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Chapter 10: Problem 15

Sketch the appropriate traces, and then sketch and identify the surface. $$z=\cos x$$

Short Answer

Expert verified
The surface is a cosine wave, extending indefinitely along the y-axis. The wave oscillates between -1 and 1, peaking at points where \(x = 2\pi n\) and reaching its valleys at points where \(x = \pi + 2\pi n\), with n is any integer.

Step by step solution

01

Graphing the function in 2D

The first step involves making a 2D graph of the function \(z = \cos x\). In a 2D setting, this function is a periodic function that oscillates between -1 and 1, with a period of \(2\pi\). Draw the x-axis and plot the values of \(\cos x\), marking the points where x is a multiple of \(\pi/2\) as these are key points where the function reaches its maximum, minimum, and zero-crossing points.
02

Extending to 3D

To extend the function into three dimensions, imagine the 2D graph of \(\cos x\) drawn in the x-z plane. Now, this graph will be duplicated along the y-axis. Given \(z = \cos x\) is not dependent on y, the graph will be the same for all values of y. When you sketch this, you will see a series of wave-like surfaces extending in the positive and negative y direction.
03

Identify the surface

The 3D surface for the function \(z = \cos x\) represents a wave-like form called a cosine wave, extending indefinitely along the y-axis. The peaks of this wave are located at points where \(x = 2\pi n\), with n is any integer, and the valleys occur at points where \(x = \pi + 2\pi n\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function is a fundamental trigonometric function represented by the symbol \ \( \cos(\theta) \ \). It is part of the family of sine and tangent functions used commonly in mathematics, especially within geometry and calculus. \It's important to understand a few key characteristics of the cosine function: \
    \
  • Its values range from -1 to 1. This means on a graph, the cosine function oscillates between these two points creating a wave-like shape.
    • \
  • Cosine belongs to a group of functions known as even functions. This simply implies that \(\cos(-x) = \cos(x)\), denoting symmetry about the y-axis.
    • \
  • Key points on the graph occur at multiples of \(\pi/2\), where the function reaches its peaks and troughs. \(\cos(0) = 1\ ext{ and } \cos(\pi) = -1\). Between these points, it crosses the axis at \(x = \pi/2\) and \(x = 3\pi/2\).
    • \
\Understanding the regular, wave-like behavior of the cosine function is essential when it comes to sketching 3D graphs, as you'll begin to see repetitive patterns extend through additional dimensions.
Periodic Function
Periodic functions are a category of functions that repeat their values over regular intervals or periods. The cosine function is an outstanding example of a periodic function, and understanding its nature helps with sketching in 3D. \Some critical features of these functions include: \
    \
  • Periodicity: For the cosine function \(z=\cos x\), the period is \(2\pi\). This implies that every \(2\pi\) units along the x-axis, the pattern repeats, creating cyclical waves.
    • \
  • Regularity: Because the function returns to its starting value after one period, new cycles emerge seamlessly in both 2D and 3D visualizations.
    • \
  • Applications: These functions are often found in natural phenomena such as sound waves, light waves, and ocean waves, embodying repetitive cycles.
    • \
\Recognizing that the cosine function is periodic ultimately leads to insights about how it appears in multi-dimensional graphs. It demonstrates consistent intervals and patterns that help when conceptualizing complex sketches like 3D surfaces.
Wave-Like Surfaces
Wave-like surfaces in mathematics are fascinating structures often described using periodic functions like cosine. When extending a 2D cosine graph into a 3D space, wave-like surfaces are beautifully manifested. \Here's how these structures take shape and behave: \
    \
  • The Basic Sketch: In a 2D graph, \(z = \cos x\) displays a series of undulating curves. By adding a third dimension, primarily the y-axis, these curves stretch infinitely, resulting in comprehensive wave-like formations.
    • \
  • Independence from the y-axis: For functions such as \(z = \cos x\), the value of \(z\) doesn鈥檛 change with variations in \(y\). Hence, any slice parallel to the x-z plane will show identical cosine waves.
    • \
  • Visualizing the Surface: Think of an arrangement of repeated sine waves, stacked alongside each other in the y-direction, producing a rippling effect through space.
    • \
  • Peaks and Valleys: As these waves form, peaks align where \(x = 2\pi n\) and valleys appear at \(x = \pi + 2\pi n\), creating symmetrical high and low points across the structure.
    • \
\Sketching wave-like surfaces enhances one's ability to comprehend complex, three-dimensional forms. The repetitive and predictable nature of these surfaces allows for easier identification and understanding in both theoretical and applied contexts.

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