Problem 8
Question
What is the name of the surface defined by the equation \(y=\frac{x^{2}}{4}+\frac{z^{2}}{8} ?\)
Step-by-Step Solution
Verified Answer
Answer: Elliptic paraboloid
1Step 1: Analyze the equation
The given equation is \(y = \frac{x^2}{4} + \frac{z^2}{8}\). Notice that it can be also written as \(\frac{x^2}{4} + \frac{z^2}{8} - y = 0\).
2Step 2: Recognize the surface type
The equation can be rewritten as follows: \(\frac{x^2}{4} + \frac{z^2}{8} - y = 0\). We observe that this expression involves the squares of \(x\) and \(z\) variables, both are positive. Since there are no other terms, we deduce that it is an elliptic paraboloid.
3Step 3: Conclusion
The surface defined by the equation \(y = \frac{x^2}{4} + \frac{z^2}{8}\) is an elliptic paraboloid.
Other exercises in this chapter
Problem 8
Write the differential \(d w\) for the function \(w=f(x, y, z)\)
View solution Problem 8
Consider the function \(f(x, y)=2 x^{2}+y^{2}\) whose graph is a paraboloid (see figure). a. Fill in the table with the values of the directional derivative at
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How many axes (or how many dimensions) are needed to graph the level surfaces of \(w=f(x, y, z) ?\) Explain.
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Use Theorem 7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$d z / d t, \text { where } z=x y^{2}
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