Question

Identify the surface with parametrization x = 3 cos θ sin φ, y = 3 sin θ sin φ, z = cos φ where 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ π. Hint: Find an equation of the form F(x, y, z) = 0 for this surface by eliminating θ and φ from the equations above. (b) Calculate a parametrization for the tangent plane to the surface at (θ, φ) = (π/3, π/4).

Answer #1

Parametererized surface and tangent plane

3. Let P = (a cos θ, b sin θ), where θ is not a multiple of π/2
be a point on the ellipse (x 2/ a2 )+ (y 2/ b 2) = 1, where a ≥ b
> 0; and let P1 = (a cos θ, a sin θ) the corresponding on the
circle x 2 /a2 + y 2/ a2 = 1. Prove that the tangent to the ellipse
at P and the tangent to the circle at...

Compute equations of tangent plane and normal line to the
surface z = x cos (x+y) at point (π/2, π/3, -√3π/4).

Evaluate C (y + 6 sin(x)) dx + (z2 + 2 cos(y)) dy + x3 dz where
C is the curve r(t) = sin(t), cos(t), sin(2t) , 0 ≤ t ≤ 2π. (Hint:
Observe that C lies on the surface z = 2xy.) C F · dr =

Find the tangent plane to the surface z = cos(xy) when (x, y) =
(π, 0).

4)
Consider the polar curve r=e2theta
a) Find the parametric equations x = f(θ), y =
g(θ) for this curve.
b) Find the slope of the line tangent to this curve when
θ=π.
6)
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the...

Let Θ ∼ Unif.([0, 2π]) and consider X = cos(Θ) and Y =
sin(Θ).
Can you find E[X], E[Y], and E[XY]?
clearly, x and y are not independent
I think E[X] = E[Y] = 0 but how do you find E[XY]?

(a) Is the vector field F = <e^(−x) cos y, e^(−x) sin y>
conservative?
(b) If so, find the associated potential function φ.
(c) Evaluate Integral C F*dr, where C is the straight line path
from (0, 0) to (2π, 2π).
(d) Write the expression for the line integral as a single
integral without using the fundamental theorem of calculus.

7. For the parametric curve x(t) = 2 − 5 cos(t), y(t) = 1 + 3
sin(t), t ∈ [0, 2π) Part a: (2 points) Give an equation relating x
and y that represents the curve. Part b: (4 points) Find the slope
of the tangent line to the curve when t = π 6 . Part c: (4 points)
State the points (x, y) where the tangent line is horizontal

An implicitly defined function of x, y and z is given along with
a point P that lies on the surface: sin(xy) + cos(yz) = 0, at P =
(2, π/12, 4)
Use the gradient ∇F to:
(a) find the equation of the normal line to the surface at
P.
(b) find the equation of the plane tangent to the surface at
P.

We are given a level surface F ( x , y , z ) = 0 where F ( x , y
, z ) = x^3 - y^2 + z^4 - 20 . Find the equation of the tangent
plane to the surface at the point P ( 2 , 2 , 2 ) . Write the final
answer in the form a x + b y + c z + d = 0

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