What Is Calculus? 42 CHAPTER 1 Limits and Their Properties

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1 60_00.qd //0 : PM Pge CHAPTER Limits nd Their Properties The Mistress Fellows, Girton College, Cmridge Section. STUDY TIP As ou progress through this course, rememer tht lerning clculus is just one of our gols. Your most importnt gol is to lern how to use clculus to model nd solve rel-life prolems. Here re few prolemsolving strtegies tht m help ou. Be sure ou understnd the question. Wht is given? Wht re ou sked to find? Outline pln. There re mn pproches ou could use: look for pttern, solve simpler prolem, work ckwrds, drw digrm, use technolog, or n of mn other pproches. Complete our pln. Be sure to nswer the question. Verlize our nswer. For emple, rther thn writing the nswer s.6, it would e etter to write the nswer s The re of the region is.6 squre meters. Look ck t our work. Does our nswer mke sense? Is there w ou cn check the resonleness of our nswer? GRACE CHISHOLM YOUNG (868 9) Grce Chisholm Young received her degree in mthemtics from Girton College in Cmridge, Englnd. Her erl work ws pulished under the nme of Willim Young, her husnd. Between 9 nd 96, Grce Young pulished work on the foundtions of clculus tht won her the Gmle Prize from Girton College. A Preview of Clculus Understnd wht clculus is nd how it compres with preclculus. Understnd tht the tngent line prolem is sic to clculus. Understnd tht the re prolem is lso sic to clculus. Wht Is Clculus? Clculus is the mthemtics of chnge velocities nd ccelertions. Clculus is lso the mthemtics of tngent lines, slopes, res, volumes, rc lengths, centroids, curvtures, nd vriet of other concepts tht hve enled scientists, engineers, nd economists to model rel-life situtions. Although preclculus mthemtics lso dels with velocities, ccelertions, tngent lines, slopes, nd so on, there is fundmentl difference etween preclculus mthemtics nd clculus. Preclculus mthemtics is more sttic, wheres clculus is more dnmic. Here re some emples. An oject trveling t constnt velocit cn e nlzed with preclculus mthemtics. To nlze the velocit of n ccelerting oject, ou need clculus. The slope of line cn e nlzed with preclculus mthemtics. To nlze the slope of, ou need clculus. A tngent line to circle cn e nlzed with preclculus mthemtics. To nlze tngent line to generl grph, ou need clculus. The re of rectngle cn e nlzed with preclculus mthemtics. To nlze the re under generl curve, ou need clculus. Ech of these situtions involves the sme generl strteg the reformultion of preclculus mthemtics through the use of limit process. So, one w to nswer the question Wht is clculus? is to s tht clculus is limit mchine tht involves three stges. The first stge is preclculus mthemtics, such s the slope of line or the re of rectngle. The second stge is the limit process, nd the third stge is new clculus formultion, such s derivtive or integrl. Preclculus mthemtics Limit process Clculus Some students tr to lern clculus s if it were simpl collection of new formuls. This is unfortunte. If ou reduce clculus to the memoriztion of differentition nd integrtion formuls, ou will miss gret del of understnding, self-confidence, nd stisfction. On the following two pges some fmilir preclculus concepts coupled with their clculus counterprts re listed. Throughout the tet, our gol should e to lern how preclculus formuls nd techniques re used s uilding locks to produce the more generl clculus formuls nd techniques. Don t worr if ou re unfmilir with some of the old formuls listed on the following two pges ou will e reviewing ll of them. As ou proceed through this tet, come ck to this discussion repetedl. Tr to keep trck of where ou re reltive to the three stges involved in the stud of clculus. For emple, the first three chpters rek down s shown. Chpter P: Preprtion for Clculus Preclculus Chpter : Limits nd Their Properties Limit process Chpter : Differentition Clculus

2 60_00.qd //0 : PM Pge SECTION. A Preview of Clculus Without Clculus With Differentil Clculus Vlue of f = f() Limit of f s when c pproches c c c = f() Slope of line Slope of d d Secnt line to Tngent line to Averge rte of chnge etween t nd t Instntneous t = t = rte of chnge t t c t = c Curvture of circle Curvture of Height of curve when c c Mimum height of on n intervl Tngent plne to sphere Tngent plne to surfce Direction of motion long line Direction of motion long

3 60_00.qd //0 : PM Pge CHAPTER Limits nd Their Properties Without Clculus With Integrl Clculus Are of rectngle Are under Work done constnt force Work done vrile force Center of rectngle Centroid of region Length of line segment Length of n rc Surfce re of clinder Surfce re of solid of revolution Mss of solid of constnt densit Mss of solid of vrile densit Volume of rectngulr solid Volume of region under surfce Sum of finite numer of terms... n S Sum of n infinite numer of terms... S

4 60_00.qd //0 : PM Pge SECTION. A Preview of Clculus = f() Tngent line P The tngent line to the grph of f t P Figure. The Tngent Line Prolem The notion of limit is fundmentl to the stud of clculus. The following rief descriptions of two clssic prolems in clculus the tngent line prolem nd the re prolem should give ou some ide of the w limits re used in clculus. In the tngent line prolem, ou re given function f nd point P on its grph nd re sked to find n eqution of the tngent line to the grph t point P, s shown in Figure.. Ecept for cses involving verticl tngent line, the prolem of finding the tngent line t point P is equivlent to finding the slope of the tngent line t P. You cn pproimte this slope using line through the point of tngenc nd second point on the curve, s shown in Figure.(). Such line is clled secnt line. If P c, f c is the point of tngenc nd Q c, f c is second point on the grph of points is given f, the slope of the secnt line through these two m sec f c f c c c f c f c. Q(c +, f(c + )) P(c, f(c)) f(c + ) f(c) Q Secnt lines P Tngent line () The secnt line through c, f c nd c, f c Figure. () As Q pproches P, the secnt lines pproch the tngent line. As point Q pproches point P, the slope of the secnt line pproches the slope of the tngent line, s shown in Figure.(). When such limiting position eists, the slope of the tngent line is sid to e the limit of the slope of the secnt line. (Much more will e sid out this importnt prolem in Chpter.) EXPLORATION The following points lie on the grph of f. Q., f., Q., f., Q.0, f.0, Q.00, f.00, Q.000, f.000 Ech successive point gets closer to the point P,. Find the slope of the secnt line through Q nd P, Q nd P, nd so on. Grph these secnt lines on grphing utilit. Then use our results to estimte the slope of the tngent line to the grph of f t the point P.

5 60_00.qd //0 : PM Pge 6 6 CHAPTER Limits nd Their Properties Are under Figure. = f() The Are Prolem In the tngent line prolem, ou sw how the limit process cn e pplied to the slope of line to find the slope of generl curve. A second clssic prolem in clculus is finding the re of plne region tht is ounded the grphs of functions. This prolem cn lso e solved with limit process. In this cse, the limit process is pplied to the re of rectngle to find the re of generl region. As simple emple, consider the region ounded the grph of the function f, the -is, nd the verticl lines nd, s shown in Figure.. You cn pproimte the re of the region with severl rectngulr regions, s shown in Figure.. As ou increse the numer of rectngles, the pproimtion tends to ecome etter nd etter ecuse the mount of re missed the rectngles decreses. Your gol is to determine the limit of the sum of the res of the rectngles s the numer of rectngles increses without ound. = f() = f() HISTORICAL NOTE In one of the most stounding events ever to occur in mthemtics, it ws discovered tht the tngent line prolem nd the re prolem re closel relted. This discover led to the irth of clculus. You will lern out the reltionship etween these two prolems when ou stud the Fundmentl Theorem of Clculus in Chpter. Approimtion using four rectngles Figure. Approimtion using eight rectngles EXPLORATION Consider the region ounded the grphs of f, 0, nd, s shown in prt () of the figure. The re of the region cn e pproimted two sets of rectngles one set inscried within the region nd the other set circumscried over the region, s shown in prts () nd (c). Find the sum of the res of ech set of rectngles. Then use our results to pproimte the re of the region. f() = f() = f() = () Bounded region () Inscried rectngles (c) Circumscried rectngles

6 60_00.qd //0 : PM Pge 7 SECTION. A Preview of Clculus 7 Eercises for Section. In Eercises 6, decide whether the prolem cn e solved using preclculus, or whether clculus is required. If the prolem cn e solved using preclculus, solve it. If the prolem seems to require clculus, eplin our resoning nd use grphicl or numericl pproch to estimte the solution.. Find the distnce trveled in seconds n oject trveling t constnt velocit of 0 feet per second.. Find the distnce trveled in seconds n oject moving with velocit of v t 0 7 cos t feet per second.. A icclist is riding on pth modeled the function f 0.0 8, where nd f re mesured in miles. Find the rte of chnge of elevtion when. Figure for Figure for. A icclist is riding on pth modeled the function f 0.08, where nd f re mesured in miles. Find the rte of chnge of elevtion when.. Find the re of the shded region. (, ) (0, 0) ( ) f() = (, 0) 6 Figure for Figure for 6 6. Find the re of the shded region. 7. Secnt Lines Consider the function f nd the point P, on the grph of f. () Grph f nd the secnt lines pssing through P, nd Q, f for -vlues of,., nd 0.. () Find the slope of ech secnt line. (c) Use the results of prt () to estimte the slope of the tngent line of f t P,. Descrie how to improve our pproimtion of the slope. 8. Secnt Lines Consider the function f nd the point P, on the grph of f. () Grph f nd the secnt lines pssing through P, nd Q, f for -vlues of,, nd. () Find the slope of ech secnt line. f() = See for worked-out solutions to odd-numered eercises. (c) Use the results of prt () to estimte the slope of the tngent line of f t P,. Descrie how to improve our pproimtion of the slope. 9. () Use the rectngles in ech grph to pproimte the re of the region ounded, 0,, nd. () Descrie how ou could continue this process to otin more ccurte pproimtion of the re. 0. () Use the rectngles in ech grph to pproimte the re of the region ounded sin, 0, 0, nd. () Descrie how ou could continue this process to otin more ccurte pproimtion of the re. Writing Aout Concepts. Consider the length of the grph of f from, to,. (, ) (, ) () Approimte the length of the curve finding the distnce etween its two endpoints, s shown in the first figure. () Approimte the length of the curve finding the sum of the lengths of four line segments, s shown in the second figure. (c) Descrie how ou could continue this process to otin more ccurte pproimtion of the length of the curve. (, ) (, )