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curvature and surface equation generated by revolving curves

1 Find the point on the graph of y= e^x at which the curvature is the greatest.

2 Write the equation for the surface generated by revolving the curve x^2 - 2y^2 = 1 about the y-axis. Describe the surface

3 The parabola z = y^2, x =0 is rotated around the z-axis. Write a cylindrical-coordinate equation for the surface.

4 Write the equation of the plane that contains P(1,3,5) and the line L: x=4t, y= 6+5t, z=3-2t.

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Please see the step-by-step solution in the attached file.

3. Let P(x, y, z) be a generic point on the surface of revolution. Fix a point Q(0, y1, z) on the parabola with the same z-coordinate as P. Then we have
z = y1^2 (1)
The square of the ...

Solution Summary

The solution includes a detailed explanation on how find the curvature of the curve. It also contains the step-by-step solutions on how to solve the surface equation generated by revolving the curve in both rectangular and cylindrical coordinates.