Surfaces
Identify and sketch the surface:
x2+z2=1
Circular cylinder with radius of 1 in the x-z plane!
Find the tangent plane to the surface at the point (1,1)
z = xy3 - x2y
x = t2 + 1, y = t2 - 1
Find dz/dt
dz/dt = (y3 -2xy)(2t) + (3xy2 - x2)(2t)
f(x,y) = x/y
Find the directional derivative of f(x,y) at (2,1) in the direction <3/5,4/5>
-1
Find the critical points of f(x,y) = xy - 2x -2y - x2 - y2
(-2,-2)
Identify and sketch the surface:
x = y2 + 4z2
Elliptic paraboloid opening towards the positive x-axis!
f(x,y) = (x + 2)2 -2(y - 1)2 - 5
Find the tangent plane to the function
z = 3 + 8(x - 2) - 8(y - 3) = 8x - 8y + 11
z = sin(x)cos(y)
x = sqrt(t), y = 1/t
Find dz/dt
dz/dt = (cos(x)cos(y))(1/(2sqrt(t))) + (sin(x)(sin(y))(1/t2)
f(x,y,z) = x2yz - xyz3
Find the rate at which f(x,y,z) is changing at (2,-1,1) in the direction of (2,3,-2)
2/5
f(x,y) = x3 + y3 - 3x2 - 3y2 - 9x
Given the critical points (-1,0), (-1,2), (3,0), (3,2), classify each as either a local max, local min, saddle point, or cannot tell.
(-1,0) is local max
(3,2) is a local min
(-1,2), (3,0) are saddle points
Identify and sketch the surface:
4x2 + 9y2 + 9z2 = 36
Find the linear approximation of f(x,y) = x/y2 at (-4,2)
z = -1 + (x + 4)/4 + (y - 2)
f(x,y) = ln(3x + 2y)
x = h*sin(t), y = t*cos(h)
Find ft
ft = (3h*cos(t) + 2cos(h)) / (3x + 2y)
f(x,y,z) = y2exyz
Find the directional derivative at (0, 1, -1) in the direction <3, 4, 12>
5/13
f(x,y) = x2 + xy + y2 + y
Find and classify all local extrema of this function
Only critical point (1/3, -2/3), a local minimum
Identify and sketch the surface:
z2 - 4x2 - y2 = 4
Vertical hyperboloid of two sheets!
Find the linearization of f(x,y) = x*sin(x + y) at (-1,1)
L(x,y) = - (x + 1) - (y - 1)
w = xy + xz + yz
x = r*cos(θ), y = r*sin(θ), z = rθ
Find the value of ∂w/∂θ when r = 2 and θ = π/2
∂w/∂θ = (y + z)(-r*sin(θ)) + (x + z)(r*cos(θ)) + (y +x)(r)
x = 0, y = 2, z = π
∂w/∂θ =-2π
f(x,y,z) = x*ln(yz)
Find the maximum rate of change of the function at (1,2,0.5), and specify a vector in the direction of that maximum.
Max rate of change: sqrt(17)/2
Direction: <0,1,4> or equivalent
Find the extreme values of the function along the given boundary.
f(x,y) = 3x + y
x2 + y2 = 10
Max is 10 at (3,1)
Min is -10 at (-3,-1)
Identify and sketch the surface:
y = z2 - x2
Hyperbolic paraboloid facing away from positive y axis
Parallel resistance given by 1/Req = 1/R1 + 1/R2 + 1/R3
The nominal values of the resistors are as follows:
R1 = 120Ω R2 = 180Ω R3 = 360Ω
If there is a 5% tolerance in the values (+/- 6Ω, 9Ω, and 18Ω respectively), what is the tolerance of the equivalent resistor, Req? Hint: Use the tangent plane approximation!!
+/- 3.75Ω -> 6.25% tolerance for Req = 60Ω! The uncertainty propagated! EEs beware!
z = tan(u/v)
u = 2s +3t, v = 3s - 2t
Find ∂z/∂s
∂z/∂s = ((2v - 3u)/v2)*sec2(u/v)
f(x,y) = y*cos(xy)
Find the rate of change of f(x,y) at (0,1) in the direction of θ = π/4
sqrt(2)/2
Find the max and min of the function on the given region. Include all interior and border points in your search.
f(x,y) = e-xy
x2 + 4y2 = 1
Crazy stuff that is probably better done on a computer lowkey