MAT 7 Tst Rviw (Spring ) Tst on April in PSA Sction.7 Implicit Drivativ Rmmbr: Equation of t tangnt lin troug t point ( ab, ) aving slop m is y b m( a ). dy Find t drivativ y d. y y. y y y. y 4. y sin( y ) 4. Find t quation of t tangnt lin to 5. Find t quation of t tangnt lin to y, y at t point =. y y at t point (, ). Sction.8 Drivativ of Logaritmic and Eponntial Functions d d Rmmbr: ( ) ( a ) a ln a d d d Not: (ln p),, p is a constant d d (ln ), d Practic problms. f ( ). / ( ) f. ( ) f ln 4. f ( ) ln ln 5. f ( ) ln( ) ( ) f ln 5 4 7. f ( ) 8. ( ) ( ) f 9. f( ) 9.. 4 f ( ) ln 4. f( ) 5 f ( ) ln Sction.9 Drivativ of Invrs Trigonomtric Functions Rmmbr t following: 4. f( ).. d (sin ) d d (tan ) d d d. (cos ) (sin ) d d d d 4. (cot ) (tan ) d d
d 5. (sc ) d d d (csc ) (sc ) d d Practic problms:. f ( ) sin ( ) sin (/ ) cos ( ).. 4. 5. f ( ) cos (ln ) csc (/ ) tan ( ) f ( ) sin (cos ) tan (/ ) f ( ) sin ( ) f ( ) sc ( ) sin (cos ) cos (cos ) f ( ) tan, simplify t answr. 7. Givn y f ( ), find ( f ()) 8. T function f ( ) ln, (.68, ) as an invrs function g(). Eplain wy. And find g() 5 9. T function f ( ) 6 as an invrs function g(). Eplain wy. Find g (7). Find quation of t tangnt lin to y = g() at (7, ).. Find ( g ) (), t drivativ of t invrs function of g if g( ) 9. Sction. Rlatd Rats Practic problms:.a puddl is vaporating in suc a way tat its diamtr is dcrasing at rat of. cm/min. At wat rat is t ara of t puddl dcrasing wn t diamtr is 8 cm.. T volum of a cub is incrasing at a rat of cm/min. How fast is t surfac ara incrasing wn t lngt of t dg is cm.. Two cars start moving from t sam point. On travls Sout at 6mi/ and t otr travls Wst at 5mi/. At wat rat is t distanc btwn t two cars incrasing two ours latr? 4. A car is travling at 5 mp du sout at a point / mil nort of an intrsction. A polic car is travling at 4 mp du wst at a point /4 mil ast of t sam intrsction. At tat instant, t radar in t polic car masurrs t rat at wic t distanc btwn t cars is canging. Wat dos t radar gun rgistr? 5. A -foot laddr lans against t sid of a building. If t top of t laddr bgins to slid down t wall at t rat of ft/s, ow fast is t bottom of t laddr sliding away from t wall wn t top of t laddr is 8 ft off t ground.
An oil tankr as an accidnt and oil pours at t rat of 5 gallons pr minut. Suppos tat t oil sprads onto t watr in a circl at a ticknss of. Givn tat ft 7.5 gallons. Dtrmin t rat at wic t radius of t spill is incrasing wn t radius racs 5 ft. Solution: T sap is lik a cylindr. W considr t volum quation dv dr V r r, wr incs ft dv 5 Also givn tat 5 gallons / m ft ft / m 7.5 dv dr dr dr Tus r (5) 5 7. Gravl is bing dumpd from a convyor blt at a rat of cubic ft pr minut, and its coarsnss is suc tat it forms a pil in t sap of a con wos bas diamtr and igt ar always qual. How fast is t igt of t pil incrasing wn t pil is ft ig? dv Solution: W ar givn, wr v d Tn w gt. 8. 8. A particl movs along t curv y. As it racs t point (, ), t y-coordinat is incrasing at a rat of 4 cm/s. How fast is t -coordinat of t point canging at tat instanc? dy d Solution: Givn y, 4, and y. Find. dy d W writ y. By diffrntiation w gt y. Using t givn valus d w gt cm / sc. 4.-4.4 Application/Optimization. Find t opn intrval wr is t function down?. Find t local trma of t function r dv 4 d f ( ) 6 ln( ) concav up, concav / f ( ) ( ). Find all critical numbrs of f ( ) 7 / / 8 and also maimum and minimum valus if any 4. Grap t function f ( ) 4 4 5 9sowing all maimum, minimum valus, opn intrval(s) wr t function is incrasing or dcrasing..
/ 5. Find t opn intrval wr t function f ( ) ( ) is incrasing or dcrasing. Find also local maimum and local minimum if any. A farmr as 5 ft of fncing and wants to fnc of a rctangular fild tat bordrs a straigt rivr. H nds not to fnc along t rivr. Wat ar t dimnsions of t fild tat as t largst ara? 7. Find t radius of t rigt circular cylindr of largst volum tat can b inscribd in a rigt circular con wit radius 6 incs and igt incs. 8. An 8 ft tall fnc runs paralll to t sid of a ous ft away. Wat is t lngt of t sortst laddr tat clars t fnc and racs t ous? Assum tat t vrtical wall of t ous and t orizontal ground av infinit tnt. Answr: Lngt approimatly 5 ft. 9. T managr of a -unit apartmnt compl knows from princ tat all units will b occupid if t rnt is $8 pr mont. A markt survy suggsts tat, on avrag, on additional unit will rmain vacant for ac $ incras in rnt. Wat rnt sould t managr carg to maimiz rvnu?. A rigt circular cylindr is inscribd in a spr of radius r. To find t largst possibl surfac ara of suc a cylindr.. A 8 cubic foot tank wit a squar bas and an opn top is to b constructd of a st of stl of a givn ticknss. Find t lngt of a sid of t squar bas of t tank wit minimum surfac ara.. Find t -coordinat(s) for t points on t grap of y tat ar closst to t point (, 5).. Find t maimum ara of t circl inscribd in an quilatral triangl wos sids av lngt mtrs. Round your answr corrct to two dcimal placs if ndd. Sction 4.5 Linar Approimation W av t linar approimation f ( ) f ( ) f ( )( ) Practic problms:. Find t linar approimation of f ( ) at and tus approimat t valus of 8. and 5.. Evaluat 8 using Linar Approimation. Prov tat for clos to, and illustrat tis approimation by drawing t graps of y and y on t sam scrn. W considr f ( ) f '( ) '(), as f a Now using linar approimation formula w find f ( ) f ( a) f '( a)( a) f ( ) f () f '()( )
Sction 4.6 Man Valu Torm First of all w discuss t spcial cas of Man Valu Torm. T Roll s Torm: Suppos tat f is continuous on t intrval [ ab,, ] diffrntiabl on ( ab, ) and f ( a) f ( b) tn tr is a numbr c ( a, b) suc tat f ( c ) T Man Valu Torm: Suppos tat f is continuous on t intrval [ ab,, ] diffrntiabl on ( ab, ) tn tr is a numbr c ( a, b) suc tat f ( b) f ( a) f () c b a Constant Torm: Suppos tat f ( ) for all on an opn intrval I, tn f must b a constant, tat is f () c, a constant Corollary: Suppos tat f ( ) g ( ) for all in som opn intrval I, tn for som constant c, f ( ) g( ) c, I. Dtrmin wtr Roll s Torm applis to t function f ( ) ( 9) on [, ]. If so, find t points tat ar guarantd to ist by Roll s Trom.. Dtrmin wtr Roll s Torm applis to t function f ( ) ( ) on [, ]. If so, find t points tat ar guarantd to ist by Roll s Trom.. Dtrmin wtr Man Valu Torm applis to t function f ( ) ( 9) on [, 9]. Sction 4.7 L Hopitals Rul W av noticd t following indtrminat forms,,,,,,. T f( ) L Hopital rul is applicabl for lim a g if tis limit as t form. T L Hopital rul ( ) f ( ) f ( ) is as follows lim lim. On may apply t rul rpatdly as long as t a g( ) a g ( ) indtrminat form appars. Eampls wit t abov indtrminat forms:. lim lim
. lim lim, dos not ist (DNE). lim lim 4. lim lim, DNE as t rsult is not a finit numbr 5. lim lim lim lim lim 7. ln lim csc 8. lim ln( ) 9. lim ln. lim. lim (sin ). lim( / ). lim 4 cos sin 4. lim sin sin 5. lim lim.5 7. lim sin sin(sin ) 8. lim sin 9. lim ln = sin. lim DNE
Sction 4.8 Intgration. Find f() if. Find f() if. Find g() wn g ( ), g() 4, g () 4. Find f(), givn f ( ) sin, f (), f (), f () Find t following indfinit intgral 7. 8. 9.... 5sin( ) cos( ) cos C 4 d 4 6 6 d ( / 6 6 ) d / ( 5( ) ) d sin sc d C d C Answr: Sction.7:. / y. y/. Sction.8:. (4 ). 4. 6ln 7 5. 8 648 ( ln ) / 6 ( / ) y y y 4. y/ 5. y. (/ ln ) ln 7. ( ln ) 8.
8 9. (9 ).. 4 /. 5. ( / ) 5ln( / ) /( ) ( ) ( / ) Sction.9:. Sction.:. 4. (ln ) ( ) 4. 5. 4 8. ½ 9. y / ( 7). /76.4 cm / min. 4 / cm / min. 65 mp. 7. / / Sction 4.-4.4:. Up on (/, ). Ma (.8,.). Ma (-,.4); min(, -.4) 4. Min(.6, -.46), do t rst by yourslf 5. Incrass on its domain, no trmum 65 ft. by 5ft 7. Radius = inc, igt 67 inc. 8. 5 ft. 9. $9 Sction 4.5: Look at your class not. Sction 4.6:. No.. Ys, / only. You try