Limits Webwork
Problem I
flex
consider si
I c
5
co y I
1,0
I c
Curves Webwork
Problem I
2
Parametrize 64 x 10 4 7 ra x a y b
8cost 10 y 8sint 7 a 10 b 7
Problem 2
t t 75 y 3 t 75
a tangent is horizontal when
8 o
i tangent is horizontal when 0
36 25
i tangent is horizontal when t o
b tangent is vertical when o
3t
i when
f 0 3 2 75 0
III
t
Problem 3
2 1 3
y t In to 11
y t If i t 350
y 11 34172
11 it 363
1 y 1 11 544635
Problem 4
Length of parametric curves s t
If It
int d tti
IF
s
fiFEET.at
s
fFÉE at
s
f Effy dt
t.IE
Id s
fEITa s
fihFE
Graidents Webwork
Problem 1
d
If 5in s 4in s r 50in 4 20in
If
hr Product rule
F Gfr 2hrdef
If f4 50 2120 50 s
10000 10000
I
Problem 2
F FEME t intern's derivation
In
g t f Uct U t U t fa u t Lt v't
E 8
81 x ty 2xycost 2x 2xcost
2 2y
2FE LXYCOS
I.IE
xs 8f f x2ty2 2xycosQ 2y 2xcost
IE FE
fi E st E.IE
xst
T 2 3 5057
1
Problem 3
f xy z x y4 z and x s't y s t 2 53 t
2
4 3 354 352
8 3 4 1st 3
Problem 4
Flu u Fu u v G u v F u v H usu find f x for
a f x F x 3
f x G x 3
b f x F 4 x
f x H 4
c f x F X X
f x G x x H x x
d f x F 2x
3 3 2
f x G 2x 2 H 2x 3
Problem5
Of 22 it it 2yxk P C 2 6,9 lies on f x y z 0
Off 2 6,9 181 20 24k f x y z 0
Tangent plane to surface
f Xo Yo 2 x Xo fa Xo yo Zo y y tfs Xo yo 2 2 20 0
18 2 y 6
2 24 2 9 0
18 36 2x 12 242 216 0
18 2y 192 242 0
242 21 92
284
z
9xt
Problem 6
f x y z x yz xyz P G1 2,1
Tf x yz 2xyz 423 x z 230 x y 3 422 k
f 1 2 1 Git 20 8k
find rate of change in the direction of the vector u 0
Daf 152 1 u Of 1 2,1 Find unit vector
20 8k
Def 1 2 1 j 3 61
Ot
4
I
Problem 7
f s t test
f s t test est stest
Of 0 2 4iti
max rate of change will be the magnitude of the gradient
FE
Problem 8
f x xe Y P 2,4
y
2
f x ex Y 2 2 Y e
y
f 2,4 I 212 1 i 2 1 j
Of 2,4 9 20
a 110811 TE TH 85
9
Duf 2,4 f 2,4
11
b u a
2
si Fsi i 20 si Fsu
5
8 5
5.5
8f
5
Optimization Webwork
Problem I
f x y x y 4 y
f x x 4x x
4y xy
y
f x y 4 2 yty falx y x2 4 2 4
2 2 4 x2 2 4
4 0 4 0
x2 4 4 0
41,414
6
I 1
4 8 6 4 4
4
3 2 16
4
Problem 2
f x y 8 3 7 2
3x
24 2 14
f x y x 3 f x y
24 2 14 y 0
5 2 12 7 0
y o and y
CP 0,0 and 3,72
or min
IQeveropenwecqfen.IT say anything about global max
Problem 3
f x x x y 6x and D is the closed region with vertices
6,0 0,6 and 0 6
10,6
F x y 2 6 f x x 2x
y 6
É
6,0 CP at 3,0
0
f 3,0 9
y x 6
2
For X o flo y on 6 y 6
0 6 max 36 min 0
2
For y 6 f x x 6 x x 6 6
EJ
EIIY.IE 36 6xglE E
2 18 36
1 16
x 4 18
g E 36
g
g E
4 18 0
4
I g E
1 min max 0
For 4 6
Integration Webworn
Problem 2
f x y ax by has average value 30 on the rectangle 0 42
y 5
a F
f da
30 s axtby dydx
30 to axy 1 dx
300
f sax 25 dx
2
300 5
252 1
300 10 a 250
Problem 3
f 2x y da where R is the region in the first quadrant
enclosed by the circle y 25 and the lines o
and y x
roost yersino
5
roost rsint rardt
Év rs.int drdt
g f 2r cost
Iii cost
Isino do
cost 12 sint do
4
5 sino cost 1T
12s aint cost
125 25in cos 25in cos
24 0
EE EE
12 SE
3756516
255
Problem 5
f f la r r drdt
f f ear r dr do
f 1 do
1
19
0
19 14 IT 14 17
Problem 6
I t y 22 dxdydz
I 4 If y 22 did z
F 4 9 3 2 322 dydz
I 9y y 32 y da
I 4 36 64 1222 dz
I 100 1222 dz
4
f 4 1002 423
f 232
Problem 8
x x e and
set up using cylindrical coordinates
2 22 8 x2 22
2 2 8
22 22
8
242t222
x21 22 4
KZ 22 r
r 4 r 2
fff du
fififfixardt
2 g r
ff ry I drdt
r 8 r2 r ra drdt
f f or 2 r ³ drdt
f 4r ² -
I i do
f 8 at
801
1617
