Question

Consider the surfaces x^2 + y^2 + z^2 = 1 and (z +√2)^2 = x^2 +
y^2, and let (x0, y0, z0) be a point

in their intersection. Show that the surfaces are tangent at this
point, that is, show that they

have a common tangent plane at (x0, y0, z0).

Answer #1

4. Consider the function z = f(x, y) = x^(2) + 4y^(2)
(a) Describe the contour corresponding to z = 1.
(b) Write down the equation of the curve obtained as the
intersection of the graph of z and the plane x = 1.
(c) Write down the equation of the curve obtained as the
intersection of the graph of z and the plane y = 1.
(d) Write down the point of intersection of the curves in (b)
and...

prove that the equation of the plane tangent to the sphere x^2 +
y^2 + z^2 = a^2 at the point (x0, y0, z0) on the sphere is x*x0 +
y*y0 + z*z0 = a^2

Topic: Calculus 3 / Differential Equation
Q1) Let (x0, y0,
z0) be a point on the curve C described by the following
equations
F1(x,y,z)=c1 , F2(x,y,z)=c2 .
Show that the vector [grad F1(x0,
y0, z0)] X [grad F2(x0, y0,
z0)] is tangent to C at (x0, y0,
z0)
Q2) (I've posted this question before but
nobody answered, so please do)
Find a vector tangent to the space circle
x2 + y2 + z2 = 1 , x + y +...

Anyone have any skills with Mathematica? If not, show how you'd
solve it normally, maybe that will help me understand this problem
better.
We are going to start by computing the tangent plane to the
ellipsoid
x^2 + 2y^2 + 3z^2 = 1
at the point (x0, y0, z0) = (1/2, 1/6, 5/(6*(Sqrt [3])))
Define the Mathematica function f(x, y, z) = x^2 + 2y^2 + 3z^2,
the values x0, y0, z0, as above (stored in variables x0, y0, z0),...

Let Q1 be a constant so that Q1 = L(−3, 2), where z = L(x, y) is
the equation of the tangent plane to the surface z = ln(5x − 7y) at
the point (x0, y0) = (2, 1). Let Q = ln(3 + |Q1|). Then T = 5 sin2
(100Q)
satisfies:— (A) 0 ≤ T < 1. — (B) 1 ≤ T < 2. — (C) 2 ≤ T
< 3. — (D) 3 ≤ T < 4. —...

Let Q1 be a constant so that Q1 = L(5, 17), where z = L(x, y) is
the equation of the tangent plane to the surface z = x 6 + (y − x)
4 at the point (x0, y0) = (3, 4). Let Q = ln(3 + |Q1|). Then T = 5
sin2 (100Q)
satisfies:— (A) 0 ≤ T < 1. — (B) 1 ≤ T < 2. — (C) 2 ≤ T
< 3. — (D) 3 ≤...

Let Q1 be a constant so that Q1 = L(20, 12), where z = L(x, y)
is the equation of the tangent plane to the surface z = ln(19x +
8y) at the point (x0, y0) = (7, 11). Let Q = ln(3 + |Q1|). Then T =
5 sin2 (100Q)
satisfies:— (A) 0 ≤ T < 1. — (B) 1 ≤ T < 2. — (C) 2 ≤ T
< 3. — (D) 3 ≤ T < 4. —...

1. Consider x=h(y,z) as a parametrized surface in the natural
way. Write the equation of the tangent plane to
the surface at the point (5,3,−4) given that ∂h/∂y(3,−4)=1 and
∂h/∂z(3,−4)=0.
2. Find the equation of the tangent plane to the surface
z=0y^2−9x^2 at the point (3,−1,−81). z=?

Consider the function F(x, y, z) =x2/2−
y3/3 + z6/6 − 1.
(a) Find the gradient vector ∇F.
(b) Find a scalar equation and a vector parametric form for the
tangent plane to the surface F(x, y, z) = 0 at the point (1, −1,
1).
(c) Let x = s + t, y = st and z = et^2 . Use the multivariable
chain rule to find ∂F/∂s . Write your answer in terms of s and
t.

1. Let T(x, y, z) = (x + z, y − 2x, −z + 2y) and S(x, y, z) =
(2y − z, x − z, y + 3x). Use matrices to find the composition S ◦
T.
2. Find an equation of the tangent plane to the graph of x 2 − y
2 − 3z 2 = 5 at (6, 2, 3).
3. Find the critical points of f(x, y) = (x 2 + y 2 )e −y...

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