Calculus

Linear 2nd order non homogeneous differential equations we don't have zero on the right head grde ay t by Cy g X f Zero tunetron Ca b c all constant a 40 Him find gumerur Sorurron assume Yp is a fixed particular sorutron ay 1 by'p t Cyp SCH Let y be another serration Consider y YP a Cy yp t boy yp tely Yp acy xp b y y 1 Cy y s q y t b y't Cy ayp t byp t Cyp gey gcy O 7 Y Yp is a solution it y

MATH | 1b | Calculus | Linear 2nd order non homogeneo

Review 2 core ingenues Xrdx ftp.txrt to cry c in 1 1 tC Cra 1 g b dx I bx te b o b 1 2 ncb since dx coscxstc S Casc dx Sina to S sectx dx tuna tc S 4tx2dx arctancx ta ftp.jdx arcsincxstc core integration techniques Substitution old variable to new variable

MATH | 1b | Calculus | Review 2 core ingenues Xrdx ft

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

Trigonometric substitution Him Integrate certain algebraic taution by exploiting trigonometric identities Example area of a circle at radius r n Fix area at 2 a cinae PEX DX observatory r since Trix p Koso dx rcasoda 7 do rdfosoc 72 ftp.xzdx zgiriosos do Ty 2 r 050 zr qltcosz 2 r2 Ot Isin 20 iq More generally 1 4 asano tax alcoso Oeta 2 Katana taxi alsecol 3 X a Seco q

MATH | 1b | Calculus | Trigonometric substitution Him

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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