Question

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

Answer #1

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).

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...

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 +...

Find the equation of the tangent plane (in terms of x, y and z)
to the surface given by x = u, y = v and z = uv at the point (3, 2,
6).

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

Find an equation of the tangent plane to z=32-3(x^2)-4(y^2) at
the point (2,1,16)

Find the equation for the tangent plane to the surface
z=(xy)/(y+x) at the point P(1,1,1/2).

say that point (x, y, z) is the plane tangent to the
paraboloid z = x ^ 2 + 3y ^ 2 parallel to the plane z = x + y

8).
a) Find an equation of the tangent plane to the surface z = x at
(−4, 2, −1).
b) Explain why f(x, y) = x2ey is differentiable at (1, 0). Then
find the linearization L(x, y) of the function at that point.

Write an equation of a sphere with center (0,8,0), and with the
sphere tangent to the plane with equation x=y. Write an equation
for a sphere centered at the point (-6,5,-7) with radius 3.

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