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Chapter 12: Problem 33

Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. $$z=\sqrt{x^{2}+4 y^{2}} ;[-8,8] \times[-8,8]$$

Short Answer

Expert verified
Question: Graph the level curves for the function $$z = \sqrt{x^2+4y^2}$$ within the window $$[-8, 8] \times [-8, 8]$$ and label at least two of them with their z-values. Answer: After plotting the level curves for $$z = 4$$ and $$z = 8$$, we find that these level curves are ellipses centered at (0,0) with different radii. The $$z = 4$$ level curve has a major radius of 2 and a minor radius of 1, while the $$z = 8$$ level curve has a major radius of 4 and a minor radius of 2.

Step by step solution

01

Write the function in terms of x and y with z as a constant

Firstly, we want to rewrite the function in terms of x and y while z is constant (c). So, we can write the equation as follows: $$c = \sqrt{x^2+4y^2}$$
02

Find the equation of level curve

Next, we need to find the expression of the level curve. To do this, we square both sides of the equation and solve for c: $$c^2=x^2+4y^2$$
03

Plot and label level curves using the window $$[-8,8]\times[-8,8]$$

Now, we will plot some level curves using the given window. Let's start with two z-values, say $$c_1=4$$ and $$c_2=8$$. For $$c_1=4$$, the level curve is: $$16=x^2+4y^2$$ This is an ellipse with center at (0,0), major axis on the y-axis, and a major radius of 2 and a minor radius of 1. For $$c_2=8$$, the level curve is: $$64=x^2+4y^2$$ This is another ellipse with center at (0,0), major axis on the y-axis, and a major radius of 4 and a minor radius of 2. After plotting these two level curves in the given window, don't forget to label two level curves with their z-values (4 and 8). If you want to graph additional level curves, you can choose different values of c and follow the same process.

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