24 Written exams and pseudo–exams[1T8]

E442

[1T9] Note:reworked from the written exam held January 26th, 2016.

Let \((q_ n )_{n≥1}\) be an enumeration of the rationals of \((0, 1)\) and define

\[ f(t) {\stackrel{.}{=}}∑_{n: q_ n {\lt}t} 2^{ −n} \]

and

\[ g (t) {\stackrel{.}{=}}∑_{n: q_ n ≤ t} 2^{ −n} \]

for \(t ∈ (0, 1)\).

Show that \(f,g\) are strictly increasing.
Calculate limits for \(t ↓ 0\) and \(t ↑ 1\).
Show that \(f\) is left continuous, \(g\) is right continuous, and that

\[ \lim _{𝜏→ t+} f (𝜏) =g(t) \quad ,\quad \lim _{𝜏→ t-} g (𝜏) =f(t) \quad . \]
Also show that \(f\) is discontinuous in \(t\) if and only if \(t ∈ ℚ ∩ (0, 1)\); and similarly for \(g\).
What changes if we replace \(2^{ −n}\) with the term \(a_ n\) of an absolutely convergent series?

Hidden solution: [UNACCESSIBLE UUID ’1TB’][UNACCESSIBLE UUID ’1TC’]

E442

[1TD]Prerequisites:6.Note:written exam, June 23th, 2012.

Let f be a \(C^ 1\) class function on \(ℝ\), with \(f (0) ≠ 0\). Prove that \(x ∈ ℝ\) exists such that the two vectors

\[ v = (x, f (x)) \quad , \quad w = (− f' (x), 1) \]

are linearly dependent. (Note that the vector \(w\) is orthogonal to the tangent of the graph of \(f\).) Discuss the possibility that this condition is verified for every \(x ∈ ℝ\).

Hidden solution: [UNACCESSIBLE UUID ’1TF’]

E442

[1TG]Note:adapted from the written exam, April 9th, 2011.

Let \(f:[0,∞)→ℝ\) be a continuous function such that

\[ \lim _{x→+∞} f(x)/x=+∞\quad . \]

Fixed \(a{\lt}f(0)\), let \(M_ a\) be the set of \(m∈ℝ\) such that the line \(y=mx+a\) intersects the graph \(y=f(x)\) of the function \(f\) at least in one point: show that \(M_ a\) admits minimum \(\hat m=\hat m(a)\);
show that \(\hat m\) depends continuously on \(a\), ¹
and that \(\hat m(a)\) is monotonic strictly decreasing.
If \(f\) is differentiable, show that the line \(y=\hat m(a) x+a\) is tangent to the graph at all points where it encounters it.
Suppose further that \(f\) is of class \(C^ 2\) and that \(f''(x){\gt}0∀ x{\gt}0\) ² . Show that there is only one point \(x\) where the line \(y=\hat m(a) x+a\) meets the graph \(y=f(x)\); name it \(\hat x=\hat x(a)\);
and show that the functions \(a↦ \hat x(a)\) and \(a↦ \hat m(a)\) are differentiable.

Hidden solution: [UNACCESSIBLE UUID ’1TH’]

E442

[1TJ] Topics:osculating circle. Note:adapted from the written exam, April 9th 2011.

Let \(f:ℝ→ℝ\) be twice differentiable in \(0\), with \(f(0)=0\) and \(f''(0)≠ 0\). Prove that there is an unique point \(P=(a,b)\) in the plane and an unique constant \(r{\gt}0\), such that

\[ d\big(\, P,\, (x,f(x))\, \big) = r + o(x^ 2), \]

determining \(a,b,r\) as a function of \(f'(0),f''(0)\). Here \(d(P,Q)\) is the Euclidean distance between two points \(P,Q\) in the plane.

Hint. First, study the case in which also \(f'(0)=0\).

(The graph of the function \(f\) is a curve in the plane; by hypothesis this curve passes through the origin. In this exercise we have determined the circle, of radius \(r\) and center \(P\), which best approximates the curve near the origin. This circle is called the ”osculating circle”, and its radius is called the ”radius of curvature”, and the inverse of the radius is the ”curvature” of the curve at the origin.)

Hidden solution: [UNACCESSIBLE UUID ’1TK’][UNACCESSIBLE UUID ’1TM’]

E442

[1TN]Note:Exercise 2, written exam 4 April 2009.

Verify that for every \(t{\gt}1\) the equation

\[ \sin x = x ^ t \]

admits one and only one solution \(x{\gt}0\).
Call \(f(t)\) this solution, determine the image of the function \(t\) and show that it is strictly increasing and continuous on \((1,+∞)\).
Prove that \(f\) is extended by continuity to \(t=1\) and discuss the existence of the right derivative of the prolonged function at that point.

Hidden solution: [UNACCESSIBLE UUID ’1TP’]

E442

[1TS]Suppose that \(f:ℝ→ℝ\) is a continuous fuction such that \(\cos (f(x))\) is differentiable: can it be deduced that \(f\) is differentiable? If it is true, prove it. If it is not true, produce an example.

E442

[1TT]Suppose that \(f:ℝ→ℝ\) is a function such that \(f{\gt}0\) and \(\log (f(x))\) is convex: can it be deduced that \(f\) is convex? If it is true, prove it. If it is not true, produce an example.

E442

[1TV]Let \(f,g:ℝ→ℝ\) be class \(C^∞\) function, with \(g{\gt}0\): show that \(f/g\) is a class \(C^∞\) function.

E442

[1TW]Let \(f(x)=∑_{n=0}^∞ a_ n x^ n\) with radius of convergence \(𝜌{\gt}0\), and let \(f(0)=f'(0)=\ldots =f^{(n)}(0)=0\); show that the function \(g(x)=f(x) / x^ n\) is extendable to \(x=0\); show that (the extension of) \(g\) coincides with an appropriate power series \(g(x)=∑_{n=0}^∞ b_ n x^ n\). What can be said about the radius of convergence of \(g\)?

E442

[1TX]Note:Dirichlet criterion for integrals.

Let \(f,g:[0,∞)→ℝ\) be continuous, where \(f\) is positive and monotonic decreasing with \(\lim _{x→∞} f(x)=0\), while

\[ \sup _{x{\gt}0} |∫_ 0^ x g(t)\, {\mathbb {d}}t| {\lt}∞\quad . \]

Then prove that

\[ \lim _{x→∞} ∫_ 0^ x f(t) g(t)\, {\mathbb {d}}t \]

converges.

E442

[1TY]Note:written exam 12/1/2013.

Given a subset \(E\) of \(ℕ\) and an integer \(n ∈ ℕ\), the expression

\[ \frac{\mathrm{card}( E ∩ \{ 0, 1, . . . , n\} )}{n+1} \]

indicates which fraction of the segment \(\{ 0, 1, . . . , n\} \) is contained in E. The notion of "density" in \(ℕ\) of \(E\) refers to the behavior of such fractions as n tends to infinity. Precisely, we define the upper density \( \overline d(E)\) of E and its lower density \(\underline d(E)\) as

\[ \overline d(E) = \limsup _{n→∞} \frac{\mathrm{card}( E ∩ \{ 0, 1, … , n\} )}{n+1}\quad , \]

\[ \underline d(E) = \liminf _{n→∞} \frac{\mathrm{card}( E ∩ \{ 0, 1, … , n\} )}{n+1}\quad . \]

If \(\overline d(E) = \underline d(E) = d ∈ [0, 1]\), E is said to have density d. (See also [ 62 ] .)

Prove that, for every \(α ∈ ℝ, α ≥ 1\), the set \(E_α = [nα] : n ∈ N\) has density \(d = 1/α\) (the symbol \([x]\) indicates the integer part of \(x ∈ R\)).
Let \(E = \{ m_ 0 , m_ 1 , … , m_ k , … \} \) be an infinite set, with \(m_ 0 {\lt} m_ 1 {\lt} … {\lt} m_ k {\lt} …\). Prove that \(\overline d(E) = \limsup _{ k→∞} \frac{k}{m_ k}\) and \(\underline d(E) = \liminf _{ k→∞} \frac{k}{m_ k}\).
Find a set E with \(\overline d(E) = \overline d(ℕ ⧵ E) = 1\).

E442

[1TZ] Note:exercise 6 in the written exam 13/1/2011.

Each integer \(n ≥ 1\) decomposes uniquely as \(n = 2^ k d\), with \(k ∈ N\) and \(d\) odd integer. Consider the sequence \(a_ n = d/2^ k\) and compute

its upper and lower limit;
the set of limit points.

E442

[1V0] Topics:matrix, determinant. Note:exercise 4 in the pseudo-homework of 14/3/2013.

Let \(A∈ℝ^{2× 2}\) be a 2 by 2 matrix. Identifying \(ℝ^{2× 2}\) with \(ℝ^{4}\), calculate the gradient of the determinant, and verify that it is nonzero if and only if the matrix is nonzero.
Let \(Z\) be the set of matrices \(ℝ^{2× 2}\) with zero determinant. Show that it is a closed set with an empty interior.

Hidden solution: [UNACCESSIBLE UUID ’1V1’]

E442

[1V2]Topics:matrix,determinant.Difficulty:*.

Prove Jacobi’s formula:

\[ \frac{d}{d a_{i,j}} \det (A) = C_{i,j}\quad , \]

where \(a_{i,j}\) is the element of \(A\) in row \(i\) and column \(j\), and \(C\) is the matrix of cofactors of \(A\), which is the transpose of the adjoint matrix \({\operatorname {adj}}(A)\). Consequently, if \(F:ℝ→ℂ^{n × n}\) is differentiable, then

\[ {\frac{d}{dt}}\det F(t)={\operatorname {tr}}\left({\operatorname {adj}}(F(t))\, {\frac{dF(t)}{dt}}\right) \]

where \({\operatorname {tr}}(X)\) is the trace of \(X\).

Hint: use Laplace’s expansion for the determinant.

Hidden solution: [UNACCESSIBLE UUID ’1V3’]

E442

[1V4]Topics:matrix,determinant.Prerequisites:14.Difficulty:*.

We want to generalize the results of the previous exercise 13 to the case of matrices \(n× n\).

Recall the following properties of the determinant of matrices \(A∈ℝ^{n× n}\).

The rank is the dimension of the image of \(A\) (considered as a linear application from \(ℝ^ n\) to \(ℝ^ n\)) and is also the maximum number of linearly independent columns in \(A\).
\(A\) has rank \(n\) if and only \(\det (A)≠ 0\).
If you exchange two columns in \(A\), the determinant changes sign;
if you add a multiple of another column to a column, the determinant does not change.
The characterization of rank through minors, "The rank of A is equal to the highest order of an invertible minor of A".
Laplace’s expansion of the determinant, and Jacobi’s formula (cf 14).
The determinant of \(A\) is equal to the determinant of the transpose; So every previous result holds, if you read ”row” instead of ”column”.