MATH 251: Calculus 3, SET8

14: Partial Derivatives

14.1: Functions of Several Variables

These problems are done with the CAS. See Hand Solutions for details.

1. [900/10]

The domain of F is , an infinite horizontal strip in the xy-plane. The range of F is , a closed interval.
syms x y
F(x,y) = 1 + sqrt(4-y^2)
F(x, y) = 
F_3_1 = F(3,1)
F_3_1 = 
%
figure
fill([-4 4 4 -4 -4], [-2 -2 2 2 -2], 'r'); grid on; hold on
plot([-4 4], [-2 -2], 'b')
plot([-4 4], [2 2], 'b')
xlabel('x'); ylabel('y')
axis equal; axis([-4 4 -4 4])
xticks(-4:2:4); yticks(-2:2:2)
title('SET8, 900/10: domain of F')
%
figure
fsurf(x, y, F(x,y), [-4 4 -2 2], 'MeshDensity', 12)
xlabel('x'); ylabel('y'); zlabel('z')
axis equal; axis([-4 4 -4 4 0 4])
xticks(-4:2:4); yticks(-2:2:2); zticks(0:2:4)
title('SET8, 900/10: graph of F')

2. [900/12]

The domain of g is the half-space bounded by the plane .
syms x y z
g(x,y,z) = x^3 * y^2 * z * sqrt(10 - x - y - z)
g(x, y, z) = 
g_1_2_3 = g(1,2,3)
g_1_2_3 = 24
figure
fsurf(10-x-y, 'g', [-4 4 -4 4], 'MeshDensity', 8)
xlabel('x'); ylabel('y'); zlabel('z')
axis equal; axis([-4 4 -4 4 2 18])
xticks(-4:4:4); yticks(-4:4:4); zticks(2:4:18); view(50,50)
title('SET8, 900/12: domain of g: on and below plane')

3. [900/18]

The domain of g is , all points in the half-plane except those on the circle .
figure
ts = linspace(0,2*pi); xc = cos(ts); yc = sin(ts);
fill([-2 2 2 -2 -2], [-2 -2 2 2 -2], 'c', 'EdgeColor', 'none'); grid on; hold on
plot(xc, yc, 'k--')
plot([2 2], [-2 2], 'k--')
plot([-2 3], [0 0], 'k')
plot([0 0], [-2 2], 'k')
xlabel('x'); ylabel('y')
axis equal; axis([-2 3 -2 2])
xticks(-2:3); yticks(-2:2)
title('SET8, 900/18: domain of g')

4. [900/28]

Here is the graph of , a circular paraboloid that opens downward.
syms r t
%
figure
fsurf(r*cos(t), r*sin(t), 2 - r^2, 'm', [0 2 0 2*pi], 'MeshDensity', 16)
xlabel('x'); ylabel('y'); zlabel('z')
axis equal; axis([-2 2 -2 2 -2 2])
xticks(-2:2); yticks(-2:2); zticks(-2:2)
% view(44,33)
title('SET8, 900/28: graph of f, a circular paraboloid')

5. [901/54]

Here is a contour map and a graph of , the upper half of an ellipsoid with center at the origin.
syms x y phi theta
%
figure
x = linspace(-2,2); y = linspace(-3,3);
[X,Y] = meshgrid(x,y); Z = real(sqrt(36 - 9*X.^2 - 4*Y.^2));
contour(X,Y,Z); grid on; hold on
plot([-2 2], [0 0], 'k')
plot([0 0], [-3 3], 'k')
xlabel('x'); ylabel('y')
axis equal; axis([-2 2 -3 3])
xticks(-2:2); yticks(-3:3)
title('SET8, 901/54: contour map of f')
%
figure
fsurf(2*sin(phi)*cos(theta), 3*sin(phi)*sin(theta), 6*cos(phi), [0 pi/2 0 2*pi], 'MeshDensity', 16)
xlabel('x'); ylabel('y'); zlabel('z')
axis equal; axis([-2 2 -3 3 0 6])
xticks(-2:2); yticks(-2:2); zticks(0:6)
axis equal
title('SET8, 901/54: graph of f, a semiellipsoid')

6. [902/58]

A dog saddle is graphed.
syms x y
f = x*y^3 - y*x^3
f = 
fsurf(f, [-1.5 1.5 -1.5 1.5], 'MeshDensity', 20)
axis equal; % axis([-2 2 -3 3 0 6])
axis equal
xlabel('x'); ylabel('y'); zlabel('z')
xticks(-1:1); yticks(-1:1); zticks(-2:2)
title('SET8, 902/58: graph of f, a dog saddle')

7. [903/68]

The level surfaces of f are where . These are ellipsoids centered at the origin.
syms phi theta
%
figure
fsurf(sqrt(15)*sin(phi)*cos(theta), sqrt(5)*sin(phi)*sin(theta), sqrt(3)*cos(phi), ...
'r', [0 pi 0 2*pi], 'MeshDensity', 20)
xlabel('x'); ylabel('y'); zlabel('z')
axis equal;
xticks(-3:3); yticks(-2:2); zticks(-1:1)
axis equal
title('SET8, 903/68: an ellipsoid, a typical level surface of f')

8. [903/74]

There are two maximums of 0.2 at (0.7, 0.7) and (-0.7, 0.7), approximately.
There are two minimums of -0.2 at (-0.7, 0.7) and (0.7, -0.7), approximately.
In Section 14.7, we will learn methods to determine these values exactly.
(NOTE: There is what is called a "saddle point" at the origin at which the function value is 0. It is neither a maximum nor minimum.)
figure
x = linspace(-2,2); y = linspace(-2,2);
[X,Y] = meshgrid(x,y); Z = X.*Y.*exp(-X.^2-Y.^2);
contour(X,Y,Z); grid on; hold on
plot([-2,2], [0 0], 'k')
plot([0 0], [-2,2], 'k')
xlabel('x'); ylabel('y')
axis equal; axis([-2 2 -2 2])
xticks(-2:2); yticks(-2:2)
title('SET8, 903/74: contour map of f')
%
figure
x = linspace(-2,2,41); y = linspace(-2,2,41);
[X,Y] = meshgrid(x,y); Z = X.*Y.*exp(-X.^2-Y.^2);surf(X,Y,Z)
xlabel('x'); ylabel('y'); zlabel('z')
axis([-2 2 -2 2 -0.2 0.2])
xticks(-2:2); yticks(-2:2); zticks(-0.2:0.1:0.2)
title('SET8, 903/74: graph of f')
view(18,23)

9. [903/76]

The limiting behavior of f as approaches the origin depends on the direction of approach along straight lines in the xy-plane through the origin.
The same is true as both x and y become large.
To see this, use polar coorindates, where it is immediately apparent that the limiting value depends upon , the direction of approach!
figure
x = linspace(-2,2); y = linspace(-2,2);
[X,Y] = meshgrid(x,y); Z = X.*Y ./ (X.^2 + Y.^2);
contour(X,Y,Z); grid on; hold on
plot([-2,2], [0 0], 'k')
plot([0 0], [-2,2], 'k')
xlabel('x'); ylabel('y')
axis equal; axis([-2 2 -2 2])
xticks(-2:2); yticks(-2:2)
title('SET8, 903/76: contour map of f')