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

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