Question

Find the maximum rate of change of the function f (x, y) = x^2y^2 + xy −10 at the
point (1, 1), and the direction in which it occurs.

Answer 1

Answer:

The maximum rate of change of the given function is equal to 3√2.

The direction at which the vector points is given by the unit vector $\displaystyle \frac{\sqrt{2}}{2} \hat{\i} + \frac{\sqrt{2}}{2} \hat{\j}$.

General Formulas and Concepts:
Pre-Calculus

Vectors

• Unit Vector:
  $\displaystyle \frac{\overline{v}}{|\overline{v}|}$

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Addition/Subtraction]:
$\displaystyle (u + v)' = u' + v'$

Derivative Rule [Basic Power Rule]:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Multivariable Calculus

Partial Derivatives

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 given.

$\displaystyle f(x, y) = x^2y^2 + xy - 10$

$\displaystyle \text{Point} \ (1, 1)$

Step 2: Find Maximum Rate of Change

1. [Function] Take gradient:
   $\displaystyle \nabla f(x, y) = \frac{\partial}{\partial x} \bigg[ x^2y^2 + xy - 10 \bigg] \hat{\i} + \frac{\partial}{\partial y} \bigg[ x^2y^2 + xy - 10 \bigg] \hat{\j}$
2. [Gradient] Rewrite [Gradient Property - Addition/Subtraction]:
   $\displaystyle \nabla f(x, y) = \bigg[ \frac{\partial}{\partial x} x^2y^2 + \frac{\partial}{\partial x} xy - \frac{\partial}{\partial x} 10 \bigg] \hat{\i} + \bigg[ \frac{\partial}{\partial y} x^2y^2 + \frac{\partial}{\partial y} xy - \frac{\partial}{\partial y} 10 \bigg] \hat{\j}$
3. [Gradient] Differentiate [Derivative Rule - Basic Power Rule]:
   $\displaystyle \nabla f(x, y) = \bigg( 2xy^2 + y \bigg) \hat{\i} + \bigg( 2yx^2 + x \bigg) \hat{\j}$
4. [Gradient] Substitute in point:
   $\displaystyle \nabla f(1, 1) = 3 \hat{\i} + 3 \hat{\j}$
5. [Gradient] Take magnitude:
   $\displaystyle \bigg| \nabla f(1, 1) \bigg| = \boxed{3 \sqrt{2}}$

∴ the maximum rate of change of the given function f(x, y) is equal to 3√2.

Step 3: Find Direction

1. [Gradient] Convert to unit vector:
   $\displaystyle \text{Direction} = \boxed{ \frac{\sqrt{2}}{2} \hat{\i} + \frac{\sqrt{2}}{2} \hat{\j} }$

∴ we have found the direction of the gradient.

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Learn more about gradient: https://brainly.com/question/6158243

Learn more about multivariable calculus: https://brainly.com/question/2407209

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Topic: Multivariable Calculus

Unit: Directional Derivatives