0302 - Math 31A 2010.03.02 MATH 31A DISCUSSION JED YANG 1...

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Unformatted text preview: Math 31A 2010.03.02 MATH 31A DISCUSSION JED YANG 1. Theorems 1.1. Comparison Theorem. If g ( x ) ≤ f ( x ) on an interval [ a, b ], then integraldisplay b a g ( x ) dx ≤ integraldisplay b a f ( x ) dx. 1.2. Fundamental Theorem of Calculus, I. Assume that f ( x ) is continuous on [ a, b ] and let F ( x ) be an antiderivative of f ( x ) on [ a, b ]. Then integraldisplay b a f ( x ) dx = F ( b )- F ( a ) . 1.3. Fundamental Theorem of Calculus, II. Assume that f ( x ) is continuous on [ a, b ]. Let A ( x ) = integraldisplay x a f ( t ) dt. Then A is an antiderivative of f , that is, A ′ ( x ) = f ( x ), or equivalently d dx integraldisplay x a f ( t ) dt = f ( x ) . Furthermore, A ( x ) satisfies the initial condition A ( a ) = 0. 2. Fundamental Theorem of Calculus 2.1. Exercise 5.3.39. Write the integral integraltext π | cos x | dx as a sum of integrals without absolute values and evaluate....
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0302 - Math 31A 2010.03.02 MATH 31A DISCUSSION JED YANG 1...

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