T07-Linearization,Directional Derivatives PDF

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MATH 2011 Multivariable Calculus Tutorial 7: Linearization and Directional Derivative

Example 1. Consider the function f (x, y) = x2 + y 2 . What is the directional derivative of f in the direction of v=(1, 2) at point (0, 0)? Find the direction in which f has its maximum and minimum rate of change at point (0, 0). Do this same for f (x, y, z) = x2 + y 2 + 3z 2 at point (0, 1, 0).

Definitions and Formulas Definition 1. The linearization of a function y = f (x) at (x0 , y0 ) is the linear approximation of f at that point. The tangent line at (x0 , y0 ) is determined as y = f ′ (x0 )x + f (x0 ) − x0 f ′ (x0 ). Notice that this is equivalent to (f ′ (x0 ), −1) • (x − x0 , y − f (x0 )) = 0. Formula 2. The linearization of a function z = f (x, y) at point (x0 , y0 , z0 ) is similarly (fx (x0 ), fy (y0 ), −1) • (x − x0 , y − y0 , z − z0 ) = 0. This is the tangent plane. Formula 3. Directional derivative of f (x, y) at (x0 , y0 ) in the direction of unit (x0 ,y0 ) vector v=(u, w) = limt→0 f (x0 +tu,y0 +tw)−f . This is also equivalent to t (fx (x0 , y0 ), fy (x0 , y0 ) • (u, w)=∇f (x0 , y0 ) • v. Remember that v is a UNIT vector. Definition 4. The gradient of f (x, y) is the vector (fx (x, y), fy (x, y)) = ∇f Result 5. Directions of Change: 1. f has its maximum rate of increase at (a, b) in the direction of the gradient ∇f (a, b). The rate of increase in this direction is |∇f (a, b)|. 2. f has its maximum rate of decrease at (a, b) in the direction of −∇f (a, b). The rate of decrease in such direction is −|∇f (a, b)|. 3. The directional derivative is zero in any direction orthogonal to ∇f (a, b).

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Example 2. Find the linearization of the following functions. 2

Example 3. Suppose the temperature of a room is calculated by T (x, y, z) = 3xy − 4z 2 + zx. If a particle is positioned at (1, 1, 0), which direction should it travel to attain the greatest increase in temperature? Which direction should it travel to attain no temperature increase?

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1. f (x, y) = 4x − 2xy + y at point (−1, −5) 2. f (x, y, z) = exy zatpoint(3, 1, 1)

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