Your task is to produce a report comparing the surfaces of
the paraboloid z=x^2+y^2 and the cone z=sqrt(x^2+y^2).
Your finished product will be a word-processed document including complete sentence explanations, clearly-labeled equations, and graphs in two and three dimensions, all organized into a cohesive discussion addressing the points below. Your audience is your peers. Think of how you would explain these concepts to a student starting multivariable calculus.
Part 1: Traces
Create a contour map of each surface. On each map, include a curve in each of the planes z=0,1,2,3,4, and label each curve with its z-value. What are the similarities between these contour maps? How are they different? What do their differences tell you about the shape of the two surfaces?
Part 2: Directional Derivatives
The point (1,0,1) lies on both of these surfaces. Use directional derivatives to find the direction of greatest increase at this point on each surface, and also find the magnitude of the directional derivative in that direction. In each of your contour maps from Part 1, include a vector located at (x,y) = (1,0) and pointing in the direction of greatest increase. Does the direction of greatest increase for each surface make sense in terms of the contour map? Why or why not? What do the values of the directional derivative in that direction tell you about the shape of the surface?
Part 3: Tangent Planes
The point (0,0,0) lies on both of these surfaces. Find the tangent plane at this point for each surface, if it exists. If a tangent plane exists at this point, explain why the equation you found makes sense. If a tangent plane does not exist at this point, explain why it does not exist. For each surface, plot the surface and the tangent plane (if it exists) using a 3D plotting device such as this one. Include a screenshot of the 3D graph in your report.