Question

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

Answer 1

Answer:

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

General Formulas and Concepts:
Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Rule [Basic Power Rule]:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Multivariable Calculus

Differentiation

• Partial Derivatives
• Derivative 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

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

Unit: Directional Derivatives