Math 126, Winter 2022 Final Examination Page 1 of 10
1. (2 points per part) Each of the following multiple choice problems has one correct answer.
Circle it. You do not need to show any reasoning.
(a) Suppose |a ⇥ b| > a · b > 0. Then the angle between a and b is between...
(i) 0 and 45 . (ii) 45 and 90 .
☐
(iii) 90 and 135 . (iv) 135 and 180 .
If
HH/ sino
-
oi.to
>
> 0
-
,
then
latlbtcoso ,
E. b-
so
< 0 so
,
tanto)<
0
-1
isobt
.
(b) Suppose proja b = h1, 1, 1i. Then b could be...
(i) h2, 2, 2i. (ii) h 1, 1, 1i. (iii) h2, 2, 2i.
☐
(iv) h2, 3, 4i.
)
>y, =*§' 41 -1,1> =L ! 1. D. so yt =3
-
projg , ,
× >
-
,
,
(c) The intersection of the hyperboloid x2 + y 2 z 2 = 1 and the xy-plane is...
(i) a line. (ii) a circle. (iii) a hyperbola. (iv) the empty set.
☐
' '
ok , I
✗
-
+
y
(d) The surface z = f (x, y) = x3 + y 3 3x 3y has a local maximum of...
fxCx.yj-3x2-3fxxGyk6Ihgshov1dbotLbenegativet@fyCx.y
(i) f (1, 1). (ii) f (1, 1). (iii) f ( 1, 1). (iv) f ( 1, 1).
)= 3yd -3 fyyltsy )=6y
fxylx,y)=0
(e) A lamina occupies the disc x + y 1, and the density at (x, y) is ⇢(x, y) = x3 + y 2 + 2.
2 2
The center of mass of the lamina is...
(i) at the origin. (ii) on the x-axis. (iii) on the y-axis.
r (iv) none of these.
p↳y)=p↳ y)
-
posy) pay )
>
symmetric But
about both axes
, Math 126, Winter 2022 Final Examination Page 2 of 10
2. (4 points per part) For each part, consider the space curve of the vector function
r(t) = ht2 + 1, cos(t) + 4t, 3ti.
(a) Find parametric equations for the line tangent to the space curve at t = 0.
I G) =
4 ,
1
,
D
F ( o) =
'
(2g -
si - ( t) +4 ,
3)
FED =L 0,4 , 3)
: y
=
I + 4T
(b) Find the unit tangent vector to the space curve at t = 0.
in = =
(c) Find the curvature of the space curve at t = 0.
F ( t) {2 D
"
=
-
cos A)
, ,
F "( o) =
42 ,
-
1
, D
ñ (o) ✗ F (o)
40,4 3) K2 I D= 43 8)
' "
= -
,
6 -
, , ,
*
Y÷÷÷='
1. (2 points per part) Each of the following multiple choice problems has one correct answer.
Circle it. You do not need to show any reasoning.
(a) Suppose |a ⇥ b| > a · b > 0. Then the angle between a and b is between...
(i) 0 and 45 . (ii) 45 and 90 .
☐
(iii) 90 and 135 . (iv) 135 and 180 .
If
HH/ sino
-
oi.to
>
> 0
-
,
then
latlbtcoso ,
E. b-
so
< 0 so
,
tanto)<
0
-1
isobt
.
(b) Suppose proja b = h1, 1, 1i. Then b could be...
(i) h2, 2, 2i. (ii) h 1, 1, 1i. (iii) h2, 2, 2i.
☐
(iv) h2, 3, 4i.
)
>y, =*§' 41 -1,1> =L ! 1. D. so yt =3
-
projg , ,
× >
-
,
,
(c) The intersection of the hyperboloid x2 + y 2 z 2 = 1 and the xy-plane is...
(i) a line. (ii) a circle. (iii) a hyperbola. (iv) the empty set.
☐
' '
ok , I
✗
-
+
y
(d) The surface z = f (x, y) = x3 + y 3 3x 3y has a local maximum of...
fxCx.yj-3x2-3fxxGyk6Ihgshov1dbotLbenegativet@fyCx.y
(i) f (1, 1). (ii) f (1, 1). (iii) f ( 1, 1). (iv) f ( 1, 1).
)= 3yd -3 fyyltsy )=6y
fxylx,y)=0
(e) A lamina occupies the disc x + y 1, and the density at (x, y) is ⇢(x, y) = x3 + y 2 + 2.
2 2
The center of mass of the lamina is...
(i) at the origin. (ii) on the x-axis. (iii) on the y-axis.
r (iv) none of these.
p↳y)=p↳ y)
-
posy) pay )
>
symmetric But
about both axes
, Math 126, Winter 2022 Final Examination Page 2 of 10
2. (4 points per part) For each part, consider the space curve of the vector function
r(t) = ht2 + 1, cos(t) + 4t, 3ti.
(a) Find parametric equations for the line tangent to the space curve at t = 0.
I G) =
4 ,
1
,
D
F ( o) =
'
(2g -
si - ( t) +4 ,
3)
FED =L 0,4 , 3)
: y
=
I + 4T
(b) Find the unit tangent vector to the space curve at t = 0.
in = =
(c) Find the curvature of the space curve at t = 0.
F ( t) {2 D
"
=
-
cos A)
, ,
F "( o) =
42 ,
-
1
, D
ñ (o) ✗ F (o)
40,4 3) K2 I D= 43 8)
' "
= -
,
6 -
, , ,
*
Y÷÷÷='
