EXAM OF CALCULUS I - 03/12/2008
Exercise 1
a) Represent in the complex plane the set
A={z∈C:z¯z−Re(¯z(2 −i)) <−1
4,Im(z)>0}.
b) Solve the equation 4iz3−8 = 0.
Exercise 2 Establish a, b ∈Rsuch that the function
f(x) =
xeax +a x < 0
−bx + log(cos 2x) 0 ≤x < π
4
is differentiable at x= 0.
Exercise 3 Let f(x) = 2(e3x2−cos x)−7x2.
a) Compute the infinitesimal order and the principal part of fat x= 0.
b) Establish whether f(x) has a maximum, a minimum or an inflection
point at x= 0.
Exercise 4 Consider the function
f(x) = e−1
x(4
x2−1).
a) Study fand draw its graph (Note: the study of f00 is not required).
b) Establish the maximal interval containing x= 2 on which fis invert-
ible.
c) Draw the graph of |f(x+ 2)|.
Exercise 5 Study the convergence of the numerical series
+∞
X
n=1
log(n+ 2
n)
and
+∞
X
n=1
log(n2+ 2
n2).
