WW Prob Lib1 WeBWorK, Version Demo Course

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1 Tom Robbis WW Prob Lib WeBWorK, Versio.7 - Demo Course.( pt) If f x 5 xl x, fid f x. Fid f..( pt) If f x 8log 6 x, fid f 5. 3.( pt) Let f x.( pt) Let f x l si x f x Use logarithmic differetiatio to determie the derivative. f x f 5.( pt) Let x 6 f x l 3x 6 f x x x 6.( pt) Evaluate the defiite itegral. 5x 7 7.( pt) Evaluate the defiite itegral. e x lx [NOTE: Remeber to eter all ecessary *, (, ad )!! Eter arcta(x) for ta x, si(x) for six. ] 8.( pt) Verify that x x x ad use this equatio to evaluate 6 x 9.( pt) Evaluate the idefiite itegral. l x x.( pt) Evaluate the idefiite itegral. cosx 3six.( pt) Evaluate the idefiite itegral. x 5 x x.( pt) I 86 days ukow radioactive substace decay to percet of its size. (a) What is the half life of this substace? t (days) (b) How log will it take for a sample of mg to decay to 63 mg? T 3.( pt) A cell of some bacteria divides ito two cells every 5 miutes.the iitial populatio is bacteria. (a) Fid the populatio after t hours y t (fuctio of t) (b) Fid the populatio after hours. y (c) Whe will the populatio reach? T.( pt) A bacteria culture starts with 8 bacteria ad grows at a rate propotioal to its size.after hours there are 36 bacteria. (a) Fid the populatio after t hours y t (fuctio of t) (b) Fid the populatio after 8 hours. y 8 (c) Whe will the populatio reach 7? T Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, c UR

2 Tom Robbis.( pt) If f x 8arcsi x 3, fid f x..( pt) Let f x 8 cos x si x f x NOTE: The webwork system will accept arcsi x si x as the iverse of si x. 3.( pt) If f x 6arcta 3si x, fid f x..( pt) Let f x ta si 3x f x 5.( pt) Evaluate the defiite itegral. 5 x ad ot WW Prob Lib WeBWorK, Versio.7 - Demo Course [NOTE: Remeber to eter all ecessary *, (, ad )!! Eter arcta(x) for ta x, si(x) for six. ] 6.( pt) Evaluate the idefiite itegral. 6x x [NOTE: Remeber to eter all ecessary *, (, ad )!! Eter arcta(x) for ta x, si(x) for six. ] 7.( pt) Evaluate the idefiite itegral x 9x [NOTE: Remeber to eter all ecessary *, (, ad )!! Eter arcta(x) for ta x, si(x) for six... ] 8.( pt) Solve the iital value problem for y x ; xy 3y 5x 5 with the iitial coditio: y. y x Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, c UR

3 Tom Robbis.( pt) Evaluate the idefiite itegral. 9cos 7x.( pt) Evaluate the defiite itegral. π 8 π 8 ta 6x 3.( pt) Evaluate the defiite itegral. si π 3 x 3 x [NOTE: Remeber to eter all ecessary *, (, ad )!! Eter arcta(x) for ta x, si(x) for six... ].( pt) Evaluate the defiite itegral. x [NOTE: Remeber to eter all ecessary *, (, ad )!! Eter arcta(x) for ta x, si(x) for six... ] WW Prob Lib WeBWorK, Versio.7 - Demo Course 5.( pt) Evaluate the idefiite itegral. x x [NOTE: Remeber to eter all ecessary *, (, ad )!! Eter arcta(x) for ta x, si(x) for six... ] 6.( pt) Use itegratio by parts to evaluate the itegral. xe x 7.( pt) Use itegratio by parts to evaluate the itegral. 5xsix 8.( pt) Use itegratio by parts to evaluate the itegral. 3xcos3x 9.( pt) Use itegratio by parts to evaluate the itegral. 5xl 6x Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, c UR

4 Tom Robbis.( pt) Determie whether the sequeces are icreasig, decreasig, or ot mootoic. If icreasig, eter as your aswer. If decreasig, eter - as your aswer. If ot mootoic, eter as your aswer.. a 5 7 cos. a a. a 5 5.( pt) Determie the sum of the followig series ( pt) Determie the sum of the followig series. 7.( pt) If the followig series coverges, compute its sum. Otherwise, eter INF if it diverges to ifiity, MINF if it diverges to mius ifiity, ad DIV otherwise. (Hit: try breakig the summads up partial fractios-style.) 5.( pt) Determie the sum of the followig series. si si 6.( pt) Give: 7 A 3 For both of the followig aswer blaks, decide whether the give sequece or series is coverget or diverget. If coverget, eter the it (for a sequece) or the sum (for a series). If WW Prob Lib WeBWorK, Versio.7 - Demo Course diverget, eter INF if it diverges to ifiity, MINF if it diverges to mius ifiity, or DIV otherwise. (a) The series A. (b) The sequece A. 7.( pt) Match each of the followig with the correct statemet. C stads for Coverget, D stads for Diverget l 8.( pt) Match each of the followig with the correct statemet. C stads for Coverget, D stads for Diverget.. 3. e l ( pt) Compute the value of the followig improper itegral if it coverges. If it diverges, eter INF if it diverges to ifiity, MINF if it diverges to mius ifiity, or DIV otherwise. 3l x x 6 Determie whether is a coverget series. Eter C if the series is coverget, or D if it is diverget. 3l 6 Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, c UR

5 Tom Robbis.( pt) Match each of the followig with the correct statemet. A. The series is absolutely coverget. C. The series coverges, but is ot absolutely coverget. D. The series diverges ! 7 3..! 5.! 3! 6.( pt) Match each of the followig with the correct statemet. A. The series is absolutely coverget. C. The series coverges, but is ot absolutely coverget. D. The series diverges ! !3 3 si ( pt) Fid the iterval of covergece for the give power series. x 6 The series is coverget from x =, left ed icluded (Y,N): to x =, right ed icluded(y,n):.( pt) Match each of the power series with its iterval of covergece. x 8. 8 x 8.! 8 WW Prob Lib WeBWorK, Versio.7 - Demo Course 9x 3. 8! 9x 8. 8 A. 6 B. 8 9 C. D ( pt) Fid the iterval of covergece for the give power series. x 8 3 The series is coverget: from x =, left ed icluded (Y,N): to x =, right ed icluded (Y,N): x c x 8x 6.( pt) Suppose that Fid the first few coefficiets. c c c c 3 c Fid the radius of covergece R of the power series. R. 7.( pt) The fuctio f x is represeted as a power 3x series f x c x Fid the first few coefficiets i the power series. c c c c 3 c Fid the radius of covergece R of the series. R. Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, c UR

6 Tom Robbis.( pt) Decide if the poits give i polar coordiates are the same. If so, eter T. If ot, eter F. π π 3 59π 59π 7π 3π 77π π 8π π 3 3 3π 3 3π.( pt) For each set of Cartesia coordiates x y, match the equivalet set of Polar coordiates r θ, with θ π. ( 6.5, 9. ). ( -7.6,.9 ) 3. ( -8.8,.9 ). ( -6.9, 7.3 ) A B C D ( pt) For each set of Polar coordiates, match the equivalet Cartesia coordiates A. 5 9 B C D E. 6 F ( pt) Match each polar equatio below to the best descriptio. Possible aswers are C,E,H,L,P,R,S,V,ad Z. DESCRIPTIONS C. Circle cetered at origi, E. Ellipse, H. Hyperbola, L. Lie either vertical or horizotal, P. Parabola, R. Circle ot cetered at origi, S. Spiral, V. Vertical Lie, Z. Horizotal Lie WW Prob Lib WeBWorK, Versio.7 - Demo Course POLAR EQUATIONS. r siθ. r cosθ 3. r 7 siθ cosθ. r 7 siθ 5. r 5.( pt) Match each polar equatio below to the best descriptio. Each aswer should be C,F,I,L,M,O,or T. DESCRIPTIONS C. Cardioid, F. Rose with four petals, I. Iwardly spiralig spiral, L. Lemaco, M. Lemiscate, O. Outwardly spiralig spiral, T. Rose with three petals POLAR EQUATIONS. r 8 8siθ. r 3siθ 3. r 8θ r. r 3 3cosθ 5. r 8cos3θ 6. r 6cosθ 6.( pt) Fid the area of the regio iside: r 8siθ but outside: r 3 7.( pt) Fid the area of the regio bouded by the give curve: r 9e θ o the iterval 9 π θ π. 8.( pt) Fid the area of the regio bouded by: r 6 siθ 9.( pt) A circle C has ceter at the origi ad radius 6. Aother circle K has a diameter with oe ed at the origi ad the other ed at the poit The circles C ad K itersect i two poits. Let P be the poit of itersectio of C ad K which lies i the first quadrat. Let r θ be the polar coordiates of P, chose so that r is positive ad θ Fid r ad θ. r θ.( pt) Fid the area iside the loop of the followig aco: r 5 siθ Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, c UR

7 Tom Robbis WW Prob Lib WeBWorK, Versio.7 - Demo Course.( pt) Evaluate the itegral. x x.( pt) Write out the form of the partial fractio decompositio of the fuctio appearig i the itegral: x 3 x x 63 Determie the umerical values of the coefficiets, A ad B, where A B. A deomiator B deomiator A = B = 3.( pt) Evaluate the itegral. 5 8x 3 x 3x.( pt) Evaluate the idefiite itegral. x x Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, c UR

8 Tom Robbis.( pt) Evaluate the it x x 8x 9 x.( pt) Evaluate the it usig L Hopital s rule if ecessary x x x x 3.( pt) Evaluate the it usig L Hopital s rule if ecessary.( pt) Evaluate the it 5.( pt) Evaluate x xe x x 5x 3 9x x x 7 6x 7x 3 t 9 t t t 5 6.( pt) Determie the ifiite it of the followig fuctios. Eter I for ad -I for.. x 3 x 3. x 7 3. x x x 7 x x 7 x 3 5. x 5 7.( pt) Fid the area uder the curve y x 3 from x to x t ad evaluate it for t t. The fid the total area uder this curve for x. (a) t = (b) t = WW Prob Lib WeBWorK, Versio.7 - Demo Course (c) Total area 8.( pt) Determie whether the itegral is diverget or coverget. If it is coverget, evaluate it. If ot, state your aswer as diverget. 6 x ( pt) Determie whether the itegral is diverget or coverget. If it is coverget, evaluate it. If it diverges to ifiity, state your aswer as INF (without the quotatio marks). If it diverges to egative ifiity, state your aswer as MINF. If it diverges without beig ifiity or egative ifiity, state your aswer as DIV. x 8 7.( pt) Determie whether the itegral is diverget or coverget. If it is coverget, evaluate it. If it is diverget, eter your aswer as -. x.( pt) Determie whether the itegral is diverget or coverget. If it is coverget, evaluate it. If it diverges to ifiity, state your aswer as INF (without the quotatio marks). If it diverges to egative ifiity, state your aswer as MINF. If it diverges without beig ifiity or egative ifiity, state your aswer as DIV. 5 x.( pt) Determie whether the itegral is diverget or coverget. If it is coverget, evaluate it. If it diverges to ifiity, state your aswer as INF (without the quotatio marks). If it diverges to egative ifiity, state your aswer as MINF. If it diverges without beig ifiity or egative ifiity, state your aswer as DIV. 3 x 5 5 Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, c UR

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