Department of Mathematics ISEG
Bachelor in Economics, Finance and Managment
2nd semester - Mathematics I - 2021/2022
Resit Period - 1st of July 2022 - Duration 2h
1. Consider the linear subspace of R3 given by
V = span {(1, 0, 1); (0, 2, 1); (3, 3, 4); (0, 1, 0)}.
(a) [1.5 values] Give a basis of V and indicate its dimension.
(b) [1 value] Compute the coordinates of v = (−1, 2, 1) in the basis you found.
If you have not found a basis of V , use this one B{(1, 0, 0); (0, 3, 0); (3, 3, 1)}.
2. Consider the matrix with real entries given by


1 2 λ
Aλ = 2 2 0 
6 9 3
where λ denotes a real parameter.
(a) [1 values] Determine for which values of λ the matrix Aλ is invertible.
(b) [1.5 values] Set λ = 2 and compute A−1
2 .
3. Consider the following system of linear equations AX = B where


 
 
0 2 2a
x
8
A = 2 2a a  ; X = y  and B = 4
1 a 1
z
2
where a ∈ R denotes a real parameter.
(a) [2 values] Classify the system according to the parameter a ∈ R.
(b) [1.5 values] Set a = 2 and determine the general solution of the system.
4. Let f be the real function defined by
f (x) = ln(x4 ) − 2x2
(a) [1 value] Write the domain of f and justify if it is an open or closed, bounded or
unbounded subset of the real line.
(b) [1.5 values] Write the Taylor polynomial of order two around the point x = 1 of f .
(c) [1 value] By means of this second order approximation, estimate the value of f (0.8).
Hint: Pn f (x) =
Pn
k=0
f (k) (x0 )
(x
k!
− x0 )k for all x in a neighbourhood of x0 .
5. [2 values] Verify that the following series is a geometric series and determine its ratio,
then study its convergence and (if possible) compute its sum.
+∞
X
(−1)n × 5n+1
32n+1
n=2
.
6. [2 values] Compute the following limit
Rx
lim
x→1
1
(et − e)dt
ln2 (x)
.
7. [2 values] Use the substitution ln(x) = t to compute the following integral
Z e
ln(x)
dx.
1 x(ln(x) + 3)
8. Denote by A the region of the plane delimited by the lines of equations:
y =x+1
and
y = −x + 1
and
y = 0.
(a) [0.5 values] Geometrically represent the region A.
(b) [1.5 values] Compute the area of A by means of defined integrals.
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