Math 100 Review Sheet

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1 Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s n ddition to your notes, homework ssignments, nd the book. 1 Derivtives nd Integrls 1.1 L Hôpitl s Rule f(x) If lim f(x) = 0 nd lim g(x) = 0, then lim x x x g(x) = lim f (x) x g (x). Sme if lim f(x) = lim g(x) =. 1.2 Definite Integrls f(x) dx = lim n n f(x i ) x, where x = b, where the intervl x b is divided into n pieces of n width x, nd where x i is in the i th piece. i=1 1.3 Fundmentl Theorems of Clculus Prt I: Prt II:Define F (x) = f(x) dx = F (b) F (), where F (x) is n ntiderivtive of f(x). x f(t) dt. Then F (x) = f(x). 1

2 2 Applictions of Definite Integrls 2.1 Volumes by Slicing V = A(x) dx, 2.2 Volumes of Revolution where A(x) = cross-sectionl re. Rotte region under y = f(x) with x b round x-xis. V = 2.3 Length of Curve Length of y = f(x) for x b. 2.4 Surfce Are L = πf(x) 2 dx. 1 + f (x) 2 dx. Rotte curve y = f(x) for x b round x-xis. A = 2πf(x) 1 + f (x) 2 dx. 3 Methods of Integrtion 3.1 Substitution Substitute x = g(u) to get f(x) dx = f ( g(u) ) g (u) du. 3.2 Integrtion by Prts u dv = uv v du. 2

3 3.3 Prtil Frctions Exmple: Set x 2 + 3x 2 x 2 (x 1)(x 2 + x + 2) equl to A x + B x 2 + C x 1 + Dx + E x 2 + x + 2. Cler denomintors, equte powers of x, solve for A, B, C, D, E. 3.4 Integrl Tbles Mnipulte given integrl to mke it mtch n integrl in the tble. 3.5 Numericl Integrtion Trpzoidl Rule Simpson s Rule f(x) dx x 2 (y 0 + 2y 1 + 2y y n 1 + y n ). f(x) dx x 3 (y 0 + 4y 1 + 2y 2 + 4y y n 2 + 4y n 1 + y n ). 3.6 Improper Integrls f(x) dx = lim b f(x) dx, nd similrly for f(x), nd similrly for f(x) dx if f(x) is not defined t some point in the intervl. Cn check convergence or divergence by comprison. If 0 f(x) g(x) for ll x nd if If 0 g(x) f(x) for ll x nd if g(x) dx converges, then f(x) dx lso converges. g(x) dx diverges, then f(x) dx lso diverges. 3

4 4 Sequences nd Series 4.1 Sequences A sequence is list of numbers 1, 2, 3, Series The infinite series n = is the limiting vlue of the sequence of prtil sums s 1 = 1, s 2 = 1 + 2, s 3 = , Some Specil Series Geometric Series { if r < 1, r n = + r + r 2 + r 2 + = 1 r diverges if r 1. n= p-series 1 n = p p 3 + p { converges if p > 1, diverges if p 1. (The p-series with p = 1 is clled the hrmonic series.) 4.4 Convergence Tests n th Term Test If lim n n 0, then n diverges. 4

5 4.4.2 Integrl Test n = f(n) with f(x) 0. Then n nd 1 f(x) dx either both converge or both diverge. 4.5 Comprison Tests If 0 n b n nd If 0 b n n nd b n converges, then n converges. b n diverges, then n diverges. 4.6 Rtio Test For n 0, compute Then 4.7 Root Test For n 0, compute Then n n ρ = lim n n+1 n. converges if ρ < 1, diverges if ρ > 1, test is inconclusive if ρ = 1. ρ = lim n n n. converges if ρ < 1, diverges if ρ > 1, test is inconclusive if ρ = 1. 5

6 4.8 Alternting Series Test If then u n 0, nd u n u n+1, nd lim n u n = 0, ( 1) n 1 u n = u 1 u 2 + u 3 u Power Series 5.1 Power Series A power series is series of the form n (x ) n. n=0 converges. It will generlly converge for ll x in some intervl centered t, which my or my not include the endpoints. The rtio nd root tests re good for finding the intervl, then use other tests for the endpoints. 5.2 Tylor Series The Tylor series of f(x) round x = is the power series f (k) () k! (x ) k = f()+f ()(x )+ f () 2! When = 0, it is lso clled the Mclurin series f (k) (0) k! x k = f(0) + f (0)x + f (0) 2! (x ) 2 + f () (x ) ! x 2 + f (0) x ! 6

7 5.3 Some Common Tylor Series e x x k = k! sin(x) = ( 1) k x 2k+1 (2k + 1)! cos(x) = ( 1) k x2k (2k)! x k ln(1 x) = k (converges for ll x) (converges for ll x) (converges for ll x) (converges for x < 1) 5.4 Mnipulting Tylor Series If you know Tylor series f(x) = f() + f ()(x ) + f () 2! (x ) 2 + f () (x ) 3 +, 3! you cn compute the series for f (x) by differentiting ech term, nd you cn compute the series for f(x) dx by integrting ech term. 5.5 Tylor Polynomils nd Error Estimtes The n th Tylor polynomil is n f (k) () T n (x) = (x ) k k! The error = f() + f ()(x ) + f () 2! R n (x) = f(x) T n (x) in using T n (x) to estimte f(x) is bounded by Rn (x) x n+1 M, (n + 1)! 7 (x ) f (n) () (x ) n. n!

8 where M is ny number so tht f (n+1) (t) M for ll t between x nd. 5.6 Euler s Identity e iθ = cos θ + i sin θ, where i = 1. 6 Differentil Equtions A differentil eqution is n eqution tht involves x, y, nd one or more derivtives of y. The order of differentil eqution is the highest derivtive tht ppers. A function y = f(x) is solution to differentil eqution if substituting y = f(x) mkes the eqution true. 6.1 Slope Fields A differentil eqution of the form dy dx = f(x, y) cn be studied by drwing the slope field, which mens t ech point (x 0, y 0 ), drwing smll line segment through the point hving slope f(x 0, y 0 ). 6.2 First Order Liner Differentil Equtions A first order liner differentil eqution hs the form dy + P (x)y = Q(x). dx To solve, compute the integrting fctor v(x) = e P (x) dx. Then multiplying by v(x) mkes the left-hnd side of the eqution equl to d ( ) v(x)y, so the solution is dx y = 1 v(x)q(x) dx. v(x) 8

9 6.3 Applictions of First Order Liner Differentil Equtions Popultion growth nd rdioctive decy problems led to seprble differentil equtions of the form dy dx = ky with solution y = Cekx. Mixing problems in which liquid contining chemicl is pouring into nd drined out of continer often led to generl first order liner differentil equtions. 6.4 Euler s Method To pproximtely solve dy = f(x, y), dx strt t point (x 0, y 0 ), choose (smll) increment vlue, nd then for n = 1, 2, 3,... let x n = x n 1 + dx nd y n = y n 1 + f(x n 1, y n 1 ) dx. An improved method is to let x n = x n 1 + dx, z n = y n 1 + f(x n 1, y n 1 ) dx, ( ) f(xn 1, y n 1 ) + f(x n, z n ) y n = y n 1 + dx Second Order Liner Differentil Equtions A second order liner differentil eqution hs the form P (x) d2 y dx + Q(x)dy + R(x)y = G(x). 2 dx If G(x) = 0, the eqution is homogeneous. 9

10 6.6 Homogeneous Second Order Liner Differentil Equtions with Constnt Coefficients To solve y + by + cy = 0 when nd b re constnts, first find the roots of r 2 + br + c = 0. The roots re r 1, r 2 = b ± b 2 4c. 2 There re three cses. Cse I: b 2 4c > 0: Then r 1 nd r 2 re rel, nd the generl solution is y = c 1 e r 1x + c 2 e r 2x. Cse II: b 2 4c < 0: Then r 1 nd r 2 re complex numbers, so we cn write them s r 1 = α + βi nd r 2 = α βi. Then the generl solution is y = e αx( c 1 cos(βx) + c 2 sin(βx) ). Cse III: b 2 4c = 0: Then r 1 = r 2 = b, nd the generl solution is 2 y = c 1 e r 1x + c 2 xe r 1x = (c 1 + c 2 x)e r 1x. 6.7 Non-Homogeneous Second Order Liner Differentil Equtions with Constnt Coefficients To find the generl solution to y + by + cy = G(x), first find the solution y c (x) to the complementry homogeneous eqution y + by + cy = 0. Next find prticulr solution y p (x) to the non-homogeneous eqution. Then the generl solution to the non-homogeneous eqution is y(x) = y c (x)+y p (x). 10

11 6.8 Method of Undetermined Coefficients If G(x) is polynomil, or e rx or trig function multiplied by polynomil, one cn guess the form of the solution, leving the coefficients s unknowns, substitute into the differentil eqution, nd solve for the coefficients. 6.9 Method of Vrition of Prmeters Let y c (x) = c 1 y 1 (x) + c 2 y 2 (x) be the solution to the complementry homogeneous eqution. Solve the equtions v 1(x)y 1 (x) + v 2(x)y 2 (x) = 0 nd v 1(x)y 1(x) + v 2(x)y 2(x) = G(x)/ for v 1(x) nd v 2(x). Integrte to find v 1 (x) nd v 2 (x). Then y p (x) = v 1 (x)y 1 (x) + v 2 (x)y 2 (x) Hrmonic Motion If weight of mss m is ttched to spring hving spring constnt k, nd if it moves through medium with friction equl to δ times its instntneous velocity, then the eqution of motion is my + δy + ky = 0. Depending on the vlues of m, δ, nd k, the solution will involve exponentils nd/or trig functions Series Solutions to Differentil Equtions To find series solution to substitute P (x)y + Q(x)y + R(x)y = 0, y(x) = c n x n n=0 into the eqution, set ll power of x equl to zero, nd find reltions on c 0, c 1, c 2,.... In generl, there should be two coefficients of y(x) tht cn be chosen rbitrrily, nd then ll of the other coefficients cn be expressed in terms of those two. 11

12 7 Polr Coordintes Polr coordintes (r, θ) describe point whose distnce from the origin is r nd whose line segment mkes n ngle θ with the positive x-xis. Polr nd Crtesin coordintes re relted by the formuls: x = r cos θ, y = r sin θ, x 2 + y 2 = r 2, 7.1 Grphing in Polr Coordintes y x = tn θ. Methods to use include: Check for symmetries. Wht hppens when you replce r by r nd/or replce θ by θ? Mke tble of vlues. Convert to Crtesin coordintes. 7.2 Ares in Polr Coordintes The re inside the curve r = f(θ) with α θ β is A = β 8 Prmetric Equtions α 1 2 f(θ)2 dθ. We cn describe curve using prmetric equtions x = f(t) nd y = g(t). The slope t t = t 0 is dy dx = (dy/dt) t=t0 t=t0 (dx/dt). t=t0 12