Calculus

Direction tends goal Find solutions to as many differential eucratrons Problem It is impossible to explicitly Serve most differentiae eaucetroas There are partum solutions in some situations Partial solution Find a graph for certain solutions y f X y an Expression involving X and y Examples y Xty Y Hy y y l y g when it only involves a single variable we call it a

MATH | 1b | Calculus | Direction tends goal Find solu

Taylor and Madaurin series o Lets assume 1 Cx E Ch X ash on some rat containing a no A can we express Cn directly in terms of f observations x a 3 3Cx at O x a d x a 3 32 x a o x a 7 2 d X a 3 3 2 I 3 Tx's d 4 x a 3 0 Tx more generally k a a 0 Ken h K h f a oddk ykcncx.at a Ck K no conclusion 1 Cx Ch CX 9 on an open internet containing 11 9 no cu f ch a nth derivative n Definition Taylor series of CCa t tY x a

MATH | 1b | Calculus | Taylor and Madaurin series o L

Representing functions as Power series a Given f X can we express it as a power series at least near x a x ie f Cx 2 calx up on some interval containing a h o Remark preys Polynomial tan Enocacx at him cncx.at h 0 ie Pucx far as Nso Example I txt x2tx3 t i Geo series with RX 1 on C ll l x careful f maxes sauce for a X1 2 however it is represented by the Power series 1 7 7 3 only in C ll E

MATH | 1b | Calculus | Representing functions as Powe

recall Y sexy hcy separable Constant we broke them into two types y x k hCKJ 0 non constant we treat dyldy as a fraction dye box hCY dx dYthy Gandy Sw Sm y Interesting application Orthogonal trajectories Orthoguhal i right angle Perpendicular 3d land scuba t.EE oo.i t angles orthogonal trajectory to the family of curve Instructive example Family X2ty2 K2 K is an

MATH | 1b | Calculus | recall Y sexy hcy separable Co

Improper integrals aim extend concept of definite integral to deal with Potentially infinite length intervals and vertical asymptotes Instructive example what is the area under Y over IX Y 7 since is not a number we cannot directy use 1 1 1 the fundamental theorem Y 2 area 3 fingy area s a cans ax It I 1 I I t area 3 lying I la L Definition type 1 improper integral T A fa

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