MATH 135 Sample Review for the Final Exam

Transcription

1 MATH 5 Sample Review for the Final Eam This review is a collection of sample questions used by instructors of this course at Missouri State University. It contains a sampling of problems representing the material covered throughout the semester and may not contain every type of question of the final eam. Any material listed on the lecture schedule and/or the assignment sheet may be on the final eam. Please also be aware that a few questions on the final eam, while requiring knowledge and understanding of the content covered in the course, may be presented in a form different than the problems in the tet. PART I: Short Answer Section ) True or False: a b a b If false, what would make the statement true? ) True or False: 4 If false, what would make the statement true? ) True or False: 4) True or False: e e e If false, what would make the statement true? e e e If false, what would make the statement true? 5) True or False: log log 5 log 5 6) True or False: If false, what would make the statement true? ln ln ln 6 If false, what would make the statement true? ln 6 7) What is the relationship of the lines y and y 5? 8) What do you know about the lines y and 9) Can a function have two y-intercepts? Eplain. y? 0) If for function f, f 0 9, then to what point does 0, 0 ) Determine the y-intercept for ) Determine the -intercept for y 6, if any. y 6, if any. ) Determine the y-intercept for f log, if any. 4) Determine the -intercept for f log 5) Can a parabola of the form or why not? 6) If f 5, then f 5., if any. f correspond? y a b c have an inverse function over its entire domain? Why 7) If the point, 0 lies on the graph of f, give a point of 8) If 9) Is 0 f has domain in the domain of f,, 4, 5 and range 4, 9, 6, 5, what is the domain of f f?.?

2 0) What is the domain of y 4? ) What is the range of y 4? ) Does the graph to the right represent a function? Eplain. ) Give the domain of the graph to the right. 4) Give the range of the graph to the right. 5) Sketch any one-to-one function. 6) Sketch any function that is not one-to-one. 7) How does the graph of y 5 compare to y? 8) What is the minimum value of 9) Does the graph of 0) What are the asymptotes of f? Eplain. 8 y cross the line? Eplain. y 5? ) Does y a b c have a maimum or minimum value when a 0? Eplain. What is that value? ) Write a quadratic equation in standard form a b c 0 ) Write log4a log4b as a single logarithm. 4) Evaluate log b b 5) Evaluate log a a 6) Evaluate log m 7) What is the relationship of f log and whose solution set is,7 g?. 8) Without a calculator, determine between what two consecutive integers will you find log t 9) Does the model A t 90e represent eponential growth or decay? Eplain. 40) Write an equation to represent the following statement: y varies jointly with and z. 4) Write an equation to represent the following statement: m varies inversely with n. 4) The imaginary number i has the property i. 4) For the comple number 4i the real part is and the imaginary part is. 44) If a system of linear equations in two variables has no solution, then the lines are. 45) If a system of linear equations has eactly one solution, then the lines. 46) If a system of linear equations has infinitely many solutions, then the lines are.

3 PART II: Constructed Response Section problems require work to support answers. Problems Give the domain of each function. 47) f 48) h 5 49) g 50) L log 8 5) k 4 5 5) R 5 5) d 4 54) P 5 Problems Evaluate without a calculator. 55) log55 56) log9 8 57) log8 58) log ) ln e 60) log7 7 6) log6 log4 6) log7log9 Problems Perform the indicated operation and write each epression in the form a + bi. 6) 4 i 4i 64) 8i5 9i 65) i 4 7i 66) 7i 7i 67) 5 4i Problems Solve each equation. Leave all answers eact. DO NOT ROUND. 68) 69) 8 (Solve without using logarithms.) (Solve without using 8 logarithms.) 5 70) 5 7) 7) ) p p p p ) 5 5 log 8 log log 4 76) ) log 78) ) ) log 64 8) 8 log 7 8) log 5 log 4 log 8) log log ) 5 5 y 5y 0

4 Problems Solve each equation. Leave all answers eact. DO NOT ROUND. 84) log log 4 87) ln 6 ln 6 ln 85) 86) ) 5e 5 89) log log Problems Solve each system of equations. Write solutions as ordered pairs. 90) 9) y9 y y 5 54y 94) 95) y 6 8y 4 y y y ) y y y y 0 9) y 9 y 96) 6y 7 9y0 y 0 9) y 6y 5 97) y 5 4 y 9y 5 Problems Solve each inequality. Write solutions using interval notation. 99) ) 4 0) 8 0) 6 7 0) 04) ) ) 0 Problems 07. Write the equation of a line that satisfies the given conditions. 07) Write the equation of the line through the points, 7 and 8, 4 08) Write the equation of the line through 09) Write the equation of a line through 0) Write the equation of a line perpendicular to 8 ) Write the equation of a line parallel to , and perpendicular to y 5., and parallel to y 5. y and passing through, 5 y and passing through, 5..

5 Problems 5. Find the average rate of change of the following functions from one -value to another. ) f 5 from to 4 4) h ) g 7 from to 6 5) k log 6 from 5 to 0 Problems 6 5. Evaluate and simplify the given values of these functions. g 5 f if 0 h if if 0 6) h 8) gt 0) f h ) g f 7) h 0 9) f ) f g 4) f g ) f g 5) g Problems 6 4. Find the given values of these functions f() ) f g 7) fg 6 g f 8) 5 9) f g ) f 4 ) g 4 ) f g 0 4) f f 0) f g0 5

6 Problems 5 8. Find the equation of the inverse function for each one-to-one function. 5) f 8 6) f log 5 7) f 8) f Problems 9 4. Write the equation that relates the quantities given. DO NOT SOLVE. 9) Epress the area, A, of a rectangle as a function of the width,, if the length is twice the width of the rectangle. 40) A commissioned sales person earns $00 base pay plus $0 per item sold. Epress her gross salary, G, as a function of the number of items sold,. 4) The illumination, I, produced on a surface by a source of light varies directly with the candlepower, c, of the source and inversely with the square of the distance, d, between the source and the surface. Problems 4 5. Solve each problem. Be sure to identify the variables and give the equation used to solve each problem. 4) The difference between the squares of two real numbers is. Twice the square of the first number increased by the square of the second number is 9. Find the numbers. 4) The product of two real numbers is 0 and the difference of their squares is. Find the numbers. 44) The illumination provided by a car s headlight varies inversely with the square of the distance from the headlight. A car s headlight produces an illumination of.75 foot candles at a distance of 40 feet. What is the illumination when the distance is 50 feet? 45) The electrical resistance R of a wire varies directly with its length L and inversely with the square of its diameter d. A wire 00 feet long of diameter 0.0 inch has a resistance of 5 ohms. Find the resistance of a wire made of the same material that has a diameter of 0.05 inch and is 50 feet long. 46) A pool measuring 0 meters by 0 meters is surrounded by a walkway of uniform width. If the area of the pool and the walkway combined is 600 square meters, what is the width of the walkway? 47) Find the length and width of a rectangle whose perimeter is 6 feet and whose area is 77 square feet. 48) One pan pizza and two beef burritos provide 980 calories. Two pan pizzas and one burrito provide 670 calories. Find the caloric content of each item. 49) A riverboat travels 46 km downstream in hours. It travels 5 km upstream in hours. Find the speed of the boat in still water and find the speed of the stream. 50) A 000 acre farm in Illinois is used to grow corn and soybeans. The cost per acre for raising corn is $65, and the cost per acre for raising soybeans is $45. If $54,5 has been budgeted for costs and all the acreage is to be used, how many acres should be allocated for soybeans? 5) Bronze which costs $9.0/kg is made by combining copper which costs $8.90/kg, with tin which costs $9.50/kg. Find the number of kilograms of copper and tin required to make 5. kg of bronze. 6

7 Problems 5 6. Solve each problem. Be sure to identify the variables and give the equation used to solve each problem. 5) The net income, y, (given in millions of dollars) of Pet Products Unlimited from 00 to 0 is modeled by the function y 9 5 5, where represents the number of years after 00. Assume this trend continues and predict the year in which Pet Products Unlimited s net income will be $598 million. 5) Suppose that the manufacturer of a gas clothes dryer has found that the revenue R, in dollars, can be modeled by a function of the unit price, R p 4000 p 4p, where the unit price is p dollars. What unit price should be established for the dryer in order to maimize the revenue? What is the maimum revenue? 54) Since 950, the growth in the world population in millions closely fits the eponential function t P t 600e, where t is the number of years since 950. Estimate the population in the year , to the nearest million. 0.04h 55) The formula D 6e can be used to find the number of milligrams D of a certain drug in a patient s bloodstream h hours after the drug has been given. When the number of milligrams reaches, the drug is given again. What is the number of hours between injections? 56) If Andrew has $800 to invest at 6% per year compounded monthly, how many years will it be before he has $700? How many years will it take if it is compounded continuously? Round your answers to decimal places. 57) Selina estimates that she will require $0,000 in four years in order to return to college to get an MBA degree. How much money should she invest now if it to earn 4.5% compounded continuously? 58) How long will it take Quenton to double his investment if he plans to put it in an account earning 5% annually and is compounded quarterly? 59) The half-life of silicon- is 70 years. If 00 grams is present now, how much will be present in 600 years? Round your answer to decimal places. 60) The life epectancy of a newborn child in Madagascar can be modeled by a linear function. In 985 the life epectancy of a newborn child in Madagascar was 50 years. By 000 the life epectancy had risen to 58 years. a) Let 0 Lt represent the life epectancy, to t represents the number of years after 975 and write a linear function that models the life epectancy of a newborn child in Madagascar t years after 975. b) Interpret the meaning of the slope of the linear model from part a). c) Interpret the meaning of the L-intercept of the linear model from part a). 6) For individuals filing their 04 income taes as Single and having no more than $89,50 of taable income, the ta bill was calculated as follows: 0% of their taable income up to $9075, plus 5% of any taable income between $9075 and $6,900, plus 5% of any taable income between $6,900 and $89,50. T that computes the 04 ta bill for a single filer with a) Write a piecewise-defined function taable income less than or equal to $89,50, as a function of the taable income. b) What is the ta bill for a single filer with taable income of $8,000, of $,000, of $89,000? 7

8 Problems Graph each of the following functions. Be sure to identify key points and all asymptotes, if they eist. 6) 5y0 6) 9 y 64) y 5 65) 5 y 66) f e 67) y ) f 4 69) f 70) f 5 4 7) f log 7) f 7) f log 74) f ln 4 75) f 76) f 4 9 if 4 if Problems Given the graph of y = f() to the right, describe the transformation(s) used to graph the following functions. Then graph the functions. 77) y f 78) y f 5 79) y f 80) y f 4 Problems Give all asymptotes, -intercepts, and y-intercepts for the following functions. Write asymptotes as equations of lines. Write intercepts as ordered pairs. If there is none, write none. 8) f 8) 54 4 f 8 6 8) f 84) f ) f 86) f 6 log 8 8

9 Problems Matching. Identify each of the following graphs as a type of function. 87) 9) 88) 9) 89) 94) 90) 95) 9) 96) 9

10 Problems Match each function to its graph, A U, on one of the net two pages. 97) f 5 f 08) f 98) 5 f 99) 09) f log 0) f 00) f 9 ) f 5 0) f log ) f 9 f 0) f ) 5 f 0) 6 f 4) f 04) 5 05) f f 5) 6 6) f 06) f f 7) 6 07) f 5 0

11 Graphs A U. Match some of these graphs to the functions in Problems A. B. C. D. E. F. G. H. I. J.

12 K. L. M. N. O. P. Q. R. S. T. U.

13 SOLUTIONS PART I: SHORT ANSWER SECTION ) False ) False ) True 4) False 5) True 6) False 7) They are parallel. 8) They are perpendicular. 9) No. The number 0, from the domain, would be paired with two different values of the range. 0) 0, 9 ) 0, 6 ),0 and,0 ) 0, 4),0 5) No. A parabola is not a one-to-one function. 6) 7) 0, 8) 4,9,6,5 9) No. The range contains only real numbers. 0), ) 4, ) Yes ), 4) 0, 5) Any graph that passes both the vertical line test and horizontal line test. 6) Any graph that passes the vertical line test but not the horizontal line test. 7) It is shifted to the right units and shifted up 5 units. 8) Minimum = ; eplanation required 9) No. is not in the domain of the function because of division by 0. 0) HA: y ; VA: 5 ) Maimum ) an eample is 54 0 ) log ab 4) 5) 6) 0 4 7) They are inverse functions of each other. 8) and 4 because log 8 log 0 log 6 9) Decay because the coefficient of t is negative. 40) y = kz 4) 4) k m n 4) ; 4 44) parallel 45) intersect 46) the same line PART II: THESE ANSWERS MUST BE SUPPORTED WITH APPROPRIATE WORK 47), 48),5 5, 49), 50) 8, 5), 5),,, 5), 54), 55) 56) 57) 5 58) 59)

14 60) 6) 6) 6) 5 i 64) 5 7i 65) i 66) 5 67) 9 40i 68) 69) 70) 8, 7), 7) 5, 64 7),,, 5 74) 75) 5 4, 4 76) 5 i, 5i 77) 8 78) 0 79) 80) 4 8), 4 8) No solution 8), 8 84) 5 85) ln7 ln7 ln 4 ln 4 86) 7ln 6 ln8 ln 6 87) No solution 88) 89) 5 ln 5 90) 5,4 9), 9) 0, 9) 94) 5 7 0,,,, y y 6 95) 7,, 7,,,,, 96) No solution 97) 4,,, 98), 99), 00) 5, 0) 7,4 0), 6, 0), 9 7, 04),8 05), 7, 06) 0, 4, 07) 08) 44 y 5 5 y 4 09) y 0) ) y 5 ) 8 ) 8 5 4) 5) 0 6) 5 7) 8) t 9) 0) ) ) 9 ) 0 4) 5) 7 6) 7) ) undefined 9) 4 0) 8 ) 6 ) ) 4) 8 f 4 5) 6) f 7) f 5

15 8) f 9) A 40) G 00 0 kc 4) I d 4) and, and, and, and 4) 5,, 5, 44).4 foot candles 45) ohms 46) 5 meters 47) feet by 7 feet 48) Pan pizza 0 calories Beef burrito 40 calories 49) Boat 0 kph, Stream kph 50) 5.75 acres 5) Copper 0. kg, Tin 5. kg 5) During year 07 5) $500; $,000,000 54),007 million 55) 7.5 hours 56) years; years 57) $6, ).95 years 59) grams 60) a) L or L b) The life epectancy of a newborn in Madagascar increased by approimately 0.5 years each year after 975. c) In 975 the life epectancy of a newborn in Madagascar was approimately years. 6) see answer after # 7 6) 6) 64) 65) HA: y 0 VA: 66) HA: y 67) 68) 69) 70) HA: y 0 VA: 4, 7) VA: 7) HA: y 0 5

16 7) VA: 0 74) VA: 4 75) HA: y VA:, 76) 77) 78) 79) 80). 8) HA: y ; VA:, 7 ; Int:,0, 0,0 8) HA: none; VA: none; Int: 0, ) HA: y 8; VA: none; Int: 0, 9 84) HA: none; VA: 0; Int:,0 85) HA: y 0 ; VA: none; Int: 0,0 86) HA: none; VA: ; Int: 64,0, 8 0, 6 log or ln 0, 6 ln8 87) J 88) H 89) B 90) K 9) F 9) G 9) L 94) E 95) I 96) C 97) O 98) F 99) M 00) B 0) T 0) C 0) J 04) G 05) D 06) N 07) P 08) K 09) U 0) R ) Q ) A ) E 4) L 5) I 6) S 7) H 6