MATA37H3 Lecture Notes - Lecture 5: Telescoping Series, Antiderivative, Guidonian Hand
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Mata37 - lecture 5 - fundamental theorem of calculus, and mvt for de nite integrals. = f (b) f (a) f (x)dx = F (cid:48)(x)dx a: f (x) a can also be written as [f (x)]b a (cid:12)(cid:12)(cid:12)(cid:12)b a. F (cid:48)(xi) f (cid:48)(xi 1: #1: suppose f is integrable on [a, b], and #2: suppose f is an anitderivative of f, i. e. f (cid:48)(x) = f (x) x [a, b]. [xi 1, xi] [a, b], by mvt (applied f (x) on [xi 1, xi]), ci (xi 1, xi) such that. F (x) = sin(3x) is continuous on r, so f is continuous on [0, 3 i=1 f (ci) x = lim lim n n (f (b) f (a)) = f (b) f (a) n (f (b) f (a)) (cid:90) lim: example 1: find. 3 sin(3x)dx integrability , f is integrable on [0, (cid:12)(cid:12)(cid:12)(cid:12) . 3 (cos(3x))(cid:48) ( sin(3x)) 3: example 2: compute (cid:90) 1 (x2(cid:112)(x) + ex + (cid:90) 1.
