MATHEMATICS 1B FINAL EXAM PRACTICE PROBLEMS

MATHEMATICS 1B FINAL EXAM PRACTICE PROBLEMS DISCLAIMER: The practice problems below are not intended to be a complete review of the material covered this quarter. They are intended only to give you practice in applying SOME of the concepts you learned. For a complete review (strongly recommended), you should review old exams/quizzes/HW and try new problems from each section covered in class. You are also encouraged to review lecture handouts (e.g., Survival Guide material). Finally, some problems below may be from sections NOT covered this quarter. Ask your instructor about any problems that you think fall into this category. DIRECTIONS: Read each question carefully at least twice before answering it. Choose the BEST of the answers provided by circling the letter of your choice. Beware: Some questions may require you to circle more than one letter. For all True/False questions, answer “True” if the statement is always true; answer “False” if it is ever false. 1. What is the area enclosed between the two curves y  and y  x 2 ? (A) 2 3 (B) 1 6 (C) 12 (D) 1 3 (E) 5 6 2. If   5x  10 dx   A dx   B dx , what is the value of A? 2x 2  3x  2 2x  1 x  2 (A)  4 (B) 1 2 (C)  1 3 (D) 4 (E) 3 3. Suppose you want to numerically estimate 12  f xdx 0 using n = 3 subdivisions. What does the expression 2 f 0 4 f 4 4 f 8 2 f 12 represent? (A) L3 (B) R3 (C) T3 (D) M3 (E) S3 4. Solve the differential equation dy  2 y  50 if dx y0 100 . (A) (D) y  100e 2x y  25  75e 2x (B) (E) y  50x  x 2 100 y  50  50e 2x (C) y  25  75e 2x 5. Suppose you want to use The Comparison Test to prove that 4  f xdx either 0 converges or diverges, where f x 1 . Your next step should be to find: (A) A function g such that 0  4 f x gx over [0,4] and  g xdx 0 4 diverges (B) A function g such that 0  gx f x over [0,4] and  g xdx 0 4 converges (C) A function g such that 0  f x gx over [0,4] and  g xdx 0 4 converges (D) A function g such that 0  gx f x over [0,4] and  g xdx 0 diverges 6. Continuing problem 5, a good choice for function g is: (A) gx  1 (B) gx  1 1 (C) gx  (D) gx  7.  x3  x  2 x 1 dx equals: x 3 x2 (A) x2  x  2ln x  1  C (B)   2ln 3 2 x  1  C (C) x  x2 3 2 x 3  2x  ln x  1  C (D) x  2x  ln x  1  C 3 (E)  x2  x  2ln x  1  C 3 8. Which number best approximates the arc length of the curve y  2 tan x from x  0 to x   ? 4 (A) 2.154 (B) 1.509 (C) 1.182 (D) 2.014 (E) 1.386 9. The table below shows the velocity, v(t) , in meters per second, of a body falling downward through a viscous liquid versus time, t, in seconds. Find the TRAP(2) estimate of the total distance traveled by the falling body over the interval 6  t  22 . t 6 10 14 18 22 v( t) 3.0 3.4 3.9 4.6 5.6 (A) 64.0 (B) 62.3 (C) 56.0 (D) 55.2 (E) 65.6 10. Continuing problem 9, for EACH method listed below, circle whether the method should overestimate or underestimate the total distance traveled by the falling body in the 16-second interval. (A) T4 : Overestimates Underestimates (B) M 2 : Overestimates Underestimates (C) L4 : Overestimates Underestimates (D) R4 : Overestimates Underestimates 11. A car is driving along a straight highway. The acceleration of the car is given by at  30 cos t , where the units of acceleration are [ft/sec2], and the units of time are [sec]. At t  0 , the velocity of the car is 30 [ft/sec]. Approximately how many feet does the car travel in the interval 0  t   ? (Hint: Integrate acceleration to find velocity, integrate velocity to find position.) (A) 30.0 (B) 173.5 (C) 60.0 (D) 154.2 (E) 97.4 2 d 2 y 12. Circle ALL solutions of the differential equation x dx 2  6 y . (A) y  e  x (B) y  xe x (C) y  x 2 (D) y  3x 2 (E) y  x3 13. Suppose that you numerically integrate the function f x  sin 2x over 0,  . Circle ALL values below that will give you the exact value of the integral. (A) T2 (B) L3 (C) R1 (D) M4 (E) S2 14. Suppose that F ' x e x2 . Estimate F 2 if F 0 1.7 . (A) 2.582 (B) 0.818 (C) 1.764 (D) 0.882 (E) 1.499 d 2  15. Find dx  sin 2t dt . (A) x sin 2x  (B)  sin 2x (C)  2 cos 2x (D) cos 2x (E) 2 cos 2x 16. A stone is thrown upward with an initial velocity of 128 feet/second from the edge of a 500 ft high cliff. How many seconds after the stone is thrown does it finally hit the beach below? Recall that the acceleration of gravity is approximately 32 feet/sec2. (A) 8.32 (B) 4.00 (C) 15.63 (D) 21.34 (E) 10.87 17. Suppose that region R, bounded by the lines x  0 , y  1 and y  x  2 is rotated about the x-axis to create a solid with a hollow center. Find the volume of this solid. (A) 3.254 (B) 2.073 (C) 4.189 (D) 3.826 (E) 2.954 18. Use the substitution w  sin x to find  cossin xcos x dx . (A) sincos x C (B) sinsin x C (C)  cossin x C (D)  sincos x C (E) coscos x C 19. Suppose rt  models the water flow rate [gallons per hour] in and out of the San Francisco Bay versus time t [hours]. Assuming that positive r means water is T flowing into the Bay, what does the quantity T  0 . F T   rt dt 0 represent? Assume that (A) The total number of gallons in the Bay at time T. (B) The change in the number of gallons in the Bay between time 0 and T. (C) The average flow rate into the Bay between time 0 and T. (D) The area of the water flowing into the Bay between time 0 and T. (E) The rate of change of the number of gallons in the Bay between time 0 and T. 20. Suppose a 50-foot length of chain hangs down over the top edge of a tall building. If the chain weighs 2 pounds per linear foot, how much work, in foot-pounds, is required to pull 30 feet of the chain up over the edge (still leaving 20 feet hanging down)? (A) 2500 (B) 2000 (C) 1800 (D) 2100 (E) 2125 21. One curve in the family of solutions of the differential equation dy  2x  y dx is a straight line. Use the slope field of the differential equation (or any other method you wish) to find the line and determine the equation for it. (A) y  2x  2 (B) y  2x 1 (C) y  0.5x 1 (D) y  0.5x 1 (E) y  x  1 22. Consider the following statement about definite integrals: 1  f xdx 1 1   f x dx . 1 Which one of the functions below can be used to prove that this statement is false? (A) f x x (B) f x sin x (C) f x e x (D) f x 2 (E) f x x 2 5 23. Suppose that  2 f y gydy  10 2 5 and  4gydy  6 . What is the value of 2 2  f ydy ? (Carefully examine the limits of integration.) 5 (A) 11.50 (B) 8.50 (C) 11.50 (D)  4.25 (E)  7.00 24.  x 6 ln x dx 7 equals: x7 x 7 x6 x 7 x7 (A) x x7 (D) 7 ln x  ln x   C (B) 7 x  C (E) 56 ln x  7 7 x ln x  x 7 49  C (C)  C ln x   C 7 7 25. Evaluate  x cos 3x2 dx . (A) (D) sin3k2   1 cos3k2  6 (B) (E) sin3k2   1 cos3k2   1 6 (C) sin3k 2  6 26. 26.  cos e2 xdx  sin e2x  2  C . (A) False (B) True 27. In the case of an object undergoing a constant acceleration, a, due to gravity, recall at 2 that the equations of motion of the object are v  at  v0 and y  2  v0t  y0 . Also recall that near the surface of the earth a = – 9.8 m/sec2. Suppose an object is thrown upward from the surface of the earth at t = 0 with an initial velocity of 100 meters/sec. What is the object’s average velocity, in meters per second, over the interval 1  t  5 seconds? (A) 64.8 (B) 70.6 (C) 82.4 (D) 50.7 (E) 76.2 28. Continuing problem 27, what is the greatest height, in meters, the object attains? (A) 510 (B) 348 (C) 625 (D) 841 (E) 722 29. Let f be an increasing differentiable function on [a, b]. Suppose that L5  21.76 and L20  27.36 . Which value below should best approximate the exact value of b  f xdx ? a (A) 21.76 (B) 27.36 (C) 29.23 (D) 28.51 (E) 27.61 30. Recall that  3 x dx  2x 2  and 3 3 1 dx  ln x  C . Which of the following x antiderivatives equals 2x 2  3  C ? (A) 1  x dx (B) 3  x dx (C)  x dx x2 x 3  x dx (E) 1 1 x 2 x dx  2 dx 31 Find the value of improper integral  . 4 (A) 8 (B) 4 (C) 2 (D) 1 (E) The integral diverges 32 Circle ALL of the following improper integrals that converge.  dx 1 dx  (A)  2 1  (B)  (C) 0   e2 x dx 4 (D) dx 1 (E)  1 33. The differential equation dy  y dx 2 with initial condition y0 10 has solution y  10ex 2 (you can easily verify this). What is the error when Euler’s Method is used with a step size x  0.2 to approximate the exact solution y2? (A)  .835 (B) 1.245 (C) 1.678 (D) .588 (E)  2.385 x 34. Suppose that you define F x   e 0 t 2 dt . Find F ''1 – carefully notice that this is the second derivative of F evaluated at x = 1. (A)  1 e (B) 1 e (C)  2 e (D) 2 e 35. Continuing problem 34, use your calculator to estimate the value of F2  F1 . Hint: First draw a sketch to interpret what this quantity represents. (A) .320 (B) .073 (C) .284 (D) .164 (E) .135 36. Suppose that the density,  year , of material that falls to the ground from a  smokestack is a function of the radial distance from the smokestack, rkm, and is given by   200 . How many kilograms of material does this model predict will fall r 2 between r = 1 km and r = 5 km in one year? (A) 3570 (B) 7900 (C) 2022 (D) 1580 (E) 4130 b b 37. 37.  2  sin xdx   3 dx  3 b  a for all a  b . a a (A) True (B) False 38. Solve the following equation for x: x  1dt  ln2 . The solution is: e t e2 (A) No solution (B) 2e (C) 2 (D) e2 (E) e 2 39. The population, P[bacteria], of a bacteria colony is observed to grow at a continuous rate of 1% per hour. Which of the following differential equations most accurately reflects this model if the units of time are hours? (A) dP  .01P dt (B) dP  .01t dt (C) dP  t .01 dt (D) dP  .01 P dt (E) dP  .01 dt 40. Use the fact that  dx  1 arctan x  C to find  dx by “completing the square.” 1  x 1 x2  a2 a a 1 2x 2  4 x  10 (A) arctan 4  2   C (B)  arctanx 1  C (C) 2arctanx  1  C 2  x 1 1 (D) 2arctan    C (E) 2  arctan 2x  2  C 2 41. At the place a continuous function has a local minimum, its antiderivative has: (A) A zero (B) An inflection point (C) A critical point (D) A change of sign 42. Suppose the density of a one-meter long rod decreases linearly from   4 kg/m on the left to   2 kg/m on the right. Where is the center of gravity of the rod, in meters, measured from its left end? (A) 4 9 (B) 3 7 (C) 1 3 (D) 2 5 (E) 11 43. Consider the differential equation 1 dz  2 with initial condition z 2 dt z0 10 . Find the exact value of z2. (A) 4 3e (B) (B) (C) 3 (D) 10 41 44. Suppose the force of gravitation on an object above the earth’s surface is modeled by  r0  2 F  k   r  where r is the radial distance from the object to the center of the earth and r0 is the earth’s radius. How much work is required to lift an object from the earth’s surface ( r  r0 ) to a height above the surface equal to the earth’s radius ( r  2r0 )? (A) x 45.  e 0 kr0 3 dt  0 (B) kr0 for all x. (C) 2kr0 (D) 3kr0 (E) kr0 2 (A) False (B) True ANSWER KEY FOR PRACTICE EXAM 1. D 2. E 3. C 4. D 5. C 6. D 7. B 8. A 9. E 10. A) Over B) Under C) Under D) Over 11. D 12. C, E 13. A,B,C,D,E 14. A 15. B 16. E 17. C 18. B 19. B 20. D 21. A 22. B 23. D 24. E 25. B 26. A 27. B 28. A 29. C 30. D 31. C 32. B, C 33. B 34. C 35. E 36. C 37. A 38. B 39. A 40. A 41. B 42. A 43. D 44. E 45. A