Review Guie or MAT Finl Em Prt II. Mony Decemer th 8:.m. 9:5.m. (or the 8:3.m. clss) :.m. :5.m. (or the :3.m. clss) Prt is worth 5% o your Finl Em gre. NO CALCULATORS re llowe on this portion o the Finl Em. You my NOT use your own scrtch pper or this Multiple Choice Em. Inste, your instructor will give you two sheets o scrtch pper tht you cn use to o whtever work you eel tht you nee to in orer to otin the nswer to ech question. This portion o the Finl Em consists o ive pges. Ech pge hs lnk sie tht cn lso e use or scrtch work i neee. You will turn in BOTH sheets o scrtch pper with your Finl Em (even i they hve nothing written on them). NO PARTIAL CREDIT will e given on this portion o the Finl Em. You will hve minutes to complete this portion o the Finl Em (ssuming tht you show up on time). There re 5 questions on this prt o your inl em. MANY o these questions will e very quick n require little to no written work! Things you shoul mke sure tht you cn o! Note: Section numers hve een provie y ech topic so tht you cn go ck through your NOTES, HOMEWORK n OLD TESTS to in prolems to prctice. You cn lso go ck to the clss HELP pge n view some o the relevnt supplementl reings n vieos. Some prolems re not necessrily speciic to section, rther will test how well you unerstn n ie tht is covere over possily multiple sections. BE SURE THAT YOU HAVE LOOKED AT, THOUGHT ABOUT AND TRIED THE SUGGESTED PROBLEMS ON THIS REVIEW GUIDE PRIOR TO LOOKING AT THESE COMMENTS!!! -. Mke sure you unerstn ew sic properties o ntierivtives. For emple, is n ntierivtive o ierence o two unctions, g, the sme s n ntierivtive o minus n ntierivtive o g?(section 3.) True. g g. Does this property hol or ition n multipliction? 3-4. Be sure tht you unerstn the Funmentl Theorem o Clculus (prt I) n wht MUST e true out, in orer or to eist. I look like? (sections 3., 3.4) oes eist, wht oes the nswer look like? Wht oes the nswer to One o these is mily o unctions n the other is numer. Which is which? Also e sure tht you know WHEN the FTC cn NOT e use to evlute einite integrl! 5-6 I g, oes g (True)? Wht out the converse o tht sttement(flse)? Ask yoursel this sme question (oth wys) out ineinite n einite integrls. i.e I g oes g? (True) I g oes g?(flse) Now nswer oth questions i the integrl ws ineinite rther thn einite (True) (True). Try to gin some insight into wht ech o these sttements re sying!
7. I the erivtive o t cis zero wht oes tht men hppens t cin the grph o? (sections.9,.). Think creully! It mens tht the grph hs horizontl tngent line t = c (oes this men tht the grph HAS to hve reltive m or min t = c?). 8. I the secon erivtive o t p is zero wht oes tht men hppens t p in the grph o? (sections.,.). Think creully! It mens tht there is POSSIBLE point o inlection with n coorinte o p (i.e. it is POSSIBLE tht the grph chnges concvity t this loction ut there is no gurntee). 9 -. Be sure tht you unerstn the secon prt o the Funmentl Theorem o Clculus (section 3.4). For emple F i F cos t t. Wht i F cos t t? cn you in emples? 3 3 3 3 Cn you in F 3 3 F cost t F cost t cos F cos sin F cost t F cost t cos cos or oth o these F cos sin cos sin cos. Wht is the mimum numer o verticl symptotes tht unction coul hve? Wht oes horizontl symptotes represent in Clculus (how mny o those coul unction hve)? (section.,.) A unction coul hve unlimite verticl symptotes (y=tn). A horizontl symptote or is limit t ininity... i.e. y y Lim n y Lim. Be le to in the vlues where unction hs n ctul point o inlection. (section.,.) 3. Be le to in n ineinite integrl (section 3.). Emple: Fin 3 4 3 4 u 3 4 u 3 u 3 3 3 4 u u u c c 3 3 3 9 3
4. I n oject unergoing rectiliner motion is moving right n slowing own, wht woul the grph o its position unction look like (i.e incresing concve up, ecresing concve own etc.)? (section.9) Moving right mens the velocity is positive (thus the slope o the tngent line to the position grph woul e positive n consequently the position grph must e incresing). I the oject is ALSO slowing own then the ccelertion must hve the opposite sign o velocity so in THIS emple tht woul men negtive. I the ccelertion is negtive the grph o the position unction must e concve DOWN s ccelertion is the secon erivtive o position. So in THIS emple the grph o the position unction woul e incresing concve own. Cn you igure out wht the grph woul look like i the oject ws moving let n slowing own, moving let n speeing up n moving right n speeing up? 5. How woul you in the secon erivtive o unction? How woul you in the secon ntierivtive o unction? To in the secon erivtive o unction just tke the erivtive o the erivtive. To in the secon ntierivtive just tke the ntierivtive o the ntierivtive (or the integrl o the integrl). 6. Be sure tht you unerstn when the is positive, when it is negtive n when it is zero (section 3.4) I () > n < then the integrl is positive. Wht i ()> n >? Wht i ()< n <? Wht i ()> n >? 7. A simple einite integrl prolem or you. (section 3.4) 8. Given the grph o some unction. I g t t where is some numer (like or or emple). Be le to look t the grph o n in things like g n g or emple (rememer you re given the grph o NOT the grph o g ). g() woul e the net signe re uner the grph o (t) etween n. o (t) t (recll the secon prt o the FTC). Think out this! g woul e the y vlue o the grph 9. Another einite integrl prolem or you BUT this one is not s simple s #7. This time the ntierivtive will e n inverse trig. unction (so e sure tht you cn evlute things like section 4.3 HW # 5 n 6).. An ineinite integrl involving sic trigonometric unction. Cn e solve y sustitution (section 4.). Be le to evlute einite integrls using sustitution (section 4.). Note: Be sure to prctice VARIETY o prolems involving ierent integrns (some involving trig unctions (like HW #7 section 4. n some involving e like e ) e u u u u u e ] e u e e e e e 3. Given unction know how to tell things like.where it is continuous, where it hs m or min, where it is incresing or ecresing, where the erivtive eists, where the secon erivtive is positive n negtive. (sections.9,.,.)
4. I you re given the erivtive o unction, e le to tell where the originl unction hs reltive m(s) n min(s) (section.9). See Test #4 question #3,4 or n emple. 5. Given the ccelertion unction o some prticle long with two initil conitions (one or velocity n one or position), e le to in the position unction. (section 3.) Rememer tht in orer to in the velocity unction just integrte the ccelertion unction (use the initil conition given or velocity to HELP you in the constnt o integrtion tht you get in your velocity unction). THEN integrte your velocity unction to otin your position unction AND now use the initil conition given or position to in the constnt o integrtion or the position unction. 6. Be le to in out where given unction is ecresing (section.9). 7. Mke sure tht you unerstn L Hopitl s Rule n WHEN it pplies! (section.) I you otin or when you o irect sustitution then it pplies so long s the unction meets the criteri spelle out in L Hopitl s rule (go re it) 8. Be sure tht you unerstn the vrious properties o the einite integrl tht we covere (section 3.4) 9. Be le to evlute einite integrls or sic sine n cosine unctions (section 3.4) 3. Be le to evlute einite integrl or piece-wise eine unction (section 3.4) Rememer tht IF the piece-wise eine unction is eine ierently on one prt o the intervl tht you re integrting over thn n nother then you must split the integrl up using properties o the einite integrl! Fin ln = ln ln = 4 ln ln Note: Wht is e pproimtely equl to? 3. Mke sure tht you review the properties o the einite integrl tht we went over (section 3.3) See Test #5 prolems 5 n 6! 3. Be le to in points o inlection or given unction (section.). Rememer just ecuse n -vlue is possile point o inlection, it oesn t men tht it ctully is. 5? Emple: Wht is the Point o Inlection or 5 A. 3, B.,3 C., D.,3 E. 3,
5 4 3 5 5 5 only "possile" p.o.i is t = Set up tle n check to see there is sign chnge in s you move cross =. 5,3 5 5 3 Answer 33. Be le to in limits t ininity! (section.) Here re couple o prolems or you to prctice BUT e sure to review your notes n HW!!! 4 Fin lim 3 6 3 5 5 3 4 A. B. C. D. E. 3 5 9 4 9 9 Fin lim A. B. C. D. 3 E. 3 4 5 5 5 Be sure to review your notes on limits t ininity. There re QUICK n esy wys to o these type o prolems! 34. Be le to pply the etreme vlue theorem to in n solute m o solute min vlue o unction on given intervl. (section.7) Emple: Wht is the mimum vlue o the unction 3 3 on,3 A. 3 B. C. D. 6 E. 8? 3 3 6 6 6 6 criticl numers re, Note tht BOTH o these re in our intervl (tht is not lwys the cse) 3 8 9 3 8 So the solute mimum o the unction on the given intervl is 8 35. Be le to etermine intervls on which unction is incresing or ecresing (section.9). 3 3 36 ecresing? C. 3, E Emple: On wht intervl(s) is A., 3 B.,3 D.,3.,
Tke the erivtive n in the criticl numers n 3. Brek up the omin o (), your tle will hve 3 intervls (since you hve two criticl numers). The sign o the erivtive is only negtive on the intervl rom -3 to so tht is the only intervl on which the originl unction is ecresing. 36 n 37. Review over the rectiliner motion prolems rom section.9 Emple: A prticle moving long horizontl line hs position unction s t t t t 4 3 4 8. Fin it s ccelertion unction. When oes the oject spee up n slow own? When oes the oject chnge irection? Note: The one on your inl em will work out MUCH nicer (n quicker) thn this one BUT this still gives you one to prctice tht you hven t seen (you cn in others in your notes n HW etc.). Criticl numers will e, ½, n (plces where the velocity is zero.i.e. the only plces where the oject MIGHT chnge irection). Your PPOI will e 3 3 3 3.789 n. (your ccelertion unction is qurtic tht oesn t ctor so use the qurtic 6 6 ormul oviously you won t hve THIS issue on the Finl Em s no clcultors re llowe on this prt). 46 6 t t t Spees up on,. ;.5,.789 ;, Slows own on.,.5 ;.789, Chnges irection t n secon. 38. Be le to in the vlue o c gurntee y the Men Vlue Theorem or erivtives (section.8) c 39. y y I cos wht oes y equl? I cot wht oes y equl? You re GIVEN the erivtive o y, just in the ntierivtive o it to otin y. More ormlly you coul solve the ierentil eqution y seprtion o vriles. 4. Be le to evlute einite integrl involving n solute vlue unction. (like section 3.4 HW # 8) Be sure to check the einition o the integrn to see IF the prolem nees to e split into prts! 4 45. Be sure tht you unerstn Newton s Metho (section.6), the irst erivtive test (section.9), the secon erivtive test (section.), the men vlue theorem or erivtives (section.8) n the men vlue theorem or integrls (section 3.4). In prticulr mke sure tht you know ectly wht ech o these is use or in Clculus! 46 5. Severl ineinite integrl prolems. Most o these re VERY quick n require little i ny work t ll! Be sure to know ALL o the integrtion ormuls n rules tht we hve covere in this course! THIS TEST WILL FAVOR THE EXTRAORDINARILY PREPARED STUDENT!