Examination of Mathematical Analysis II
University of Warsaw
Department of Economics
year 2013/14, summer semester
9 September 2014 r.
Attention: Each question should be solved on a separate sheet. Each sheet should be clearly signed (first name, surname, index, teacher name, question number). Exam time: 2.5
hours. Do not use calculators or other electronic devices. Each answer should be carefully
justified.
1.
(a) Calculate the indefinite integral:
Z
(b) Calculate the improper integral:
ln(x) dx
.
x3
Z∞
2x
dx.
1 + x4
0
2. Let f : R2 → R be given by the formula
f (x, y) = x2 (1 + y)3 + y 3 .
Find all points (x, y) ∈ R2 such that grad f (x, y) = (0, 0) and determine if the function f has a
local extremum at these points.
3. Let g : R3 → R be given by the formula
g(x, y, z) = 3z + 4xyez + x2 − 3.
(a) Show that there is a neighborhood of the point (x, y) = (0, 0) on which one can define
a function z = z(x, y) of class C 1 such that z(0, 0) = 1 and g(x, y, z(x, y)) = 0. Calculate
∂z
∂z
(0, 0) and ∂y
(0, 0).
∂x
(b) Find the equation of the tangent plane to the surface
M = {(x, y, z) ∈ R3 : g(x, y, z) = 0}.
at the point (0, 0, 1).
4. Among the points which belong to the intersection of the paraboloid z = x2 + y 2 and the
plane x + y + z = 6, find those with maximal and minimal z coordinate.
5. Calculate the double integral:
ZZ
y dxdy,
A
where
A = {(x, y) ∈ R2 : x3 ≤ y ≤ x2 , x ≥ 0}.