MATH 101 EXAM 4 REVIEW I Rational Functions 1 Find the domain of the following rational functions x 8 x 7 b g x a f x 2 x 25 x 4 c h x 3x2 x2 3x 88 2 Write the equations for the vertical horizontal and oblique asymptotes for each of the following rational functions 2x2 3 a f x 2 x 1 c f x x2 3 x3 9x b f x x 4 x 7 d f x x2 2x 8 x 2 3 Find the x and y intercepts for the graph of the each rational function a f x x 8 3x 14 b g x x 3 x 1 x2 5x 6 c h x 50 x 10 x 2 d j x x2 8x x2 3x 4 4 Determine algebraically if the graph of f x oblique asymptote If so at what point x2 7x 10 intersects the horizontal or x2 8x 16 5 Make a careful sketch of the graph of each rational function Label all intercepts asymptotes and at least 3 additional points a f x 2x x 3 x 1 b g x x 3 x 2 c h x x2 1 x 2 6 The illumination of provided by a car s headlight varies inversely as the square of the distance from the headlight If a car s headlight produces an illumination of 5 footcandles at a distance of 44 feet what is the illumination when the distance is 55 feet 7 When the temperature stays the same the volume of a gas is inversely proportional to the pressure of the gas If a balloon is filled with 210 cubic inches of a gas at a pressure of 14 pounds per square inch find the new pressure of the gas if the volume is decreased to 70 cubic inches 8 A yak is directly proportional to the square of a zak and inversely proportional to the square root of a dak If a yak is 10 when a zak is 5 and a dak is 16 what is a zak when a yak is 32 and a dak is 25 II Exponents Logarithms Be able to convert between exponential logarithmic forms 9 Use a calculator to evaluate the following expressions correct to 3 decimal places e3 2 ln 5 a log5 13 b log0 6 2 c d 5 ln 5 ln 2 10 Use transformations to graph the exponential and logarithmic functions in the following problems Identify the domain range and asymptotes for each Label at least 3 key points on the graphs a f x 3x 2 b g x 2x 1 c h x log5 x 1 d k x log3 x 2 III Properties of Logarithms In addition to these sample exercises you should know the basic properties of logarithms 11 Use the properties of logarithms to rewrite each logarithm as a sum and or difference of logarithms Express powers as factors a log ab2 c3 p5 c ln 4 q w b log5 x x 5 2 9 y d log3 w3 12 Use the properties of logarithms to rewrite each expression as a single logarithms a 2 log3 x 4 log3 y 3 log3 z b 2 ln 5 1 ln x 3 ln x 4 2 IV Equations Applications Solve for x in the following equations Give both an exact answer and an approximation correct to 3 decimal places 13 93x 27x 5 x 2 1 14 324 x 2 15 5x 1 42x 16 42x 7 17 ln x 3 5 4 18 log7 2x 1 1 2 19 2 log x log 5 log 15 log 2x 20 log5 x 3 log5 x 4 log5 8 21 Use u substitution to solve a 2e2x 7ex 3 0 b 52x 3 5x 4 0 22 Suppose you borrow 15 000 at 6 75 interest compounded continuously How much will you owe in 4 years Assuming you make no payments how many years to the nearest year will it take for the amount due to be 22 500 23 Sue just inherited 35 000 which she decides to invest a If the interest is compounded quarterly at 3 how much will be in the account in 3 years How long will it take to double the investment b If the interest is compounded continuously at 2 75 how much will be in the account in 3 years How long will it take to double the investment 24 Find the number of years to the nearest tenth needed for an original principal investment of 48 000 to become 65 500 if the interest is compounded at 2 5 semiannually 25 How much should be invested now to obtain 3500 in 7 years if it can be invested in an account paying 5 3 compounded monthly 26 In 2003 the United Nations estimated the population of Ethiopia to be about 70 7 million with an annual growth rate of 2 9 Find an exponential model for the population of Ethiopia x years since 2003 When does the model predict the population to be 90 million Use the model A x C ax 27 The population of Lawrence was 80 000 in 2000 and 89 000 in 2006 Use the continuous growth model with base e to find the rate of growth k in exact form and predict the population for 2015 Assume the growth continues at the same rate 28 Assume fish populations obey the exponential growth model A t A0 ekt There are 15 angel fish in a fish tank at the beginning of the month Thirty days later there are 55 angel fish a Find an exact representation for the rate of growth k b How many fish will there be in the tank in 2 weeks note the units c Use the exact value to determine how many days will it be before there are 100 angel fish 29 A nuclear power plant produces radioactive plutonium 239 which has a half life of 24 100 years Because of its long half life plutonium 239 must be disposed of safely How long does it take 100 grams of radioactive waste from the nuclear power plant to decay to 10 grams Give the answer to the nearest tenth of a year 30 The half life of a certain radioactive substance is 33 5 years An initial sample weighs 850 grams Find a formula for the decay function for this substance Find an exact expression for the rate of decay k to use in determining your final answer How long will it take for the original amount to decay to 180 grams 31 A metal object is heated to 110 F and then allowed to cool in a room with temperature 60 F If the object is 100 F after 10 minutes when will the temperature of the object measure 80 F Use Newton s Law of Cooling and find an exact expression for the rate of cooling k first Answers 1 a x x 6 5 or 5 5 5 5 b x x 6 4 or 4 4 c x x …