WW Prob Lib1 Math course-section, semester year

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1 WW Prob Lib Math course-section, semester year WeBWorK assignment Tarea due /6/03 at :00 PM..( pt) Let a = (4, 9, -7) and b = (-8, -6, -3) be vectors. Compute the following vectors. A. a + b = (,, ) B. 7a= (,, ) C. a - b= (,, ) D. a =.( pt) Find a b if a =, b =, and the angle between a and b is π 4 radians. a b = 3.( pt) If a = (0,, -7) and b = (, -9, -6), find a b =. 4.( pt) What is the angle in radians between the vectors a = (-4, -7, -8) and b = (-8, -3, 8)? Angle: (radians) 5.( pt) Find a unit vector in the same direction as a = (-, 9, -8). (,, ) 6.( pt) A constant force F i 9j 8k moves an object along a straight line from point (-3, -6, ) to point (9, -, -3). Find the work done if the distance is measured in meters and the magnitude of the force is measured in newtons. Work: Nm 7.( pt) A woman exerts a horizontal force of 7 pounds on a box as she pushes it up a ramp that is feet long and inclined at an angle of 30 degrees above the horizontal. Find the work done on the box. Work: ft-lb 8.( 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, 3). If he began walking in the direction of the vector v 5i 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 WW Prob Lib Math course-section, semester year WeBWorK assignment Tarea due /6/03 at :00 PM..( pt) You are looking down at a map. A vector u with u = 5 points north and a vector v with v = 8 points northeast. The crossproduct u v points: A) south B) northwest C) up D) down Please enter the letter of the correct answer: Enter an equation for the magnitude of u v =.( pt) Let a = (7, 0, ) and b = (9, 3, 9) be vectors. Compute the cross product a b. (,, ) 3.( pt) If a i j 5k and b i j 3k Compute the cross product a b. i + j + k 4.( pt) If a i j k and b i 8j k, find a unit vector with positive first coordinate orthogonal to both a and b. i + j + k 5.( pt) Find the area of the parallelogram with vertices (4,), (5, 4), (8, 8), and (9, ).

3 WW Prob Lib Math course-section, semester year WeBWorK assignment Tarea3 due /3/03 at :00 PM..( pt) Find a unit vector with positive first coordinate that is orthogonal to the plane through the points P = (,, 4), Q = (5, 4, 7), and R = (5, 4, 0). (,, ).( pt) A bicycle pedal is pushed straight downwards by a foot with a Newton force. The shaft of the pedal is 0 cm long. If the shaft is π 5 radians past horizontal, what is the magnitude of the torque about the point where the shaft is attached to the bicycle? Nm 3.( 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 planes either intersect or are parallel.. A plane and a line either intersect or are parallel. 3. Two lines parallel to a third line are parallel. 4. Two planes parallel to a third plane are parallel. 5. Two lines perpendicular to a plane are parallel 6. Two lines perpendicular to a third line are parallel. 7. Two lines parallel to a plane are parallel. 8. Two planes perpendicular to a line are parallel. 9. Two lines either intersect or are parallel. 0. Two planes parallel to a line are parallel.. Two planes perpendicular to a third plane are parallel. 4.( pt) Find a vector equation for the line through the point P = (0, 5, -) and parallel to the vector v = (-, -4, -5). Assume r 0 0i 5j k and that v is the velocity vector of the line. r(t) = i + j + k 5.( pt) Given a the vector equation r(t) = (3 + 4t)i + ( + 4t)j + (5 + 3t)k, rewrite this in terms of the parametric equations for the line. x(t) = y(t) = z(t) = 6.( pt) Consider the two lines L : x t y t z 3t and L : x 5 s y 3s z 5s Find the point of intersection of the two lines. P = (,, ) 7.( pt) Find an equation of the plane through the point (-5, -3, ) and parallel to the plane 5x y 0z. Do this problem in the standard way or Web- Work may not recognize a correct answer. x + y + z = 8.( pt) Find the point P where the line x = + t, y = t, z = -3t intersects the plane x + y - z =. P = (,, ) 9.( 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, 0), and (0,, 3). By what angle does the tower now deviate from the vertical? radians.

4 WW Prob Lib Math course-section, semester year WeBWorK assignment Vmultivariable6Gradient due 3//03 at 0:00 PM..( pt) If f x y x 4y, find the value of the directional derivative at the point 3 3 in the direction given by the angle θ π 5..( pt) Suppose f x y x xy y, P 5, 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 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 x 3.( pt) Suppose f x y y, P 3 3 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. 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 x y 4.( pt) Suppose f x y z y z, P 3. 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 5.( pt) Suppose that distances are measured in lightyears and that the temperature T of a gaseous nebula is inversely proportional to the distance from a fixed point, which we take to be the origin. Suppose that the temperature lightyear from the origin is 00 degrees celsius. Find the gradient of T at x y z. f i j k Note: Your answers should be expressions of x, y and z; e.g. 3x - 4y 6.( pt) Consider the surface 6x y 5z 4 and the point P on this surface. A. Starting with the equation x 3t, find equations for y and z which combine with this equation to give parametric equations for the normal line through P. y z Note: Your answers should be expressions of t; e.g. 3x - 4y B. Find an equation for the tangent plane through P. z Note: Your answers should be expressions of x and y; e.g. 3xy + y 7.( pt) 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 5 degree(s). What is its maximum angle of elevation? degrees 8.( pt) You are hiking the Inca Trail on the way to Machu Picchu. 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.

5 (B) The path is tangent to a contour line. (C) The path follows a contour line. (D) The path crosses a contour line. possibilities:

6 WW Prob Lib Math course-section, semester year WeBWorK assignment Vmultivariable5ChainRule due 3/3/03 at 0:00 PM. x y.( pt) Suppose w y z, x et, y sin t, z cos 4t. 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..( pt) Suppose z x siny, x s 3t, y 0st. A. Use the chain rule to find s and t as functions of x, y, s and t. s t B. Find the numerical values of s s t 3 3. and t when s t ( pt) The radius of a right circular cone is increasing at a rate of 4 inches per second and its height is decreasing at a rate of 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 4.( pt) In a simple electric circuit, Ohm s law states that V IR, where V is the voltage in volts, I is the current in amperes, and R is the resistance in ohms. Assume that, as the battery wears out, the voltage decreases at 0.0 volts per second and, as the resistor heats up, the resistance is increasing at 0.04 ohms per second. When the resistance is 00 ohms and the current is 0.03 amperes, at what rate is the current changing? amperes per second

7 WW Prob Lib Math course-section, semester year WeBWorK assignment Vmultivariable7MaxMin due 3/3/03 at 0:00 PM..( pt) Suppose f x y x y 4x 4y (A) How many critical points does f have in R? (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? If there is none, type N. (F) What is the minimum value of f on R? If there is none, type N..( pt) 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? (B) How many local maxima does f have in R? (C) How many saddle points does f have in R? 3.( pt) You are to manufacture a rectangular box with 3 dimensions x, y and z, and volume v 78. 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 4x 5y on the disk D: x y. maximum value: minimum value: 5.( pt) Find the maximum and minimum values of f x y 5x y on the ellipse x 4y maximum value: minimum value: 6.( pt) Find the maximum and minimum values of f x y z x y 5z on the sphere x y z. maximum value = minimum value = 7.( pt) Find the maximum and minimum values of f x y xy on the ellipse 8x y 9. maximum value = minimum value =

8 # & # Young-Seon Lee WW Prob Lib Math course-section, semester year WeBWorK assignment TareaIntDoble due 4//03 at 0:00 PM. 4 0.( pt) Evaluate the iterated integral 0 6x y 3 dxdy.( pt) Calculate the double integral! R 4x 0y 40 da where R is the region: 0 x 5 0 y. " 3.( pt) Calculate the double integral R xcos x y da where R is the region: 0 x π 6 0 y π 4 4.( pt) Calculate the volume under the elliptic paraboloid z 4x 8y and over the rectangle R 3 3$% $. 5.( pt) Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 8. 6.( pt) Evaluate the integral by reversing the order of integration. 0 y ex dxdy 7.( 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 ' 3y 0 ( 4x 3y dxdy 4)+* 4 x 4 4x 3y dydx * x * x x y dydx * y ( y 4x 3y dxdy 4 y dydx 5. A. Solid bounded by a circular paraboloid and a plane. B. Solid under an elliptic paraboloid and over a planar region bounded by two parabolas. C. One half of a cylindrical rod. D. One eighth of an ellipsoid. E. Solid under a plane and over one half of a circular disk. 8.( pt) Using polar coordinates, evaluate the integral, R sin x y da where R is the region x y ( pt) Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles x y 00 and x 0x y 0. 0.( pt) Use the polar coordinates to find the volume of a sphere of radius 6..( pt) A cylindrical drill with radius 5 is used to bore a hole throught the center of a sphere of radius 0. Find the volume of the ring shaped solid that remains.