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Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingInstructionsShow all w...
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LAC2013 UNIT preTESTs!

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  1. 1. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingInstructionsShow all work justifying each answer in the space provided.Your answers must be supported analytically, graphically, numericallyand/or verbally! All answers must be stated in exact and simplest form ifpossible.An answer, N, that cannot be so stated, should be estimated to threedecimal places, for example:{∀∀∀∀N | (|N|=0.001) ∩∩∩∩ (|N| < 1000)}-.0861 = -0.086,125.2335 = 125.234.Any approximations, N, such that |N| ∉ [0.001, 1000) must be stated inscientific or engineering notation rounded to three decimal places, for example:{∀∀∀∀N | (|N|<0.001) ∪∪∪∪ (|N|=1000}.000123 = 1.230 x 10-4 or 123.000 x 10-6,-34567.8 = -3.457 x 10 4 or -34.568 x 10 3.All graphs must be complete sketches including neatly labeled axes,origin, scales, intercepts, asymptotes and function(s) or relation(s).
  2. 2. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpageTBQ(1) A function f is defined on the closed interval from -4 to 3and has the graph shown above.(1a) Sketch the entire graph of y = |f(x)|.
  3. 3. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(1b) Sketch the entire graph of y = f(|x|).
  4. 4. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(1c) Sketch the entire graph of y = f(–x).
  5. 5. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(1d) Sketch the entire graph of y = 2f(x + 1).
  6. 6. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(1e) Is the graph of |f(x)| continuous at x = 1?
  7. 7. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(1f) Does the derivative of f(–x) exist at x = 2?
  8. 8. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpageTBQ(3) Let f(x) = x4 – 4x2 and g(x) = x3 – x.(3a) Write an expression for h(x) =f(x)g(x).(3b) Find all the asymptotes of h(x). Justify your answer analytically.
  9. 9. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(3c) State the domain and range of h.(3d) Make a complete sketch of h.
  10. 10. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(3e) Test h(x) analytically for all three symmetries.
  11. 11. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(5) Equal squares of side length x are removed from eachcorner of a 20” by 30” piece of aluminum. The sides are turned up toform a box with no top.(5a) Make a drawing of the box and label each dimension in terms of x.(5b) Write an expression for the volume V as a function of x.(5c) Make a complete sketch of V(x). Use your sketch to show when thevolume of the box is 750 in3.
  12. 12. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(5d) State the domain and range of V(x).(5e) Find the maximum volume possible in this model. Describe yourprocedure.(5f) What sub domain and range make sense in the context of this model?
  13. 13. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(7) The following implicitly defined relation describes a conicsection on the XY-plane:f(x,y) = Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.Let: f(x,y) = 9x2 + 0xy + 4y2 – 18x + 16y – 11 = 0.(7a1) Use factoring and completing the square to rewrite f(x,y) = 0 as:(x – h)2a2 +(y – k)2b2 = 1.(7a2) State the following: Center Point: __________Horizontal VerticalTangents: __________ Tangents: __________Semi-Minor Semi-MajorAxis Length: __________ Axis Length: __________Domain: __________ Range: __________(7a3) Make a complete sketch of f(x,y) = 0 using function mode on your GC.State y1 and y2 as explicit functions of x for the labels of your curve.
  14. 14. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(7b1) Let B = 8, D = E = 0 and F = –36 such thatf(x,y) = 9x2 + 8xy + 4y2 – 36 = 0.Rewrite f(x,y) = 0 in polar form as r = f(θ).(7b2) Make a complete sketch of r = f(θ) using polar mode on your GC.(7b3) Calculate B2 – 4AC.
  15. 15. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(7c1) Let B = 12 such that f(x,y) = 9x2 + 12xy + 4y2 – 36 = 0.Rewrite f(x,y) = 0 in polar form as r = f(θ).(7c2) Make a complete sketch of r = f(θ) using polar mode on your GC.(7c3) Calculate B2 – 4AC.
  16. 16. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(7d1) Let B = 16 such that f(x,y) = 9x2 + 16xy + 4y2 – 36 = 0.Rewrite f(x,y) = 0 in polar form as r = f(θ).(7d2) Make a complete sketch of r = f(θ) using polar mode on your GC.(7d3) Calculate B2 – 4AC.
  17. 17. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(7e1) Make a complete sketch of 9x2 + 4y2 – 36 = 0.(7e2) What happens when you change the values of D and E?(7e3) Explain what the value of B2 – 4AC predicts for the behavior of a conic.
  18. 18. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpageTBQ(9) Use the Difference Quotient to find the derivative off(x) = cos(x).(9a) State the classic definition of the derivative as a limit.(9b) Apply the definition of the derivative to f(x).Show that f ’(x) =limh→0 cos(x)cos(h) – 1h – sin(x)sin(h)h .NB: cos(α + β) = cos(α)cos(β) – sin(α)sin(β)
  19. 19. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(9c) Make a numerical argument to findlimh→0sin(h)h.
  20. 20. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(9d) Make a numerical argument to findlimh→0cos(h) – 1h.(9e) Evaluatelimh→0 cos(x)cos(h) – 1h – sin(x)sin(h)h and state f ’(x).
  21. 21. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(9f) Find an equation, in point-slope form, of the line tangent to y = f(x) atx =π2. Restate this equation in slope-intercept form.(9g) Find an equation, in point-slope form, of the line normal to y = f(x) atx =π2. Restate this linear equation in slope-intercept form.
  22. 22. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(9h) Make a complete sketch, ∀x ∈ [-2π, 2π], of f(x) as well as the tangent andnormal lines at x =π2.(9i) Find the area of the region enclosed by the y-axis, the tangent and thenormal. Give both the exact analytical answer and the numericalapproximation.
  23. 23. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(11) Let f be the piece-wise defined function:f(x) =-x2 + C for x < 1-2x + 5 for x ≥≥≥≥ 1(11a) For what value of C will f be continuous at x = 1? Use the definition ofcontinuity at a point to justify your answer.
  24. 24. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(11b) Use the value of C you found in part (11a) to determine whether f isdifferentiable at x = 1. Justify your answer using the definition of thederivative at a point.(11c) Let C = 3. Determine whether f is differentiable at x = 1. Justify youranswer.
  25. 25. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(13) Consider the relation defined implicitly byx2 + xy + y2 = 50.(13a) Write an expression of the formdydx = f(x, y) for the slope of the curve atany point (x, y).(13b) Find all x and y intercepts. Write an equation of the tangent line ateach of these points.
  26. 26. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(13c) Find all points of vertical and horizontal tangency. Write the equationof each tangent line.
  27. 27. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(13d) Convert the relation to polar coordinates: r = f(θ). Use polar mode tomake a complete sketch of the curve. Include all the points you found in(b) and (c) in your sketch.
  28. 28. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpageTBQ(15) On Monday, you started your career as a new employee inthe marketing division at Pepsi Cola’s Worldwide Headquarters. OnTuesday, the VP of manufacturing asks you to redesign the 12 ouncesoda can.(15a) Write an equation for A (the amount of aluminum used to manufactureany cylindrical can) in terms of R (the radius of the base) and H (theheight).(15b) Write an equation for V (the amount of soda in any cylindrical can) alsoin terms of R and H.(15c) On Wednesday, marketing research found that the most popular 12ounce soda can has the following approximate dimensions: R = 3.068 cmand H = 12 cm. Using this data, find the volume of the can currently inproduction to the nearest cubic centimeter.
  29. 29. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(15d) Now, manufacturing wants to save money by minimizing the aluminumused, but marketing needs to sell the same amount of soda as they dowith the present can. Rewrite A in terms of the single variable R.(15e) FinddAdR and use it to find the value of R which minimizes A. Confirmthat you have minimized A and state the new dimensions, R and H, ofthe can.
  30. 30. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(15f) A week from Thursday, a survey was conducted while test marketingthe new can on LI. One question on the survey was:Describe the look and feel of the new can.Do you like the new design? Why or why not?Based on the dimensions of the can you found in (e), how do you thinkmost test subjects answered this question?(15g) By next Thursday, manufacturing has implemented your new designnationwide. You’re feeling pretty proud of yourself. You may already beup for a promotion or even a raise! The next day, you get email from theVP of Advertising with the subject line: “YOU’RE FIRED!!!” Explainwhy advertising is so upset with you that your career at Pepsi Cola isdoomed after only 3 weeks on the job!
  31. 31. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(17) Let f(x) = 1 – tan(x -ππππ2), ∀∀∀∀x ∈∈∈∈ [0, ππππ].(17a) State the general form of the recursive equation describing Newton’sMethod for Roots.(17b) Apply the Quotient Rule of Differentiation to findddx (tan(x)).(17c) Apply the recursion formula to f(x).
  32. 32. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(17d) Let x0 =π2. Write an equation of the tangent line to f(x) at x0. Use thisequation to find x1.(17e) Use prgmNEWT and Mode Fix 9 to approximate the root of f(x) nearestto x0 by completing the following convergence table.n xn0 1.570796327
  33. 33. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(17f) Find43of your best estimate to the root from part (e).(17g) State the exact root based on part (f).(17h) Make a complete sketch of f(x) and the tangent line at x0. Label x0, x1andlimn ∞xn on the x-axis.
  34. 34. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpageTBQ(19) The volume V of a cone is increasing at the rate of 28ππππft3s.At the instant when the radius r of the cone is 3 ft, its volume is 12ππππ ft3and the radius is increasing at12fts.(19a) At the instant when the radius of the cone is 3 ft, what is theinstantaneous rate of change of the area of its base?(19b) At the instant when the radius of the cone is 3 ft, what is theinstantaneous rate of change of its height?(19c) At the instant when the radius of the cone is 3 ft, what is theinstantaneous rate of change of the area of its base with respect to itsheight.
  35. 35. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(21) Let F(x) = ⌡⌠0xsin(t2)dt , ∀∀∀∀x ∈∈∈∈ [0, 1].(21a) Write out the trapezoidal rule with four equal subdivisions (TRAP4) onthe closed interval [0, 1] and use it to approximate F(1).(21b) Make a complete sketch of F(1) and the trapezoids in question.
  36. 36. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(21c) Complete the following convergence table using program lrte() and FIX4to estimate F(1).N LSUM RSUM TRAP141664(21d) Discuss the behavior of LSUM vs. RSUM.(21e) Discuss the behavior of TRAP.
  37. 37. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(21f) Theorem: If f(x) is continuous, monotonic (increasing or decreasing, notboth) and positive on the closed interval [a, b], then the maximum errorin the Trapezoidal Rule approximation for the area under the graphof f is:|f(b) - f(a)|∆t2 where ∆t =(b - a)n .Use the given theorem to find the minimum value of n such that 3decimal place accuracy in the TRAP column is assured.(21g) On what interval is F increasing?(21h) If the average rate of change of F on the closed interval [1, 3] is k, find⌡⌠13sint(t2)dt in terms of k.
  38. 38. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpageTBQ(23) Let f be a function defined ∀∀∀∀x > -5 having the followingproperties:(i) f ’’(x) =13 x + 5(ii) the line tangent to the graph of f at (4, 2) has an angle ofinclination of 45°.Find an expression for f(x).
  39. 39. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(25) Consider the curve y = x3 on the closed interval x ∈∈∈∈ [0,12].(25a) Let l represent the length of the curve. Set up, but do not integrate, anintegral expression of a single variable to calculate l.(25b) Rewrite the integral for arc length in terms of the numerical integratorbuilt-in to your GC.. Use your GC to estimate the arc length.
  40. 40. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(25c) Let S represent the area of the surface generated by revolving the givencurve about the x-axis. Find S analytically by setting up andintegrating a definite integral. Your solution must include an anti-derivative.(25d) Rewrite the integral for surface area in terms of the numericalintegrator built-in to your GC. Use your GC to confirm your calculationfor S.
  41. 41. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(27) A solid is constructed so that it has a circular base ofradius r centimeters and every plane section perpendicular to acertain diameter of the base is a square, with a side of the square beinga chord of the circle.(27a) Find the volume of the solid, V = f(r). Hint: use the relation x2 + y2 = r2.
  42. 42. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(27b) If the solid described in part (a) expands so that the radius of the baseincreases at a constant rate of12cmmin, how fast is the volume changingwhen the radius is 4cm.
  43. 43. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(29) A particle moves on the x–axis so that its position at anytime t ≥≥≥≥ 0 is given by x(t) = 2te-t.(29a) Find the acceleration of the particle at t = 0.(29b) Find the velocity of the particle when the acceleration is 0.(29c) Find the total distance traveled by the particle from t = 0 to t = 5.
  44. 44. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpageTBQ(31) Let P(t) represent the number of wolves in a population attime t years, t ≥≥≥≥ 0. The population P(t) is increasing at a rate directlyproportional to 800 – P(t), where the constant of proportionality is k.(31a) If P(0) = 500, P(t) in terms of t and k.(31b) If P(2) = 700, find k.(31c) Findlimt→∞P(t).
  45. 45. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(BC33) Consider the following differential equation.dPdt=11 + t2 t ≥≥≥≥ 0(BC33a) If a solution curve includes the point (0, 0), complete the tablebelow to estimate the value of P(1). Use Euler’s method with∆t = 0.1 and FIX6.t P ∆P0.0 0.0000000.10.20.30.40.50.60.70.80.91.0
  46. 46. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(BC33b) Refine your estimate of P(1) by completing the followingconvergence table (use FIX6).∆t P(1)0.10.010.001State your best estimate for P(1): _______________.
  47. 47. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(BC33c) Solve the given differential equation analytically for P = f(t).State the exact value of P(1): _______________.State the error, in FIX6, of your best estimate: _______________.
  48. 48. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpageTBQ(BC35) A certain rumor spreads through the community at theratedydt= 2y(1 – y), where y is the proportion of the population that hasheard the rumor at time t.(BC35a) What proportion of the population has heard the rumor when it isspreading the fastest?(BC35b) If, at time t = 0, ten percent of the people have heard the rumor,find y as a function of t.(BC35c) At what time t, is the rumor spreading the fastest?
  49. 49. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(BC35d) Graph y(t) and describe the long-term behavior of the population.(BC35e) Graph y’(t). What does y’ tell you about the population?(BC35f) Graph y’’(t). What does y ’’ tell you about the population?
  50. 50. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpageTBQ(BC37) Let the Gamma function be defined as follows.ΓΓΓΓ(n) = ⌡⌠0∞∞∞∞tn-1e-tdtn≥≥≥≥1, n∈∈∈∈Z(BC37a) Find Γ(1) Numerically.
  51. 51. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(BC37b) Find Γ(2) Analytically.
  52. 52. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(BC37c) Find Γ(3) Numerically.
  53. 53. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(BC37d) Find Γ(4) Analytically.
  54. 54. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(BC37e) Restate Γ(n) as a function of n based on any pattern you haveobserved in the behavior of Γ(1) .. Γ(4).
  55. 55. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(BC39) Let R be the region enclosed by the graphs of y = e-x, x = kwith k > 0 and the coordinates axes.(BC39a) Write an improper integral that represents the limit of the area ofthe region R as k increases without bound and find the value ofthe integral if it exists.(BC39b) Find the volume, in terms of a finite k, of the solid generated ifthe region R were rotated about the y–axis.
  56. 56. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(BC39b) Find the volume, in terms of a finite k, of the solid whose base isR and whose cross sections perpendicular to the x–axis aresquares.
  57. 57. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(BC41) Let w(t) represent the population of worms (in millions)and r(t) the population of robins (in thousands) on an isolated island.A model for the interaction of these populations is given by thefollowing system of simultaneous, first order, ordinary differentialequations (NB: [t] = years).dwdt = w – wrdrdt = –r + wr(BC41a) Solve for w(t) assuming the robins have died out.(BC41b) Describe the population growth for worms in the absence ofrobins: (circle one)(1) Exponential Growth(2) Exponential Decay(3) Heating Curve(4) Cooling Curve(5) Logistic Growth(6) Logistic Decay(7) Stable Equilibrium(8) Unstable Equilibrium
  58. 58. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(BC41c) Solve for r(t) assuming the worms have died out.(BC41d) Describe the population growth for robins in the absence ofworms: (circle one)(1) Exponential Growth(2) Exponential Decay(3) Heating Curve(4) Cooling Curve(5) Logistic Growth(6) Logistic Decay(1) Stable Equilibrium(2) Unstable Equilibrium(BC41e) Find all Equilibrium points for the interacting populations.
  59. 59. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(BC41f) Using a Slope Field fordrdwin the window [0, 4]x[0, 4] to plot atrajectory for the initial populations of 2x106 worms and 2x103 robins. Useyour trajectory to estimate the minimum and maximum populations for bothspecies.Wmin = _______________Wmax= _______________Rmin = _______________Rmax = _______________
  60. 60. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(BC41g) Using a Slope Field fordrdwin the window [0, 4]x[0, 4] to plot atrajectory for the initial populations of 2x106 worms and 3x103 robins. Useyour trajectory to estimate the minimum and maximum populations for bothspecies.Wmin = _______________Wmax= _______________Rmin = _______________Rmax = _______________
  61. 61. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(BC41h) Describe the interaction of these two species: (circle one)(1) Symbiotic(2) Competitive(3) Predator Prey(4) Non-interactive(BC41i) The people on this island do not usually interact with the robinsor the worms. However, they do love robins! Does it make senseto introduce the additional 1000 robins at t = 0? Please explainwhy or why not in paragraph form.
  62. 62. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpageTBQ(BC43) Let P(t) represent population of fish in one pond of a fishfarm. The following differential equation models the growth of the fishpopulation when 75 fish are harvested at the end of each year to be soldto the fish market (NB: [t] = years).dPdt = 2P – 0.01P2 – 75(BC43a) When isdPdt= 0?(BC43b) Describe the population growth for these fish: (circle one)(1) Exponential Growth(2) Exponential Decay(3) Heating Curve(4) Cooling Curve(5) Logistic Growth(6) Logistic Decay(7) Stable Equilibrium(8) Unstable Equilibrium(BC43c) Find the general solution for the differential equation: P = f(t).
  63. 63. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(BC43d) Given the initial condition P(0) = 40, find the particular solutionto the given differential equation: P = f(t). Graph your solution.Explain what this graph predicts for the future of your business.
  64. 64. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(BC43e) Given the initial condition P(0) = 60, find the particular solutionto the given differential equation: P = f(t). Graph your solution.Explain what this graph predicts for the future of your business.
  65. 65. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(BC43f) Given the initial condition P(0) = 140, find the particular solutionto the given differential equation: P = f(t). Graph your solution.Explain what this graph predicts for the future of your business.
  66. 66. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(BC43g) Given the initial condition P(0) = 160, find the particular solutionto the given differential equation: P = f(t). Graph your solution.Explain what this graph predicts for the future of your business.
  67. 67. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(BC45) Consider the series ∑n=2∞∞∞∞1npln(n)(BC45a) Show that this series converges when p > 1.Use the Comparison Test.
  68. 68. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(BC45b) Does this series converge for p = 1?Use the Integral Test.
  69. 69. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(BC45c) Show divergence when 0 ≤ p < 1!Choose a convergence test.
  70. 70. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpageTBQ(BC47) Let f(x) = x –x22+x33– … = ∑n=1∞∞∞∞(-1)n-1xnn(BC47a) Show that the series for f(1) converges.(BC47b) Estimate f(1).
  71. 71. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(BC49) Let f(x) =x1 + x(BC49a) Generate the first six non–zero terms, and the general term, ofthe Taylor Series Expansion for f(x) near x = 0. Find the intervalof convergence.(BC49b) Generate the first six non–zero terms, and the general term, ofthe Taylor Series Expansion for ⌡⌠f(x)dx near x = 0. Find theinterval of convergence.
  72. 72. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(BC49c) Use the first five terms of the antiderivative to estimate ⌡⌠014f(x)dx(use FIX12).(BC49d) Find the error in your approximation (use FIX12). Howaccurately can you state the answer?
  73. 73. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(BC51) Consider the area: A = ⌡⌠01sin(x2)dx.(BC51a) Write the first four non–zero terms of the MacLaurin Expansionfor sin(x2) and ⌡⌠sin(x2)dx. What are the general terms?(BC51b) Use the first three non–zero terms of your expansion for theantiderivative to estimate ⌡⌠01sin(x2)dx (use FIX12).(BC51c) Find the error in your approximation (use FIX12). Howaccurately can you state the answer?
  74. 74. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpageTBQ(BC53) A particle moves in the XY-plane so that at any time t ≥≥≥≥ 0 itsposition (x, y) is given by x(t) = e t + e -t and y(t) = e t – e -t.(BC53a) Find the velocity vector for any t ≥ 0.(BC53b) Findlimt → ∞dydtdxdt.
  75. 75. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(BC53c) The particle moves on a hyperbola. Find an equation for thishyperbola in terms of x and y.(BC53d) Sketch the path of the particle showing the velocity vector andinitial position when t = 0.
  76. 76. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpageTBQ(BC55) Consider the curves defined by r1 = 3cosθθθθ and r2 = 1 + cosθθθθ.(BC55a) Sketch the curves and shade–in the region inside the curve r1 andoutside the curve r2.(BC55b) Find the area of the shaded region by setting up and evaluating adefinite integral. Your work must include an anti–derivative andhow you arrived at it.
  77. 77. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsultingTBQ(BC41) Givend2ydx2 = xy, y(0) = 1, y ’(0) = -1(BC41a) Use a MacLaurin series to find a sixth degree Taylor Polynomialsolving the given second order ordinary differential equation.
  78. 78. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(BC41b) Let P4(x) be the fourth degree Taylor Polynomial approzimatingy=f(x). Find the roots of y = P4(x).(BC41c) Let P4(x) be the fourth degree Taylor Polynomial approzimatingy=f(x). Find the absolute max of y = P4(x).
  79. 79. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(BC41d) Complete the following table and make a sketch of y=P4(x).x P4(x) |P4(x) – f(x)|-2- 2-1-12012122
  80. 80. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpageTBQ(BC43) The equation for the charge Q(t) on a capacitor in a circuitwith inductance L, capacitance C and resistance R satisfies thedifferential equation:Ld2Qdt2 + RLdQdt +QC = 0.NB: [Q, L, R, C, t] = Coulomb, Henry, Ohm, Farad, and second(BC43a) Given L=1, R=2 and C=4, use a characteristic equation to find thegeneral solution for Q(t).
  81. 81. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(BC43b) Given L=1, R=1 and C=4, use a characteristic equation to find thegeneral solution for Q(t).
  82. 82. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(BC43c) Given L=8, R=2 and C=4, use a characteristic equation to find thegeneral solution for Q(t).
  83. 83. Calculus AB&BC TechBook Questions Name:Copyright  2000 CALCPAGE@aol.com Imprint of MNAConsulting(BC43d) How did reducing the resistance of the circuit affect theaccumulated charge on the capacitor? Compare the results fromparts a and b.
  84. 84. Calculus AB&BC TechBook Questions Name:Copyright  2000 http://members.aol.com/calcpage(BC43e) How did increasing the inductance of the circuit affect theaccumulated charge on the capacitor? Compare the results fromparts a and c.

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