Problem 58 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the second partial derivatives \(f_{x x^{\prime}} f_{x y^{\prime}} f_{y y^{\prime}}\) and \(f_{y x}\) at the point. $$ f(x, y)=x^{2} e^{y} $$

Step-by-Step Solution

Answer

The second order partial derivatives are: \(f_{xx} = 2e^{y}\), \(f_{xy}=f_{yx} = 2x e^{y}\), \(f_{yy} = x^{2} e^{y}\)

Step 1: Find the first-order partial derivatives

The first order partial derivatives are found by differentiating the function with respect to one variable while holding the other variable constant. \n- Differentiating with respect to x while holding y constant, we get: \(f_{x} = 2x e^{y}\) \n- Differentiating with respect to y while holding x constant, we get: \(f_{y} = x^{2} e^{y}\)

Step 2: Find the second-order partial derivatives

The second order partial derivatives are found by differentiating the first order partial derivatives again with respect to the other variable. \n- Differentiating \(f_{x}\) with respect to x while holding y constant, we get: \(f_{xx} = 2e^{y}\) \n- Differentiating \(f_{x}\) with respect to y while holding x constant, we get: \(f_{xy} = 2x e^{y}\) \n- Differentiating \(f_{y}\) with respect to y while holding x constant, we get: \(f_{yy} = x^{2} e^{y}\) \n- Differentiating \(f_{y}\) with respect to x while holding y constant, we get: \(f_{yx} = 2x e^{y}\)

Step 3: Evaluate the second-order partial derivatives

The second-order partial derivatives don't have any x or y in the function, except for \(f_{xy}\), \(f_{yy}\), and \(f_{yx}\), so their values will remain the same wherever they are.evaluated.

Key Concepts

Calculus

Calculus is a branch of mathematics that studies continuous change. It helps us understand the behavior of functions and enables us to work out a variety of mathematical problems like those involving rates of change and areas under curves.
Calculus is fundamentally composed of two main concepts:

• **Differentiation**: This involves calculating the derivative of a function, which is a measure of how a function changes as its input changes. It's used for finding slopes of curves and solving optimization problems.


• **Integration**: This involves calculating the integral of a function, which is a measure that aggregates values over an interval, like finding the total area under a curve.

Calculus is indispensable in various fields such as physics, engineering, and economics, where it's used to model and interpret dynamic systems.

Multivariable Functions

Multivariable functions are functions with more than one input variable, forming the base of multivariable calculus. These functions are usually denoted as \(f(x, y, z,...)\).

• These functions are essential in expressing relationships in systems with more than one changing variable.


• The domain of a multivariable function is a subset of \(b{R}^n\), and the range is a subset of \(\bb{R}\).

For example, a function \(f(x, y) = x^2 e^y\) is a multivariable function that takes two inputs, \x\ and \y\, and outputs a single value. To explore their behavior, we use partial derivatives, which involve differentiating these functions with respect to one variable while treating others as constants.

Differentiation

Differentiation is the process of finding the derivative of a function, which indicates how much the function's output changes in response to changes in its input. In the context of multivariable functions, we use:

• **Partial Differentiation**: This is used to find the rate of change of a multivariable function with respect to one variable, keeping the others constant.

To differentiate a function like \(f(x, y) = x^2 e^y\), you would find:

• **Partial derivative with respect to \x\**: \(f_x = \frac{\partial f}{\partial x} = 2x e^y\).


• **Partial derivative with respect to \y\**: \(f_y = \frac{\partial f}{\partial y} = x^2 e^y\).

These derivatives help us understand how changing individual variables affects the function, which is crucial in real-world applications like economics and physics.

Second-order Derivatives

Second-order derivatives are simply the derivatives of the first-order derivatives. In the realm of multivariable functions, calculating these provides deeper insights into the function's curvature and interaction between variables.

• **Second-order partial derivatives**: These include derivatives like \(f_{xx},\) \(f_{yy},\) \(f_{xy},\) and \(f_{yx}\).


• **Mixed partial derivatives**: These derivatives, such as \(f_{xy}\) and \(f_{yx},\) are especially interesting because, under certain conditions, they equal each other. This is known as Clairaut's Theorem.

In the function \(f(x, y) = x^2 e^y\):

• The second-order derivative with respect to \(x\) is \(f_{xx} = 2e^y\).


• With respect to \(y\), the second-order derivative is \(f_{yy} = x^2 e^y\).


• The mixed derivatives are \(f_{xy} = f_{yx} = 2x e^y\), illustrating the symmetry.

These derivatives enhance our ability to understand complex surfaces, which is invaluable in fields requiring precise modeling.