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Multivariable calculus, 2011-10-28.
Per-Sverre Svendsen, Tel.035 - 16 76 15/0709 - 398 526.
Minimum requirements: Grade 5: 27p, 4: 21p, 3: 15p.You are
allowed to use one unmarked, ordinary (non-mathematical) dictionary
of your choice.No additional written material is allowed. Also, no
calculators or any other electronic equipment.
1. Find and classify all critical points of the function
g(x, y) = x2y + 2xy + y2. (3p)
2. The equation exz + z + x− y2 = 2
implicitly defines a function z = f (x, y)
for which f (2, 1) = 0.(a) Calculate f x(2,
1) and f y(2, 1). (3p)
(b) Use the result above to calculate the
linear approximation of f (95
, 1110). (2p)
3. Find the points on the curve x2−xy + y2 = 3 that are
closest to and farthest from the origin. (3p)
4. Calculate
A
x√ y
dx dy , A = {(x, y) ∈
R2 | 1 ≤ y ≤ 4− x2, x ≥ 0}. (3p)
5. Calculate
D
ln(x− y)(x + y)2
dx dy , D = {(x, y) ∈
R2 | 1 ≤ x − y ≤ 2, 1 ≤ x + y ≤
2}. (3p)
6. Calculate the surface integral
S
z3 dS , S = {(x ,y ,z) ∈
R3 | x2 + y2 + z2 = 1, z ≥ 0}.
(3p)
7. Find the absolute minimum and maximum values of
f (x, y) = xy e−2x2−2y2
on the set ∆ = {(x, y) ∈ R2 | x2 + y2 ≤
1}. (5p)
8. Calculate
K
1
x2 + y2 + z2 + 1 dx dy dz ,
K = {(x ,y ,z) ∈ R3 | x2 + y2
+ z2 ≤ 1, z ≥
x2 + y2}. (5p)
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Reference formulas and equations in Multivariable Calculus
Trigonometry and Logarithms
sin(x± y) = sin x cos y ± cos x sin y sin x± sin y =
2 sin x ± y2
cos x ∓ y2
cos(x± y) = cos x cos y ∓ sin x sin y cos x− cos y =
−2sin x + y2
sin x − y
2
tan(x± y) = tan x ± tan y1∓ tan x tan y cos
x + cos y = 2cos
x + y
2 cos
x − y2
cot(x± y) = cot x cot y ∓ 1± cot x + cot y
2sin x sin y = cos(x− y)− cos(x + y)
sin2x = 2sin x cos x 2cos x cos y = cos(x− y) +
cos(x + y)
cos2x = cos2 x− sin2 x = 2cos2 x− 1 = 1− 2sin2 x
2sin x cos y = sin(x− y) + sin(x + y)
ln x + ln y = ln xy ln x
−ln y = ln
x
yln xa = a ln x (x, y > 0)
Standard limits
limx→0+
xα loga x = 0 (a > 1, α > 0)
limx→∞
ax
xα = ∞ (a > 1)
limx→0
sin x
x = 1 lim
x→∞
xα
loga x = ∞ (a > 1, α
> 0)
limx→0
ln(1 + x)
x = 1 lim
n→∞
an
n! = 0
limx→0
ex − 1
x
= 1
Basic derivatives
f (x) f (x)
xa axa−1
ax ax ln a
ln |x| 1x
sin x cos x
cos x − sin x
tan x 1 + tan2 x = 1
cos2 x
arcsin x 1√
1− x2arccos x − 1√
1− x2arctan x
1
1 + x2
lnx + x2 + α
1√ x2 + α
12x√ x
2
+ α + α
2 lnx +√ x
2
+ α
x2
+ α
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Taylor Series
Taylor’s formula for a function f : R→
R
f (a + h) = f (a) +
f (a) h + f (a)
2 h2 + · · · =
∞n=0
f (n)(a)
n! hn
Table of particular expansions (a = 0, h → x)
1. 1
1− x =∞k=0
xk = 1 + x + x2 + x3 + · · ·
(−1 < x
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Double Integrals
General substitution
Assume a one-to-one mapping between a region D in
the xy-plane and a region Duv in the uv
-plane
x = x(u, v)
y = y(u, v)⇔
u = u(x, y)
v = v(x, y)
Then
D
f (x, y) dxdy =
Duv
f (x(u, v), y(u, v))∂ (x, y)
∂ (u, v)
dudv,
with ∂ (x, y)
∂ (u, v) =
xu xv
yu yv
= 0.
Polar coordinates
x = r cos θ
y = r sin θ
∂ (x, y)
∂ (r, θ) = r,
D
f (x, y) dxdy =
Drθ
f (r cos θ, r sin θ) rdr dθ
Triple Integrals
General substitution
As above assume a one-to-one mapping between points (x ,y ,z) in
∆ and (u ,v ,w) in ∆uvw.
∆
f (x ,y ,z) dxdydz =
∆uvw
f (x(u ,v ,w), y(u ,v ,w), z(u ,v ,w)) ∂ (x ,y ,z)
∂ (u ,v ,w)
dudvdw,
with ∂ (x ,y ,z)
∂ (u ,v ,w) = 0
Spherical coordinates
x = ρ sin φ cos θ
y = ρ sin φ sin θ
z = ρ cos φ
∂ (x ,y ,z)
∂ (ρ,φ,θ) = ρ2 sin φ
∆
f (x ,y ,z) dxdydz =
∆ρφθ
f (ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ) ρ2 sin φ dρ
dφ dθ
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Line Integrals
Tangent line integral
Given a parametrized curve C : r(t) =
(x(t), y(t), z(t)), a ≤ t ≤ b and a vector field
F = (P,Q,R).
C
F
· dr
=
b
a
(P,Q,R) · (x
(t), y
(t), z
(t)) dt =
b
a
P x
(t) + Q y
(t) + R z
(t)
dt
Line integral with respect to arc length
Assume a curve C as above and a function
f : R3 → R.
C
f (x ,y ,z) ds =
b a
f (x(t), y(t), z(t))
(x(t))2 + (y(t))2 + (z(t))2 dt, (ds =
d|r|)
Green’s theorem
Given a plane, closed, positively oriented curve
C that encloses a region D and a
field F = (P, Q).
C
F · dr =
D
∂Q∂x − ∂P
∂y
dxdy
Surface Integrals
General parametrized surface
S : r = r(u, v) = (x(u, v), y(u,
v), z(u, v)), (u, v) D.
S
f (x ,y ,z) dS =
D
f (r(u, v))∂ r
∂u × ∂ r
∂v
dudv
Function graph z = h(x, y)
S : r = (x ,y ,z) = (x ,y ,h(x,
y)).
S
f (x ,y ,z) dS =
D
f (x ,y ,h(x, y))
1 + h2x + h2y dxdy
Gauss’ theorem
Given a space region K with (closed) surface
boundary S and a field F .n is the
outer unit normal vector of the surface.
S
F
·n
dS = K
∇ ·F
dx dy dz.
