LFM Pure

grandes-ecoles 2016 QIV.A.1 Proof of Differentiability Class for Parameterized Integrals View

For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application $$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
Show that $\delta(f)$ is of class $\mathcal{C}^\infty$ on $\mathbb{R}_+^*$. Compare $(\delta(f))'$ and $\delta(f')$.

grandes-ecoles 2016 QIV.A.3 Limit Evaluation Involving Composition or Substitution View

For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application $$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
Explain why, for every $x > 0$, there exists a $y_1 \in \left]0, 1\right[$ such that $$(\delta(f))(x) = f'(x + y_1)$$

grandes-ecoles 2016 QIV.A.4 Proof of Differentiability Class for Parameterized Integrals View

For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application $$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
Deduce that for every $x > 0$, for every $n \in \mathbb{N}^*$, there exists a $y_n \in \left]0, n\right[$ such that $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ One may proceed by induction on $n \in \mathbb{N}^*$ and use the three preceding questions.

grandes-ecoles 2020 Q2 Derivative of Inverse Functions View

Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. That is, for every real $x \geqslant -\mathrm{e}^{-1}$, $W(x)$ is the unique solution of $f(t) = x$ with $t \in [-1,+\infty[$. Justify that $W$ is continuous on $\left[ - \mathrm { e } ^ { - 1 } , + \infty \left[ \right. \right.$ and is of class $\mathcal { C } ^ { \infty }$ on $] - \mathrm { e } ^ { - 1 } , + \infty [$.

grandes-ecoles 2020 Q3 Derivative of Inverse Functions View

Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. Explicitly determine $W ( 0 )$ and $W ^ { \prime } ( 0 )$.

grandes-ecoles 2023 Q18 Derivative of Composite Function in Applied/Modeling Context View

We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, the endomorphism $u : X \mapsto (I_N - K)X$, the orthogonal projection $p : E \rightarrow E$ onto $\ker(u)$, and for $X \in E$, the function $\varphi_X : t \mapsto \|H_t X\|^2$. We set $Y = X - p(X)$. We denote by $\lambda$ the smallest nonzero eigenvalue of $u$. Show that for all real $t \in \mathbf{R}_+$, $\varphi_Y'(t) \leq -2\lambda \varphi_Y(t)$. Deduce that $\forall t \in \mathbf{R}_+, \|H_t X - p(X)\|^2 \leq e^{-2\lambda t} \|X - p(X)\|^2$.

grandes-ecoles 2023 Q18 Proof of Differentiability Class for Parameterized Integrals View

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Show that for all $k \geqslant 1$ we have
$$\left\|\psi_{k}^{(k)}\right\|_{\infty} \leqslant \left\|\psi_{0}\right\|_{\infty} \frac{1}{\mu_{1} \cdots \mu_{k}}$$

grandes-ecoles 2023 Q19 Derivative of Composite Function in Applied/Modeling Context View

We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, the endomorphism $u : X \mapsto (I_N - K)X$, the orthogonal projection $p : E \rightarrow E$ onto $\ker(u) = \operatorname{Vect}(U)$, and $\lambda$ the smallest nonzero eigenvalue of $u$. We have established that $\forall t \in \mathbf{R}_+, \|H_t X - p(X)\|^2 \leq e^{-2\lambda t} \|X - p(X)\|^2$. Let $i \in \llbracket 1;N \rrbracket$ and $t \in \mathbf{R}_+$. Show that $\|H_t E_i - \pi[i] U\| \leq e^{-\lambda t} \sqrt{\pi[i]}$.

grandes-ecoles 2023 Q19 Proof of Differentiability Class for Parameterized Integrals View

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Let $f = \lim_{n \rightarrow \infty} \psi_{n}$. Show that $f$ has compact support and that if $\psi_{0}$ is positive and not identically zero, then so is $f$.

grandes-ecoles 2023 Q20 Derivative of Composite Function in Applied/Modeling Context View

We consider the matrix $H_t$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, H_t[i,j] = e^{-t} \sum_{n=0}^{+\infty} \frac{t^n K^n[i,j]}{n!}$, and $\pi$ the stationary probability. Show that for all $(i,j) \in \llbracket 1;N \rrbracket^2$ and all $t \in \mathbf{R}_+$, $$H_t[i,j] - \pi[j] = \sum_{k=1}^{N} \left(H_{t/2}[i,k] - \pi[k]\right)\left(H_{t/2}[k,j] - \pi[j]\right)$$ One may use question 5.

grandes-ecoles 2023 Q20 Proof of Differentiability Class for Parameterized Integrals View

We seek to show that if $(M_{n})_{n \geqslant 0}$ is a sequence of strictly positive real numbers such that the series $\sum_{n \geqslant 1} \frac{M_{n-1}}{M_{n}}$ converges, there exists a function $f \in \mathcal{C}_{c}^{\infty}(\mathbb{R})$ not identically zero such that for all $n \geqslant 0$, $\|f^{(n)}\|_{\infty} \leqslant M_{n}$.
Conclude regarding the initially posed question.

grandes-ecoles 2024 Q7 Proof of Differentiability Class for Parameterized Integrals View

Show that for all functions $f, g \in C^{2}(\mathbf{R})$ such that the functions $f, f^{\prime}, f^{\prime\prime}$ and $g$ have slow growth, we have
$$\int_{-\infty}^{+\infty} L(f)(x) g(x) \varphi(x) \mathrm{d}x = -\int_{-\infty}^{+\infty} f^{\prime}(x) g^{\prime}(x) \varphi(x) \mathrm{d}x$$
where $\forall x \in \mathbf{R}, \quad L(f)(x) = f^{\prime\prime}(x) - x f^{\prime}(x).$

grandes-ecoles 2024 Q7 Proof of Differentiability Class for Parameterized Integrals View

Show that for all functions $f, g \in C^2(\mathbf{R})$ such that the functions $f, f', f''$ and $g$ have slow growth, we have $$\int_{-\infty}^{+\infty} L(f)(x)\,g(x)\,\varphi(x)\,\mathrm{d}x = -\int_{-\infty}^{+\infty} f'(x)\,g'(x)\,\varphi(x)\,\mathrm{d}x,$$ where $L(f)(x) = f''(x) - x f'(x)$ and $\varphi(x) = \frac{1}{\sqrt{2\pi}}\mathrm{e}^{-x^2/2}$.

grandes-ecoles 2024 Q8 Proof of Differentiability Class for Parameterized Integrals View

Show that if $f \in C^{1}(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f^{\prime} \in CL(\mathbf{R})$ and $x \in \mathbf{R}$, then $t \in \mathbb{R}_{+} \mapsto P_{t}(f)(x)$ is of class $C^{1}$ on $\mathbb{R}_{+}$ and show that for all $t > 0$, we have
$$\frac{\partial P_{t}(f)(x)}{\partial t} = \int_{-\infty}^{+\infty} \left(-x \mathrm{e}^{-t} + \frac{\mathrm{e}^{-2t}}{\sqrt{1 - \mathrm{e}^{-2t}}} y\right) f^{\prime}\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y$$

grandes-ecoles 2024 Q8 Proof of Differentiability Class for Parameterized Integrals View

Show that if $f \in C^1(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f' \in CL(\mathbf{R})$ and $x \in \mathbf{R}$, then $t \in \mathbf{R}_+ \mapsto P_t(f)(x)$ is of class $C^1$ on $\mathbf{R}_+^*$ and show that for all $t > 0$, we have $$\frac{\partial P_t(f)(x)}{\partial t} = \int_{-\infty}^{+\infty} \left(-x\mathrm{e}^{-t} + \frac{\mathrm{e}^{-2t}}{\sqrt{1-\mathrm{e}^{-2t}}}\,y\right) f'\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y.$$

isi-entrance 2011 Q1 Limit Evaluation Involving Composition or Substitution View

The limit $\lim \left[ \left\{ 1 - \cos \left( \sin ^ { 2 } a x \right) \right\} / x \right]$ as $x -> 0$
(a) Equals 1
(b) Equals a
(c) Equals 0
(d) Does not exist

isi-entrance 2013 Q16 4 marks Iterated/Nested Exponential Differentiation View

Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is equal to
(A) $f _ { n } ( x )$
(B) $f _ { n } ( x ) f _ { n - 1 } ( x )$
(C) $f _ { n } ( x ) f _ { n - 1 } ( x ) \cdots f _ { 1 } ( x )$
(D) $f _ { n + 1 } ( x ) f _ { n } ( x ) \cdots f _ { 1 } ( x ) e ^ { x }$

isi-entrance 2014 Q25 Limit Involving Derivative Definition of Composed Functions View

Let $f$ be a differentiable function with $f(3) \neq 0$. Evaluate $\displaystyle\lim_{x \to \infty} \left(\frac{f(3 + 1/x)}{f(3)}\right)^x$.
(A) $e^{f'(3)/f(3)}$ (B) $e^{f(3)}$ (C) $e^{f'(3)}$ (D) 1

isi-entrance 2015 Q11 4 marks Iterated/Nested Exponential Differentiation View

Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is:
(a) $f _ { n } ( x )$
(b) $f _ { n } ( x ) f _ { n - 1 } ( x )$
(c) $f _ { n } ( x ) f _ { n - 1 } ( x ) \ldots f _ { 1 } ( x )$
(d) $f _ { n + 1 } ( x ) f _ { n } ( x ) \ldots f _ { 1 } ( x ) e ^ { x }$.

isi-entrance 2015 Q11 4 marks Iterated/Nested Exponential Differentiation View

Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is:
(a) $f _ { n } ( x )$
(b) $f _ { n } ( x ) f _ { n - 1 } ( x )$
(c) $f _ { n } ( x ) f _ { n - 1 } ( x ) \ldots f _ { 1 } ( x )$
(d) $f _ { n + 1 } ( x ) f _ { n } ( x ) \ldots f _ { 1 } ( x ) e ^ { x }$.

isi-entrance 2016 Q16 4 marks Iterated/Nested Exponential Differentiation View

Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is equal to
(A) $f _ { n } ( x )$
(B) $f _ { n } ( x ) f _ { n - 1 } ( x )$
(C) $f _ { n } ( x ) f _ { n - 1 } ( x ) \cdots f _ { 1 } ( x )$
(D) $f _ { n + 1 } ( x ) f _ { n } ( x ) \cdots f _ { 1 } ( x ) e ^ { x }$

isi-entrance 2016 Q16 4 marks Iterated/Nested Exponential Differentiation View

Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is equal to
(A) $f _ { n } ( x )$
(B) $f _ { n } ( x ) f _ { n - 1 } ( x )$
(C) $f _ { n } ( x ) f _ { n - 1 } ( x ) \cdots f _ { 1 } ( x )$
(D) $f _ { n + 1 } ( x ) f _ { n } ( x ) \cdots f _ { 1 } ( x ) e ^ { x }$

isi-entrance 2018 Q3 Derivative of Composite Function in Applied/Modeling Context View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function such that for all $x \in \mathbb { R }$ and for all $t \geq 0$, $$f ( x ) = f \left( e ^ { t } x \right)$$ Show that $f$ is a constant function.

isi-entrance 2020 Q3 Derivative of Inverse Functions View

If $f , g$ are real-valued differentiable functions on the real line $\mathbb { R }$ such that $f ( g ( x ) ) = x$ and $f ^ { \prime } ( x ) = 1 + ( f ( x ) ) ^ { 2 }$, then $g ^ { \prime } ( x )$ equals
(A) $\frac { 1 } { 1 + x ^ { 2 } }$
(B) $1 + x ^ { 2 }$
(C) $\frac { 1 } { 1 + x ^ { 4 } }$
(D) $1 + x ^ { 4 }$.

isi-entrance 2021 Q30 Iterated/Nested Exponential Differentiation View

Let $$\begin{gathered} p ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 2 x , x \in \mathbb { R } \\ f _ { 0 } ( x ) = \begin{cases} \int _ { 0 } ^ { x } p ( t ) d t , & x \geq 0 \\ - \int _ { x } ^ { 0 } p ( t ) d t , & x < 0 \end{cases} \\ f _ { 1 } ( x ) = e ^ { f _ { 0 } ( x ) } , \quad f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) } , \quad \ldots \quad , f _ { n } ( x ) = e ^ { f _ { n - 1 } ( x ) } \end{gathered}$$ How many roots does the equation $\frac { d f _ { n } ( x ) } { d x } = 0$ have in the interval $( - \infty , \infty ) ?$
(A) 1 .
(B) 3 .
(C) $n + 3$.
(D) $3n$.

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LFM Pure

Straight Lines & Coordinate Geometry 247

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Simultaneous equations 79

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Proof by induction 25

Sine and Cosine Rules 195

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Differential equations 352

3x3 Matrices 144