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Chapter 12: Problem 32

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

Short Answer

Expert verified
Question: Graph the level curves of the function z = 2x - y within the window [-2, 2] x [-2, 2], and label two of them with their z-values. Answer: We graphed three level curves within the given window: y = 2x + 2 (z = -2), y = 2x (z = 0), and y = 2x - 2 (z = 2). We labeled the level curves y = 2x + 2 as z = -2 and y = 2x as z = 0.

Step by step solution

01

Understanding level curves

Level curves are the points in the xy-plane where a function, in this case, z = 2x - y, has a constant height (z-value). To graph some level curves, we'll need to find the equation that represents the level curve for a specific z-value.
02

Find the equation for the level curve

We're given that z = 2x - y. For a constant z-value, let's say z = k, we'll have the equation for the level curve as: k = 2x - y
03

Rearrange the level curve equation

We'll rearrange the level curve equation to make it easier to plot. We'll make y the subject of the formula: y = 2x - k
04

Choose specific z-values to find level curves

To graph a few level curves, we'll need to use specific values of z or k. We'll use k = -2, 0, and 2 for this exercise. You can choose any other k values for more level curves. For k = -2: y = 2x - (-2) <=> y = 2x + 2 For k = 0: y = 2x - 0 <=> y = 2x For k = 2 y = 2x - 2
05

Plot the level curves within the given window

Now we have three level curves, we can plot them within the given window ([-2, 2] x [-2, 2]) in the xy-plane. You can use graphing software or simply draw them on graph paper. Also, label two of them with their z-values to indicate the corresponding height or k value. So, in this case, we'll label the following level curves: - Level curve y = 2x + 2 as z = -2 - Level curve y = 2x as z = 0 That's it! We have successfully graphed and labeled several level curves for the given function.

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