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

Short Answer

Expert verified

The directional derivative of the

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

Step by step solution

Given data

The given function is $f (x, y) = \frac{x}{y^{2}}$

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

Solution

Consider directional derivative

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

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

Therefore

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

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

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

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

Step 3

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

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

and

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

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

Short Answer

Step by step solution

Given data

Solution

Step 3

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

Short Answer

Step by step solution

Given data

Solution

Step 3

One App. One Place for Learning.

Most popular questions from this chapter

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