Chapter 12: Q. 21 (page 953)

In Exercises $21 - 28$ , find the directional derivative of the given
function at the specified point $P$ and in the direction of the
given unit vector $u$ .
$f (x, y) = x^{2} - y^{2} at P = (2, 3), u = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

Short Answer

Expert verified

The directional derivative of the given

function is $- \sqrt{2}$

Step by step solution

Given data

$f (x, y) = x^{2} - y^{2}$

$P = (x_{0}, y_{0}) = (2, 3) u = (α, β) = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

Solution

Consider directional derivative

$D_{w} f (x_{0}, y_{0}) = {Lim}_{h \to 0} \frac{f (x_{0} + α h, y_{0} + β h) - f (x_{0}, y_{0})}{h}$

$D_{w} f (2, 3) = {Lim}_{h \to 0} \frac{f (2 + \frac{\sqrt{2}}{2} h, 3 + \frac{\sqrt{2}}{2} h) - f (2, 3)}{h}$

Therefore

$f (2 + \frac{\sqrt{2}}{2} h, 3 + \frac{\sqrt{2}}{2} h) = {(2 + \frac{\sqrt{2}}{2} h)}^{3} - {(3 + \frac{\sqrt{2}}{2} h)}^{2}$

$= (4 + 2 \sqrt{2} h + \frac{1}{2} h^{2}) - (9 + 3 \sqrt{2} h + \frac{1}{2} h^{2})$

Equation $2$

$f (x_{0} + y_{0}) = 2^{2} - 3^{2} = - 5$ Equation $3$

Substitute

Substituting equation $2$ and $3$

$D_{v} f (2, 3) = {Lim}_{h \to 0} \frac{- 5 - \sqrt{2} h + 5}{h}$

$= {Lim}_{b \to 0} - \sqrt{2}$

$D_{v} f (2, 3) = - \sqrt{2}$

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In Exercises $21 - 28$ , find the directional derivative of the given
function at the specified point $P$ and in the direction of the
given unit vector $u$ .
$f (x, y) = x^{2} - y^{2} at P = (2, 3), u = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

Short Answer

Step by step solution

Given data

Solution

Substitute

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In Exercises21-28 , find the directional derivative of the givenfunction at the specified point Pand in the direction of thegiven unit vector u.f(x,y)=x2−y2atP=(2,3),u=22,22

Short Answer

Step by step solution

Given data

Solution

Substitute

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

In Exercises $21 - 28$ , find the directional derivative of the given
function at the specified point $P$ and in the direction of the
given unit vector $u$ .
$f (x, y) = x^{2} - y^{2} at P = (2, 3), u = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$