Math 3B Midterm Review
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1 Mth 3B Midterm Review Writte by Victori Kl SH 643u Office Hours: R 11:00 m - 1:00 pm Lst updted /15/015 Here re some short otes o Sectios i your ebook. The best idictio of wht will be o the test is wht you hve see i lecture d o the homework. No clcultors will be llowed. You will be llowed otecrd. Plese remember your TARDIS code. I suggest redig your syllbus for more iformtio. You should... Kow how to evlute itegrl usig itegrtio by prts Kow how to evlute itegrl with trigoometric fuctios Kow how to evlute itegrl usig trigoometric substitutio Kow how to evlute itegrl of rtiol fuctio usig prtil frctio decompositio Kow how to estimte itegrl usig Midpoit Rule, Trpezoidl Rule, d Simpso s Rule d the ssocited errors Kow how to evlute improper itegrl d determie why it is improper Kow how to use compriso theorem to determie if itegrl is coverget or diverget Kow how to fid Tylor Series/Polyomil for give fuctio Itegrtio by Prts The followig is the itegrtio by prts formul: udv = uv vdu To determie wht fuctio you should choose for u is helpful by the memoic LIATE : Logrithmic, Iverse Trig, Algebric, Trig, Expoetil. Wheever you hve itegrl of the form p(x) r(x)dx where p(x) is polyomil d r(x) is repetitive fuctio (i.e. e x, si x, cos x, sih x, cosh x, etc.), it is helpful to itegrte by prts usig the tbulr method. 1
2 7. - Trigoometric Itegrls Trig itegrls re usully mixed. For exmple, you my see itegrls with sie d cosie, tget d sect, cotget d cosect. Most of the time you will use u substitutio d simplify your itegrl usig trig idetities. Note: wheever you hve si xdx or cos xdx where is eve umber, you will eed to use the hlf gle idetity. Some helpful trig idetities will be listed i the lst sectio of this review Trigoometric Substitutio Trig substitutios re clled iverse substitutio. It my pper t first tht these substitutios mke the itegrl more complicted, but by usig trig idetities, the itegrl ctully becomes lot simpler. To use trig substitutio, you eed to choose x, the fid dx, d substitute both ito your itegrl. Here is summry of wht trig substitutios to use i wht scerio: x = x = si θ + x = x = t θ x = x = sec θ Itegrtio of Rtiol Fuctios by Prtil Frctios A rtiol fuctio looks like f(x) = p(x) q(x) where p(x) d q(x) 0 re polyomils. To evlute itegrl of the form p(x) p(x) q(x) dx we eed to use prtil frctio decompositio o q(x). Steps for prtil frctio decompositio: 1. Mke sure the degree of the umertor is smller th the degree of of the deomitor. If it is t, you will eed to use log divisio.. Fctor the deomitor. 3. Split up the fctors. If it is lier fctor, costt (like A or B) goes o top. If it is irreducible qudrtic fctor, somethig like Ax + B will go o top. 4. Solve for the costts.
3 7.5 - Strtegies for Itegrtio Here re some helpful questios to sk yourself whe decidig how to evlute itegrl: 1. C I simplify the itegrl? Try distributig terms, see if stuff ccels out, etc.. C I use u-substitutio? Try it out, d if it does t work the try somethig else. 3. Does my itegrl hve... () Two differet types of fuctios multiplied (mybe with l x, e x,...)? Try itegrtio by prts. (b) Trig fuctios? Try rewritig the itegrl i wy where you c use u-substitutio or hlf gle idetity. (c) Rdicls, irreducible qudrtics or both? Use trig substitutio. This my ivolve completig the squre. (d) Rtiol frctios? Use prtil frctio decompositio the try d itegrte. You my eed to use trig substitutio if it ivolves irreducible qudrtics. Sometimes your first ttempt t solvig itegrl my ot be successful, so try it gi other wy. The more you prctice the clerer the steps t solvig itegrls will become Approximte Itegrtio Midpoit Rule: b f(x)dx M = [f( x 1 ) + f( x ) f( x )] x where x = b d x i = x i 1 + x i where x = b d x i = 1 (x i 1 + x i ) (just the midpoit [x i 1, x i ]). The error for the Midpoit Rule is give by K(b )3 E M 4 where f (x) K. Trpezoidl Rule: b f(x)dx T = x [f(x 0) + f(x 1 ) f(x 1 ) + f(x )] where x = b d x i = + i x. The error for the Trpezoidl Rule is give by where f (x) K. E K K(b )3 1 3
4 Simpso s Rule: b f(x)dx S = x 3 [f(x 0) + 4f(x 1 ) + f(x ) + 4f(x 3 ) f(x ) + 4f(x 1 ) + f(x )] where is eve d x = b. The error for the Trpezoidl Rule is give by where f (4) (x) K. E S K(b ) Improper Itegrls Improper itegrls re t tht differet from wht we hve bee studyig, the oly differece is tht we eed to split them up i certi wy d pply limit. There re two types of improper itegrls: 1. Ifiite Itegrls () 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 y rel umber (you decide). The pply () d (b) bove.. Discotiuous Itegrls () If f(x) is discotiuous t b, the rewrite b f(x) dx s lim t t b f(x) dx. (b) If f(x) is discotiuous t, the rewrite b f(x) dx s lim b t + f(x) dx. (c) If f(x) is discotiuous t c where < c < b, the rewrite b f(x) dx s c f(x) dx + b f(x) dx. The pply () d (b) bove. c If the limit exists (ot ifiite), the the itegrl is coverget. If the limit does ot exist (ifiite), the the itegrl is diverget. We c lso mke use of the Compriso Theorem: If 0 g(x) f(x) where f d g re cotiuous for x, the: () If (b) If Tylor s Series f(x) dx is coverget, the g(x) dx is coverget. g(x) dx is diverget, the f(x) dx is diverget. The Tylor s Series for fuctio f(x) cetered t poit is give by f(x) = =0 f () ()! (x ) = f() + f () 1! (x ) + f ()! t (x ) + f (3) () (x ) ! 4
5 Useful Trig Idetities d Itegrls More trig idetities c be foud uder Referece Pges of your ebook. Below re the idetities I fid most useful: si x + cos x = 1 t x + 1 = sec x 1 + cot x = csc x si x = si x cos x cos x = cos x si x si x = 1 (1 cos x) cos x = 1 (1 + cos x) sec x = l sec x + t x + C csc x = l csc x cot x + C 5
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