8. Find the area of the surface given by z - f(x, y) over the region R. f(x,y)- 42-x2-y2, R = {(x,y): x2 +y2 29
8. Find the area of the surface given by z - f(x, y) over the region R. f(x,y)- 42-x2-y2, R = {(x,y): x2 +y2 29
6 (20 pts). Let F(x, y, z) = x2 + y2 + x2 - 6xyz. (1) Find the gradient vector of F(x, y, z); (2) Find the tangent plane of the level surface F(x, y, z) = x2 + y2 + x2 - 6xyz = 4 at (0, 0, 2); (3) The level surface F(x, y, z) = 4 defines a function z = f(x,y). Use linear approxi- mation to approximate z = = f(-0.002,0.003).
1. Find the absolute maximum and minimum values of f(r,y) = x2+y2+5y on the disc {(x, y) | x2+y2 < 4}, and identify the points where these values are attained 2. Find the absolute maximum and minimum values of f(x, y) = x3 - 3x - y* + 12y on the closed region bounded by the quadrilateral with vertices at (0,0), (2,2), (2,3), (0,3), and identify the points where these values are attained. 3. A rectangular box is to have...
5. (7 points) Let f: R3 → R be the function f(x,y,z) = x2 + y2 +3(2-1)2 Let EC R3 be the closed half-ball E = {(x, y, z) e R$: x² + y2 +< 9 and 2 >0}. Find all the points (x, y, z) at which f attains its global maximum and minimum on E.
Question 3. Consider the function h: R3 → R h(x, y, 2) = (x2 + y2 + 2) +3/(x2 + 2xy + y) (a) What is the maximal domain of h? Describe it in words. (it may help to factor the denominator in the second term) > 0 for any a, (b) It is difficult to immediately find the range of h. Using the fact that a show that h cannot take negative values. Can h be an onto function?...
All of 10 questions, please.
1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the cylinder x2 + y2- 16 below z-2x2 + 2y2 and above the xy - plane. 4. Determine the surface area of the portion of 2x + 3y + 6z - 9 that is in the 1st octant. » 5. Evaluate JSxz...
1. Consider the surface of revolution that is given by the equation Z-R= -(x2 + y2)/R where [x],[y] < R/V2 . (a) Find the volume enclosed between the surface and the x-y plane. (b) Find the normal vector în and an equation for the tangent plane to the surface at i = ? (î+ ſ + Â). (Hint: Choose appropriate coordinate systems in each part).
The gravitational field F(x,y,z) =cx /(x2 + y2 + z2)3/2 e1+ cy /(x2 + y2 + z2)3/2 e2+ cz/ (x2 + y2 + z2)3/2 e3 is a gradient field, where c is a constant, such that the field is rotation free. If we define f(x,y,z) = −c /(x2 + y2 + z2)1/2 , then show that (a) F = grad(f). (b) curl(F) = 0.
Find the area of the surface given by z = f(x, y) over the region R. (Hint: Some of the integrals are simpler in polar coordinates.) f(x, y) = x2 + y2, R = {(x, y): 0 = f(x,y) 3}
(x2 + y2 + z?)1/2, and e, = r1(x, y, z) is the unit radial vector. Let F = r"e, where n is any number, r= (a) Calculate div(F). (2+n)"-1 (b) Calculate the flux of F through the surface of a sphere of radius R centered at the origin. 4TR"+2 F. ds, where C is a closed curve that does not pass through the origin? (c) What is the value of (d) A function o satisfying Ap = 0 is...