Mathematics, 11.07.2019 05:00 sherlock19

Let f(x, y)=x2 −y2. find the gradient of f at the point (√2,1). sketch the level curve of f through this point, together with the gradient at that point. g

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Answer from: hetdashadia123

$\displaystyle \nabla f(\sqrt{2}, 1) = 2\sqrt{2} \hat{\i} - 2 \hat{\j}$

General Formulas and Concepts:
Calculus

Differentiation

DerivativesDerivative Notation

Derivative Rule [Basic Power Rule]:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Multivariable Calculus

Differentiation

Partial DerivativesDerivative Notation

Gradient: $\displaystyle \nabla f(x, y, z) = \frac{\partial f}{\partial x} \hat{\i} + \frac{\partial f}{\partial y} \hat{\j} + \frac{\partial f}{\partial z} \hat{\text{k}}$

Gradient Property [Addition/Subtraction]: $\displaystyle \nabla \big[ f(x) + g(x) \big] = \nabla f(x) + \nabla g(x)$

Step-by-step explanation:

Step 1: Define

Identify.

$\displaystyle f(x, y) = x^2 - y^2$

$\displaystyle P(\sqrt{2}, 1)$

Step 2: Find Gradient

[Function] Differentiate [Gradient]: $\displaystyle \nabla f(x, y) = \frac{\partial f}{\partial x} \bigg[ x^2 - y^2 \bigg] \hat{\i} + \frac{\partial f}{\partial y} \bigg[ x^2 - y^2 \bigg] \hat{\j}$ [Gradient] Rewrite [Gradient Property - Addition/Subtraction]: $\displaystyle \nabla f(x, y) = \bigg[ \frac{\partial f}{\partial x}(x^2) - \frac{\partial f}{\partial x}(y^2) \bigg] \hat{\i} + \bigg[ \frac{\partial f}{\partial y}(x^2) - \frac{\partial f}{\partial y}(y^2) \bigg] \hat{\j}$ [Gradient] Differentiate [Derivative Rule - Basic Power Rule]: $\displaystyle \nabla f(x, y) = 2x \hat{\i} - 2y \hat{\j}$ [Gradient] Substitute in point: $\displaystyle \nabla f(\sqrt{2}, 1) = 2\sqrt{2} \hat{\i} - 2(1) \hat{\j}$ [Gradient] Simplify: $\displaystyle \nabla f(\sqrt{2}, 1) = 2\sqrt{2} \hat{\i} - 2 \hat{\j}$

∴ the gradient of the given f(x, y) function is equal to <2√2, -2>.

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Learn more about gradient: link

Learn more about multivariable calculus: link

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Topic: Multivariable Calculus

Unit: Directional Derivatives

Let f(x,y)=x2 −y2. find the gradient of f at the point (√2,1). sketch the level curve of f through t

Answer from: Quest

i can not answer the question due to the fact that you spelled "carton" as "cartoon", a cartoon is something you watch on tv, a carton is a box that holds things. and also you've been waiting for this question for a while so why don't you get off your lazy a** and try to figure it out for yourself for once!

step-by-step explanation:

===============================================================^

look above, and leave your comments down below, and don't for get a five stars and a and brainliest xd

sorry for the stars in the middle of words its because it wouldn't let me submit. xd

How many cartoon of food an how many cartoons of clothing should be sent with each plane? including

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Let f(x, y)=x2 −y2. find the gradient of f at the point (√2,1). sketch the level curve of f through...