# Week 10 small class notes

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Week 10 Small Class Topics: Sketching functions of two variables Learning Objectives Plot points on the x - y - z axes. Given a function of two variables sketch level curves by setting z to be a constant. Given a relation between three variables, sketch traces by setting one variable to be a constant. A. Problems and takeaways (Plotting in 3D): 1. Fold a lose piece of paper to represent the x - y - z axes. (Note: Pictorial instructions are at the end of this document). Draw lines in the creases to represent the x axis, y axis, and z axis. 2. Plot the points (1 , 0 , 0), (0 , 1 , 0), and (0 , 0 , 1), marking them with dots. Once teams are done with those points have them plot the additional points (2 , 2 , 0), (2 , 0 , 2), and (0 , 2 , 2). 3. Using a pen or other small object represent the point (3 , 2 , 1) in space, for example, by holding the base of the pen at the origin and the writing end at the point. B. Problems and takeaways (Sketching level curves and traces): 1. Consider z = x 2 + y 2 . Set z = 4 and plot the resulting equation in 2D. Do the same with z = 1, z = 0, and z = 1. 2. Definition : A level curve is 2D horizontal slice of a larger 3D surface. We find these by setting z equal to a constant. 3. Takeaway : We can get a good sense of what a function or surface in 3D looks like by plotting level curves. 4. So far, we know our surface doesn't have points for z < 0, is a single point at z = 0 and is a tower of increasingly larger circles for z > 0. How do the circles change? Is our surface (a) a bowl (b) a cone (c) a wormhole? 5. To get a sense of which it is, let's plot 2D curves while fixing values of x and y this time. Sketch when y = 0, x = 0, x = 1, and y = 2. 6. Definition : A trace is just like a level curve only we can set any variable equal to a constant. 1
Pictoral folding instructions : open z x y 2