Lagrange Multipliers
Use Lagrange multipliers to find the maximum of
f(x,y) = x+y
subject to the constraint
xy = 16
Show that (4,4) and (-4,-4) are candidates, but f has no maximum on the hyperbola xy=16.
What is: f has no maximum because along y = 16/x:
f(x, 16/x) = x + 16/x -> oo as x -> oo
Candidates (4,4) give f = 8 and (-4,-4) give f = -8, but neither is a global max on the unbounded curve xy = 16.
Find the volume of the solid bounded by the plane z = 0 and the paraboloid
z = 4 - x^2 - y^2
What is: Convert to polar. The paraboloid meets z=0 at r=2. Volume =
int_0^(2pi) int_0^2 (4 - r^2) r dr d theta = 8pi
Let
f(x,y) = (x^3 y)/(x^6 + 2y^2)
Show the limit equals 0 along any line y = mx, but find the limit along y = x^3, proving the overall limit does not exist as (x,y) -> (0,0).
What is: Along y = mx:
f(x,mx) = (mx^4)/(x^6 + 2m^2 x^2) -> 0
But along y = x^3:
f(x,x^3) = x^6/(x^6 + 2x^6) = 1/3 != 0
So the limit does not exist.
Calculate the double integral
int int_R (xy^2)/(x^2+1) dA
where
R = [0,1] xx [-3,3]
What is: The integral separates:
int_0^1 x/(x^2+1) dx * int_(-3)^3 y^2 dy = (ln 2)/2 * 18 = 9 ln 2
Evaluate the line integral
int_C (2 + x^2 y) ds
where C is the upper half of the unit circle
x^2 + y^2 = 1
parameterized as x = cos t, y = sin t, 0 <= t <= pi.
What is: With ds = dt and x^2 y = cos^2(t) sin(t):
int_0^pi (2 + cos^2 t sin t) dt = 2pi + 2/3
Use Lagrange multipliers to find the absolute maximum and minimum values of
f(x,y) = x^2 + y^2
subject to the constraint
2x + y = 5
What is: The minimum value is 5, achieved at (2, 1). The constraint is a line (unbounded), so there is no maximum value.
Find the volume of the solid that lies under
z = x^2 + y^2
above the xy-plane, and inside the cylinder
x^2 + y^2 = 2x
Describe the region in polar/cylindrical coordinates.
What is: In polar, the cylinder is r = 2cos(theta), theta in [-pi/2, pi/2]. Volume =
int_(-pi/2)^(pi/2) int_0^(2cos theta) r^2 * r dr d theta = (3pi)/2
Let
f(x,y) = (xy^2)/(x^2 + y^4)
Show the limit = 0 along any line y = mx, but that a parabolic approach gives a different result, proving the limit does not exist as (x,y) -> (0,0).
What is: Along y = mx:
f(x,mx) = (m^2 x^3)/(x^2 + m^4 x^4) -> 0
But along x = y^2:
f(y^2, y) = y^4/(y^4 + y^4) = 1/2 != 0
So the limit does not exist.
Find the volume of the tetrahedron bounded by the planes
x + 2y + z = 2, x = 2y, x = 0, z = 0
Determine the shadow on the xy-plane to find limits of integration.
What is: The shadow on xy-plane is bounded by x=0, x=2y, x+2y=2. Volume =
int_0^1 int_(x/2)^((2-x)/2) (2-x-2y) dy dx = 1/3
Evaluate the piecewise line integral
int_C 2x ds
where C consists of the arc of y = x^2 from (0,0) to (1,1), followed by the vertical segment from (1,1) to (1,2).
What is: C1: r(t)=(t,t^2), ds=sqrt(1+4t^2) dt, t in [0,1]. C2: r(t)=(1,t), ds=dt, t in [1,2].
int_0^1 2t sqrt(1+4t^2) dt + int_1^2 2 dt = (5sqrt(5)-1)/6 + 2
Find the extreme values of
f(x,y,z) = 2x + 3y - z
subject to the spherical constraint
x^2 + y^2 + z^2 = 16
using Lagrange Multipliers.
What is: Maximum value is
2sqrt(14)
minimum is
-2sqrt(14)
Achieved at
(8/sqrt(14), 12/sqrt(14), -4/sqrt(14))
and its negative.
Find the volume of the solid that lies within the sphere
x^2 + y^2 + z^2 = 4
above the xy-plane, and below the cone
z = sqrt(3(x^2 + y^2))
using spherical coordinates.
What is: The cone z = sqrt(3)r gives phi = pi/6 in spherical. Volume =
int_0^(2pi) int_0^(pi/6) int_0^2 rho^2 sin phi d rho d phi d theta = (8pi)/3 (1 - sqrt(3)/2)
Let
f(x,y) = |x + y|
Determine whether the partial derivatives
(del f)/(del x)(0,0) and (del f)/(del y)(0,0)
exist, and if so, find their values.
What is: By definition:
(del f)/(del x)(0,0) = lim_(h->0) |h|/h
This limit does not exist (left = -1, right = 1). Similarly for the y-partial. So neither partial derivative exists at (0,0).
Evaluate the double integral
int int_D (x+y) dA
where D is the region bounded by the parabola y = x^2 and the line y = x.
What is: The region is 0 <= x <= 1, x^2 <= y <= x. The integral =
int_0^1 int_(x^2)^x (x+y) dy dx = int_0^1 [xy + y^2/2]_(x^2)^x dx = 3/10
Determine whether the vector field
F(x,y) = <3 + 2xy, x^2 - 3y^2>
is conservative. If del Q / del x = del P / del y, find its potential function f(x,y).
What is: del P / del y = 2x = del Q / del x, so F is conservative. The potential function is:
f(x,y) = 3x + x^2 y - y^3 + C
A rectangular box without a lid is to be made from 12 m² of cardboard. Use Lagrange multipliers to find the maximum volume. The surface area constraint (no lid) is:
xy + 2xz + 2yz = 12
What is: Maximum volume = 2 m³, achieved when x = y = 2 m and z = 1 m (base 2×2, height 1).
Set up a triple integral in spherical coordinates to evaluate
int int int_E (x^2 + y^2 + z^2) dV
where E is the unit ball
x^2 + y^2 + z^2 <= 1
What is: In spherical coords, x^2+y^2+z^2 = rho^2, dV = rho^2 sin(phi) d rho d phi d theta:
int_0^(2pi) int_0^pi int_0^1 rho^4 sin phi d rho d phi d theta = (4pi)/5
If
z = e^x sin(y), x = st^2, y = s^2 t
find
(del z)/(del s) and (del z)/(del t)
using the Chain Rule.
What is:
(del z)/(del s) = e^x sin(y) * t^2 + e^x cos(y) * 2st
(del z)/(del t) = e^x sin(y) * 2st + e^x cos(y) * s^2
Use polar coordinates to find the volume of the solid bounded by the paraboloid
z = 18 - 2x^2 - 2y^2
and the xy-plane (z = 0).
What is: The paraboloid meets z=0 at r=3. In polar:
int_0^(2pi) int_0^3 (18 - 2r^2) r dr d theta = int_0^(2pi) [9r^2 - r^4/2]_0^3 d theta = 81pi
Use the Fundamental Theorem for Line Integrals to evaluate
int_C F * dr
for the conservative field
F(x,y) = <3+2xy, x^2-3y^2>
over
r(t) <e^tsint,e^tcost> = , quad 0 <= t <= pi
What is: Endpoints: at t=0, r(0)=(0,1) ; at t=pi , r(pi)=(0,-e^pi) . Using
f(x,y) = 3x + x^2y - y^3
:
int_C F * dr = f(0, -e^pi) - f(0, 1) = e^(3pi) - (-1) = e^(3pi) + 1
Find the points on the sphere
x^2 + y^2 + z^2 = 4
that are closest to and farthest from the point (3, 1, -1) using Lagrange Multipliers.
What is: Closest point:
(3/sqrt(11), 1/sqrt(11), -1/sqrt(11))
at distance sqrt(11) - 2. Farthest point:
(-3/sqrt(11), -1/sqrt(11), 1/sqrt(11))
at distance sqrt(11) + 2.
Express the volume of the solid bounded by the parabolic cylinder
y = x^2
and the planes z = 0, z = 3, and y = 4 as an iterated triple integral in Cartesian coordinates.
What is: Shadow on xy-plane: -2 <= x <= 2, x^2 <= y <= 4. Volume =
int_(-2)^2 int_(x^2)^4 int_0^3 dz dy dx = 64
Let x = s+t and y = s-t. Show that for any differentiable function f(x,y):
(del f)/(del s) * (del f)/(del t) = ((del f)/(del x))^2 - ((del f)/(del y))^2
What is: By Chain Rule:
(del f)/(del s) = f_x + f_y, quad (del f)/(del t) = f_x - f_y
So their product is:
(f_x + f_y)(f_x - f_y) = f_x^2 - f_y^2 = ((del f)/(del x))^2 - ((del f)/(del y))^2
Sketch the region of integration and reverse the order of integration for:
int_0^4 int_(sqrt(x))^2 f(x,y) dy dx
What is: The region is 0 <= x <= 4, sqrt(x) <= y <= 2, equivalently 0 <= y <= 2, 0 <= x <= y^2. Reversed order:
int_0^2 int_0^(y^2) f(x,y) dx dy
Evaluate
int_C F*dr
where
F(x,y) = <(-y)/(x^2+y^2), x/(x^2+y^2)>
and C is any positively oriented simple closed path enclosing the origin. (Hint: Use Green's Theorem with a hole.)
What is: F is not defined at the origin, but del Q/del x = del P/del y away from it. Using Green's Theorem with a small circle C_r around the origin:
int_C F*dr = int_(C_r) F*dr = int_0^(2pi) 1 d theta = 2pi