79. Parametrised Surface 1. Consider the parametrised surface defined by: x 2u, y u2+ v (a) Find a vector normal to the...
5. [12 Marks) Consider the level surface of the function f(x, y, z) defined by f(x, y, z) = x2 + y2 + x2 = 2a?, (1) where a is a fixed real positive constant, and the point u = (0,a,a) on the surface f(x, y, z) = 2a. a) Find the gradient of f(x, y, z) at the point u. b) Calculate the normal derivative of f(x, y, 2) at u. c) Find the equation of the tangent plane...
9a. Find a normal vector to the tangent plane to the surface x = y2zs at (1,-1,-1). 35 b. Find the equation of the tangent plane to the surface x=y'7 at the point (1,-1,-1).
4. (4 pts) Consider the
surface z=x2y+y3.(a) Find the normal direction of the tangent plane
to the surface through (1,1,2).(b) Find the equation of the tangent
plane in (a).(c) Determine the value a so that the vector−→v=−−→i+
2−→j+a−→k is parallel to the tangent plane in (a).(d) Find the
equation of the tangent line to the level curve of the surface
through (1,1).
4. (4 pts) Consider the surface z = z2y + y). (a) Find the normal direction of the...
Consider the surface given as a graph of the function g(x, y) = x∗y 2 ∗cos(y). The gradient of g represents the direction in which g increases the fastest. Notice that this is the direction in the xy plane corresponding to the steepest slope up the surface, with magnitude equal to the slope in that direction. 1. At the point (2, π), find the gradient, and explain what it means. 2. Use it to construct a vector in the tangent...
3 4. (4 pts) Consider the surface z = z = x²y + y3. (a) Find the normal direction of the tangent plane to the surface through (1,1,2). (b) Find the equation of the tangent plane in (a). (e) Determine the value a so that the vector 7= -7+27 +ak is parallel to the tangent plane in (a). (d) Find the equation of the tangent line to the level curve of the surface through (1,1).
Find an equation of the tangent plane to the following parametric surface, r(u, v) = (u2 + 9) i + (v3 + 6u) j + (u + 2v) k , at the point (10, 5, −1). Write the equation in the form ax + by + cz + d = 0, where a, b, c, and d have no common factors. Then enter the values of a, b, c, and d (in that order) into the answer box below, separated...
12. A surface is given parametrically by the equations -u2-u y lue, = (3u-20)". Find an equation of the plane tangent to the surface at the point where u u = 1
Given f(x, y): 10-2x2-y, find a) The equation of the tangent plane to the surface at the point (2.2,6) b) The parametric equations of the normal line at the point (2, 2, 6) c) The outward unit normal vector to the surface at the point (2, 2,6) d) Sketch the surface and the outward unit normal vector at the point (2, 2,6). 1.
Given f(x, y): 10-2x2-y, find a) The equation of the tangent plane to the surface at the...
TOTAL MARKS: 25 QUESTION 4 (a) Find a normal vector and an equation for the tangent plane to the surface at the point P: (-2,1,3). Determine the equation of the line formed by the intersection of this plane with the plane z = 0. 10 marks (b) Find the directional derivative of the function F(r, y, z)at the point P: (1,-1,-2) in the direction of the vector Give a brief interpretation of what your result means. 2y -3 [9 marks]...
all questions are related and need help answering!
rough the surface 4. o pm) What is the value of the flux of the vector field F(x,y)j+z ioriented with upward- pointing normal vector? (A) 0 (B) 2n/3 (C) π (D) 4T/3 (E) 2π Use Stokes, Theorem to evaluateⅡcurl F.dS, where F(x, y, z)-(x2 sin Theorem to evaluate Jceun F'.asS , where Fl.e)(', ») and 5. (5pts.) F,y, sin z, y', xy) and s is the part of the paraboloid : -...