5. (1 pt) set1/p1-7.pg. Let T be the triangle with vertices at (9, 1),(3, 8),( 6, 2). The area of T is

Transcription

1 Hsiang-Ping Huang Math , Spring 2008 WeBWorK Assignment due 0/7/2008 at 0:59pm MST Vectors Geometry, Dot and Cross Products This assignment will cover the material from Chapters..4.. ( pt) set/p-.pg 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 6 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) ( pt) set/p-2.pg Find a b if a = 7, b = 7, and the angle between a and b is π 9 radians. a b = ( pt) set/p-3.pg Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (0, -3) and arrived in the Iron Hills at the point with coordinates (, 2). 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. (, ) ( pt) set/p-4.pg An object is at rest on the plane. Three forces, V,W,X are acting on the object. If V = 9I + 6J,W = 7I + 2J, then X must be I+ J ( pt) set/p-7.pg Let T be the triangle with vertices at (9, ),(3, 8),( 6, 2). The area of T is Hint: Use the projection formula to find the length of an altitude orthogonal to any chosen base ( pt) set/p-8.pg Find the vector V which makes an angle of 0 degrees with the vector W = 3I + 0J and which is of the same length as W and is counterclockwise to W. I+ J ( pt) set/p2-.pg If a = 3I + 8J 3K and b = 5I + 7J + 9K, find a b = ( pt) set/p2-2.pg What is the angle in radians between the vectors a = 9I + 4J + K and b = 4I + 2J + 2K Angle: (radians) ( pt) set/p2-3.pg Find a unit vector in the same direction as a = I+0J+3K. I+ J+ K (Note that, a unit vector is a vector whose length is. Multiplying any non-zero vector V by / V produces a unit vector, and multiplying any unit vector by - shows that there are two unit vector which are multiples of any non-zero vector. ) ( pt) set/p2-8.pg What is the distance from the point (9, 2, ) to the xz-plane? Distance = 2

2 . ( pt) set/p2-5.pg Find a unit vector with positive first coordinate that is orthogonal to the plane through the points P = (-, -5, -), Q = (3, -, 3), and R = (3, -, 8). I+ J+ J ( pt) set/p2-7.pg Given vectors u and v such that u v = 4I 5J 2K, find: a) v u = I+ J+ K b) (u v) (u v) =. c) (u v) (u v) = I+ J+ K ( pt) set/p2-0.pg Consider the planes 2x + 4y + z = and 2x + z = 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 the vectors found in parts (A) and (B) to find a vector equation for the line of intersection of the two planes,r(t) = I + J + K t*/sqrt( 2**2 + **2 ) /4 t*(-2)/sqrt( 2**2 + **2 ) 4. ( pt) set/p4-7.pg The vectors U = cosθi + sinθj,v = sinθi + cosθj, for any θ, form an orthonormal basis for the plane; that is, they are orthogonal vectors of length. Let θ = 2π 3, and X = 8I + 4J. Then we can write with, X = uu + vv u = v = ( pt) set/p5-5.pg The axis of a light in a lighthouse is tilted. When the light points east, it is inclined upward at 3 degree(s). When it points north, it is inclined upward at 7 degree(s). What is its maximum angle of elevation? (Hint: The maximum angle of elevation of plane of the beam above the horizontal plane is the same as the angle between the normal to the plane of the beam and the normal to the horizontal plane.) degrees Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 2

3 Hsiang-Ping Huang Math , Spring 2008 WeBWorK Assignment 2 due 0/25/2008 at 0:59pm MST Lines, Curves, Velocity, Acceleration, Curvature, Tangents and Normals This assignment will cover the material from Chapters ( pt) set2/p-5.pg The distance between the two parallel lines L : 6x 9y = 5,L 2 : 6x 9y = 0 is ( pt) set2/p-6.pg Find the distance of the point (9,8) from the line through (9, 8) which points in the direction of I 3J ( pt) set2/p2-6.pg F T F F F T T T T F T 5. ( pt) set2/p3-2.pg The position of a particle in motion in the plane at time t is X(t) = exp(8.4t)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 X(t): I+ J (c) The tangential acceleration: (d) The normal acceleration: The equation X (t) = A + tl is the parametric equation of a line through the point P : (2, 3, ). The parameter t represents distance from the point P, directed so that the I component of L is positive. We know that the line is orthogonal to the plane with equation 5x y 8z = 6. Then A= I+ J+ K L= I+ J+ K ( pt) set2/p2-9.pg 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.) Note, all questions assume that you are in R 3!. Two planes parallel to a line are parallel. 2. Two planes either intersect or are parallel. 3. Two lines either intersect or are parallel. 4. Two lines perpendicular to a third line are parallel. 5. Two planes perpendicular to a third plane are parallel. 6. Two planes perpendicular to a line are parallel. 7. Two lines perpendicular to a plane are parallel 8. A plane and a line either intersect or are parallel. 9. Two lines parallel to a third line are parallel. 0. Two lines parallel to a plane are parallel.. Two planes parallel to a third plane are parallel ( pt) set2/p3-4.pg The position of a particle in motion in the plane at time t is X(t) = ti + ln(cos(t))j. At time t = (0.π)/2, determine the following: (a) the unit tangent vector: I+ J (b) the unit normal vector to X(t): I+ J (c) the acceleration vector: I+ J (d) the curvature:

4 7. ( pt) set2/p3-5.pg Consider the vector functions X(t) = 3I + cos(0t)j, Y(t) = sin(2t)j + 8K. Let Z(t) = X(t) Y(t) ( pt) set2/p3-9.pg Find the curvature κ(t) of the curve X(t) = (5 sint) I + (5sint)j + (cost)k Then dz (t)= I+ J+ K. dt -8*0*sin(0*t).0 3*2*cos(2*t) 8. ( pt) set2/p3-6.pg Given that the acceleration vector is a(t) = ( 6cos(4t))I + ( 6sin(4t))J+(5t)K, the initial velocity is v(0) = I + K, and the initial position vector is X(0) = I + J + K, compute: A. The velocity vector v(t) = I+ J+ K B. The position vector X(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) - 4 * sin( 4 * t ) + 4 * cos( 4 * t ) - 4 ( 5 * t**2 ) / 2 + cos( 4 * t ) + t sin( 4 * t ) - 4 * t + ( 5 * t**3 ) / 6 + t + 9. ( pt) set2/p3-7.pg If X(t) = cos( t)i + sin( t)j 0tK, 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) - - * sin( - * t ) - * cos( - * t ) -0 - (-)**2 * cos( - * t ) - (-)**2 * sin( - * t ) 0. ( pt) set2/p3-8.pg Consider the helix X(t) = (cos( 4t), sin( 4t), 2t). Compute, at t = π 6 : A. The unit tangent vector T = I+ J+ K B. The unit normal vector N = I+ J+ K Hint, to compute κ, you can use the formula listed on page 599 of the text. (2**(/2))*abs(5*)/( 2*(5*cos(t))**2+(*sin(t))**2 )ˆ(3/2) 2. ( pt) set2/p3-0.pg (A) Find the parametric equations for the line through the point P = (4, -4, -5) that is perpendicular to the plane 5x+y 4z =. Use t as your variable, t = 0 should correspond to P, and the velocity vector of the line should be the same as the normal vector to the plane found directly from its equation. x = y = z = (B) At what point Q does this line intersect the yz-plane? Q = (,, ) 4 + t * t * -5 + t * ( pt) set2/p2-4.pg Find a unit vector orthogonal to 6I 6J+0K and 0I+J+0K: I+ J+ K ( pt) set2/p3-.pg While a planet P rotates in a circle about its sun, a moon M rotates in a circle about the planet, and both motions are in a plane. Let s call the distance between M and P one lunar unit. Suppose the distance of P from the sun is lunar units; the planet makes one revolution about the sun every 5 years, and the moon makes one rotation about the planet every years. Choosing coordinates centered at the sun, so that, at time t = 0 the planet is at ( ,0), and the moon is at ( ,), then the location of the moon at time t, where t is measured in years, is (x(t),y(t)), where x(t)= y(t)=

5 (Note that, a unit vector is a vector whose length is. Multiplying any non-zero vector V by / V produces a unit vector, 4.6*E3*cos(2* /5*t)-sin(2* / *t) 4.6*E3*sin(2* /5*t)+cos(2* / *t) and multiplying any unit vector by - shows that there are two unit vector which are multiples of any non-zero vector. The unit tangent vector to X(t) is defined as V (t)/ V (t). ) 5. ( pt) set2/p3-3.pg The position of a particle in motion in the plane at time t is X(t) = 8t I + sin(3t) J. At time any t, determine the following: (a) the speed of the particle is: (b) the unit tangent vector to X(t) is: I+ J sqrt( (-8*-8)+(3*3)*(cos(3*t)*cos(3*t)) ) -8/sqrt( (-8*-8)+(3*3)*cos(3*t)*cos(3*t) ) 3*cos(3*t)/sqrt( (-8*-8)+(3*3)*cos(3*t)*cos(3*t) ) Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 3

6 Hsiang-Ping Huang Math , Spring 2008 WeBWorK Assignment 3 due 02/0/2008 at 0:59pm MST Surfaces, Cylindrical and Spherical Coordinates This assignment will cover the material from Chapters ( pt) set3/p4-.pg Match the function with the description of its level sets (the sets z =constant). A. a collection of ellipses. B. a collection of hyperbolas. C. a collection of parallel lines. D. a collection of circles centered at the origin. E. a collection of parabolas.. z = 5 x 2 y 2 2. xy + z 2 = 3. z = + x + y 4. z = 5x 3y 5. z = (x + y ) 6. z = ( x 2 + 2y 2) 2 7. z = x 2 2xy + y 2 + x + y D B E C C A E 2. ( pt) set3/p4-3.pg Match the surfaces with the appropriate descriptions.. z = 4 2. z = y 2 2x 2 3. x 2 + 2y 2 + 3z 2 = 4. z = x 2 5. z = 2x + 3y 6. z = 2x 2 + 3y 2 7. x 2 + y 2 = 5 A. nonhorizontal plane B. elliptic paraboloid C. circular cylinder D. hyperbolic paraboloid E. parabolic cylinder F. horizontal plane G. ellipsoid F D G E A B C 3. ( pt) set3/p4-4.pg Match the given equation with the verbal description of the surface: A. Half plane B. Elliptic or Circular Paraboloid C. Plane D. Sphere E. Circular Cylinder F. Cone. φ = π 3 2. r = 4 3. ρcos(φ) = 4 4. ρ = 2cos(φ) 5. r = 2cos(θ) 6. ρ = 4 7. r 2 + z 2 = 6 8. z = r 2 9. θ = π 3 F E C D E D D B A 4. ( pt) set3/p4-5.pg Let P be the point ( 9, 5,4) in cartesian coordinates. A. The cylindrical coordinates of P are r =, θ =, z =. B. The spherical coordinates of P are ρ =, θ =, φ = ( pt) set3/p4-6.pg Let P be the point with the spherical coordinates ρ = 7, φ = π/2, θ = π/6. A. The cylindrical coordinates of P are r =, θ =, z =. B. The cartesian coordinates of P are x =, y =, z =

7 E E-9 6. ( pt) set3/p4-8.pg Let L be the line y = 8, x = 4z. If we rotate L around the x-axis, we get a surface whose equation is Ax 2 + By 2 +Cz 2 =, where A=, B =, C = ( pt) set3/p4-9.pg Match the curves with the appropriate descriptions. A. Two parallel lines. B. ellipse. C. parabola. D. hyperbola.. x 2 2xy + y 2 + x + y = x 2 3xy 2y 2 + x + y = 2 3. x 2 2xy = 6 4. x 2 2xy + 2y 2 = 6 5. x 2 2xy + y 2 = 6 C D D B A Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 2

8 Hsiang-Ping Huang Math , Spring 2008 WeBWorK Assignment 4 due 02/4/2008 at 0:59pm MST Functions of Two or More Variables, Partial Derivatives, Limits, Continuity, Differentiability This assignment will cover the material from Chapters ( pt) set4/p5-.pg Find the first partial derivatives of f (x,y) = sin(x y) at the point (8, 8). A. f x (8,8) = B. f y (8,8) = - 2. ( pt) set4/ur VC 5 5.pg The level curves of a function f (x,y) consist of a collection of hyperbolas and two lines. If the lines intersect at a point P, what are the possibilities for P? Type the letters of all possibilities, with no punctuation, in alphabetical order. A. P is a local maximum, that is, f (P) f (Q) for all Q near P. B. P is a local minimum, that is, f (P) f (Q) for all Q near P. C. P is neither a local maximum nor a local minimum. ABC 3. ( pt) set4/ur VC 5 F.pg On a map showing the grave of George Mallory, the contour lines are: A. closely spaced B. far apart A 4. ( pt) set4/ur VC 5 3.pg Find the limit, if it exists, or type N if it does not exist. lim (x,y) (5, 4) e 4x 2 +y 2 = eˆ(sqrt(4*(5)ˆ2 + *(-4)ˆ2)) 5. ( pt) set4/ur VC 5 4.pg Find the limit, if it exists, or type N if it does not exist. 2x 2 lim (x,y) (0,0) 4x 2 + 4y 2 = N 6. ( pt) set4/ur VC 5 5.pg Find the limit, if it exists, or type N if it does not exist. (x + 25y) 2 lim (x,y) (0,0) x y = 2 N 7. ( pt) set4/ur VC 5 6.pg Find the limit, if it exists, or type N if it does not exist. (Hint: use polar coordinates.) 3x 3 + 4y 3 lim (x,y) (0,0) x 2 + y 2 = 8. ( pt) set4/ur VC 5 7.pg Find the limit, if it exists, or type N if it does not exist. 5ze x2 +y 2 lim (x,y,z) (5,4,4) 5x 2 + 4y 2 + 4z 2 = ( pt) set4/ur VC 5 8.pg Find the limit, if it exists, or type N if it does not exist. 5xy + 3yz + 5xz lim (x,y,z) (0,0,0) 25x 2 + 9y z 2 = N 0. ( pt) set4/ur VC 5 0.pg Find the first partial derivatives of f (x,y) = x 2y x+2y at the point (x,y) = (4, ). f x (4,) = f y (4,) =

9 . ( pt) set4/ur VC 5 2.pg Find the first partial derivatives of f (x,y) = sin(x y) at the point (8, 8). A. f x (8,8) = B. f y (8,8) = - 2. ( pt) set4/ur VC 5 3.pg If sin( 3x y + z) = 0, find the first partial derivatives z x and z y at the point (0, 0, 0). A. z x (0,0,0) = B. z y (0,0,0) = 3 Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 2

10 Hsiang-Ping Huang Math , Spring 2008 WeBWorK Assignment 5 due 02/2/2008 at 0:59pm MST Directional Derivatives, the Gradient, Tangent Planes, the Chain Rule, Implicit Differentiation, Approximation with Differentials This assignment will cover the material from Chapters ( pt) set5/p5-2.pg If f (x,y) = 4x 2 + 3y 2, find the value of the directional derivative at the point ( 3,2) in the direction given by the angle θ = 2π 3. 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 /y -x / y** ( pt) set5/p5-3.pg Suppose f (x,y) = 3x 2 +3xy 2y 2, P = (,0), and u = ( 5 3, 2 3). 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 2*3*x + 3*y 3*x + 2*-2*y ( pt) set5/p5-4.pg Suppose f (x,y) = x y, P = (4, 2) and v = I + 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, where the direction u of a vector v is the unit vector obtained by normalizing that vector, i.e., u = v 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 4. ( pt) set5/p5-6.pg Consider the equation xz 2 4yz + 4lnz = 9 as defining z implicitly as a function of x and y. The values of z z and x y at ( 5,, ) are and. (This problem used to have log instead of ln, but the answer was the same, because in webwork log means the natural logarithm ( pt) set5/p6-3.pg Let w = 4xy 0x + 9y, 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 =, t = 2. w r =. w s =. w =. t ( pt) set5/ur vc 6 5.pg The dimensions of a closed rectangular box are measured as 00 centimeters, 60 centimeters, and 80 centimeters, respectively, with the error in each measurement at most.2 centimeters. Use differentials to estimate the maximum error in calculating the surface area of the box. square centimeters 92

11 7. ( pt) set5/p5-7.pg Find the equation of the tangent plane to the surface z = y 2 4x 2 at the point (4,0, 64). z = Note: Your answer should be an expression of x and y; e.g. 3x - 4y + 6 ( -2 * 4 * 4 * x ) + (2 * * 0 * y ) + ( 4 * 6 - * 0 ) 8. ( pt) set5/p5-8.pg The intensity of light at a distance r from a source is given by L = Ir 2, where I is the illumination at the source. Starting with the values I = 70, r = 0, suppose we increase the distance by 4 and the illumination by 3. By (approximately) how much does the intensity of light change? dl = ( pt) set5/p6-2.pg Consider the surface x 2 + y z 2 = 27 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 ( pt) set5/ur VC 5 4.pg Find all the first and second order partial derivatives of f (x,y) = 8sin(2x + y) + 8cos(x y). A. f x = f x = B. f y = f y = C. 2 f = f x 2 xx = D. 2 f = f y 2 yy = E. 2 f x y = f yx = F. 2 f y x = f xy = 2*8*cos(2*x+y) - 8*sin(x-y) 8*cos(2*x+y) + 8*sin(x-y) -4*(8 * sin(2*x + y)) - 8*cos(x-y) - 8*sin(2*x+y) - 8*cos(x-y) -2*(8*sin(2*x+y)) + 8*cos(x-y) -2*(8*sin(2*x+y)) + 8*cos(x-y) Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 2

12 Hsiang-Ping Huang Math , Spring 2008 WeBWorK Assignment 6 due 02/28/2008 at 0:59pm MST Maxima and Minima, Lagrange s Method This assignment will cover the material from Chapters ( pt) set6/p6-4.pg Suppose f (x,y) = x 2 + y 2 0x 8y + 4 (A) How many critical points does f have in R 2? (Note, R 2 is the set of all pairs of real numbers, or the (x,y)- plane.) (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. 4 N N N 4-5**2-4**2 2. ( pt) set6/p6-5.pg Suppose f (x,y) = xy( 5x 8y). 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 S.25 S MA.2 S 3. ( pt) set6/p6-6.pg You are to manufacture a rectangular box with 3 dimensions x, y and z, and volume v = 8. Find the dimensions which minimize the surface area of this box. x = y = z = ( pt) set6/p6-7.pg Find the coordinates of the point (x, y, z) on the plane z = 2 x + y + which is closest to the origin. x = y = z = ( pt) set6/p6-8.pg Find the maximum and minimum values of f (x,y) = 7x + y on the ellipse x y 2 = maximum value: minimum value: ( pt) set6/ur vc 7 2.pg 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?

13 7. ( pt) set6/ur vc 7 3.pg Consider the function f (x, y) = x sin(y). In the following questions, enter an integer value or type INF for infinity. (A) How many local minima does f have in R 2? (B) How many local maxima does f have in R 2? (C) How many saddle points does f have in R 2? INF 8. ( pt) set6/ur vc 7 5.pg Each of the following functions has at most one critical point. Graph a few level curves and a few gradiants and, on this basis alone, decide whether the critical point is a local maximum (MA), a local minimum (MI), or a saddle point (S). Enter the appropriate abbreviation for each question, or N if there is no critical point. (A) f (x,y) = e 2x2 4y 2 Type of critical point: (B) f (x,y) = e 2x2 4y 2 Type of critical point: (C) f (x,y) = 2x 2 + 4y Type of critical point: (D) f (x,y) = 2x + 4y + 4 Type of critical point: MA S MI N 9. ( pt) set6/ur vc 7 8.pg Find the maximum and minimum values of f (x,y) = 8x 2 + 9y 2 on the disk D: x 2 + y 2. maximum value: minimum value: 9 (C) f (x,y) = 4x 2 5y 2 : maximum value = minimum value = ( pt) set6/ur vc 7.pg For each of the following functions, find the maximum and minimum values of the function on the rectangular region: 4 x 4, 5 y 5. Do this by looking at level curves and gradients. (A) f (x,y) = x + y + 2: maximum value = minimum value = (B) f (x,y) = 2x 2 + 3y 2 : maximum value = minimum value = (C) = f (x,y) = (5) 2 x 2 (4) 2 y 2 : maximum value = minimum value = ( pt) set6/ur vc 7 2.pg Find the maximum and minimum values of f (x,y,z) = 3x+y+ 5z on the sphere x 2 + y 2 + z 2 =. maximum value = minimum value = ( pt) set6/ur vc 7 0.pg For each of the following functions, find the maximum and mimimum values of the function on the circular disk: x 2 + y 2. Do this by looking at the level curves and gradients. (A) f (x,y) = x + y + 4: maximum value = minimum value = (B) f (x,y) = 4x 2 + 5y 2 : maximum value = minimum value = 2 3. ( pt) set6/ur vc 7 3.pg Find the maximum and minimum values of f (x,y) = xy on the ellipse 2x 2 + y 2 = 3. maximum value = minimum value =

14 4. ( pt) set6/ur vc 7 4.pg You are hiking the Inca Trail on the way to Machu Piecho. When you arrive at the hightest point on the trail, which of the following are possibilities? In alphabetical order without punctuation or spacing, list the letters which indicate possibilities. (A) The path passes through the center of a set of concentric contour lines. (B) The path is tangent to a contour line. (C) The path follows a contour line. (D) The path crosses a contour line. possibilities: ABCD Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 3

15 Hsiang-Ping Huang Math , Spring 2008 WeBWorK Assignment 7 due 03/3/2008 at 0:59pm MDT Double Integrals and Applications This assignment will cover the material from Chapters ( pt) set7/p7-.pg Evaluate the iterated integral R 2 R 0 0 2x2 y 3 dxdy 6 2. ( pt) set7/p7-2.pg Calculate the double integral R R R (4x + 0y + 40) da where R is the region: 0 x 5,0 y ( pt) set7/p7-3.pg Calculate the volume under the elliptic paraboloid z = 2x 2 + 3y 2 and over the rectangle R = [ 2, 2] [ 4, 4]. E C A D B 6. ( pt) set7/p7-6.pg Using polar coordinates, evaluate the integral RR R sin(x2 + y 2 )da where R is the region x 2 + y ( pt) set7/p7-7.pg A lamina occupies the part of the disk x 2 +y 2 25 in the first quadrant and the density at each point is given by the function ρ(x,y) = 3(x 2 + y 2 ). A. What is the total mass? B. What is the moment about the x-axis, R xρ(x,y)dxdy? C. What is the moment about the y-axis, R yρ(x,y)dxdy? Please note, the notation [-2,2] x [-4,4] refers to the Cartesian product of these two closed intervals, that is, all pairs (x,y) with x in [-2,2] and y in [-4,4] which is a rectangle with corners at (-2,-4) (2,-4) (-2,4) and (2,4) ( pt) set7/p7-4.pg Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = ( pt) set7/p7-5.pg 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.. R R x 2 x 2 x2 y 2 dydx 2. R 3 R 2 3y x 2 3y 2 dxdy 3. R R y 0 4x 2 + 3y 2 dxdy y 2 4. R 2R y 2 dydx R 4+ 4 x 2 5. R x + 3y dydx A. Solid under an elliptic paraboloid and over a planar region bounded by two parabolas. B. Solid under a plane and over one half of a circular disk. C. One eighth of an ellipsoid. D. One half of a cylindrical rod. E. Solid bounded by a circular paraboloid and a plane. D. Where is the center of mass? (, ) E. What is the moment of inertia about the origin, R (x 2 + y 2 )ρ(x,y)dxdy? ( pt) set7/p7-8.pg find R R R x yda, where R is the region in the first quadrant bounded above by the curve y = 25 x ( pt) set7/p8-2.pg 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? ft 3 perhour B. What is the total amount of water that goes throught the sprinkler per hour? ft 3 perhour

16 0. ( pt) set7/p8-4.pg Electric charge is distributed over the disk x 2 + y 2 4 so that the charge density at (x,y) is σ(x,y) = 3 + x 2 + y 2 coulombs per square meter. Find the total charge on the disk ( pt) set7/p8-5.pg Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles x 2 + y 2 = 6 and x 2 4x + y 2 = ( pt) set7/p8-7.pg Use single variable calculus methods to find the area of the region in the first quadrant bounded by the curves y 2 = 3x,, y 2 = 4x, x 2 = 8y, x 2 = 9y. In the next set, we will do the same problem using multivariable calculus methods Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 2

17 Hsiang-Ping Huang Math , Spring 2008 WeBWorK Assignment 8 due 04/03/2008 at 0:59pm MDT Surface Area, Triple Integrals This assignment will cover the material from Chapters ( pt) set8/p8-.pg A cylindrical drill with radius 4 is used to bore a hole throught the center of a sphere of radius 8. Find the volume of the ring shaped solid that remains ( pt) set8/p8-3.pg (See the formulas regarding the surface area element for parametrically defined surfaces around Example 9, on pages of the text.) ( pt) set8/p9-3.pg Find the mass of the region (in cylindrical coordinates) r 3 z 5, where the density function is ρ(r,θ,z) = 8z. Answer: ( pt) set8/p9-4.pg Evaluate the triple integral Z Z Z E xyzdv where E is the solid: 0 z, 0 y z, 0 x y. 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 = 9x + 8y + 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, lim c ψ? ( pt) set8/p8-7.pg Use the multivariable calculus change of variables method to find the area of the region in the first quadrant bounded by the curves y 2 = 2x,, y 2 = 3x, x 2 = 2y, x 2 = 3y by mapping the region to a rectangle by a transformation whose Jacobian is constant. You may use (but it is also good to check!) that the Jacobian of the inverse transformation is the inverse of the Jacobian of the forward transformation ( pt) set8/p9-.pg Find the surface area of the part of the plane 2x+4y+z = 4 that lies inside the cylinder x 2 + y 2 = ( pt) set8/p9-2.pg The vector equation r(u,v) = ucosvi + usinvj + vk, 0 v 5π, 0 u, describes a helicoid (spiral ramp). What is the surface area? ( pt) set8/p9-5.pg Find the average value of the function f (x,y,z) = x 2 + y 2 + z 2 over the rectangular prism 0 x 3, 0 y 2, 0 z ( pt) set8/p9-6.pg 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 = 9 4 ( x 2 + y 2) and the xy-plane ( pt) set8/p9-7.pg 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 ( pt) set8/p9-8.pg 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. R y 2 0 x dx dy da where D is: x 2 + y 2 4 x 2 +y 2 3. RRR E z2 dv where E is: 2 z 2, x 2 + y 2 2. R 0 2. RR D 4. RRR E dv where E is: x2 +y 2 +z 2 4,x 0,y 0,z 0 5. RRR E z dv where E is: x 2,3 y 4,5 z 6 A. spherical coordinates B. polar coordinates

18 C. cylindrical coordinates D. cartesian coordinates D B C A D Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 2

19 Hsiang-Ping Huang Math , Spring 2008 WeBWorK Assignment 9 due 04/0/2008 at 0:59pm MDT Vector Fields and Line Integrals This assignment will cover the material from Chapters ( pt) set9/p0-.pg Compute the gradient vector fields of the following functions: A. f (x,y) = 6x 2 + 5y 2 f (x,y) = I+ J B. f (x,y) = x 4 y 7, f (x,y) = I+ J C. f (x,y) = 6x + 5y f (x,y) = I+ J D. f (x,y,z) = 6x + 5y + 4z f (x,y) = I+ J+ K E. f (x,y,z) = 6x 2 + 5y 2 + 4z 2 f (x,y,z) = I+ J+ K 2*6*x 2*5*y (4*xˆ(4 -))*yˆ(7) (7*yˆ(7 -))*xˆ(4) *6*x 2*5*y 2*4*z 2. ( pt) set9/p0-2.pg D. ellipsoids E. lines F. hyperboloids G. planes H. spheres I. hyperbolas I B I F H E E D G A C 3. ( pt) set9/p0-3.pg 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 (5, 25). (Compare your answers! Compute the line integrals directly from the definition. In the next set, you will do the same integrals using the Fundamental Theorem for Line Integrals.) A. If C is the parabola: x = t, y = t 2, 0 t 5, then R C F dx = B. If C 2 is the straight line segment: x = 5t 2, y = 25t, 0 t, then R C 2 F dx = 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 = yi + xj 2. F = xi + yj 3. F = xi yj 4. F = xi + yj zk 5. F = xi + yj + zk 6. F = yi + xj 7. F = 2I + J 8. F = 2xI + yj + zk 9. F = 2I + J + K 0. F = 2xI + yj. F = xi + yj K A. ellipses B. circles C. paraboloids 4. ( pt) set9/p0-4.pg 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 F dx are positive, negative, or zero and type P, N, or Z as C 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: Z P N Z

20 5. ( pt) set9/p0-5.pg If ( C is the curve given by X(t) = ( + 5 sint) I + + 5sin 2 t ) J + ( + sin 3 t ) K, 0 t π 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 ( pt) set9/p0-6.pg Let R be the rectangle with vertices (0,0), (8,0), (0,3), (8,3), and let C be the boundary of R traversed counterclockwise. For the vector field F(x,y) = 4yI + xj, find -72 Z C F dx. 7. ( pt) set9/p0-8.pg Suppose C is any curve from (0,0,0) to (,,) and F(x,y,z) = (3z + 2y)I + (z + 2x)J + (y + 3x)K. Compute the line integral R C F dx ( pt) set9/p0-9.pg Find the work done by the force field F(x,y,z) = 2xI+2yJ+ K on a particle that moves along the helix X(t) = 5cos(t)I + 5sin(t)J + 5tK,0 t 2π ( pt) set9/p0-0.pg Calculate the divergence and curl of these vector fields: A. F (X) = ( x 3 3xy 2) I + ( 3x 2 y + y 3) J curl (F)= I+ J+ K div (F)= B. G(X) = x 3 yi + x 2 zj + yz 3 K curl (F)= I+ J+ K div (G)= zˆ3-xˆ2 2*x*z-xˆ3 3*xˆ2*y+3*zˆ2*y Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 2

21 Hsiang-Ping Huang Math , Spring 2008 WeBWorK Assignment 0 due 04/7/2008 at 0:59pm MDT Independence of Path, Green s Theorem This assignment will cover the material from Chapters ( pt) set0/p0-7.pg 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) = ( 4x + 5y)I + (5x + 4y)J f (x,y) = B. F(x,y) = 2yI xj f (x,y) = C. F(x,y,z) = 2xI yj + K f (x,y,z) = D. F(x,y) = ( 2siny)I + (0y 2xcosy)J f (x,y) = E. F(x,y,z) = 2x 2 I + 5y 2 J + 2z 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 -2*x**2 + 5*x*y + 2*y**2 N -2*x**2/2 + (-2+)*y**2/2 + z -2*x*sin(y) + 5*y**2 (/3)*(-2*x**3 + 5*y**3 + 2*z**3) 2. ( pt) set0/p-.pg Let C be the positively oriented circle x 2 + y 2 =. Use Green s Theorem to evaluate the line integral R C 9ydx + 7xdy ( pt) set0/p-2.pg Let F = 5yI + 5xJ. Use the tangential vector form of Green s Theorem to compute the circulation integral R C F dx where C is the positively oriented circle x 2 + y 2 = ( pt) set0/p-3.pg Let F = 4xI + 4yJ and let n be the outward unit normal vector to the positively oriented circle x 2 + y 2 = 4. Compute the flux integral R C F nds ( pt) set0/p-4.pg 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 (0, 00). (Use the Fundamental Theorem for Line Integrals instead of computing the line integral from the definition, as you did in the previous set. This way shows why the answers to the two parts must be the same - independence of path!) Z A. If C is the parabola: x = t, y = t 2, 0 t 0, then F dx = C B. If C 2 is Z the straight line segment: x = 0t 2, y = 00t 2, 0 t, then F dx = C ( pt) set0/p-5.pg Let F(x,y) = yi+xj x 2 +y 2 and let C be the circle X(t) = (cost)i + (sint)j, 0 t 2π. A. Compute R C F dx Note: Your answer should be a number B. Is F conservative? Type Y if yes, type N if no N 7. ( pt) set0/p-7.pg Let C be the positively oriented square with vertices (0, 0), (,0), (,), (0,). Use Green s Theorem to evaluate the line integral R C 5y2 xdx + 4x 2 ydy ( pt) set0/p-8.pg Find a parametrization of the curve x 2/3 + y 2/3 = and use it to compute the area of the interior ( pt) set0/p-0.pg Let F = x 3 I+5y 3 J and let n be the outward unit normal vector to the positively oriented circle x 2 + y 2 = 9. Compute the flux integral R C F nds Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

22 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) setdemo/demo pr.pg Evaluate the expression (2 3) = ( pt) setdemo/demo pr2.pg Evaluate the expression 2/(3 + ) =. 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.).5 3. ( pt) setdemo/demo pr3.pg Let r = 9. Evaluate 4/π r =. Next, enter the expression 4/(π r) = WorK compute the result ( pt) setdemo/demo pr4.pg Enter here the expression a + b. Enter here the expression a+b. /a+/b /(a+b) and let WeB- 5. ( pt) setdemo/demo pr5.pg Enter here the expression a b Enter here the expression a + b c + d If WeBWorK rejects your answer use the preview button to see what it thinks you are trying to tell it. (a+)/(2+b) (a+b)/(c+d) 6. ( pt) setdemo/demo pr6.pg Enter here the expression a + b Enter here Enter here sqrt(a+b) a/sqrt(a+b) (a+b)/sqrt(a+b) 7. ( pt) setdemo/demo pr7.pg Enter here Enter here Enter here a a + b a + b a + b x 2 + y 2 x x 2 + y 2 sqrt(x**2+y**2) x*sqrt(x**2+y**2) (x+y)/sqrt(x**2+y**2) 8. ( pt) setdemo/demo pr8.pg Enter here the expression the expression the expression the expression the expression x + y x 2 + y 2 the expression b + b 2 4ac 2a Note: this is an expression that gives the solution of a quadratic equation by the quadratic formula. (-b+sqrt(b**2-4*a*c))/(2a)

23 Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 2

24 Hsiang-Ping Huang Math , Spring 2008 WeBWorK Assignment due 04/24/2008 at 0:59pm MDT Surface Integrals, Divergence and Stokes Theorems This assignment will cover the material from Chapters ( pt) set/p2-.pg Let F = 2xI+4yJ+0zK. Compute the divergence and the curl. A. div F = B. curl F = I+ J+ K 6 2. ( pt) set/p2-2.pg Let F = (yz)i + (xz)j + (7xy)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 (7 - )*x ( - 7)*y ( - )*z 3. ( pt) set/p2-3.pg A fluid has density 5 and velocity field v = yi + xj + 2zK. Find the rate of flow outward through the sphere x 2 +y 2 +z 2 = ( pt) set/p2-4.pg Use Stokes theorem to evaluate Z Z S curlf ds where F(x,y,z) = 9yzI+9xzJ+9(x 2 +y 2 )zk and S is the part of the paraboloid z = x 2 + y 2 that lies inside the cylinder x 2 + y 2 =, oriented upward ( pt) set/p2-5.pg Z Use Stokes Theorem to evaluate F dr where F(x,y,z) = xi + yj + 3(x 2 + y 2 )K and C is the boundary of the part of the paraboloid where z = 4 x 2 y 2 which lies above the xy-plane and C is oriented counterclockwise when viewed from above. C 6. ( pt) set/p2-6.pg Suppose F = F(x,y,z) is a gradient field with F = f, S is a level surface of f, and C is a curve on S. What is the value of the line integral R C F dr? 7. ( pt) Zset/p2-7.pg Z Evaluate + x 2 + y 2 ds where S is the helicoid: r(u,v) = S ucos(v)i + usin(v)j + vk, with 0 u 2,0 v 2π (See the formulas regarding the surface area element for parametrically defined surfaces around Example 9, on pages of the text.) ( pt) set/p2-8.pg Find the surface area of the part of the sphere x 2 + y 2 + z 2 = that lies above the cone z = x 2 + y ( pt) set/p2-0.pg Let S be the part of the plane 2x + 2y + z = 4 which lies in the first octant, oriented upward. Find the flux of the vector field F = 2I + 2J + 3K across the surface S ( pt) set/p6-.pg Find an equation of the tangent plane to the parametric ( surface x = 3r cosθ, y = 4r sinθ, z = r at the point 3 2, 4 ) 2,2 when r = 2, θ = π/4. z = Note: A normal to a surface defined parametrically by x(r, θ), y(r, θ), z(r, θ) is given by < x r,y r,z r > < x θ,y θ,z θ >. Your answer should be an expression of x and y; e.g. 3x - 4y * (x- ((3)*sqrt(2))) Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester