WW Prob Lib1 Math course-section, semester year

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1 WW Prob Lib Math course-section, semester year WeBWorK assignment due /25/06 at :00 PM..( pt) Consider the parametric equation x = 7(cosθ + θsinθ) y = 7(sinθ θcosθ) What is the length of the curve for θ = 0 to θ = 5 2 π? 2.( pt) Let a = (0, 3, 0) and b = (5, 9, 0) be vectors. Compute the following vectors. A. a + b = (,, ) B. -a= (,, ) C. a - b= (,, ) D. a = 3.( pt) A child walks due east on the deck of a ship at 5 miles per hour. The ship is moving north at a speed of 8 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 = 8, b = 7, π and the angle between a and b is 0 radians. a b = 5.( pt) Find a unit vector in the same direction as a = (2, 3, 0). (,, ) 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 (5, 6). If he began walking in the direction of the vector v = 4i + 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 demonstration assignment The main purpose of this WeBWorK set is to familiarize yourself with WeBWorK. Here are some hints on how to use WeBWorK effectively: After first logging into WeBWorK change your password. Find out how to print a hard copy on the computer system that you are going to use. Print a hard copy of this assignment. Get to work on this set right away and answer these questions well before the deadline. Not only will this give you the chance to figure out what s wrong if an answer is not accepted, you also will avoid the likely rush and congestion prior to the deadline. The primary purpose of the WeBWorK assignments in this class is to give you the opportunity to learn by having instant feedback on your active solution of relevant problems. Make the best of it!.( pt) Evaluate 5(5 2) =. 2.( pt) Evaluate 2/(4 + ) =. Enter you answer as a decimal number listing at least 4 decimal digits. (WeBWorK will reject your answer if it differs by more than one tenth of percent from what it thinks the answer is.) 3.( pt) Let r = 9. Evaluate 4/π r =. Next, enter 4/(π r) = and let WeBWorK compute the result. b. 4.( pt) a + a+b. 5.( pt) a b a + b c + d If WeBWorK rejects your answer use the preview button to see what it thinks you are trying to tell it. 6.( pt) a + b 7.( pt) 8.( pt) a a + b a + b a + b x 2 + y 2 x x 2 + y 2 x + y x 2 + y 2 b + b 2 4ac 2a Note: this is an expression that gives the solution of a quadratic equation by the quadratic formula.

3 Math 220-4, Spring 2006 WeBWorK Assignment 2 due /30/06 at :00 PM This assignment will cover the material from Sections.4, 2. and 2.2..( pt) If a = (-8, -8, -5) and b = (4, 6, 2), find a b =. 2.( pt) What is the angle in radians between the vectors a = (5, -7, -0) and b = (-, -2, -0)? Angle: (radians) 3.( pt) Let a = (6, 2, -5) and b = (7, 0, -2) be vectors. Find the scalar, vector, and orthogonal projections of b onto a. Scalar Projection: Vector Projection: (,, ) Orthogonal Projection: (,, ) 4.( pt) Let a = (7, 8, 3) and b = (8, 2, 6) be vectors. Compute the cross product a b. (,, ) 5.( pt) Find the area of the parallelogram with vertices (4,), (8, 5), (3, 9), and (7, 3). 6.( pt) Find the tangential and normal components (a T and a N ) of the acceleration vector at t for r(t) = t 2 i +tj. a T = a N =

4 WW Prob Lib Math course-section, semester year WeBWorK assignment 3 due 2/8/06 at :00 PM..( pt) Find the distance from the point (, 5, 3) to the line x = 0,y = 5 + 5t,z = 3 + 2t. 2.( pt) A million years ago, an alien species built a vertical tower on a horizontal plane. When they returned they discovered that the ground had tilted so that measurements of 3 points on the ground gave coordinates of (0, 0, 0), (3, 2, 0), and (0, 3, 3). By what angle does the tower now deviate from the vertical? radians. 3.( pt) Find parametric equations for the tangent line at the point (cos( 2π 6 ),sin( 2π 6 ), 2π 6 )) on the curve x = cost, y = sint, z = t x(t) = y(t)= z(t)= 4.( pt) Match the surfaces with the appropriate descriptions.. z = x 2 2. x 2 + 2y 2 + 3z 2 = 3. z = 4 4. z = 2x + 3y 5. x 2 + y 2 = 5 6. z = y 2 2x 2 7. z = 2x 2 + 3y 2 A. parabolic cylinder B. nonhorizontal plane C. elliptic paraboloid D. horizontal plane E. hyperbolic paraboloid F. circular cylinder G. ellipsoid

5 WW Prob Lib Math course-section, semester year WeBWorK assignment 4 due 2/5/06 at :00 PM..( pt) What are the rectangular coordinates of the point whose cylindrical coordinates are (r = 8, θ = 4π 5, z = 0)? x = y = z = 2.( pt) What are the spherical coordinates of the point whose rectangular coordinates are (4, 3, )? ρ = θ = φ = 3.( pt) What are the cylindrical coordinates of the point whose spherical coordinates are (5, 4, 3π 6 )? r = θ = z= 4.( pt) Match the given equation with the verbal description of the surface: A. Cone B. Elliptic or Circular Paraboloid C. Half plane D. Circular Cylinder E. Plane F. Sphere. r = 4 2. r 2 + z 2 = 6 3. ρ = 2cos(φ) 4. φ = π 3 5. z = r 2 6. r = 2cos(θ) 7. θ = π 3 8. ρcos(φ) = 4 9. ρ = 4 5.( pt) Find the first partial derivatives of f (x,y) = 4x 4y 4x+4y at the point (x,y) = (3, 3). f x (3,3) = f y (3,3) = 6.( pt) Find the first partial derivatives of f (x, y, z) = z arctan( y x ) at the point (5, 5, -5). A. f x (5,5, 5) = B. f y (5,5, 5) = C. f z (5,5, 5) =

6 WW Prob Lib Math course-section, semester year WeBWorK assignment 5 due 3/3/06 at :00 PM..( pt) If sin( 4x + 4y + z) = 0, find the first partial derivatives z z x and y at the point (0, 0, 0). A. z x (0,0,0) = B. z y (0,0,0) = 2.( pt) Suppose w = y x + y z, x = et, y = 2+sin(2t), z = 2 + cos(5t). A. Use the chain rule to find dw dt as a function of x, y, z, and t. Do not rewrite x, y, and z in terms of t, and do not rewrite e t as x. dw dt = Note: Use exp() for the exponential function. Your answer should be an expression in x, y, z, and t; e.g. 3x - 4y B. Use part A to evaluate dw dt when t = 0. 3.( pt) The axis of a light in a lighthouse is tilted. When the light points east, it is inclined upward at 7 degree(s). When it points north, it is inclined upward at 3 degree(s). What is its maximum angle of elevation? degrees 4.( pt) Suppose f (x,y) = x 2 + y 2 4x 2y + 2 (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 ax by. (A) How many local minimum points does f have in R 2? (The answer is an integer). (B) How many local maximum points does f have in R 2? (C) How many saddle points does f have in R 2?

7 WW Prob Lib Math course-section, semester year WeBWorK assignment 6 due 3/8/06 at :00 PM..( pt) The radius of a right circular cone is increasing at a rate of 2 inches per second and its height is decreasing at a rate of 4 inches per second. At what rate is the volume of the cone changing when the radius is 30 inches and the height is 40 inches? cubic inches per second 2.( pt) Suppose f (x,y) = xy( 9x 7y). 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 3.( pt) You are to manufacture a rectangular box with 3 dimensions x, y and z, and volume v = 728. Find the dimensions which minimize the surface area of this box. x = y = z = 4.( pt) Find the maximum and minimum values of f (x,y) = 7x + y on the ellipse x y 2 = maximum value: minimum value:

8 WW Prob Lib Math course-section, semester year WeBWorK assignment 7 due 3/29/06 at :00 PM..( pt) Evaluate the integral by reversing the order of integration. Z Z 2 0 2y e x2 dxdy = 2.( 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. Z 2 Z 2. 4 y 2 dydx Z Z x 2 Z 3 Z 2 3y 2 x2 y 2 dydx x 2 4x 2 3y 2 dxdy 0 0 Z 2 Z 4+ 4 x Z Z y 0 y 2 4x + 3y dydx 4x 2 + 3y 2 dxdy A. One half of a cylindrical rod. B. Solid under a plane and over one half of a circular disk. C. Solid under an elliptic paraboloid and over a planar region bounded by two parabolas. D. One eighth of an ellipsoid. E. Solid bounded by a circular paraboloid and a plane. 3.( pt) Z ZUsing polar coordinates, evaluate the integral sin(x 2 + y 2 )da where R is the region 4 x 2 + R y ( pt) Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles x 2 +y 2 = 64 and x 2 8x+y 2 = 0. 5.( 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

9 WW Prob Lib Math course-section, semester year WeBWorK assignment 8 due 4/7/06 at :00 PM..( pt) Find the surface area of the part of the plane 5x + 5y + z = 2 that lies inside the cylinder x 2 + y 2 = ( pt) Find the mass of the region (in cylindrical coordinates) r 3 z 3, where the density function is ρ(r,θ,z) = 6z. Answer:. 3.( pt) Evaluate the triple integral Z Z Z xyzdv E where E is the solid: 0 z 3, 0 y z, 0 x y. 4.( pt) Find the average value of the function f (x,y,z) = x 2 + y 2 + z 2 over the rectangular prism 0 x 4, 0 y 2, 0 z 5.( pt) Use cylindrical coordinates to evaluate the triple integral RRR E x 2 + y 2 dv, where E is the solid bounded by the circular paraboloid z = 4 4 ( x 2 + y 2) and the xy-plane. 6.( pt) Use spherical coordinates to evaluate the triple integral RRR E x2 + y 2 + z 2 dv, where E is the ball: x 2 + y 2 + z 2 36.

10 WW Prob Lib Math course-section, semester year WeBWorK assignment 9 due 4/4/06 at :00 PM..( pt) Compute the total mass of a wire bent in a quarter circle with parametric equations: x = cost, y = sint, 0 t π 2 and density function ρ(x,y) = x2 + y 2. 2.( pt) Let C be the curve which is the union of two line segments, the first going from (0, 0) to (3, -2) and the second going from Z (3, -2) to (6, 0). Computer the line integral 3dy + 2dx. 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 (9, 8). (Compare your answers!) A. ZIf C is the parabola: x = t, y = t 2, 0 t 9, then F dr = C B. If C 2 is the straight Z line segment: x = 9t 2, y = 8t 2, 0 t, then F dr = C 2 C 4.( 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 + 7y)i + (7x + 6y)j f (x,y) = B. F(x,y) = 3yi 2xj f (x,y) = C. F(x,y,z) = 3xi 2yj + k f (x,y,z) = D. F(x,y) = ( 3siny)i + (4y 3xcosy)j f (x,y) = E. F(x,y,z) = 3x 2 i + 7y 2 j + 3z 2 k f (x,y,z) = Note: Your answers should be either expressions of x, y and z (e.g. 3xy + 2yz ), or the letter N 5.( pt) Suppose C is any curve from (0, 0, 0) to (,,) and F(x,y,z) = (3z + 2y)i + (2z + 2x)j + (2y + 3x)k. Compute the line integral R C F dr.