2.
Partial Differentiation
2A. Functions and Partial Derivatives
2A1
Sketch five level curves for each of the following functions. Also, for adl 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.
a)
1

x

y
b)
Jw
c) x2
+
Y2
d)
1

x2

y2
e) x2

y2
2A2
Calculate the first partial derivatives of each of the following functions:
x
w
=
x3y

3xy2
+
2y2
b) z
=

c) sin(3x
+
2y)
ex2y
Y
e) z
=
x ln(2x
+
y)
f) x2z

2yz3
2A3
Verify that
f,,
=
fyx
for each of the following:
x
a) xmyn, (m,n positive integers)
b'
zfy
c) cos(x2
+
y)
f
(x)g(y), for any differentiable
f
and g
2A4
By using
fxy
=
fy,,
tell for what value of the constant a there exists a function
f
(x, y) for which
f,
=
axy
+
3y2,
fy
=
+
6xy, and then using this value, find such a
function by inspection.
2A5
Show the following functions
w
=
f
(x,y) satisfy the equation
w,,
+
wyY
=
0 (called
the twodimensional Laplace equation):
w
=
eax
sin ay
(a constant)
b) w
=
ln(x2
+
y2)
2B.
Tangent Plane; Linear Approximation
2B1
Give the equation of the tangent plane to each of these surfaces at the point indicated.
a) z
=
xy2, (Ill,1)
b) w
=
y2/x, (1,2,4)
2B2
a) Find the equation of the tangent plane to the cone z
=
d
w
at the point
Po
:
(xo
,
yo, zo) on the cone.
b) Write parametric equations for the ray from the origin passing through
Po,and
using them, show the ray lies on both the cone and the tangent plane at Po.
2B3
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
.
2B4
The combined resistance R of two wires in parallel, having resistances R1 and R2
respectively, is given by
1
1


+
1
R
R1
R2
If the resistance in the wires are initially
1 and 2 ohms, with a possible error in each
of
f
.l ohm, what is the value of R, and by how much might this be in error? (Use the
approximation formula.)
2B5
Give the linearizations of each of the following functions at the indicated points:
a) (x+ y +2)2 at (0,O); at (1,2)
ex
cosy at (0,O); at (017r/2)
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E.
18.02
EXERCISES
a) (x+ y +2)2 at (0,O); at (1,2)
b) excosy at (0,O); at (0, n/2)
2B6
To determine the volume of a cylinder of radius around 2 and height around 3, about
how accurately should the radius and height be measured for the error in the calculated
volume not to exceed .1
?
2B7
a) If x and y are known to within .01, with what accuracy can the polar coordinates
r and 8 be calculated? Assume x
=
3, y
=
4.
b) At this point, are
r and 8 more sensitive to small changes in x or in y? Draw a
picture showing x, y, r, 8 and confirm your results by using geometric intuition.
2B8*
Two sides of a triangle are a and b, and 8 is the included angle. The third side is c.
a) Give the approximation for Ac in terms of a, b, c, 8, and Aa, Ab, Ad.
b) If a
=
1, b
=
2, 8
=
n/3, is c more sensitive to small changes in a or b?
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 Spring '08
 Auroux
 Chain Rule, Derivative, Laplace, Partial differential equation

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