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Unformatted text preview: Math 1a It’s Fundamental Fall, 2009 ’ & $ % The Fundamental Theorem: Suppose f ( x ) is continuous on an interval I = [ a,b ]. 1. If g ( x ) = Z x a f ( t ) dt , then g ( x ) = f ( x ). 2. Z b a f ( x ) dx = F ( b )- F ( a ), where F ( x ) is any anti-derivative of f ( x ) (that is, F = f ) Calculate the following derivatives of integrals in two ways: by using the first part of the fundamental theorem and by computing the integral first, then differentiating. 1 f ( x ) = Z x √ t- 1 dt 2 g ( x ) = Z x 5 ( t 3- 4 t + 2 ) dt Often we are unable to integrate, but the derivative is still computable using the first part of the fundamental theorem. Find derivatives of the following functions, all of which involve integrals that we currently are unable to compute: 3 f ( x ) = Z x √ t 2 + 1 dt 4 g ( x ) = Z x tan( θ ) dθ 5 S ( x ) = Z x sin( πt 2 / 2) dt 6 erf( x ) = 2 √ π Z x e- t 2 dt Sometimes we might be required to use the chain rule or properties of integrals rather than simply...
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.
- Fall '09