AP Calculus

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Special Focus: The Fundamental Theorem of Calculus The error here is that although the answer is correct, the second and third items, 3 1 namely ∫ 0 u du 3 and ln u , are meaningless. One correct way to evaluate the 0 definite integral follows: t= 3 2t u= 10 1 10 ∫ dt = 10 0 2 1 t 1 u du = u = t= ∫ ln ln( ). + u= 1 Here the variable of integration is shown not only by the differential, but also in the bounds on the integrand, which give the interval over which the definite integral is being computed. Substitution for Definite Integrals ∫ b a g( b) f ( g( t)) ⋅ g′ ( t) dt = f ( u) du. ∫ g( a) Note that with u=g(t), then t=a corresponds to u=g(a), and t=b corresponds to u=g(b). Indeed, if h′ = f , then the derivative of h(g(t)) is h′ ( g( t)) ⋅ g′ ( t), or f ( g( t)) ⋅ g′ ( t) . Thus the left-hand side is b ∫ f ( g( t)) ⋅ g′ ( t) dt = h( g( t)) = h( g( b)) − h( g( a) ) . a We have applied the evaluation part of the FTC to f (( g( t)) ⋅ g′ ( t) on the interval [a,b]. Now on the right-hand side, g( b) ∫ f ( u) du= h( u) = h( g( b)) − h( g(( a)) . g( a) g( b) g( a) Here we have applied the evaluation part of the FTC to f(u) on the interval [g(a),g(b)]. Since both sides are equal to h(g(b))–h(g(a)), we have ∫ b a b a g( b) f ( g( t)) ⋅ g′ ( t) dt = f ( u) du. ∫ g( a) AP® Calculus: 2006–2007 Workshop Materials 61
Special Focus: The Fundamental Theorem of Calculus Exploring the FTC from Numerical and Graphical Points of View Mark Howell Gonzaga College High School Washington, D.C. In this activity, students will explore the Fundamental Theorem of Calculus from numerical and graphical perspectives. The exploration will give students additional practice with x functions of the form F( x) = ∫ f ( t) dt. The given instructions are for students using a 0 TI-83 calculator. A version of this activity using the TI-89 is available on AP Central. Define Y1 and Y2 in the Y= editor, and set up the viewing window as shown. When defining Y2, select 9:fnInt from the MATH menu, and select 1:Y1 from the VARS-YVARS(1:Function)menu. Set your calculator to radian mode. Note that Y2 is a function defined as a definite X ⎛ 2 T ⎞ integral of Y1. That is, Y2( X) = ∫ cos⎜ ⎟ dT . 0 ⎝ 2 ⎠ Make sure that Y1 is the only function selected for graphing, and look at the graph. Your graph should look like the one shown: If your graph is different, check that you’ve entered Y1 correctly, that your window is correct, and that you are in radian mode. 62 AP® Calculus: 2006–2007 Workshop Materials
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AP Calculus

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