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10. The temperature T in a metal ball is inversely proportional to the distance from the center o...
The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2) is 110° (a) Find the ra...
The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2) is 110° (a) Find the rate of change of T at (1, 2, 2) in the direction toward the point (3, 5, 3) (b) Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that points...
Suppose f is a differentiable function of x and y, and g(r, s) f(8r - s, s2 - 5). Use the table of values below to calc...
Suppose f is a differentiable function of x and y, and g(r, s) f(8r - s, s2 - 5). Use the table of values below to calculate g(4, 6) and s(4, 6) fx (26, 16) 6198 (4, 6)1652 (4, 6) g(4, 6)- The temperature Tin a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (2, 1, 2) is 140°K Find DuT, the...
6.24Consider a medium in which the refractive index n is inversely proportional tor2: that is, n ...
6.24Consider a medium in which the refractive index n is inversely proportional tor2: that is, n a/r2, where r is the distance from the origin. Use Fermat's principle, that the integral (6.3) is stationary, to find the path of a ray of light travelling in a plane containing the origin. [Hint: Use two- dimensional polar coordinates and write the path as φ = φ(r). The Fermat integral should have the form .ff(ф, ф'.r) dr, where f(d, ф. r) is actually...
5. The temperature of a metal plate at (x,y) is e rate of 8ft/min (.e. -2). degrees. A bug is wal...
5. The temperature of a metal plate at (x,y) is e rate of 8ft/min (.e. -2). degrees. A bug is walking northeast at a dx dy dt dt a) From the bug's point of view, at what rate is the temperature changing with respect to distance as it crosses the origin? Label your answer with correct units. b) If the bug is sitting at the origin, in what direction should it start moving to experience the greatest rate of decrease...
11. The figure shows the contour map of the function T (in "C giving the temperature at each poin...
11. The figure shows the contour map of the function T (in "C giving the temperature at each point (r, y) on an 8cm x 5 cm rectangular metal plate. Use it to estimate the rate of change of the temperature at the point (3,2)in the positive r-direction and in the positive y-direction. 4 75 80 105 00 9508:s 0
11. The figure shows the contour map of the function T (in "C giving the temperature at each point (r,...
Problem 3. (1 point) The temperature at a point (X,Y,Z) is given by T(x, y, z)...
Problem 3. (1 point) The temperature at a point (X,Y,Z) is given by T(x, y, z) = 200e-x=y+14–2–19, where T is measured in degrees Celsius and x,y, and z in meters. There are lots of places to make silly errors in this problem; just try to keep track of what needs to be a unit vector. Find the rate of change of the temperature at the point (-1, 1,-1) in the direction toward the point (-4,-5, -5). In which direction...
(8 points) The temperature at a point (x, y, z) is given by T(x, y, z)...
(8 points) The temperature at a point (x, y, z) is given by T(x, y, z) = 1300e-x-2y-2? where T is measured in °C and x, y, and z in meters. 1. Find the rate of change of the temperature at the point P(2, -1, 2) in the direction toward the point Q(3,-3,3). Answer: Dp S(2.-1, 2) = 2. In what direction does the temperature increase fastest at P? Answer: 3. Find the maximum rate of increase at P. Answer:
Suppose E is the half-cylinder described by 2y2 between z and the ry-plane where y 2 0. Suppose further that the density at each point in E is proportional to the distance from the z-axis. (a) (6 poi...
Suppose E is the half-cylinder described by 2y2 between z and the ry-plane where y 2 0. Suppose further that the density at each point in E is proportional to the distance from the z-axis. (a) (6 points) Find an expression for the mass of E as a triple integral. Then briefly explain why this integral is difficult to compute Suppose E is the half-cylinder described by r' +ґ-1 between z = 4 and the xy-plane where y > 0....
2.1 2.2 2.3 2 BALL AND PLANE Consider a spherical shell of radius R and charge...
2.1 2.2 2.3
2 BALL AND PLANE Consider a spherical shell of radius R and charge per unit area ơi sitting at the origin. There is also an infinite plane parallel to the x - y plane sitting at zzo with charge per unit area σ2. We will take z02R. Compute the electric field at the following locations: 2.1 10 POINTS The origin. 2.2 15 POINTS The point (xo.0,0) with xo>R 2.3 15 POINTS The point (x1,0, z) with 0...
Hi, I need help solving number 13. Please show all the steps, thank you. :) Consider the solid Q bounded by z-2-y2;z-tx at each point Р (x, y, z) is given by mass of Q [15 pts] 9. x-4. The density Z/...
Hi, I need help solving number 13. Please show all the steps,
thank you. :)
Consider the solid Q bounded by z-2-y2;z-tx at each point Р (x, y, z) is given by mass of Q [15 pts] 9. x-4. The density Z/m 3 . Find the center of (x, y, z) [15 pts] 10. Evaluate the following integral: ee' dy dzdx [15 pts] 11. Use spherical coordinates to find the mass m of a solid Q that lies between the...
The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2) is 110° (a) Find the rate of change of T at (1, 2, 2) in the direction toward the point (3, 5, 3) (b) Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that points...
Suppose f is a differentiable function of x and y, and g(r, s) f(8r - s, s2 - 5). Use the table of values below to calculate g(4, 6) and s(4, 6) fx (26, 16) 6198 (4, 6)1652 (4, 6) g(4, 6)- The temperature Tin a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (2, 1, 2) is 140°K Find DuT, the...
6.24Consider a medium in which the refractive index n is inversely proportional tor2: that is, n a/r2, where r is the distance from the origin. Use Fermat's principle, that the integral (6.3) is stationary, to find the path of a ray of light travelling in a plane containing the origin. [Hint: Use two- dimensional polar coordinates and write the path as φ = φ(r). The Fermat integral should have the form .ff(ф, ф'.r) dr, where f(d, ф. r) is actually...
5. The temperature of a metal plate at (x,y) is e rate of 8ft/min (.e. -2). degrees. A bug is walking northeast at a dx dy dt dt a) From the bug's point of view, at what rate is the temperature changing with respect to distance as it crosses the origin? Label your answer with correct units. b) If the bug is sitting at the origin, in what direction should it start moving to experience the greatest rate of decrease...
11. The figure shows the contour map of the function T (in "C giving the temperature at each point (r, y) on an 8cm x 5 cm rectangular metal plate. Use it to estimate the rate of change of the temperature at the point (3,2)in the positive r-direction and in the positive y-direction. 4 75 80 105 00 9508:s 0
11. The figure shows the contour map of the function T (in "C giving the temperature at each point (r,...
Problem 3. (1 point) The temperature at a point (X,Y,Z) is given by T(x, y, z) = 200e-x=y+14–2–19, where T is measured in degrees Celsius and x,y, and z in meters. There are lots of places to make silly errors in this problem; just try to keep track of what needs to be a unit vector. Find the rate of change of the temperature at the point (-1, 1,-1) in the direction toward the point (-4,-5, -5). In which direction...
(8 points) The temperature at a point (x, y, z) is given by T(x, y, z) = 1300e-x-2y-2? where T is measured in °C and x, y, and z in meters. 1. Find the rate of change of the temperature at the point P(2, -1, 2) in the direction toward the point Q(3,-3,3). Answer: Dp S(2.-1, 2) = 2. In what direction does the temperature increase fastest at P? Answer: 3. Find the maximum rate of increase at P. Answer:
Suppose E is the half-cylinder described by 2y2 between z and the ry-plane where y 2 0. Suppose further that the density at each point in E is proportional to the distance from the z-axis. (a) (6 points) Find an expression for the mass of E as a triple integral. Then briefly explain why this integral is difficult to compute Suppose E is the half-cylinder described by r' +ґ-1 between z = 4 and the xy-plane where y > 0....
2.1 2.2 2.3
2 BALL AND PLANE Consider a spherical shell of radius R and charge per unit area ơi sitting at the origin. There is also an infinite plane parallel to the x - y plane sitting at zzo with charge per unit area σ2. We will take z02R. Compute the electric field at the following locations: 2.1 10 POINTS The origin. 2.2 15 POINTS The point (xo.0,0) with xo>R 2.3 15 POINTS The point (x1,0, z) with 0...
Hi, I need help solving number 13. Please show all the steps,
thank you. :)
Consider the solid Q bounded by z-2-y2;z-tx at each point Р (x, y, z) is given by mass of Q [15 pts] 9. x-4. The density Z/m 3 . Find the center of (x, y, z) [15 pts] 10. Evaluate the following integral: ee' dy dzdx [15 pts] 11. Use spherical coordinates to find the mass m of a solid Q that lies between the...
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