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Looking ahead—tangent planes Consider the following
Chapter 11, Problem 65E(choose chapter or problem)
Looking ahead-tangent planes Consider the following surfaces f(x, y, z) = 0, which may be regarded as a level surface of the function w = f(x, y, z). A point P(a, b, c) on the surface is also given.
a. Find the (three-dimensional) gradient of f and evaluate it at P.
b. The heads of all vectors orthogonal to the gradient with their tails at P form a plane. Find an equation of that plane (soon to be called the tangent plane to the surface ar f).
\(f(x, y, z)=x^{2}+y^{2}+z^{2}-3=0 ; P(1,1,1)\)
Questions & Answers
QUESTION:
Looking ahead-tangent planes Consider the following surfaces f(x, y, z) = 0, which may be regarded as a level surface of the function w = f(x, y, z). A point P(a, b, c) on the surface is also given.
a. Find the (three-dimensional) gradient of f and evaluate it at P.
b. The heads of all vectors orthogonal to the gradient with their tails at P form a plane. Find an equation of that plane (soon to be called the tangent plane to the surface ar f).
\(f(x, y, z)=x^{2}+y^{2}+z^{2}-3=0 ; P(1,1,1)\)
ANSWER:Solution 65E
Step 1:
Given that
Consider the following surfaces f(x, y, z) = 0, which may be regarded as a level surface of the function w = f (x, y, z). A point P(a, b, c) on the surface is also given.
