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(a) Find an equation of the plane tangent to the surface f(x, y) = xy² – 5x at the point (1, 2, – 1). | (b) Find the linearization L(x, y) of the function in part (a) near the given point. Use this function to approximate the value of f(0.99, 2.02) by hand.

(a) Find an equation of the plane tangent to the surface f(x, y) = xy² – 5x at the point (1, 2, – 1). | (b) Find the linearization L(x, y) of the function in part (a) near the given point. Use this function to approximate the value of f(0.99, 2.02) by hand.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Complete each of the following. (a) Find an equation of the plane tangent to the surface \( f(x, y) = xy^2 - 5x \) at the point \( (1, 2, -1) \). (b) Find the linearization \( L(x, y) \) of the function in part (a) near the given point. Use this function to approximate the value of \( f(0.99, 2.02) \) by hand. (c) Suppose that \( F : \mathbb{R}^3 \to \mathbb{R} \) is a function such that the level surface \( F(x, y, z) = -1 \) is the unit sphere of \( \mathbb{R}^3 \) centered at the origin. Identify the function \( F(x, y, z) \) and find both the tangent plane to this level surface at the point \( (0, -1, 0) \) and the parametric equations of the line normal to this surface through the given point.
Transcribed Image Text:Complete each of the following. (a) Find an equation of the plane tangent to the surface \( f(x, y) = xy^2 - 5x \) at the point \( (1, 2, -1) \). (b) Find the linearization \( L(x, y) \) of the function in part (a) near the given point. Use this function to approximate the value of \( f(0.99, 2.02) \) by hand. (c) Suppose that \( F : \mathbb{R}^3 \to \mathbb{R} \) is a function such that the level surface \( F(x, y, z) = -1 \) is the unit sphere of \( \mathbb{R}^3 \) centered at the origin. Identify the function \( F(x, y, z) \) and find both the tangent plane to this level surface at the point \( (0, -1, 0) \) and the parametric equations of the line normal to this surface through the given point.
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