Math 129 Past Exam Questions

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1 Math 9 Past Exam Questions Here are some questions that appeared on common exams in past semesters. This is not a sample exam, but it is a reasonable guide to the style and level of common exam given by the U of A Mathematics Department.. Simplify to the form +Ð>Ñ,Ð>Ñ3 / ( $ %3 ) > )3. Write $ $3 in the form V / with V and ) real numbers. 3. Let and be the complex numbers œ $ 3ß and œ 3Þ Write each of the following complex numbers (exactly) in the form +,3. a) b Ñ/ cñ / Ð$ 3 Ñ > 4. Write $ in the form +Ð>Ñ 3,Ð>Ñ where +Ð>Ñ and,ð>ñ are real. È$ a) Write the complex number Š 3 in the form V/ ) b) Write the complex number ˆ / % in the form C3 6. Write / > &> 3 in the form +Ð>Ñ 3,Ð>Ñ where + ( >Ñ and,ð>ñ are real valued functions. $ $ 7 Find the exact value of a) /. and b) sin( Ñ sin ( Ñ. 8. Find $ Ð ln Ñ sin Ð ln Ñ. (Hint: Use the substitution A œ ln ) 9. Find the Taylor polynomial of degree that approximates the function 0ÐÑ œ / near œ Þ cos$ Ð ln ÐÑÑ 0. Find. Hint: Start with the substitution A œ lnðñ. Write 5 5 3in the form V/ 3). Your answer must be exact Þ Sketch a graph of the region between the graph of C œ/ / sin and the -axis bounded by œ and œ $ and find its exact area. Spring 003

2 3 Þ A biologist hypotheses that a certain fungus spreads by growing in a rough circle. The radius of the circle grows at a rate inversely proportional to the area of the circle. If the fungus starts in a circle with a radius ft.. and has a radius of. ft one week later, find a formula for the radius of the circle after > weeks. 4. Let + 0. Find a).. + b) + + 5Þ Find a value for + so that the area between the -axis and the curve C œ Ð +ÑÐ +Ñ is exactly 8. X -a a Y 6. Find the exact volume of the solid obtained by revolving the region between the graph of Cœ% and the -axis about the line Cœ&ÞHint: use slices perpendicular to the -axis Þ Find the radius of convergence of the following power series: œ 5œ a) b) 8. Find a function C œ CÐÑ so that œ ÐC %Ñ and CÐÑ œ.c. (Do not worry about the endpoints) 9. Sketch the parabola Cœ( ÑÐ Ñand the curve Cœcos( Ñshowing their points of intersection. Find the area of the region between these two graphs. 0. Here is a slope field for the differential equation Spring 003

3 .C. œjðßcñ You must explain your answer for credit. a) Could Cœ be a solution to the differential equation? Yes No Explain: b) Could Cœ $ be a solution to the differential equation? Yes No Explain: c) Could Cœ sin( Ñ be a solution to the differential equation? Yes No Explain:. Circle the improper integrals that converge: Ð5 points off for each incorrectly marked integral up to a maximum of 0 points off.) È È a). b). c). d). e). f). Þ a) For what values of : does the improper integral.> converge? 0 b) For what values of : does the improper integral.> converge? > : > : 3. Mathematicians have found that the improper integral below converges and have determined its value ln(sin Ñ. œ lnðñ a) Explain why the integral in this formula is improper. b) The graph of 0ÐÑ œ ln(sin Ñbetween œ and œ is 4. Find the exact area of the region between the graphs of the two functions 0ÐÑ œ sin ÐÑ and ÐÑ œ Ð Ñ 5. Find sin Ð+Ñ. where + is a nonzero constant. 6. Find: Ðsin ÐÑ cos( Ñ cos ÐÑ sinðñ Ñ. 7 Þ Evaluate the following integral exactly: $ /. Spring 003

4 8. Find: 4 /. 9. Find Ð Ñ /. 30. Find a formula for the volume of the solid obtained by revolving one arc Ð0 ŸŸ Ñ of the curve Cœ + sin Ð+Ñabout the -axisþ Assume + Þ Shade in the area determined by this integral, and explain why the sign is negative in the formula. c) Use substitution and the formula to evaluate > ln(sin Ñ.> œ % $ 3. a) Evaluate the following integral exactly: /.. (You must show work to earn credit. Poor approximations will not earn partial credit.) b) Evaluate the following integral exactly: % ). (You must show work to earn credit.) Spring 003

5 3. Match the graph with the approximation it illustrates: A a b Right Hand Rule a b Left hand Rule C a b Midpoint Rule D a b Trapezoid Rule E a b Simpsons Rule 33. Evaluate the following integral exactly: / $. / $ $. Spring 003

6 34 Þ Does the following improper integral converge or diverge? % $ d) Ð $Ñ ) $ Give a reason for your answer; give a precise argument if you can. 35 Þ Does the following improper integral converge or diverge? You must give a complete justification of your answer. (Hint: You do not need to evaluate the integral to answer the question.) d ) È) You are given the following mathematical fact /. œ Use this formula and substitution to find an exact value for / 3. $ 37. The region bounded by Cœßœßand Cœ is revolved about the line, Cœ )Þ Sketch a picture of the solid, and find its volume. 38. Sketch a picture of the solid obtained by revolving the region bounded by the curve Cœ 4 and the line Cœ about the C-axis. Also find the volume of this solid. (Hint: you may want to use slices perpendicular to the C axis.) 39. Solve the following initial value problem for =Ð>Ñ:.= =.> > œ => where =ÐÑœ/ 40. Does the following improper integral converge or diverge? If it converges, evaluate it exactly. If it diverges, explain why. (Approximate answers may earn partial credit, but only if accurate and completely justified.) $ %. 4 Dead leaves accumulate on the floor of a forest at a continuous rate of 4 grams per square centimeter per year. At the same time, these leaves decompose at the rate of 60% per year. Write a differential equation for the quantity of leaves (in grams per square centimeter) at time >. Use this differential equation to find the amount of dead leaves (in grams per square centimeter) which represents equilibrium in the system. 4. Solve the following initial value problem for =Ð>Ñ: œ => where =ÐÑœ/.= =.> > Spring 003

7 43Þ The function 0ÐÑ œ È $ has a graph that looks like Approximate È $.to two decimal digits of accuracy. Give an argument that shows that your answer has the required accuracy that includes an upper bound and a lower bound on the exact answer. 44. In this problem, you will approximate the value of the definite integral.þ a) Use 30 divisions to find an approximation using: Right hand sums, Left hand sums, The midpoint rule, The trapezoid rule b) Use the above to give an upper and a lower bound on the exact value of the integral. c) Justify your answer to part b. 45. Recall that the arc length of the curve C œ 0ÐÑ from ( +, 0Ð+Ñ ) to (,, 0Ð,Ñ) is given by, + É ( 0 w ( ) ). Use this to approximate the length of the curve Cœln from (, ) to (%, ln %). The approximation should include a strict lower bound and a strict upper bound, and it should be accurate to at least two decimal places. Explain how you determined your answer. 46. A room with a southern exposure heats up during the morning. The temperature of the room increases linearly all morning so that it rises F every 5 minutes. Early in the morning, a cup of coffee with a temperature of 80 F is placed in the room when the room temperature is 60 F. Newtons law of cooling states that the rate of change in the temperature of the coffee should be proportional to the difference in temperature between the coffee and the room. a) Write a formula for the temperature of the room > minutes after the coffee placed there. b) Write a differential equation that the temperature of the coffee satisifies. c) Give specific initial conditions necessary to solve this problem. You do not need to solve the differential equation. Spring 003

8 47. iologists have introduced a new variety of fish into a lake. They began by releasing 000 fish. Their model predicts that the population will double in 8 months and then double again ( months later (45 months after the start.) Do you think that the biologists are using the exponential model.e.> œ5e or the logistic model.e.> œ 5EÐG EÑ for the population of the fish? Explain your answer 5=0 48. Find the exact value of Þ (Hint: Write the sum in terms of geometric series.) 49Þ a) Give an example of a convergent infinite geometric series. Explain why it converges, and say what it converges to. b) Give an example of a divergent geometric series. Explain why it diverges. 50. Find the Taylor polynomial of degree that approximates the function 0ÐÑ œ È $ near œ Þ You must show your work for full credit. 5. Find the Taylor polynomial of degree that approximates the function 0ÐÑ œ/ near œ Þ 5Þ As you should know, the Taylor Expansion of the sine function is $ & ( * $ sinðñ œ $x &x (x *x x $x ÞÞÞÞÞÞÞ Consider the function sinðñ if Á ÐÑ œ œ if œ ÐÑ Find ÐÑ, the tenth derivative of ÐÑ evaluated at œ. (You must show your work or explain your answer for credit.) 53. What is the radius of convergence of the power series $ % & 0ÐÑ œ % * & $ ÞÞÞÞÞÞ.C C. 54. Solve œ / sin Ð$Ñ for C œ 0ÐÑ when 0Ð Ñ œ Þ 55. What functions have the Taylor expansions given below? x Ð5Ñx 5œ 5œ 5œ 5 $ 5 $ Ð Ñ 5+ ( )( ) ( ) 5 Ð5+Ñx 5x 5œ 5œ a) b) c) Ð Ñ 5 d) e) 56. Find Taylor polynomials of degree 6 that approximate the following functions near œþ a) 0ÐÑ œ / b) ÐÑ œ $È cos c) ÐÑ œ Spring 003

9 5 $ 57. a )Does the series converge? You must give a mathematically valid reason for 5œ your answer to receive any credit. b) Does the series cos Ð5Ñ converge? You must give a mathematically valid reason 5œ for your answer to receive any credit. 5 $ 58. Give the Taylor series expansion of the following functions about œ. Give as complete an answer as possible. For example, the Taylor expansion of 0ÐÑ œ lnð Ñ should be written as either $ % & 8 8 0ÐÑ œ $ % & ÞÞÞ Ð Ñ 8 ÞÞÞ or 0ÐÑ œ Ð Ñ 8œ a) 0ÐÑ œ b) 0ÐÑ œ / c) 0ÐÑ œ sin d) 0ÐÑ œ cos 59ÞFind a function 0ÐÑso that 0ÐÑ œ 0 ÐÑ œ 0 ÐÑ œ 0 ÐÑ œ 0 ÐÑ œ 0 ÐÑ œ w ww www wwww wwwww. 60. Give the Taylor series expansion of the following functions about œ. Give as complete an answer as possible. For example, the Taylor expansion of 0ÐÑ œ lnð Ñ should be written as either $ % & 8 8 0ÐÑ œ $ % & ÞÞÞ Ð Ñ 8 ÞÞÞ or 0ÐÑ œ Ð Ñ 8œ a) 0ÐÑ œ bñ0ðñ œ cos cñ0ðñ œ sin ÐÑ d) 0ÐÑ œ 3 / 6. a) Give the Taylor series expansion about œ of the function 0ÐÑœsin e sure to give an expression for the 8-th term of the series. b) What is the 5-th degree Taylor polynomial approximation of 0ÐÑ œ sin near œ? 6. Consider the initial value problem,. œcln where CÐ/ÑœÞ Find a function CœCÐÑthat satisfies these conditions..c 63. A metallic rod 0 cm in length is made from a mixture of several materials so that its density changes along its length. Suppose that the density of the rod at a point cm from one end is $ÐÑ œ Þ& cos ( Ñ grams per cm of length a) Where is the rod the most dense? Where is it the least dense? b) What is the total mass of the rod? Spring 003

10 3 64. Simplify / to the form +,3. Give an exact answer. 0 > 65Þ Consider the function JÐÑ œ /.> w a) What is J ( )? b) Give the first 5 nonzero terms of the power series expansion of the function JÐÑnear œ. 66Þ A book of formulas states that for any : with : ß :. œ sin( Use this formula and the substitution œ+? to obtain a formula for..c 67. The following slope field for the differential equation. œ0ðßcñ was drawn on a hand calculator with the window set to &ŸßCŸ&Þ : Ñ : + Which of the following functions is most likely to be a solution to the differential equation? Please explain your answer. (If you can rule out some of the answers, include that in your explanation to increase your chances for partial credit.) $ $ a) Cœ b) Cœ/ c) Cœ d) Cœ e) Cœsin 68Þ a) For what values of : does the improper integral > :.> converge? b) For what values of : does the improper integral 0 >.> : converge? c) For what values of : does the improper integral.> converge? Ð> Ñ :.= = œ 69. Solve the following initial value problem for =Ð>Ñ:.> > * where =ÐÑ œ / $ 70 Find the exact value of sin( Ñsin ( Ñ. 7 Þ For what values of the parameter : does the following improper integral converge? 0 > :.> Give a reason for your answer; you must give an explanation to receive credit. Spring 003

11 7Þ A social scientist models the spread of a rumor using a differential equation. She will let TÐ>Ñ stand for the fraction of people who know the rumor at time >. She wants the reasonable properties that if this fraction is ever ß then it always will be. Also when the fraction is between and, then the fraction should grow. Finally if ever it is, then the fraction will remain. She intends to find a differential equation that models this behavior. a) Which of the following slope fields is most appropriate to the model? b) Which of the following differential equations is most appropriate to the model? (Whenever they occur, 5 and are positive constants.).t.t.t i) œ5t ii) œ5tð TÑ iii) œ5t Ð TÑ.>.>.>.T 5.T.T.> TÐ TÑ.>.> iv) œ vi) œ 5 T ÐT Ñ v) œ 5T ÐT Ñ 73. Suppose that at :00, noon, one summer afternoon, there is a power failure in your home in Tucson, and your cooling does not work. When the power goes out, it is 73 F in your house; the outside temperature is 08. At :00 pm, the temperature in your house has climbed to 85. Assume that the outside temperature is constant from :00 pm until 6:00 pm, and that your house obeys Newtons law of cooling. (Newtons law of cooling says that the rate of change of temperature of an object is directly proportional to the difference in temperature between the object and the ambient temperature.) Write a differential equation that the temperature of your home satisfies, and use it to predict the temperature of the house at 6:00 pm. Spring 003

12 .C 74 Consider the differential equation.c = JÐßCÑ. Suppose that its slope field is The slope field is representitive of all the features of the differential equation. Let 0ÐÑbe a solution to the differential equation. Which of the following statements are true.: w a) If 0Ð Ñ œ, then 0 Ð Ñ Þ w b) No matter what 0Ð Ñis, 0 Ð Ñ. w c) No matter what is, 0 ÐÑ. ww d) If 0ÐÑ œ, then 0 ÐÑ Þ (Note the change in ww e) No matter what 0ÐÑ is, 0 ÐÑ. ww f) No matter what is, 0 ÐÑ. value.) 75. Hydrocodone bitartrate is used as a cough suppressant. After the drug is fully absorbed, the quantity of drug in the body decreases at a rate proportional to the amount left in the body. The half life of hydrocodone bitartrate in the body is 3.8 hours. A dose of 0 mg. is administered. a) Write an initial value problem (a differential equation and a particular value) for the quantity U of hydrocodone bitartrate in the body at time > measured in hours after the drug is administered. b) Use the differential equation to find how much of the 0 mg. dose is still in the body after hours. 76. Give the Taylor series expansion of the following function about œ. Give as complete an answer as possible including an expression for the 8-th term. ÐThe first few terms correctly given may earn partial credit.) 0ÐÑ œ > /.>Þ 77. Find the Taylor expansion of 0ÐÑœ sin ÐÑto degree 7 about the point œþ You must show work to receive credit. 78Þ Let 0ÐÑ be a function so that 0ÐÑ œ, 0 ÐÑ œ, 0 ÐÑ œ, 0 ÐÑ œ ) and wwww 0 ÐÑœ%. Find the Taylor approximation of degree 4 of 0ÐÑnear œþ w ww www Spring 003

13 .C 79. A first order differential equation of the form. œjðßcñ has the slope field given below Which of the following could be the graph of a solution equation. (There may be more than one.) - CœCÐÑto the differential Þ A water tank has the shape of a right circular cone. The top of the tank is a circle with a radius of 8 ft. and the tank has a depth of 5 ft. The tank is filled to a depth of 0 ft. How much work is required to pump all the water out of the tank to a level equal to the top of the tank? ( The density of water is 6.4 lbs/ft $ Þ) (You must show your work to obtain full credit even for correct answer.). 8Þ Use the integration formula cos œ tanð Ñ G to evaluate exactly. % % cos Ð $ Ñ 8. Find a solution =Ð>Ñ to the initial value problem: w $ =Ð>Ñœsin Ð%>Ñ where =ÐÑœ Spring 003

14 83. Consider the initial value problem.c. œ 0 C ; CÐÑ œ $ Use Eulers method to fill in the following table of values for a solution CœCÐÑ. Please give some explanation of your calculations. (There are extra cells in the table for your convenience.) Þ Þ Þ Þ$ C 84. Find the volume of the solid obtained by revolving the shaded region about the axis y= - x+ 3 y= x The rate at which barometric pressure decreases with altitude is proportional to the pressure at that altitude. If the barometric pressure is measured in inches of mercury and & the altitude in feet, then the constant of proportionality is Þ a) Find a differential equation that expresses the relationship described above. b) Suppose that the barometric pressure at sea level is 9.9 lbs/in. What is the pressure outside an airplane flying at 0,000 ft? 86. A cylindrical water tank is half filled with water. (The tank is standing on its circular base.) The tank has a radius of 5 ft. and a height of 30 ft. How much work is required to pump the water to a level 6 ft. above the top of the tank? The density of water is 6.4 lbs/ft $ 87. A banner in the shape of an isosceles triangle is hung from the roof over the side of a building. As a triangle, the banner has a base of 5 ft. and a height of 0 ft. The banner is made from material with a density of 5 lbs per ft. Set up an integral to compute the work required to lift the banner onto the roof of the building. Evaluate the integral to find the work. Spring 003

15 88Þ Match the slope field with the differential equation: C.C.C $ $... A) œ ) œðc Ñ C) œ( C ÑÐC Ñ.C.C C. C. œ D) œ E) 89. Newtons Law of Gravitation says that the force of gravity between two objects of mass Qand 7 a distance apart is Jœ K7Q where K is the gravitation constant. Assuming that there are only two objects some distance apart, does it take an infinite amount of work to move one of the objects an infinite distance from the other? Explain your answer completely; a Yes or No answer will not earn credit unless there is a mathematical explanation. Spring 003

16 .C 90Þ A differential equation. œ 0Ðß CÑ has C œ sin( Ñ as a solution. Which of the following slope fields could be the slope field of the differential equation? YES NO YES NO YES NO YES NO YES NO 9. Match the function with the correct Taylor expansion a) Ð Ñ 5 5 % $ Ð Ñ i) * $ %* $ Ð5 Ñx 5œ 5 %5 (5+) x 5œ0 5 b) Ð Ñ Ð Ñ ii) sin Ð Ñ c) Ð Ñ iii) sin( Ð Ñ Ñ d) 5œ % Ð Ñ Ð Ñ 5œ 5 5 iv) Spring 003

17 9. Find the volume of the solid whose base is the region in the C-plane bounded by the curves Cœ and Cœ) and whose cross sections perpendicular to the -axis are squares with one side in the C-plane. 93 An engineer estimates that the amount of work necessary for a certain task is given by 3 ln. Does this require a finite or an infinite amount of work? YOU MUST JUSTIFY YOUR ANSWER TO RECEIVE ANY CREDIT. 94Þ There are a number of functions in Mathematics named after the Russian Mathematician Chebyshev. One is usually written as XÐÑÞ & Its domain is all real numbers. A part of its graph, for from to Þ&, looks like True or False: Circle the correct answer Þ Þ a) XÐÑ.Ÿ & XÐÑ. & XVYI JEPWI Þ4 0Þ5 b) XÐÑ.Ÿ & XÐÑ. & XVYI JEPWI Þ Þ c) XÐÑ.Ÿ & axðñ & b. XVYI JEPWI d) XÐÑ. & 0 XVYI JEPWI e) lx ÐÑl. XVYI JEPWI & 95Þ Evaluate ( / /% /. Hint: Use the substitution? œ/ Þ You must show each step in the work to receive credit 96. Show your work as you find a) /. b) /. c) /. Spring 003

18 97. An object moves along the real line. Let =Ð>Ñ be the position of the object at time > seconds. Match different assumptions about the motion of this object with the Differential Equation that correctly reflects the assumptions. Throughout, 5, 6and 8are taken as positive constants. The velocity of the object ww is directly proportional to the a) = Ð>Ñ œ 5 time it has been in motion. The acceleration of the object is directly proportional to the time it has been in motion. The velocity of the object is directly proportional to its position The acceleration of the object is directly proportional to its position w b) = Ð>Ñœ5> w c) = Ð>Ñ œ 5 =Ð>Ñ ww d) = Ð>Ñ œ 5 =Ð>Ñ The acceleration of the object ww is a linear function of its e) = Ð>Ñ œ 5> velocity. The acceleration of the object is ww w a linear function of its velocity f) = Ð>Ñ œ 5 = Ð>Ñ 6 and its position. The acceleration of the object ww w is constant. g) = Ð>Ñ œ 5 = Ð>Ñ 6 =Ð>Ñ A cylindrical barrel, standing upright on its circular end, contains muddy water. The top of the barrel, which is open, has a diameter of meter. The height of the barrel is.8 meters, and the depth of the water in the barrel is.5 meter. The density of the muddy water varies with the depth of the water, and is given by 3ÐÑ œ Ð 5 Ñ kg m $ where is the depth measured as the distance to the surface (from the top to the bottom), and 5 is a positive constant. Find the work necessary to pump the muddy water to the top rim of the barrel. (You may leave constants like 5, and (the acceleration due to gravity) in your answer unevaluated.) Spring 003

19 99. Find the radius of convergence of the power series: 5 a) 5 b) ) 5 Ð Ñ $ $ 5œ 5œ 5 5.C. C 00. Solve œ where CÐÑ œ and describe the graph of the solutionþ $ 0. The graph of the function 0ÐÑœ $ is If the region bounded by this curve, the -axis, œ and œ is revolved about the - axis, what is the volume of the resulting solid? Give exact volume for full credit. 0. Do these improper integrals converge or diverge: a) 0. Converges Diverges b) È. Converges Diverges 3 $ c) È. Converges Diverges $ 0 $ d) È. Converges Diverges $ 03Þ Give the series expansions of the following functions include the radius of convergence with your answer: a) 0ÐÑ œ sinðñ b) 0ÐÑ œ cosðñ c) 0ÐÑ œ d) 0ÐÑ œ / Spring 003