L1 - Bilingual Group - 2012/2013 - 15 May 2013 - Session 1
Examination in Mathematics for Sciences (MS2)
Duration: 3 hours Documents, calculators, cell phones, smartphones and tablets are not allowed The total score: 7+3+5+5 is given in an informal manner and may be modied
Exercise 1 :
1. (a) Decompose the rational fraction X4−6
(X+ 2)(X2+ 1) into simple elements onR.
(b) From this deduce the calculation of the integral : I = Z 1
0
x4−6
(x+ 2)(x2+ 1)dx.
2. Give the domain of denition of the following primitives (antiderivatives), next nd their expres- sions :
(a) Z
ln(x)dx, thenZ
ln(x) ln(x2)dx. (Hint : IBP) (b)
Z sinx dx
sin2(x) − 5. (Hint : Bioche's rules)
Exercise 2 : One wants to nd the set S of functions y : R → R, solutions of the dierential equation :
(E) y00+ 2y0−3y =xe2x.
1. If (E0) denotes the associated homogeneous equation, write down the characteristic equation and nd the setS0 of solutions y0 of (E0).
2. Find the set S of solutionsy of (E).
Exercise 3 : Letf :R2 →Rbe dened byf(x, y) =x2+ 2xy2. Let us denote byGf the graph of f, namely, the setGf ={(x, y, z)∈R3 |(x, y)∈R2 etz=f(x, y)}.
1. Draw in the plane the curve of level z= 0 for f.
2. Compute the partial derivatives of f (to be denoted by ∂f
∂x and ∂f
∂y) in an arbitrary point (x, y)∈R2.
3. Give the expression of the gradient vector off in such a point (to be denoted bygradf(x, y)).
4. Find a cartesian equation of the tangent plane toGf at the pointM(1; 1; 3). 5. Find the set of critical points off.
6. By examining f(−y2, y), show that f does not have a local minimum in the origin.
7. Show thatf does not have a local maximum in the origin.
8. Make a complete study for the extrema off. Exercise 4 : Consider the dierential equation :
(E) x3y0+ 3x2y=−sinx
1) Find the set of solutions of(E)and give the intervals on which these solutions are valid.
2) Give the solution which satisesy(1) = 2. 3) Does(E)have solutions onR?
Questions related to the lecture.
1) Recall the denitions of the following functions and give the expression of their derivatives : Arcsin, Arccos, Arctan
2) Let f : [0,1] → R be a function. Suppose f continuous, strictly increasing on [0,1]; let g be the inverse function off (in particular, ∀x∈[0; 1], g(f(x)) =x).
Let Φ : [0,1]→Rbe a function dened as follows :
∀x∈[0,1], Φ(x) = Z x
0
f(t)dt+ Z f(x)
0
g(t)dt−xf(x).
Show that if f is moreover dierentiable thenΦis dierentiable too andΦ0 = 0.
