LFM Pure

grandes-ecoles 2018 Q34 Limit involving transcendental functions View

We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
Deduce that $\lim _ { \substack { x \rightarrow 0 \\ x > 0 } } \left| \theta ^ { ( n ) } ( x ) \right| = 0$. One may perform the change of variable $y = - \ln x$.

grandes-ecoles 2018 Q35 Regularity and smoothness of transcendental functions View

We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
Demonstrate that $\theta$ is of class $C ^ { \infty }$ on $\mathbb { R }$.

grandes-ecoles 2019 Q2 Higher-order or nth derivative computation View

Let $f$ be defined on $I = ]-\pi/2, \pi/2[$ by $f(x) = \frac{\sin x + 1}{\cos x}$. Show that there exists a sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ with real coefficients such that $$\forall n \in \mathbb{N}, \forall x \in I, \quad f^{(n)}(x) = \frac{P_n(\sin x)}{(\cos x)^{n+1}}$$ Make explicit the polynomials $P_0, P_1, P_2, P_3$ and, for every natural integer $n$, express $P_{n+1}$ as a function of $P_n$ and $P_n^{\prime}$.

grandes-ecoles 2019 Q12 Prove inequality or sign of transcendental expression View

Let $g : \mathbb{R}_+ \rightarrow \mathbb{R}$ be the function defined by $g(x) = \ln\left(1 - p + pe^x\right)$ for all $x \geq 0$.
a. Show that $g$ is well defined and of class $C^2$ on $\mathbb{R}_+$. For $x \geq 0$, express $g''(x)$ in the form $\frac{\alpha\beta}{(\alpha+\beta)^2}$ where $\alpha$ and $\beta$ are positive reals that may depend on $x$.
b. Show that $g''(x) \leq \frac{1}{4}$ for all $x \geq 0$.
c. Show that $$\ln\left(1 - p + pe^x\right) \leq px + \frac{x^2}{8} \text{ for all } x \geq 0$$

grandes-ecoles 2020 Q23 Differentiation under the integral sign with transcendental kernels View

If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Let $N$ be a natural integer and let $F _ { N }$ be the function defined, for all real $x$, by $F _ { N } ( x ) = \int _ { - N } ^ { N } K _ { a , x } ( t ) \mathrm { d } t$. Show that $F _ { N }$ is of class $C ^ { 1 }$ on $\mathbb { R }$ and that, for all real $x , F _ { N } ^ { \prime } ( x ) = N \operatorname { sinc } ( N x )$.

grandes-ecoles 2020 Q35 Existence and uniqueness of solutions involving transcendental equations View

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. Prove that, for every positive real $x$, $W(x)$ is a fixed point of $\phi_x$, that is, a solution of the equation $\phi_x(t) = t$.

grandes-ecoles 2020 Q36 Compute derivative of transcendental function View

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ Prove that, for every positive real $x$, the function $\phi_x$ is of class $\mathcal{C}^2$ on $\mathbb{R}$ and that $$\forall t \in \mathbb{R}, \quad 0 \leqslant \phi_x'(t) \leqslant \frac{x}{\mathrm{e}}.$$

grandes-ecoles 2021 Q19a Higher-order or nth derivative computation View

Let $v(x) = \frac{1}{\cos(x)}$ on $]-\frac{\pi}{2}, \frac{\pi}{2}[$ and $E_{2k} = v^{(2k)}(0)$ for $k \in \mathbb{N}$.
Show that, for $n \in \mathbb{N}^*$, $$\sum_{k=0}^{n} (-1)^k \binom{2n}{2k} E_{2k} = 0$$ and deduce the values of $E_0$, $E_2$ and $E_4$.

grandes-ecoles 2021 Q27 Regularity and smoothness of transcendental functions View

Let $\varphi$ be the function defined on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \quad \varphi(t) = \begin{cases} 0 & \text{if } t \leqslant 0 \\ \mathrm{e}^{-1/t} & \text{otherwise} \end{cases}$$
Show that $\varphi$ is of class $\mathcal{C}^\infty$ on $\mathbb{R}$.
One may show that: $\forall k \in \mathbb{N}, \exists P_k \in \mathbb{R}[X], \forall t > 0, \varphi^{(k)}(t) = P_k(1/t)\mathrm{e}^{-1/t}$.

grandes-ecoles 2021 Q27 Regularity and smoothness of transcendental functions View

Let $\varphi$ be the function defined on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \quad \varphi(t) = \begin{cases} 0 & \text{if } t \leqslant 0 \\ \mathrm{e}^{-1/t} & \text{otherwise} \end{cases}$$ Show that $\varphi$ is of class $\mathcal{C}^\infty$ on $\mathbb{R}$.
One may show that: $\forall k \in \mathbb{N}, \exists P_k \in \mathbb{R}[X], \forall t > 0, \varphi^{(k)}(t) = P_k(1/t)\mathrm{e}^{-1/t}$.

grandes-ecoles 2021 Q28 Regularity and smoothness of transcendental functions View

Let $\psi$ be the function defined on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \quad \psi(t) = \begin{cases} 0 & \text{if } t \notin ]-1,1[ \\ \mathrm{e}^{1/(t^2-1)} & \text{otherwise.} \end{cases}$$
Show, by expressing it in terms of $\varphi$, that $\psi$ is of class $\mathcal{C}^\infty$.

grandes-ecoles 2021 Q28 Regularity and smoothness of transcendental functions View

Let $\psi$ be the function defined on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \quad \psi(t) = \begin{cases} 0 & \text{if } t \notin \left]-1,1\right[ \\ \mathrm{e}^{1/(t^2-1)} & \text{otherwise.} \end{cases}$$ Show, by expressing it in terms of $\varphi$, that $\psi$ is of class $\mathcal{C}^\infty$.

grandes-ecoles 2021 Q29 Regularity and smoothness of transcendental functions View

Let $\theta$ be the unique antiderivative of $\psi$ vanishing at 0. Show that $\theta$ is of class $\mathcal{C}^\infty$, constant on $]-\infty, -1]$ (we denote this constant by $A$) and constant on $[1, +\infty[$ (we denote this constant by $B$). Verify that $A \neq B$.

grandes-ecoles 2021 Q29 Regularity and smoothness of transcendental functions View

Let $\theta$ be the unique antiderivative of $\psi$ vanishing at 0. Show that $\theta$ is of class $\mathcal{C}^\infty$, constant on $]-\infty, -1]$ (we denote this constant by $A$) and constant on $[1, +\infty[$ (we denote this constant by $B$). Verify that $A \neq B$.

grandes-ecoles 2023 Q13 Compute derivative of transcendental function View

We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by: $$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$
Show that $\Psi$ is of class $\mathcal{C}^1$ on $\mathbf{R}$, then that for all $x \in \mathbf{R}$, $$\Psi'(x) = 4\sum_{k=1}^{+\infty} \rho^k \sin(2kx)$$

grandes-ecoles 2023 Q14 Prove inequality or sign of transcendental expression View

Let $A \in S_n^{++}(\mathbf{R})$ and let $f : t \mapsto \ln(\operatorname{det}(I_n + tA))$. Show that $$\forall t \in \mathbf{R}_+, \quad \ln(\operatorname{det}(I_n + tA)) \leq \operatorname{Tr}(A) t.$$

grandes-ecoles 2023 Q14 Prove inequality or sign of transcendental expression View

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and let $f : t \mapsto \ln \left( \operatorname { det } \left( I _ { n } + t A \right) \right)$. Show that
$$\forall t \in \mathbf { R } _ { + } , \quad \ln \left( \operatorname { det } \left( I _ { n } + t A \right) \right) \leq \operatorname { Tr } ( A ) t$$

grandes-ecoles 2023 Q15 Regularity and smoothness of transcendental functions View

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let the application $f_A$ defined on $\mathbf{R}$ by $$f_A(t) = \operatorname{det}(A + tM).$$ Show that $f_A$ is of class $C^\infty$ on $\mathbf{R}$.

grandes-ecoles 2023 Q15 Regularity and smoothness of transcendental functions View

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
Show that $f _ { A }$ is of class $C ^ { \infty }$ on $\mathbf { R }$.

grandes-ecoles 2024 QII Compute derivative of transcendental function View

Exercise II

II-A- The function $f$ defined on $\mathbb { R } ^ { * }$ by $f ( x ) = e ^ { \frac { 1 } { x } }$ has derivative $f ^ { \prime } ( x ) = e ^ { \frac { 1 } { x } }$. II-B- The function $F$ defined on $[ 0 ; + \infty [$ by $F ( x ) = x \sqrt { x }$ is an antiderivative of the function $f$ defined by $f ( x ) = \frac { 3 } { 2 } \sqrt { x }$. II-C- The function $f$ defined on $] 0 ; + \infty [$ by $f ( x ) = ( \ln ( 3 x ) ) ^ { 2 }$ has derivative $f ^ { \prime } ( x ) = \frac { 2 } { 3 x } \ln ( 3 x )$. II-D- $\quad \lim _ { x \rightarrow 0 } ( x \ln ( x ) - x ) = - \infty$. II-E- $\quad \lim _ { x \rightarrow + \infty } \left( x e ^ { x } - \ln ( x ) \right) = 0$.
For each statement, indicate whether it is TRUE or FALSE.

grandes-ecoles 2025 Q18 Differentiation under the integral sign with transcendental kernels View

Justify that the function $\psi_n$ is differentiable on $\mathbb{R}_+$ and that $\psi_n' = m_n$.
We admit that, when $\psi$ is differentiable on $\mathbb{R}_+^*$, then $(\lim \psi_n)' = \lim \psi_n'$, that is $\psi' = m$, on $\mathbb{R}_+^*$.

grandes-ecoles 2025 Q24 Compute derivative of transcendental function View

In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV.
Deduce an expression for the function $m$ and conclude that $m^+ = 0$.
Recall that $m = \psi'$ when $\psi$ is differentiable on $\mathbb{R}_+^*$.

isi-entrance 2010 Q4 Limit involving transcendental functions View

$\lim_{x \to 2} \left[\frac{e^{x^{2}} - e^{2x}}{(x-2)e^{2x}}\right]$ equals
(a) 0
(b) 1
(c) 2
(d) 3

isi-entrance 2010 Q7 Piecewise function analysis with transcendental components View

Let $f(x) = |x|\sin x + |x - \pi|\cos x$ for $x \in \mathbb{R}$. Then
(a) $f$ is differentiable at $x = 0$ and $x = \pi$
(b) $f$ is not differentiable at $x = 0$ and $x = \pi$
(c) $f$ is differentiable at $x = 0$ but not differentiable at $x = \pi$
(d) $f$ is not differentiable at $x = 0$ but differentiable at $x = \pi$

isi-entrance 2012 Q7 Regularity and smoothness of transcendental functions View

Let $f(x) = e^{-1/x}$ for $x > 0$ and $f(x) = 0$ for $x \leq 0$. Which of the following is true?
(A) $f$ is not differentiable at $x = 0$
(B) $f$ is differentiable at $x = 0$ but $f'$ is not differentiable at $x = 0$
(C) $f$ is differentiable at $x = 0$ and $f'$ is differentiable at $x = 0$
(D) $f$ is differentiable everywhere and $f'$ is also differentiable everywhere

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LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

Simultaneous equations 79

Proof 711

Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

Trig Graphs & Exact Values 75

Standard trigonometric equations 106

Trig Proofs 53

Quadratic trigonometric equations 36

Trigonometric equations in context 18

Modulus function 49

Matrices 1553

Linear transformations 26

Invariant lines and eigenvalues and vectors 188

Reciprocal Trig & Identities 44

Addition & Double Angle Formulae 98

Harmonic Form 12

Small angle approximation 20

Chain Rule 113

Product & Quotient Rules 18

Differentiating Transcendental Functions 183

Exercise II

Connected Rates of Change 56

Standard Integrals and Reverse Chain Rule 43

Integration by Substitution 116

Integration by Parts 74

Implicit equations and differentiation 43

Differential equations 352

3x3 Matrices 144