Problem 14 - Active Calculus Multivariable

Question

Let \(f(x, y)=8-x^{2}-y^{2}\) and \(g(x, y)=8-x^{2}+4 x y-y^{2}\). a. Determine \(f_{x}, f_{y}, f_{x x}, f_{y y}, f_{x y},\) and \(f_{y x}\) b. Evaluate each of the partial derivatives in (a) at the point (0,0) . c. What do the values in (b) suggest about the behavior of \(f\) near (0,0)\(?\) Plot a graph of \(f\) and compare what you see visually to what the values suggest. d. Determine \(g_{x}, g_{y}, g_{x x}, g_{y y}, g_{x y},\) and \(g_{y x}\) e. Evaluate each of the partial derivatives in (d) at the point (0,0) . f. What do the values in (e) suggest about the behavior of \(g\) near (0,0)\(?\) Plot a graph of \(g\) and compare what you see visually to what the values suggest. g. What do the functions \(f\) and \(g\) have in common at (0,0)\(?\) What is different? What do your observations tell you regarding the importance of a certain second-order partial derivative?

Step-by-Step Solution

Answer

In summary, both functions \(f\) and \(g\) have zero values for their first-order partial derivatives and negative values for their pure second-order partial derivatives at the point (0,0), indicating no increasing or decreasing behavior and concavity in both x and y directions. However, function g has non-zero mixed second-order partial derivatives, which suggests a saddle-shaped surface near (0,0), whereas function f has a purely parabolic behavior with zero mixed second-order partial derivatives. This highlights the importance of considering mixed second-order partial derivatives when analyzing the behavior of functions near a point.

Step 1: Calculate first-order partial derivatives of \(f(x, y)\)

\[ f_{x}(x, y) = \frac{\partial{(8 - x^2 - y^2)}}{\partial{x}} = -2x \] \[ f_{y}(x, y) = \frac{\partial{(8 - x^2 - y^2)}}{\partial{y}} = -2y \]

Step 2: Calculate second-order partial derivatives of \(f(x, y)\)

\[ f_{xx}(x, y) = \frac{\partial{f_{x}(x, y)}}{\partial{x}} = \frac{\partial{(-2x)}}{\partial{x}} = -2 \] \[ f_{yy}(x, y) = \frac{\partial{f_{y}(x, y)}}{\partial{y}} = \frac{\partial{(-2y)}}{\partial{y}} = -2 \] \[ f_{xy}(x, y) = \frac{\partial{f_{x}(x, y)}}{\partial{y}} = \frac{\partial{(-2x)}}{\partial{y}} = 0 \] \[ f_{yx}(x, y) = \frac{\partial{f_{y}(x, y)}}{\partial{x}} = \frac{\partial{(-2y)}}{\partial{x}} = 0 \] b. Evaluate each of the partial derivatives in (a) at the point (0,0)

Step 3: Evaluate the partial derivatives of \(f(x, y)\) at point (0, 0)

To evaluate the partial derivatives at the point (0, 0), plug in x=0 and y=0 into the partial derivatives found in part (a): \[ f_{x}(0, 0) = -2(0) = 0 \] \[ f_{y}(0, 0) = -2(0) = 0 \] \[ f_{xx}(0, 0) = -2 \] \[ f_{yy}(0, 0) = -2 \] \[ f_{xy}(0, 0) = 0 \] \[ f_{yx}(0, 0) = 0 \] c. What do the values in (b) suggest about the behavior of \(f\) near (0,0)\(?\) Plot a graph of \(f\) and compare what you see visually to what the values suggest. Since both \(f_x(0,0)\) and \(f_y(0,0)\) are 0, there is neither increasing nor decreasing behavior of the function in either the x or y direction at the point (0,0). The second-order partial derivatives \(f_{xx}(0,0)\) and \(f_{yy}(0,0)\) are both negative, which indicates that the function shows concave down behavior (like an inverted bowl) along both x and y axes near (0,0). The graph of the function f would show a parabolic surface with downward curvature. d. Determine \(g_x, g_y, g_{xx}, g_{yy}, g_{xy},\) and \(g_{yx}\) Let's find the first and second order partial derivatives of the function g(x, y).

Step 4: Calculate first-order partial derivatives of \(g(x, y)\)

\[ g_{x}(x, y) = \frac{\partial{(8 - x^2 + 4xy - y^2)}}{\partial{x}} = -2x + 4y \] \[ g_{y}(x, y) = \frac{\partial{(8 - x^2 + 4xy - y^2)}}{\partial{y}} = 4x - 2y \]

Step 5: Calculate second-order partial derivatives of \(g(x, y)\)

\[ g_{xx}(x, y) = \frac{\partial{g_{x}(x, y)}}{\partial{x}} = \frac{\partial{(-2x + 4y)}}{\partial{x}} = -2 \] \[ g_{yy}(x, y) = \frac{\partial{g_{y}(x, y)}}{\partial{y}} = \frac{\partial{(4x - 2y)}}{\partial{y}} = -2 \] \[ g_{xy}(x, y) = \frac{\partial{g_{x}(x, y)}}{\partial{y}} = \frac{\partial{(-2x + 4y)}}{\partial{y}} = 4 \] \[ g_{yx}(x, y) = \frac{\partial{g_{y}(x, y)}}{\partial{x}} = \frac{\partial{(4x - 2y)}}{\partial{x}} = 4 \] e. Evaluate each of the partial derivatives in (d) at the point (0,0)

Step 6: Evaluate the partial derivatives of \(g(x, y)\) at point (0, 0)

To evaluate the partial derivatives at the point (0, 0), plug in x=0 and y=0 into the partial derivatives found in part (d): \[ g_{x}(0, 0) = -2(0) + 4(0) = 0 \] \[ g_{y}(0, 0) = 4(0) - 2(0) = 0 \] \[ g_{xx}(0, 0) = -2 \] \[ g_{yy}(0, 0) = -2 \] \[ g_{xy}(0, 0) = 4 \] \[ g_{yx}(0, 0) = 4 \] f. What do the values in (e) suggest about the behavior of \(g\) near (0,0)\(?\) Plot a graph of \(g\) and compare what you see visually to what the values suggest. Unlike the function f, for function g, it has non-zero mixed partial derivatives \(g_{xy}\) and \(g_{yx}\). The presence of these mixed second-order partial derivatives suggests that the function g has a saddle behavior near (0,0), i.e., it has concave and convex behavior simultaneously in different directions. The graph of the function g would visually show a saddle-shaped surface. g. What do the functions \(f\) and \(g\) have in common at (0,0)\(?\) What is different? What do your observations tell you regarding the importance of certain second-order partial derivatives? At the point (0,0), both functions \(f\) and \(g\) have zero values for their first-order partial derivatives (\(f_x, f_y, g_x,\) and \(g_y\)), as well as negative values for their pure second-order partial derivatives (\(f_{xx}, f_{yy}, g_{xx},\) and \(g_{yy}\)). This indicates that both functions don't have increasing or decreasing behavior and are concave down in both the x and y directions. However, they differ in their mixed second-order partial derivatives (\(f_{xy}, f_{yx}, g_{xy}\) and \(g_{yx}\)). The function f has zero mixed second-order partial derivatives, which indicates a purely parabolic (bowl-shaped) behavior. On the other hand, the function g has non-zero mixed second-order partial derivatives, which indicates a saddle-shaped surface. This observation tells us that the mixed second-order partial derivatives can provide useful information about the presence of saddle behavior, which cannot be determined from the first-order or pure second-order partial derivatives alone.

Key Concepts

Partial Derivatives

In multivariable calculus, partial derivatives represent how a function changes as one variable changes while keeping others constant. For a function with two variables, like our function \( f(x, y) = 8 - x^2 - y^2 \), the partial derivative with respect to \( x \) is denoted as \( f_x \) and with respect to \( y \) as \( f_y \).

• Representation: The partial derivative \( f_x(x, y) \) indicates the rate of change of the function as \( x \) changes with \( y \) held constant.
• Illustration: Similarly, \( f_y(x, y) \) represents the rate of change of the function as \( y \) changes with \( x \) held constant.

For our function \( f(x, y) = 8 - x^2 - y^2 \), the derivatives are simple because each component that depends on the variable being differentiated is separate and independent of the other. So, we find: - \( f_x(x, y) = -2x \)- \( f_y(x, y) = -2y \). When plugging \( x = 0 \) and \( y = 0 \) into these derivatives, both become zero, indicating that there is no immediate incline or decline in any direction at point (0,0).

Second-Order Derivatives

Second-order derivatives in multivariable calculus provide deeper insights into the function's curvature. They examine how the rate of change (given by the first derivative) itself changes. For a function \( f(x,y) \), the pure second-order derivatives are:

• \( f_{xx} \): the rate at which \( f_x \) changes with changes in \( x \).
• \( f_{yy} \): the rate at which \( f_y \) changes with changes in \( y \).

Mixed second-order derivatives like \( f_{xy} \) and \( f_{yx} \) show how the rate of change in one direction is affected by changes in the other direction:

• \( f_{xy} \): changes in \( f_x \) with changes in \( y \).
• \( f_{yx} \): changes in \( f_y \) with changes in \( x \).

For our function, these derivatives are clearly determined:- \( f_{xx} = -2 \), showing consistent concave down curvature in the \( x \)-direction.- \( f_{yy} = -2 \), showing consistent concave down curvature in the \( y \)-direction.- \( f_{xy} = f_{yx} = 0 \), indicating no interaction between the directions regarding their rates of change. At point (0,0), the behavior reflected by these derivatives confirms a symmetric concave down curvature.

Concavity in Functions

Concavity, or how a function curves, is essential in understanding how maxima, minima, or flat regions occur. It's largely determined by the second-order partial derivatives:- A function is concave up (like a bowl) if the second-order partial derivatives are positive.- A function is concave down (like a dome) if they are negative.For \( f(x, y) = 8 - x^2 - y^2 \), both \( f_{xx} \) and \( f_{yy} \) are -2. This negative value indicates that the function is concave down along both axes. It resembles an inverted bowl shape and suggests the potential for a maximum point at (0,0) for this region.With function \( g(x, y) = 8 - x^2 + 4xy - y^2 \), \( g_{xx} \) and \( g_{yy} \) are also -2, suggesting a similar concave down nature. However, the presence of non-zero mixed derivatives \( g_{xy} = 4 \) and \( g_{yx} = 4 \) introduces a twist or saddle shape, altering the overall curvature experience.

Saddle Points

Saddle points are intriguing features in multivariable functions. They occur where a function behaves like it's increasing in one direction and decreasing in another, forming a surface akin to a saddle on a horse.The key to identifying saddle points lies in the mixed second-order derivatives. For \( g(x, y) = 8 - x^2 + 4xy - y^2 \):- Both \( g_{xy} \) and \( g_{yx} \) are non-zero, suggesting mixed behavior between the \( x \) and \( y \) directions.- The presence of this "cross curvature" indicates a saddle structure rather than a maximum or minimum.The saddle point at (0,0) suggests that while changes in one direction tend to make the function larger, simultaneous changes in another direction make it smaller. Unlike \( f(x,y) \), which only shows a bowl or dome-like structure, \( g(x,y) \) exhibits this complex saddle shape at the point. Understanding saddle points is crucial for visualizing the topology of functions and their implications in real-world systems or optimizations.