Math 112 Fall 2015 Midterm 2 Review Problems Page 1. has a maximum or minimum and then determine the maximum or minimum value.

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1 Math Fall 05 Midterm Review Problems Page f has a maimum or minimum and then determine the maimum or minimum value.. Determine whether Ma = 00 Min = 00 Min = 8 Ma = 5 (E) Ma = 84. Consider the function f TRUE?. Which one of the following statements is The graph opens up. The y-coordinate of the verte is 9. The graph opens down. The graph has a minimum at the verte.. Determine the formula for the quadratic function in standard form for the parabola graphed below y y y y 4 (E) y 4

2 Math Fall 05 Midterm Review Problems Page 4. Determine the formula for the quadratic function in standard form that fits the data eactly H ( t) ( t 0) 0 H ( t) ( t ) 0 64 H ( t) ( t 0) H ( t) ( t 7) 0 69 (E) None of these t H t Determine the -intercepts of f. The largest -intercept is:, 0, 0 7, 0, 0 7 (E) None of these 6. Determine the verte for the quadratic function 8 g. 6, 6,, 5, (E), Determine a possible formula for the quadratic function that passes through the points, 0, 0,. 4,, and 6 y 4 y 4 y 4 y 4 (E) y 4 5

3 Math Fall 05 Midterm Review Problems Page 8. Determine a possible formula for the quadratic function that has a verte at 5, passes through the point, y 5 y 5 y y 5 (E) y 4 5 and 9. A mathematical model for the profit, P, in thousands of dollars, that a company makes in one year from producing units of its product is: P , 750. What is the maimum profit the company can make? $00,000 $50,000 $75,000 $00,000 (E) $5, A woman is shooting a basketball toward a basketball hoop. The starting height of the ball is 5.5 feet. The path of the basketball is parabolic in shape, and the ball reaches its maimum height of.5 feet when the ball is 0 feet from the player. The ball hits the front of the rim, which is 0 feet high. Determine the horizontal distance from the shooter to the rim. The distance is: Less than 0 feet Between 0 and feet Between and 4 feet Between 4 and 6 feet (E) More than 6 feet. A homeowner wants to fence in a rectangular garden net to a very long stream. Since the garden will border the stream, that side of the garden will not need fencing. What is the largest area that can be enclosed with 0 feet of fencing?,00 sq ft 6,050 sq ft 8,5 sq ft,00 sq ft (E) None of these

4 Math Fall 05 Midterm Review Problems Page 4. A horticulturist has determined that the number of inches a young redwood tree grows in one year is a function of the annual rainfall, r (in inches), given by g r 0.0r r What is the maimum number of inches a young redwood can grow in a year? The maimum number of inches is: less than between.9 and. between. and.6 between 4.4 and 5. (E) between 4.7 and 5.. When an apple orchard owner plants 65 trees on an acre of ground, he gets an average yield of 500 apples per tree per year. For each additional tree planted per acre, the annual yield per tree drops by 0 apples. Calculate the number of trees that should be planted per acre to produce the maimum crop of apples and use that value to determine the maimum yield? The maimum yield is: 7,000 apples 0,000 apples 97,500 apples 98,000 apples (E) 00,000 apples Bob s high school is performing a musical. If each ticket is priced at $9.50 then 600 tickets will be sold. For each $0.0 increase in price, 6 fewer tickets will be sold. Use this information to answer the net two questions. 4. Determine the price that should be charged for each ticket to maimize the revenue. (Hint: Revenue is quantity times price.) Less than $9.60 Between $9.60 and $9.80 Between $9.80 and $0.00 Between $0.00 and $0.0 (E) More than $ Determine the maimum possible revenue from the musical ticket sales. More than $600 Between $6000 and $600 Between $5900 and $6000 Between $5800 and $5900 (E) Less than $5800

5 Math Fall 05 Midterm Review Problems Page 5 A company has determined that the demand function for a dining room table is given by p 00 where p is the price per table and is the number of tables sold. The fied cost of producing a line of dining room tables is $00,000 and each table costs $00 to make. Use this information to answer the net three questions. 6. Determine a simplified formula for the revenue function. (Hint: Revenue is quantity times price.) R 00 R 00 R 00 R 00 (E) R Determine the quantity that should be manufactured to maimize revenue. The quantity is: Less than 950 tables Between 950 and 050 tables Between 050 and 50 tables Between 50 and 50 tables (E) More than 50 tables 8. Determine the quantity that should be manufactured to maimize profit. The quantity is: Less than 950 tables Between 950 and 050 tables Between 050 and 50 tables Between 50 and 50 tables (E) More than 50 tables

6 Math Fall 05 Midterm Review Problems Page 6 Consider the graph of the polynomial three questions. S shown below. Use this graph to answer the net 9. What is the smallest degree that S could have? (E) 7 0. What is the sign of the leading coefficient of S? Positive Negative. What is the sign of the constant term of S? Positive Negative Zero

7 Math Fall 05 Midterm Review Problems Page 7. Consider the polynomial function f 4 5 the polynomial f?. What is the degree of 5 6 (E) None of these Use the graph of y g to answer the net two questions The graph of y g.. Determine the multiplicity of the zero at on the graph of g y. 6 (E) 0 4. Determine the multiplicity of the zero at on the graph of g y. 0 (E) 5

8 Math Fall 05 Midterm Review Problems Page 8 5. The polynomial function T has a zero at. What is the multiplicity of this zero? (E) 6. Consider the polynomial function 4 5 behavior of the graph. P. Determine the end P approaches as approaches P approaches as approaches P approaches as approaches P approaches as approaches P approaches as approaches P approaches as approaches P approaches as approaches P approaches as approaches

9 Math Fall 05 Midterm Review Problems Page 9 7. Which one of the following could be the graph of a polynomial of degree 4 with a negative leading coefficient?

10 Math Fall 05 Midterm Review Problems Page 0 8. Determine an equation for the polynomial graphed below y ( ) ( )( ) y ( ) ( )( ) y ( )( )( ) y ( ) ( )( ) (E) y ( ) ( )( ) 9. Determine a possible equation for the polynomial function that has the following zeros. 0,,, A possible equation is: y y (E) y y y

11 Math Fall 05 Midterm Review Problems Page 0. Determine the equation for the polynomial function in factored form that has the following characteristics. Zeros at (multiplicity ) and (multiplicity ), 8 Passes through the point y y y y 7 (E) y 4. Determine all the real zeros of P ( ) 5 6. The smallest real zero is: (E) None of these 4. Determine all the real zeros of h 0. The largest real zero is: 6 6 (E) None of these. Which one of the following statements is/are equivalent to 4 is a factor of the polynomial f? 4 f 4, 0 is an -intercept of y f f 4 will have a non-zero remainder 4 f is a zero of is a zero of

12 Math Fall 05 Midterm Review Problems Page 4. Consider the polynomial function f 4 8 statements is FALSE? is a zero of f. Which one of the following f has only one real zero The end behavior of the graph is f approaches as approaches approaches as approaches The y-intercept of the graph is 0, 8 (E) is a factor of f and f 5. Determine all the real zeros of y The SUM of all the real solutions is: (E) None of these 6. Write the polynomial in factored form: 6 4 f. f f f f (E) None of these 7. Determine the remainder when 0 6 is divided by (E) None of these

13 Math Fall 05 Midterm Review Problems Page f by and write the answer in the form. Use this to answer the net two questions. Divide 7 f q r f by 8. After dividing 7 f q r, what is the formula for and writing the answer in the form q? 5 q 9 q 4 q 9 6 q 5 (E) None of these f by 9. After dividing 7 f q r, what is the value of r Less than 5 Between 5 and 0 Between 0 and 5 Between 5 and 0 (E) More than 0 and writing the answer in the form? The value of r is: f by Divide 5 Use this to answer the net two questions. 40. After dividing 5 f q r, what is the formula for and write the answer in the form f q r f by. and writing the answer in the form q? q q 8 6 q 4 q 8 (E) q 4 f by 4. After dividing 5 f q r, what is the value of r and writing the answer in the form? The value of r is: (E) None of these

14 Math Fall 05 Midterm Review Problems Page 4 4. Determine the value of k that makes value of k is: More than.5 Between 0.5 and.5 Between 7.5 and 0.5 Between 4.5 and 7.5 (E) Less than 4.5 a factor of f k 6. The 4. Which one of the following functions is not a rational function? 5 y 7 y y 7 5 y 44. Determine the domain of the function f. 5 0,,,,,,,,, (E),,, 45. Determine the real zeros of k. 0 (E) k () has no real zeros 46. Determine the real zeros of k., only, only only, only (E),,, only

15 Math Fall 05 Midterm Review Problems Page Determine the horizontal and vertical asymptotes for the rational function 5 4 y. 7 5 Horizontal y 0 Vertical 5 and 5 y 0 and y and 7 7 y 0 and 7 (E) 4 y 7 5 and 5 Consider any vertical asymptotes and removable discontinuities (holes) for the rational function 5 6 f to answer the net two questions Determine the vertical asymptotes for the graph of f y. 4 only 4, only only, only (E) only 49. Determine the removable discontinuities (holes) for the graph of f y. 4 only 4, only only, only (E) only

16 Math Fall 05 Midterm Review Problems Page Determine the slant asymptote of y 0 y 8 y y (E) y 0 5 h. 5. Determine an equation to represent the rational function graphed below: 6 5 4, (E) g g g g g 4-6

17 Math Fall 05 Midterm Review Problems Page 7 5. Which one of the following represents a rational function whose graph has a vertical asymptote at, a vertical asymptote at, and a horizontal asymptote at y 0? (E) f f f f f Which one of the following represents a rational function whose graph has a vertical asymptote at 5 and has a removable discontinuity (hole) at? (E) k k k k k

18 Math Fall 05 Midterm Review Problems Page The graph of 8 f was created from transformations of the graph of 5 Which one of the following lists the transformations that were performed? y 5. Horizontal compression by and shift right 8 units Horizontal compression by and shift left 8 units Vertical stretch by and shift right 8 units Vertical stretch by and shift left 8 units (E) Vertical stretch by and shift up 8 units Determine the asymptote for y y 7 y y 5 (E) Determine the y-intercept for y 5. 0, 0 0, 0, 0, (E) There is no y-intercept

19 Math Fall 05 Midterm Review Problems Page Determine an eponential function of the form 8 y Cb which has the graph shown below , 6 y y -8 y 6 y 6 (E) y 58. A piece of machinery, initially purchased for $5,000, decreases in value by.4% per year. t Determine a model of the form v k b that can be used to predict the value of this machinery if t is measured in years. (E) v 5,000 (0.04) v 5,000 (.04) v 5,000 ( 0.04) v 5,000 (.986) v 5,000 (0.986) t t t t t 59. Suppose $0,000 was invested in a trust fund for a child at birth. After 0 years, the value of the fund is $7,6.40. Determine the interest rate the fund has been earning if interest has been compounded monthly. The interest rate is: Less than 4% Between 4% and 6% Between 6% and 8% Between 8% and 0% (E) More than 0%

20 Math Fall 05 Midterm Review Problems Page How much money should you invest at 7.5% compounded quarterly so that you have $0,000 after 5 years? $4, $98.88 $9.0 $ What is the range of y e 5? 0,, 0, 5 5, (E),5 6. Solve for : e 5 e e. The answer is: More than 7. Between 5.4 and 7. Between.6 and 5.4 Between.8 and.6 (E) Less than.8 6. Solve for. 0 e e 7 ln() (E) There is no real solution 64. Samantha invested $00 in an investment account that earned 4.6% interest compounded continuously for 40 years. How much money is in the account after 40 years? The amount of money is: Between $700 and $000 Between $000 and $00 Between $00 and $600 Between $600 and $900 (E) Between $900 and $00

21 Math Fall 05 Midterm Review Problems Page 65. Zhang Liu needs to have $85,000 in an investment account in 5 years. How much should be invested today at an interest rate of 7.8% compounded continuously to have $85,000 in 5 years? The minimum amount that should be invested is: More than $5,000 Between $,000 and $5,000 Between $7,000 and $,000 Between $,000 and $7,000 (E) Between $9000 and $, A species of snake was introduced in an area 0 years ago. It is estimated that there are 500 snakes in the area now, and the population has a relative eponential growth rate of 7% per year. How many snakes will there be 0 years from now? Less than 4,000 Between 4,000 and 5,000 Between 5,000 and 6,000 Between 6,000 and 7,000 (E) More than 7, Write in logarithmic form. 4 5 log / 5 4 log 4 log / log 4 (E) log Write in eponential form. log B B B 5 5 B 5 4 B 4 5 (E) B 5 4

22 Math Fall 05 Midterm Review Problems Page 69. Evaluate 4 log at. (E) 70. What is the range of log g? (, ) ( 0, ) (, ) (,) (E) (, ) 7. What is the domain of f log 4? ( 0, ),,, (E), 7. Determine the -intercepts, if any, of f log 4., 0 7, 0, 0,0 0 (E) f has no -intercepts 7. Determine values for a and b so that a blog f precisely models the data: 9 7 f.5.5 a, b a, b a, b a, b (E) a, b

23 Math Fall 05 Midterm Review Problems Page 74. Determine the inverse function for t g 7 t t t g t 7 ln( t ) g t ln(7) ln( ) g t t ln(7) (E) g t ln( t) ln(7) g t Determine the inverse function of f log f f f log ( ) / (E) f f log. 76. Solve the equation and epress your answer to the nearest log (E) Use properties of logarithms to simplify the epression. 0 log y log y y y y (E) y

24 Math Fall 05 Midterm Review Problems Page Write as a sum/difference and/or multiple of logarithms. log 5 ab log ( a) log ( ) 5 5 b log 5() log 5( a) log 5( b log 5() log 5( a) log 5(b log 5( a) log 5(b log 5() log 5( a) log 5( b ) ) ) (E) ) 79. Write as a sum/difference and/or multiple of logarithms. ln ln e y y ln e ln lny ln lny ln y lne (E) None of these 80. Write in the form log b P, where P is an algebraic epression. log y log b b log y log b b y b y log log log y b b (E) None of these

25 Math Fall 05 Midterm Review Problems Page 5 8. Write in the form log P b, where P is an algebraic epression. log log y log z b b b z( ) log b log b 6yz y log b z( ) y z( ) log b (E) 6y log y z b 8. Is the following statement true or false? ln ln e e ln True False e 8. Is the following statement true or false? ln ln True False 84. Is the following statement true or false? ln c M c ln M True False 85. Which one of the following is equivalent to log 5? log5 log log log 5 log 5 5 log (E) log log 5

26 Math Fall 05 Midterm Review Problems Page Use natural logarithms to evaluate log ln7 ln ln 6 ln 7 ln7 ln6 ln 7 ln 6 (E) None of these 87. Determine the zero of ln ln ln4 f (E) None of these 88. Solve for using the natural logarithm. 4 ln ln(4) ln 0 ln(4) ln ln(4) 4 ln (E) ln() ln Solve for t using the common logarithm. t log(4) t 8log(.05) log(.5) t log(.) t log(4) log(4) log(.5) t (E) log(.05) t log(.5) log(.05)

27 Math Fall 05 Midterm Review Problems Page Solve for t. e 0.0t 0.04 t 0.04 e t 0.0 ln 0.04 t ln0.04 t (E) None of these 9. Solve for. e ln(4) 0.05 ln ln() 0.05 ln() 0.05 ln( ) (E) 0.5 ln() Solve for using the natural logarithm ln4 4 9ln5 7ln 4 4 ln 7ln 9ln 5 4ln4 9ln5 ln4 (E) None of these

28 Math Fall 05 Midterm Review Problems Page 8 9. Solve for. 4 or only 4 only only (E) 4 or log log 94. Solve for. log log 5 (E) None of these Solve for y. log The smallest real solution is a number: Less than Between and Between and 4 Between 4 and 6 (E) Greater than 6 y 6 logy logy 96. Solve for. ln ln e e e e e (E) None of these

29 Math Fall 05 Midterm Review Problems Page Assuming an eponential growth model, the number of years it takes the world population ln to double is t, where r is the growth rate. What growth rate is necessary for r doubling to occur in 0 years? 4.7% 5.% 5.%.47% (E).5% 98. How long would it take to double your investment if you invest $,000 at 7.5% compounded quarterly? Less than 8 years Between 8 and 9 years Between 9 and 0 years Between 0 and years (E) More than years 99. The median price of a home rose from $50,000 in 970 to $00,000 in 990. Assuming an eponential growth model, what would be the median price of a home in 996? Less than $0,000 Between $0,000 and $5,000 Between $5,000 and $0,000 Between $0,000 and $5,000 (E) More than $5, The population, P, of a city is given by P 5, 000e where t is the time in years, with t 0 corresponding to 99. In 97 the population was 0,000. Determine the value of r and use it to predict the population in the year 000. The population in the year 000 was approimately: 5,4 45, 4,54 rt,095 (E) None of these

30 Math Fall 05 Midterm Review Problems Page t 0. The population of a town is given by P t 0,000e, where t is given in years with t 0 corresponding to January, 000. In what year is the population projected to reach 0,000? (E) A population of lemmings declined from 00,000 to 75,000 in months. Assuming an eponential pattern of decline, approimately how many will be remaining after another months pass? 56,50 8,750 64,50 50,000 (E) 5,50 0. Suppose 8 ounces of a radioactive substance eponentially decays to 8 ounces in 6 hours. What is the half-life of the substance? The half-life is: Less than hour Between hour and hours Between hours and hours Between hours and 4 hours (E) Greater than 4 hours 04. Radium has a half-life of,60 years. What percentage of the original amount of a radium sample would remain after 000 years? 4% 78% 58% 7% (E) 65%

31 Math Fall 05 Midterm Review Problems Page 05. Suppose after 500 years an initial amount of 000 grams of a radioactive substance has decayed to 75 grams. What is the half-life of the substance? The half-life is: Less than 600 years Between 600 and 700 years Between 700 and 800 years Between 800 and 900 years (E) Greater than 900 years 06. The concentration of a pollutant in the atmosphere increases according to the eponential rt growth model A Pe, where time is measured in years and r In 990, the concentration was 56 parts per million. Clean-up procedures will be initiated when the concentration reaches 70 parts per million. According to the model, in what year will that occur? (E) Air pressure, P, decreases eponentially with the height above the surface of the earth. At the top of Mount Everest, height 8848 meters, the air pressure is about 4.6% of the air pressure at sea level. Approimate the air pressure (as a percentage of the sea level value) at the top of Mount Kilimanjaro, height 5895 meters..% 4.7% 49.% 5.8% (E) 56.4% 08. A population of ladybugs grows according to the limited growth model 0.04t A e where t is measured in weeks, t. After how many weeks will the population be approimately 00 ladybugs? Less than 5 weeks Between 5 and 40 weeks Between 40 and 45 weeks Between 45 and 50 weeks (E) More than 50 weeks

32 Math Fall 05 Midterm Review Problems Page 09. The ph of a solution is defined as H ph log, where H is the concentration of the hydronium ion (H O) in a solution. An acid solution has a ph lower than 7; a basic solution has a ph above 7. In a region noted for its acid rain, you take a sample of lake water and test its ph. What is the ph of the lake if the concentration is H 5.0? Round your answer to the nearest tenth (E) New employees are given an initial eam and then retested monthly with an equivalent eam. The average score for the employees can be epressed as S t 76 6log t where t is measured in months since the initial eam. After approimately how many months would the average score be 54? Less than 4 months Between 4 and 6 months Between 6 and 48 months Between 48 and 60 months (E) More than 60 months. The temperature of a cup of hot tea, in F, can be modeled by the function 0.0 t Ht 7 0e, where t represents time in minutes since the cup of tea was placed on a kitchen counter. Use this function to determine how long it will take for the cup of tea to cool to 0 F. The amount of time it will take is: Less than 0 minutes Between 0 and 4 minutes Between 4 and 8 minutes Between 8 and 4 minutes (E) More than 4 minutes

33 Math Fall 05 Midterm Review Problems Page Fall 05 Midterm Study Guide Answers Question Answer Question Answer Question Answer E 8 D 75 A B 9 C 76 C E 40 E 77 D 4 A 4 D 78 E 5 A 4 D 79 C 6 E 4 B 80 B 7 A 44 E 8 E 8 A 45 E 8 B 9 B 46 A 8 A 0 D 47 C 84 A B 48 A 85 E C 49 E 86 D D 50 B 87 B 4 B 5 A 88 A 5 E 5 C 89 E 6 C 5 E 90 C 7 D 54 D 9 E 8 B 55 C 9 A 9 C 56 A 9 B 0 A 57 E 94 D B 58 E 95 C D 59 B 96 A B 60 D 97 D 4 C 6 C 98 C 5 C 6 E 99 B 6 D 6 A 00 D 7 C 64 D 0 B 8 E 65 E 0 A 9 B 66 B 0 C 0 B 67 D 04 E D 68 E 05 B A 69 B 06 D A 70 A 07 C 4 E 7 C 08 A 5 D 7 C 09 A 6 A 7 B 0 D 7 B 74 D B