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lec20 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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Lecture 20 18.01 Fall 2006 Lecture 20: Second Fundamental Theorem Recall: First Fundamental Theorem of Calculus (FTC 1) If f is continuous and F = f , then b f ( x ) dx = F ( b ) F ( a ) a We can also write that as b x = b f ( x ) dx = f ( x ) dx x = a a Do all continuous functions have antiderivatives? Yes. However... What about a function like this? 2 e x dx =?? Yes, this antiderivative exists. No, it’s not a function we’ve met before: it’s a new function. The new function is defined as an integral: x 2 F ( x ) = e t dt 0 2 It will have the property that F ( x ) = e x . sin x 1 / 2 Other new functions include antiderivatives of e x 2 , x e x 2 , , sin( x 2 ) , cos( x 2 ) , . . . x Second Fundamental Theorem of Calculus (FTC 2) x If F ( x ) = f ( t ) dt and f is continuous, then a F ( x ) = f ( x ) Geometric Proof of FTC 2 : Use the area interpretation: F ( x ) equals the area under the curve between a and x .
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