Test 1 Review MATH 176 Part 1: Computer Part

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1 / Test Review MATH 76 Part : Computer Part. Daniel buys a new car for $54,000. The car is epected to last 0 years, at which time it will be worth $7,000. a) Write a function that describes the value of the car at any time during its lifetime. b) What is the domain of the function? c) Find the value when Daniel has the car for nine years. d) When will the car be worth $0,000? Write the complete sentence for the conclusion.. Lucy bought an ipod for $5. It is epected to last for 0 years, at which time it will be worth $80. a) Write a function that describes the value of the ipod at any time during its lifetime. b) What is the domain of the function? c) Find the value of the ipod when Lucy had it for three years. d) When will the ipod be worth $50? Write the complete sentence for the conclusion.. A manufacturer produces automatic cameras to retail outlets throughout the U.S. The cost function is C( ) thousand dollars. The revenue function for the manufacturer is R( ) thousand dollars where is thousand of cameras. a) Sketch the two functions on a graph paper. Use the same aises. Use Window: min: 0, ma: 0, ymin: 0, yma: 500. b) Show the equation for the break-even point(s). c) Determine the break-even point(s). d) Find the revenue and cost at the break-even point(s). e) Interpret the meaning from d) above. 4. The demand for a DVD player is given by D( ) dollars and the supply function for a DVD is given by S( ) 50. dollars where is the number of DVD in millions. a) Sketch the two functions. Use window: min: 0, ma: 5, ymin: 0, yma: 50. b) Find the equilibrium point. c) Find the demand and supply at equilibrium point. d) Interpret the meaning from c) above. 5. Market supply and market demand for calculators are given by S( ) 00 5 dollars and D( ).5 00 dollars where is the number of calculators produced in hundreds. Find the equilibrium point and interpret its meaning. Use window: min: 0, ma: 5, ymin: 0, yma: 00.

2 / 6. The power function model for increasing in iphone use is given by users since the year of 006. a) Estimate how many users will be at the year of 05? b) Approimate in what year there will be 45 thousand users?.5 f ( ) 0.78 thousand For the following rational functions, find a) sketch each function. b) the domain of each function. c) the vertical asymptote(s) if any. d) the horizontal asymptote if any. 7. f( ) f( ) 0 8. f( ) f( ) 4. You are an employee in the summer at a souvenir shop. The souvenir shop owner wants to purchase 650 printed sweatshirts from a company. The catalog contains a table of costs which are shown in the table below. The shop owner who has tried unsuccessfully for a week to contact the company, asks you to estimate the cost for 650 shirts. Assume the rate of total cost is constant. a) Find the rate of the change for the total cost from 00 purchased to 50 purchased. b) Interpret the answer in complete sentence from a) above. c) Find a linear model to fit the data. Make sure the units are included. d) Use the model from c) above to predict the cost for 650 shirts. e) Interpret the answer from d) above. f) How many shirts would have to purchase if the total cost is $550?. The per capita consumption of bottled water in the United States has increased dramatically in the past years. The accompanying table below shows selected years and the per capita bottled water consumption in these years. a) Based on the table at the right, find the change per capita bottled water consumption between 000 and 007. Interpret this number. Make sure including the unit. Number Purchased Total Cost (dollars) Year Bottled Water Consumption (Gallons Per Person)

3 / b) What is the percentage change per capita bottled water consumption between 995 and 00? c) Give the average rate of change of bottled water consumption between 99 and 00. Interpret the practical meaning of the result. d) Find power function model for the data since 990. e) Use the power function model from d) above to estimate bottled water consumption in 05. f) According to the power function model above, when will per capita bottled water consumption 5 or more gallons per person per year? Interpret the practical meaning of this result.. Consider the average hospital stays for selected years between 000 and 007 as shown in the following table. a) Find a cubic model for average hospital stay since 000. b) Estimate the average hospital stay in 00? 4. Suppose total cost in dollars to produce units of microwave oven is given by C( ) dollars where is the numbers of microwave oven produced. Year Average Stay (Days) a) Find the average rate of change of total cost when production changes from 0 to 00 units of microwave oven. Interpret this answer in complete sentence. b) Find the average rate of change of total cost when production changes from 00 to 00 units of microwave oven. Interpret this answer in complete sentence. c) Numerically estimate (use table) the slope of the cost with 00 units of microwave oven. Make sure you have two tables, one is from left and the other one is from right. Be sure to have the conclusion from these two tables. 5. For the function f( ) 4 For Converge: Key in ( ^ + ) [For ]; (4 ) [For > ] Use Window: min: 8, ma: 6, ymin: 8, yma: 5 a) Copy the graph. Make sure to label all the tic marks. b) Is f () continuous at =? Why or why not? Eplain clearly. c) Use numerically to show whether f () has derivative at =. 6. For the equation f ( ) 4 5 a) find the slope of the equation when = b) find the equation of the tangent line to the equation above when =

4 / 7. For the function Window: min: 4. ma: 4, ymin:, yma: a) Copy the graph. Make sure to label all the tic marks. b) Does limit eist at = for this function? Use numerical table to show work. c) Is f() continuous at =? Eplain clearly. d) Does f() have derivative at =? Eplain and show work. 8. During the first 4 months of employment, the monthly sales S (in thousands of dollars) for a new salesperson depends on the numbers of hours of training. Find lim S( ) and interpret the meaning of this result. 9 S( ) 0 where 4 4 Window: min:, ma: 0, ymin: 6, yma: 0 9. Suppose that the profit function for the monthly sales of a car by a dealership is P( ) dollars of profit where is the number of cars are sold. Window: min: 00, ma: 400, ymin: 0,000, yma: 75,000 a) What is the profit when 00 cars are sold for the month? b) What is the marginal profit when 00 cars are sold? Interpret the answer in complete sentence. c) What is the marginal profit when 00 cars are sold? Interpret the result The cost and revenue functions for production of table saws are C( ) and R( ) where is number of saws produced. a) Find the marginal cost. b) Find the marginal revenue. c) Evaluate R (500) and interpret the answer. d) Evaluate R (4500) and interpret the answer. e) Find the break-even point(s) and the regions of gain and loss. f) Find the profit function. g) Find the marginal profit. h) Evaluate P (500) and interpret the answer. i) Evaluate P (4500) and interpret the answer. 4

5 /. Find the slope and equation of the tangent line to f ( ) at = 8.. Use numerical tables on Converge to find each of the following limits. a) Find lim 0 4 c) Find lim b) Find lim d) Find 4 8 lim 4 6. Let f( ) 4 0. Use numerically to find f ()? 4. A company analyzes the production costs for one of the products and determines the hourly operating costs when units are produced each hour. The function for hourly cost in terms of production level is at the below. C( ) dollars when the production level is units per hour. Find the marginal cost when 40 units are produced. 5

6 / MATH 76 Test Review Part : Non Computer or Graphing Calculator Part. Sketch each graph of the following functions on the graph paper if the graph of the function ( ) f is follows, sketch the following graphs by shifting. a) f ( ). b) f ( ) 4. c) f ( ). For 5 4 if 0 f ( ) if if 5 find a) f ( ) =? b) f (0) =? c) f (4) =? d) f (7) =?. A truck rental company rents a moving truck for one day by charging $4 plus $0. per mile. a) Write the cost function of renting the truck where is miles driven. b) What is the cost of renting the truck if the truck is driven 76 miles? 4. An dishwasher repair service charges a $95 service call fee plus $45 per hour. a) Find the equation of total charge to a customer for a service call. b) How many hours the repair service used when the charge was $50? For each of the following functions, find a) domain. b) range. 5. f ( ) 6. f ( ) 4 7. f ( ) 5 6

7 / 8. Funds awarded to arthritis researchers by the Arthritis Foundation for selected years are given in the table on the right. a) Use the data to find the change in research Year funding from 996 to 006. b) Interpret the answer a) above. Make sure you include the unit. c) Use the data to find the percentage change in research funding from 996 to 006. d) Interpret the answer c) above. Make sure you include the unit. e) Use the data to find the average rate of change in researching funding from 996 to 006. f) Interpret the answer e) above. Make sure you include the unit. Research Funds (Millions of Dollars) Union membership as a percentage of the labor force can be modeled by M ( ) where is the number of years after 000 and M is membership as a percentage of the labor force. Find the rate at which membership is changing in Use limits of difference quotients (four steps) to find the derivative of the function such that a) f ( ) b) f ( ) 5 7 For the following problems 6, find the limit of each one.. lim lim 00. lim ( 8) 5. lim lim 4 6. lim 4 7

8 / 7. For the graph below, find the following a) lim f( )? b) lim f( )? c) lim f( )? d) f ( ) =? e) lim f( )? f) lim f( )? g) f () =? h) lim f( )? i) lim f( )? j) lim f( )? k) At what point(s) this function would not be continuous? Eplain clearly why or why not. l) At what point(s) this function would not have a derivative? Eplain clearly why or why not. 8. Based the graph on the right, a) Find lim f( )? b) Find f ( ) =? c) Is this function f () continuous at =? Eplain clearly. d) Is this function f () continuous at =? Eplain clearly. e) Is this function differentiable at =? Eplain clearly. f) Is this function differentiable at =? Eplain clearly. 8

9 / 9. Based on the graph below. a) List all the values which will make the function discontinuous. b) List all the values which will make the function not differentiable. c) List the intervals where the function has positive derivative. d) List the intervals where the function has zero derivative. e) List the intervals where the function has negative derivative. 0. For the function f() on the right, a) what interval would make a derivative to be positive? b) what interval would make a derivative to be zero? c) what interval would make a derivative to be negative?. Suppose that the cost in dollars for a company to produce pairs of a new line of jeans is C( ) dollars a) Find the marginal cost function. b) Find C (00). Interpret the meaning of this answer. 9

10 /. The cell phone manufacturing company finds that the profit from producing phones is given by P( ) dollars. a) Find the marginal profit function. b) Find the marginal profit when 00 phones are produced. Interpret your answer in complete sentence. For problems 0, find the first derivative of each following function f ( ) f ( ) 5 f ( ) f ( ) f ( ) f ( ) 9.8 f( ) f( ) f ( ) 5 6 For problems, evaluate the first derivative of the given function at the indicated value... f ( ) 4. 8 find f ( 5). f ( ) find f (4). 0

11 / Test Review Sheet MATH 76 Answer Key Part : Computer Part a) y( t) 50t b) domain: 0 0 c) y(9) = 50 (9) = $,850. In nine years, the car is worth $,850. d) y( t) 50t , t 0.. After 0 years, the car is worth approimately $0,000. a) y( t) 5 4.5t b) domain: 0 0 c) In three years, the ipod is worth $8.50. d) t After 9 years, the ipod is worth $50. a) sketch C( ) R( ) b) c).490 and.5.49 y d) when.5 y e) When produced.49 thousand (490) cameras, the cost and revenue are the same which is thousand dollars ($05,050). When produced.5 thousand (50) cameras, the cost and the revenue are the same which is thousand dollars ($40,840). 4 a) sketch b) c) D(5) =66; S(5) = 66 d) when produced 5 million DVD, the supply and demand will be $66 per DVD y 6.90 when sold 46 calculators, the demand and supply will be $6.90 per calculator. 6 a) f (9).49 There will be approimate thousand users in b) a) sketch.5 At the year 00, there will be about 45 thousand users

12 / b) domain: all real numbers c) no vertical asymptote d) horizontal asymptote: y 8 a) sketch b) domain: all real numbers and c) no vertical asymptote d) no horizontal asymptote 9 a) sketch b) domain: all real numbers but cannot equal to and 5 c) vertical asymptote at = 5 and = d) no horizontal asymptote 0 a) sketch b) domain: all real numbers but cannot equal to and c) vertical asymptote at = and = d) horizontal asymptote at y = 0 Total cost (dollars). 5 9 Number of shirts purchased a) b) This means that the total cost increased $.50 per shirt form 00 purchased to 50 purchased. c) Y( ) dollars cost where is number of shirts purchased. d) Y(650) = (since the output is whole number, we keep it whole number.) e) It would cost $69 to purchase 650 shirts. f) Y( ) = 550 and find. You have to use algebra to solve it and use TI-8 to check the answer where = shirts. You need to purchase 66 shirts if the total cot is $550.

13 / a) The bottled water consumption increased 6. gallons per person from 000 to 007. b) The percentage rate of change increased 6% gallons per person from 995 to 00. c) The average rate of change increased.4 gallons per person per year from 99 to d) Power function model: f ( ).589 bottled water consumption gallons per person where is the year since 990. e) Y(5) = 9.. In 05, we estimate the consumption for bottled water would be 9. gallons per person. f) You have to use algebra find the answer and you may use computer or TI-8 to check the answer. Y () = 5 and So, in the year of 00, we estimate the consumption of bottled water would be 5 gallons per person per year. a) The cubic model as follows Y ( ) days stay in a hospital where is the year since 000. b) Y(0) = 4.6 The average hospital stay would be about4 days in a) C(00) C(0) $6/ microwave This means that the total cost has increased $6 per microwave oven when produced from 0 to 00 units of microwave oven. b) C(00) C(00) $0/ microwave The total cost has increased $0 per microwave oven when produced from 00 to 00 units of microwave oven. c) From the left hand derivative: h 0 f ( h) f ( ) h Last est..000 C (00) $/ microwave oven From the right hand derivative: h 0 f ( h) f ( ) h Last est..000 C (00) $/ microwave oven

14 / Conclusion: C (00) $/ microwaveoven This means the total cost has increased $ per microwave oven when produced 00 units of microwave oven. 5 a) b) Yes, it is a continuous function since there is no gap and no jump at =. When = from the left hand side, we use the function f ( ) and f () = 5. When = from the right hand side, we use the function f ( ) 4 and f () = 5. c) Find the derivative from the left hand side. From the left hand side Derivative of f() The derivative from the left hand side: f () 4 Find the derivative from the right hand side. From the right hand side Derivative of f() The derivative from the right hand side is f () 4 The function when > is a straight line with the slope of 4. Since the derivative of the function from the left hand side and the right hand side all equal to 4 when =, f () has a derivative at = and f () 4 4

15 / f ( ) f 4 5 So, it passes the point (, 6) 6 We have to find the slope f ( ) 4 f 4 5 So the equation of the tangent line is y y m y 6 5 y a) b) lim f ( ) 8 lim f ( ) 0 Since the limit from the left hand side and the right hand side are NOT equal, the limit does not eist at =. c) f() is not a continuous function at = since there is a jump at the point. d) f() does not have derivative at = since it is not a continuous function at the point. 8. lim S( ) 5 8 After 8 hours of training for a new employee, the monthly sales approach to 5 thousands of dollars. 9 a) P(00) = When 00 cars are sold for the month, the profit for the car dealership is $59,900. b) P (00) 00 The profit increases $00 per car when 00 cars are sold. 5

16 / c) P (00) 00 The profit decreases $00 per car when 00 cars are sold. There is no profit to make more cars. As matter of fact, you lost money for making 00 cars. It could be the dealer has to pay more employees to sell the cars. 0 a) C( ) 6. The additional cost of producing one more saw is $6. b) R ( ) c) R (500) 0.. When 500 saws are produced, the additional revenue epected by producing one more saw is $0.0. d) R (4500) 9.7. When 4500 saws are produced, the epected revenue for producing one more saw decreases by $9.70. e) Break-even point. set C() = R() and solve for and (599, $0,794) and (644, $9,064). When 599 saws are produced, the cost and revenue are approimately the same as $0,794. When 644 saws are produced, the cost and revenue are approimately the same as $9,064. The company shows a loss if less than 599 saws or more than 644 saws are produced. f) P( ) R( ) C( ) P( ) g) P ( ) h) P (500) 4.. When 500 saws are produced, the company s profit is epected to increase by $4.0 with the production of one more saw. i) P (4500) 5.7. When 4500 saws are produced, the company s profit is epected to decrease by $5.70 with the production of one more saw.. f ( ) f ( 8) f ( ) f ( 8) 8 y 8 a) 0 lim.788 e b) 4 c) lim 0 d) lim 4 8 lim 4 6. f () = 5.97 from the derivative tables. C ( ) C (40).4 6

17 / When 40 units of product are produced, the cost of producing one more unit is $.4. MATH 76 Test Review Part : Non-Computer/Graphing Calculator Part Answer Key. Sketch a) b) 5 c) d) a) C( ) 4 0. b) C (76) 4.6. It cost $4.6 when driving 76 miles 4 a) C( ) b) The repair service used about hours. 5. domain: range: y 0 6. domain: 4 range: y 0 7. domain: 5 range: y 8a) = 7.7 b) The Research Funding for arthritis researchers increased $7.7 millions of dollars from 996 to 006. c) % 8.7% d) The Research Funding for arthritis researchers increased 8.7% between 996 and e) millions of dollars per year f) Between 996 and 006, the research funding increased at an average rate of $.77 millions of dollars per year. 9. The rate of change is M ( ) For 007, which is 7 years after 000, M (7).76. Union membership as a percentage of the labor force was increasing by.76% per year in Has to use the definition of limit of difference quotients to find the derivative. Do not use the power rules. a) f ( ) b) f ( ) 5.. lim lim ( 8) ( ) ( )

18 /. lim lim lim lim 4 4( ) ( ) 4 7 a) b) 5 c) does not eist d) does not eist e) f) g) h) does not eist i) j) k) =, = since there are gaps at these points, so it is not continuous at these points. l) =, = since it is not continuous at these points, there is no derivative at these points. 8 a) b) c) yes, there is no gaps nor open circle. The limit does eist and the function is defined at the point and they both equal to. d) yes, there is no gaps nor open circle. The limit does eist and the function is defined at the point and they both equal to 4. e) no, there is a sharp corner at the point. The derivative from left is negative and the derivative from the right is positive. The left derivative and right derivative are not equal. f) yes, there is no sharp corner and it is continuous at the point = so the derivative eists at that point. 9 a) =, 0, b) =, 0, c), 0, d) 0 e) there is no negative derivative. All the piece functions are increasing. 0. Derivative is positive at the interval 0 < <.5,.5 < < Derivative would be zero at = 0, =.5 =.5 Derivative is negative at the interval < < 0,.5 < <.5 a) C ( ) b) C (00). The marginal cost of 00 pairs of jeans would increase $ per jean. a) P ( ) b) P (00) (00) 50.0 When 00 phones are produced, the profit from one additional sale is $50.0. f ( ) f ( ) 4 4. f ( ) 7. f ( ) f ( ) 5 8. f ( ).7 8 8

19 / f ( ) 5 f ( ) 8 4 f ( ) f ( 5) 45. f (4)