Math 080 Final Exam Review

Transcription

1 Math 080 Final Ea Review Note: If you have difficulty with any of these proles, get help, then go ack to the appropriate sections in the tet and work ore proles! For proles through 0, solve for, writing solutions of inequalities in interval notation: 7. ( 5) = ( ). = 5 0. < 7. 6 = 5. a = t 6. 5 < 7 7. = 5 8. ln = 9. = <. 8 =. = 6. log8 () log8( ) = = 0 6. = = 9. or 0. e = For proles through, solve each syste:. 5 y =. y z =. y z = 57 y = 5y z = y = 6 7y z = z = 6 For proles through 7, perfor the indicated operations and/or siplify as uch as possile:. ( 8 ) 5. (8 9 6) ( 6) 6. ( )( ) 7. a a 8. 5 ( ) ( ) 6 ( ) y y. a a

2 ( ) ( 8) ( a) ( y ). ( 8 9 ) ( ). ( 6) (7) a 7 7. a 8 For proles 8 through 5, siplify y rationalizing the denoinator: 6 a For proles 5 through 5, find the eact value of each logarith: 5. log 8 5. log log ( ) Find, algeraically, the verte of the paraola f() = 5. Find the aiu or iniu value of the function and indicate whether the value is the aiu or whether it is the iniu. 56. Find, algeraically, the -intercept(s) of the graph of f() =. Give eact, siplified values and then round your answers to two decial places. 57. Solve y copleting the square: 9 5 = Find the doain (in interval notation) of f() =. 59. Find the doain (in interval notation) of g () =. For proles 60 through 65, factor copletely: y 6y y 65. 5c 6d 000 y

3 ***For proles 66 through 7, graph on a rectangular coordinate syste (use graph paper). Additionally, for proles 67, 68, 69 and 70, give the doain and range of each function y reading it fro your graph. 66. y = 67. f() = ( ) g() = 69. p() = 70. F() = 7. = 7. Find the slope and y-intercept of y =. 7. Find the slope of the line passing through the points ( 6,) and (8,). 7. Write the equation of the line passing through the point (,5) and a) parallel to y = 5 ) perpendicular to y =. 75. If f() = 7 and g() =, find ( f o g)(). Siplify. 76. If f() =, find a forula for f (). 77. Find f(5) if f() =. Siplify Find f( h) if f () =. Siplify. 79. Use properties of logariths to write as a single logarith: 5 log a log log c. 80. Use properties of logariths to write as a su/difference of sipler logariths: log y For proles 8 and 8, perfor the indicated operations with the cople nuers: i 8. (7 i)(5 i) 8. i For proles 8 through 00, use algeraic ethods consistent with what your instructor epects. 8. An airplane can travel 00 iles against the wind in the sae tie it travels 00 iles with the wind. If the speed of the wind is 5 ph, what is the speed of the plane in still air?

4 8. Using straight line depreciation, the value V of a particular piece of office equipent years after purchase is given y the linear function V() = 5, for 0 8. a) Identify the independent and dependent variales and eplain what each represents. ) What is the doain of this function? c) What is the value of this office equipent years after it is purchased? d) After how long will the value of this equipent e $0? 85. An oject falls fro a window 00 feet high. Its distance aove the ground is given y the function d(t) = 00 6t, where t is in seconds and d(t) is in feet. a) Find the height of the oject after seconds. ) How long does it take the oject to hit the ground? 86. Find the diensions of a rectangular floor if its width is foot less than half its length and if the floor can e covered y square feet of carpeting. 87. Suppose that a 750 square foot apartent rents for $550 and that a 050 square foot apartent rents for $95. Suppose that the relation etween the size (nuer of square feet) of an apartent and rent is linear. a) Find a linear function that epresses the cost C of renting an apartent as a function of its size s. ) Interpret the slope. c) Predict the rent of a 900 square foot apartent. d) If the rent of an apartent is $800, how ig (nuer of square feet) would we epect it to e? 88. A student drives 0 iles at a constant speed, ut she will e late for her Interediate Algera final ea unless she drives 0 ph faster for the last 0 iles. If her total travel tie is hour, what is her original speed? 89. Using data fro 960 to 000, the function N () = represents the approiate nuer of registered cliers at Mt. Rainier years after 960. (Source: National Parks Service, U.S. Dept of the Interior) a) Identify the independent and dependent variales and eplain what each represents. ) Evaluate N() and eplain what it represents. c) There were 97 registered clier on Mr. Rainier in 00. Copare this value to your result in part () and coent on any differences. 90. A an rows 0 iles downstrea and return in hours and 0 inutes. If he can row ph in still water, what is the rate of the current? 9. A usinessan received a total of $0 in interest for the year on two investents. If one of the investents paid 7% annual interest and the other paid % annual interest, how uch was invested at each rate of the total aount of oney invested was $,800?

5 9. If each works alone, it takes one clerk twice as long as another clerk to stuff a atch of envelopes. Together, the two clerks can do the jo in hours. How long does it take each clerk, alone, to stuff the envelopes? 9. A copany offers two types of health plans to its eployees. Plan A pays 90% of an eployee s edical ills after a $00 deductile, while Plan B pays 80% of an eployee s edical ills after a $50 deductile. For what aounts of edical ills is Plan A etter for an eployee than Plan B? 9. A new car has a sticker price of $,590. Suppose that the dealer arkup on this car is 5%. What was the dealer s cost, to the nearest dollar? 95. The outside diensions of a picture frae are 0 inches y inches. The area of the picture within the frae is 008 square inches. Find the width of the frae. 96. An investor split $60,000 aong three anks. He received 5%, 6%, and 7% in interest on the three deposits. In the account earning 7% interest, he deposited twice as uch as in the account earning 5% interest. If his total earnings were $760, how uch did he deposit in each ank? 97. It takes hours for a oat to travel 8 iles downstrea. The sae oat can travel 8 iles upstrea in hours. Find the speed of the current and the speed of the oat in still water. 98. A sall copany found that when their product is sold for a price of dollars, the weekly revenue in dollars as a function of the price is R() = For what selling price will the weekly revenue e aiized, and what is the aiu weekly revenue? 99. Based on data fro the Kelly Blue Book, the value V of a Dodge Stratus that is t t years old can e odeled y the function V (t) = 9,8(0.8). According to the odel, when will the car e worth $,000? Round answer to the nearest hundredth. 00. A wood craftsan akes children s rocking horses. He sells each rocking horse for $95. His onthly fied costs of operating his usiness are $500, and it costs $5 in aterials for each rocking horse. a) Find the revenue function R treating the nuer of rocking horses as the independent variale. ) Find the cost function C treating the nuer of rocking horses as the independent variale. c) Find the reak-even nuer of rocking horses that ust e anufactured and sold. What is the revenue and cost of this nuer of rocking horses?

6 Solutions:. 6. no solution. (, )., , 8... (, ] [, ) 6. log e 9. ± i 0. ( 9,]. U 5. log or log.585) 9. (, 8] [ 7, ) ±, ± i 6 a 6. (,) t. U 8. ± 9 ( 7. (, ] [, ) U 0. ln (.85). (,). (,,). ( 60,8, 66) a ( )( ).. ( ) ( 5)( y). 5a (a )(a ) i a y y y a 0a 7. a a verte = (,7), aiu value = and ; approiately.8 and ± 58. (, ) U (,) U (, ) 59. [, ) 60. ( )( )( ) 6. ( 5 y) 6. (0 y)(00 0y y ) 6. ( )( ) 6. (8 y )(8 y ) 65. (c d)(9c 6cd d )

7 For proles 66 through 7, see instructor for graphs. 7. slope =, y-intercept = or (0, ) 7. 7a) y = ) y = 75. (f o g)() = 76. f () = h a log c 80. log log log y i 8. i speed is 75 ph 8. a) independent variale is, which represents the tie in nuer of years; dependent variale is V, which represents the value of the office equipent ) [0,8] c) $75 d) after 8 years 85. a) 6 ft ) 5 seconds 86. length is 6 ft, width is 7 ft 87. a) C =.5s 6.5 ) slope =.5, and this eans that the cost of renting an apartent is increasing y $.5 for each square foot increase in its size c) $77.50 d) 90 square feet 88. original speed = 0 ph 89. a) independent variale is, which represents the nuer of years since 960; dependent variale is N, which represents the nuer of registered cliers at Mt. Rainier ) approiately,507; this represents the nuer of registered cliers in the year rate of current = ph 9. $590 at 7% and $6880 at % 9..5 hours and 9 hours 9. edical ills ust eceed $ dealer s cost was $8, 95. width is inches 96. $6,000 at 5%, $,000 at 6%, $,000 at 7% 97. speed of current = ph and speed of oat in still water = 0 ph 98. selling price to aiize revenue is $0; aiu revenue is $8, in aout.7 years 00. a) R() = 95 ) C() = c) 75 rocking horses; $75