Math 122 Final Review

Transcription

1 Math Final Review I. Tech. Of Integration x e x ln(x + ) sin / x cos x x x x + (x )(x )(x ) (x ) cos x 9 + sin x 8. e x x + e 9. (arctan x)( + x ). x + x... π/ sin 4 x sin 4 (θ/) cos (θ/) dθ sin θ cos θ dθ tan x sec 6 x (tan x sec x) 4 π/4 π/6 tan (x/) sec 4 (x/) x 5 4x (x + 4) / x 6 x(x ) ln x (x + ) x/ sec x tan x x + x x cos x 6. Use numerical integration (by trapezoidal and Simpson s methods) to approximate the value of the following integral with N = 6 sub-intervals. x4 + x.5 7. Use Trapezoidal and Simpson s method to approximate 4 4x with n = 8.

2 5 8. Use Simpson s method to approximate 9. Use Simpson s method to approximate x x + x. Use Simpson s method with n = 4, to approximate with n = 4. with n = 4. (x + ) x If f(x) 4 5 approximate f(x) using Simpson s method and n = 8. Find x and ȳ for the region bounded by y = 4 x and y = II. Application of Integration Find the length of the curve:. f(x) = (x 7)/ on [7, 4] 4. f(x) = (x 6)/ on [6, ] 5. y = x + on [/, ] 6 x 6. y = x/ + on [, ] 7. y = x x on [, ] 8. Find the force on a 44 foot wide by 9 foot deep wall of a swimming pool filled with water. ρ =6.4 lb/ft. 9. A triangle with sides 5,5 and 6 feet is submerged vertically in water (ρ =6.4 lb/ft ) with the 6 foot side at the surface. Find the force on the plate. 4. A flat plate in the form of a semicircle m in diameter is submerged in water (ρ =98 N/m ). Find the force on the plate. 4. A triangle with sides,, and 4 feet is submerged vertically in water (ρ =6.4 lb/ft ) with the point up and the long side is parallel to the surface. If the vertex is 4 feet below the surface, find the force on the plate. 4. Find the center of mass of the triangle with vertices (, ), (6, ) and (, ). 4. Find the center of mass of the region bounded by y = x + and y = x Find the center of mass of the region bounded by y = 8 x and y = x from x 45. Find the center of mass of the region bounded by y = 4 x and y = Find the n-th order Taylor Polynomial for: 46. f(x) = e x centered at a =, n = f(x) = sin x centered at a = π, n = 5

3 48. f(x) = cos x centered at a =, n = 49. f(x) = x centered at a =, n = 4 5. f(x) = ln x centered at a =, n = 5 5. f(x) = e x centered at a =, n = 5. f(x) = arctan x centered at a =, n = 5. f(x) = ln(x + 4) centered at a =, n = 54. f(x) = ln( x) centered at a =, n = III. Differential Equations: Sketch the slope field and a likely solution curve for: dy = y + x y() = dy = x y y() = 57. dy = x y y() = Use Euler s method with h =. to approximate y() for the initial value problem: 58. dy = y x y() = 59. dy = x + y y() = 6. dy = x y y() = Solve the following differential equations: 6. y = 6. y = 6. y + y = 64. y + y = 4 + x x x 65. y + y = e x 66. xy + y = 6x 67. y + x y = sin x 68. y = xy 69. (x + )y = y 7. x(ln x)y + y = ln x 7. dy 4xy = x 7. x dy y = x 7. dy = y x 74. dy = x (+x 4 )y 75. dy + y = e x 76. (x + )y y = x

4 Find the solution of each of the following differential equations with boundary values (= initial conditions): 77. y = x y, y() = 78. xy = y + x cos x, y(π) = π 79. If dy dt = y ( y ) 5 y() =, find y(t) 8. If dy dt =.8y ( y ) y() = a. Find y(4) b. Findt where y(t) = Biologists stock a lake with 5 fish and the carrying capacity of the lake is,. The number of fish triple during the first year. If the population of fish follow the logistic differential equation, how long will it take for the population of fish t reach 4,? 8. Ebola spreads through a population of 5, people. Suppose people start with the disease and 5 people have the disease after days. If the number of people that have get the disease follows the logistic differential equation, how long will it take for half the people to have ebola? 8. Suppose that a tank contains gal of a solution of a certain chemical and that 5 lb of the chemical are in the solution. Starting at a certain instant, a solution of the same chemical, with concentration of lb/gal, is allowed to flow into the tank at the rate of gal/min. The mixture is drained off at the same rate so that the volume of the solution in the tank remains constant. How many gallons of solution should be pumped into the tank to raise the amount of dissolved chemical to 5 lb? IV. Sequences and Series 84. If a n = n n +, find lim n a n 85. If S n = n i= a i = n n+, does i= a i converge or diverge? 86. Find the Taylor series for f(x) = x x at x = 87. Find the third degree Taylor polynomial P (x) at x = for f(x) = ln x 88. Does the sequence given by a n = n + 6n + n + 6 n + n + converge or diverge? If it converges, find its limit. 89. Determine if the sequence given by a n = n sin n find its limit. converges or diverges? If it converges, Determine whether the following series converge or diverge; if a series converges, find its sum.

5 9. 9. e n (n + )(n + ) n(n + ) n= [ 5 6 ] n n Determine whether the following series converge or diverge: n= n e n n n arctan n + n n! n n n= ( + ) n n (n + )[ln(n + )] ln n n e n ( n ) n n n n + n(n + )(n + ) n ln n n + Determine whether the following series converge absolutely, conditionally, or diverge: ( ) n+ n ( ) n n + n.. ( ) n ln n n ( ) n n e n. ( ) n+ n + n(n + ) Find the interval of convergence for the following series:. 4. (x ) n n n!x n n ( ) n x n e n n n (x ) n 7. Find the Maclaurin series for f(x) = ln( + x) 8. Find the Taylor series for f(x) = ln(x ) about c =. 9. Use an appropriate series to approximate the cos to two decimal place accuracy.

6 . Use an appropriate series to approximate the cos o to four decimal place accuracy.. Find the first four nonzero terms of the Maclaurin series for e x cos x.. Use a series to approximate / V. Parametric Equations cos(x ) to four decimal place accuracy.. Express the equation (x + y) = x y in polar form. 4. Express the polar equation r = 4 sec θ tan θ in rectangular form, identify and sketch. 5. Express x(x + y ) = (x y ) in polar form. 6. Express (x + y ) / = x y xy in polar form. 7. Find the area of the region enclosed by r = θ from θ = to θ = π 8. Find the area of the region that is inside r = and outside r = cos θ. 9. Find the area of the region enclosed by r = + sin θ.. Find the area of the region enclosed by r = cos θ.. Find the length of the curve x = t +, y = t, from (,-) to (,-).. Find dy/ and d y/ at the point where t = without eliminating t for x = t, y = + t. Find the length of the curve x = + sin θ, y = cos θ, θ π/ 4. Sketch the curve x = cos t, y = sin t, t π/4. by eliminating the parameter t and label the direction of increasing t. 5. Find the length of the curve r = e 4θ from θ = to θ = Find the area of one loop of r = cos θ VI. Conic Sections Sketch the parabola, and label the focus, vertex and directrix. 7. x + 4x + 4y 4 = 8. x + 8y = 9. y = 4x + 8x + 5

7 Sketch the ellipse, and label the foci and vertices. 4. x + 4y = 4 4. (x ) (y 5) = 4. 9x + 4y + 6x 4y + 6 = Sketch the hyperbola, and label the foci, vertices and asymptotes. 4. y x 4 = 44. (x ) (y 5) 9 5 = 45. x 9y + x 54y 8 = VII. Vectors 46. Solve for U if U (i + j) = i + U 47. Given the vectors V =,, and W =, 4, compute V W, and the cosine of the angle between V and W. 48. If A B = A C and A does B = C 49. Find the equation of the line of intersection of the two planes x + y + z = x y + z = 5. Find an equation of a plane that contains the three points (,,), (,,), and (,,). 5. Show that (,7,-), (-,8,), and (-,4,-) are vertices of an isosceles triangle. 5. Find the standard equation of the sphere with a diameter whose endpoints are (4, 6, ) and (-,, ). 5. Find the perimeter of the triangle whose vertices are (6,,5), (,,), and (6,,-7) 54. Find the distance between P(,7,8) and Q(,9,7), and find the midpoint of the line segment joining P and Q. 55. Find the length of u = i j + k and find a unit vector in a direction opposite to that of u. 56. Find the length of P Q if P (4, ) and Q(, 5). 57. Find a unit vector in the direction from P (, 9, ) to P (, 7, 8). Express your answer in component form. 58. Find the cosine of the angle between u = i j k and v = i + j + k.

8 59. Find the cosine of the angle between u =,, and v = 6, 4, 6. Find unit vectors that are perpendicular to both a =,, and b =,,. 6. Find unit vectors that are perpendicular to both a =,, and b =,,. 6. Find the sine of the angle between a =,, and b =,,. 6. Find all unit vectors parallel to the yz-plane that are perpendicular to the vector i j k. 64. Find the point of intersection of L : x = t, y = t, z = t and the plane x + y z = Find the cosine of the angle between the lines L : x = + t, y = + t, z = + t and L : x = + t, y = t, z = + t 66. Find the parametric equations of the line through (,,-) that is parallel to the line of intersection of the planes x + y + z + 4 = and x y z 5 =. 67. Find the equation of the plane through P(,,), Q(,4,), and R(,, ). 68. Find the equation of the plane through (,,-) that is perpendicular to x = + t, y = + t, z = 5t. 69. Find the projection of u =,, along b =,,. 7. Show that u =, and v =, 6 are perpendicular. 7. Let a =, 4,, b =,,, and c =,, ; find c ( a b). 7. Find the parametric equation of the line which passes through P (4,, 5) and P (,, ). 7. Do the lines that pass through (,,) and (,,8) and through (-,-,) and (-,,) intersect? If so, find their point of intersection. 74. Find the point where the line which passes through (, 4, ) that is parallel to,, intersects the xy-plane. 75. Find the equation of the plane through P (,, ), Q(,, 5), and R(,, 6). 76. Find the equation of the plane through (,, ) that is parallel to x+y +6z +8 =. VIII. 77. Misc. cot 4 (6t) sec(6t) dt

9 cos x tan x sec 5 x tan 5 x sin 4 x cos 5 x 8. Find the area in the first and rd quadrants that lies inside r = cos θ and r = 8. Find the area inside the star if the function is r = sin(5θ) 8. Find the area outside r = cos θ but inside r = cos θ 84. A swimming pool company is designing a pool to be installed at a hotel catering to mathematicians and heart surgeons. The shape of the pool will be formed by two intersecting cardioids with equations r = cos θ and r = 5 5 cos θ with the distance in meters. There are to be two depths to the pool. The first depth, which is the diving area, is meters and the entire area of one of the cardioids. The remaining area is to be a meter deep wading area. Find the volume of the swimming pool in cubic meters. 85. Christian D. Vulter wishes to gives his mother-in-law something special for Mother s Day, just because he loves spending holidays with her and enjoys he cooking. Graph the curve given by the polar equation r = sin θ to see what Christian will give his adoring mother-in-law. However, Christian would like to make the gift even more special (he really loves his mother-in-law), by pasting the figure he obtained before on to rectangular piece of cardboard. Find the area of the region outside of and inside the rectangle of area 6 m. r = sin θ 86. The MasterCard Company is printing advertisements and they need to determine the amount of ink needed in the area of the logo where the two circles intersect. The Company determines that the two circles (both with radius of inch) can be described by the equations r = and r = cos θ Find the shared area. 87. Find the area of one petal of r = cos(θ)

10 88. It is Valentines Day and Joey-Joe-Joe can t decide what to get his girl friend Bambi. Being a bit short on cash and wise to the fact that Bambi just loves Math (Math is Easy, Math is Fun), specially polar coordinates. Joey-Joe-Joe decides to make a personal card in the shape of a cardioid (r = sin θ). Wanting to write some nice saying on the front of the card, Joey-Joe-Joe must know what area he has to work with. 89. Find the area outside r = sin θ and inside r = For problems 9 to 98, use the Comparison Test or Limit Comparison Test to determine if the series converges or diverges n + n + 5 n + n n n / n 5 n n sin n sin 6n n 5 + n + cos n 9. 4n n + 6 8n 7 + n n + 4 n + n n 6 + 4n 4 n + In problems 99 to 6, find the radius and interval of convergence for each power series n n!(x )n ( ) ( ) n xn n n n ( ) n n (x ) n n ) n (x + 4) n ( 6 8 (ln n) x n n n! n (x + 4)n x n ln n n!(n + ) (x ) n n n 7. x + x 8. x 8 e x / (x ). Determine if the series converges absolutely, converges conditionally, or diverges. ( ) n+ (5n)! + n In problems to, determine whether the series converges, if so find its sum. ( ) 7 n + n. ( ) n 6 n. n + n. 9n + n 4. n n n!

11 n n 9 n n (n + 7n + ) (n + )(n + 4) n + 7 n n +. A geometric series has the following properties: The st and nd terms have a sum of -4, and the 4th and 5th terms have a sum of 8. Find the first term and the common ratio of the series.. A math-magical potato wandered down to his favorite watering hole. He sat down at his favorite table and waited for the waitress. The petite carrot approached the table. What ll you have? she asked. The potato responded, I ll have some bacon. How much will you be having today? I will have the sum of the infinite series: ( ) n 7 6 n The carrot glared at him with a puzzling loon on her face. She wrote down the series and took it back to the chef. The Tomato looked at it and promptly prepared the meal. How many strips of bacon will the potato be having today? In problems to, use the integral test to determine whether the following series converges or diverges.. n ln n 4. n ln n 5. n(ln n).. n= sin x + cos x x e x + x7 x 5 + x x n + 9n + 4 n + 5. Determine whether the following series converges: 9.., 45 n n n n(n + )(n + ) (n + )(n + 4)(n + 5)(n + 6)n! 4. Find the Taylor series for f(x) = sin x at x = π in sigma notation. 5. Find the Maclaurin series for f(x) = sinh x in sigma notation.

12 6. Find the Taylor series for f(x) = ln(x + ) at x = in sigma notation. 7. Find the Taylor series for f(x) = sin x at x = in sigma notation. 8. Find the Taylor series for f(x) = x at x = 4 in sigma notation. 9. Find the Maclaurin series for e x + x. 4. Find the Taylor series for f(x) = x at x = in sigma notation. 4. Does the series converge of diverge? Find the general solution: ( n ) n + n n xy + y = x 44. y y tan x = 4. xy y = x 45. y = (y + ) Find the particular solution: 46. y cos x + y = ; y() = y + y tan x = sec x + cos x; y() = 48. y + y sin x = sin x; y(π/) = 49. y = x + cos x; y() = 5. y = x y ; y() = 5. sin x cos x 6 5. x + 6x + 4 x + 4x + 4x 5. x + 5 x + 4x x x x 4x x + x 4x + 4x 7x (x )(x 4) In problems 57 to 69, use integration by parts. 57. e x sin x 6. sin(ln x) 64. x + x tan x sec x e x cos x 6. xe x 65. x e x 6. x ln x 6. x e x 66. e x + sec x

13 67. π/ x sin 4x 68. x 5 e x 69. 7x sin x 7. Your name is Doe: John Q. Doe and you work for the company of Math is Our Life Inc. Your boss has a special assignment for you. He wants you to crack the code on a new super computer the rival company is using. You know the code is in the form of a polar equation written in a typed format, and you also know that it will work with equations in polar form r =. You know from code cracking that you have already done that the x, y equation is Can you crack the code? x + x y + y = 7. Josie is sitting in math class and she is thinking about the guy two rows in front of her. She is really into math so instead of using a flower, she draws one using polar coordinates. If she uses r = 9 sin 4θ and starts with He loves me not in the he love me, he loves me not game, does he love her? 7. Find the polar equation for (x + 4) + (y + ) = 5 + 8x 7. Find the area between r = θ π and r = θ π 74. Find the polar equation for x y = Find the rectangular equation for r = Name the type of polar coordinate graphs: a. r = sin(θ) b. r = cos(θ) c. r = sin(θ) d. r = cos(θ) for θ π 77. Approximate e to five decimal place accuracy using a Maclaurin polynomial. 78. Use a second degree Maclaurin polynomial, P (x) to estimate the value of 6 e and estimate the error in your approximation. 79. Approximate e to four decimal place accuracy using a Maclaurin polynomial. 8. Use an appropriate Taylor series to approximate cos o to four decimal place accuracy. 8. You are using a computer program to crack a top secret password. Unfortunately, before it can give you the answer, it crashes. But you were able to find out some information. First of all,it is{ a four } number password. The first three numbers are the first three terms of the sequence n ln n rounded to the first two decimal places. If the term is undefined, the password number is. The last number can be found by figuring out whether the sequence converges or diverges. If it converges, it is the number it converges to, if it diverges, the number is. What is the password?

14 Determine whether the sequence converges or diverges, and if it converges, what is its limit: { ln n(n + 4) n!(5n + 7) } { } e n n n { e n n } } 85. { n + sin n n { 86. 4n + n + 7n + n + 7n + 4n + } 87. For what values of r does the sequence { n } 6n r + converge, and when it converges, what is the limit. 88. Find the 5th degree Maclaurin polynomial for f(x) = e x + cos x 89. Find the third degree Taylor polynomial for f(x) = x centered at x = π 9. Find the fourth degree Taylor polynomial for f(x) = 5 (x ) centered at x = 5 9. Find the third degree Taylor polynomial for f(x) = x 4 x + x centered at x = 9. Find the third degree Taylor polynomial for f(x) = + x centered at x = x x x + 5 x 97. x cos x 96. x + x x 49 x 99. Find the arc length of r = e θ for θ. Find the fourth degree Maclaurin polynomial for f(x) = sin 5x. Find the Maclaurin series for f(x) = x x

15 Answers. ex + C. x ln(x + ) x + arctan x + C. 4 sin4/ x sin/ x + C [ ] arcsin x x x + C ln(x ) 9 ln(x ) + ln(x ) 6. D.N.E. 7. arctan ( sin x 8. arctan(e x ) + C 9. ln arctan x + C ) + C. 5 ( + x)5/ ( + x)/ + C x. sin 4x + sin 8x + C sin5 (θ/) 7 sin7 (θ/) + C. π tan8 x + 6 tan6 x + 8 tan4 x + C 5. x + C 6. tan6 (x/) + tan4 (x/) + C (5 4x ) / + 48 (5 4x ) / + C 8. x 4 x +4 + C 9. 6 ln x + 7 ln(x ) C x. ln x + ln [ ] x (x+) x+ + C. Diverges.. Diverges 4. / 5. π Trapezoid:.4449, Simpson s: Trap: 86, Simpson s: (, 8/5). ( ) 6 4. ( 7 7 ) ( ) ,96.8 lb lb 4. 87,5 N 4. 7,456 lb 4. (, ) 4. (, ) 44. (.47857, ) 45. (, 8/π) 46. T ( x) = + x + x + x! + x4 4! 47. T 5 (x) = (x π) + 6 (x π) π) T 4 (x) = x 49. T 4 (x) = + (x ) 5 (x )4 6 8 (x ) (x ) 8 5. T 5 (x) = (x ) (x ) + (x ) 4 (x ) (x )5 5. T (x) = e + e (x ) + 9e (x ) + 9e (x ) 5. T (x) = x x 5. T (x) = ln 4 + x T (x) = x x x 58. y() y() 7.8 +

16 6. y() C 6. y = t + C 6. y = ce t 64. y = x + x + C x 65. y = ex + Ce x 66. y = x + Cx 67. y = sin x cos x + C x x 68. y = x +c 69. y = c(x + ) 7. y = ln x + c ln x 7. y = 4 + Cex 7. y = x ln x + Cx 7. y = Cx 74. y = ± ln( + x4 ) + C 75. y = e x + Ce x 76. y = c(x + ) + + (x + ) ln(x + ) 77. y = ( x + ) y = x (sin x + ) 79. y(t) = 5et 4 + e t 8. a. 7.6 b. t = years days gal Converge 86. x + x + x P (x) = (x ) (x ) + (x ) 88. / e 9. / 9. / 9. -5/ Converges 95. Converges 96. Converges 97. Diverges 98. Diverges 99. Diverge. Converge. Diverge. Converge. Diverges 4. Diverges 5. Diverges 6. Diverges 7. Diverges 8. Converges Absolutely 9. Converges conditionally. Converges conditionally. Converges absolutely. Converge conditionally. [, 4) 4. x = 5. ( e, e) 6. (5/, 7/) ( ) n+ x n n ( ) n+ (x ) n. x + x x cos θ sin θ. r = + sin θ cos θ 4. x = 4y n

17 5. r = ( cos θ sin θ tan θ) 6. r = cos θ sin θ 7. 9π 6 8. π π.. 7 ( 8). dy/ = /, d y/ = /4. π x 4. + y = (e6 ) 6. / 46., / 47., /5 48. NO 49. x = + t, y = t, z = 5t 5. x y z = 5. Two sides have length (x ) + (y 4) + (z ) = , (5/, 8, 5/) 55.,,, ,, /9 59. cos θ = ± 6,, 4 6. ± 6,, ±,, 64. (7,7/,/) x = + t, y = 7t, z = 5t 67. y z = 68. x + y 5z = , 9, u v = x = 4 t, y = t, z = 5 4t 7. NO 74. (4, 6, ) 75. x 5y + 4z = 76. x + y + 6z = [ ] sin (6t) + sin(6t) + C 78. cos x + cos x + C 79. sec 9 x 9 8. sin 5 x π 4 8. π sec7 x 7 sin7 x 7 8. π meters m square inches. 87. π 88. π 89. 9π 9. Diverges 9. Converges 9. Converges 9. Diverges 94. Converges 95. Converges 96. Converges + sec5 x 5 + sin9 x 9 + C + C

18 97. Converges 98. Converges 99. r = interval x = only.. r = interval: (, ). r = / interval: (5/, 7/). r = 8/6 interval: ( /6, 6/6). r = interval: [, ] 4. r = interval: (, ) 5. r = interval: (, ) 6. r = interval: [, ) 7. ln ( + ). Converges Absolutely. Converges, 6. Converges, 6. Converges, 4. Diverges 5. Converges, 7 6. Converges, Converges, /6 8. Converges, /4 9. Diverges. Diverges. a = and r =. 6. Diverges 4. Diverges 5. Converges 6. Converges 7. Diverges 8. Diverges 9. Converges. Diverges. ln( + cos x) + C. e x + x6 6 x4 4 + x + C. Converges ( ) (x π n )n n= (n)! x n (n )! n+ xn ( ) n ( ) n+ x n (n )! n (x 4)n ( ) n= n x4 x6 + x n (x )n ( ) n= 4. Converge. n+ 4. 4x y = x 4 + C 4. y = x (ln x + C) 44. y = tan x + C sec x 45. y = x + C 46. y = + 4e tan x 47. y = sin x + (x + ) cos x 48. y = + e cos x 49. y = x + sin x 5. y = x 5. 8 ln cos cos x x C 5. ln x + ln x + x C 5. ln x + ln x + + C 54. ln x + ln x + C 55. ln x + ln x x + C

19 56. ln x + 5 ln x 4 + C 57. e x sin x e x cos x + C 58. [sec x tan x + ln sec x + tan x ] + C ( e x ) sin x + ex cos x 9 + C 6. [ x / ln x x/] + C 6. [x sin(ln x) x cos(ln x)] + C e x tan x + ln cos x + C 65. x e x xe x + e x + C 66. ( xe x e x ) + [sec x tan x + ln sec x + tan x ] + C 67. π x cos x [ x sin x + ( x cos x + sin x)] + C 7. r = 7. Yes. 7. r = 6 sin θ 7. cos θ sin θ cos θ sin θ 74. r cos θ = x + y = a. a petal rose b. a 4 petal rose c. a limacon with inner loop d. a cardioid 77. You need a 9th degree, e and error < You need a 5th degree, e You need a rd degree, cos o , 5.77, 8.9, 8. converges, 8. converges, diverges 85. converges, 86. converges, / 87. r < diverges, r = converges with limit /6, r > converges with limit 88. P 5 (x) = + x + x! + x4 4! + x5 5! 89. P (x) = π + π (x π) + π (x 9 π) + π 4! (x π) 9. P 4 (x) = (x 5) (x 5) 4 (x 5) (x 5)4 9. P (x) = 4 + (x ) + (x ) + 8(x ) 9. P (x) = 4 (x ) + 8 (x ) 6 (x ) 9. ln 9 + x + x + C 94. sin x sin x + C x + 5 x + C 96. x + (ln x 9 + x C [ (x4 ) ( x ) 6 ( ) ( ) ] 4 + ln x4 + x C x + C 99. (e 6 ). P 4 (x) = 5x 5x!. x x x 5 x 7 x