section 4.3 fundamental theorem of calculus math 1231: single-variable calculus
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Section 4.3 Fundamental Theorem of Calculus
Math 1231: Single-Variable Calculus
(Signed) Area function
Signed area function: example
Example If f is a function whose graph is shown in the Figure and , find the values of g(0), g(1), g(2), g(3), g(4), g(5).
Signed area function: example
Fundamental Theorem of Calculus: Part I
The signed area function defined by f is an anti-derivative of f.
Examples
Fundamental Theorem of Calculus: Part II
The Fundamental Theorem of Calculus, Part 2 If f is continuous on [a, b], then
where F is any anti-derivative of f .
Property
Examples
Differentiation and Integration: Inverse processes
The Fundamental Theorem of Calculus If f is continuous on [a, b], then
1. : If f is integrated and then is differentiated, we arrive back to f.
2. : If we take a function F, first differentiate it then integrate it, we arrive back to F, but in the form of F(b) – F(a).
Importance of FTC
Examples
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