AP Calculus

Recommendations

Info

Special Focus: The Fundamental Theorem of Calculus Antiderivative Step 2: Writing an expression for the antiderivative of a function that is continuous on a open interval and that has a given value at a given point in the interval The concept is very useful but is not (yet) typically emphasized in textbooks. We have b already developed the idea that ∫ f ( t) dt = F( b) − F( a) if F′ ( x) = f ( x). Thus we can a x write the following relationship: ∫ f ( t) dt = F( x) − F( a) if F′ ( x) = f ( x). The concept a x I want students to see and understand is that F( a) + ∫ f ( t) dt = F( x) if F′ ( x) = f ( x). a Students have no difficulty following the algebra that leads to the desired conclusion. The following discussion illustrates how I try to help students understand the significance of looking at the Fundamental Theorem in this way. The first two of the following examples look quite similar, but students find them to be very different. My goal is for x students to see how both examples can be solved using F( a) + ∫ f ( t) dt = F( x) . Example 1 Write the antiderivative F for f ( x) = − sin x so that F(0)=3. Typically, students mentally run through the derivatives that they know or the antidifferentiation technique that might apply. For this problem, students quickly hit upon the following solution: a F′ ( x) = −sin x and F( 0) = 3, F( x) = cos x + C , 3 = cos0 + C ⇒ C = 2 , F( x) = 2 + cos x. Solutions to similar problems appear in every calculus textbook. Example 2 2 Write the antiderivative F for f ( x) = −sin( x ) so that F(0)=3. This problem appears to be very much like Example 1. Typically, the students attempt to come up with the formula of the function with the desired derivative. However, this time the student is not successful! There is no such “nice” function. A frequent response is, “I cannot do this problem!” AP® Calculus: 2006–2007 Workshop Materials 13
Special Focus: The Fundamental Theorem of Calculus However, the use of the antiderivative part of the Fundamental Theorem gives the function directly: F x x 2 ( ) = 3 + −sin( w ) dw. ∫ 0 It is a simple task to verify that F meets both of the given conditions, 0 2 F( 0) = 3 − ∫ sin( w ) dw = 3 and F′ ( x) = −sin( x 2 ) , directly from the statement of the 0 Fundamental Theorem. If possible, the integral should be simplified. However, simplified or not, this is an expression for the function F. With the technology that is currently available, obtaining numerical values for F is not difficult. When we try this technique for Example 1 (see below), we get the same result as before, since in this case the integral can be simplified: ∫ x F( x) = 3 + −sin( w) dw, x 0 0 F( x) = 3 + (cos w) = 3 + (cos x − cos 0) = 2 + cos x. The scoring guidelines for AP free-response questions frequently use this approach. (See 2001 AB3/BC3, 2001 BC1, 2002 AB3, 2003 AB2, 2003 AB4/BC4, 2004 AB3, and 2004 BC3.) I believe that we will continue to see this application assessed on future AP Exams. A recent example using this technique follows. Example 3: 2004 AB3 This application of the Fundamental Theorem was useful in solving part (d) of the 2004 AB3 free-response question. A calculator was allowed on this problem. A particle moves along the y-axis so that its velocity v at time t ≥ 0 is given by −1 v( t) = 1− tan ( e t − ). At time t=0, the particle is at y=–1. (Note tan 1 x = arctan x.) (a) Find the acceleration of the particle at time t=2. (b) Is the speed of the particle increasing or decreasing at time t=2? Give a reason for your answer. (c) Find the time t ≥ 0 at which the particle reaches its highest point. Justify your answer. (d) Find the position of the particle at time t=2. Is the particle moving toward the origin or away from the origin at time t=2? Justify your answer. 14 AP® Calculus: 2006–2007 Workshop Materials
Page 1 and 2: AP ® Calculus 2006-2007 Profession
Page 3 and 4: The College Board: Connecting Stude
Page 5 and 6: Table of Contents Stage Four: Confi
Page 7 and 8: Special Focus: The Fundamental Theo
Page 15: Special Focus: The Fundamental Theo
Page 67 and 68:
Special Focus: The Fundamental Theo
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Contact Us College Board Regional O
show all

AP Calculus

Create successful ePaper yourself

Delete template?

Save as template?