Math 241
Demo: Partial Derivatives
April 10, 2009
The Geometry
Load the plots package
with plots :
Define a function f
f x, y d 4
x y cos x y 2
x, y /4 x y cos x y 2
Plot its graph, save the plot with the nameP1
plot3d f x, y , x = 0 ..1, y = 0 ..1, axes = normal, orientation = K40, 40 ,
style = patch ; P1 d % :
2
y
0.5
1
1.0
0.0
0.00
x
0.5
1.0
M ake plots of two planes (as parametrized surfaces), save the plots with the names
P2, P3
plot3d
plot3d
x, 0.5, z , x = 0 ..1, z = 0 ..2.5, style = patchnogrid, color = blue,
transparency = 0.4 : P2 d % :
0.5, y, z , y = 0 ..1, z = 0 ..2.5, style = patchnogrid, color = green,
transparency = 0.4 : P3 d % :
Display the three plots together
(1.1)
display P1, P2, P3
2
y
0.5
1
1.0
0.0
0.00
0.5
x
1.0
M ake plots of the curves of intersection of the surface with the planes, save them as
L2, L3
L2 d intersectplot f x, y = z, y = 0.5, x = 0 ..1, y = 0 ..1, z = 0 ..2.6, thickness = 3, color = blue :
L3 d intersectplot f x, y = z, x = 0.5, x = 0 ..1, y = 0 ..1, z = 0 ..2.6, thickness = 3, color
= green :
Display the survace and the curves of intersection
display P1, L2, L3
2
1.0
y
0.5
1
0.0
0.00
x
0.5
1.0
M ake the same picture by obtaining curves of intersections as parametrized curves named
S2, S3
S2 d spacecurve x, 0.5, f x, 0.5 , x = 0 ..1, color = blue, thickness = 3 :
S3 d spacecurve 0.5, y, f 0.5, y , y = 0 ..1, color = green, thickness = 3 :
display P1, S2, S3
2
y
0.5
1
0.0
0.00
x
1.0
0.5
1.0
The Calculus
f x, y = 4
x y cos x y 2
Partial derivative, D operator
D 2 f
0.5, 0.5
2.718200446
D2, 1 f
(2.1)
x, y
2 cos x y 2
x
x sin x y 2 y 2 K 8 x 3 / 2 y 4 cos x y 2
K 16
(2.2)
Partial derivative, Leibniz notation
v
f x, y
vx
2 y cos x y 2
x
K4
x y 3 sin x y 2
Section 15.4, The Tangent Plane
(2.3)
Recall that the tangent line approximation to the curvey = f x at the point a, f a
is the function
T x = f a Cf ' a $ xKa .
The tangent plane approximation to the surfacez = f x, y at the point a, b, f a, b
T x, y = f a, b C D1 f
a, b $ x K b C D2 f
is the function
a, b $ y K b .
For example
f x, y = 4
x y cos x y 2
T x, y d f 0.5, 0.5 C D1 f
x, y /f 0.5, 0.5 C D1 f
0.5, 0.5 $ x K 0.5 C D2 f
0.5, 0.5
x K 0.5 C D2 f
0.5, 0.5 $ y K 0.5
0.5, 0.5
f 0.6, 0.6 = 1.815832858
T 0.6, 0.6 = 1.810909465
The tangent plane to the surface, namedTP
TP d plot3d T x, y , x = 0 ..1, y = 0 ..1, color = brown, transparency = 0.4,
grid = 2, 2 :
Display the surface, the two trace curves, and the tangent plane.
display P1, S2, S3, TP
3.4
y
0.5
1.4
0.0
0.0
-0.6
1.0
0.5
x
1.0
T x, y = K0.6354709360 C 1.359100222 x C 2.718200446 y
y K 0.5
(3.1)