Chapter 12: Q. 27 (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.
$f (x, y) = \frac{x + y}{x^{2} + y^{2}}, P = (1, 2), v = ⟨ 3, - 2 ⟩$

Short Answer

Expert verified

$D_{u} f (1, 2) = \frac{11}{25 \sqrt{13}}$

Step by step solution

Step 1. Given Information

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

Step 2. Solution

$The unit vector in the direction of the given vector v = ⟨ 3, - 2 ⟩ is given by \begin{array}{l} u = \frac{1}{∥ v ∥} v \\ = \frac{1}{\sqrt{3^{2} + (- 2)^{2}}} ⟨ 3, - 2 ⟩ \\ = \frac{1}{\sqrt{13}} ⟨ 3, - 2 ⟩ \\ Hence the directional derivative is given by \\ \begin{array}{l} D_{u} f (1, 2) = lim_{h \to 0} \frac{f (1 + \frac{3 h}{\sqrt{13}}, 2 - \frac{2 h}{\sqrt{13}}) - f (1, 2)}{h} \\ = lim_{h \to 0} \frac{\frac{1 + \frac{3 h}{\sqrt{13}} + 2 - \frac{2 h}{\sqrt{13}}}{{(1 + \frac{3 h}{\sqrt{13}})}^{2} + {(2 - \frac{2 h}{\sqrt{13}})}^{2}} - \frac{1 + 2}{1^{2} + 2^{2}}}{h} \end{array} \\ = lim_{h \to 0} \frac{\frac{3 + \frac{h}{\sqrt{13}}}{5 + h^{2} - \frac{2 h}{\sqrt{13}}} - \frac{3}{5}}{h} \\ S i m p l i f y i n g N u m e r a t o r g i v e s \\ \begin{array}{l} D_{u} f (1, 2) = lim_{h \to 0} \frac{\frac{11 h}{\sqrt{13}} - 3 h^{2}}{5 h (5 + h^{2} - \frac{2 h}{\sqrt{13}})} \\ = lim_{h \to 0} \frac{\frac{11}{\sqrt{13}} - 3 h}{5 (5 + h^{2} - \frac{2 h}{\sqrt{13}})} \\ = \frac{11}{25 \sqrt{13}} \\ H e n c e D_{u} f (1, 2) = \frac{11}{25 \sqrt{13}} \end{array} \end{array}$

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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.
$f (x, y) = \frac{x + y}{x^{2} + y^{2}}, P = (1, 2), v = ⟨ 3, - 2 ⟩$

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)=x+yx2+y2,P=(1,2),v=⟨3,−2⟩

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solution

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Discrete Mathematics

Decision Maths

Statistics

Calculus

Pure Maths

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.
$f (x, y) = \frac{x + y}{x^{2} + y^{2}}, P = (1, 2), v = ⟨ 3, - 2 ⟩$