ARNOLD PIZER rochester problib from CVS Summer 2003

Transcription

1 WeBWorK assignment VmultivariableFunctions due 3/3/08 at 2:00 AM.( pt) setvmultivariablefunctions/ur VC 5.pg 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 xy 2. z 25 x 2 2 y 3. z x 4. z x 2 y 2 5. z 2x 3y 6. z 2x 2 3y z x 2 y A. a collection of concentric ellipses B. a collection of equally spaced parallel lines C. a collection of unequally spaced parallel lines D. a collection of equally spaced concentric circles E. a collection of unequally spaced concentric circles F. two straight lines and a collection of hyperbolas 2.( pt) setvmultivariablefunctions/ur VC 5 2.pg 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 2y 3z 2. w x 2 2 2y 3z 2 3. w 2 2 x 2 y z 4. w x 2 2y 3z 5. w x 2 y z 2 6. w x 2y 3z 7. w x 2 y 2 z 2 A. a collection of equally spaced parallel planes B. a collection of concentric ellipsoids C. a collection of unequally spaced concentric spheres D. a collection of unequally spaced parallel planes E. a collection of equally spaced concentric spheres F. two cones and two collections of hyperboloids 3.( pt) setvmultivariablefunctions/ur VC 5 9.pg Each of the following functions has a set on which it is continuous and that set has a boundary. Match the verbal description of this boundary with the function by putting the letter of the boundary to the left of the letter of the function.. f x y z x 2 y 2 z 2 2. f x y 4 x 2 y 2 3. f x y z xyz x 2 y 2 z 4. f x y z z x 2 y 2 5. f x y xlny A. a circular cylinder B. a circular parabaloid C. one point D. a circle E. a straight line 4.( pt) setvmultivariablefunctions/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. 5.( pt) setvmultivariablefunctions/ur VC 5 F.pg On a map showing the grave of George Mallory, the contour lines are: A. far apart B. closely spaced

2 WeBWorK assignment Vmultivariable2Limits due 3/4/08 at 2:00 AM.( pt) setvmultivariable2limits/ur VC 5 3.pg 5x x y 0 e 2 2y 2 2.( pt) setvmultivariable2limits/ur VC 5 4.pg 2x 2 2x 2 3y 2 x y ( pt) setvmultivariable2limits/ur VC 5 5.pg 2 x y x 5y 0 0 x y 2 4.( pt) setvmultivariable2limits/ur VC 5 6.pg (Hint: use polar coordinates.) 9x x y y 3 x 2 y 2 5.( pt) setvmultivariable2limits/ur VC 5 7.pg x y z x2 2ze y x 2 5y 4z 2 6.( pt) setvmultivariable2limits/ur VC 5 8.pg x y z 2xy yz 2xz x 2 y 4z 2

3 WeBWorK assignment Vmultivariable3ParDer due 3/5/08 at 2:00 AM B. f y 0 0.( pt) setvmultivariable3parder/ur VC 5 0.pg 4.( pt) setvmultivariable3parder/ur VC 5 3.pg x y Find the first partial derivatives of f x y x y (x,y) = (4, 4). f f ( pt) setvmultivariable3parder/ur VC 5.pg at the point Find the first partial derivatives of f x y z y z arctan x at 4 4 the point (4, 4, ). A. f B. f C. f ( pt) setvmultivariable3parder/ur VC 5 2.pg Find the first partial derivatives of f x y sin x y at the A. f x 0 0 point (0, 0). If sin 2x 2y z 0, find the first partial derivatives and A. B. at the point (0, 0, 0) ( pt) setvmultivariable3parder/ur VC 5 4.pg Find all the first and second order partial derivatives of f x y 8sin 2x y 2cos x y. A. f B. f f x f y C. 2 f 2 f xx D. 2 f 2 f yy E. 2 f F. 2 f f yx f xy

4 WeBWorK assignment Vmultivariable4Linearization due 3/5/08 at 2:00 AM.( pt) setvmultivariable4linearization/ur vc 6.pg Find the equation of the tangent plane to the surface z 6y 2 x 2 at the point Note: Your answer should be an expression of x and y; e.g. 3x - 4y ( pt) setvmultivariable4linearization/ur vc 6 2.pg Find the equation of the tangent plane to the surface z e x 7 ln 2y at the point Note: Your answer should be an expression of x and y; e.g. 5x + 2y ( pt) setvmultivariable4linearization/ur vc 6 3.pg Find the linearization L x y of the function f x y 98 6x 2 9y 2 at 3. L x y Note: Your answer should be an expression in x and y; e.g. 3x - 5y ( pt) setvmultivariable4linearization/ur vc 6 4.pg Find the differential of the function w xsin 2yz 3. dw = dx + dy + dz Note: Your answers should be expressions of x, y and z; e.g. 3xy + 4z 5.( pt) setvmultivariable4linearization/ur vc 6 5.pg The dimensions of a closed rectangular box are measured as 70 centimeters, 90 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 6.( pt) setvmultivariable4linearization/ur vc 6 6.pg Find an equation of the tangent plane to the parametric surface x 5r cosθ, y r sinθ, z r at the point when r 2, θ π 4. Note: Your answer should be an expression of x and y; e.g. 3x - 4y

5 WeBWorK assignment Vmultivariable5ChainRule due 3/6/08 at 2:00 AM.( pt) setvmultivariable5chainrule/ur vc 6 7.pg Suppose w x y y z, x e 4t, y 2 sin t, z 2 cos 3t. 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 4t 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. 2.( pt) setvmultivariable5chainrule/ur vc 6 8.pg Suppose z x 2 siny, x 2s 2 2t 2, y 4st. 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 and t when s t 2 0. s t ( pt) setvmultivariable5chainrule/ur vc 6 9.pg 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 2 inches per second. At what rate is the volume of the cone changing when the radius is 40 inches and the height is 0 inches? cubic inches per second 4.( pt) setvmultivariable5chainrule/ur vc 6 0.pg 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 0.0 ohms per second. When the resistance is 400 ohms and the current is 0.04 amperes, at what rate is the current changing? amperes per second

6 WeBWorK assignment Vmultivariable6Gradient due 3/7/08 at 2:00 AM.( pt) setvmultivariable6gradient/ur vc 6.pg If f x y x 2 y 2, find the value of the directional derivative at the point 3 3 in the direction given by the angle θ 2π. 2.( pt) setvmultivariable6gradient/ur vc 6 2.pg Suppose f x y 3x 2 xy y 2, P 3, 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 3 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 3.( pt) setvmultivariable6gradient/ur vc 6 3.pg x Suppose f x y y, P 3 and v i 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 4.( pt) setvmultivariable6gradient/ur vc 6 4.pg x Suppose f x y z y y z, P 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) setvmultivariable6gradient/ur vc 6 5.pg 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) setvmultivariable6gradient/ur vc 6 6.pg Consider the surface 2 x 2 4y 9z 2 4 and the point P on this surface. A. Starting with the equation x 2t, 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 + 2y 7.( pt) setvmultivariable6gradient/ur vc 6 7.pg The axis of a light in a lighthouse is tilted. When the light points east, it is inclined upward at degree(s). When it points north, it is inclined upward at 5 degree(s). What is its maximum angle of elevation? degrees 8.( pt) setvmultivariable6gradient/ur vc 7 4.pg You are hiking the Inca Trail on the way to Machu Picchu. When you arrive at the highest 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:

7 WeBWorK assignment Vmultivariable7MaxMin due 3/8/08 at 2:00 AM.( pt) setvmultivariable7maxmin/ur vc 7.pg Suppose f x y x 2 y 2 6x 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. (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. 2.( pt) setvmultivariable7maxmin/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? 3.( pt) setvmultivariable7maxmin/ur vc 7 3.pg Consider the function f x y xsin 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? 4.( pt) setvmultivariable7maxmin/ur vc 7 4.pg Suppose f x y xy x 4y. 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 5.( pt) setvmultivariable7maxmin/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 x 2 y 2 Type of critical point: (B) f x y e x2 y 2 Type of critical point: (C) f x y x 2 y 2 Type of critical point: (D) f x y x y Type of critical point: 6.( pt) setvmultivariable7maxmin/ur vc 7 6.pg You are to manufacture a rectangular box with 3 dimensions x, y and z, and volume v 297. Find the dimensions which minimize the surface area of this box. x = y = 7.( pt) setvmultivariable7maxmin/ur vc 7 7.pg Find the coordinates of the point (x, y, z) on the plane 4 x + y + which is closest to the origin. x = y = 8.( pt) setvmultivariable7maxmin/ur vc 7 8.pg Find the maximum and minimum values of f x y 6x 2 7y 2 on the disk D: x 2 y 2. maximum value: minimum value: 9.( pt) setvmultivariable7maxmin/ur vc 7 9.pg Find the maximum and minimum values of f x y 2x y on the ellipse x 2 36y 2 maximum value: minimum value: 0.( pt) setvmultivariable7maxmin/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 gradiants. (A) f x y x y 3: (B) f x y 3x 2 4y 2 : (C) f x y 3x 2 4y 2 :

8 .( pt) setvmultivariable7maxmin/ur vc 7.pg For each of the following functions, find the maximum and minimum values of the function on the rectanglar region: 4 x 4 5 y 5. Do this by looking at level curves and gradiants. (A) f x y x y 3: minimum value = (B) f x y 3x 2 4y 2 : minimum value = (C) = f x y 5 2 x y 2 : 2.( pt) setvmultivariable7maxmin/ur vc 7 2.pg Find the maximum and minimum values of f x y z x y 2z on the sphere x 2 y 2 z 2. 3.( pt) setvmultivariable7maxmin/ur vc 7 3.pg Find the maximum and minimum values of f x y xy on the ellipse 7x 2 y ( pt) setvmultivariable7maxmin/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: 2