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Chapter 12: Problem 37

Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. $$z=3 \cos (2 x+y) ;[-2,2] \times[-2,2]$$

Short Answer

Expert verified
Question: Sketch the level curves for the function z = 3cos(2x + y) in the window [-2, 2] 脳 [-2, 2] and label at least two level curves with their corresponding z-values. Answer: To graph the level curves for z = 3cos(2x + y), we first set z equal to a constant, say z = c, and solve for y to obtain y = arccos(c/3) - 2x. We can choose different z-values between the amplitude range (-3 to 3), such as -3, -1.5, 0, 1.5, and 3, and plot their respective level curves in the given window. The level curves are the set of points (x, y) that satisfy the equations obtained for each z-value. For labeling, we can label the level curve for z = 1.5 with "1.5" and the level curve for z = -1.5 with "-1.5".

Step by step solution

01

Understand the Function

We are given a function z = 3cos(2x + y), and we need to find its level curves. The function describes a three-dimensional surface that is periodic in both the x and y directions. The level curves are the intersections of this surface with planes parallel to the xy-plane.
02

Find the Level Curves

To find the level curves, we can set z equal to a constant and solve for the curve that results. We will do this for various values of z. $$z = c = 3\cos(2x+y)$$
03

Solve for y

We will solve for y in terms of x and c. $$y= \arccos(\frac{c}{3}) - 2x$$
04

Choose z-Values to Plot Level Curves

Choose various z-values (c) between the amplitude range of the function, which is -3 to 3. In this case, we can pick -3, -1.5, 0, 1.5, and 3.
05

Plot the Level Curves

Plot the level curves for the chosen z-values within the given window, which is between -2 and 2 for both x and y. The level curves will be the set of points (x, y) that satisfy the equations obtained in Step 3 for each z-value.
06

Labeling the Level Curves

Label at least two level curves with their corresponding z-values. For example, we can label the level curve for z=1.5 with "1.5" and the level curve for z=-1.5 with "-1.5". This way, we have successfully graphed the level curves for the given multivariable function in the given window and labeled them with their z-values.

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