Math 225: Calculus III    Assignment 5

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Instructions

This assignment contains problems involving functions of several variables: the chain rule,  tangent planes, the gradient of a function, local maxima/minima & saddle points on the graph of a function, see Chapter 13 in Finney & Thomas or Chapter 3 in the Lecture Notes. Before starting this assignment you may want to review the notebooks Derivatives, TangentsNormals, and AdvancedCalculusDemo (in the Notebook folder or the back of the Lecture Notes). The functions Grad[], CriticalPoints[], and Discriminant[] are used for the first time in this assignment.

Recall that to substitute a value into an expression use: expr /. var -> value.
For example,
    x^2 - y^2 /. x->2
    4 - y^2
If an expression has already been defined as a function, then just use the function notation to get values. For example
    f[x_,y_] = x^2 - y^2;
    f[2,y]
    4 - y^2

To plot the level curve f[x,y] == c, use
ContourPlot[f[x,y],{x,minx,maxx},{y,miny,maxy},
    Contours->{c}, ContourShading->False]
To plot a level surface f[x,y,z] == c, use
ContourPlot3D[f[x,y,z],{x,minx,maxx},{y,miny,maxy},
    {z,minz,maxz}, Contours->{c}]
To get a line segment from p to q representing a 3D vector use Vector[p,q] and  combine it with other plots with the Show[] command.

Be sure to type in comments explaining what you are doing.

Remember to uncheck Show In/Out Names in the Kernel menu and close this group before printing.

Initialization

<<Calculus`Math225`

Needs["Graphics`ContourPlot3D`"]

i = {1, 0, 0}; j = {0, 1, 0}; k = {0, 0, 1};

Problem 1

Let
    w[x_,y_] = Tan[x^2 + y^2]
and
    x[r_,s_] = E^(-r) + E^s; y[r_,s_] = r/s

Find the derivatives of w with respect to r and s when r == 2 and s == 1.
(The dependencies between the variables must be made explicit: use
D[w[x[r,s],y[r,s]],r], and  D[w[x[r,s],y[r,s]],s]) .

Solution

Problem 2

Let f[x_,y_] = x^3 y - x^2 y^2 - y^4.
a) Find the gradient of f[x,y],  Grad[f[x,y],{x,y}].
b) Find the direction in which f[x_,y_] increases most rapidly at the point {1,2} and find the derivative of f[x,y] in that direction.
c) Plot the level curve of f[x,y] through the point {1,2}.

Solution

Problem 3

Let f[x_,y_,z_] = y^3 + z^2 - x^2. 
a) Plot the level surface f[x,y,z] == 1
b) Find Grad[f[x,y,z]] at the point {1,1,1} and plot this vector with the level surface from part a). ({1,1,1} is on this level surface).

Solution

Problem 4

a) Find the equation for the plane tangent to the level surface z - x^2 + y^2 == 1 at the point {3,2,6}.
b) Graph the surface together with this tangent plane. (Suggestion: choose a domain for the tangent plane smaller than the domain for the surface so that the point of tangency is easy to see.)

Solution

Problem 5

Let f[x_,y_] = 3 x y  - x^3 - y^2
a) Test the function f[x,y] for local maxima, local minima, and saddle points.
b) Find the value of f[x,y] at the points found in a).
c) Produce a graph of f[x,y]that includes the points found in a).

Solution