Question

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 P1 meet on the x-axis.

Answer #1

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).

Find the equation of the tangent line to the curve
r = 2 sin θ + cos θ
at the point
( x 0 , y 0 ) = ( − 1 , 3 )

Assume sin(θ)=19/36 where π2<θ<π, and cos(ϕ)=−11/38 where
π<ϕ<3π/2. Use sum or difference formula to compute:
sin(θ +ϕ)=
sin(θ -ϕ)=
cos(θ +ϕ)
cos(θ -ϕ)=

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]?

1) Find the values of the trigonometric functions of θ from the
information given.
cot(θ) = − 3/5, cos(θ) > 0
sin(θ)
=
cos(θ)
=
tan(θ)
=
csc(θ)
=
sec(θ)
=
2)
The point P is on the unit circle. Find
P(x, y) from the given information.
The x-coordinate of P is −√5/4, and P
lies below the x-axis.
P(x, y) = ( )
3) Find the terminal point P(x, y) on the unit circle
determined by the given value of...

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...

For r = f(θ) = sin(θ)−1
(A) Find the area contained within f(θ).
(B) Find the slope of the tangent line to f(θ) at θ = 0
,π,3π/2
.

Let c(t) = (t^2, t sin(π t), t cos(π t)). Find the intersection
point of the tangent line to c at t = 3 with the yz-plane?

Find f. f ''(θ) = sin(θ) + cos(θ), f(0) = 3, f '(0) = 2

Consider the linear system x' = x cos a − y sin a
y'= x sin a + y cos a
where a is a parameter. Show that as a ranges over [0, π], the
equilibrium point at the origin passes through the sequence stable
node, stable spiral, center, unstable spiral, unstable node.

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