# sample final exam for calculus 3

i have 13 questions with the correct answers bellow of the question. can you solve it step by step writing it down and send it for me, thanks

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MTH 277 Sample Final Exam
———————————–1. Given the points
a. Find the projection of
b. Find the angle, in radians, between
.
2. Let
be a point in
Let
a. Show that is not on the line.
b. Find the distance between the point and the line.
2.687
3. Let
be a space curve in .
a. Find the unit tangent vector
at
.
b. Find a set of parametric equations for the tangent line to at .
x = 3, y = 5s + 10, z = 12s + 12
4. Let
,
be a path in .
a. Find the arc length of between
.
b. Find the curvature and radius of curvature of
at .
5. Let
be a surface in .
a. Sketch the surface.
hyperbolic paraboloid (saddle surface)
b. Sketch a contour map of the surface using
hyperbolas and lines (asymptotes)
6. Let
define a surface S in
a. Find the gradient of
.
.
.
b. Find the directional derivative of at the point
in the direction of
c. Find an equation of the tangent plane to at
.
.
7. Let
.
Use the second partial derivatives test to find any relative extrema or saddle points of .
8. Let
with domain
9. Find
with vertices
be the parametric surface in
. Find the surface area of .
on the closed path
defined as the boundary of the trapezoid in
, oriented in the counterclockwise direction.
10. Let
be a vector field in
Let
oriented by the outward unit normal vector .
Calculate the flux of
11. Let
a. Let
.
be a force field in .
be the path parametrized by
oriented from
b. Let
over :
be the closed surface given by
to
. Find
for
,
.
be the simple closed path (the “knot” shown below) parametrized for
by
.
Find
.
12. Find the volume of the solid bounded by
13. Let
be the surface
Let
be the surface
in
oriented with upward normal
.
oriented with upward normal
Let
be the ellipse
,
Let F be any vector field with continuous 1st partials in
Show:
.
=
.
.
.

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