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
1.007 radians
2. Let
be a point in
a. Show that is not on the line.
by contradiction
b. Find the distance between the point and the line.
3. Let
be a space curve in .
a. Find the unit tangent vector
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
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
. Find
be the simple closed path (the “knot” shown below) parametrized for
12. Find the volume of the solid bounded by
13. Let
be the surface
be the surface
oriented with upward normal
oriented with upward normal
be the ellipse
Let F be any vector field with continuous 1st partials in

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