CALCULUS III LAB 4 (Max score: 50 pts)
The path of steepest descent
Due at noon on Wednesday, November 9, 2006.
Goal: The goal of this lab is to explore some of the properties of the gradient of f(x,y), f, the
level curves of f, f ( x, y )  c , and the directional derivative of f(x,y), Du f ( x, y ) .
The  Icon: Whenever you see the  icon in this lab, there is some question to answer or task to
be performed. Hopefully, this will give you a quick way of checking over your lab report to ensure
that you didn't miss anything in your write-up. Let me know if it helps.

x2 y 2

3 2
Part 1: You are now ready to Author 200(1  e
) . (Remember to use the ê button in the
lower right of the Algebra window when you Author this expression.) Plot this expression in a 3D
window with x and y ranging from –3 to 3 and z varying from 0 to 200. Then set the Number of

x2 y 2

3 2
Panels to 40 and copy the graph to the worksheet. Now, Author f(x,y) : = 200(1  e
) . (NB:
For the remainder of this lab, whenever I write f(x,y) I am referring to this function.) Now compute
z1= f(1,-2) by Authoring f(1,-2). Simplify to get the “exact” value of z1 and Approximate to get an
approximate value of z1 to three decimal places.  What did you get for z1? (exact and
approximate values)
 Use the exact value of z1 to plot the z1= f(x,y) level curve in a 2D window. You do not need to
solve for y in terms of x because DERIVE can do implicit plotting. Just Author the appropriate
expression, go to Options, Display, Grids, and set Horizontal to 12 and Vertical to 7. Then plot the
level curve.  What is the name and Cartesian equation (in standard form) of this curve?
Now Author, grad(f(x,y)) and Simplify. This yields an expression for f, which is valid for any
point (x,y). Evaluate f at P1(1,-2) using the SUB icon.  What did you get for f (P1)? (Exact
and approximate to three decimals.)
What I’d really like is a unit vector pointing in the opposite direction as the gradient. So, Author the
appropriate unit vector, let’s call it u1. (Reminder: On DERIVE, you can use either abs(v) or |v| for
the length of a vector.)  What did you get for u1?
Plot the point P1(1,-2) in the 2D window and then  get a printout (=hardcopy) of the z1= f(x,y)
level curve in the 2D window. Once you have the printout,  draw in the unit vector, u1, by hand
with its “tail” at P1. It should look  to the level curve and be one unit long.
Part 2: From the graph of z=f(x,y), we see there's a minimum at (x,y)=(0,0). Hence, one might be
lead to think that the fastest way to the bottom, from some point on the surface, say P0, would be to
make a beeline for (0,0). However, you will notice from your 2D plot that the u1 vector does not
point toward (0,0). Draw a dashed line from the point P1 to the origin to see this more clearly.
Recall that u1 is pointing in the direction of  f, which is the direction of most rapid decrease in
the function, f(x,y). Hence, the direction of most rapid decrease in f at P1 is NOT in the direction
toward the origin.
To investigate this further, calculate the directional derivative of f at P1 in the direction u1, namely,
calculate Du1 f(P1).  What did you get? Next, calculate the directional derivative of f at P1 in the
direction toward the origin. Call this direction u2.  What is u2? What is Du2 f(P1)?
Based on the directional derivatives you just calculated, does f decrease more rapidly in the
direction of u1 or u2?  Round Du1 f(P1) and Du2 f(P1) to three decimal places in your lab report.
Part 3: Wouldn’t you know it, just when you thought you were almost done with this lab, along
comes Lisa the lightning bug and alights upon the surface, z=f(x,y). As Lisa looks down through the
transparent surface, she notices that she is directly above the point P1(1,-2). Incidentally, the
surface is very slippery. In fact, the surface is a solid sheet of ice!
Lisa, like most lightning bugs, is not always as careful as she should be. As you may imagine, a
sheet of ice in the shape of our surface is a precarious place to be for a lightning bug that
experiences vertigo. Well, no sooner had Lisa landed than she started experiencing an acute attack
of vertigo and her feet went right out from under her. As Lisa slides down the surface on her back
with her legs thrashing wildly about, she will proceed in the direction of steepest descent due to the
attractive force of gravity.
Question: What would Lisa’s path look like if it were projected onto the xy-plane?
To answer this question graphically,  use DERIVE to plot a family of level curves. To plot a
family of level curves on DERIVE, Author vector(z=f(x,y), z, 5, 195, 20) and Simplify. The
simplified expression should now be highlighted so just  go to the 2D-plot window and hit Plot.
Get a hardcopy, start at the point P1(1,-2) and pencil in Lisa’s path by hand, always proceeding
perpendicular to the level curves and in the direction of  f.
To answer this question analytically,  use pencil and paper to find the Cartesian equation of
Lisa’s path in the xy-plane? To do this, (1) set the slope of  f(x,y) equal to dy/dx, (2) separate the
variables, (3) integrate, and (4) determine the arbitrary constant by using the fact that the curve must
pass through the point P1(1,-2).  Once you’ve found the Cartesian equation for Lisa’s path in the
xy-plane, find the parametric representation for her path, and then use this to find her position
vector in 2D (use the parameter t). Once you have the position vector, use it to plot Lisa’s path in
2D on the same set of axes as the level curves starting at (1, 2) and ending at (0, 0) . Does Lisa’s
path look  to the level curves? Get a printout for your lab report. Make sure the Horizontal and
Vertical grids are set to 12 and 7, respectively. Also label the starting point and ending point, and
put arrows on Lisa’s path (by hand) to indicate her direction of motion.
 What is a parametric representation of Lisa’s path in 3D, i.e. the path of steepest descent? 
Find the position vector of Lisa's path in 3D and then plot it in the 3D window. Mark&Copy the
graph over to the worksheet. Be sure to label the starting point and ending point, and put arrows on
Lisa’s path (by hand) to indicate her direction of motion as she slides down the icy surface.