5_3 - Section 5.3: The Fundamental Theorem of Calculus...

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Section 5.3: The Fundamental Theorem of Calculus Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu) What is the Fundamental Theorem of Calculus? The Fundamental Theorem of Calculus establishes a connection between the two branches of calculus: differential calculus and integral calculus. Differential calculus (derivative) arose from the tangent problem, where integral calculus arose from the area problem. Newton and Leibniz exploited the relationship of derivative and integral in the Fundamental Theorem of Calculus to compute areas and integrals without having to compute them as limits of Riemann sums. The first part of the Fundamental Theorem of Calculus Define an equation of the form g ( x ) = Z x a f ( t ) dt where f is a continuous function on [ a, b ] and x varies between a and b . Here, x is a variable upper limit in the integral; so, g ( x ) is a function of x . If x is a fixed number, then the integral R x a f ( t ) dt is a definite number. Consider any continous function
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This note was uploaded on 05/23/2011 for the course MAC 2311 taught by Professor Noohi during the Fall '08 term at FSU.

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5_3 - Section 5.3: The Fundamental Theorem of Calculus...

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