Description
Need help on Math exam, Calculus III.
Need to be online with me anytime 05/22/2018 13:30-15:45 CA, US time.
Explanation & Answer
Attached.
Problem 1:
We have
d
( 2sin t ) = 2 cos t
dt
d
y = ( 2sin 2t ) = 4 cos 2t
dt
d
z = ( 2sin 3t ) = 6 cos 3t
dt
x =
(
)
At ( x, y, z ) = 1, 3, 2 , we can solve for t
2 sint = 1
2sin 2t = 3
2sin 3t = 2
1
sint = 2
sin 2t = 3 / 2
sin 3t = 1
t =
3
Hence,
( x, y, z ) = 2 cos
, 4 cos 2 , 6 cos 3
3
3
3
3
1
( x, y, z ) = 2
, 4 , 0
2
2
( x, y, z ) =
(
3,1, 0
)
Hence, the equation of the tangent line is
( x, y, z ) = (
Or
) (
3,1, 0 t + 1, 3, 2
)
x = 3t + 1
y=t+ 3
z=2
Attached.
Problem 4:
We have
P Q R
+
+
x y z
divF = ( y 2 z 3 ) + ( 2 yz ) + ( 4 z 2 )
x
y
z
divF = 0 + 2 z + 8 z
divF = 10 z
divF =
By Divergence theorem, we have
F .ndS = divFdV = 10 zdV
2 3 9
=
10 z rdzdrd
0 0 r2
2 3
=
5z
2 9
r2
rdrd
0 0
2
=
3
d r ( 405 − 5r ) dr
4
0
0
= ( 2 )(1215 )
= 2430
Attached.
Problem 5:
We can calculated the value of b as follow
lim
x , y →0
3 xy
4 x2 + 4 y 2
= lim
x , y →0
= lim
=
3 x ( kx )
4 x 2 + 4 ( kx )
2
3kx 2
2 k 2 + 1x
3k
=
lim x = 0
2 k 2 + 1 x , y →0
x , y →0
Hence,
b=0
We have
F .d r =
C
(
)
= 1, x + yz, z − xy ( 3,3,1) dt
Problem 3:
We want to minimize
D = f ( x, y , z ) = ( x − 4 ) + ( y − 2 ) + ( z − 0 )
2
2
2
f ( x, y , z ) = ( x...