Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the level curve \(z=k\) for the indicated values of \(k\). \(z=y-\sin x, k=-2,-1,0,1,2\)

Step-by-Step Solution

Answer

The level curves are sine waves shifted vertically by \(k\). Draw them at \(-2, -1, 0, 1, 2\).

Step 1: Understand the Equation

The equation given is for a level curve, which is in the form \(z = y - \sin x\). We're asked to sketch the level curves for different values of \(k\), which means we need to set \(z = k\). Thus, for each \(k\), we have the equation \(k = y - \sin x\). Rearranging gives us \(y = k + \sin x\).

Step 2: Identify the Level Curves for each k

Each \(k\) value gives us a different equation for the level curve. For \(k = -2\), the equation is \(y = -2 + \sin x\). For \(k = -1\), \(y = -1 + \sin x\). For \(k = 0\), \(y = \sin x\). For \(k = 1\), \(y = 1 + \sin x\). And for \(k = 2\), \(y = 2 + \sin x\).

Step 3: Analyze the Equation Forms

Since the level curves are in the form \(y = k + \sin x\), each curve is a vertical shift of the sine wave \(y = \sin x\). The shift corresponds to the specific \(k\) value. Thus, each curve represents the sine wave shifted up or down on the graph.

Step 4: Sketch the Level Curves

Position the sine wave \(y = \sin x\) on your graph as the base reference. For each \(k\) value, shift this wave vertically. For \(k = -2\), shift down by 2 units; for \(k = -1\), shift down by 1 unit; for \(k = 0\), no shift is needed; for \(k = 1\), shift up by 1 unit; and for \(k = 2\), shift up by 2 units. This pattern shows how the level curves are distributed across different vertical levels in the plane.

Key Concepts

Sine Wave Transformation

The sine wave is a smooth, periodic oscillation that is fundamental in trigonometry and the study of waves. When we talk about sine wave transformation, we often refer to changes made to the basic sine function, which is typically \(y = \sin x\). These transformations can include:

• Vertical shifts, which move the wave up or down along the y-axis.
• Horizontal shifts, which move the wave left or right along the x-axis.
• Amplitudes changes, which adjust the height of the wave peaks and troughs.
• Frequency changes, which affect how many waves fit in a given length of the x-axis.

In the context of the given exercise, the sine wave transformation specifically involves vertical shifts resulting from varying the constant \(k\). This impacts where the sine wave appears in relation to the y-axis.

Vertical Shifts

In mathematics, a vertical shift involves moving a graph up or down on a coordinate plane. This is particularly relevant in our exercise where the equation \(y = k + \sin x\) showcases a sine wave vertically shifted by varying values of \(k\). Each different \(k\) results in a unique graph position.To break it down:

• When \(k = -2\), the sine wave shifts down by 2 units.
• When \(k = -1\), it shifts down by 1 unit.
• When \(k = 0\), there's no shift, keeping it centered on the x-axis at zero level.
• When \(k = 1\), it shifts up by 1 unit.
• When \(k = 2\), it moves further up by 2 units.

These shifts indicate that the overall wave maintains its shape and properties, while only its position changes vertically.

Level Curve Sketching

Sketching level curves, particularly for multivariable functions, involves graphically representing curves along which a function attains specific values. In our exercise, these curves are described by the equation \(y = k + \sin x\), where each unique \(k\) leads to a different level curve. To effectively sketch these curves:

• Start with the basic sine curve, \(y = \sin x\).
• For each value of \(k\), shift the sine curve vertically to reach \(y = k + \sin x\).
• Pay attention to maintaining the curve's periodic nature during each shift.
• Spacing between curves on your graph should reflect the magnitude of each \(k\) shift, ensuring clarity in their distinct vertical positions.

These sketches can help visualize how varying parameters affect the graph of a function, making them crucial for understanding multivariable behavior in mathematics.

Multivariable Functions

Multivariable functions involve two or more variables and provide vast opportunities to explore complex phenomena across different dimensions. In our exercise, the multivariable function is expressed as \(z = y - \sin x\), with \(z\) being determined by the other two variables, \(y\) and \(x\).Key characteristics include:

• The function \(z\) depends on changes in both \(y\) and \(x\), offering insights into how small variations affect the larger system.
• Level curves, like those we sketched from this function, reveal additional layers of understanding by showing constant value settings of \(z\).
• Analyzing multivariable functions helps in determining relationships and dependencies between variables.

In this exercise, plotting level curves for specific values of \(z=k\) gives us a powerful tool to explore such relationships practically by observing how the sinusoidal patterns vary with the constant \(k\).