Q. 69 Let (a,聽b) be a point in the do... [FREE SOLUTION]

Chapter 12: Q. 69 (page 954)

Let $(a, b)$ be a point in the domain of the function of two variables, role="math" localid="1650475276808" $f (x, y)$ , and $u$ be a unit vector for which $D_{u} f (a, b)$ exists. Prove that $D_{- u} f (a, b) = - D_{u} f (a, b) .$

Short Answer

Expert verified

Hence proved is $= - D_{u} f (a, b)$

Step by step solution

Variable function.

Let $(x, y)$ be a two-variable function and $u = 鉄� 伪, 尾鉄�$ , be a unit vector for $D_{- u} f (a, b)$ The objective is to prove that

$D_{- u} f (a, b) = - D_{u} f (a, b)$

The direction derivative of a function $f (x, y)$ at $(x_{0}, y_{0})$ in the direction of $u = 鉄� 伪, 尾鉄�$ is given by

$D_{u} f (x_{0}, y_{0}) = \lim_{h 鈫� 0} \frac{f (x_{0} + 伪 h, y_{0} + 尾 h) - f (x_{0}, y_{0})}{h}$

Thus,

D_{u} f (a, b) = \lim_{h 鈫� 0} \frac{f (a + 伪 h, b + 尾 h) - f (a, b)}{h}

(1)

Find the Equations.

For $- u = 鉄� - 伪, - 尾鉄�$

$D_{- u} f (a, b) = \lim_{畏鈫� 0} \frac{f (a - 伪畏, b - 尾畏) - f (a, b)}{畏}$

$= \lim_{畏鈫� 0} \frac{f (a + 伪 (- 畏), b + 尾 (- 畏)) - f (a, b)}{畏}$

Let $h = - 畏$ then

$D_{- u} f (a, b) = \lim_{h 鈫� 0} \frac{f (a + 伪 h, b + 尾 h) - f (a, b)}{- h}$

$= - D_{u} f (a, b)$ from $(1)$

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Let $(a, b)$ be a point in the domain of the function of two variables, role="math" localid="1650475276808" $f (x, y)$ , and $u$ be a unit vector for which $D_{u} f (a, b)$ exists. Prove that $D_{- u} f (a, b) = - D_{u} f (a, b) .$

Short Answer

Step by step solution

Variable function.

Find the Equations.

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91影视

Let (a,b) be a point in the domain of the function of two variables, role="math" localid="1650475276808" f(x,y), and u be a unit vector for which Duf(a,b) exists. Prove thatD-uf(a,b)=-Duf(a,b).

Short Answer

Step by step solution

Variable function.

Find the Equations.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Let $(a, b)$ be a point in the domain of the function of two variables, role="math" localid="1650475276808" $f (x, y)$ , and $u$ be a unit vector for which $D_{u} f (a, b)$ exists. Prove that $D_{- u} f (a, b) = - D_{u} f (a, b) .$