Math 152 Final Exam Review

Transcription

1 Math 5 Fial Eam Review Problems Math 5 Fial Eam Review Problems appearig o your i-class fial will be similar to those here but will have umbers ad fuctios chaged. Here is a eample of the way problems selected may be chaged for your i-class fial eam. If problems ad were selected, they might appear like this: # modified A water tak has the shape of the surface geerated by revolvig the parabola segmet,0 4, about y the y-ais. If the tak is filled to a depth of 6 feet with a fluid weighig 80 lbs per cubic foot, fid the work W required to pump the cotets of the tak to a height of 4 feet above the top of the tak. # modified... below by the curve y cos, 0 /,... with the remaider of the questio as prited.. A water tak has the shape of the surface geerated by revolvig the parabola segmet y,0 6 about the y-ais. If the tak is filled to a depth of 8 feet with a fluid weighig 80 lbs. per cubic foot, fid the work W required to pump the cotets of the tak to a height of feet above the top of the tak.. Fid the eact value of the volume geerated by rotatig the regio i the first quadrat bouded above by the lie y, o the left by the y-ais, ad below by the curve y = ta, 0 π/, about (a) the -ais; (b) the lie y.. The desig of a ew airplae requires a gasolie tak of costat cross-sectio area i each wig. The tak must hold 5000 lb. of gasolie that weighs 4 lb/ft. A scale drawig of a cross sectio of the wig is show. Estimate the legth of the tak. y 0 =. ft., y =. ft., y =.0 ft, y =.9 ft, y 4 =.8 ft, y 5 =.6 ft, y 6 =.5 ft.; Horizotal spacig = ft. Use h b a T y0 y y y... y y (for subitervals of legth h )

2 Math 5 Fial Eam Review Problems b. A cross sectio of a wig is i the shape of the area below y6e si o the iterval [0, 7.5]. The presece of wires ad other mechaisms mea the tak must fit i the iterval from [0., 6.]. Use si trapezoids to estimate the area ad calculate the legth of the tak. If the legth of the tak (ad the wig) is to be accurate to the earest teth of a ich (ad all other values are assumed to be eact), how may trapezoids would be eeded to estimate the area of the tak cross sectio? Hit: To have the desired accuracy, the area of a cross sectio must have sigificat figures, or accuracy to the earest hudredth. 4. Fid the eact value of the legth of the curve: Completely simplify your aswer. e t, y e t for 0 t l. 8 t 5. Evaluate the followig itegrals. Show the details of substitutios you make i reachig your aswer. b. d. 4 si ( ) cos ( ) si ( ) cos ( ) 4 e e ta l e. arccot ( )

3 Math 5 Fial Eam Review Problems f. si( ) g. 9 (l ) h. i. 0 ta ( t) e 4t dt arcta( ) j. 0 4 k Sketch the curve y arccos. Fid the area betwee this curve ad the -ais from = - to =. 7. Compute the volume V of the solid formed by rotatig the area betwee the curve y ad the -ais (for < ) aroud the -ais. Is the volume a fiite umber? Sketch. 8. Iterpret the itegrals as areas ad use the result to epress the 0 4 sum above as oe defiite itegral. Evaluate the ew itegral. 9. Suppose b is a sequece of positive umbers covergig to m 0, with b =. Prove that the series b b coverges to a sum epressible i terms of m.

4 Math 5 Fial Eam Review Problems 0. Fid the Taylor series epasio of f ( ) cos( ) about π/. Write the series i sigma otatio i two forms: the first with ide value startig at zero, ad the secod with ide value startig at oe.. Determie whether the followig series are absolutely coverget, coverget, or diverget. Specify which tests you use ad show all relevat reasoig. b l a! ( ) d. e. f. ( ) ( ) ta e. The base of a certai solid is the regio of the y-plae bouded by y = ad y =. Every cross sectio perpedicular to the -ais is a semicircle with a diameter i the base. Fid the eact volume of this solid.. Determie whether coverges. If it coverges, fid its eact sum. 4 4

5 Math 5 Fial Eam Review Problems 4. Use the miimum umber of series terms eeded to compute 0. 0 si with a error of 7 magitude less tha 0. Show how you kow whe you have used the miimum umber of terms. 5. Fid the ceter of mass coordiates for the thi plate of costat desity coverig the regio bouded by y = l, y = -4, =, ad =. 6. Fid the iterval of covergece, clearly showig testig details ad ames of tests used at iterval edpoits. 4 b. k 0 l k k 5 k 7. Fid eactly the volume geerated whe the area bouded by y ais, y-ais, ad = is revolved about the y-ais. Simplify your aswer., the - 8. Write the equatio of the plae cotaiig the poits (0,, ) ad (, -, 7) ad orthogoal to the plae + y z = Fid the distace betwee the lies y z ad y z Fid the parametric equatios of the lie through the poit (, 6, 4) that itersects the z- ais ad is parallel to the plae y + 5z = 6. 5

6 Math 5 Fial Eam Review Problems. The force F (i Newtos) of a hydraulic cylider i a press is proportioal to the square of sec, where is the distace (i meters) that the cylider is eteded i its cycle. The domai of F is [0, π/] ad F(0) = 500. Fid F as a fuctio of. b. Fid the average force eerted by the press over the iterval [0, π/].. A weight of 55 pouds is to be lifted straight up with ropes at agles of 0 ad 0 degrees to the directio of travel as show below. Fid the tesio i each rope.. Fid the poit i which the lie through the origi perpedicular to the plae y z = 4 itersects the plae 5y + z = Fid all umbers b ad c such that the vectors bi -j + k ad ci + j + k are orthogoal ad have the same magitude. 5. Epress the vector a = i + 4j + 5k as the sum of a vector p which is parallel to the vector b = i j k ad a vector which is ormal to p. 6. Fid the volume of the solid formed by rotatig the regio bouded by y 9, the - ais, ad the lies = ± about the -ais. 6

7 Math 5 Fial Eam Review Problems 7. Use the Maclauri series represetatio of to fid a Maclauri represetatio of f ( ) l( ) which coverges for. Write the series i sigma otatio i two forms: the first with ide value startig at zero, ad the secod with ide value startig at two. 8. Fid the Maclauri series for of 0.6 e with error of magitude less tha f ( ) e. b. Use the series to estimate the value π ft-lbs MATH 5 Fial Eam Review Aswers. cubic uits b feet b. 4.4 feet ; uits 5. b. C cos ( ) cos ( ) d. 56. cubic uits. coverges to (.5, -.80) 6.,0 b. [,4) 7. cubic uits y + z = 5 e. 7

8 Math 5 Fial Eam Review Problems f. g = t + ; y = 6t + 6; z = t + 4 h. i. j. coverges to k. diverges 6. π 7. The solid has a fiite volume of π cubic uits.. F() = 500 sec b. 87 ewtos. rope at 0 : T 4 lb, T 57i 99 j rope at 0 : T 6 lb, T 57i 56 j. (4/, -/,-/) b = c = ; b = c = coverges absolutely because b. diverges coverges absolutely d. coverges e. coverges absolutely f. coverges absolutely b