Question

Let \(f(x, y)=3 x^{2}+y^{3}\) a. Compute \(f_{x}\) and \(f_{y}\) b. Evaluate each derivative at (1,3) c. Find the four second partial derivatives of \(f\)

Short answer

Question: Compute the partial derivatives of the function \(f(x, y) = 3x^2 + y^3\) and evaluate them at the point (1, 3). Also, find the second partial derivatives. Answer: The partial derivatives of the function are \(f_x = 6x\) and \(f_y = 3y^2\). When evaluated at the point (1, 3), we get \(f_x = 6\) and \(f_y = 27\). The second partial derivatives are \(f_{xx} = 6\), \(f_{xy} = 0\), \(f_{yx} = 0\), and \(f_{yy} = 6y\).

Step-by-step solution

(a) Computing Partial Derivatives: \(f_x\) and \(f_y\)#

To compute the partial derivative of \(f\) with respect to \(x\), we treat \(y\) as a constant and differentiate \(f\) with respect to \(x\). Similarly, to compute the partial derivative of \(f\) with respect to \(y\), we treat \(x\) as a constant and differentiate \(f\) with respect to \(y\). $$f_x(x, y) = \frac{\partial}{\partial x} (3x^2 + y^3) = 6x$$ $$f_y(x, y) = \frac{\partial}{\partial y} (3x^2 + y^3) = 3y^2$$

(b) Evaluating Partial Derivatives at (1, 3)#

Now that we have computed \(f_x\) and \(f_y\), let's evaluate them at the point (1, 3): $$f_x(1, 3) = 6(1) = 6$$ $$f_y(1, 3) = 3(3^2) = 27$$

(c) Computing Second Partial Derivatives: \(f_{xx}\), \(f_{xy}\), \(f_{yx}\), and \(f_{yy}\)#

We will now compute the second partial derivatives by differentiating the first partial derivatives with respect to the corresponding variables: $$f_{xx}(x, y) = \frac{\partial^2}{\partial x^2} (3x^2 + y^3) = \frac{\partial}{\partial x} (6x) = 6$$ $$f_{xy}(x, y) = \frac{\partial^2}{\partial x \partial y} (3x^2 + y^3) = \frac{\partial}{\partial x} (3y^2) = 0$$ $$f_{yx}(x, y) = \frac{\partial^2}{\partial y \partial x} (3x^2 + y^3) = \frac{\partial}{\partial y} (6x) = 0$$ $$f_{yy}(x, y) = \frac{\partial^2}{\partial y^2} (3x^2 + y^3) = \frac{\partial}{\partial y} (3y^2) = 6y$$ The four second-order partial derivatives are \(f_{xx} = 6\), \(f_{xy} = 0\), \(f_{yx} = 0\), and \(f_{yy} = 6y\).

Key concepts

Multivariable Calculus

Multivariable calculus is an extension of traditional calculus that involves more than one variable. In this field, we deal with functions that have multiple inputs. Unlike single-variable calculus, where we differentiate or integrate with respect to just one variable, multivariable calculus allows us to consider changes in two or more directions.

When working with functions of multiple variables, we often encounter concepts such as partial derivatives. These are crucial for understanding how the function behaves in a multi-dimensional space. For example, if we have a function of two variables, say \( f(x, y) \), we find its partial derivatives with respect to each variable to analyze how the function changes as each input varies separately.

Multivariable calculus is essential in various fields like physics, engineering, and economics, where processes depend on several factors. Mastery of this area can lead to a deeper understanding of systems influenced by multiple dimensions.

Second-Order Derivatives

Second-order derivatives involve taking the derivative of a derivative. In the context of functions of two variables, this means we first find the partial derivative of a function and then differentiate that result with respect to one of the variables again.

In our example with the function \( f(x, y) = 3x^2 + y^3 \), we computed the first-order partial derivatives \( f_x \) and \( f_y \), which represent how the function changes with a small change in \( x \) or \( y \), respectively.

• \( f_{xx} \) is the second derivative with respect to \( x \), meaning we differentiate \( f_x \) again with respect to \( x \).
• \( f_{yy} \) is the second derivative with respect to \( y \), meaning we differentiate \( f_y \) again with respect to \( y \).
• Cross derivatives like \( f_{xy} \) and \( f_{yx} \) provide information about how the function changes when both \( x \) and \( y \) vary.
  In many cases, these mixed derivatives are equal, meaning \( f_{xy} = f_{yx} \).

Second-order derivatives help reveal properties such as concavity or convexity and are particularly useful in optimizing functions subject to constraints.

Functions of Two Variables

Functions of two variables, like \( f(x, y) = 3x^2 + y^3 \), are a cornerstone of multivariable calculus. These functions take in two inputs and provide a single output.

Understanding these functions involves exploring how changes in one or both of the input variables affect the function's output. To do this, we use partial derivatives to understand the rate of change in multiple directions.

• A function like \( f(x, y) \) can be visualized as a surface in three-dimensional space, where \( x \) and \( y \) are horizontal axes and \( f(x, y) \) is the vertical axis.
• Exploring the function involves examining cross-sections along the \( x \) and \( y \) planes, understanding how it behaves as you move in these directions.

With functions of two variables, we can model complex situations where multiple factors influence outcomes, making them widely applicable in realistic scenarios.