1 Tangent Line Problem

Transcription

1 October 9, 018 MAT18 Week Justi Ko 1 Tget Lie Problem Questio: Give the grph of fuctio f, wht is the slope of the curve t the poit, f? Our strteg is to pproimte the slope b limit of sect lies betwee poits, f d b, fb. The pproimtio improves s b gets closer d closer to. b b Defiitio 1. The slope of sect lie pproimtio for f betwee poits, f d + h, f + h for h > 0 is give b f + h f. h The sect lie pproimtio c be visulized below f h f + h f }{{} f + h h + h Pge 1 of 7

2 October 9, 018 MAT18 Week Justi Ko Defiitio. The slope f of the tget lie to f t poit is give b f d f + h f lim. d h 0 h This is the limit of sect of sect lies betwee the poits, f d + h, f + h s h 0. If the umber f eists, the we s f is differetible t d we cll the qutit f the derivtive of f t. 1.1 Applictio to Velocit Let st be the positio of prticle t time t. I this cotet, Defiitio 1 d Defiitio hve the followig iterprettios 1. Sect Lie: The verge velocit v v of the prticle is give b the sect lie pproimtio of the fuctio st o the itervl t b, v v sb s b. Tget Lie: The istteous velocit v ist is the tget lie of the fuctio st t the poit s + h s v ist lim h 0 h 1. Emple Problems Useful Formuls: The equtio of tget lie pproimtio of the fuctio f t the poit is give b f f. Problem 1. Let f, fid the sect lie betwee the poits 1, f1 d 4, f4 Solutio 1. Tkig d h 3 i our formul, we hve f4 f Problem. Suppose tht the positio of prticle movig horizotll o the -is is give b st t 3 1 for t [0, 10]. Fid the verge velocit of the object o the time itervl [0, 5]. b Fid the istteous velocit t time t. Solutio. Prt Tkig 0 d h 5 i our formul, the verge velocit is give b t s5 s Prt b The istteous velocit is give b ds s1 + h s1 1 + h 3 1 h 3 + 3h + 3h lim lim lim 3. dt t1 h 0 h h 0 h h 0 h Pge of 7

3 October 9, 018 MAT18 Week Justi Ko Are Problem Questio: Give the grph of fuctio f, wht is the et re the re bove the -is d uder the curve f mius the re below the -is d bove the curve of f of the grph betwee the poits d b? b Our strteg is to divide the regio [, b] ito subitervls d pproimte the re b limit of rectgles pproimtig our fuctio. The pproimtio improves b tkig lrger d lrger. b b Defiitio 3. The Riem sum pproimtio of b f d o the itervl [, b] with uiform subitervls is give b S [,b] f f i where b d i [ + i 1, + i ]. The pproimte et re of the grph f is give b the Riem Sum pproimtio. Pge 3 of 7

4 October 9, 018 MAT18 Week Justi Ko Remrk: We usull smple our fuctio f t the right edpoit, midpoit, or left edpoit of ech itervl: 1. Right Riem Sum: Tke i + i. Midpoit Riem Sum: Tke i + i 1 3. Left Riem Sum: Tke i + i 1 The midpoit pproimtio c be visulized below f 5 5 b Defiitio 4. The et re of the grph f o the itervl [, b] is give b the defiite itegrl of f o [, b]. We cll the qutit b f d the defiite itegrl of f o [, b], d it is defied b b f d lim f i where b d i [ + i 1, + i ]. This is the limit of Riem sum pproimtios s. If the umber b f d eists1, the we s f is itegrble o [, b]..1 Applictio to Velocit Let vt be the velocit of prticle t time t. I this cotet, Defiitio 4 hs the followig iterprettios 1. Defiite Itegrl of f : The distce trveled b the prticle is give b the defiite itegrl of v o the itervl t b, which is give eplicitl b the formul 1 The limit hs to eist d must ll be ideticl for ll choices of smples i. b vt dt. Pge 4 of 7

5 October 9, 018 MAT18 Week Justi Ko. Defiite Itegrl of f: The et distce trveled or displcemet d v of the prticle is give b the defiite itegrl of v o the itervl t b, which is give eplicitl b the formul. Emple Problems b vt dt. Useful Formuls: The followig formuls for the prtil sums of umber will be useful to compute the Riem Sums of certi fuctios 1. Sum of first costts:. Sum of first itegers: 3. Sum of first squres: 4. Sum of first cubes: 1. 1 i i i 3 4 Problem 1. Approimte the re uder the curve f bove the -is o the itervl [0, 10] usig 10 uiform subitervls d smplig f t the right edpoit of ech itervl. Solutio 1. We tke 0, b 0, d 0 i Defiitio 3. Sice we re smplig t the right edpoits, we choose b i i [i 1, i ] where. Therefore, usig our formul, we hve S [0,10] f fi i i sice i + 1. Problem. Approimte the re uder the curve f bove the -is o the itervl [0, 10] usig uiform subitervls d smplig f t the right edpoit of ech itervl. Wht does the re coverge to whe we tke. Solutio. We tke 0, b 0, with vrible i Defiitio 3. Sice we re smplig t the right edpoits, we choose 10 0 i i [i 1, i ] where. Therefore, usig our formul, we hve S [0,10] f f i 10 10i i sice i + 1. Pge 5 of 7

6 October 9, 018 MAT18 Week Justi Ko Tkig, we hve lim S [0,10]f lim Note: The fil swer is the sme s 10 0 d Problem 3. Approimte the re uder the curve f bove the -is o the itervl [0, 1] usig 100 uiform subitervls d smplig f t the left edpoit of ech itervl. Solutio 3. We tke 0, b, d 00 i Defiitio 3. Sice we re smplig t the left edpoits, we choose 1 i i 1 [i 1, i ] where 100. Therefore, usig our formul, we hve S [0,1] f f i i i 1 99 i j b reideig j i 1. sice j j0 j j Problem 4. Approimte the re uder the curve f bove the -is o the itervl [1, 5] usig 100 uiform subitervls d smplig f t the right edpoit of ech itervl. Solutio 4. We tke, b 5, d 00 i Defiitio 3. Sice we re smplig t the right edpoits, we choose 5 1 i + i [1 + i 1, 1 + i ] where Therefore, usig our formul, we hve S [0,1] f f 1 + i i i i + i i + i formuls 1,, 3 Pge of 7

7 October 9, 018 MAT18 Week Justi Ko Problem 5. Approimte the re uder the curve f bove the -is o the itervl [0, 1] usig uiform subitervls d smplig f t the midpoit of ech itervl. Wht does the re coverge to whe we tke. Solutio 5. We tke 0, b, with vrible i Defiitio 3. Sice we re smplig t the midpoits of the itervls, we choose i i 1 [i 1, i ] where. Therefore, usig our formul, we hve S [0,1] f f i 1 i i As, we hve 4i 4i i 4 i Remrk: The fil swer is the sme s lim d usig formuls 1,, 3 Problem. Approimte the vlue of l d b usig left edpoit Riem sum d 4 1 uiform subitervls. Solutio. We tke, b, d 4 i Defiitio 3. Sice we re smplig t the left edpoits, we choose i + i 1 [1 + i 1, 1 + i ] where b 4. Therefore, usig our formul, we hve S [1,] f f 1 + i l 1 + i 1 4 l1 + l1.5 + l1.5 + l Pge 7 of 7