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math 126 final exam with answers 2022 : Very important and wonderful questions for success with ease
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  • June 7, 2022
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  • 2021/2022
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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÷÷÷='

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