MATH 3090 Problem Set #5
Directions: Work each problem on clean paper. Your solutions should be clear, error-free, cogent, and didactic. You should assume that you are writing so that a C calculus student can understand how to solve the problem by reading your solution. To that end, writing short explanations and notes are a good.
- Draw a contour map for for . Then sketch the shape of the solid base on what you found. You can check your drawing against calcplot3D or geogebra to see how close you came. Make sure you understand how the contour map leads to the 3-dimensional sketch.
- List all regions of discontinuity for the function in the domain region given by
- Sketch the solid produced by . You may find online graphing tools such as https://c3d.libretexts.org/CalcPlot3D/index.html or https://www.geogebra.org/3d?lang=en However, when test time comes, you should be able to reason out the graph of a function of this type. So, make certain that you understand why it makes the shape that it does.
- Sketch the solid produced by . You should be able to draw paraboloids of this level of difficulty on the test.
- Suppose that . Find , .
- Suppose that . Find all points in the xy-plane where .
- Suppose that . Find .
- . Find .
- In the previous problem, find an ordered pair, , with , where .
- Use implicit differentiation to find of the surface at the point .
- Use implicit differentiation to find all points on the surface given by where it holds that .
- Suppose that . Find .
- , , Find .
- Suppose that , where k is a constant. Find .
- , , Find and .
- . Find and .
- Find the equation of the tangent plane to the surface given by at the point .
- Find the equation of the tangent plane to the surface given by at the point .
- Find the directional derivative of in the direction of .
- Find the directional derivative of in the direction of the vector .
