Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 17-22, 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 vertical translations of the sine wave \( y = \sin x \).

Step 1: Understand the Level Curve

A level curve for a given function is a curve along which the function has a constant value. Here, we will consider the function \( z = y - \sin x \) and set \( z = k \) for different values of \( k \).

Step 2: Express y in terms of x and k

Rearrange the equation \( z = y - \sin x \) to express \( y \) explicitly in terms of \( x \) and the constant \( k \). This gives \( y = k + \sin x \).

Step 3: Level Curve for k = -2

Substitute \( k = -2 \) into the equation \( y = k + \sin x \). This gives \( y = -2 + \sin x \), which is the level curve for \( k = -2 \).

Step 4: Level Curve for k = -1

Substitute \( k = -1 \) into the equation \( y = k + \sin x \). This gives \( y = -1 + \sin x \), which is the level curve for \( k = -1 \).

Step 5: Level Curve for k = 0

Substitute \( k = 0 \) into the equation \( y = k + \sin x \). This gives \( y = 0 + \sin x \) or simply \( y = \sin x \), which is the level curve for \( k = 0 \).

Step 6: Level Curve for k = 1

Substitute \( k = 1 \) into the equation \( y = k + \sin x \). This gives \( y = 1 + \sin x \), which is the level curve for \( k = 1 \).

Step 7: Level Curve for k = 2

Substitute \( k = 2 \) into the equation \( y = k + \sin x \). This gives \( y = 2 + \sin x \), which is the level curve for \( k = 2 \).

Step 8: Sketch the Level Curves

For each value of \( k \), the level curve is a sine wave shifted vertically in the y-direction. For \( k=-2 \), sketch \( y = -2 + \sin x \). Repeat similarly for \( k=-1, 0, 1, \) and \( 2 \). Each curve is a vertical translation of the sine wave \( y = \sin x \).

Key Concepts

Level Curve Analysis

In mathematical analysis, **level curves** are an essential concept that helps us visualize functions of two variables. A level curve, sometimes called a contour line, is a curve along which a function of two variables remains constant. In simpler terms, if you have a function like \( z = y - \sin x \), then a level curve for a specific value of \( z = k \) is the set of all points \((x, y)\) where the function equals the constant \( k \).

To better understand level curves, let's consider a typical scenario where the function is given, and we need to sketch these curves for various values of \( k \). For the function \( z = y - \sin x \), the level curves are defined as \( y = k + \sin x \) for different values of \( k \).

• When \( k = -2 \), the level curve is \( y = -2 + \sin x \).
• When \( k = -1 \), the level curve is \( y = -1 + \sin x \).
• And similarly for other values of \( k \), such as 0, 1, and 2.

Each curve reveals how the function \( z = y - \sin x \) behaves for those constant values, making level curves a powerful tool in visualizing and understanding the nature of functions of two variables.

Function Graphing

Function graphing is a crucial method for visually exploring mathematical functions. By graphing a function, we can identify patterns, symmetries, and behaviors that might not be immediately obvious from a simple equation.

In our example, the function \( z = y - \sin x \) is of interest. When graphing the level curves for various values of \( k \), such as -2, -1, 0, 1, and 2, it's important to understand the impact each \( k \) value has.

These values affect the vertical position of the sine wave. The graph of each level curve is a sine wave shifted vertically depending on the value of \( k \):

• For \( k = 0 \), the graph is simply \( y = \sin x \), a standard sine wave.
• If \( k = 2 \), the curve shifts up by two units, producing \( y = 2 + \sin x \).

By sketching these graphs, you can visualize how the structure of the waves changes with different \( k \) values. Overall, graphing functions reveals essential properties and offers insights that are not immediately apparent from the formula alone.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics. They describe relationships within triangles and, importantly, periodic phenomena such as waves. In our exercise, the sine function \( \sin x \) plays a key role.

The function \( z = y - \sin x \) means that for any curve \( z = k \), the y-value is dependent on the sine of \( x \). This dependency creates familiar wave patterns. The sine function is periodic with a period of \( 2\pi \), which means it repeats every \( 2\pi \) units.

• The basic sine wave, \( y = \sin x \), oscillates between -1 and 1.
• By adding a constant \( k \), as in \( y = k + \sin x \), the whole wave is shifted vertically by \( k \) units.
• Regardless of \( k \), the wave retains its basic shape and periodicity.

Understanding trigonometric functions like \( \sin x \) is crucial for interpreting wave-like phenomena and solving a wide range of mathematical problems. They provide a bridge between abstract mathematical concepts and real-world applications, such as sound and light waves.