Review Handout For Math 2280
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1 Review Hout For Mth 80 si(α ± β siαcos β ± cos α si β si θ 1 [1 cos(θ] cos si α cos β 1 [si(α + β + si(α β] si cos α cos β 1 [cos(α + β + cos(α β] si x +siy si ( ( x+y cos x y Trigoometric Ietites cos(α ± β cosα cos β si α si β θ 1 [1+cos(θ] α si β 1 [cos(α β cos(α + β] cos x +cosy cos ( x+y ( cos x y Here, ω is the lower cse Greek letter omeg A cos(ωt+bsi(ωt C si(ωt + φ wherec { A + B φ is the gle such tht si φ A C t cos φ B,orφ 1 ( A, if B>0 B C π +t 1 ( A, if B<0 B Complex Expoetil (Euler s Formul: Polr Form of Complex Numbers: De Moivre s Formul: e iθ cosθ + i si θ z + bi re iθ r cos θ + irsi θ where r + b θ is the gle such tht si θ { b cos θ,or r r t 1 ( b, if >0 θ π +t 1 ( b, if <0 For z + bi re iθ,itsth power is z r e iθ r cos(θ+ir si(θ Here, w is the lower cse of the letter ouble-u The -th roots of z + bi re iθ,re w k re i ( θ+(k 1π ( r cos θ+(k 1π + i ( r si θ+(k 1π for k 1,, Or, the solutios of w + bi re iθ re w k re i ( θ+(k 1π ( r cos θ+(k 1π + i ( r si θ+(k 1π for k 1,, z r w b / r w 1
2 Mth 80 - Review Hout Pge Determits b c bc The etermit of the three colums o the left is the sum of the proucts log the soli igols mius the sum of the proucts log the she igols * 11 1(j 1 1j 1(j+1 1 (i 11 (i 1(j 1 (i 1j (i 1(j+1 (i 1 For mtrix A i1 i(j 1 ij i(j+1 i,leta ij (i+11 (i+1(j 1 (i+1j (i+1(j+1 (i+1 1 (j 1 j (j+1 be the mtrix with the ith row jth colum remove: 11 1(j 1 1(j+1 1 A ij (i 11 (i 1(j 1 (i 1(j+1 (i 1 (i+11 (i+1(j 1 (i+1(j+1 (i+1 1 (j 1 (j+1 The, for y row i, DetA A ( 1 i+j ij A ij j1 Also, for y colum j, DetA A ( 1 i+j ij A ij i1 * The picture i the right is from Wikipei
3 Mth 80 - Review Hout Pge 3 (c 0 x [cf(x] cf (x x Differetitio Rules [f(xg(x] f (xg(x+f(xg (x x f(g(x f (g(x g (x x x ex e x l x 1 x x (si x cosx x (t x x sec x (sec x secxt x x x (si 1 x 1 1 x x (x x 1 [f(x ± g(x] f (x ± g (x x [ ] f(x f (xg(x f(xg (x x g(x g (x x x (l x x log x 1 (l x (cos x si x x (cot x x csc x (csc x csc x cot x x x (cos 1 x 1 1 x x (t 1 x 1 1+x x (cot 1 x 1 1+x x (sec 1 1 x x x 1 Fumetl Theorem of Clculus: For cotiuous fuctio f, x (csc 1 x 1 x x 1 [ x ] f(t t f(x x Strtegies For Itegrtio Metho Exmple Expig Completig the squre Usig trigoometric ietity Elimitig squre root Reucig improper frctio Seprtig frctio Multiplyig by form of 1 (e x e x e x +e x 8x x 16 (x 4 si x 1 [1 cos(x] ( x ++ 1 x x + 1 x x + 1 x x 3 7x x x +x 3 6 x 3x + 1 x 3x 1 x + 1 x sec x +tx sec x sec x sec x +tx sec x + sec x t x sec x +tx
4 Mth 80 - Review Hout Pge 4 Metho u-substitutio f (g(x g (x x f(u u Itegrtio Techiques Exmple For u x +1wehvexx u x x +1 x u u so Itegrtio by prts uv uv vu For u x, v e x x we hve u x, v e x so xe x x xe x e x x For x tθ with π/ <θ<π/ Trigoometric Substitutio For expressios ±x ± we hve sec θ>0, x sec θθ 1+x 1+t θ sec θ sec θ sec θ x sec Thus θ 1+x sec θ θ Prtil Frctios P (x x where P D re D(x polyomils, eg P<eg D D(x hs istict fctors x x 3 + x x x (x +1 ( 1 x + 1 x + 1 x x +1 Tble of Itegrtio Formuls x x x C,for 1 x l x + C +1 x e x x e x + C x x x l + C si xx cos x + C cos xxsix + C sec xxtx + C csc xx cot x + C sec x t xx sec x + C csc x cot xx csc x + C sec xxl sec x +tx + C csc xxl csc x cot x + C t xxl sec x + C cot xxl si x + C x x + 1 ( x x ( x t 1 + C x si 1 + C x x 1 l x x + + C x x ± l x + x ± + C x x x 1 ( x x sec 1 + C + x x ( x + + l x + x + + C
5 Mth 80 - Review Hout Pge 5 For the power series c (x x 0 exctly oe of the followig three cses will hol for ech 0 cse rius of covergece ρ is efie 1 0 c (x x 0 coverges oly for x x 0 ρ 0 0 c (x x 0 coverges for ll x ρ 3 0 c (x x 0 coverges if x x 0 <R iverges if x x 0 >R, for some positive umber R, ρ R The rius of covergece the itervl of covergece of the power series fou s follows Let c (x x 0 cosier lim +1 < 1 c (x x 0 c be 0 1 If lim +1 < 1 hols oly for x x 0, the ρ 0theitervlofcovergece is {x 0 } If lim +1 < 1 hols for ll x, the ρ the itervl of covergece is (, 3 ρ>0 is the rius of covergece if lim +1 < 1 hols for x x 0 < ρ the itervl of covergece is (x 0 ρ, x 0 + ρ plus oe, oe or both epoits x x 0 ± ρ which must be checke iiviully Equivletly, the rius of covergece c be fou s follows If, c 0 for lrge, lim c c +1 L, where0 L,theρ L I this versio, for L positive fiite, the itervl of covergece is (x 0 L, x 0 + L plus oe, oe or both epoits x x 0 ± L which must be checke iiviully
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