Problem 19 - Calculus with Concepts in Calculus

Question

Find the direction in which \(f\) increases most rapidly at the given point, and find the maximal directional derivative at that point. $$ f(x, y, z)=e^{x}+e^{y}+e^{2 z} ;(1,1,-1) $$

Step-by-Step Solution

Answer

The direction is along \( \nabla f = (e, e, \frac{2}{e^2}) \); rate is \( e\sqrt{2} \).

Step 1: Determine the Gradient

The gradient vector \( abla f \) of a function \( f(x, y, z) \) gives the direction in which the function increases most rapidly. First, find the partial derivatives of \( f \) with respect to \( x \), \( y \), and \( z \):\(\frac{\partial f}{\partial x} = e^{x}, \\frac{\partial f}{\partial y} = e^{y}, \\frac{\partial f}{\partial z} = 2e^{2z}.\)Thus, the gradient is:\( abla f = \left( e^{x}, e^{y}, 2e^{2z} \right).\)

Step 2: Evaluate the Gradient at the Given Point

Substitute the given point \((1,1,-1)\) into the gradient to find its components:\(abla f(1,1,-1) = \left( e^{1}, e^{1}, 2e^{2(-1)} \right) = \left( e, e, \frac{2}{e^2} \right).\)

Step 3: Find the Magnitude of the Gradient Vector

The magnitude of the gradient vector gives the rate of increase of the function in that direction. Calculate the magnitude:\[\|abla f(1,1,-1)\| = \sqrt{e^2 + e^2 + \left(\frac{2}{e^2}\right)^2} = \sqrt{2e^2 + \frac{4}{e^4}}.\]

Step 4: Simplify the Maximum Rate of Increase

Simplify the expression for the magnitude obtained:Since \( e^2 \gg \frac{2}{e^2} \), the dominant terms simplify the magnitude to:\(\|abla f(1,1,-1)\| \approx \sqrt{2e^2} = e\sqrt{2}.\)

Step 5: State the Direction and Maximal Derivative

The direction in which \( f \) increases most rapidly is given by the unit vector in the direction of \( abla f \), and the maximum rate of increase is the magnitude itself:- The direction is \( \frac{1}{\|abla f\|} abla f = \left( \frac{e}{e\sqrt{2}}, \frac{e}{e\sqrt{2}}, \frac{2/(e^2)}{e\sqrt{2}} \right) \).- The maximal directional derivative is \( e\sqrt{2} \).

Key Concepts

Directional Derivative

The directional derivative of a function measures how the function changes as you move in a particular direction from a point. It's essential for understanding how a function behaves along a path, rather than just along the coordinate axes. To calculate the directional derivative at a point, you need two main components:

• The gradient vector, which gives the direction of the steepest ascent.
• A unit vector in the desired direction of movement.

The gradient vector, denoted as \( abla f \), consists of the partial derivatives of the function. When you dot this gradient with a unit vector, you obtain the directional derivative. This value reflects the rate of change of the function in that specific direction.
In the provided exercise, the gradient vector at the point \((1,1,-1)\) is evaluated, and its magnitude indicates the maximal rate of increase. Thus, the directional derivative not only tells us how much the function changes but also how fast it changes in a specific direction.

Partial Derivatives

Partial derivatives help us understand how a function changes as one of its variables changes, while the other variables are held constant. They are fundamental in multivariable calculus, where functions depend on more than one variable. For a function \( f(x, y, z) \), the partial derivatives with respect to \( x \), \( y \), and \( z \) are defined as follows:

• \( \frac{\partial f}{\partial x} \): Represents the rate of change of \( f \) with respect to \( x \) while keeping \( y \) and \( z \) constant.
• \( \frac{\partial f}{\partial y} \): Represents the rate of change of \( f \) with respect to \( y \) while keeping \( x \) and \( z \) constant.
• \( \frac{\partial f}{\partial z} \): Represents the rate of change of \( f \) with respect to \( z \) while keeping \( x \) and \( y \) constant.

In the given exercise, these partial derivatives are computed as \( e^x \), \( e^y \), and \( 2e^{2z} \). They come together to form the components of the gradient vector \( abla f \). By understanding each partial derivative, we gain insight into how the function behaves locally around a point.

Function Optimization

Function optimization involves finding maxima, minima, or saddle points of a function. It's a critical concept in fields like economics, engineering, and data science. By determining points where a function reaches its highest or lowest values, one can make informed decisions or improve certain processes.
Optimization often involves the gradient vector, as it naturally points in the direction of steepest ascent. At critical points where \( abla f = 0 \), the function may have a maximum, minimum, or saddle point. However, finding the maximum rate of increase, as shown in the step-by-step solution, is key in identifying where and how the function grows fastest at a given point.
This process includes calculating the gradient at a specific point, evaluating its magnitude, and using this information to find the maximal directional derivative. The maximal directional derivative indicates the steepest growth path, while the unit vector of the gradient gives the precise direction for this optimal change. Understanding these elements in function optimization can help solve complex problems involving multivariable functions.