Chapter 12: Q. 36 (page 953)

In exercise $35 - 38$ , find the directional derivative of the given function at the specified point $P$ and in the direction of the given vector $v$ .
role="math" localid="1650433937626" $f (x, y) = \frac{x}{y^{2}} at P = (9, - 3), v = 2 i + 7 j$

Short Answer

Expert verified

The directional derivative of the function is $\nabla f (P) . u = \frac{44}{9 \sqrt{53}}$ .

Step by step solution

Directional derivation.

For a given function $P = (x_{0}, y_{0}) = (9, - 3)$ and $v = 2 i + 7 j$ , we must find the directional derivative $f (x, y) = \frac{x}{y^{2}}$ .

role="math" localid="1650434437663" $||v|| = \sqrt{2^{2} + 7^{2}} = \sqrt{53}$

$∴ u = (α, β) = (\frac{2}{\sqrt{53}}, \frac{7}{\sqrt{53}})$

Directional unit vector.

The directional derivative of a variable at point $P$ with directional unit vector u is calculated as follows:

localid="1650642364216" $\nabla f (P) \cdot u = \nabla f (9, - 3) \times u = ({\frac{d f}{d x}|}_{(9, - 3)} i + {\frac{d f}{d y}|}_{(9, - 3)} j) \times (\frac{2}{\sqrt{53}} i + \frac{7}{\sqrt{53}} j)$

localid="1650642387307" $= [{(\frac{1}{y^{2}})}_{(9, - 3)} i + {(\frac{- 2 x}{y^{3}})}_{(9, - 3)} j \times (\frac{2}{\sqrt{53}} i + \frac{7}{\sqrt{53}} j)$

localid="1650642404428" $= (\frac{1}{9} i + \frac{18}{27} j) \times (\frac{2}{\sqrt{53}} i + \frac{7}{\sqrt{53}} j)$

$= (\frac{1.2}{9 . \sqrt{53}} + \frac{18.7}{27 . \sqrt{53}})$

localid="1650642421175" $= (\frac{2}{9 . \sqrt{53}} + \frac{126}{27 . \sqrt{53}})$

$= \frac{2}{9 . \sqrt{53}} (1 + \frac{63}{3})$

$\nabla f (P) . u = \frac{44}{9 \sqrt{53}}$

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In exercise $35 - 38$ , find the directional derivative of the given function at the specified point $P$ and in the direction of the given vector $v$ .
role="math" localid="1650433937626" $f (x, y) = \frac{x}{y^{2}} at P = (9, - 3), v = 2 i + 7 j$

Short Answer

Step by step solution

Directional derivation.

Directional unit vector.

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In exercise 35-38, find the directional derivative of the given function at the specified point Pand in the direction of the given vector v.role="math" localid="1650433937626" f(x,y)=xy2atP=(9,−3),v=2i+7j

Short Answer

Step by step solution

Directional derivation.

Directional unit vector.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Mechanics Maths

Pure Maths

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

In exercise $35 - 38$ , find the directional derivative of the given function at the specified point $P$ and in the direction of the given vector $v$ .
role="math" localid="1650433937626" $f (x, y) = \frac{x}{y^{2}} at P = (9, - 3), v = 2 i + 7 j$