Vector Madness
The gradient vector of the function:
f(x,y,z) = e^(5xyz) + cos(x)y^2 - xsin(z)
grad f = <<5yze^(5xyz) - sin(x)y^2 - sin(z), 5xze^(5xyz) + 2ycos(x), 5xye^(5xyz) - xcos(z)>>
The ranges of p, phi, and theta that would be used to describe a solid sphere of radius 4 in the second octant.
What is
0 <= rho <= 4, 0 <= phi <= pi/2, pi/2 <= theta <= pi
The work required to pull a wagon a distance of 100 meters along a horizontal path by a constant force of 70 N if the handle of the wagon is held at an angle of 35 degrees above the horizontal.
What is 5,734 J?
The normal vector to the plane:
5x + 4y - z = 18
What is
vecn = <<5,4,-1>>
The velocity of a particle with acceleration:
veca(t) = <4t, 6t, 1>
with initial velocity
vecv(0) = <1,-1,1>
What is
vecv(t) = <2t^2 + 1, 3t^2 - 1, t + 1>
The curl F if
vec F = <<2x, x^2y, x^2y^2z^2>>
What is
curl vecF = <<2x^2yz^2, -2xy^2z^2, 2xy>>
The center and radius of the sphere
(x - 4)^2 + y^2 + (z + 2)^2 = 10
What is center: (4, 0, -2)
radius =
sqrt(10)
The magnitude of the torque about the center of a bolt that is tightened by applying a 30 N force (at an angle of 60 degrees from the line of the wrench) to a 0.3 m long wrench.
What is 7.8 N*m?
The vector equation of a line through the point (1,2,-3), parallel to the vector <5,-4,3>.
What is
vecr(t) = <<1 + 5t, 2 - 4t, -3 + 3t>>
dz/dt at the point (0,1), if
z = x^2y+3xy^4
,
x = sin(2t)
, and
y = cos(t)
.
What is 6?
The vectors <-2,5,-6> and <4,4,2> are orthogonal or not orthogonal.
What is orthogonal?
The parametric equations of the sphere:
x^2 + y^2 + z^2 = 25
What is
x = 5cos(theta)sin(phi)
y = 5sin(theta)sin(phi)
z = 5cos(phi)
0<=phi<=pi
0<=theta<=2pi
The velocity and acceleration of a particle with position
vecr(t)=<<t^2,e^t,te^t>>
What is
vec v(t) = <<2t, e^t, e^t + te^t>>
veca(t) = <<2, e^t, 2e^t + te^t>>
The vector equation of a line that is orthogonal to the plane
2x - 3y + 5z = 12
and passes through the point (2, 1, -5).
What is
vecr(t) = <<2 + 2t, 1 - 3t, -5 + 5t>>
The local maximum and minimum value(s) and saddle point(s) of
f(x,y) = x^4 + y^4 - 4xy + 1
What is a saddle point at (0,0,1) and f(1,1) = -1 and f(-1,-1) = -1 are local minimum values?
The angle between the vectors <5,-1, 2> and <2,3,-4>.
What is 91.9 degrees?
The volume of the portion of a sphere of radius 6 centered centered at the origin that is below the xy-plane. (You must use a triple integral to solve.)
What is 144 pi units^3
The amount of time it takes for a projectile to hit the ground if it is fired with an initial speed of 150 m/s and angle of elevation of 45 degrees from a height of 10 m above ground level.
What is 21.7 seconds?
The parametric equations of the line that passes through the points: (5, -2, -3) and (4, 3, 7).
What is
x = 5 - t
y = -2 + 5t
z = -3 + 10t
The volume of the solid that lies within the cylinder:
x^2 + y^2 = 1
, above the xy-plane, and below the cone:
z^2 = 4x^2 + 4y^2
.
What is
(4pi)/3
(a) The direction in which the function
f(x,y) = xe^y
has the maximum rate of change at the point (2,0).
(b) The maximum rate of change of
f(x,y) = xe^y
at the point (2,0).
What is f increases the fastest in the direction of <1,2> and the maximum rate of change is
sqrt(5)
The center and radius of the Sphere:
x^2 + y^2 + z^2 - 2x - 4y + 8z = 15
What is center: (1, 2, -4) radius = 6
The work done in moving an object along the curve
y = x^2
from the point (0,0) to the point (2,4) in the presence of the force
vecF(x,y)=<<x^2,y^2>>
What is 24 J?
The tangent plane to the surface
x^2 + 2xyz + 3z^2 = 6
at the point (1,1,1).
What is
2x+ y + 4z = 7
The flux of F = <z,y,x> over the surface of the region E bounded by the cylinder:
x^2 + y^2 = 16
, bounded below by the plane z = 0, and bounded above by the paraboloid:
z = 4 - x^2 - y^2
.
What is
-64pi