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Chapter 12: Problem 34

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

Short Answer

Expert verified
Question: Graph several level curves of the function $$z = e^{-x^2 - 2y^2}$$ within the window $$[-2, 2] \times [-2, 2]$$. Label at least two level curves with their respective z-values. Answer: We graphed the level curves for z-values $$0.5$$ and $$0.25$$. The level curve for z = $$0.5$$ is given by the equation $$y^2 = \frac{1}{2}(-\ln(0.5) - x^2)$$, and the level curve for z = $$0.25$$ is given by the equation $$y^2 = \frac{1}{2}(-\ln(0.25) - x^2)$$. These level curves are graphed within the window $$[-2, 2] \times [-2, 2]$$.

Step by step solution

01

Find the equation for the level curves

A level curve is a curve of constant z value on a graph. To find the level curves for the given function, we set z equal to a constant, which we will call $$c$$: $$c = e^{-x^2 - 2y^2}$$
02

Solve for y

Next, we will solve this equation for $$y$$, which will give us a series of contour lines for different c values: $$y^2 = \frac{1}{2}(-\ln(c) - x^2)$$
03

Choose level curves by varying c

We can obtain different level curves by selecting different values for c. Let's take c = $$0.5$$ and c = $$0.25$$: For c = $$0.5$$: $$y^2 = \frac{1}{2}(-\ln(0.5) - x^2)$$ For c = $$0.25$$: $$y^2 = \frac{1}{2}(-\ln(0.25) - x^2)$$
04

Graph the level curves in the [-2, 2] 脳 [-2, 2] window

Now, we graph the level curves determined in Step 3 within the given window: For c = $$0.5$$: $$y^2 = \frac{1}{2}(-\ln(0.5) - x^2)$$, which is labeled with a z-value of $$0.5$$ For c = $$0.25$$: $$y^2 = \frac{1}{2}(-\ln(0.25) - x^2)$$, which is labeled with a z-value of $$0.25$$ These are our level curves for the given function within the specified window. We have successfully graphed the level curves and labeled them with their z-values.

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