Question

Consider the function \(z=x / y^{2}.\) a. Compute \(z_{x}\) and \(z_{y}\) b. Sketch the level curves for \(z=1,2,3,\) and 4 c. Move along the horizontal line \(y=1\) in the \(x y\) -plane and describe how the corresponding \(z\) -values change. Explain how this observation is consistent with \(z_{x}\) as computed in part (a). d. Move along the vertical line \(x=1\) in the \(x y\) -plane and describe how the corresponding \(z\) -values change. Explain how this observation is consistent with \(z_{y}\) as computed in part (a).

Short answer

In conclusion, this exercise focused on finding partial derivatives \(z_x\) and \(z_y\) of the function \(z = x / y^2\). We analyzed how the function changes when moving along horizontal and vertical lines in the xy-plane. Additionally, we sketched the level curves for the given values of z. Our analysis revealed that, along the horizontal line \(y=1\), the function \(z\) increases at a constant rate with respect to \(x\). On the other hand, along the vertical line \(x=1\), the function \(z\) decreases as \(y\) increases, consistent with the partial derivative \(z_y\).

Step-by-step solution

Find the partial derivatives #\(z_x\) and \(z_y\)

The function \(z\) is given by \(z = x / y^2\), or \(z = x \cdot y^{-2}\). We compute the following partial derivatives: \(z_x = \frac{\partial z}{\partial x} = y^{-2}\) \(z_y = \frac{\partial z}{\partial y} = -2xy^{-3}\) So, \(z_x = y^{-2}\) \(z_y = -2xy^{-3}\) #b. Sketch the level curves for \(z=1,2,3,\) and 4#:

Calculate curves for the given \(z\) values

We will find equations for level curves for \(z=1,2,3\) and \(4\) by substituting these values into the original function, \(z = x / y^2\): For \(z=1\): \(1 = x / y^2 \Rightarrow x = y^2\) For \(z=2\): \(2 = x / y^2 \Rightarrow x = 2y^2\) For \(z=3\): \(3 = x / y^2 \Rightarrow x = 3y^2\) For \(z=4\): \(4 = x / y^2 \Rightarrow x = 4y^2\)

Sketch the level curves

Plot the equations obtained in the previous step in the \(xy\)-plane: \(z=1 \Rightarrow x = y^2\), a parabola opening to the right. \(z=2 \Rightarrow x = 2y^2\), a parabola opening to the right, wider than the previous one. \(z=3 \Rightarrow x = 3y^2\), a parabola opening to the right, wider than the previous ones. \(z=4 \Rightarrow x = 4y^2\), the widest parabola opening to the right. As we notice, levels are found by parabolas opening to the right. #c. Move along the horizontal line \(y=1\) in the \(x y\) -plane and describe how the corresponding \(z\) -values change. Explain how this observation is consistent with \(z_{x}\) as computed in part (a).#:

Analyze how \(z\) values change along the line \(y=1\)

Since \(y = 1\) on this horizontal line and \(z = x / y^2\), the corresponding \(z\) values will be equal to \(x\) values as we move along this line. Now, let's look at our partial derivative \(z_x\): \(z_x = y^{-2}\), for the line \(y=1\), we get: \(z_x = 1\). This means that, along the line \(y = 1\), the function \(z\) is increasing at a constant rate with respect to \(x\). #d. Move along the vertical line \(x=1\) in the \(x y\) -plane and describe how the corresponding \(z\) -values change. Explain how this observation is consistent with \(z_{y}\) as computed in part (a).#:

Analyze how \(z\) values change along the line \(x=1\)

Since \(x = 1\) on this vertical line and \(z = x / y^2\), we have \(z = 1 / y^2\), which implies that, as \(y\) increases, the corresponding \(z\) values decrease. Now, let's evaluate our partial derivative \(z_y\) along the line \(x=1\): \(z_y = -2xy^{-3}\), and for the line \(x=1\), we get: \(z_y = -2y^{-3}.\) This means that, along the line \(x=1\), the function \(z\) is decreasing as \(y\) increases, which is consistent with the previous observation.

Key concepts

Level Curves

Level curves give us a powerful way to visualize functions of two variables. Imagine slicing through a 3D surface with a horizontal plane at a certain height. The intersection creates a curve on the plane that we call a level curve. In this exercise, we deal with the function \( z = \frac{x}{y^2} \). Each level curve corresponds to a specific value of \( z \), for instance, 1, 2, 3, or 4. By substituting these values back into the function, we get different curves in the \( xy \)-plane: \( x = y^2 \), \( x = 2y^2 \), \( x = 3y^2 \), and \( x = 4y^2 \). These are parabolas opening to the right. Each curve tells us where the given \( z \) value is constant across the \( xy \)-plane.

Function Analysis

Function analysis involves examining the behavior and properties of a mathematical function. For \( z = \frac{x}{y^2} \), it's helpful to break down how \( z \) changes with small changes in \( x \) or \( y \). This is done by calculating partial derivatives: \( z_x \) and \( z_y \). When we differentiate \( z \) with respect to \( x \), thinking that \( y \) is constant, we get \( z_x = y^{-2} \). This measures how \( z \) changes as \( x \) changes. Similarly, differentiating with respect to \( y \) gives us \( z_y = -2xy^{-3} \), indicating how \( z \) changes as \( y \) changes. This derivative is more complex due to the inverse and its negative sign, revealing that increases in \( y \) generally lead to decreases in \( z \).

Rate of Change

The rate of change tells us how fast a function increases or decreases. It is fundamental when measuring how values evolve in a function. For our function, moving along lines such as \( y=1 \) changes how \( z \) behaves. With \( y=1 \), \( z = x \). Here, by directly varying \( x \), \( z \) increases steadily since \( z_x = 1 \). Thus, \( z \) changes at a consistent, linear rate. When exploring \( x=1 \), things differ: \( z = \frac{1}{y^2} \). Here, as \( y \) increases, \( z \) trends downward rapidly, informed by the negative partial derivative \( z_y = -2y^{-3} \). Such insights into the rate of change are crucial for predicting and understanding the behavior of functions.

Parabolas

Parabolas are essential curves that appear frequently during level curve analysis. In our context, the function \( z = \frac{x}{y^2} \) yields level curves like \( x = y^2 \), forming parabolas in the \( xy \)-plane. Each parabola opens to the right, and as \( z \) values increase (for example, from 1 to 4), the width of the parabola increases too. This means that for larger \( z \) values, the set of points \((x, y)\) satisfying the equation becomes broader. Understanding the geometry of parabolas in this way aids in visualizing the function's characteristics and how various level curves relate to each other.