Problem. Consider the function f(x, y) = e-²-y? (a) Compute the tangent plane to the graph of f(x, y) at a the point (1, 2, e-5) (b) Compute the gradient of the function of three variables g(x, y, z) = f(x, y) – z at the point (1, 2, e-5) and show that every vector in the tangent plane in part (a) is perpendicular to this vector. (c) The above situation is part of a more general statement. Fill in the blanks to complete the geeral statement. Given any (differentiable) function f(x, y), the level surface g(x, y, z) = f(x, y) – - z = 0 can be identified with the f(r, y) as of the function. The gradient can be computed in terms of the partials of Vg = Since the gradient at a point (xo, yo, 2o) is always we see that Vg(xo, yo, zo) is perpendicular to the recovers the general equation of the tangent plane to the level surface passing through this point, of f at the point (ro, yo, f(ro, yo)). This 0 = Vg(x, y, z)· (x – xo, y - Yo, z - f(xo, yo))

Problem. Consider the function f(x, y) = e-²-y? (a) Compute the tangent plane to the graph of f(x, y) at a the point (1, 2, e-5) (b) Compute the gradient of the function of three variables g(x, y, z) = f(x, y) – z at the point (1, 2, e-5) and show that every vector in the tangent plane in part (a) is perpendicular to this vector. (c) The above situation is part of a more general statement. Fill in the blanks to complete the geeral statement. Given any (differentiable) function f(x, y), the level surface g(x, y, z) = f(x, y) – - z = 0 can be identified with the f(r, y) as of the function. The gradient can be computed in terms of the partials of Vg = Since the gradient at a point (xo, yo, 2o) is always we see that Vg(xo, yo, zo) is perpendicular to the recovers the general equation of the tangent plane to the level surface passing through this point, of f at the point (ro, yo, f(ro, yo)). This 0 = Vg(x, y, z)· (x – xo, y - Yo, z - f(xo, yo))

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.3: Vectors

Problem 60E

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