Problem 24 - Advanced Engineering Mathematics

Question

Find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate. $$ f(x, y)=x y e^{x-y} ;(5,5) $$

Step-by-Step Solution

Answer

Gradient vector: (30, -20); Max rate: 10√13.

Step 1: Understanding the Problem

To find the vector that gives the direction in which the function increases most rapidly at the point, we need to calculate the gradient vector at this point. The gradient vector of a function points in the direction of steepest ascent.

Step 2: Calculate Partial Derivatives

Let's find the partial derivatives of the function \( f(x, y) = x y e^{x-y} \).- First, the partial derivative with respect to \( x \):\[ \frac{\partial f}{\partial x} = y e^{x-y} + xy e^{x-y} \]- Next, the partial derivative with respect to \( y \):\[ \frac{\partial f}{\partial y} = x e^{x-y} - xy e^{x-y} \]

Step 3: Evaluate Partial Derivatives at the Given Point

Substitute \((x, y) = (5, 5)\) into the partial derivatives to find:- \( \frac{\partial f}{\partial x} \biggr|_{(5, 5)} = 5 e^{5-5} + 5 \times 5 \times e^{5-5} = 5 + 25 = 30 \)- \( \frac{\partial f}{\partial y} \biggr|_{(5, 5)} = 5 e^{5-5} - 5 \times 5 \times e^{5-5} = 5 - 25 = -20 \)

Step 4: Compute the Gradient Vector at (5, 5)

The gradient vector \( abla f \) at the point \((5, 5)\) is:\[ abla f(5, 5) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \biggr|_{(5, 5)} = (30, -20) \]

Step 5: Find the Maximum Rate of Increase

The maximum rate of increase of the function is the magnitude of the gradient vector. Calculate it using:\[ \lVert abla f(5, 5) \rVert = \sqrt{30^2 + (-20)^2} = \sqrt{900 + 400} = \sqrt{1300} = 10\sqrt{13} \]

Key Concepts

Partial Derivatives

In multivariable calculus, partial derivatives play a vital role in understanding how a function changes with respect to each of its variables. Imagine you have a surface (like a mountain or a hill) represented by a function \( f(x, y) \). The partial derivative with respect to \( x \) tells us how the surface is rising or falling as you move in the \( x \)-direction, while keeping \( y \) constant. Similarly, the partial derivative with respect to \( y \) indicates the change as you move in the \( y \)-direction, regardless of changes in \( x \). By computing these derivatives, we can determine how small changes in each variable affect the overall function.

To find these derivatives for a function \( f(x, y) = xy e^{x-y} \), you calculate:

• Partial derivative with respect to \( x \): \( \frac{\partial f}{\partial x} = ye^{x-y} + xye^{x-y} \)
• Partial derivative with respect to \( y \): \( \frac{\partial f}{\partial y} = xe^{x-y} - xye^{x-y} \)

These derivatives help us explore how the function behaves near a specific point. In our exercise, these were evaluated at the point (5,5), which provided the values needed for further analysis.

Direction of Steepest Ascent

The direction of steepest ascent for a function \( f(x, y) \) at a given point is the direction that leads to the most rapid increase in the function value. This is where the gradient vector, denoted as \( abla f(x, y) \), comes into play. The gradient vector is composed of the partial derivatives of the function and provides a systematic way to find this direction.

At any point, the gradient vector \( abla f(x, y) \) is given by:

• \( abla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right) \)

For the function \( f(x, y) = xy e^{x-y} \), when calculated at the point (5,5), the gradient vector is \( abla f(5, 5) = (30, -20) \). This vector points in the direction where the function increases most quickly at that specific point. Essentially, if you were standing on a hill, the gradient vector would tell you which way to start walking to climb the steepest path.

It's interesting that this vector doesn't only inform us about direction but also relates to the concept of function maximization. Understanding and calculating the gradient vector serve many practical applications, from maximizing functions to finding critical points in optimization problems.

Function Maximization

Maximizing a function often involves finding where it increases the fastest and what that maximum rate is. In our context of multivariable calculus, the step of finding the gradient vector naturally leads us to understand the rate at which a function increases as well as its maximum increase rate.

The rate of increase in the direction of the gradient vector is represented by the magnitude (or length) of the gradient vector. For our function \( f(x, y) = xy e^{x-y} \) at the point (5,5), this rate is calculated as:

• Magnitude of \( abla f(5, 5) \): \( \left\lVert abla f(5, 5) \right\rVert = \sqrt{30^2 + (-20)^2} = \sqrt{1300} = 10\sqrt{13} \)

Thus, the value \( 10\sqrt{13} \) represents the maximum rate of increase at that point. It is key because it tells us not just the direction but also how steeply the function will ascend if we proceed in that optimal direction.

Comprehending function maximization through gradient vectors is crucial in fields like economics, engineering, and mathematics where optimization is required. It allows one to efficiently find the best direction and speed for increasing gains, which can be extrapolated to many real-world scenarios.