s = 1 2 at2 + v 0 t + s 0

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1 Mth A UCB, Sprig A. Ogus Solutios for Problem Set 4.9 # 5 The grph of the velocity fuctio of prticle is show i the figure. Sketch the grph of the positio fuctio. Assume s) =. A sketch is give below. Note i prticulr tht i the regio where the velocity fuctio vt) is costt d positive, your positio grph should be stright lie with positive slope. 4.9 # 64. Show tht for motio i stright lie with costt ccelertio, iitil velocity v, d iitil displcemet s, the displcemet fter time t is s = t + v t + s If is costt the v, the tiderivtive of, is t + v. The the displcemet s is the tiderivtive of v = t + v, or t + v t + s. 4.9 # 7. The lier desity of rod of legth m is give by ρx) = / x, i grms per cetimeter, where x is mesured i cetimeters from oe ed of the rod. Fid the mss of the rod. As i Exmple.7. o pge, we re met to ssume tht lier desity is the derivtive of mss. More precisely, let mx) be the fuctio tht tells you the mss i grms of the [, x] portio of the rod where x is mesured i cm. The m) = d m is the tiderivtive of the lier desity ρx) = / x. The tiderivtive of / x = x / is x / = x. The mss of the whole rod is the simply m) = = grms. 5. # 4. Estimte the distce trveled usig the give velocity dt see text). Usig left edpoit estimtes for the velocity ft/s) durig ech time itervl s), we get the estimte )+855 )+9 5)+447 )+7459 )+56 59) ) =, 98 c 9 by Michel Christ. modified by A. Ogus All rights reserved.

2 Altertively, usig right edpoit velocity estimtes, we d get: 85 )+95 )+447 5)+74 )+559 ) )+455 6) = 6, 7 5. # 8. Write expressio usig Defiitio defiig the re uder the grph of the fuctio fx) = lx)/x s limit. Recll from p. 6 tht this defiitio uses right edpoits, so the expressio is: lim fx i ) x 5. #. Determie regio whose re is equl to: lim 5 + i ) f + ) i ) l + 7i)/) + 7i)/ l + 7i) l) + 7i This is right edpoit computtio of the re uder the grph of fx) = x betwee x = 5 d x = 7 so the = 5 7 i = x d 5 + = x i). 5. # 6. ) Let A be the re of polygo with equl sides iscribed i circle of rdius r. Divide the polygo ito cogruet trigles with cetrl gle π/ so show tht A = r si ) π b) Show tht lim A = πr ) The re of trigle hvig sides of legths,b with gle θ betwee them is b si θ. Here ech trigle hs = b = r d gle θ = π, d there re of them, so the totl re is r r si )) π = r si ) π b) Isted of the hit, which ivolves usig substitutio e.g. x = π/), we c use l Hospitl s rule thikig of s cotiuous vrible):

3 ) π lim r si = siπ/) r lim / = cosπ/) π/ ) r lim / = r lim cosπ/) π) = ) π) r = πr 5. #. Use Theorem 4 to evlute 4 x + x 5)dx. This theorem uses right edpoits : 4 x + x 5)dx f + 4 ) 4 i f + i) + i ) + + i ) 5 9i + i ) 9 i ) + i) ) 9 + ) + ) ) ) smller powers = lim 7 + smller powers = lim = 7 = ) 5. #. Express x 4 l x) s limit of Riem sums do ot evlute). Usig right edpoit sums, we get:

4 x 4 l x) + ) i) 4 l + i) + 9 i) 4 l + 9 ) 9 i) 5. # 4c. Use the grph i text) to evlute the itegrl 7 gx)dx 7 gx)dx = gx)dx + 6 gx)dx + 7 = trigle + semicircle + trigle = )4) + π) + )) = 9 + 4π 6 gx)dx 5. # 48. If 5 fx)dx = d 5 4 fx)dx =.6, fid 4 fx)dx. 4 fx)dx = 5 fx)dx fx)dx = -.6= # 5. Suppose f hs bsolute miimum vlue m d bsolute mximum vlue M. Betwee wht two vlues must fx)dx lie? Which property of itegrls llows you to mke your coclusio? fx)dx must lie betwee m d M by Property # 59. Use Property 8 to estimte xe x dx. The bsolute miimum vlue of xe x o [, ] is. The bsolute mximum vlue of xe x o [, ] is /e. Apply property 8, we get xe x dx /e. 5. # 65. If f is cotiuous o [, b], show tht fx)dx From fx) fx) fx) d Property 7, we get fx) dx fx)dx 4 fx) dx. fx) dx.

5 So fx)dx fx) dx. 5. # 68. Let f) = d fx) = /x if < x. Show tht f is ot itegrble o [, ]. Recll tht i the of defiite itegrl Pge 66), we divided the itervl [, ] ito subitervls of equl legth /. The the first term i the Riem sum, fx )/ = /x ) c be mde rbitrry lrge by mkig x rbitrrily close to. So the limit does ot exist. or oe c sy tht the limit is ot fiite umber.) So f is ot itegrble o [, ]. 5. # 9. Use Prt of the Fudmetl Theorem of Clculus to fid the derivtive of gy) = y t sitdt. By Prt of the Fudmetl Theorem of Clculus, g y) = y siy. 5. # 6. Use Prt of the Fudmetl Theorem of Clculus to fid the derivtive of y = cosx + v ) dv. Let gx) = x + v ) dv, the y = gcosx), pply the chi rule, y = g cosx) six). By Prt of the Fudmetl Theorem of Clculus, g cosx) = + cos x). Filly, y = g cosx) six) = + cos x) six) 5. #. Evlute x4/5 dx. By Prt of the Fudmetl Theorem of Clculus, x4/5 dx = 5 9 x9/5 ] = #. Evlute π/4 sec tdt. By Prt of the Fudmetl Theorem of Clculus, π/4 sec tdt = tx] π/4 = tπ/4) t =. 5. # 44. Wht is wrog withe equtio 4 dx = ] x x =. 4 is ot defied ot itegrble, ot cotiuous) over [-,]. So oe c ot pply Prt of the x Fudmetl Theorem of Clculus. 5. # 5 Evlute the itegrl x dx d iterpret it s differece of res. Illustrte with grph. 5

6 Apply Prt of the Fudmetl Theorem of Clculus, x dx = 4 x4 ] = )4 = 4-4 =

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