Given The Information And Content Studied About Partial Derivatives, Analyze The Following Statements And Mark V For True And F For False: I. ( ) The Second Derivative In X Of The Function F(x,y) = X^3 + X^2y^3 - 2y^2 Is F_xx(x,y) = ?

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In the realm of multivariable calculus, partial derivatives stand as a cornerstone concept, extending the familiar notion of derivatives from single-variable calculus to functions of multiple variables. This article delves deep into the world of partial derivatives, exploring their definition, computation, interpretation, and applications. We will unravel the intricacies of this mathematical tool, equipping you with the knowledge and skills to confidently navigate the landscape of multivariable functions.

At its core, a partial derivative measures the rate of change of a multivariable function with respect to one of its variables, while holding all other variables constant. This concept allows us to isolate the influence of each variable on the function's output, providing a nuanced understanding of its behavior. To grasp this idea, let's consider a function f(x, y) of two variables, x and y. The partial derivative of f with respect to x, denoted as ∂f/∂x, represents the instantaneous rate of change of f as x varies, assuming that y remains constant. Similarly, the partial derivative of f with respect to y, denoted as ∂f/∂y, captures the rate of change of f with respect to y, while keeping x constant.

Formal Definition

Mathematically, partial derivatives are defined using limits, mirroring the definition of ordinary derivatives. The partial derivative of f(x, y) with respect to x at a point (x₀, y₀) is given by:

∂f/∂x (x₀, y₀) = lim (h→0) [f(x₀ + h, y₀) - f(x₀, y₀)] / h

This limit represents the slope of the tangent line to the curve obtained by intersecting the surface z = f(x, y) with the vertical plane y = y₀ at the point (x₀, y₀, f(x₀, y₀)). Analogously, the partial derivative of f(x, y) with respect to y at (x₀, y₀) is defined as:

∂f/∂y (x₀, y₀) = lim (k→0) [f(x₀, y₀ + k) - f(x₀, y₀)] / k

This limit corresponds to the slope of the tangent line to the curve formed by intersecting the surface z = f(x, y) with the vertical plane x = x₀ at the point (x₀, y₀, f(x₀, y₀)).

Computation

In practice, computing partial derivatives often involves applying differentiation rules from single-variable calculus, while treating the other variables as constants. For instance, to find ∂f/∂x, we differentiate f with respect to x as if y were a constant. Conversely, to find ∂f/∂y, we differentiate f with respect to y, treating x as a constant. This straightforward approach simplifies the process of finding partial derivatives, making it accessible even for complex functions.

Let's illustrate this with an example. Consider the function f(x, y) = x³ + x²y³ - 2y². To find ∂f/∂x, we differentiate with respect to x, treating y as a constant:

∂f/∂x = 3x² + 2xy³

Similarly, to find ∂f/∂y, we differentiate with respect to y, treating x as a constant:

∂f/∂y = 3x²y² - 4y

These examples demonstrate the ease with which partial derivatives can be computed using familiar differentiation rules.

The concept of partial derivatives extends beyond first-order derivatives. We can further differentiate partial derivatives to obtain higher-order partial derivatives. For a function f(x, y), the second-order partial derivatives are:

  • ∂²f/∂x²: The partial derivative of ∂f/∂x with respect to x.
  • ∂²f/∂y²: The partial derivative of ∂f/∂y with respect to y.
  • ∂²f/∂x∂y: The partial derivative of ∂f/∂y with respect to x.
  • ∂²f/∂y∂x: The partial derivative of ∂f/∂x with respect to y.

These higher-order derivatives provide valuable information about the function's concavity and the interaction between its variables. For instance, ∂²f/∂x² represents the concavity of the function in the x-direction, while ∂²f/∂y² represents the concavity in the y-direction. The mixed partial derivatives, ∂²f/∂x∂y and ∂²f/∂y∂x, reveal how the rate of change of the function with respect to one variable changes as the other variable varies.

Clairaut's Theorem

A fundamental theorem in the study of partial derivatives, known as Clairaut's Theorem (also called Schwarz's Theorem), states that if the mixed partial derivatives ∂²f/∂x∂y and ∂²f/∂y∂x are continuous at a point, then they are equal at that point. This theorem simplifies the computation of higher-order partial derivatives, as it implies that the order of differentiation does not matter for continuous mixed partial derivatives.

For example, if we have a function f(x, y) with continuous second-order partial derivatives, then:

∂²f/∂x∂y = ∂²f/∂y∂x

This equality reduces the number of calculations required when finding all second-order partial derivatives.

Example: Second Derivative in x

Let's revisit our previous example, f(x, y) = x³ + x²y³ - 2y², and compute the second partial derivative with respect to x, denoted as f_xx(x, y) or ∂²f/∂x². We already found the first partial derivative with respect to x:

∂f/∂x = 3x² + 2xy³

Now, we differentiate this expression again with respect to x, treating y as a constant:

f_xx(x, y) = ∂²f/∂x² = ∂/∂x (3x² + 2xy³) = 6x + 2y³

This result gives us the rate of change of the slope in the x-direction, providing further insight into the function's behavior.

Partial derivatives are not merely theoretical constructs; they have a wide range of applications in various fields, including:

  • Optimization: Partial derivatives are crucial in finding local maxima and minima of multivariable functions. These critical points occur where all first-order partial derivatives are zero or undefined. By analyzing the second-order partial derivatives, we can determine whether a critical point corresponds to a local maximum, local minimum, or a saddle point.
  • Tangent Planes: Partial derivatives are used to find the equation of the tangent plane to a surface at a given point. The tangent plane is a linear approximation of the surface near that point, providing a valuable tool for analyzing the surface's local behavior.
  • Directional Derivatives: Partial derivatives are the building blocks for computing directional derivatives, which measure the rate of change of a function in a specific direction. This concept is essential in understanding how a function changes along different paths.
  • Gradient Vector: The gradient vector, formed by the partial derivatives of a function, points in the direction of the function's steepest ascent. This vector is instrumental in optimization algorithms and understanding the function's overall behavior.
  • Related Rates: Partial derivatives are used to solve related rates problems involving functions of multiple variables. These problems involve finding the rate of change of one quantity in terms of the rates of change of other related quantities.
  • Physics and Engineering: Partial derivatives are indispensable in physics and engineering, appearing in equations governing heat flow, fluid dynamics, electromagnetism, and many other phenomena. They allow us to model and analyze complex systems involving multiple interacting variables.
  • Economics: In economics, partial derivatives are used to analyze marginal cost, marginal revenue, and other economic concepts. They help economists understand how changes in one variable affect other economic variables.

Let's consider an example that demonstrates the application of partial derivatives. Suppose we have the function:

f(x, y) = x³ + x²y³ - 2y²

We want to analyze the following statements regarding its partial derivatives:

Statement I: The second derivative in x of the function f(x, y) is f_xx(x, y) = 6x + 2y³.

To verify this statement, we first find the first partial derivative with respect to x:

∂f/∂x = 3x² + 2xy³

Next, we differentiate this expression again with respect to x:

f_xx(x, y) = ∂²f/∂x² = ∂/∂x (3x² + 2xy³) = 6x + 2y³

Therefore, Statement I is true.

This example illustrates how we can use partial derivatives to analyze and verify statements about functions of multiple variables.

Partial derivatives are a fundamental tool in multivariable calculus, providing a means to analyze the rate of change of functions with respect to individual variables. Their applications span various fields, making them an indispensable concept for students and professionals alike. By understanding the definition, computation, and interpretation of partial derivatives, you can unlock a deeper understanding of multivariable functions and their behavior. From optimization to physics and economics, partial derivatives empower us to model and analyze complex systems with greater precision and insight.

Throughout this article, we have explored the intricacies of partial derivatives, from their formal definition to their practical applications. We have seen how they allow us to isolate the influence of each variable on a function's output, providing a nuanced understanding of its behavior. We have also examined higher-order partial derivatives and Clairaut's Theorem, which simplifies their computation. Finally, we have highlighted the diverse applications of partial derivatives in various fields, demonstrating their importance in both theoretical and practical contexts. With a solid grasp of partial derivatives, you are well-equipped to tackle a wide range of problems in multivariable calculus and beyond.