Math 3B Final Review


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1 Mth 3B Finl Review Written by Victori Kl SH 6432u Office Hours: R 9:4510:45m SH 1607 Mth Lb Hours: TR 12pm Lst updted: 12/06/14 This is continution of the midterm review. Prctice problems re locted in seprte documents. You will be expected to know the formule nd how to use them. I highly recommend completing your homework before the finl. Here re some things you should know how to do: Know how to evlute n integrl using integrtion by prts Know how to evlute n integrl filled with plethor of trigonometric functions Know how to evlute n integrl using trigonometric substitution Know how to evlute n integrl using prtil frction decomposition Know how to evlute n integrl by recognizing its form (strtegies for integrtion) Know how to evlute n improper integrl Know how to use the Comprison Theorem to decide if n integrl is convergent or divergent Know how to find the length of curve Know how to find the surfce re of curve Know how to find the Tylor Polynomil or Series of function Integrtion by Prts The following is the integrtion by prts formul: udv = uv vdu A helpful mnemonic device to remember this formul is the sying ultrviolet voodoo. To determine wht function you should choose for u is helpful by nother mnemonic LIATE: Logrithmic, Inverse Trig, Algebric, Trig, Exponentil. Whenever you hve n integrl of the form r(x)dx where is polynomil nd r(x) is repetitive function (i.e. e x, sin x, cos x, sinh x, cosh x, etc.), it is helpful to integrte by prts using the tbulr method. 1
2 7.2  Trigonometric Integrls Trig integrls re usully mixed. For exmple, you my see integrls with sine nd cosine, tngent nd secnt, cotngent nd cosecnt. Most of the time you will use u substitution nd simplify your integrl using trig identities. Note: whenever you hve sin n xdx or cos n xdx where n is n even number, you will need to use the hlf ngle identity. Here re some trig identities listed below: sin 2 x + cos 2 x = 1 tn 2 x + 1 = sec 2 x 1 + cot 2 x = csc 2 x sin 2x = 2 sin x cos x cos 2x = cos 2 x sin 2 x = 2 cos 2 x 1 = 1 2 sin 2 x cos 2 x = 1 (1 + cos 2x) 2 sin 2 x = 1 (1 cos 2x) 2 sin(a ± B) = sin A cos B ± sin B cos A cos(a ± B) = cos A cos B sin A sin B Trigonometric Substitution Trig substitutions re clled n inverse substitution. It my pper t first tht these substitutions mke the integrl more complicted, but by using trig identities, the integrl ctully becomes lot simpler. To use trig substitution, you need to choose x, then find dx, nd substitute both into your integrl. Here is summry of wht trig substitutions to use in wht scenrio: 2 x 2 = x = sin θ 2 + x 2 = x = tn θ x2 2 = x = sec θ 2
3 7.4  Integrtion of Rtionl Function by Prtil Frctions A rtionl function f(x) = q(x) where nd q(x) 0 re polynomils. To evlute n integrl of the form q(x) dx we need to use prtil frction decomposition on q(x). Steps for prtil frction decomposition on q(x) : 1. Mke sure deg < deg q(x). (If, however, deg deg q(x), use long division first to mke sure the degree of the top polynomil is less thn the degree of the bottom.) 2. Fctor q(x). (By the Fundmentl Theorem of Algebr, q(x) cn be fctored into liner or irreducible qudrtic fctors.) 3. Write out the prtil frction decomposition using the fctors of q(x) nd the cses below: I. If q(x) hs distinct liner fctors, i.e., q(x) = ( 1 x b 1 )( 2 x b 2 ) ( n x b n ) where n, then where A 1, A 2,..., A n re constnts. q(x) = A 1 A 2 A n x b 1 2 x b 2 n x b n II. If q(x) hs distinct irreducible qudrtic fctors, i.e. q(x) = ( 1 x 2 + b 1 x + c 1 )( 2 x 2 + b 2 x + c 2 ) ( n x 2 + b n x + c n ), then q(x) = A 1 x + B 1 A 2 x + B 2 A n x + B n 1 x b 1 x + c 1 2 x b 2 x + c 2 n x 2 + b n x + c n where A 1, A 2,..., A n, B 1, B 2,..., B n re constnts. III. If q(x) hs repeted liner fctors, i.e., q(x) = (x b) n, then where A 1, A 2,..., A n re constnts. q(x) = A 1 x b + A 2 (x b) A n (x b) n IV. If q(x) hs repeted qudrtic fctors, i.e., q(x) = (x 2 + bx + c) n, then q(x) = A 1x + B 1 x 2 + bx + c + A 2 x + B 2 (x 2 + bx + c) A nx + B n (x 2 + bx + c) n where A 1, A 2,..., A n, B 1, B 2,..., B n re constnts. V. q(x) my be mix of Cses I  IV. How fun! 4. Solve for the constnts in the numertors of the prtil frction decomposition. 5. Integrte q(x) dx using the prtil frction decomposition. Your nswer will most likely hve inverse tngent nd/or nturl log in your nswer. Note: You don t need to memorize exctly wht it is bove, but you should intuitively know wht to do when you see prtil frctions, i.e., if you see liner fctor just constnt goes on top, wheres if it is qudrtic fctor something like Ax + B goes on top. 3
4 7.5  Strtegies for Integrtion There is very helpful tble of integrtion formuls on pge 495 of your textbook. You should know formuls 112, 15, 16 by hert. The other formuls you cn derive using usubstitutions nd trig substitutions. Here re some helpful questions to sk yourself when deciding how to evlute n integrl: 1. Cn I simplify the integrl? Try distributing terms, see if stuff cncels out, etc. 2. Cn I use usubstitution? Try it out, nd if it doesn t work then try something else. 3. Does my integrl hve... () Two different types of functions multiplied (mybe with ln x, e x,...)? Try integrtion by prts. (b) Trig functions? Try rewriting the integrl in wy where you cn use usubstitution or hlf ngle identity. (c) Rdicls, irreducible qudrtics or both? Use trig substitution. This my involve completing the squre. (d) Rtionl frctions? Use prtil frction decomposition then try nd integrte. You my need to use trig substitution if it involves irreducible qudrtics. Sometimes your first ttempt t solving n integrl my not be successful, so try it gin nother wy. The more you prctice the clerer the steps t solving integrls will become Improper Integrls Improper integrls ren t tht different from wht we hve been studying, the only difference is tht we need to split them up in certin wy nd pply limit. There re two types of improper integrls: 1. Infinite Integrls () Rewrite f(x) dx s lim t t f(x) dx. (b) Rewrite f(x) dx s lim t f(x) dx. t (c) Rewrite f(x) dx s f(x) dx + f(x) dx where is ny rel number (you decide). Then pply () nd (b) bove. 2. Discontinuous Integrls () If f(x) is discontinuous t b, then rewrite b f(x) dx s lim t t b f(x) dx. (b) If f(x) is discontinuous t, then rewrite b f(x) dx s lim b t + f(x) dx. t (c) If f(x) is discontinuous t c where < c < b, then rewrite b f(x) dx s c f(x) dx + b f(x) dx. Then pply () nd (b) bove. c 4
5 If the limit exists (not infinite), then the integrl is convergent. If the limit does not exist (infinite), then the integrl is divergent. We cn lso mke use of the Comprison Theorem: If 0 g(x) f(x) where f nd g re continuous for x, then: () If (b) If f(x) dx is convergent, then g(x) dx is convergent. g(x) dx is divergent, then f(x) dx is divergent Arc Length If f is continuous on [, b], then the length of the curve y = f(x), x b is b b ( ) 2 L = 1 + [f dy (x)] 2 dx = 1 + dx dx Both of the bove equtions re the sme. It is similr for functions of y where x = g(y), c y d: d d L = 1 + [g (y)] 2 dy = Are of Surfce of Revolution c c ( ) 2 dx dy dy If we rotte curve bout the xxis, then the surfce re is given by S = 2πy ds where ds = 1 + (dy/dx) 2 dx or ds = 1 + (dx/dy) 2 dy. You pick ds depending on wht bounds you re given. For exmple, if you re told x b, then you would pick the first option. About the yxis, the surfce re is given by S = 2πx ds Tylor Series (k Tylor Polynomils) If you would like to red more bout Tylor Polynomils, I would suggest reding Section of your textbook nd/or the chpter the professor uploded on GuchoSpce. (Wrning: my red little bit hevy since you hven t covered much mteril on series.) 5
6 The Tylor Polynomil T n (x) of function f(x) centered bout is given by T n (x) = f() + f () 1! (x ) 1 + f () 2! (x ) f (n) () (x ) n n! The lrger the n, the better T n (x) will pproximte f(x). In fct, the Tylor Series is given by f(x) = n=0 We cn estimte the error using Tylor s inequlity: f (n) () (x ) n n! R n (x) M (n + 1)! x n+1 where f (n+1) M We cn lso use Binomil Series to estimte some functions. If k is ny rel number nd x < 1, then ( ) k (1 + x) k = x n k(k 1) = 1 + kx + x 2 k(k 1)(k 2) + x n 2! 3! n=0 6
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