Mathias André
Maxime Tô
M1 EPP 2010
Maths Summer Camp
Exercises Sessions 5 and 6
Master Economics and Public Policies, Maths Summer Camp
Derivatives
Proove the derivability and calculate the derivative of the following functions :
√
1. ln(x + x2 + 1)
√
2. x2 x2 + 4
2
3. x(x )
√
2 −1
4. ln √1+x
2
1+x +1
5.
xx
ex (x ln x
− x − 1)
6. ln(ln x)(ln ln ln x − 1)
Polynomials
1. Find P ∈ R[X] s.t. P − XP 0 − X.
2. Find P ∈ R[X] s.t. P (X 2 ) = P (X)2 .
3. Decompose P (X) = X 8 + X 4 + 1.
4. P ∈ R[X] with an odd degree. Show that P has at least one root.
Misc.
1. f : [0, 1] → R continuous, (p, q) ∈ (R∗+ )2 . Show that it exists x0 ∈ [0, 1], pf (0)+qf (1) = (p+q)f (x0 ).
Hint : Use the mean-value theorem with a well-chosen function.
2. a > 0. Solve the system :
xy = y x
1
y = ax
Solutions Sessions 5 and 6
Master Economics and Public Policies, Maths Summer Camp
Derivatives
1. x ∈ R, f 0 (x) =
2. x ∈ R, f 0 (x) =
√ 1
x2 +1
√
x2 + 4
3. x > 0, f 0 (x) = (1 + 2 ln x)xx
2
+1
2
4. x 6= 0, f 0 (x) = x√1+x
2
0
( u ) (x)
u(x)
f (x) = ln v(x) implies that f 0 (x) = vu(x) . Furthermore,
v(x)
u0 (x) = v 0 (x) =
Thus, f 0 (x) =
5. x > 0, f 0 (x) =
0
6. x > 0, f (x) =
√ x
.
x2 +1
√
1+x2√
+1
2x
√
√
1+x2 ( 1+x2 +1)2 ( 1+x2 −1)
xx+1
ex ln x(ln x
ln ln ln x
x ln x
=
√2
x 1+x2
u 0
v
(x) =
u0 (x)v(x)−v 0 (x)u(x)
v 2 (x)
and
√
√
because ( 1 + x2 + 1)( 1 + x2 − 1) = x2 .
− 1)
Polynomials
1. No solution (use degrees).
2. P = X n (use roots).
√
√
3. P (X) = (X 2 +X +1)(X 2 −X +1)(X 2 + 3X +1)(X 2 − 3X −1) (use Y 4 +Y 2 +1 = (Y 2 +1)2 −Y 2 )
4. Use mean value theorem.
Misc.
1. Use g(x) = (p + q)f (x) − pf (0) − qf (1)
2. Take logs.
2