Tom Robbins WW Prob Lib1 Math , Fall 2001

Transcription

1 Tom Robbins WW Prob Lib Math 220-2, Fall 200 WeBWorK assignment due 9/7/0 at 6:00 AM..( 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 7 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) 2.( pt) Find a b if a = 2, b =, and the angle between a and b is π 0 radians. a b = 3.(2 pts) 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, 6). If he began walking in the direction of the vector v 5i 2j and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn. (, ) 4.( pt) The position of a particle in motion in the plane at time t is R t exp 0 t I exp t J At time t 0, determine the following: (a) The speed of the particle is: (b) Find the unit tangent vector to R t : I J (c) The tangential acceleration: (d) The normal acceleration: 5.( pt) The position of a particle in motion in the plane at time t is R t 0t I sin 6t J At time any t, determine the following: (a) the speed of the particle is: (b) the unit tangent vector to R t : I J 6.(2 pts) The position of a particle in motion in the plane at time t is R t ti ln cos t J At time t 0 4π 2, determine the following: (a) the unit tangent vector: I J (b) the unit normal vector to R t : I J (c) the acceleration vector: I J (d) the curvature:

2 Tom Robbins WW Prob Lib Math 220-2, Fall 200 WeBWorK assignment 2 due 9/7/0 at 6:00 AM..( pt) If a = (4, 0, -8) and b = (8, -9, -0), find a b =. 2.( pt) What is the angle in radians between the vectors a = (-9, 3, 9) and b = (-4, -4, -0)? Angle: (radians) 3.( pt) Find a unit vector in the same direction as a = (-7, 3, 6). (,, ) 4.( pt) Find a unit vector orthogonal to 6 5 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 = (-2, 4, -2), Q = (, 7, ), and R = (, 7, 4). (,, ) 6.( pt) Find a vector equation for the line through the point P 2 0 and parallel to the vector v 5 2. Assume r 0 2I 0J K 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 7I 3J 9K, find: a) v u,,. b) u v u v. c) u v u v,,.

3 Tom Robbins WW Prob Lib Math 220-2, Fall 200 WeBWorK assignment 3 due 9/24/0 at 6:00 AM..( pt) What is the distance from the point (0, 9, -4) to the xz-plane? Distance = 2.( pt) Enter T or F depending on whether the statement is true or false. (You must enter T or F True and False will not work.). Two lines parallel to a plane are parallel. 2. Two lines parallel to a third line are parallel. 3. Two planes either intersect or are parallel. 4. A plane and a line either intersect or are parallel. 5. Two lines either intersect or are parallel. 6. Two lines perpendicular to a plane are parallel 7. Two lines perpendicular to a third line are parallel. 8. Two planes perpendicular to a third plane are parallel. 9. Two planes parallel to a third plane are parallel. 0. Two planes parallel to a line are parallel.. Two planes perpendicular to a line are parallel. 3.( pt) Consider the planes 3x 2y 3z and 3x 3z 0 (A) Find the unique point P on the y-axis which is on both planes. (,, ) (B) Find a unit vector u with positive first coordinate that is parallel to both planes. I + J + K (C) Use parts (A) and (B) to find a vector equation for the line of intersection of the two planes,r t I + J + K 4.( pt) Given that the acceleration vector is a t cos t I sin t J t K, the initial velocity is v 0 I K, and the initial position vector is r 0 I J K, compute: A. The velocity vector v t I J K B. The position vector r t I J K Note: the coefficients in your answers must be entered in the form of expressions in the variable t; e.g. 5 cos(2t) 5.( pt) If r t cos 9t I sin 9t J 8tK, compute: A. The velocity vector v t I J K B. The acceleration vector a t I J K Note: the coefficients in your answers must be entered in the form of expressions in the variable t; e.g. 5 cos(2t) 6.( pt) Consider the helix r t cos 4t sin 4t 2t. Compute, at t π 6 : A. The unit tangent vector T (,, ) B. The unit normal vector N (,, ) C. The unit binormal vector B (,, ) D. The curvature κ 7.( pt) Find the curvature κ t of the curve r t sint I sint j 5cost K 8.( pt) (A) Find the parametric equations for the line through the point P = (-4, 3, -) that is perpendicular to the plane x 2y z Use t as your variable, t = 0 should correspond to P, and the velocity vector of the line should be the same as the standard normal vector of the plane. x = y = z = (B) At what point Q does this line intersect the yzplane? Q = (,, )

4 Tom Robbins WW Prob Lib Math 220-2, Fall 200 WeBWorK assignment 4 due 0/8/0 at 6:00 AM..( pt) Match the surfaces with the verbal description of the level curves by placing the letter of the verbal description to the left of the number of the surface.. z 2x 2 3y 2 2. z x 2 y 2 3. z x 2 y 2 4. z x 5. z 2x 3y 6. z 25 x 2 y 2 7. z xy A. two straight lines and a collection of hyperbolas B. a collection of unequally spaced parallel lines C. a collection of equally spaced parallel lines D. a collection of concentric ellipses E. a collection of unequally spaced concentric circles F. a collection of equally spaced concentric circles Note: Some answers may be correct more than once, or not at all. 2.( pt) Match the functions with the verbal description of the level surfaces by placing the letter of the verbal description to the left of the number of the function.. w x 2 y 2 z 2 2. w x 2 y 2 z 2 3. w x 2 2y 2 3z 2 4. w x 2y 3z 5. w x 2 y 2 z 2 6. w x 2y 3z 7. w x 2 2y 2 3z 2 A. a collection of unequally spaced concentric spheres B. two cones and two collections of hyperboloids C. a collection of unequally spaced parallel planes D. a collection of equally spaced parallel planes E. a collection of equally spaced concentric spheres F. a collection of concentric ellipsoids Note: Some answers may be correct more than once, or not at all. 3.( pt) Find the first partial derivatives of f x y z y x z arctan at the point (4, 4, -2). A. f x B. f y C. f z ( 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 x y and. 5.( pt) a) Find the limit, if it exists, or type N if it does not exist. x 7y 2 x y 2 lim x y 0 0 b) Find the limit, if it exists, or type N if it does not exist. (Hint: use polar coordinates.) 5x lim x y y 3 x 2 y 2 6.( pt) Find the equation of the tangent plane to the surface z 6y 2 6x 2 at the point z = Note: Your answer should be an expression of x and y; e.g. 3x - 4y ( pt) Find an equation of the tangent plane to the parametric surface x 3r cosθ, y 3r sinθ, z r at the point when r 2, θ π 4. z = Note: Your answer should be an expression of x and y; e.g. 3x - 4y 8.( pt) If f x y 3x 2 3y 2, find the value of the directional derivative at the point 4 in the 2π direction given by the angle θ 5.

5 9.( pt) Suppose f x y 3x 2 3xy 2y 2, P 0, and u A. Compute the gradient of f. f i j Note: Your answers should be expressions of x and y; e.g. 3x - 4y B. Evaluate the gradient at the point P. f 0 i j Note: Your answers should be numbers C. Compute the directional derivative of f at P in the direction u. D u f P Note: Your answer should be a number 0.( pt) Suppose f x y x y, P 3 and v 2i 2j. 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.( pt) The axis of a light in a lighthouse is tilted. When the light points east, it is inclined upward at 6 degree(s). When it points north, it is inclined upward at 2 degree(s). What is its maximum angle of elevation? degrees 2.( pt) Consider the surface 25x 2 y 2 9z 2 35 and the point P on this surface. a) The outward unit normal at the point P is I + J + K. b) The equation of the tangent plane at the point P is z x y. 2

6 Tom Robbins WW Prob Lib Math 220-2, Fall 200 WeBWorK assignment 5 due 0/9/0 at 6:00 AM..( pt) Consider the surface x 2 25y 2 z 2 27 and the point P on this surface. Find an equation for the tangent plane through P. z Note: Your answers should be expressions of x and y; e.g. 3xy + 2y 2.( pt) Let w 0xy 4x 5y, x r s t, y r s, and z s t. Find the partial derivatives of w with respect to r, s and t at the point r 2, s 2, t. w r =. w s =. w =. t 3.( pt) Find the point of intersection of the tangent lines to the parabola y 5x 2 at the points 3 45 and 5 Intersection point: (, ) 4.( pt) Suppose f x y x 2 y 2 8x 4y 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. (D) If there is a saddle point, what is the value of the discriminant D at that point? If there is none, type N. (E) What is the maximum value of f on R 2? If there is none, type N. (F) What is the minimum value of f on R 2? If there is none, type N. 5.( pt) Suppose f x y xy 6x 6y. f x y has 4 critical points. List them in increasing lexographic order. By that we mean that (x, y) comes before (z, w) if x z or if x z and y w. Also, describe the type of critical point by typing MA if it is a local maximum, MI if it is a local minimim, and S if it is a saddle point. First point (, ) of type Second point (, ) of type Third point (, ) of type Fourth point (, ) of type 6.( pt) You are to manufacture a rectangular box with 3 dimensions x, y and z, and volume v 729. Find the dimensions which minimize the surface area of this box. x = y = z = 7.( pt) Find the coordinates of the point (x, y, z) on the plane z = 4 x + 4 y + which is closest to the origin. x = y = z = 8.( pt) Find the maximum and minimum values of f x y 3x y on the ellipse x 2 6y 2 maximum value: minimum value:

7 Tom Robbins WW Prob Lib Math 220-2, Fall 200 WeBWorK assignment 6 due /2/0 at 6:00 AM. 3 0.( pt) Evaluate the iterated integral 4 0 4x2 y 3 dxdy 2.( pt) Calculate the double integral R 0x 8y 80 da where R is the region: 0 x 4 0 y 5. 3.( pt) Calculate the volume under the elliptic paraboloid z 3x 2 5y 2 and over the rectangle R ( pt) Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 6. 5.( pt) Match the following integrals with the verbal descriptions of the solids whose volumes they give. Put the letter of the verbal description to the left of the corresponding integral x 2 4 4x 3y dydx y 4x 2 3y 2 dxdy y 2 x 2 x 2 x2 y 2 dydx y x 2 3y 2 dxdy y 2 dydx A. Solid under a plane and over one half of a circular disk. B. One eighth of an ellipsoid. C. One half of a cylindrical rod. D. Solid under an elliptic paraboloid and over a planar region bounded by two parabolas. E. Solid bounded by a circular paraboloid and a plane. 6.( pt) Using polar coordinates, evaluate the integral R sin x2 y 2 da where R is the region 4 x 2 y ( pt) Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles x 2 y and x 2 20x y ( pt) A cylindrical drill with radius 2 is used to bore a hole throught the center of a sphere of radius 4. Find the volume of the ring shaped solid that remains. 9.( 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 2? ft 3 perhour B. What is the total amount of water that goes throught the sprinkler per hour? ft 3 perhour 0.( pt) You are getting married and your dearest relative has baked you a cake which fills the volume between the two planes, z 0 and z 3x 7y c, and inside the cylinder x 2 y 2. You are to cut it in half by making two vertical slices from the center outward. Suppose one of the slices is at θ 0 and the other is at θ ψ. What is the limit, c lim ψ?

8 Tom Robbins WW Prob Lib Math 220-2, Fall 200 WeBWorK assignment 7 due /2/0 at 6:00 AM..( pt) Electric charge is distributed over the disk x 2 y 2 6 so that the charge density at (x,y) is σ x y 20 x 2 y 2 coulombs per square meter. Find the total charge on the disk. 2.( pt) A lamina occupies the part of the disk x 2 y 2 4 in the first quadrant and the density at each point is given by the function ρ x y x 2 y 2. A. What is the total mass? B. What is the moment about the x-axis? C. What is the moment about the y-axis? D. Where is the center of mass? (, ) E. What is the moment of inertia about the origin? 3.( pt) Find the surface area of the part of the plane 5x 3y z 3 that lies inside the cylinder x 2 y 2. 4.( pt) The vector equation r u v ucosvi usinvj vk, 0 v π, 0 u, describes a helicoid (spiral ramp). What is the surface area? 5.( pt) Find the mass of the region (in cylindrical coordinates) r 3 z 3, where the density function is ρ r θ z 7z. Answer:. 6.( pt) Evaluate the triple integral xyzdv E where E is the solid: 0 z 4, 0 y z, 0 x y. 7.( pt) Find the average value of the function f x y z x 2 y 2 z 2 over the rectangular prism 0 x 5, 0 y 2, 0 z 5 8.( pt) Use cylindrical coordinates to evaluate the triple integral E x 2 y 2 dv, where E is the solid bounded by the circular paraboloid z 9 4 x 2 y 2 and the xy-plane. 9.( pt) Use spherical coordinates to evaluate the triple integral E x2 y 2 z 2 dv, where E is the ball: x 2 y 2 z ( pt) Match the integrals with the type of coordinates which make them the easiest to do. Put the letter of the coordinate system to the left of the number of the integral. y x dx dy 2. E z dv where E is: x 2 3 y 4 5 z 6 3. D x da where D is: x 2 y 2 2 y E dv where E is: x2 y 2 z 2 4 x 0 y 0 z 0 5. E z2 dv where E is: 2 z 2 x 2 y 2 2 A. polar coordinates B. cartesian coordinates C. spherical coordinates D. cylindrical coordinates

9 Tom Robbins WW Prob Lib Math 220-2, Fall 200 WeBWorK assignment 8 due /30/0 at :59 PM..( pt) Compute the gradient vector fields of the following functions: A. f x y 9x 2 3y 2 f x y i j B. f x y x 7 y 4 f x y i j C. f x y 9x 3y f x y i j D. f x y z 9x 3y 7z f x y i j k E. f x y z 9x 2 3y 2 7z 2 f x y z i j k 2.( pt) Match the following vector fields with the verbal descriptions of the level curves or level surfaces to which they are perpendicular by putting the letter of the verbal description to the left of the number of the vector field.. F xi yj zk 2. F 2i j k 3. F 2xi yj 4. F yi xj 5. F xi yj k 6. F xi yj zk 7. F xi yj 8. F yi xj 9. F xi yj 0. F 2i j. F 2xi yj zk A. hyperboloids B. spheres C. planes D. lines E. paraboloids F. ellipses G. circles H. ellipsoids I. hyperbolas 3.( 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 C F dr B. If C 2 is the straight line segment: x 2t 2 y 4t 0 t, then C2 F dr 4.( 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 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: 5.( 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 x y 6x y i x 0y j f x y B. F x y 8yi 9xj f x y C. F x y z 8xi 9yj k f x y z D. F x y 8siny i 2y 8xcosy j f x y E. F x y z xzi yzj xyk f x y z Note: Your answers should be either expressions of x, y and z (e.g. 3xy + 2yz ), or the letter N 6.( pt) If C is the curve given by r t 3sint i sin 2 t j sin 3 t k, 0 π t 2 and F is the radial vector field F x y z xi yj zk, compute the work done by F on a particle moving along C. 7.( pt) Suppose C is any curve from to and F x y z 3z y i z x j y 3x k. Compute the line integral C F dr.

10 8.( pt) Let C be the positively oriented circle x 2 y 2. Use Green s Theorem to evaluate the line integral C 2ydx 2xdy. 9.( pt) Find a parametrization of the curve x 2 3 y 2 3 and use it to compute the area of the interior. 0.( pt) Let F 5xi 7yj 5zk. Compute the divergence and the curl. A. div F B. curl F i j k.( pt) Let F 7yz i 5xz j 2xy 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 2.( pt) Let F yi 2xj. Use the tangential vector form of Green s Theorem to compute the circulation integral C F dr where C is the positively oriented circle x 2 y ( pt) Let F xi yj and let n be the outward unit normal vector to the positively oriented circle x 2 y 2 9. Compute the flux integral C F nds. 2