MATH 140 Lecture Notes - Lecture 30: Mean Value Theorem, Antiderivative
17 Nov 2015
School
Department
Course
Professor
Document Summary
0 sin x dx with riemann sums: to prepare for the fundamental theorem, let f be continuous on [a, b], define g( x)= x a f (t )dt for all x in [a, b], note 1: g( a)=9= a a f (t)dt, note 2: if f is greater than or equal to 0, then g(x) is the area below the graph of f and the x axis. Theorem 5. 12: let f be continuous on [a, b] and let g( x)= x a f (t) dt for a x b . Examples: let g( x)= , let g( x)= x. 1 x t 3 dt g"( x)=x3 cost dt g ( x)= x cost dt g"( x)= cos x: using the chain rule, if y = h(x) and h"(x) exists, then if g( x )= h( x ) a f (t) dt. 0 et dt g"( x)=e x3(3 x2)
