2313 MAC HW Problems Part 2 Answers
Section 91
A. p.173 (1) dz =
dt
1
1+x2 +2y 2
6xt2 +
2y
t
(2) z = ex (y cos(xy ) sin(xy )) t + xex cos(xy ) s2s+t2 ;
s
z
= ex (y cos(xy ) sin(xy )) s + xex cos(xy ) s2t+t2
t
z
z
z
(4) u = 23, v = 32, and w = 39
(5) Note: the problem should state when (u, v, w) = (1/3, 2, 0). f = 18fx (4, 0, 1),
u
f
f
2
= fy (4, 0, 1), and w = fx (4, 0, 1) + 3 fy (4, 0, 1) + fz (4, 0, 1)
v
(6) OMIT this problem
9
(7) temperature is rising by 2 degrees Celsius/sec as the insect ies through the point
(2, 6, 8)
(8) fu = 2a + 2b + 3c, fv = 2a 2b + c, fw = b + c
(9) 6, 10, and 0.
B.
C.
F
z
= Fx and y = F y , where F (x, y, z ) = arctan(x, y, z ) + x2 + y 2 xz + yz ,
Fz
z
y
z
z
Fx = 1+(yz )2 + 2x 2xz+yz , Fy = 1+(xz )2 + 2y 2xz+yz , and Fz = 1+(xy )2 2x++yz .
xyz
xyz
xyz
xz
z
x
w
t
= 4s2 t + 4t3 and
w
s
= 4st2 + 4s3 .
D. gu (0, 0) = 2; gv (0, 0) = 5
E.
z
z
1 z
z
1
=
=;
=
= .
x
1+z
2 y
y (1 + z )
2
F. Use the previous problem,
dz
= 1 3/2 = 1/2.
dt
dz
dt
=
z dx
x dt
+
z dy
y dt
=
z
x
1+z
+
z
y
y (1+z )
. z = 1 when t = 0, so
Section 92
A. (1) (i) df = 2xy 2 sin(x2 y )dx + [cos(x2 y ) x2 y sin(x2 y )]dy ]
(ii) df = (1 + y 2 zexyz )dx + (z + xyzexyz + exyz )dy + (y + xy 2 exyz )dz
x
(iii) df = m 2 j 2 dxj
j =1
ar
(5) 30 m2 . Relative error is
99.4% accurate.
B.
30
5000
= .006 = 0.6% so accuracy of the approximation is
675
4
Section 93
A. p. 191 (2) (i) 46, < 1, 3, 6 >, (ii) 60 away from f , (iii) 13/3
(4) f changes faster at P0 in the direction of P2 . The direction of greatest increase at P0
is (1, 1/2, 1/2)
(5) (i) 2, (ii) 1, (iii) /4
(6) 2x 4y 6z = 10
(7) towards the cave *see class notes
1
(8) 2x0 (x x0 ) + 2y0 (y y0 ) 8z0 (z z0 ) = 0 (10) at the point (2, 3, 1), re the laser in
the direction C =< 4, 12, 6 > and escape in the direction C =< 4, 12, 6 >
(14) G =< 1, 3, 1 > *see class notes
Section 94
A. p. 200 (1) (ii) four critical points (1, 1), (1, 1), (1, 1), (1, 1) are all local minimums
(iv) saddles at (2k, 0) and local minimums at ( + 2k, 0), k = . . . , 1, 0, 1, . . .
(vi) saddle at (0, 0) and local minimum at (2/3, 2/3)
(viii) local minimum at (1, 1)
(x) points critical (/2 + k, /2 + k ), (/6 + 2k, /6 + 2k ), (5/6 + 2k, 5/6 +
2k ), (/2 + k, 3/2 k )
(xii) (0, 0) is a saddle, (1/4, 1/2) is a local maximum
B. This exercise p. 200 (2) relates the Second Derivative Test in our textbook to the one in
the Stewart text, it is not necessary to do this exercise so I will list this as an optional
exercise.
Section 95
A. (4) (i) D is the triangle with vertices (0, 0), (1, 2), and(4, 2), on the closed and bounded
set D, the absolute maximum value is 3 and occurs at (4, 2), the absolute minimum value
is 3 and occurs at (1, 2)
(ii) the absolute maximum value is 8 and occurs at (1, 2), (1, 2), the absolute minimum
value is 1 and occurs at (0, 0)
(iii) the absolute maximum value is 4 and occurs at ( 2, 2), the absolute minimum
value is 0 and occurs on the boundary where x = 0, y = 0
(5) (4/3, 7/3, 8/3)
1
2
3
(6) the points ( 1 , 1 , 1+ ) where = 73 5
2
B. x = 4 3 15, y = 4 3 15, z = 2 3 15
C. x = 4, y = 2, z = 8, minimum cost is $192
Section 96
A. (1) max, min values are (i)4, 4, (iii) 5, 5, (v)
(4) x = c/3, y = c/3, z = c/3
2
, 2
3
3
B. Use the Lagrange multiplier method to answer questions (6), B., and C. in Section 95
Section 97
A. (1) (i) k (b a)(d c) , (ii) k1 + k2
(2) (i) 28
(3) (i) k
2
(ii) 23 (iv) k2a
(v) 12
2
Section 98
A. (1) (i) k (16 ), (ii) 3 , (iii) 6 , (iv)
8
3
Section 99
4
A. (1) (i) 4, (iii) 1, (vii) 15 ((2 + e)5/2 35/2 e5/2 + 1)
(2) (i) 40, (ii) 16, (iii) 3
(3) 1/4
Section 100
A. (1) (i) 1/24, (ii) 8/3, (iv) 18, (v) 8/5, (vi) 1/3
(2) (i) 1/6, (ii) 148/105, (v) 32
42
2 x2
(3) (i) 1 y f (x, y )dxdy = 1 1 f (x, y )dydx
(iii)
(v)
3y
x
1
1
f (x, y )dydx = 0 y2 f (x, y )dxdy
0 x3
1 ex
e1
f (x, y )dydx = 1 ln y f (x, y )dxdy
01
6 6y
3 y /3
f (x, y )dxdy + 3 0 f (x, y )dxdy
00
x2
3
6x
= 0 3x f (x, y )dydx
(vii)
2
(4)(i)(x, y ) (x, y ) and f (x, y ) = e sin y 3 = f (x, y ) so
ex sin y 3 dA = 0
D
1
(iii) 3 ba2
Section 101
A. (1) (i)3/2, (ii) a2
1
(2) (i) 1 (b4 a4 ), (ii) (1 cos a2 ), (iii) 48 (b2 a2 ) 2 , (iv) (a2 b2 + b2 ln b2 a2 ln a2 )
8
4
(3) (i) 2 (e 1), (ii) 2/3, (iii) 16/9, (iv) 15/4
1
(4) (i) 3/2, (iii) 8 (9 3 2 )
(5) (i) 4/3, (ii) 14/3, (iii) 8 , (v) 3/2
3