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MATH.APP.220 Multivariable Calculus / Hirvonen
Exam 27.02.2023

No calculators, no written material. A collection of formulas is on the Hipside

1. (a) Show that the limit does not exist.
, aa?
(x,y) (0,0) x4 + 344

(b) Consider the set T' that contains that part of the unit sphere z? + y?+ 2? < 1 where
T<0y<Oandz<0. Write the following integra! in spherical coordinates.

VA 1 dzdydz.
JIIT

Note: You don't have to evaluate the integral, just change the variables.

2. Find the extreme values of f(zn,y) = z?y — ry? + ry+1in a triangle bounded by the

x-axis, the y-axis and the line y = x + 2.
3. Consider the functions
ij 105 007) = 0/4 G (v, v) = (u? SORT u”v?) 5

(a) Find the partial derivatives of f o G using the chain rule.
Note: If you don't remeber how to do it using the chain rule, you can do it without
the chain rule, but you will get less points.

(b) Find all second order partial derivatives of f oG.

4. A solid is bounded by the paraboloid 2 = 2 — x? — y? and the cone 2 = 4/42 +3?. Its
density is given by ;
N) [s====<3
0 30
The solid is cut in half along the xrz-plane. Calculate the y-coordinate of the center of
mass for one half of the solid. (Choose yourself which half, they are symmetrical.)
MATH.App
.2
Exam Formia ha ivarinbl Calculus

x b
T (x) P (a) 4 (a) (x a)

2. (Po G) (x) = P(G (2) C' (x)
20 Dafi(x) <. Dnfilx)
3. F'(x)= 20 Dafa (x) «++ Dnfa(x)

Difm (2) Daf (x) +<. Dnfmlx)

4. D.f (x) = Vf (x)-e
8 rr2(0)
s. [[ samaan [ J J G'oa0rdn) rä
R a Jril6)
1 = psindcos)
6. 4 y=psingdsin/ dzdydz = p? sin 6 dpdodf

z = pcosd

g: m= [J] ota.v.2)

T= % [| 200.09 dV, 7= * []] 200.02 dV,

1. [J] (2? + 77) p (2,4, 2) dV
;

8. J] 3 te.v avan = N) (0

9. j SW (2)g (2)2 = [3 (00)

(7,9)
d(u,v)

b
a

dudv

 

 

 

[ 104 - [7 (o)9 0] - [1099 0) 4

* f'(x)
a (7)

 

42 = [misco]

10. sin?? = 3 (1— cos(20)), — cos*t= 5 (1 +005 (20)