[rsin(u)] = u x [ros(u)] = u x u [rtn(u)] = + x [rot(u)] = u + x [rse(u)] = u x x [rs(u)] = u x x Exm Formul Sheet tn(u) u = ln os(u) + C ot(u) u = ln sin(u) + C se(u) u = ln se(u) + tn(u) + C s(u) u = ln s(u) + ot(u) + C u ( u ) u u = rsin + C u + u u = ( u u u u = ) ( u rtn + C ) ( ) rse ( ) u + C
Volume Formuls b b b π[r(x)] π([r(x)] [r(x)] ) πr(x)h(x) π[r(y)] y π([r(y)] [r(y)] ) y πr(y)h(y) y Ar Length b + [f (x)] + [g (y)] y b πr(x) + [f (x)] Are of Surfe of Revolution πr(y) + [g (y)] y Theorem of Pppus Let R be region in plne n let L be line in the sme plne suh tht L oes not interset the interior of R (see figure). If r is the istne between the enter of mss ( or entroi) of R n the line L, then the volume of V of the soli of revolution forme by revolving R bout the line is V = πra where A is the re of the region R. Integrtion by prts uv = uv vu sin(mx) sin(nx) = (os[(m n)x] os[(m + n)x]) sin(mx) os(nx) = (sin[(m n)x] + sin[(m + n)x]) os(mx) os(nx) = (os[(m n)x] + os[(m + n)x])
Powers of sine n osine. Power of sine is o. Sve one sine n onvert the remining to osines using sin (x) + os (x) =.. Power of osine o n positive. Sve one osine n onvert the remining to sines using sin (x) + os (x) =. 3. Powers re both even n nonnegtive. Mke use of the following formuls to onvert the integrn to o powers of osine. sin (x) = ( os(x)) os (x) = ( + os(x)). Powers of sent n tngent. Power of sent is even n positive. Sve sent-squre n onvert remining sents to tngents using + tn (x) = se (x).. Power of tngent is o n positive. Sve sent-tngent n onvert remining tngents to sents using + tn (x) = se (x). 3. tn n (x), n even n positive. Convert tngent-squre to sent-squre (repet if neessry). 4. se m (x), m o n m 3. Use integrtion by prts with u = se m (x) n v = se (x). Geometri Series A geometri series is of the form, onverges to r. n th Term Test If lim n 0 then Integrl Test n iverges. n= r n. If r, then it iverges. If 0 < r <, then it n=0 If f is positive, ontinuous, n eresing for x n n = f(n) then either both onverge or both iverge. n n n= f(x) p-series The p-series n= onverges for p > n iverges for p. np
Diret Comprison Test Let 0 < n b n for ll n.. If b n onverges, then n onverges.. If n iverges, then b n iverges. Limit Comprison Test n If n > 0, b n > 0, n lim b n onverge or both iverge. Alternting Series Test Consier ( ) n n with n > 0. If n=. n+ n for ll n. lim n = 0 then the series onverges. = > 0 ( is finite number), then n n b n either both Theorem If the series n onverges, then the series n lso onverges. Rtio Test. If lim n+ n <, then the series n onverges.. If lim n+ n >, then the series n iverges. 3. If lim n+ n =, then the test is inonlusive. Root Test. If lim n n <, then the series n onverges.. If lim n n >, then the series n iverges. 3. If lim n n =, then the test is inonlusive.
f n () Defn. The series (x ) n is lle the Tylor series for the funtion entere t n! n=0 x = (or expne t x = ). If = 0, then the series is lle the Mlurin series for f. Prmetri form of the erivtive ( ) y y = t ( ). t Ar length in prmetri form: x = f(t), y = g(t) b ( ) ( ) y b + t = [f (t)] t t + [g (t)] t Are of surfe of revolution - prmetri form: x = f(t), y = g(t) b ( ) ( ) y π g(t) + t revolve bout x-xis t t b ( ) ( ) y π f(t) + t revolve bout y-xis t t Polr form of the erivtive Polr Retngulr: x = r os(θ), y = r sin(θ) Retngulr Polr: tn(θ) = y x, r = x + y y = f (θ) sin(θ) + f(θ) os(θ) f (θ) os(θ) f(θ) sin(θ) Are in polr β α [f(θ)] θ = β α r θ Ar length in polr β [f(θ)] + [f (θ)] θ α
Power Series for Elementry Funtions Intervl of Convergene 3 4 n n = ( x ) + ( x ) ( x ) + ( x )... + ( ) ( x ) +... x 0 < x < + x 3 4 5 ( ) n n = x + x x + x x +... + x +... < x < ( ) ( x ) ( x ) 3 ( x ) 4 ( ) ( x ) n n ln x = x + +... + +... 3 4 n 0 < x 3 4 5 x x x x x x e = + x + + + + +... + +...! 3! 4! 5! n! n < x < 3 5 7 9 ( n ) x n ( n + ) + x x x x sin x = x + +... + +... 3! 5! 7! 9!! < x < ( ) ( n ) n n 4 6 8 x x x x x os x = + +... + +...! 4! 6! 8!! < x < 3 5 7 9 n n + ( ) x x x x x rtn x = x + +... + +... 3 5 7 9 n + x ( ) 3 5 7 x 3 x 3 5x n! x rsin x = x + + + +... + +... 3 3 5 4 6 7! + n + n ( n ) ( n ) x ( ) ( ) ( )( ) ( )( )( ) 3 4 3 k k k x k k k x k k k k x + x = + kx + + + +... < x <! 3! 4! The onvergene t x =± epens on the vlue of k. AUSTIN COMMUNITY COLLEGE RIVERSIDE LEARNING LAB Lst revise -Aug-06
Coni Setions. Prbol The stnr form of the eqution of prbol with vertex (h, k) n iretrix y = k p is (x h) = 4p(y k). For iretrix x = h p, the eqution is (y k) = 4p(x h). The fous lies on the xis of symmetry p units from the vertex.. Ellipses The stnr form of the eqution of n ellipse with enter (h, k) n mjor n minor xes of lengths n b, where > b, is (x h) (y k) + =, mjor xis is horizontl b or (x h) (y k) + =, mjor xis is vertil. b The foi lie on the mjor xis, units from the enter, with = b. The eentriity e of n ellipse is given by the rtio e =. 3. Hyperbol The stnr form of the eqution of hyperbol with enter (h, k) is (x h) (y k) =, trnsverse xis is horizontl b or (y k) (x h) =, trnsverse xis is vertil. b The verties re units from the enter, n the foi re units from the enter, where = + b. For horizontl trnsverse xis, the equtions of the symptotes re y = k ± b (x h). For vertil trnsverse xis, the equtions of the symptotes re y = k ± (x h). b The eentriity e of hyperbol is given by the rtio e =.