Lab05 - Partial derivatives - part 2/2

[Your name(s) here...]

We’ve been working with the function defined (in function syntax) below, and evaluating partial derivatives at (x,y)=(π/2,1).

z[x_,y_]:=y Cos[x (y^3)]

Your next step will be to make a contour plot of this function, and then estimate some of the second derivatives of the function.  Here, for you reference, is the picture of the heat-seeking kitten contour plot, which will serve as an example of some of the things you can do to complete this lab:

Problem 5 4 pts

Using your single variable plots (Problem 2, from part 1 of this lab), estimate visually whether these second derivatives (related to the curvature of your graph) are positive, negative, or zero:

(Subscript[f, xx])(π/2,1)?

(Subscript[f, xx])(1,1)?

(Subscript[f, yy])(π/2,1)? 

(Subscript[f, yy])(π/2, 1/2)?

Problem 6 4 pts

Make a contour plot of z(x,y).  See the kitten example for how to place a dot on your contour plot.  Include a dot at the coordinate (x,y)=(π/2, 1).

Using your contour plot, estimate visually whether these second derivatives are positive, negative, or zero:

(Subscript[f, xy])(π/2,1)?

(Subscript[f, yx])(π/2,1)?

Problem 7 4 pts

To take a 2nd derivative in Mathematica you can “chain” D[...] functions together.  For example, taking the partial derivative of a function with respect to x or y. And then taking the derivative of *that* with respect to x or y.

Use Mathematica to calculate the following second derivatives exactly and compare to your visual estimates above.

(Subscript[f, xx])(π/2,1)?

(Subscript[f, yy])(π/2,1)?

(Subscript[f, xy])(π/2,1)?

(Subscript[f, yx])(π/2,1)?