Understanding the Fundamental Theorem of Calculus: Integral and Derivative Relationship

Jan 05, 2025

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Discover the relationship between integrals and derivatives through the Fundamental Theorem of Calculus. Learn its applications, properties, and the connection between function values and slopes.

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Understanding the Fundamental Theorem of Calculus: Integral and Derivative Relationship

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4.4 The Fundamental Theorem of Calculus
Essential Question: How are the integral & the derivative related?
The Fundamental Thm, Part 1: If f is continuous on [a, b], then where F = f.
Ex 1: Find the area under the parabola y = x2 from 0 to 1.
Ex 2: Evaluate the integral.
Ex 3: Evaluate the integral.
The Fundamental Thm, Part 2: IF
The Fundamental Thm, Part 2: AND f is continuous on [a, b], THEN g is continuous on [a, b] AND differentiable on (a, b), AND …
The Fundamental Thm, Part 2: g (x) = f (x) WOW!
Ex 4: Sketch the graph of g.
Ex 5: Find the derivative of g.
Ex 6: Find
HW: 4.4 pg. 291 # 5 – 25 EOO, #27 – 37 odds, #51, 55 – 59 all, #67 – 93 odds, 74

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