262 9462 m 1 We should get that because of the Pythagorean Theorem 27 The population of the city of Watsonopolis obeys the law of uninhibited growth Initially the popula tion is 13 000 After 149 years the population is 1000 000 When will the population hit 2 000 000 1n2000000 0 t 0038880 17278 39 l 3000 J vim 9 NEW If c056 I and 6 lies in quadrant IV find 51n6 and tan6 Iknow that 51n26 c0526 1 So this gives me 51n26 1amp2 1 or 51n2616 1 Now I solve for 51n6 16 16 51n26 51n6 i S015 51116 lg 0174 Since 6 15 in Quadrant IV then the sine of 6 15 negative Recall the starred chart from class Then 51n6 7 g 7 15 tame 42 74 74 7m
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