Understanding the Maximum Rate of Change

Oct 13, 2025 by wbrum.org 113 views

Introduction

In this article, we will explore the concept of the maximum rate of change of a multivariable function. We will use the given function $f(x, y) = \left(x^1 + y^2\right) e^{3x - y}$ and determine its maximum rate of change at the point $(1, 3)$ . Additionally, we will calculate the directional derivative of the function at the same point.

Understanding the Maximum Rate of Change

The maximum rate of change of a function is a measure of how fast the function changes as its input changes. In the case of a multivariable function, the maximum rate of change is a vector that points in the direction of the greatest increase of the function. This vector is called the gradient of the function.

Calculating the Gradient

To calculate the gradient of a multivariable function, we need to find the partial derivatives of the function with respect to each of its variables. The partial derivative of a function $f(x, y)$ with respect to $x$ is denoted as $\frac{\partial f}{\partial x}$ , and the partial derivative with respect to $y$ is denoted as $\frac{\partial f}{\partial y}$ .

For the given function $f(x, y) = \left(x^1 + y^2\right) e^{3x - y}$ , we can calculate the partial derivatives as follows:

Partial Derivative with Respect to x

\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left(x^1 + y^2\right) e^{3x - y}

Using the product rule and the chain rule, we get:

\frac{\partial f}{\partial x} = \left(x^1 + y^2\right) e^{3x - y} \cdot 3e^{3x - y} + e^{3x - y} \cdot 1

Simplifying the expression, we get:

\frac{\partial f}{\partial x} = 3\left(x^1 + y^2\right) e^{3x - y} + e^{3x - y}

Partial Derivative with Respect to y

\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left(x^1 + y^2\right) e^{3x - y}

Using the product rule and the chain rule, we get:

\frac{\partial f}{\partial y} = \left(x^1 + y^2\right) e^{3x - y} \cdot -e^{3x - y} + e^{3x - y} \cdot 2y

Simplifying the expression, we get:

\frac{\partial f}{\partial y} = -\left(x^1 + y^2\right) e^{3x - y} + 2y e^{3x - y}

Calculating the Gradient at the Point (1, 3)

Now that we have the partial derivatives, we can calculate the gradient at the point $(1, 3)$ .

Gradient at the Point (1, 3)

\nabla f(1, 3) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)_{(1, 3)}

Substituting the values of the partial derivatives at the point $(1, 3)$ , we get:

\nabla f(1, 3) = \left(3\left(1^1 + 3^2\right) e^{3\cdot 1 - 3} + e^{3\cdot 1 - 3}, -\left(1^1 + 3^2\right) e^{3\cdot 1 - 3} + 2\cdot 3 e^{3\cdot 1 - 3}\right)

Simplifying the expression, we get:

\nabla f(1, 3) = \left(3\left(1 + 9\right) e^{0} + e^{0}, -\left(1 + 9\right) e^{0} + 6 e^{0}\right)

\nabla f(1, 3) = \left(3\cdot 10 + 1, -10 + 6\right)

\nabla f(1, 3) = \left(30 + 1, -4\right)

\nabla f(1, 3) = \left(31, -4\right)

Calculating the Maximum Rate of Change

The maximum rate of change of the function at the point $(1, 3)$ is the magnitude of the gradient at that point.

Magnitude of the Gradient

\left|\nabla f(1, 3)\right| = \sqrt{\left(31\right)^2 + \left(-4\right)^2}

Simplifying the expression, we get:

\left|\nabla f(1, 3)\right| = \sqrt{961 + 16}

\left|\nabla f(1, 3)\right| = \sqrt{977}

Calculating the Directional Derivative

The directional derivative of a function at a point in a specific direction is the rate of change of the function in that direction.

To calculate the directional derivative, we need to find the unit vector in the direction of the gradient.

Unit Vector in the Direction of the Gradient

\mathbf{u} = \frac{\nabla f(1, 3)}{\left|\nabla f(1, 3)\right|}

Substituting the values of the gradient and its magnitude, we get:

\mathbf{u} = \frac{\left(31, -4\right)}{\sqrt{977}}

Simplifying the expression, we get:

\mathbf{u} = \left(\frac{31}{\sqrt{977}}, \frac{-4}{\sqrt{977}}\right)

Directional Derivative

D_{\mathbf{u}}f(1, 3) = \nabla f(1, 3) \cdot \mathbf{u}

Substituting the values of the gradient and the unit vector, we get:

D_{\mathbf{u}}f(1, 3) = \left(31, -4\right) \cdot \left(\frac{31}{\sqrt{977}}, \frac{-4}{\sqrt{977}}\right)

Simplifying the expression, we get:

D_{\mathbf{u}}f(1, 3) = \frac{31\cdot 31}{\sqrt{977}} + \frac{-4\cdot -4}{\sqrt{977}}

D_{\mathbf{u}}f(1, 3) = \frac{961}{\sqrt{977}} + \frac{16}{\sqrt{977}}

D_{\mathbf{u}}f(1, 3) = \frac{977}{\sqrt{977}}

D_{\mathbf{u}}f(1, 3) = \sqrt{977}

Conclusion

$In this article, we calculated the maximum rate of change of the function $f(x, y) = \left(x^1 + y^2\right) e^{3x - y}$ at the point $(1, 3)$. We also calculated the directional derivative of the function at the same point. The maximum rate of change is the magnitude of the gradient at the point, which is $\sqrt{977}$. The directional derivative is the rate of change of the function in the direction of the gradient, which is also $\sqrt{977}$.<br/> **Q&A: Calculating the Maximum Rate of Change of a Multivariable Function**$

Introduction

In our previous article, we explored the concept of the maximum rate of change of a multivariable function. We used the given function $f(x, y) = \left(x^1 + y^2\right) e^{3x - y}$ and determined its maximum rate of change at the point $(1, 3)$ . Additionally, we calculated the directional derivative of the function at the same point. In this article, we will answer some frequently asked questions related to the maximum rate of change of a multivariable function.

Q: What is the maximum rate of change of a multivariable function?

A: The maximum rate of change of a multivariable function is a measure of how fast the function changes as its input changes. It is a vector that points in the direction of the greatest increase of the function.

Q: How do I calculate the maximum rate of change of a multivariable function?

A: To calculate the maximum rate of change of a multivariable function, you need to find the partial derivatives of the function with respect to each of its variables. Then, you need to calculate the gradient of the function, which is the vector of the partial derivatives. The maximum rate of change is the magnitude of the gradient.

Q: What is the directional derivative of a multivariable function?

A: The directional derivative of a multivariable function is the rate of change of the function in a specific direction. It is calculated by finding the unit vector in the direction of the gradient and then taking the dot product of the gradient and the unit vector.

Q: How do I calculate the directional derivative of a multivariable function?

A: To calculate the directional derivative of a multivariable function, you need to find the unit vector in the direction of the gradient. Then, you need to take the dot product of the gradient and the unit vector.

Q: What is the relationship between the maximum rate of change and the directional derivative?

A: The maximum rate of change and the directional derivative are related in that the maximum rate of change is the magnitude of the gradient, while the directional derivative is the rate of change of the function in a specific direction.

Q: Can the maximum rate of change and the directional derivative be equal?

A: Yes, the maximum rate of change and the directional derivative can be equal. This occurs when the direction of the gradient is the same as the direction of the unit vector.

Q: How do I use the maximum rate of change and the directional derivative in real-world applications?

A: The maximum rate of change and the directional derivative are used in various real-world applications, such as optimization problems, machine learning, and data analysis. They are used to find the maximum or minimum of a function, or to determine the rate of change of a function in a specific direction.

Conclusion

In this article, we answered some frequently asked questions related to the maximum rate of change of a multivariable function. We hope that this article has provided you with a better understanding of the concept of the maximum rate of change and its applications.

Additional Resources

For more information on the maximum rate of change of a multivariable function, please refer to the following resources:

[1] Calculus: Early Transcendentals by James Stewart
[2] Multivariable Calculus by Michael Spivak
[3] Optimization Techniques by Stephen Boyd and Lieven Vandenberghe

We hope that this article has been helpful in your understanding of the maximum rate of change of a multivariable function. If you have any further questions or need additional clarification, please do not hesitate to contact us.