Consider the function $$f(x) := \frac{1}{3}x^3 \quad \text{if } x \geq 0, \quad f(x) := 0 \quad \text{if } x < 0$$ and the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_{n+1} := x_n - \tau f'(x_n)$. We suppose in this question that $0 < x_0 < 1/\tau$. a) Justify that the sequence $\left(x_n\right)_{n \in \mathbb{N}}$ is decreasing, with strictly positive values, and satisfies $x_{n+1} = x_n(1 - \tau x_n)$ for all $n \in \mathbb{N}$. b) Justify that $x_n \rightarrow 0$ when $n \rightarrow \infty$. c) Show that $1/x_{n+1} = 1/x_n + \tau/(1 - \tau x_n)$ for all $n \in \mathbb{N}$. Deduce that $x_n \leq x_0/(1 + n\tau x_0)$.
114- What is the value of $\displaystyle\lim_{x \to 0}\left([2x]+[-2x]\right)\dfrac{1-\cos^2 x}{1-\sqrt{1+x^2}}$? (The symbol $[\,]$ denotes the floor function.)
116. Suppose $a$ is known and $a + n$ is given. Find the value of $\displaystyle\lim_{x \to 0^+} \frac{\tan^2\!\left(\dfrac{1}{\sqrt{1-x^2}}-1\right)}{\left(1-\cos(\sqrt{7x})\right)^n} = a$. What is $a+n$? (1) $\dfrac{7}{4}$ (2) $\dfrac{9}{4}$ (3) $\dfrac{15}{4}$ (4) $\dfrac{17}{4}$
119. If $\displaystyle\lim_{x \to -\infty} \frac{\sqrt[5\circ]{(a^2x^2-1)(a^4x^4-1)\cdots(a^{100}x^{100}-1)}}{a^{49}x^k - 1} = -1$, what are the values of $a$ and $k$? (1) $k = 51,\ a = -1$ (2) $k = 51,\ a = 1$ (3) $k = 49,\ a = -1$ (4) $k = 49,\ a = 1$
The number of roots of the equation $x ^ { 2 } + \sin ^ { 2 } x = 1$ in the closed interval $\left[ 0 , \frac { \pi } { 2 } \right]$ is (a) 0 (b) 1 (c) 2 (d) 3
The number of roots of the equation $x ^ { 2 } + \sin ^ { 2 } x = 1$ in the closed interval $\left[ 0 , \frac { \pi } { 2 } \right]$ is (a) 0 (b) 1 (c) 2 (d) 3
The number of roots of the equation $x ^ { 2 } + \sin ^ { 2 } x = 1$ in the closed interval $\left[ 0 , \frac { \pi } { 2 } \right]$ is (A) 0 (B) 1 (C) 2 (D) 3
The number of elements in the set $$\left\{x : 0 \leqslant x \leqslant 2,\, \left|x - x^5\right| = \left|x^5 - x^6\right|\right\}$$ is (A) 2 (B) 3 (C) 4 (D) 5
Let $f : \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right] \rightarrow \mathbb { R }$ be a continuous function such that $$f ( 0 ) = 1 \text { and } \int _ { 0 } ^ { \frac { \pi } { 3 } } f ( t ) d t = 0$$ Then which of the following statements is (are) TRUE ? (A) The equation $f ( x ) - 3 \cos 3 x = 0$ has at least one solution in $\left( 0 , \frac { \pi } { 3 } \right)$ (B) The equation $f ( x ) - 3 \sin 3 x = - \frac { 6 } { \pi }$ has at least one solution in $\left( 0 , \frac { \pi } { 3 } \right)$ (C) $\lim _ { x \rightarrow 0 } \frac { x \int _ { 0 } ^ { x } f ( t ) d t } { 1 - e ^ { x ^ { 2 } } } = - 1$ (D) $\lim _ { x \rightarrow 0 } \frac { \sin x \int _ { 0 } ^ { x } f ( t ) d t } { x ^ { 2 } } = - 1$
The real number $k$ for which the equation, $2x^3 + 3x + k = 0$ has two distinct real roots in $[0,1]$ belongs to (1) lies between - 1 and 0 . (2) does not exist. (3) lies between 1 and 2 . (4) lies between 2 and 3 .