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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section: Chapter Questions

Problem 18T

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Question

Consider the level surface defined by the equation x² - y² + z² - 4xyz = 6, find the following:
a) Find the 3D gradient to this level surface.
VF(x, y, z)=
b) Find the normal vector to this surface at the point (1, 2,-1).
c) Find the equation of the tangent plane to this surface at the point (1, 2,-1).

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Consider x=h(y,z) as a parametrized surface in the natural way. Write the equation of the tangent plane to the surface at the point (−4,4,0) given that ∂h∂y(4,0)=5 and ∂h∂z(4,0)=4.
find an equation for the plane tangent to thesurface z = ƒ(x, y) at the given point. z = 1/ (x2 + y2), (1, 1, 1/2)
Let g(x,y,z)=5xe^y+yz^2. Use the gradient vector to find the equation of the tangent plane to the surface g(x,y,z)=45 at the point (0,5,3).
If the gradient of f is ∇f=y3j−xi+zyk and the point P=(−8,6,2) lies on the level surface f(x,y,z)=0, find an equation for the tangent plane to the surface at the point P. Z=?
Consider the surface whose equation is f(x, y) = x^2 - y^2 Determine the vector function r(t) that describes the curve of intersection between f(x,y) and the cylinder of radius 3 that runs along the z axis
Find the equation of the tangent plane to the surface z = xy at the point (1, 2, 2) and to the surface x2y + y2z + z2x= 1 at the point (1, −1, 1).
Find the slope of the tangent line at the point (1,−2,−3) on the curve of intersection of the surface z = 5−4x² −y² with (a) the plane x = 1 (b) the plane y = −2.
Find the equation of the plane tangent to the surface of Z = x^2 + y^2 at point P(1, 2, 5).
Find a vector normal to the surface x2 + yz = 5 at (2, 1, 1).

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