Final Exam Review. [Top Bottom]dx =

Transcription

1 Finl Exm Review Are Between Curves See 7.1 exmples 1, 2, 4, 5 nd exerises 1-33 (odd) The re of the region bounded by the urves y = f(x), y = g(x), nd the lines x = nd x = b, where f nd g re ontinuous nd f(x) g(x) is A = [Top Bottom]dx = [f(x) g(x)]dx. The re of the region bounded by the urves x = f(y), x = g(y), nd the lines y = nd y = d, where f nd g re ontinuous nd f(y) g(y) is A = d [Right Left]dy = d Volume (Method of Disks or Wshers) See 7.2 exmples 1-6 nd exerises 1-11, (odd) [f(y) g(y)]dy. The Disk Method The Wsher Method V = π[r(x)] 2 dx = π [f(x)] 2 dx. V = π { [R(x)] 2 [r(x)] 2} dx = π Volume (Method of Cylindril Shells) See 7.3 exmples 1-4 nd exerises 1-25 (odd) [f(x)] 2 [g(x)] 2 dx. The Shell Method V = 2πr(x)h(x)dx = 2π Work See 7.4 exmples 1-4 nd exerises 1-19 (odd), 20 x The work done by vrible fore, F (x), in moving n objet from x = to x = b is W = F (x)dx.

2 Averge Vlue of Funtion See 7.5 exmple 1 nd exerises 1-5 (odd) The verge vlue of f(x) on the intervl [, b] is Avg. = 1 b Integrtion by Prts See 8.1 exmples 1-5 nd exerises 1-33 (odd) Integrtion by prts formul udv = uv vdu. Trigonometri Integrls See 8.2 exmples 1-9 nd exerises 1-31 (odd) Integrls of the form sin n x os m xdx. 1. If n is odd, ftor out one sin x, use the identity sin 2 x = 1 os 2 x, nd use the substitution u = os x. 2. If m is odd, ftor out one os x, use the identity os 2 x = 1 sin 2 x, use the substitution u = sin x. 3. If both n nd m re even, use the identities sin 2 x = 1 2 (1 os 2x) nd os2 x = 1 (1 + os 2x). 2 Integrls of the form tn n x se m xdx. 1. If m is odd, ftor out se x tn x, use the identity tn 2 x = se 2 x 1, nd use the substitution u = se x. 2. If n is even, ftor out se 2 x, use the identity se 2 x = 1 + tn 2 x, nd use the substitution u tn x. 3. If m is even nd n is odd, use the identity tn 2 x = se 2 x 1 nd use integrtion by prts to evlute se n xdx, where n is odd.

3 Trigonometri Substitution See 8.3 exmples 1, 3-5, 7 nd exerises 1-27 (odd) Integrtion by trigonometri substitution 1. If the integrnd involves 2 x 2, let x = sin θ, dx = os θdθ, so tht 2 x 2 = os θ. 2. If the integrnd involves x 2 2, let x = se θ, dx = se θ tn θdθ, so tht x2 2 = tn θ. 3. If the integrnd involves x 2 + 2, let x = tn θ, dx = se 2 θdθ, so tht x2 + 2 = se θ. Prtil Frtions See 8.4 exmples 1-7 nd exerises 1-51 (odd) To integrte rtionl funtion R(x) = P (x) Q(x), where P (x) nd Q(x) re polynomils, nd the degree of P (x) is less thn the degree of Q(x), deompose R(x) into the sum of prtil frtions. 1. If Q(x) is the produt of distint liner ftors x 2 + 2x 1 (x + 1)(x 1)(x + 2) = A x If Q(x) is the produt of repeted liner ftors B x 1 + C x + 2. x 2 + 2x 1 (x + 1) 3 (x + 2) = A x B (x + 1) + C 2 (x + 1) + D 3 x If Q(x) ontins distint irreduible qudrti ftors x 2 + 2x 1 (x 2 + 2)(x + 1) 2 (x + 2) = Ax + B x B x If Q(x) ontins repeted irreduible qudrti ftors C (x + 1) + D 2 x + 2. x 2 + 2x 1 (x 2 + 2) 2 (x + 1) = Ax + B x Cx + D (x 2 + 2) + E 2 x + 1. If the degree of P (x) is greter thn or equl to the degree of Q(x), then we must perform long division.

4 Improper Integrls See 8.9 exmples 1-10 nd exerises 3-41 (odd), 49, 51 Summry of improper integrls 1. If f(x) is ontinuous on [, ), then f(x)dx = lim N 2. If f(x) is ontinuous on (, b], then f(x)dx = 3. If f(x) is ontinuous on (, ), then f(x)dx = N lim N N f(x)dx + 4. If f(x) is ontinuous on [, b) nd disontinuous t b, then f(x)dx = lim N b N 5. If f(x) is ontinuous on (, b] nd disontinuous t, then f(x)dx = lim N b + N 6. If f(x) is disontinuous t, where < < b, then f(x)dx = Ar Length See 9.3 exmples 1-3 nd exerises 1-9 (odd) f(x)dx + Ar length formuls 1. If we re given urve with eqution y = f(x), x b, then L = 1 + [f (x)] 2 dx. 2. If we re given urve with eqution x = g(y), y d, then L = d 1 + [g (y)] 2 dy.

5 3. If we re given prmetri urve defined by x = x(t), y = y(t), α t β, then β (dx ) 2 ( ) 2 dy L = + dt. dt dt Surfe Are See 9.4 exmples 1-3 nd exerises 1-15, (odd) α Surfe re formuls 1. If we revolve the urve y = f(x), x b bout the x-xis, then SA = 2π f(x) 1 = [f(x)] 2 dx. 2. If we revolve the urve x = g(y), x d bout the x-xis, then SA = 2π d y 1 + [g (y)] 2 dy. 3. If we revolve the prmetri urve defined by x = x(t), y = y(t), α t β bout the x-xis, then β (dx ) 2 ( ) 2 dy SA = 2π y(t) + dt. α dt dt Similr formuls re used for rottion bout the y-xis. Sequenes See 10.1 exmples 1-11 nd exerises 2-25, (odd) A sequene is n ordered list of numbers. A sequene { n } onverges with limit L if lim n = L <. n A sequene is bounded if there exists rel number M suh tht n M for n 1. A sequene is deresing if n+1 < n for n 1 nd inresing if n+1 > n for n 1. A sequene tht is either inresing or deresing is lled monotone sequene. Any bounded, monotone sequene onverges. Series See 10.2 exmples 1-3, 5-8 nd exerises 9-27 (odd) An infinite series is sum of the terms of n infinite sequene: n =

6 The Divergene Test If lim n n 0, then the series Geometri Series Test n diverges. A geometri series is series of the form r < 1 nd its sum is r n. The series onverges if nd only if 1 r. The Integrl Test nd Comprison Tests See 10.3 exmples 1-8 nd exerises 7-29 (odd) The Integrl Test Suppose f is positive, ontinuous, deresing funtion on [1, ) nd let n = f(n). Then the series n onverges if nd only if the improper integrl f(x)dx onverges. If the series onverges, then n+1 f(x)dx R n n The Diret Comprison Test Suppose tht n nd b n re series with positive terms. 1. If n b n for ll n nd 2. If n b n for ll n nd b n onverges, then b n diverges, then n onverges. n diverges. The Limit Comprison Test Suppose tht n nd b n re series with positive terms nd n lim = L (0, ). n b n Then either both series onverge or both diverge. Other Convergene Tests See 10.4 exmples 1-9 nd exerises 3-7, 11-13, (odd) 1

7 Alternting Series Test If the lternting series ( 1) n n stisfies the onditions 1. lim n n = 0 2. { n } is deresing sequene then the series onverges. If n lternting series onverges, then R n n+1. A series is bsolutely onvergent if the series n onverges. Rtio Test Suppose tht the series n stisfies lim n+1 n = L. n 1. If L < 1, the series onverges bsolutely. 2. If L > 1, the series diverges. 3. If L = 1, the test fils. Power Series See 10.5 exmples 1-5 nd exerises 3, 4, 5-17 (odd) A power series entered t is series of the form n (x ) n. The set of ll vlues of x for whih power series onverges is lled the intervl of onvergene. The rdius of onvergene is hlf the length of the intervl of onvergene. Representing Funtions s Power Series See 10.6 exmples 1-8 nd exerises 3-25 (odd) If f(x) is defined s power series f(x) = n (x ) n,

8 with rdius of onvergene R, then f (x) = n n(x ) n 1 = n n(x ) n 1 = f(x)dx = C + The rdius of onvergene for these series is R. n x n+1 n + 1. n (n + 1)(x ) n, Tylor nd Mlurin Series See 10.7 exmples 1-9 nd exerises 1-11, 15-21, 33-39, (odd) The Tylor series for f(x) entered t is f(x) = f (n) () (x ) n. n! The Mlurin series for f(x) is the Tylor series entered t 0. Applitions of Tylor Polynomils See 10.9 exmples 1-2 nd exerises 3-7, (odd) If the Tylor series of f(x) entered t is f (n) () (x ) n, n! then the nth-degree Tylor polynomil of f t is T n (x) = n k=0 f (k) () (x ) k. n! Tylor polynomils re polynomil pproximtion of f(x) for vlues of x ner. Tylor s Inequlity provides n upper bound for the error in this pproximtion: where f (n+1) (x) M. R n (x) M (n + 1)! x n+1, Three-Dimensionl Coordinte System See 11.1 exmples 1-5 nd exerises 5-39 (odd) An eqution of the sphere with enter (x 0, y 0, z 0 ) nd rdius r is (x x 0 ) 2 + (y y 0 ) 2 + (z z 0 ) 2 = r 2.

9 Vetors nd the Dot Produt in Three Dimensions See 11.2 exmples 1-6 nd exerises 1-31 (odd), 33, 34 The dot produt of v = v 1, v 2, v 3 nd w = w 1, w 2, w 3 is the slr If θ is the ngle between v nd w, then v w = v 1 w 1 + v 2 w 2 + v 3 w 3. os θ = v w v w. The vetors v nd w re orthogonl if nd only if v w = 0. The Cross Produt See 11.3 exmples 3-5 nd exerises 7-15 (odd) The ross produt of v = v 1, v 2, v 3 nd w = w 1, w 2, w 3 is the vetor i j k v w = v 1 v 2 v 3 w 1 w 2 w 3. The ross produt is orthogonl to both v nd w nd v w is the re of the prllelogrm defined by v nd w. Note: The best wy to study for the exm is to solve the problems from exms in previous semesters, old homework problems, nd s mny problems from this review s possible. The exm will tke ple My 6th from 8:00-10:00 AM in Bloker 166. Students should bring No. 2 penil, sntron, nd vlid photo ID. The use of lultors is not llowed.