Math 122 Test 3 - Review 1

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Osborne Sydney Reed
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1 I. Sequeces ad Series Math Test 3 - Review A) Sequeces Fid the limit of the followig sequeces:. a = +. a = l 3. a = π 4 4. a = ta( ) 5. a = + 6. a = + 3 B) Geometric ad Telescopig Series For the followig series, determie if the it coverges or diverges, ad if it coverges, fid the sum. 7. ( 5 8. ( ) C) Divergece Term Test For the followig series, use the -th term test to determie if the series diverge. 0. ( D) p-series For the followig series, use the p-test to determie if the series coverges or diverges

2 E) Itegral Test For the followig series, use the itegral test to determie if the series coverges or diverges. Be sure to check that the fuctio f(x) is decreasig. 6. e 7. = (l ) 3 8. = 3 l F) Compariso ad Limit Compariso Test = si ( + ) G) Ratio ad Root Test ( ) () ( + )! H) Absolute ad Coditioal Covergece Determie if the followig series coverge absolutely, coverge coditioally, or diverge ( 3 ( l( + ) ( ( ( ( ( (

3 I) Power Series Fid the radius ad iterval of covergece for the followig power series: (3x. (x (x (x ( x x ()! x (x 3 ( + )! (x (x ) ( ) + (x 4 J) Taylor Series Fid the Maclauri series of: 47. f(x) = x e x 48. f(x) = l( + x ) 49. f(x) = l +x x 50. Use a series to show lim x 0 cos x x = Differetiate the Maclauri series for xe x ad use the result to fid + 5. Fid the Maclauri series for x si x 53. Use a series to show lim x 0 x = 54. Fid the first four ozero terms of the Maclauri series for si x + x.

4 K) Miscellaeous Determie if the followig series coverge absolutely, coverge coditioally, or diverge ( ( l e 400 ( si ) 3 3 si = l + l () ( + )! + ( =3 + si(π) ( + 7. ( 5 ) ( ( ( 7 = (l ) ( 6 7 ( 0 ( (l )(l(l )) (3 + ) 3/5 ( 3 ()! ( () ()! ( + ( 3 + ( 3 +

5 II. Parametric / Polar Equatios A) Parametric Equatios 86. Sketch ad idetify the curve x = +si t ad y = 3 cos t for 0 t π by elimiatig the parameter, ad label the directio of icreasig. 87. Fid the arc legth of the curve x = t +, y = t 3 3 for 0 t. 88. Fid dy dx at the poit where t = π/4 without elimiatig t for x = 5 cos t, y = 3 + si t. 89. Sketch the curve x = cos t, y = 3 si t for 0 t π/4 by elimiatig the parameter t ad label the directio of icreasig t. 90. Fid d y dx at the poit where t = 3π/4 without elimiatig t for x = 5 cos t, y = 3 + si t. 9. Fid the equatio of the taget lie to x = t, y = 3t 3 at the poit (4, 4) 9. For the curve give by x = t + t ad y = t 3 t, fid the slope ad cocavity at the poit (4, 3). B) Polar Equatios 93. Express the polar equatio r = 4 sec θ ta θ i rectagular form. 94. Express x(x + y ) = (3x y ) i polar form. Graph the followig polar equatios: 95. r = 3 si 3θ 96. r = si θ 97. r = θ 98. r = si θ. 99. r = + si θ. 00. r = + cos θ.

6 Fid the polar equatios for each of the equatios give i rectagular form. 0. x = 0. xy = x y = 04. x + (y ) = Fid the area commo to both r = cos θ ad r = + cos θ 06. Fid the are iside r = si θ ad outside r = cos θ. 07. Fid the area iside both r = 3 cos θ ad r = + cos θ. 08. Fid the area of the regio eclosed by r = + cos θ. 09. Fid the area of the regio eclosed by the cardioid r = ( + cos θ) 0. Fid the area of the regio that is iside the circle r = ad outside the cardioid r = cos θ. Fid the area of the regio eclosed by r = cos θ.. Fid the area of the regio that is commo to r = a( + si θ) ad r = a( si θ). 3. Fid the slope of the taget to the curve r = 3 si θ at the poit θ = π. 4. Fid the slope of the taget to the curve r = si θ at the poit θ = π/3. C) Coic Sectios 5. Fid the equatio of the parabola with vertex (, 3) ad directrix x = Fid the ceter, foci, vertices of the ellipse 8y + 6x 36x 64y + 34 = 0 7. Fid the equatio of the ellipse with a major axis legth of, a mior axis legth of 8, ceter at (0, 0), ad foci o the x-axis. 8. Fid the equatio of the ellipse with a major axis legth of 6, a mior axis legth of 4, ceter at (, 4), ad foci o y = Fid the equatio of the hyperbola with foci (, 6) ad (, 0) ad vertices 6 uits apart.

7 Aswers Diverges 8. Coverges, 6 9. Coverges, 5 0. Diverges. Diverges. Diverges 3. Diverges 4. Diverges 5. Diverges 6. Coverges 7. Coverges 8. Diverges 9. Diverge 0. Coverge. Coverge. Coverge 3. Diverge 4. Coverge 5. Coverge 6. Coverge 7. Coverge 8. Coverge absolutely 9. Coverge coditioally 30. Coverge coditioally 3. Diverge 3. Coverge absolutely 33. Coverge coditioally 34. Diverge 35. Coverge absolutely 36. ( /3, /3) 37. r =, [, 3) 38. r = 0, x = r =, x 40. (, ] 4. ( 3/, 3/) 4. (, ) 43. (, ) 44. [0, ) 45. (, 3] 46. R = 0 ad x = 4 oly. 47. x + x 4 + x6 + x8 3! x x4 + x6 3 x x + x3 3 + x e 5. + x + 4x + 8x x x + 5x3 6 5x Diverge 56. Coverge absolutely 57. Diverge 58. Diverge 59. Diverge 60. Diverge 6. Coverge absolutely 6. Diverge 63. Diverges 64. Diverge 65. Coverge absolutely 66. Coverge absolutely 67. Diverge

8 68. Coverge absolutely 69. Coverges absolutely 70. Coverge coditioally 7. Diverge 7. Diverge 73. Coverge absolutely 74. Coverge absolutely 75. Coverge absolutely 76. Coverges absolutely 77. Diverge 78. Coverges 79. Coverge coditioally 80. Diverge 8. Coverge absolutely 8. Coverge absolutely 83. Coverge coditioally 84. Coverge absolutely 85. Diverge 87. S = 7 (3 3 8) dy dx = d y dx = 9. y = 9x 9., cocave up. 93. x = 4y 94. r = (3 cos θ si θ ta θ) 0. r = sec θ 0. r cos θ si θ = r cos θ r si θ = 04. r = 4 si θ 05. Area = 3π Area = π A = 6π 09. 6π 0. π 4. A = π. A = a ( 3π 4 ) 3. m = 3 4. m = 3 5. (y 3) = 6(x ) (x 3) 6. (y 4) = ceter (3,4). foci (3 ±, 4), vertices (3 ± 8, 4) ad (3, 4 ± 6) 7. x 36 + y 6 = (x ) 64 (y ) 9 + (y 4) 4 (x ) 55 = = 07. Area = 5π 4

Review Problems Math 122 Midterm Exam Midterm covers App. G, B, H1, H2, Sec , 8.9,

Review Problems Math 122 Midterm Exam Midterm covers App. G, B, H1, H2, Sec , 8.9, Review Problems Math Midterm Exam Midterm covers App. G, B, H, H, Sec 8. - 8.7, 8.9, 9.-9.7 Review the Cocept Check problems: Page 6/ -, Page 690/- 0 PART I: True-False Problems Ch. 8. Page 6 True-False