officil wesite http://uoft.me/mat137 MAT137 Clculus! Lecture 28 Tody: Antiderivtives Fundmentl Theorem of Clculus Net: More FTC (review v. 8.5-8.7) 5.7 Sustitution (v. 9.1-9.4)
Properties of the Definite Integrl Let f nd g e integrle functions, nd let,, c e ny rel numers. 1 [order of limits] f () d = f () d + 2 [constnt multiple] cf () d = c y = cf () f () d y = f () Betriz Nvrro-Lmed L0601 MAT137 y = cf () 16 Jnury 2018
3 [sum] (f () + g()) d = f () d + y = f () + g() g() d 4 [dditivity] y = f () y = g() + = f () d = c f () d + c f () d y = f () c f ()d f ()d c c
Comprison Properties of the Integrl The following properties re true only if. 5 [integrl of non-negtive function] 6 [domintion] f () 0 on [, ] If f () g() on [, ] f () d 0. f () d g() d.
Properties of the Definite Integrl Emple 1 If 1 1 f ()d = 2, 5 1 f ()d = 3, find ech of the following integrls, if possile: () () 1 1 5 1 (2f () + g()) d f () d 1 1 (c) (d) g()d = 5, 1 0 1 0 f () d g() d 0 1 g()d = 1,
Properties of the Definite Integrl If f is continuous nd f () < 0 for ll [, ], then f ()d 1 must e negtive 2 might e 0 3 not enough informtion
Properties of the Definite Integrl Let f e continuous function on the intervl [, ]. True or Flse. There eist two constnts m nd M, such tht m( ) f ()d M( )
Antiderivtives Definition (Antiderivtive) A function F is n ntiderivtive of function f on n intervl I if F () = f () for ll in I.
Antiderivtives Theorem (Generl form of ntiderivtive) If F is n ntiderivtive of f on n intervl I, then the most generl ntiderivtive of f on I is where C R is n ritrry constnt. f ()d = F () + C Note: f ()d represents the collection of ll functions whose derivtive is f (). Emple 2 Find function f such tht f () = 3 2 nd f (0) = 1.
Antiderivtives Emple 3 Find the ntiderivtive of f () = (3 + 5) 7. Here s the generl strtegy in the form of flow digrm: guess not close check close djust check correct not quite correct write most generl ntiderivtive
Antiderivtives Emple 4 Find function f () if f () = sin + e 2, nd f (0) = 0, f (0) = 2.
Indefinite Integrl - Guess nd Check Emple 5 Evlute e e + 1 d
Indefinite Integrl - Guess nd Check Emple 6 Evlute sin d
Indefinite Integrl - Guess nd Check Emple 7 Evlute sin() cos()d
The fundmentl theorem of clculus dels with functions of the form g() = f (t) dt, where f is continuous function on [, ] nd vries etween nd. For emple, if f is non-negtive, then g() cn e interpreted s the re under the grph of f etween nd, where vries from to. You cn think of g s the re so fr function.
Are so Fr Function Let f (t) = t nd = 0, then the function g() = 0 tdt represents the re under the curve in the picture. Thus, g()
Are so Fr Function Below is the grph of function f. 4 3 2 1-1 -0.5 0.5 1 1.5 2 2.5 Let g() = 0 f (t) dt. Then for 0 < < 2, g() is 1 incresing nd concve up. 2 incresing nd concve down. 3 decresing nd concve up. 4 decresing nd concve down.
Are so Fr Function Below is the grph of function f. 4 3 2 1-1 -0.5 0.5 1 1.5 2 2.5 Let g() = 0 f (t) dt. Then 1 g(0) = 0, g (0) = 0 nd g (2) = 0 2 g(0) = 0, g (0) = 4 nd g (2) = 0 3 g(0) = 1, g (0) = 0 nd g (2) = 1 4 g(0) = 0, g (0) = 0 nd g (2) = 1
FTC Are so Fr function The fundmentl theorem of clculus dels with functions of the form g() = f (t) dt, where f is continuous function on [, ] nd vries etween nd. Remrk: t is dummy vrile For emple, if f is non-negtive, then g() cn e interpreted s the re under the grph of f etween nd, where vries from to. You cn think of g s the re so fr function.
FTC Are so Fr function Let f (t) = t nd = 0, then the function g() = 0 t dt represents the re under the curve in the picture. g()
FTC Are so Fr function Let f (t) = t nd = 0, then the function g() = 0 t dt represents the re under the curve in the picture. g() = 0 g() t dt = 1 2 = 1 2 2.
FTC Are so Fr function Let f (t) = t nd = 0, then the function g() = 0 t dt represents the re under the curve in the picture. Wht is g ()? g() = 0 g() t dt = 1 2 = 1 2 2.
FTC Prt I Theorem (Fundmentl Theorem of Clculus, Prt 1 [FTC1]) If f is continuous on n intervl [, ], then the function g defined y g() = f (t) dt for is continuous on [, ] nd differentile on (, ). Moreover, g () = f (). Roughly speking, this sys the following: when f is continuous, if we first integrte nd then differentite, we get f ck.