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Chapter 12: Problem 4
Give an equation of the plane with a normal vector \(\mathbf{n}=\langle 1,1,1\rangle\) that passes through the point (1,0,0)
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Describe the set of all points (if any) at which all three planes \(x+3 z=3, y+4 z=6,\) and \(x+y+6 z=9\) intersect.
What point on the plane \(x-y+z=2\) is closest to the point (1,1,1)\(?\)
Find the points (if they exist) at which the following planes and curves intersect. $$y=2 x+1 ; \quad \mathbf{r}(t)=\langle 10 \cos t, 2 \sin t, 1\rangle, \text { for } 0 \leq t \leq 2 \pi$$
a. Consider the function \(w=f(x, y, z)\). List all possible second partial derivatives that could be computed. b. Let \(f(x, y, z)=x^{2} y+2 x z^{2}-3 y^{2} z\) and determine which second partial derivatives are equal. c. How many second partial derivatives does \(p=g(w, x, y, z)\) have?
Find the maximum value of \(x_{1}+x_{2}+x_{3}+x_{4}\) subject to the condition that \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=16\).
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