Limits and an Introduction to Calculus

Transcription

1 Nme Chpter Limits d Itroductio to Clculus Sectio. Itroductio to Limits Objective: I this lesso ou lered how to estimte limits d use properties d opertios of limits. I. The Limit Cocept d Defiitio of Limit (Pges ) Defie limit. How to use the defiitio of limit to estimte limits Describe how to estimte the limit lim umericll. The eistece or oeistece of f () whe = c hs o berig o the eistece of... II. Limits Tht Fil to Eist (Pges ) The limit of f () s c does ot eist if of the followig coditios is true:. How to determie whether limits of fuctios eist. 3. Give emple of limit tht does ot eist. Lrso/Hostetler Preclculus/Preclculus with Limits Notetkig Guide Copright Houghto Miffli Comp. All rights reserved. 95

2 96 Chpter Limits d Itroductio to Clculus III. Properties of Limits d Direct Substitutio (Pges ) Let b d c be rel umbers d let be positive iteger. Complete ech of the followig properties of limits. How to use properties of limits d direct substitutio to evlute limits. lim b = c. lim = c 3. lim = c 4. lim = c Let b d c be rel umbers, let be positive iteger, d let f d g be fuctios with the followig limits. lim f ( ) = L d lim g( ) = K c c Complete ech of the followig sttemets bout opertios with limits.. Sclr multiple: lim[ b f ( )] = c. Sum or differece: lim[ f ( ) ± g( )] = c 3. Product: lim[ f ( ) g( )] = c f ( ) 4. Quotiet: lim = c g ( ) 5. Power: lim [ f ( )] = c Emple : Fid the limit: lim 3. 4 Lrso/Hostetler Preclculus/Preclculus with Limits Notetkig Guide Copright Houghto Miffli Comp. All rights reserved.

3 Sectio. Itroductio to Limits 97 If p is polomil fuctio d c is rel umber, the lim p( ) = c. If r is rtiol fuctio give b r() = p()/q(), d c is rel umber such tht q(c) 0, the lim r( ) = c. Emple : Fid the limit: TT 4 lim. Additiol otes Lrso/Hostetler Preclculus/Preclculus with Limits Notetkig Guide Copright Houghto Miffli Comp. All rights reserved.

4 98 Chpter Limits d Itroductio to Clculus Additiol otes Homework Assigmet Pge(s) Eercises Lrso/Hostetler Preclculus/Preclculus with Limits Notetkig Guide Copright Houghto Miffli Comp. All rights reserved.

5 Sectio. Techiques for Evlutig Limits 99 Nme Sectio. Techiques for Evlutig Limits Objective: I this lesso ou lered how to fid limits b direct substitutio d b usig the dividig out d rtiolizig techiques. I. Dividig Out Techique (Pges ) A idetermite form refers to... How to use the dividig out techique to evlute limits of fuctios Whe ou ecouter idetermite form b direct substitutio ito rtiol fuctio, ou c coclude... The vlidit of the dividig out techique stems from... Emple : Fid the followig limit: lim II. Rtiolizig Techique (Pge 865) Aother w to fid the limits of some fuctios is first to rtiolize the umertor of the fuctio. This is clled the. Recll tht rtiolizig the umertor mes... How to use the rtiolizig techique to evlute limits of fuctios Lrso/Hostetler Preclculus/Preclculus with Limits Notetkig Guide Copright Houghto Miffli Comp. All rights reserved.

6 00 Chpter Limits d Itroductio to Clculus III. Usig Techolog (Pges ) To fid limits of olgebric fuctios,... How to pproimte limits of fuctios grphicll d umericll IV. Oe-Sided Limits (Pges ) A oe-sided limit is... How to evlute oesided limits of fuctios Eistece of Limit If f is fuctio d c d L re rel umbers, the if d ol if... lim f ( ) = L c V. A Limit from Clculus (Pge 869) For -vlue, the limit of differece quotiet is epressio of the form... How to evlute limits of differece quotiets from clculus Direct substitutio ito the differece quotiet lws produces. Homework Assigmet Pge(s) Eercises Lrso/Hostetler Preclculus/Preclculus with Limits Notetkig Guide Copright Houghto Miffli Comp. All rights reserved.

7 Sectio.3 The Tget Lie Problem 0 Nme Sectio.3 The Tget Lie Problem Objective: I this lesso ou lered how to pproimte slopes of tget lies, use the limit defiitio of slope, d use derivtives to fid slopes of grphs. I. Tget Lie to Grph (Pge 873) The tget lie to grph of fuctio f t poit P(, ) is... How to defie the tget lie to grph To determie the rte t which grph rises or flls t sigle poit,... II. Slope of Grph (Pge 874) To visull pproimte the slope of grph t poit,... How to use tget lie to pproimte the slope of grph t poit III. Slope d the Limit Process (Pges ) A sect lie to grph is... How to use the limit defiitio of slope to fid ect slopes of grphs A differece quotiet is... Give the defiitio of the slope of grph. Lrso/Hostetler Preclculus/Preclculus with Limits Notetkig Guide Copright Houghto Miffli Comp. All rights reserved.

8 0 Chpter Limits d Itroductio to Clculus Emple : Use the limit process to fid the slope of the grph of f ( ) = + 5 t the poit (3, ). IV. The Derivtive of Fuctio (Pges ) The derivtive of f t is the fuctio derived from... How to fid derivtives of fuctios d use derivtives to fid slopes of grphs Give the forml defiitio of the derivtive. The derivtive f () is formul for... Emple : Fid the derivtive of f ( ) = 9. Homework Assigmet Pge(s) Eercises Lrso/Hostetler Preclculus/Preclculus with Limits Notetkig Guide Copright Houghto Miffli Comp. All rights reserved.

9 Sectio.4 Limits t Ifiit d Limits of Sequeces 03 Nme Sectio.4 Limits t Ifiit d Limits of Sequeces Objective: I this lesso ou lered how to evlute limits t ifiit d fid limits of sequeces. I. Limits t Ifiit d Horizotl Asmptotes (Pges ) Defie limits t ifiit. How to evlute limits of fuctios t ifiit To help evlute limits t ifiit, ou c use the followig: If r is positive rel umber, the lim r =. If r is defied whe < 0, the lim r =. Emple : Fid the limit: lim 3 3 If f () is rtiol fuctio d the limit of f is tke s pproches or, Whe the degree of the umertor is less th the degree of the deomitor, the limit is. Whe the degrees of the umertor d the deomitor re equl, the limit is. Whe the degree of the umertor is greter th the degree of the deomitor, the limit. Lrso/Hostetler Preclculus/Preclculus with Limits Notetkig Guide Copright Houghto Miffli Comp. All rights reserved.

10 04 Chpter Limits d Itroductio to Clculus II. Limits of Sequeces (Pges ) For sequece whose th term is, s icreses without boud, if the terms of the sequece get closer d closer to prticulr vlue L, the the sequece is sid to to L. Otherwise, the sequece tht does ot coverge is sid to. How to fid limits of sequeces Give the defiitio of the limit of sequece. Emple : Fid the limit of the sequece ( 3)(4 ) =. 4 3 Homework Assigmet Pge(s) Eercises Lrso/Hostetler Preclculus/Preclculus with Limits Notetkig Guide Copright Houghto Miffli Comp. All rights reserved.

11 Sectio.5 The Are Problem 05 Nme Sectio.5 The Are Problem Objective: I this lesso ou lered how to fid limits of summtios d use them to fid res of regios bouded b grphs of fuctios. I. Limits of Summtios (Pges ) The followig summtio formuls d properties re used to evlute fiite d ifiite summtios. How to fid limits of summtios. c =. i = 3. i = 3 4. i =, c is costt. 5. ( i ± bi ) = i ± b 6. k = k, k is costt. i i i To fid the limit of summtio,... Emple : Fid the limit of S() s. S( ) = i 5 3 Lrso/Hostetler Preclculus/Preclculus with Limits Notetkig Guide Copright Houghto Miffli Comp. All rights reserved.

12 06 Chpter Limits d Itroductio to Clculus II. The Are Problem (Pges ) Describe the re problem. How to use rectgles to pproimte res of ple regios d to use limits of summtios to fid res of ple regios The bsic strteg for fidig the re of the regio R is... The ect re of ple regio R is give b... Let f be cotiuous d oegtive o the itervl [, b]. The re A of the regio bouded b the grph of f, the -is, d the verticl lies = d = b is give b Emple : Fid the re of the regio bouded b the grph of f ( ) = ( 4) = 3 d = d the -is betwee Homework Assigmet Pge(s) Eercises Lrso/Hostetler Preclculus/Preclculus with Limits Notetkig Guide Copright Houghto Miffli Comp. All rights reserved.