Question

1-Find equations of the following.

2(x − 9)^{2} + (y − 1)^{2} + (z − 8)^{2}
= 10, (10, 3, 10)

(a) the tangent plane

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2-Suppose that over a certain region of space the electrical
potential *V* is given by the following equation.

* V*(

(a) Find the rate of change of the potential at *P*(3, 2,
5) in the direction of the vector **v** =
**i** + **j** − **k**.

(b) In which direction does *V* change most rapidly at
*P*?

(c) What is the maximum rate of change at *P*?

Answer #1

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Find equations of the following.
2(x − 8)2 + (y − 9)2 + (z − 1)2
= 10, (9, 11, 3)
(a) the tangent plane
(b) the normal line
(x(t), y(t), z(t))

f(x, y, z) =
xe4yz, P(1, 0, 3),
u = <2/3, -1/3, 2/3>
(a) Find the gradient of f.
∇f(x, y, z) =
< , , >
(b) Evaluate the gradient at the point P.
∇f(1, 0, 3) = < , ,
>
(c) Find the rate of change of f at P in the
direction of the vector u.
Duf(1, 0, 3) =

Let f(x, y) = x^2 ln(x^3 + y).
(a) Find the gradient of f.
(b) Find the direction in which the function decreases most
rapidly at the point P(2, 1). (Give the direction as a unit
vector.)
(c) Find the directions of zero change of f at the point P(2,
1). (Give both directions as a unit vector.)

M := {(x, y, z) ∈ R3 : x 2 y − 4ze^x+y = −35} is a surface. Find
the equation of the tangent plane to M at p = (3, −3, 2).

The
plane y+z=2 intersects the ‘funky’ cylinder x^2 + y^4 =17 in a
curve C.
A) Find a parametric equation of the tangent line to C at the
point (4,1,1)
B) How was the direction vector found in part A and how do you
know its the right direction?

(a) Find an equation of the plane tangent to the surface xy ln x
− y^2 + z^2 + 5 = 0 at the point (1, −3, 2)
(b) Find the directional derivative of f(x, y, z) = xy ln x −
y^2 + z^2 + 5 at the point (1, −3, 2) in the direction of the
vector < 1, 0, −1 >. (Hint: Use the results of partial
derivatives from part(a))

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.

find the integral of f(x,y,z)=x over the region x^2+y^2=1 and
x^2+y^2=9 above the xy plane and below z=x+2

Find the directional derivative of the function f(x, y, z) = 4xy
+ xy3z − x z at the point P = (2, 0, −1) in the direction of the
vector v = 〈2, 9, −6〉.

The electric potential in a region of space is given by
V ( x,y,z ) = -x^2 + 2y^2 + 15. If a 5 Coulomb particle is placed
at position (x,y,z)=(-2,-2,0), what is the magnitude and direction
of the force it experiences?

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