Chapter 12: Q. 24 (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 $u$ .
localid="1650380514741" $f (x, y) = \frac{x}{y^{2}} at P = (- 2, 1), u = - \frac{5}{13} i + \frac{12}{13} j$

Short Answer

Expert verified

The directional derivative of the given

function is $\frac{43}{13}$

Step by step solution

Given data

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

$P = (x_{0}, y_{0}) = (- 2, 1) and u = (α i + β j) = (\frac{- 5}{13} i + \frac{12}{13} j)$

Solution

Consider directional derivative

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

$D_{u} f (- 2, 1) = {Lim}_{h \to 0} \frac{f (- 2 - \frac{5}{13} h, 1 + \frac{12}{13} h) - f (- 2, 1)}{h}$

Therefore

$f (- 2 - \frac{5}{13} h, 1 + \frac{12}{13} h) = \frac{- 2 - \frac{5}{13} h}{{(1 + \frac{12}{13} h)}^{2}}$

$\frac{- 2 - \frac{5}{13} h}{\frac{24}{13} h + \frac{144}{169} h^{2})}$

$= \frac{\frac{- 26 - 5 h}{13}}{\frac{24}{13} h + \frac{144}{169} h^{2})}$

$= \frac{- 26 - 5 h}{13 \times (1 + \frac{24}{13} h + \frac{144}{169} h^{2})}$

$= \frac{- 26 - 5 h}{13 \times (\frac{169 + 312 h + 144 h^{2}}{169})}$

$= \frac{- 338 - 65 h}{169 + 312 h + 144 h^{2}}$

And

$f (x_{0}, y_{0}) = f (- 2, 1) = \frac{- 2}{1^{2}} = - 2$

Substitute

Substituting

$D_{w} f (- 2, 1) = {Lim}_{h \to 0} \frac{\frac{- 338 - 65 h}{169 + 312 h + 144 h^{2}} - (- 2)}{h}$

$= {Lim}_{h \to 0} \frac{- 338 - 65 h + 338 + 624 h + 288 h^{2}}{h (169 + 312 h + 144 h^{2})}$

$= {Lim}_{h \to 0} \frac{559 h + 288 h^{2}}{h (169 + 312 h + 144 h^{2})}$

$= {Lim}_{h \to 0} \frac{h (559 + 288 h)}{h (169 + 312 h + 144 h^{2})}$

$= {Lim}_{h \to 0} \frac{(559 + 288 h)}{(169 + 312 h + 144 h^{2})} = \frac{559}{169}$
$D_{w} f (- 2, 1) = \frac{43}{13}$

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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 $u$ .
localid="1650380514741" $f (x, y) = \frac{x}{y^{2}} at P = (- 2, 1), u = - \frac{5}{13} i + \frac{12}{13} j$

Short Answer

Step by step solution

Given data

Solution

Substitute

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In Exercises 21-28, find the directional derivative of the givenfunction at the specified point Pand in the direction of thegiven unit vector u.localid="1650380514741" f(x,y)=xy2atP=(−2,1),u=−513i+1213j

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

Probability and Statistics

Statistics

Mechanics Maths

Logic and Functions

Decision Maths

Theoretical and Mathematical Physics

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 $u$ .
localid="1650380514741" $f (x, y) = \frac{x}{y^{2}} at P = (- 2, 1), u = - \frac{5}{13} i + \frac{12}{13} j$