Peter Alfeld Math , Fall 2005

Transcription

1 WeBWorK assignment due 9/2/05 at :59 PM..( pt) Consider the parametric equation x = 2(cosθ + θsinθ) y = 2(sinθ θcosθ) What is the length of the curve for θ = 0 to θ = 7 6 π? 2.( pt) Let a = (-2 4 2) and b = (-6-2 3) be vectors. Compute the following vectors. A. a + b = ) B. -5a= ) C. a - b= ) D. a = 3.( pt) A child walks due east on the deck of a ship at 3 miles per hour. The ship is moving north at a speed of 2 miles per hour. Find the speed and direction of the child relative to the surface of the water. Speed = mph The angle of the direction from the north = (radians) 4.( pt) Find a b if a = 0 b = 4 and the angle between a and b is π 5 radians. a b = 5.( pt) Find a unit vector in the same direction as a = (-9-8 9). ) 6.( pt) Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (- -3) and arrived in the Iron Hills at the point with coordinates (0 ). If he began walking in the direction of the vector v = 3i + j and changes direction only once when he turns at a right angle what are the coordinates of the point where he makes the turn. )

2 WeBWorK assignment 2 due 9/2/05 at :59 PM..( pt) Find the center and radius of the sphere x 2 8x + y 2 20y + z 2 6z = 89 Center: ) Radius: 2.( pt) If a = ( ) and b = (0 0-0) find a b =. 3.( pt) What is the angle in radians between the vectors a = (3-8 ) and b = (- 8-7)? Angle: (radians) 4.( pt) Let a = (0-5 7) and b = (7 8 6) be vectors. Find the scalar vector and orthogonal projections of b onto a. Scalar Projection: Vector Projection: ) Orthogonal Projection: ) 5.( pt) Let a = (4 3 8) and b = ( 6 9) be vectors. Compute the cross product a b. ) 6.( pt) Find the area of the parallelogram with vertices (25) (5 0) (0 5) and (3 20).

3 WeBWorK assignment 3 due 9/26/05 at :59 PM..( pt) If a = (8 0-6) and b = (-8-5 7) find a b =. 2.( pt) What is the angle in radians between the vectors a = (6-6 0) and b = (5-0 )? Angle: (radians) 3.( pt) Find a unit vector in the same direction as a = ( -3-0). ) 4.( pt) Find a unit vector orthogonal to (8 63) and (0 0): I+ J+ K 5.( pt) Find a unit vector with positive first coordinate that is orthogonal to the plane through the points P = (-5-5 ) Q = (0 0 6) and R = (0 0 8). ) 6.( pt) Find a vector equation for the line through the point P = ( 520) and parallel to the vector v = (5 5 4). Assume r(0) = 5I + 2J + 0K and that v is the velocity vector of the line. r(t) = I+ J+ K. 7.( pt) Given vectors u and v such that u v = I + 6J 7K find: a) v u = ). b) (u v) (u v) =. c) (u v) (u v) = ).

4 WeBWorK assignment 4 due 0/3/05 at :59 PM..( pt) Match the surfaces with the appropriate descriptions.. z = x 2 2. x 2 + 2y 2 + 3z 2 = 3. z = 2x + 3y 4. z = y 2 2x 2 5. z = 4 6. x 2 + y 2 = 5 7. z = 2x 2 + 3y 2 A. nonhorizontal plane B. elliptic paraboloid C. horizontal plane D. ellipsoid E. circular cylinder F. hyperbolic paraboloid G. parabolic cylinder 2.( pt) What are the rectangular coordinates of the point whose cylindrical coordinates are (r = 5 θ = 4π 9 z = 7)? x = y = z = 3.( pt) What are the spherical coordinates of the point whose rectangular coordinates are (2 )? ρ = θ = φ = 4.( pt) What are the cylindrical coordinates of the point whose spherical coordinates are (4 5 5π 6 )? r = θ = z= 5.( pt) Match the given equation with the verbal description of the surface: A. Plane B. Half plane C. Cone D. Sphere E. Elliptic or Circular Paraboloid F. Circular Cylinder. r = 2cos(θ) 2. φ = π 3 3. ρcos(φ) = 4 4. ρ = 4 5. r 2 + z 2 = 6 6. ρ = 2cos(φ) 7. z = r 2 8. r = 4 9. θ = π 3

5 WeBWorK assignment 5 due 0/7/05 at :59 PM..( pt) Find the first partial derivatives of f (x y z) = zarctan( y x ) at the point (4 4-2). A. f x (44 2) = B. f y (44 2) = C. f z (44 2) = 2.( pt) Consider the equation xz 2 + 6yz 2logz = 3 as defining z implicitly as a function of x and y. The values of z z and at ( 3) are and. x y 3.( pt) a) Find the limit if it exists or type N if it does not exist. (x+7y) lim 2 (xy) (00) x y = 2 b) Find the limit if it exists or type N if it does not exist. (Hint: use polar coordinates.) 5x lim 3 +7y 3 (xy) (00) = x 2 +y 2 4.( pt) Find the equation of the tangent plane to the surface z = 6y 2 6x 2 at the point ( 328). z = Note: Your answer should be an expression of x and y; e.g. 3x - 4y ( pt) If f (xy) = 3x 2 3y 2 find the value of the directional derivative at the point ( 4 ) in the direction given by the angle θ = 2π 5. 6.( pt) Find the first partial derivatives of f (xy) = 4x 3y 4x+3y at the point (x y) = (4 2). f x (42) = f y (42) = 7.( pt) Suppose f (xy) = y x P = (0) and v = 2i 3j. A. Find the gradient of f. f = i+ j Note: Your answers should be expressions of x and y; e.g. 3x - 4y B. Find the gradient of f at the point P. ( f )(P) = i+ j Note: Your answers should be numbers C. Find the directional derivative of f at P in the direction of v. D u f = Note: Your answer should be a number D. Find the maximum rate of change of f at P. Note: Your answer should be a number E. Find the (unit) direction vector in which the maximum rate of change occurs at P. u = i+ j Note: Your answers should be numbers 8.( pt) Suppose f (xyz) = x y + y z P = (432). A. Find the gradient of f. f = i+ j+ k Note: Your answers should be expressions of x y and z; e.g. 3x - 4y B. What is the maximum rate of change of f at the point P? Note: Your answer should be a number 9.( pt) Find the equation of the tangent plane to the surface z = 4y 2 x 2 at the point ( 0 ). z = Note: Your answer should be an expression of x and y; e.g. 3x - 4y ( pt) Suppose f (xy) = x 2 + y 2 0x 2y + 4 (A) How many critical points does f have in R 2? (B) If there is a local minimum what is the value of the discriminant D at that point? If there is none type N. (C) If there is a local maximum what is the value of the discriminant D at that point? If there is none type N.

6 (D) If there is a saddle point what is the value of the discriminant D at that point? If there is none type N. (F) What is the minimum value of f on R 2? If there is none type N. (E) What is the maximum value of f on R 2? If there is none type N. 2

7 WeBWorK assignment 6 due 0/3/05 at :59 PM..( pt) Evaluate the iterated integral Z 4 Z x 2 y 3 dxdy Z Z 2.( pt) Calculate the double integral R (0x + 6y + 60) da where R is the region: 0 x 3 0 y 5. Z 3.( Z pt) Calculate the double integral xcos(x + y) da where R is the region: 0 R x π 6 0 y 2π 4 4.( pt) Evaluate the iterated integral I = Z Z +x 0 x (6x 2 + 4y) dydx 5.( pt) Using polar coordinates evaluate the integral which gives the area which lies in the first quadrant between the circles x 2 +y 2 = 36 and x 2 6x+y 2 = 0. 6.( pt) Using polar coordinates evaluate the integral which gives the area which lies in the first quadrant between the circles x 2 +y 2 = 256 and x 2 6x+y 2 = 0. 7.( pt) A sprinkler distributes water in a circular pattern supplying water to a depth of e r feet per hour at a distance of r feet from the sprinkler. A. What is the total amount of water supplied per hour inside of a circle of radius 4? ft 3 /h B. What is the total amount of water that goes throught the sprinkler per hour? ft 3 /h 8.( pt) Electric charge is distributed over the disk x 2 + y 2 9 so that the charge density at (xy) is σ(xy) = 7 + x 2 + y 2 coulombs per square meter. Find the total charge on the disk. 9.( pt) Find the surface area of the part of the plane 4x + 2y + z = 4 that lies inside the cylinder x 2 + y 2 = 6. 0.( pt) Use cylindrical Z Z Z coordinates to evaluate the triple integral x 2 + y 2 dv where E is E the solid bounded by the circular paraboloid z = 9 9 ( x 2 + y 2) and the xy -plane.

8 WeBWorK assignment 7 due /5/05 at :59 PM..( pt) Evaluate the triple integral Z Z Z xyzdv E where E is the solid: 0 z 7 0 y z 0 x y. 2.( pt) Find the volume of the solid enclosed by the paraboloids z = 6 ( x 2 + y 2) and z = 8 6 ( x 2 + y 2). 3.( pt) Use Zcylindrical Z Z coordinates to evaluate the triple integral x 2 + y 2 dv where E is the E solid bounded by the circular paraboloid z = 6 6 ( x 2 + y 2) and the xy -plane. 4.( pt) UseZ spherical Z Z coordinates to evaluate the triple integral x 2 + y 2 + z 2 dv where E is the E ball: x 2 + y 2 + z 2. 5.( pt) Compute the total mass of a wire bent in a quarter circle with parametric equations: x = 7 cost y = 7sint 0 t π 2 and density function ρ(xy) = x2 + y 2. 6.( pt) Let C be the curve which is the union of two line segments the first going from (0 0) to (2 ) and the second going from Z(2 ) to (4 0). Computer the line integral 2dy dx. C

9 WeBWorK assignment 8 due 2/5/05 at :59 PM..( pt) Let C be the counter-clockwise planar circle with center at the origin and radius r > 0. Without computing them determine for the Z following vector fields F whether the line integrals F dr are positive negative or zero and type P N or Z as appropriate. A. F = the radial vector field = xi + yj: B. F = the circulating vector field = yi+xj: C. F = the circulating vector field = yi xj: D. F = the constant vector field = i + j: 2.( pt) For each of the following vector fields F decide whether it is conservative or not by computing curl F. Type in a potential function f (that is f = F). If it is not conservative type N. A. F(xy) = ( 2x + y)i + (x + 4y)j f (xy) = B. F(xy) = yi + 0xj f (xy) = C. F(xyz) = xi + 0yj + k f (xyz) = D. F(xy) = ( siny)i + (2y xcosy)j f (xy) = E. F(xyz) = x 2 i + y 2 j + 2z 2 k f (xyz) = Note: Your answers should be either expressions of x y and z (e.g. 3xy + 2yz ) or the letter N 3.( pt) Suppose C is any curve from (0 0 0) to () and F(xyz) = (5z + 4y)i + (4z + 4x)j + (4y + 5x)k. Compute the line integral R C F dr. C 5.( pt) Let F = (2yz)i + (4xz)j + (5xy)k. Compute the following: A. div F = B. curl F = i+ j+ k C. div curl F = Note: Your answers should be expressions of x y and/or z; e.g. 3xy or z or 5 6.( pt) Let F be the radial force field F = xi + yj. Find the work done by this force along the following two curves both which go from (0 0) to (2 4). (Compare your answers!) A. If C is the parabola: x = t y = t 2 0 t 2 then R C F dr = B. If C 2 is the straight line segment: x = 2t 2 y = 4t 0 t then R C 2 F dr = 7.( pt) Let C be the counter-clockwise planar circle with center at the origin and radius r > 0. Without computing them determine for the following vector fields F whether the line integrals R C F dr are positive negative or zero and type P N or Z as appropriate. A. F = the radial vector field = xi + yj: B. F = the circulating vector field = yi+xj: C. F = the circulating vector field = yi xj: D. F = the constant vector field = i + j: 8.( pt) Suppose C is any curve from (0 0 0) to () and F(xyz) = (3z + y)i + (z + x)j + (y + 3x)k. Compute the line integral R C F dr. 4.( pt) Let F = 3xi + 5yj + 7zk. Compute the divergence and the curl. A. div F = B. curl F = i+ j+ k 9.( pt) Let C be the positively oriented circle x 2 + y 2 =. Use Green s Theorem to evaluate the line integral R C 2ydx + 2xdy.