BIG HUMONGOUS BC PRACTICE MEMORY BOOK TEST Short Answer 1. In the limit comparison test you take the infinite limit of________________________ and then if the answer is ____________________ it means______________________. . 2. If the terms of a series go to zero, then the series________________________________. . 3. If the terms of an alternating series go to zero, then the series _______________________. . 4. Give the formula for arc length in parametric (or vector) form. . 5. Give the Trapezoidal Rule Formula for approximating area under the curve. . 6. Give the formula for any Taylor Series. . 7. Give the series for arctan x = 8. Give the series for 1/(1+x) = 9. What's the Lagrange Remainder formula? . 10. Give the formula for in terms of t . 11. Give dy/dx in terms of r and . 12. Define the range of a function. 13. Sketch y = sin x . 14. The limit is the y-value ______________________________________ and not necessarily the ___________________________________. 15. x lim 0 1 - cos x x 16. What is the definition of the slope of the tangent line at x ? . 17. If f(x) is continuous at x = c, then what do we know about the derivative of f at x = c ? . 18. If f(x) has a sharp corner at x = c, what do we know about f ' (c) ? . 19. If the derivative does not exist at x = c but it's limit is infinite, what occurs at x = c on the curve? . 20. d/dx ( mx ) = _______________ 21. d/dx ( x ) = ________________________ 22. d/dx ( f(x) + g(x) ) = __________________________________ 23. What is the Product Rule? . 24. d/dx ( sin x ) = ____________________ 25. d/dx ( tan x ) = ________________________________________ 26. d/dx ( sec x ) = _______________________________________ 27. d/dx (csc x) = ________________________________________________ 28. What is the Chain Rule (Leibniz's notation) . 29. Given dy/dx, what is the independent variable? 30. When is a particle slowing down? . 31. If v(t) > 0 then the particle is ____________________________________________. 32. What is a normal line ? . 33. Why is it important to use variable names for all variable quantities until after the derivative is taken in a related rates problem? . 34. In a related rates problem, the derivative of V is ______________________. 35. Name the three types of critical points. (1) (2) (3) 36. What two properties must f '' (x) have in order for (c,f(c)) to be an inflection point? . 37. What is the difference between a relative (or local) maximum and an absolute (or global) maximum? . 38. Besides points where f ' (c) = 0 where else may extrema occur? . 39. cost = C(x) = _____________________________+ _________________________* _____ 40. In cost/profit problems, what does x stand for? . 41. How do we verify that there is a vertical asymptote at x=a ? . 42. What are the two requirement for f(x) in order for the Mean Value Theorem to apply ? . 43. If f(x) 0, then an Lower Sum is drawn with ___________________________rectangles. 44. = lim n ____________________________________________ oo 45. What happens to the value of the integral when the limits of integration (a and b) are switched? . 46. =- 47. Under what circumstances will the integral equal the area under the curve? . 48. What is the Mean Value Theorem for Integrals ? . 49. Fill in the blanks: average value of f ' (x) on [a,b] = = -------------------------(b-a) --------------------------------(b-a) 50. What is the Mean Value Theorem for Integrals? . 51. What is the correct integral for finding the are between f(x) and g(x) from x=a to x=b if they do NOT intersect, g(x) > f(x) and you are NOT permitted to use absolute value ? . 52. What is the correct integral for computing displacement ? . 53. When using the shell method and rotating about the y-axis, what variable is used in the integral? . 54. What is the basic integral for computing work ? . 55. How do you compute average velocity? . 56. Explain the difference in computing total distance traveled verses displacement. . 57. dx = 58. ln ab = 59. Explain how to take the derivative of a function written as a variable expression raised to a variable power. . 60. e = 61. e = 62. e = 63. d/dx (e )= 64. d/dx (a )= 65. If f ' (x) = k · f(x) then f(x) = 66. lim ( 1 + c/x ) = x oo 67. What is the equation for computing compound interest ? . 68. Give the principal domain of sin x 69. Give the principal domain of cos x 70. Give the principal domain of tan x 71. d/dx ( tan x ) = 72. 1/ 73. 1/ 74. dx = dx = dx = 75. Give two forms of the antiderivative of tan x . 76. What identity is used if there is a sin x in the integrand ? . 77. What identity is used if there is a cos x in the integrand ? . 78. What identity is used if there is a sin x in the integrand ? . 79. If regular u-sub does not work, what substitution should be tried if a - b x appears in the integrand ? . 80. If regular u-sub does not work, what substitution should be tried if a + b x appears in the integrand ? . 81. If regular u-sub does not work, what substitution should be tried if a x - b appears in the integrand ? . 82. dx = 83. What identity is used if there is a cos x in the integrand ? . 84. What identity is used if there is a tan x in the integrand ? . 85. When do you use integration by parts ? . 86. How do you find a y-intercept? 87. How do you find x-intecepts? 88. Sketch y = mx + b . 89. Sketch y = . 90. Sketch y = . 91. Complete the identity: tan x + 1 = _________________ 92. Complete the identity: cos 2x = __________________________________________ 93. Complete the identity: tan x/2 = _____________________________________ 94. Draw the unit circle and lable the x and y-coordinates at the following angles: . 95. Draw the unit circle and lable the x and y-coordinates at the following angles: . 96. Explain the standard way of dealing with absolute value problems. . 97. Write the formula for the distance between two points on the plane. . 98. Write the equation of a circle with center (h,k) and radius r. . 99. What is the slope of a horizontal line? 100. ______________________________________ 101. f ' (g(x)) g ' (x) dx = _________________________________ 102. What are the three steps in solving a separable differential equation ? 1. 2. 3. 103. sec x dx = ______________________________________ 104. a - b = 105. When do we start rounding? BIG HUMONGOUS BC PRACTICE MEMORY BOOK TEST Answer Section SHORT ANSWER 1. the ratio a(n) / b(n), a positive real, both series do the same thing 2. might converge 3. converges 4. 5. 6. 7. ...uh...I forget...go look it up...or take antideriv of the series for 1/(1-(-x^2))...oh now I remember...it's like sine only no factorials 8. 1-x+x^2-x^3+.... 9. 10. 11. I think! 12. 13. 14. 15. 16. 17. 18. 19. 20. all the y-values dr the number we are getting close to, function value itself 0 lim f(x+h) - f(x) or lim f(x) - f(t) h 0 h t x x-t f ' (c) might exist as long as there is not a sharp corner. It does not exist. a vertical tangent line m 21. 22. 23. 24. nx f ' (x) + g ' (x) d/dx (f(x) · g(x)) = f(x) · g ' (x) + f ' (x) · g(x) cos x 25. sec x 26. sec x tan x 27. - csc x cot x 28. dy du dy --- · --- = ---du dx dx 29. x 30. When v(t) and a(t) have opposite signs. 31. moving right or up 32. perpendicular to the tangent line 33. Otherwise the derivatives will be zero (useless) 34. dV/dt 35. endpts, stationary pts, singular pts 36. zero (or undefined) and change signs at x=c 37. relative (local) is only higher than neighboring points wheras absolute (global) is the highest of all points 38. at endpoints OR wenever f ' (c) is undefined but f(c) is defined 39. fixed cost + price per item * x 40. the number of items bought or sold 41. Show that there is an infinite limit as x a 42. f is continuous on [a,b] and differentiable on (a,b) 43. inscribed 44. any Rieman sum or 45. It becomes the opposite of the previous value. 46. True 47. When f(x) 48. = f(c) * (b-a) for some c between a and b. 49. a,b, f ' (x) dx, f(b) - f(a) 50. dt = f(c) [ b - a ] for some c between a and b 51. dx 52. 53. x 54. 55. 56. 57. 58. 59. 60. displacement / time answers vary ln |x| + C ln a + ln b take the ln of both sides first, bring down the power, use implicit differentiation x 61. e · e 62. e / e 63. e · f ' (x) 64. a · f ' (x) · ln a 65. A e 66. e 67. A(t) = A (1 + r/n) 68. [- /2 , 69. [0, ] 70. [- /2 , /2 ] /2 ] 71. dx 72. sin x + C 73. sec x + C 74. tan x + C 75. -ln |cos x| or ln |sec x| + C 76. sin x = 1 - cos x 77. cos x = 1 - sin x 78. 79. 80. 81. 82. sin x = 1/2 - 1/2 cos 2x bx = au bx = au ax = bu ln | sec x + tan x | + C 83. cos x = 1/2 + 1/2 cos 2x 84. 85. 86. 87. 88. 89. 90. tan x = sec x - 1 When there is a simple product and/or if u-sub fails Set x=0 Set y or f(x) = 0 (drawing) dr dr 91. sec x 92. cos x - sin x 93. sin x / (1 + cos x) 94. ( , , 1/2), (- 95. ( 96. Split it into parts without absolute value. , -1/2), ( ,-1/2) 97. 98. 99. 100. 101. 102. 103. (x-h) + (y-k) = r zero -cos x + C f(g(x)) + C 1) separate the variables 2) intergrate both sides 3) solve for y tan x + C 104. (a-b)(a +ab +b ) 105. Not until the very end of the problem