lec 8 - Tuesday, January 17 - Lecture 8 : Numerical...

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Tuesday, January 17 Lecture 8 : Numerical integration . (Refers to section 5.3) After having practiced the problems associated to the concepts of this lecture the student should be able to : Apply the Trapezoidal and the Simpson's rule to approximate the value of a definite integral. 8.1 Recall Recall the statement of the Second fundamental theorem of calculus : If f is continuous on the closed interval [ a , b ] then the function F ( x ) defined as is continuous, differentiable and F ( x ) = f ( x ). Hence It says that for any continuous function f ( t ) on an interval [ a , b ] there exists an antiderivative defined on this interval. 8.2 Definition Functions which can be expressed as sums, powers, products or quotients of polynomials, trig functions, exponential functions and their inverses are called elementary functions . These are essentially all the functions which have been studied up to now in high-school. One may wonder if there exist functions which are not elementary functions. If so how can we represent them? Mathematicians have tried to find the integral for a long time without success. Since its integrand is a continuous function on the real line this integrand must have an antiderivative. It was eventually shown that this integral is not an elementary function. And so we
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lec 8 - Tuesday, January 17 - Lecture 8 : Numerical...

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