Let $y = f ( x )$ be the solution to the differential equation $\frac { d y } { d x } = x - y$ with initial condition $f ( 1 ) = 3$. What is the approximation for $f ( 2 )$ obtained by using Euler's method with two steps of equal length starting at $x = 1$ ? (A) $- \frac { 5 } { 4 }$ (B) 1 (C) $\frac { 7 } { 4 }$ (D) 2 (E) $\frac { 21 } { 4 }$
Consider the function $f$ defined on $]-1.5; +\infty[$ by $$f(x) = \ln(2x + 3) - 1$$ The purpose of this exercise is to study the convergence of the sequence $(u_{n})$ defined by: $$u_{0} = 0 \text{ and } u_{n+1} = f(u_{n}) \text{ for all natural integer } n.$$ Part A: Study of an auxiliary function Consider the function $g$ defined on $]-1.5; +\infty[$ by $g(x) = f(x) - x$.
Determine the limit of the function $g$ at $-1.5$.
We admit that the limit of the function $g$ at $+\infty$ is $-\infty$.
Study the variations of the function $g$ on $]-1.5; +\infty[$.
a. Prove that, in the interval $]-0.5; +\infty[$, the equation $g(x) = 0$ admits a unique solution $\alpha$. b. Determine an interval containing $\alpha$ with amplitude $10^{-2}$.
Part B: Study of the sequence $(u_{n})$ We admit that the function $f$ is strictly increasing on $]-1.5; +\infty[$.
Let $x$ be a real number. Show that if $x \in [-1; \alpha]$ then $f(x) \in [-1; \alpha]$.
a. Prove by induction that for all natural integer $n$: $$-1 \leqslant u_{n} \leqslant u_{n+1} \leqslant \alpha.$$ b. Deduce that the sequence $(u_{n})$ converges.
Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \frac{3}{4}x^2 - 2x + 3$$
Draw the table of variations of $f$ on $\mathbb{R}$.
Deduce that for all $x$ belonging to the interval $\left[\frac{4}{3}; 2\right]$, $f(x)$ belongs to the interval $\left[\frac{4}{3}; 2\right]$.
Prove that for all real $x$, $x \leq f(x)$. For this, one may prove that for all real $x$: $$f(x) - x = \frac{3}{4}(x - 2)^2.$$
Consider the sequence $(u_n)$ defined by a real $u_0$ and for all natural integer $n$: $$u_{n+1} = f(u_n).$$ We have therefore, for all natural integer $n$, $$u_{n+1} = \frac{3}{4}u_n^2 - 2u_n + 3.$$
Study of the case: $\frac{4}{3} \leq u_0 \leq 2$. a. Prove by induction that, for all natural integer $n$, $$u_n \leq u_{n+1} \leq 2.$$ b. Deduce that the sequence $(u_n)$ is convergent. c. Prove that the limit of the sequence is equal to 2.
Study of the particular case: $u_0 = 3$. It is admitted that in this case the sequence $(u_n)$ tends to $+\infty$. Copy and complete the following ``threshold'' function written in Python, so that it returns the smallest value of $n$ such that $u_n$ is greater than or equal to 100. \begin{verbatim} def seuil() : u = 3 n = 0 while ... u = ... n = ... return n \end{verbatim}
Study of the case: $u_0 > 2$. Using a proof by contradiction, show that $(u_n)$ is not convergent.
Consider the function $f$ defined for all real $x$ by: $$f ( x ) = \ln \left( \mathrm { e } ^ { \frac { x } { 2 } } + 2 \right)$$ It is admitted that the function $f$ is differentiable on $\mathbb { R }$. Consider the sequence $(u_n)$ defined by $u _ { 0 } = \ln ( 9 )$ and, for all natural integer $n$, $$u _ { n + 1 } = f \left( u _ { n } \right)$$
Show that the function $f$ is strictly increasing on $\mathbb { R }$.
Show that $f ( 2 \ln ( 2 ) ) = 2 \ln ( 2 )$.
Show that $u _ { 1 } = \ln ( 5 )$.
Show by induction that for all natural integer $n$, we have: $$2 \ln ( 2 ) \leqslant u _ { n + 1 } \leqslant u _ { n }$$
Deduce that the sequence $(u_n)$ converges.
a. Solve in $\mathbb { R }$ the equation $X ^ { 2 } - X - 2 = 0$. b. Deduce the set of solutions on $\mathbb { R }$ of the equation: $$\mathrm { e } ^ { x } - \mathrm { e } ^ { \frac { x } { 2 } } - 2 = 0$$ c. Deduce the set of solutions on $\mathbb { R }$ of the equation $f ( x ) = x$. d. Determine the limit of the sequence $\left( u _ { n } \right)$.
We consider $I$ a real interval of strictly positive length, $f$ a function defined on $I$ with values in $I$ and $\left(u_{n}\right)_{n \in \mathbb{N}}$ a sequence defined by $u_{0} \in I$ and $\forall n \in \mathbb{N}, u_{n+1}=f\left(u_{n}\right)$. We assume that the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ converges to an element $\ell$ of $I$ and that $f$ is differentiable at $\ell$. a) Show that $f(\ell)=\ell$. b) Show that if the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ is not stationary then it belongs to $E^{c}$. Give its convergence rate as a function of $f^{\prime}(\ell)$. c) Show that if $\left|f^{\prime}(\ell)\right|>1$, then $\left(u_{n}\right)_{n \in \mathbb{N}}$ is stationary. d) Let $r$ be an integer greater than or equal to 2. We assume that the function $f$ is of class $\mathcal{C}^{r}$ on $I$ and that the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ is not stationary. Show that the convergence rate of $\left(u_{n}\right)_{n \in \mathbb{N}}$ is of order $r$ if and only if $\forall k \in\{1,2, \ldots, r-1\}, f^{(k)}(\ell)=0$.
For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. Does the sequence of functions $(w_n)$ converge uniformly to $W$ on $[0, \mathrm{e}]$?
If $\phi : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathcal { C } ^ { 1 }$ and satisfies $$\sup \left\{ \left| \phi ^ { \prime } ( x ) \right| ; x \in \mathbb { R } \right\} < 1$$ show that $\phi$ has at least one fixed point (one may study the sign of $x - \phi ( x )$ for $| x |$ sufficiently large). Show that this fixed point is unique.
By means of the function $\psi ( x ) = \sqrt { 1 + x ^ { 2 } }$, show that in the previous question hypothesis (1) cannot be replaced by $$\forall x \in \mathbb { R } , \quad \left| \phi ^ { \prime } ( x ) \right| < 1 .$$
Let $\ell$ be a strictly positive integer. Let $F$ be a closed subset of $\mathbb { R } ^ { \ell }$ and let $\phi : F \rightarrow F$ be a map. We assume that there exists $k \in [ 0,1 [$ such that $$\forall x \in F , \quad \forall y \in F , \quad \| \phi ( y ) - \phi ( x ) \| \leqslant k \| y - x \| .$$ (a) We choose a point $x _ { 0 } \in F$. Show that the formula $x _ { n + 1 } = \phi \left( x _ { n } \right)$ defines a sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ of elements of $F$, and that this sequence is convergent in $F$. (b) Deduce that $\phi$ has a unique fixed point in $F$. (c) This fixed point being denoted $x ^ { * }$, bound $\left\| x _ { n } - x ^ { * } \right\|$ as a function of $\left\| x _ { 0 } - x ^ { * } \right\|$. (d) In what precedes, we assume that $$\phi = \underbrace { \theta \circ \cdots \circ \theta } _ { m \text { times } } ,$$ where $\theta : F \rightarrow F$ is a map and $m \geqslant 2$ is an integer. Show that $\theta$ has a fixed point, and a unique one, in $F$.
The sequence $x _ { n }$ is defined by the rules $$\begin{aligned}
x _ { 1 } & = 7 \\
x _ { n + 1 } & = \frac { 23 x _ { n } - 53 } { 5 x _ { n } + 1 }
\end{aligned}$$ The first three terms in the sequence are $7,3,1$ What is the value of $x _ { 100 }$ ? A - 5 B 0 C 1 D 3 E 7