Problem 25 - Advanced Engineering Mathematics

Question

In Problems, find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate. $$ F(x, y, z)=x^{2}+4 x z+2 y z^{2} ;(1,2,-1) $$

Step-by-Step Solution

Answer

The function increases most rapidly in the direction of the vector (-2, 2, -4) with a maximum rate of increase of \(2\sqrt{6}\).

Step 1: Compute the Gradient of the Function

The function is given as \( F(x, y, z) = x^2 + 4xz + 2yz^2 \). To find the direction of maximum increase, we first need the gradient: - For \( x \): \( \frac{\partial F}{\partial x} = 2x + 4z \) - For \( y \): \( \frac{\partial F}{\partial y} = 2z^2 \) - For \( z \): \( \frac{\partial F}{\partial z} = 4x + 4yz \) Thus, the gradient is \( abla F = (2x + 4z, 2z^2, 4x + 4yz) \).

Step 2: Evaluate the Gradient at the Given Point

We now evaluate \( abla F \) at the point \((1, 2, -1)\): - \( \frac{\partial F}{\partial x} = 2(1) + 4(-1) = 2 - 4 = -2 \) - \( \frac{\partial F}{\partial y} = 2(-1)^2 = 2 \) - \( \frac{\partial F}{\partial z} = 4(1) + 4(2)(-1) = 4 - 8 = -4 \) Thus, \( abla F(1, 2, -1) = (-2, 2, -4) \).

Step 3: Determine the Direction of Maximum Increase

The gradient vector \( abla F = (-2, 2, -4) \) gives the direction in which the function increases most rapidly at the point \((1, 2, -1)\). This vector itself points into the direction of steepest ascent.

Step 4: Calculate the Maximum Rate of Increase

The maximum rate of increase of the function at the point is the magnitude of the gradient vector \( \| abla F(1, 2, -1) \| \). Compute it as follows: \[ \| abla F \| = \sqrt{(-2)^2 + 2^2 + (-4)^2} = \sqrt{4 + 4 + 16} = \sqrt{24} = 2\sqrt{6} \]. Thus, the maximum rate is \( 2\sqrt{6} \).

Key Concepts

Partial Derivatives

Partial derivatives are a way to measure how a multivariable function changes as one particular variable changes, while keeping other variables constant. This concept is crucial when working with functions that involve several variables, like in the function \( F(x, y, z) = x^2 + 4xz + 2yz^2 \). Here, we're dealing with three variables: \( x \), \( y \), and \( z \).

To understand partial derivatives better, think of them as the "slopes" along each variable's direction. In our example:

• For \( x \), the partial derivative is \( \frac{\partial F}{\partial x} = 2x + 4z \).
• For \( y \), it is \( \frac{\partial F}{\partial y} = 2z^2 \).
• For \( z \), it's \( \frac{\partial F}{\partial z} = 4x + 4yz \).

By evaluating each partial derivative, you determine how the function \( F \) changes with respect to each variable independently. This information is used to form the gradient vector, which combines all the partial derivatives to provide a single direction of change.

Directional Derivative

The directional derivative extends the concept of partial derivatives by giving us the rate at which a function changes in any given direction. For a function \( F(x, y, z) \), the gradient \( abla F \) plays a vital role, as it points in the direction of the fastest increase.

In the exercise, the gradient \( abla F = (-2, 2, -4) \) is evaluated at the point \((1, 2, -1)\). This vector indicates the direction where the function \( F \) increases the quickest. Imagine vectors as arrows pointing outwards; the gradient vector is the longest arrow, indicating steepest ascent.

The directional derivative at this point tells us about the rate of change of \( F \) in the direction of \( abla F \). This is calculated using the dot product of the gradient and the direction vector, typically aligning them to find out how the function performs in that specific direction.

Rate of Change

Rate of change in the context of multivariable functions involves understanding how the function's value shifts as you move in the space defined by its variables. This is represented through the gradient vector. The key takeaway here is that the magnitude of the gradient at a specific point—the length of the gradient vector—tells us the maximum rate of change.

Continuing with our exercise, after deriving the gradient as \( abla F = (-2, 2, -4) \), its magnitude is computed to find that \( \|abla F\| = 2\sqrt{6} \). This value represents how rapidly the function \( F(x, y, z) \) increases as you proceed in the direction of \( abla F \) from the point \((1, 2, -1)\).

• The gradient tells us which way to move to increase or decrease our function's value.
• The magnitude gives a clear number representing the steepest slope at that point.

Thus, understanding the rate of change is about comprehending how quickly and in what direction a function's value is altering at any given point.