Lecture1(1)

# Lecture1(1) - Lecture 1 MATH 138-W12-003 January 4 2012 I...

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Lecture 1: MATH 138-W12-003 January 4, 2012 I) Review of the Fundamental Theorem of Calculus (FTC), Section 5.3 The ﬁrst theorem helps us to evaluate the following, g ( x ) = Z x a f ( t ) dt. 1) If f ( t ) is positive throughout the whole interval than this can be interpreted as computing the area between the curve and the x -axis. 2) If f ( t ) is negative than it’s is precisely the negative of the area. 3) If f ( t ) changes sign than it is the sum of the positive areas and negative areas. Fundamental Theorem of Calculus, Part 1: If f is continuous on [ a,b ] , then the function g deﬁned by g ( x ) = Z x a f ( t ) dt, a x b . is continuous on [ a,b ] and differentiable on ( a,b ) and g 0 ( x ) = f ( x ) . Therefore, to integrate f ( t ) we must ﬁnd its antiderivative (primative) . Corollary: From this we deduce that, d dx Z x a f ( t ) dt = f ( x ) , when f is continuous. Therefore, if we integrate f ( t ) then differentiate it we obtain our original functions. This is because the derivative and the integral are opposites, in a way.

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## This note was uploaded on 02/14/2012 for the course MATH 138 taught by Professor Anoymous during the Fall '07 term at Waterloo.

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Lecture1(1) - Lecture 1 MATH 138-W12-003 January 4 2012 I...

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