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 | 1. To explore Part I of the Fundamental Theorem graphically, first suppose that F(x) is increasing on the interval [a, b]. Explain why both of the expressions F(b)-F(a) and  will be positive. Further, explain why the faster F(x) increases, the larger each expression will be. Similarly, explain why if F(x) is decreasing both expressions will be negative. |
| | 2. You can think of Part I of the Fundamental Theorem in terms of position s(t ) and velocity v(t ) = s'(t ). Start
by assuming that v(t )  0. Explain why  gives the total
distance traveled, and explain why this equals s(b)-s(a). Discuss what changes if v(t ) < 0. |
| | 3. To explore Part II of the Fundamental Theorem graphically, consider the
function  . If f (t ) is positive on the interval [a, b] , explain why g' (x) will also be positive. Further, the larger f (t ) is, the larger g'(x) will be. Similarly, explain why if f (t ) is negative then g'(x) will also be negative. |
| | 4. In Part I of the Fundamental Theorem, F can be any antiderivative of f . Recall that any two antiderivatives of f differ by a
constant. Explain why F(b) - F(a) is well-defined; that is, if F1 and F2 are different antiderivatives, explain why F1(b) - F1(a) = F2(b) - F2(a). When evaluating a definite integral, explain why you do not need to include “ +c ” with the antiderivative. |
| In exercises 5-30, use Part I of the Fundamental Theorem to compute each integral exactly.
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| 5.  | 6.  |
| 7.  | 8.  |
| 9.  | 10.  |
| 11.  | 12.  |
| 13.  | 14.  |
| 15.  | 16.  |
| 17.  | 18.  |
| 19.  | 20.  |
| 21.  | 22.  |
| 23.  | 24.  |
| 25.  | 26. 
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| 27.  | 28.  |
| 29.  | 30. 
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| In exercises 31-40, use the Fundamental Theorem if possible or estimate the integral using Riemann sums. (Hint: Six problems can be worked using antiderivative formulas we have covered so far.)
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| 31.  | 32.  |
| 33.  | 34.  |
| 35.  | 36.  |
| 37.  | 38.  |
| 39.  | 40. 
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| In exercises 41-46, find the derivative f '(x).
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| 41. 
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| 42. 
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| 43.  | 44.  |
| 45.  | 46. 
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| In exercises 47-54, find the given area.
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| 47. The area above the x - axis and below y = 4 - x2
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| 48. The area above the x - axis and below y = 4x - x2
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| 49. The area below the x - axis and above y = x2 - 4
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| 50. The area below the x - axis and above y = x2 - 4x
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| 51. The area of the region bounded by y = x2, x = 2 and the x - axis |
| 52. The area of the region bounded by y = x3, x = 3 and the x - axis |
| 53. The area between y = sin x and the x - axis for 0  x 
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| 54. The area between y = sin x and the x - axis for -/2
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| In exercises 55-58, find an equation of the tangent line at the given value of x.
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| 55. 
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| 56. 
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| 57. 
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| 58. 
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| 59. Use the derivative in exercise 55 to locate and identify all relative extrema of  . |
| 60. Katie drives a car at speed  mph and Michael drives a car at speed g(t ) = 50+2t mph at time t minutes. Suppose that Katie and Michael are at the same location at time t = 0. Compute  and interpret the integral in terms of a race between Katie and Michael. |
| In exercises 61 and 62, (a) explain how you know the proposed integral value is wrong and (b) find all mistakes.
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| 61. 
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| 62. 
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| In exercises 63-70, find the position function
s(t ) from the given velocity or acceleration function and initial
value(s). Assume that units are feet and seconds.
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| 63. v(t ) = 40-sin t, s(0) = 2
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| 64. v(t ) = 30+4cos 3t, s(0) = 0
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| 65. v(t ) = 25(1-e-2t ), s(0) = 0
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| 66. v(t ) = 10e-t , s(0) = 2
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| 67. a(t ) = 4-t, v(0) = 8, s(0) = 0
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| 68. a(t ) = 16-t 2, v(0) = 0, s(0) = 30
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| 69. a(t ) = 24+e-t , v(0) = 0, s(0) = 0
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| 70. a(t ) = 3e-2t , v(0) = -4, s(0) = 0
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| 71. If  (t ) is the angle between the path of a moving object and a fixed ray (see the figure on the next column), the angular velocity of the object is  (t ) = ' (t ) and the angular acceleration of the object is  (t ) = ' (t ).
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| Suppose a baseball batter swings with a constant angular acceleration of (t ) = 10
rad/s 2. If the batter hits the ball 0.8 s later, what is the angular velocity? The (linear) speed of the part of the bat located 3 feet from the pivot point (the batter's body) is v = 3. How fast is this part of the bat moving at the moment of contact? Through what angle was the bat rotated during the swing? |
| 72. Suppose a golfer rotates a golf club through an angle of 3/2 with a constant angular acceleration of
rad/s 2. If the clubhead is located 4 feet from the pivot point (the golfer's body), the (linear) speed of the clubhead is v = 4. Find the value of that will produce a clubhead speed of 100 mph at impact. |
| 73. From 1970 to 1974, the function 16.1e0.07t closely approximated the number of billions of barrels of oil consumed in the United States, where t = 0 corresponds to 1970. When pricesrose in 1974, the consumption was approximated by the function 21.3e0.04(t-4). Show that both functions give (approximately) the same consumption value for 1974. Compute  and  and compute the amount of oil saved by the reduced consumption. |
| 74. The amount of work done by a force F(x) moving an object from x = a to x = b is given by  . As the object continues to move, the endpoint changes in time. Then b = b(t ) and the work done is a function of time. The derivative of this function defines power. Show that the power  equals F(b(t )) b' (t ). |
| In exercises 75 and 76, use the result of exercise 74 to compute the horsepower in the following situations. (Note: 1 hp = 550 ft-lb/s.)
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| 75. F(x) = 1000 lb, b' (t ) = 130
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| 76. F(x) = 1000e-x lb, b' (t ) = 100-t ft/s |
| In exercises 77-82, find the average value of the function on the given interval.
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| 77. f (x) = x2-1, [1, 3] | 78. f (x) = x2+2x, [0, 1]
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| 79. f (x) = 2x-2x2, [0, 1] | 80. f (x) = x3-3x2+2x, [1, 2] |
| 81. f (x) = cos x, [0, /2] | 82. f (x) = sin x, [0, /2]
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| In exercises 83 and 84, use the graph to list  (x)dx,  (x)dx and  (x)dx in order, from smallest to largest.
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| 83.
| 84.
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| 85. Derive Leibniz' Rule  = x 'x - x 'x. |
 | 86. Suppose that a communicable disease has an infection stage and an incubation stage (like HIV and AIDS). Assume that the infection rate is a constant f (t ) = 100 people per month and the incubation distribution is b(t ) =  te-t /30 month -1. The rate at which people develop the disease at time t = T is given by  people per month. Use your CAS to find expressions for both the rate r(T) and the number of people  who develop the disease between times t = 0 and t = x. Explain why the graph y = r(T) has a horizontal asymptote. For small x's, the graph of y = p(x) is concave up; explain what happens for large x's. Repeat this for f (t ) = 100 + 10 sin t , where the infection rate oscillates up and down. |
| | 87. When solving differential equations of the form  f (y) for the unknown function y(t ) , it is often convenient to make use of a potential function V(y). This is a function such that  . For the function f (y) = y-y3 , find a potential function V(y). Find the locations of the local minima of V(y) and use a graph of V(y) to explain why this is called a “double-well” potential. Explain each step in the calculation |
| Since  , does the function V increase or decrease as time goes on? Use your graph of V to predict the possible values of  . Thus, you can predict the limiting value of the solution of the differential equation without ever solving the equation itself. Use this technique to predict  if y' = 2-2y. |
