Homework Problem Answers
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1 Homework Problem Answers Integration by Parts. (x + ln(x + x. 5x tan 9x 5 ln sec 9x 9 8 (. 55 π π + 6 ln 4. 9 ln 9 (ln ( / 6. 6 x sin x +6cos x. ( + x e x 8. 4/e 9. 5 x [sin(ln x cos(ln x] Integrating Trigonometric Functions 0. cos 5 x 5cos x. 8cos x 4 cos x. π/4. 69π/6 4. sin / x 4 5 sin5/ x + 9 sin9/ x tan x. 44/ sec x 4 sec x sec x tan x sin(πx 6π + 4 sin(5πx 0π. 69 csc x +69cotx ln sec x +tanx Trigonometric Substitution. 6+π (9 8x +x 4 x ++C 4. x 8x + 8 arcsin(9x+c 5. (x x x + 49 arcsin ( x 6. 9 x + x + 9 ln +x + x + x + x arctan ( x 5+6x 9x 4 8. (x +9 x +8x 8 ln x +9+ x +8x 9. 4 arcsin(x + 4 x x 4 0. ln(. x 9 x arcsin ( x Integrating Rational Functions. 5 ln. ln x + ln x x 4. x 9 ln(x +9+ arctan ( x +C 5. ln x 5 ln(x +9 5 arctan ( x 6. arctan x 5 (x +. ln x 6 ln(x + x + arctan ( x+ 8. ln 9 arctan 5 π 6 9. ln(x +6x +0 9 arctan(x + x+6 (x +6x x + x +6 6 x + 6 ln 6 x 4. ln sin x sinx+
2 4. ln 9 ex arctan ( e x 4 94 ln(ex ( x ln(x x +6 x + arctan 44. ln x 4 x+ ( x 45. ln x +4 ( 46. x arctan (x x 4 ln(x ++C 48. ln ln ln(x 4x ++C Integration Strategies 5. 5 tan θ + 5 ln cos θ 5. 5e π/6 5e π/ arctan /45 ( x x +6x ln(x ln ( x+5 x x ln(8 + e x +C 5. θ tan θ θ + ln cos θ e x 5x + 0 ln ( e x e x (x ++C 60. ln 6x+49 6x+49+ ( 4x 6. ln + 4x e x + ln e x e x + ln x 0 ln(x +6 arctan(x/ (ln x x arctan x (x + a5/ 8 a(x + a8/ arctan x 6. /5 68. / e 0. ln 6. Diverges. π/8 Improper Integrals Limits and L Hôpital s Rule / e Infinite Sequences Directions: Determine whether the sequence converges or diverges. If it converges, find the limit. 9. lim 80. lim a n = 8. Diverges 8. lim 8. lim a n = 84. lim
3 85. Diverges 86. lim 8. lim a n = 88. lim a n = e 89. lim a n = ln 90. Diverges 9. lim 9. Converges to Directions: Determine (a whether the sequence is increasing, decreasing, or not monotonic; and (b whether or not the sequence is bounded. 9. (a not monotonic (b not bounded 94. (a decreasing (b bounded Infinite Series Directions: Find the sum of the series. 95. Diverges 96. 5/ 9. 40/ Directions: Determine (a whether {a n } is convergent, and (b whether a n is convergent. 98. (a yes (b no Directions: Determine whether the series converges or diverges. If it is convergent, find the sum. 99. Converges to 8/ 00. Diverges 0. Diverges 0. Converges to cos cos 0. Converges to +e e 04. Diverges 05. Converges to / 06. Converges to e Directions: Determine (a the values of x for which the series converges, and (b the sum of the series for those values. 0. (a ( 4, 4 (b x 4 x 08. No 09. (a s =/, s =5/6, s =/4, and (n +! s 4 =9/0. (b s n = (n +! (c Integral Test Directions: Determine whether the series converges or diverges. 0. Converges. Converges. Converges. Converges 4. Converges Directions: Find the values of p for which the series is convergent. 5. p>
4 Comparison Tests Directions: Determine whether the series converges or diverges. 6. Diverges. Converges 8. Converges 9. Converges 0. Diverges. Converges. Diverges. Converges 4. Diverges Alternating Series Test Directions: Determine whether the series converges or diverges. 5. Converges 6. Diverges. Converges 8. Diverges 9. Converges 0. Converges. Converges Directions: Show that the series is convergent. According to the Alternating Series Sum Estimation Theorem, how many terms of the series do we need to add in order to find the sum to the indicated accuracy?. 4terms. 4terms 4. 5terms Directions: Approximate the sum of the series correct to four decimal places Ratio and Root Tests Directions: Determine whether the series is absolutely convergent, conditionally convergent, or divergent. 6. Absolutely convergent. Divergent 8. Conditionally convergent 9. Absolutely convergent 40. Absolutely convergent 4. Absolutely convergent 4. Absolutely convergent 4. Absolutely convergent 44. Divergent 45. Conditionally convergent 46. Absolutely convergent 4. Divergent 48. Divergent 49. Absolutely convergent 50. Divergent 5. Absolutely convergent 5. Absolutely convergent 5. Divergent Directions: For each of the following series, is the Ratio Test conclusive or inconclusive? (a Inconclusive (b Conclusive (convergent (c Conclusive (divergent (d Inconclusive
5 Strategy for Testing Series Directions: Test the series for convergence or divergence. 54. Divergent 55. Convergent 56. Divergent 5. Divergent 58. Divergent 59. Divergent 60. Divergent 6. Convergent 6. Convergent 6. Convergent 64. Convergent Power Series Directions: Find the radius of convergence and interval of convergence of the series. 65. R =,I =[, 66. R =,I =(, ] 6. R =,I =[, ] 68. R =,I =(, 69. R =, I =(, ] 0. R =,I =(, ]. R =,I = [9, ]. R =,I =[, ]. R =,I =(, Directions: Suppose that c n x n converges when x = 4 and diverges when x =6.Whatcan be said about the convergence or divergence of the following series? (a Convergent (b Divergent (c Convergent (d Divergent Representations of Functions as Power Series Directions: Find a power series representation for the function and determine the interval of convergence ( n xn ; i.o.c. ( 6, 6 6n+ 9x n ; i.o.c. ( 4, 4 4n+ ( n xn+ ; i.o.c. (, 49n+ ( n 5 n x n+ ; i.o.c. ( / 5, / 5 Directions: Find a power series representation for the function and determine the radius of convergence. (a (b (c (a (b (c n= ( n (n + xn ; R =4 4n+ ( n (n +(n + xn ; R =4 n+ ( n n(n xn ; R =4 n+ ( n x n ; R = n n= ( n x n n ; R = ( n x n ; R = n
6 8. ln n= x n n n ; R = n 9 n xn ; R =9 ( n x n+ 8 n+ (n + ; R =8 Directions: Evaluate the indefinite integral as a power series and determine the radius of convergence. 8. C 8. C + 8. C + t n n ; R = ( n+ x n (n +(n ; R = ( n x 4n+ (n +(4n + ; R = Directions: Use a power series to approximate the definite integral to six decimal places Directions: Use the formula ln( x = x n n to compute the indicated value correct to five decimal places Taylor and Maclaurin Series 8. f(x = 88. f(x = ( n (x n ; R =4 4 n (n + ( n 5 n x n ; R =/5 n 89. f(x = 90. f(x = ( n π n+ x n+ ; R = n+ (n +! x n (n! ; R = 9. f(x = +8(x +8(x +8(x +(x 4 9. f(x = 9. f(x = 6(x +4 n 4 n+ 5( n+ (x 9π n (n! 94. f(x = 4 + ( n (n!(x 6 n (n! 4 n f(x =C f(x =C / 9x n n n! ( n x 4n+ (4n +(n + Curves Defined by Parametric Equations Directions: Eliminate the parameter to find a Cartesian equation of the curve. 0. y =(x +6, x> 6 0. y = e x/, x ln 6 Directions: Determine what curve is represented by the parametric equations. Be sure to indicate direction as well as any starting or ending points. 0. It s the circle x + y =4tracedout exactly once in the counterclockwise direction beginning and ending at the point (, 0.
7 04. It s the parabola y = x with x ; the point (x, y movesbackandforth infinitely many times along the parabola from (, to (,.. Calculus with Parametric Curves 05. y = x 06. y =x (a dy t + = dx (b (0, and d y dx = 4t. (a (, 54 and (, 54 (b (0, 0 (a ( /,, ( /,, (/,, and (/, (b (, 0 and (, y = x/5 andy =x/ t dt = 4 [0 0 + ln( ln(6 + ] e e 5. Polar Coordinates (a (, π/4 (b (, π/4 (c (,π/ (d (, 4π/ 6.. x +(y = 9; the circle with center (0, and radius Directions: Sketch the graph of the given polar equation.
8 (a π/4, π/4 (b 0, π/ (a 0, π/, 4π/ (b π/, π, 5π/ 0. Areas in Polar Coordinates 5. 5π /6 6. 4π/4. 50π/ π/. 0. 9π. π/6. 8π. 4π +6
9 4. 9π ( π 6. 8π +6. π π (/,π/6 and (/, 5π/6 4. (5,π/, (5, 5π/, ( 5, π/, ( 5, π/, (5, π/, (5, π/, ( 5, 9π/, and ( 5, π/ 4. π Volumes by Cylindrical Shells Directions: Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis π/e 5. 6π ln(4π 8π + π 5. 64π/ 5. 50π 54. 8π/ π/5 56. π/ 5. π/0 Areas and Volumes Directions: Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis π /4 ( 44. 9π ft-lb J (a.55 J (b 0 ft-lb J Work 45. 8π π/9 4. π/6 48. πh(r + Rr + r ft-lb J (a 50 ft-lb (b 56.5 ft-lb 6. 8cm ft-lb (a 46/ (b rectangular solid; V = b h (c square pyramid; V = b h 49. eπ 4
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