Chapter 12: Q. 35 (page 953)

In Exercises 35–38, find the directional derivative of the given
function at the specified point P and in the direction of the
given vector v.
$f (x, y) = x^{2} - y^{2} at P = (3, 3), v = (- 1, 5)$

Short Answer

Expert verified

The directional derivative of the given

function is $\frac{- 18}{13} \sqrt{26}$

Step by step solution

Given data

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

$P = (x_{0}, y_{0}) = (3, 3) and y = (- 1, 5)$

solution

Therefore

$∥ v ∥ = \sqrt{- 1^{2} + 5^{2}} = \sqrt{26}$

$u = (α, β) = (\frac{- \sqrt{26}}{26}, \frac{5 \sqrt{26}}{26})$

Directional derivation function at $P$ point is given by

$\nabla f (P) \times u = \nabla f (3, 3) \times u = ({\frac{d f}{d x}|}_{(3, 3)} i + {\frac{d f}{d y}|}_{(3, 3)} j) \times (\frac{- \sqrt{26}}{26} i + \frac{5 \sqrt{26}}{26} j)$

$= [(2 x)_{(3, 3)} i + (- 2 y)_{(3, 3)} j] \cdot (\frac{- \sqrt{26}}{26} i + \frac{5 \sqrt{26}}{26} j)$

$= (6 i - 6 j) \times (\frac{- \sqrt{26}}{26} i + \frac{5 \sqrt{26}}{26} j)$

$= (\frac{- 6 \sqrt{26}}{26} - \frac{30 \sqrt{26}}{26})$

$= \frac{- 36 \sqrt{26}}{26}$

$\nabla f (P) \cdot u = \frac{- 18}{13} \sqrt{26}$

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In Exercises 35–38, find the directional derivative of the given
function at the specified point P and in the direction of the
given vector v.
$f (x, y) = x^{2} - y^{2} at P = (3, 3), v = (- 1, 5)$

Short Answer

Step by step solution

Given data

solution

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In Exercises 35–38, find the directional derivative of the givenfunction at the specified point P and in the direction of thegiven vector v.f(x,y)=x2−y2atP=(3,3),v=-1,5

Short Answer

Step by step solution

Given data

solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Discrete Mathematics

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Geometry

Study anywhere. Anytime. Across all devices.

In Exercises 35–38, find the directional derivative of the given
function at the specified point P and in the direction of the
given vector v.
$f (x, y) = x^{2} - y^{2} at P = (3, 3), v = (- 1, 5)$