Question

Let S be a surface in the 3-D space (but we don’t have an
equation

for S). Suppose that there are two curves

r _1(t) = < cos(t), sin(t), t >

and

r_ 2(s) = < (s + 1)^2, 2s, se^s >

that both lie on S. Find an equation of the tangent plane to the
surface S at the

point (1, 0, 0).

Answer #1

Suppose you need to know an equation of the tangent plane to a
surface S at the point
P(2, 1, 4).
You don't have an equation for S but you know that the
curves
r1(t)
=
2 + 3t, 1 − t2, 4 − 5t + t2
r2(u)
=
1 + u2, 2u3 − 1, 2u + 2
both lie on S. Find an equation of the tangent plane at
P.

Let surface P containing (0, 0, 0) be the graph of the two
variable function g with domain R^2 (all real numbers squared).
Suppose the slopes of the tangent lines of curves obtained by
intersecting P with the xz−plane and yz− plane at the point (0, 0,
0) are 1 and 2 respectively. Write (with explanation) the equation
of the tangent plane to P at (0, 0, 0). This is all the information
given for the question.

The equation 4 = 2xy^3 - xyz is a level surface in
3-dimensional space. A person is standing on this surface, at the
point (1, 2, 6).
a. Write the function f for which the above surface is a level
surface, and find the gradient of this
function f. What meaning does the gradient have for the
person?
b. Find an equation for the tangent plane to this surface at
the point (1, 2, 6).
c. Find the equations of...

6) please show steps and explanation.
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 area of the triangle
PQR.

Find an equation of the tangent plane at the given point:
F(r,s)=2s^(−3)−r^3s^(−0.5) , (−2,1)

let S be the surface defined by x^4-2x^2y^2+3z^2=12, Find the
equation of the tangent plane to the surface S at (0,1,2).

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 an equation of the tangent plane to the surface given by
the equation xy + e 2xz+3yz = −5, at the point, (0, −1, 2)

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...

Find an equation of the tangent plane to the surface x y 2 + 3 x
− z 2 = 4 at the point ( 2 , 1 , − 2 ) An equation of the tangent
plane is

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