Find the linearization Upper L (x comma y )of the function f (x comma y )at Upper P 0. Then find an upper bound for the magnitud
e StartAbsoluteValue Upper E EndAbsoluteValue of the error in the approximation f (x comma y )almost equalsUpper L (x comma y )over the rectangle R. f (x comma y )equals3 ln x plus 5 ln y at Upper P 0 (1 comma 1 )R: StartAbsoluteValue x minus 1 EndAbsoluteValue less than or equals 0.4, StartAbsoluteValue y minus 1 EndAbsoluteValue less than or equals 0.3
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<h3>Answer: Bottom right corner (ie southeast corner)</h3>
This 3D solid is a strange sideways bowl shape. Each cross section is a ring to show the empty space.
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Explanation:
Check out the diagram below. The graph was created with GeoGebra. We have y = x^2 in red and x = y^2 in blue.
The gray region is the region between the two curves. We spin this gray region around the horizontal green line y = 1 to generate the answer mentioned above.
Note how (1,1) is a fixed point that does not move as this is on the line y = 1. Every other point moves to sweep through 3D space to create the solid figure. One way you can think of it is to think of propeller blades. Or you can think of a revolving door (the door is "flat" so to speak, but it sweeps out a 3D solid cylinder).