MATH 251: Calculus 3, SET8

14: Partial Derivatives

14.2: Limits and Continuity

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

1. [910/4]

syms r theta
x = r*cos(theta), y = r*sin(theta)
x = 
y = 
f = simplify(2*x*y / (x^2 + 2*y^2))
f = 
%
figure
rv = linspace(0,2,9); tv = linspace(0,2*pi,37);
[R T] = meshgrid(rv,tv); X = R.*cos(T); Y = R.*sin(T); Z = sin(2*T)./(3-cos(2*T));
contour(X,Y,Z); grid on; hold on
plot([-2 2], [0 0], 'k', 'LineWidth', 2)
plot([0 0], [-2 2], 'k', 'LineWidth', 2)
xlabel('x'); ylabel('y')
axis equal; axis([-1 1 -1 1])
xticks(-1:1); yticks(-1:1)
title('SET8, 910/4: contour map of f(x,y) = xy / (x^2 + 2y^2)')

2. [910/8]

Plug in the values and x and y approach. Voila!
syms x y
f(x,y) = exp(sqrt(2*x - y))
f(x, y) = 
L = f(3,2)
L = 

3. [910/12]

This is like #1 using polar coordinates with the pole shifted to . Again, just look at the contour plot near (1,0).
The limit does not exist (DNE) because the limiting value depends on the direction of approach.
syms r x y theta
f = (x-1)*y / ((x-1)^2 + y^2)
f = 
g = simplify( subs(f, [x y], [1+r*cos(theta) r*sin(theta)]) )
g = 
%
figure
rv = linspace(0,2,9); tv = linspace(0,2*pi,37);
[R T] = meshgrid(rv,tv); X = 1 + R.*cos(T); Y = R.*sin(T); Z = sin(2*T) / 2;
contour(X,Y,Z); grid on; hold on
plot([-1 3], [0 0], 'k', 'LineWidth', 2)
plot([0 0], [-2 2], 'k', 'LineWidth', 2)
xlabel('x'); ylabel('y')
axis equal; axis([-1 3 -2 2])
xticks(-1:3); yticks(-2:2)
title('SET8, 910/12: contour map of f(x,y) = (x*y - y) / ((x-1)^2 + y^2)')

4. [910/16]

Now as . So via the Squeeze Theorem.
syms x y
f = x*y^4 / (x^4 + y^4)
f = 
figure
fsurf(f, [-1 1 -1 1]); hold on
plot3(0,0,0, 'mo', 'MarkerFaceColor', 'm', 'MarkerSize', 12)
xlabel('x'); ylabel('y'); zlabel('z')
xticks(-1:1); yticks(-1:1); zticks(-1:1)
axis([-1 1 -1 1 -1 1])
title('SET8, 910/16')
view(-66,53)

5. [911/17]

Switch to polar coordinates and then use the single-variable limit command.
syms r x y theta
f = (x^2 + y^2) / (sqrt(x^2 + y^2 + 1) - 1)
f = 
g = simplify( subs(f, [x y], [r*cos(theta) r*sin(theta)]) )
g = 
L = limit(g, r, 0, 'right')
L = 2
%
figure
fsurf(f, [-1 1 -1 1], 'MeshDensity', 16); hold on
plot3(0,0,2, 'mo', 'MarkerFaceColor', 'm', 'MarkerSize', 12)
xlabel('x'); ylabel('y'); zlabel('z')
xticks(-1:1); yticks(-1:1); zticks(2:3)
axis([-1 1 -1 1 2 3])
title('SET8, 910/17')
view(-53,53)

6. [911/22]

Proceed as in #4. So as . So via the Squeeze Theorem.

7. [911/28]

The function is discontinuous on , for which f is undefined. That's why the graph is in two pieces.
syms r t x y
f = 1 / (1 - x^2 - y^2)
f = 
g = simplify( subs(f, [x y], [r*cos(t) r*sin(t)]) )
g = 
%
figure
fsurf(r*cos(t), r*sin(t), g, [0 0.9 0 2*pi]); hold on
fsurf(r*cos(t), r*sin(t), g, [1.1 2 0 2*pi])
xlabel('x'); ylabel('y')
axis([-2 2 -2 2 -4 4])
xticks(-2:2:2); yticks(-2:2:2); zticks(-4:2:4)
title('SET8, 911/28: graph of f(x,y) = 1 / (1 - x^2 - y^2)')
%
figure
fsurf(r*cos(t), r*sin(t), g, [0 0.9 0 2*pi], 'MeshDensity', 12); hold on
fsurf(r*cos(t), r*sin(t), g, [1.1 2 0 2*pi], 'MeshDensity', 12)
xlabel('x'); ylabel('y')
axis equal; axis([-2 2 -2 2 -4 4])
xticks(-2:2:2); yticks(-2:2:2); zticks(-4:2:4)
title('SET8, 911/28: graph of f from above')
view(0,90)