MATH Tst -SOLUTIONS Spring 5. Considr th curv dfind by x = ln( 3y + 7) on th intrval y. a. (5 points) St up but do not simplify or valuat an intgral rprsnting th lngth of th curv on th givn intrval. = 3 3y + 7 3 L = + 3y + 7 b. (3 points) St up but do not simplify or valuat an intgral rprsnting th ara of th surfac obtaind by rotating th curv on th givn intrval about th x-axis. 3 SA = πy + 3y + 7 c. (3 points) St up but do not simplify or valuat an intgral rprsnting th ara of th surfac obtaind by rotating th curv on th givn intrval about th y-axis. ( ) + SA = π ln 3y + 7 3 3y + 7
MATH Tst -SOLUTIONS Spring 5. ( points) Find th ara (if th ara is finit) btwn th curv f (x) = for x. x + and th x-axis A = = lim x + a a = lim x + a a x ( + ) Lt u = x du = du =. Also, whn x =, u =, and whn x = a, u = a. Thrfor, lim a a x + x + = lim a a/ du u + = lim ( arctanu) a a/ = lim arctan arctan a a = π = π squar units Th impropr intgral convrgs to π.
MATH Tst -SOLUTIONS Spring 5 3 3. (5 points) Find th partial fraction dcomposition of th rational xprssion. B sur to st up th ntir gnral systm of quations ndd to find th undtrmind cofficints, and thn solv this systm of quations. 3x + ( x x +) ( x +) Apply Partial Fraction Dcomposition to th rational xprssion: 3x + x x + x + B x x + ( )( x +) = A Equat Numrators: 3x + = A x ( ) x + + Cx + D = A( x ) x + ( ) + B( x +) + Cx + D (x ) (x +) ( ) + B( x +) + Cx + D ( )( x ) ( )( x ) To Solv for A, B and C w nd to Equat Cofficints and giv th gnral systm of quations: 3x + = A + C ( )x 3 + A + B C + D ( )x + A + B D ( )x + A + B + D ( ) For x 3 : = A + C For x : 3 = A + B C + D For x : = A + C D For th constant : = A + B + D Solv for A, B, C, & D: On way is th following: Obsrv that for x =, 3x + = A x ( ) x + ( ) + B( x +) + Cx + D ( )( x ) givs = B or B =. Using th quation = A + C, thn C = A. Substituting into th quation = A + C D givs = A A D or D =. Now substitut into th quation = A + B + D, thn A =. Thus C = Thus th dcomposition of th rational xprssion is: 3x + x x + x x + ( ) = x + x
MATH Tst -SOLUTIONS Spring 5. Considr th curv Γ : x = t sint y = cost, t π. a. ( points) Find th x and y coordinats of all point(s) on th curv Γ such that th tangnt lin(s) of th curv Γ passing through ths point(s) ar horizontal. Find th paramtric form of th drivativ. = cost and sin t dt dt =. Thus sin t = cost To dtrmin horizontal tangnts find thos valus of t for which = cost is not. dt sin t = for t π if and only if t = ( π ) or π. dt = sin t is and At t = π cost Thrfor thr is a horizontal tangnt whn t Howvr, whn t = (π ), cost =. = π at ( x, y) = ( π,). sint So w nd to dtrmin th bhavior of th drivativ at t = π by finding lim, which is t ± cost an indtrminat form ( ). Us L hopital s Rul. sin t lim+ t cos + t t = L cost lim sin t =. sint In a similar mannr w can show lim t + cost =. Thus thr is not a horizontal tangnt at ithr t = or t = π.
MATH Tst -SOLUTIONS Spring 5 5. Considr th curv Γ : x = t sint y = cost, t π. b. ( points) Find th lngth of th curv Γ. Hint: cost t = sin. Not that L = π ds π = + dt dt dt ( cos ) ( sin ) π = t + t dt π = cos tdt Us th formula cost t = sin π = sin t dt π = sin t dt t = cos π ( π cos) = cos L =
MATH Tst -SOLUTIONS Spring 5 6 5. ( points) Find th cntr of mass (or cntroid), ( x, y), of a thin plat of uniform dnsity covring th rgion boundd abov by th curv y = x, blow by th curv y =, on th 3 3 x lft by th lin x =, and on th right by th lin x =, shown in th graph blow. Th rgion is symmtric about th x-axis. Thrfor, y =. Find x. A = x 3 x 3 = x 3 = x = 5 6 x = 6 5 x x 3 x 3 = 6 5 x = 3 5 x x = 5 ( x, y) = 5,
MATH Tst -SOLUTIONS Spring 5 7 6. ( points) Evaluat. / = + / / Evaluat on of th intgrals. = lim a + a Lt u = ln x du =. Also, whn x =, u =, and whn x = a, u = ln a. Thrfor, x lim a + a = lim a + ln a du u = lim a + u ln a = lim + a + ln a = Th intgral divrgs. Thus divrgs. /
MATH Tst -SOLUTIONS Spring 5 7. ( points) Us calculus to find th surfac ara of a sphr with radius r. Hint: considr th function f (x) = r x shown in th graph blow. f (x) = ( r x ) / ( x) = x r x r SA = π r x + r r x r x r x + x = π r x Using symmtry. r x r =π r = πrx r SA = πr squar units. (6 points) Considr th following dcomposition of th rational xprssion: x 3 5x +7x 35 = A x x + Bx+C x x +. ( ) x + + Dx+E Is this dcomposition a corrct partial fraction dcomposition or not? Ys or No NO If not, provid th corrct partial fraction dcomposition for th rational xprssion. Do not solv for th cofficints. Th systm of quations for th undtrmind cofficints in th abov dcomposition has no solution. Th corrct partial fraction dcomposition is givn blow. x 3 5x +7x 35 = A x x + B x x + ( ) x + + Cx+D
MATH Tst -SOLUTIONS Spring 5 9 9. (6 points) Eliminat th paramtr to find a Cartsian quation of th curv dfind by x = t +, y = t for t. Sktch th curv and indicat with an arrow th dirction in which th curv is tracd as th paramtr incrass. Not that t > which implis that t + > and thus x >. x = t ln x y = ln( x ) = ln( x ) = x Thus y = ( x ) for x 3. ( ) = ln t t = ln( x ). (6 points) Idntify th conic sction by dtrmining th appropriat quation that dscribs th curv dfind by x y + x 6y = 6. Graph it and labl th vrtics and cntr. ( x + x) ( y + 6y) = 6 ( x + x + ) ( y + 6y + 9) = 6 + 9 ( x + ) ( y + 3) = This is a hyprbola with its cntr at (-, -3). It has vrtics at (-,-3) and (-3, -3).