Chapter 12: Q. 28 (page 989)

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.
$f (x, y) = \frac{x y}{x^{2} + y^{2}}, P = (- 2, 1), v = <\frac{3}{5}, - \frac{4}{5}>$

Short Answer

Expert verified

$D_{v} f (- 2, 1) = \frac{3}{25}$

Step by step solution

Step 1. Given Information

$The directional derivative of a function f (x, y) at a point P = (x_{0}, y_{0}) in the direction of a unit vector u = ⟨ a, b ⟩ is given by D_{u} f (x_{0}, y_{0}) = lim_{h \to 0} \frac{f (x_{0} + a h, y_{0} + b h) - f (x_{0}, y_{0})}{h} Here f (x, y) = \frac{x y}{x^{2} + y^{2}}, P = (- 2, 1) .$

Step 2. Solution

$Given vector is a unit vector so by definition of directional derivative we have: \begin{array}{l} D_{v} f (- 2, 1) = lim_{h \to 0} \frac{f (- 2 + \frac{3 h}{5}, 1 - \frac{4 h}{5}) - f (- 2, 1)}{h} \\ = lim_{h \to 0} \frac{\frac{(- 2 + \frac{3 h}{5}) (1 - \frac{4 h}{5})}{{(- 2 + \frac{3 h}{5})}^{2} + {(1 - \frac{4 h}{5})}^{2}} - \frac{(- 2) (1)}{(- 2)^{2} + (1)^{2}}}{h} \\ = lim_{h \to 0} \frac{\frac{- 2 + \frac{11 h}{5} - \frac{12}{25} h^{2}}{5 - 4 h + h^{2}} - \frac{2}{5}}{h} \\ Further simplifying the numerator gives: \\ \begin{array}{l} D_{v} f (- 2, 1) = lim_{h \to 0} \frac{3 h - \frac{2}{5} h^{2}}{5 h (5 h - 4 h + h^{2})} \\ = lim_{h \to 0} \frac{3 - \frac{2}{5} h}{5 (5 - 4 h + h^{2})} \\ = \frac{3}{25} \\ H e n c e D_{v} f (- 2, 1) = \frac{3}{25} \end{array} \end{array}$

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Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.
$f (x, y) = \frac{x y}{x^{2} + y^{2}}, P = (- 2, 1), v = <\frac{3}{5}, - \frac{4}{5}>$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solution

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Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.f(x,y)=xyx2+y2,P=(−2,1),v=35,−45

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solution

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Most popular questions from this chapter

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Pure Maths

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Study anywhere. Anytime. Across all devices.

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.
$f (x, y) = \frac{x y}{x^{2} + y^{2}}, P = (- 2, 1), v = <\frac{3}{5}, - \frac{4}{5}>$