MATH 281 Southwestern College Relative Maxima Math Questions In the last problem, you are allowed to use without proof that the volume inside a sphere of r

MATH 281 Southwestern College Relative Maxima Math Questions In the last problem, you are allowed to use without proof that the volume inside a sphere of radius R is 43 343πR3.attached Due May 26, 2020
Math 281 – Final
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Show your work for full credit.
1. Find and classify all critical points and any relative maxima, minima or saddle points of
f (x,y) = x 3 + 3y 2 !12x .
2. Find r(t ) if r” (t ) = 2t i + 6e3t j + 4e 4 t k and r (0) = i + 5 j + 3 k .
3. Find an equation for the tangent plane to the hyperboloid x 2 + y 2 ! z 2 = 1 at (–2, 1, 2).
Due May 26, 2020
4. Consider the integral
Math 281 – Final
25 ! x 2
5
##
0 0
page 2 of 4
xy x 2 + y 2 dy dx .
a. Sketch the region of integration of the integral.
b. Evaluate the integral, using polar coordinates.
5. a. Given that F( x, y) = x 2 + y 2 , 2 xy is conservative, find a function f such that $f = F .
& x = !3 cos t
. Use the Fundamental Theorem for
b. Let C be the parametric curve ‘
( y = 2 + 3 sin t, 0 % t % )
Line Integrals to evaluate # F * dr
C
(–3, 2)
(3, 2)
Due May 26, 2020
Math 281 – Final
page 3 of 4
6. Let r(t ) = cos t + t sin t, sin t ! t cos t , 0 % t % ) 2 describe the position of an object moving
in a plane.
a. Find the velocity, r “(t ) , of the object. (Apply the Product Rule correctly!)
b. Find the speed, r “(t ) , of the object..
b
c. Find the arc length, L, of the object’s path from t = 0 to t = )2 . (Use L = # r”(t ) dt .)
a
7. Use the Divergence Theorem to calculate
# # F * n dS , where S is the boundary of the solid
S
cylindrical region E = {( x, y, z ) : x 2 + y 2 % 25, 0 % z % 4} with outward unit normal n, and
F ( x, y, z ) = ! y i + x j + z 2 k , by evaluating a triple integral.
Due May 26, 2020
Math 281 – Final
#
page 4 of 4
$
8. Let E be the half ball of radius 8, E = ( x, y, z ) : 0 % z % 64 ! x 2 ! y 2 . Find the z-coordinate
of the centroid, z =
1 +++
3 +++
3
1024 )
2
— z dV , given that V = volume of E = 3 ) 8 = 3 .
— z dV =
1024) ,,,
V ,,,
E
Use spherical coordinates.
E

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