Algebra. x 2 y 2 = (x y)(x + y) (x + y) 2 = x 2 + 2xy + y 2. x 3 y 3 = (x y)(x 2 + xy + y 2 ) (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3

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1 Algebr Fctoring Expning x y = (x y(x + y (x + y = x + xy + y x y = (x y(x + xy + y (x + y = x + x y + xy + y x + y = (x + y(x xy + y Exponents Logrithms x y = x+y log (xy = log x + log b ( x = x y x log y y = log x log y ( x y = xy log (x b = b log x x = x ln x = log e x (b x = x b x log x = ln x ln ( b x = x b x e = x y = y x e = lim n ( + n n Ares n Volumes Are of Tringe = (bse(height = bh Are of Circle = π(rius = πr Volume of Sphere = 4 π(rius = 4 πr Surfce Are of Sphere = 4π(rius = 4πr Volume of Cyliner = π(rius (height = πr h Volume of Cone = π(rius (height = πr h Binomil Theorem n! = 4... (n n ( n n! k = k!(n k! (x + y n = n ( n k=0 k x k y n k (x y n = n k=0 ( k( n k x k y n k

2 Roots of Polynomils If x + bx + c is generl qurtic, then the roots re given by the qurtic formul: x = b ± b 4c If x + bx + cx + is generl cubic, then the roots of the cubic re given by the following formul. The cube roots refer to ny of the three possible cube roots in C. In most courses, you re never expecte to use this formul to fin roots of cubics. ( x = b 7 + bc 6 ( + b 7 + bc 6 b ( b bc 6 ( c + b 9 ( b 7 + bc 6 ( c + b 9 There is formul for the qurtic, but it is not worth writing here for resons of length. There is no formul to solve egree 5 or higher polynomils by use of roots. Fctoring Theorem for Polynomils over R The iscriminnt of qurtic x + bx + c is the vlue b 4c. A qurtic is clle irreucible if the iscriminnt if negtive. This mens tht the qurtic hs no roots in the rel numbers. All polynomils over R fctor into prouct of liner fctors (x + b n irreucible qurtic fctors (cx + x + e.

3 Trigonometry Definitions Squres tn x = sin x cos x sin + cos = cot x = cos x sin x + tn x = sec x sec x = cos x + cot x = csc x csc x = sin x Symmetry Shifts sin( x = sin x sin ( x + π cos( x = cos x cos ( x π = cos x = sin x tn( x = tn x tn ( π x = cot x Aitions Subtrctions sin(x + y = sin x cos y + cos x sin y sin(x y = sin x cos y cos x sin y cos(x + y = cos x cos y sin x sin y cos(x y = cos x cos y + sin x sin y tn(x + y = Double Angles tn x+tn y tn x tn y tn(x y = Hlf Angles tn x tn y +tn x tn y Sum to Prouct sin x = sin x cos x sin x = cos x = cos x sin x cos x = tn x = tn x tn x sin x + sin y = sin ( x+y cos x + cos y = cos ( x+y sin x sin y = sin ( x y cos x cos y = sin ( x+y sin A Sine Lw ( cos x y ( cos x y ( cos x+y sin ( x y Prouct to Sum cos x +cos x sin x sin y = cos(x y cos(x+y cos x cos y = cos(x y+cos(x+y sin x cos y = sin(x+y+sin(x y cos x sin y = sin(x+y sin(x0y Cosine Lw = sin B b = sin C c c = + b b cos C

4 Specil Tringles x sin x cos x tn x π 6 π 4 π π 0 unefine x sin x cos x tn x π π 4 5π 6 π 0 0 x sin x cos x tn x π 6 π 4 π π 0 unefine 4

5 Hyperbolics Definitions Bsics sinh x = ex e x cosh sinh = cosh x = ex +e x tnh x = sech x sech x = cosh x coth x = + csch x csch x = sinh x sinh( x = sinh x tnh x = sinh x cosh x cosh( x = cosh x coth x = cosh x sinh x tnh( x = tnh x Aitions Subtrctions sinh(x + y = sinh x cosh y + cosh x sinh y sinh(x y = sinh x cosh y cosh x sinh y cosh(x + y = cosh x cosh y + sinh x sinh y cosh(x y = cosh x cosh y sinh x sinh y tnh(x + y = Double Angles tnh x+tnh y +tnh x tnh y tnh(x y = Hlf Angles tnh x tnh y tnh x tnh y sinh x = sinh x cosh x sinh x = cosh x = cosh x + sinh x cosh x = tnh x = tnh x +tnh x cosh x cosh x+ 5

6 Inverse Trigonometric Functions function omin rnge inverse sin x [ π, ] π [, ] rcsin x cos x [0, π] [, ] rccos x ( tn x π, π R rctn x [ sec x 0, π [, rcsec x ( ] csc x 0, π [, rccsc x cot x (0, π R rccot x Inverse Hyperbolic Functions function omin rnge inverse sinh x R R rcsinh x cosh x [0, [, rccosh x tnh x R (, rctnh x sech x [0, (0, ] rcsech x csch x (, 0 (0, (, 0 (0, rccsch x coth x (, 0 (0, (, (, rccoth x 6

7 Clculus The Definition of the Derivtive f ( = f f ( = f(x f( x x = lim x x= x Common Derivtives x c = 0 x x = = lim h 0 f( + h f( h x ex = e x x x = x ln x ln x = x x log x = x ln x sin x = cos x x sinh x = cosh x x cos x = sin x x cosh x = sinh x x tn x = sec x x tnh x = sech x x sec x = sec x tn x xsech x = sech x tnh x x csc x = csc x cot x xcsch x = csch xcoth x x cot x = csc x x coth x = csc x x rcsin x = x x rcsinh x = x + x rccos x = x x rccosh x = x x rctn x = +x x rctnh x = x x rcsec x = x x x rccsc x = x x x rcsech x = x x x rccsch x = x x + xrccot x = +x x rccoth x = x 7

8 Rules of Derivtives Power Rule Linerity Prouct Rule Quotient Rule x xr = rx r f x (f + g = x + g x g x (fg = f x + g f x f x Inverses g = g f x g f g x x f (x = f (f (x f x (f g = x g x Chin Rule f xcf = c x x f(g(x = f (g(xg (x = f(u g u x u=g(x Integrl Lws b f(xx = b f(xx b cx = c(b b f(x + g(xx = b f(xx + b g(xx b cf(xx = c b f(xx b f(x g(xx = b f(xx b g(xx f(xx = 0 b f(xx + c b f(xx = c f(xx f(x g(x = b f(xx b g(xx The Funmentl Theory of Clculus x x g(tt = g(x b xf(xx = f(b f( b f(xx = F (b F ( where F (x x = f(x Integrtion by Prts f(xg (xx = f(xg(x f (xg(xx 8

9 Inefinite Integrls x n x = xn+ n+ + C x e x x = e x + C x = ln x + C x x = x ln + C ln xx = x ln x x + C log x = xlog x x ln + C sin xx = cos x + C sinh xx = cosh x + C cos xx = sin x + C cosh xx = sinh x + C sec xx = ln sec x + tn x + C sech xx = rctn sinh x + C csc xx = ln csc x cot x + C csch xx = ln tnh x + C tn xx = ln sec x + C tnh xx = ln cosh x + C cot xx = ln sin x + C coth xx = ln sinh x + C sec xx = tn x + C csc xx = cot x + C sech xx = tnh x + C csch xx = coth x + C sec x tn xx = sec x + C sech x tnh xx = sech x + C csc x cot xx = csc x + C csch xcoth xx = csch x + C x = rcsin x + C x +x x = rctn x + C x x = rccos x + C +x x = rccotx + C x = rcsinhx + C +x x x = rccoshx + C x x = rctnhx + C x x = rccothx + C cos xx = x sin x C sin xx = x sin x 4 + C 9

10 Trigonometric Substitutions Integrn Substitution Ientity Domin x + x x x = sin θ x = tn θ x = sec θ x = cos θ + x = sec θ π θ π π < θ < π x = tn θ 0 θ < π Prtil Frctions Integrls x = ln x + c x x + b x + bx + c x = ln x + bx + c + C (x + b x = b rctn ( x b + c 0

11 Arclengths n Surfce Ares Let f(x be function [, b] R. Let γ(t = (x(t, y(t be prmetric curve in R for t [, b]. Length of the grph of f: b + f (x x Length of the curve γ: b x (t + y (t t Surfce re of surfce forme by rottion the grph of f bout the x -xis: b πf(x + f (x x Surfce re of surfce forme by rottion the curve γ bout the x -xis: b πy(t x (t + y (t t Surfce re of surfce forme by rottion the grph of f bout the y -xis: f (b f ( πf (y + (f (y y Surfce re of surfce forme by rottion the curve γ bout the y -xis: b πx(t x (t + y (t t

12 Volumes Let f(x be function [, b] R. Let γ(t = (x(t, y(t be prmetric curve in R for t [, b]. Volume of surfce forme by rottion the grph of f bout the x -xis: b πf(x x Volume of surfce forme by rottion the curve γ bout the x -xis: b πy(t y (tt Volume of surfce forme by rottion the grph of f bout the y -xis: f (b f ( πf (y y Volume of surfce forme by rottion the curve γ bout the y -xis: b πx(t x (tt Volume by Cylinricl Shells If f(x is function [, b] [0, n 0 <, then the volume of the object which is circulrly symmetric roun the y xis n hs cross section f(x long the positive x xis is: b πxf(xx

13 Series n = n k= n k = k= n(n + n k n(n + (n + = 6 n ( n(n + k = k= k= n n+r k = k r = k=0 k=r n r k= r k+r n k = r + k=0 n k=0 x k = xn+ x n k=r+ Convergence of Infinite Series If we consier n infinite series expressions: n We efine the prtil sums s S k := k n Then the vlue of the infinite series is the following limit, if it exists. n := lim k S k k

14 Tylor Series Assume tht f(x is infinitely ifferentible n α R. Then on the omin (α R, α + R f(x = f (n (α (x α n n! R = lim n+ = n lim n n n n where n = f (n (α n! If either enomintor is zero, then inste the originl eqution is true on ll R, n we sy R =. If α = 0, then these series re clle McLurin series. Clculus of Tylor Series Assume we hve convergence Tylor series with rius of convergence R, possibly infinite. The the following re true within the rius of convergence. n x n x = n x n x = n x n+ n + + C x n x n = x nx n = n nx n n= 4

15 Some Common McLurin Series Function Tylor Series Rius of Convergence e x sin x cos x rctn x x ln( x ln( + x x n n! ( n x n+ (n +! ( n x n (n! ( n x n+ n + x n x n n ( ( + x r sinh x cosh x rctnh x Li k (x J k (x n+ xn ( r x n n n x n+ (n +! x n (n! x n+ n + x n n k ( n x n+k n!(n + k! n+k 5

16 Guss, Green n Stokes Funmentl Theorem of Clculus For f(x ifferentible function, the Funmentl Theorem of Cluclus sttes b f(x = f(b f( x Funmentl Theorem of Line Integrls If γ : [, b] R n is smooth curve, f : R n R is continuous ifferentible function in neighbourhoo of γ, T is the unit norml to γ, then f T = f(γ(b f(γ( C Green s Theorem If σ is region in R with bounry σ which is n oriente close piecewise-smooth curve (with unit norml T, n if F = (F, F : R R is continuously ifferentil vector fiel on neighbourhoo of σ, then ( F F T = x F y σ Stoke s Theorem σ If σ is smooth prmetric surfce in R with bounry σ which is piecewise smooth oriente curve, n if F : R R is continuously ifferentible vector fiel on neighbourhoo of σ, then (writing T for the unit norml of σ n N for the unit norml of σ: F T = ( F N σ σ Guss Theorem (Divergence Theorem If D is region in R n bounry D n outwrly oriente prmetric surfce with unit norml N, n if F : R R is continuously ifferentible vector fiel in neighbourhoo of D, then F N = ( F D D Stoke s Theorem (Differentil Forms Version If σ is smooth prmetrize submnifol of R n of imension + with bounry σ which is piecewise smooth oriente submnifol of imenion n if ω is C ifferentil -form efine in neighbourhoo of σ then ω = ω σ σ 6

17 Lplce Trnsforms Definitions A function f(t efine on (0, is si to be of exponentil type α if there exists constnt M such tht for sufficiently lrge t, f(t < Me αt. If f is of exponentil type α then the Lplce trnsform of f is efine to be function function of s with omin (α, efine s: L{f(t}(s = 0 f(te ts t If f n g re two integrble function on (0,, then the convolution of f n g is function on (0, efine by f g(t = t 0 f(τg(t ττ Lplce Trnsforms of Common Functions Function Lplce Trnsform Function Lplce Trnsform s e αt s α sin βt sinh βt e αt sin βt β s + β β s β t n n! s n+ ln t cos βt cosh β β (s + α + β e αt cos βt (ln s + γ s s s + β s s β (s + α (s + α + β δ (t e s u (t e s s 7

18 Rules for Lplce Trnsforms Assume tht F (s is the lplce trnsform of f(t, n likewise G(s is the lplce trnsform of g(t. Function Lplce Trnsform αf(t f(t + g(t αf (s F (s + G(s f(t g(t f(αt f t F (sg(s ( s α F α sf (s f(0 n f t n s n F (s s n f(0 s n f (0... sf (n (0 f (n (0 t 0 tf(t t n f(t f(t t f(τt F (s s F s ( n n F s n s F (ττ e αt f(t F (s α u α (tf(t α f(t with perio T e αs F (s T 0 f(te st t e st 8

19 Liner Algebr Determinnts ( b et = c c b = bc b c b c et e f = e f = ei fh bi + bfg + ch ceg g h i g h i Cofctor Expnsion... n b b... b n b b... b n b b b... b n et = c c... c n c c... c n r r r... r n r r... r n r r... r n b b... b n b b... b n c c... c n ( n c c... c n n r r... r n r r... r n Eigenvlues Definition: Let M be n n n mtrix. Then v R n is n eigenvector with eigenvlue λ if Mv = λv Clcultion: The eigenvlues λ of mtrix M re the root of the egree n polynomil eqution: et(m λi = 0 Given n eigenvlue λ, the corresponing eigenvectors v re the solutions to the liner system: (M λiv = 0 9

20 Mtrix Decompositions Eigenvlue Decomposition: If M is n n n mtrix with n eigenvlues λ i (counte with multiplicity n corresponing eigenvectors v i which spn R n, then M = QAQ where A is the igonl mtrix with eigenvlues on the igonl, n Q is n invertible mtrix with with eigenvectors s columns. LU Decomposition: If M is n invertible n n mtrix which cn be row reuce without exchnging ny two rows, then there exists n upper-tringulr mtrix U n lower tringulr mtrix L such tht M = LU Then L is foun by row reucing M until there re zero below the igonl n U is foun by U = L M. QR Decomposition: If M is n invertible n n mtrix, then there exists n orthogonl mtrix Q n n upper tringulr mtrix R such tht M = QR The mtrix Q is foun by pplying the Grm-Schmit process to the columns of A to prouct the columns of Q n R ij is the inner prouct of the ith columns of Q n the jth column of M for i j. SVR Decomposition: If M is ny m n mtrix, then there exists n orthogonl m m mtrix U, n orthogonl n n mtrix V n igonl mtrix Σ such tht M = UΣV T The igonl mtrix Σ hs igonl elements which re the singulr vlues of M, which re the squre roots of the non-zero eigenvlues of MM T or M T M. The colums of the mtrix U re the left singulr vectors of M, which re the eigenvectors of MM T. The columns of the mtrix V re the right singulr vectors of M, which re the eigenvectors of M t M. 0