Problem 295 - Calculus Volume 3

Question

For the following exercises, find the derivative of the function. \(f(x, y)=e^{x y}\) at point (6,7) in the direction the function increases most rapidly.

Step-by-Step Solution

Answer

The function increases most rapidly in the direction (7/sqrt(85), 6/sqrt(85)).

Step 1: Understanding the Gradient

The gradient of the function, abla f(x, y), represents the direction in which the function increases most rapidly. For a multivariable function like f(x, y) = e^{xy}, the gradient is given by the vector of partial derivatives.

Step 2: Compute Partial Derivatives

We need the partial derivatives with respect to x and y. The partial derivative with respect to x:\( f_x = \frac{\partial}{\partial x} (e^{xy}) = y e^{xy} \).The partial derivative with respect to y:\( f_y = \frac{\partial}{\partial y} (e^{xy}) = x e^{xy} \).

Step 3: Evaluating the Gradient at Point (6,7)

Evaluate the partial derivatives at the point (6,7):\( f_x(6,7) = 7 e^{6 \times 7} = 7 e^{42} \).\( f_y(6,7) = 6 e^{6 \times 7} = 6 e^{42} \).Thus, the gradient at (6,7) is \( abla f(6,7) = \langle 7 e^{42}, 6 e^{42} \rangle \).

Step 4: Determine the Direction

The direction in which the function increases most rapidly is given by the direction of the gradient vector. Therefore, the direction vector is \( \langle 7 e^{42}, 6 e^{42} \rangle \).

Step 5: Normalize the Direction Vector

To find the unit direction vector, we need to normalize the gradient vector:Magnitude of the gradient vector:\( \| abla f(6,7) \| = \sqrt{(7 e^{42})^2 + (6 e^{42})^2} = e^{42} \sqrt{7^2 + 6^2} = e^{42} \sqrt{85} \).Normalize:\( \text{Unit vector} = \left( \frac{7}{\sqrt{85}}, \frac{6}{\sqrt{85}} \right) \).

Key Concepts

Partial Derivatives

When dealing with functions of multiple variables, such as \( f(x, y) = e^{xy} \), we often need to understand how changes in each variable affect the function independently. This is where partial derivatives come in handy.

Partial derivatives measure the rate of change of the function with respect to one variable while keeping other variables constant. In simpler terms:

• A partial derivative with respect to \( x \) treats \( y \) as constant.
• A partial derivative with respect to \( y \) treats \( x \) as constant.

For the function \( f(x, y) = e^{xy} \):- The partial derivative with respect to \( x \) is \( f_x = y e^{xy} \).- The partial derivative with respect to \( y \) is \( f_y = x e^{xy} \).

These derivatives are crucial for understanding how the function behaves in different directions, helping us find the steepest ascent or descent.

Directional Derivative

The directional derivative extends the concept of partial derivatives, revealing how a multivariable function changes in any direction, not just along the axes. This helps us determine the function's rate of change in a specified direction.

To compute the directional derivative, we use the gradient vector, which is composed of all the partial derivatives, giving us a full picture of the function's behavior in all directions. The directional derivative \( D_\mathbf{u}f \) in the direction of a unit vector \( \mathbf{u} = \langle a, b \rangle \) is:

• \( D_\mathbf{u}f = abla f \cdot \mathbf{u} \)
• The dot product of the gradient \( abla f = \langle f_x, f_y \rangle \) with the vector \( \mathbf{u} \).

In practice, this allows us to explore how the function increases or decreases along any specific line or path.

Multivariable Function

Multivariable functions, like \( f(x, y) = e^{xy} \), depend on more than one variable, requiring special techniques and tools to study them effectively.

These functions are fundamental in fields such as physics, engineering, and economics, as they can model complex systems with multiple influencing factors.

• The gradient vector is a key tool, summarizing all partial derivatives in one vector, pointing in the direction of the steepest increase.
• They can encode surfaces rather than lines, requiring us to explore their behavior in \( n \) dimensions.

Understanding how these functions work helps you better analyze situations where multiple independent variables affect the outcome, much like how temperature might affect both the pressure and volume in a physics equation.