2A. Functions and Partial Derivatives
Sketch five level curves for each of the following functions. Also, for a-dl sketch the
portion of the graph of the function lying in the first octant; include in your sketch the
traces of the graph in the three coordinate planes, if possible.
Calculate the first partial derivatives of each of the following functions:
for each of the following:
a) xmyn, (m,n positive integers)
(x)g(y), for any differentiable
tell for what value of the constant a there exists a function
(x, y) for which
6xy, and then using this value, find such a
function by inspection.
Show the following functions
(x,y) satisfy the equation
the two-dimensional Laplace equation):
Tangent Plane; Linear Approximation
Give the equation of the tangent plane to each of these surfaces at the point indicated.
a) Find the equation of the tangent plane to the cone z
at the point
yo, zo) on the cone.
b) Write parametric equations for the ray from the origin passing through
using them, show the ray lies on both the cone and the tangent plane at Po.
Using the approximation formula, find the approximate change in the hypotenuse of
a right triangle, if the legs, initially of length 3 and 4, are each increased by .010
The combined resistance R of two wires in parallel, having resistances R1 and R2
respectively, is given by
If the resistance in the wires are initially
1 and 2 ohms, with a possible error in each
.l ohm, what is the value of R, and by how much might this be in error? (Use the
Give the linearizations of each of the following functions at the indicated points:
a) (x+ y +2)2 at (0,O); at (1,2)
cosy at (0,O); at (017r/2)