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42. (On the statistics of the normal distribution $\mathscr{N}\left(a, \sigma^{2}\right): a$ is unknown, $\sigma^{2}=\sigma_{0}^{2}$.)
In this and the following problem, it is assumed that $\xi_{1}, \ldots, \xi_{n}$ are independent random variables distributed according to the law $\mathscr{N}\left(a, \sigma^{2}\right... | Solution. Point (a) is established by direct verification. Point (b) follows from point (a), the Rao-Cramer inequality, and the fact that
$$
\mathrm{E}\left[\frac{\partial^{2} \ln p_{\left(a, \sigma_{0}^{2}\right)}\left(\xi_{1}\right)}{\partial a^{2}}\right]=\mathrm{E}\left[-\frac{1}{2} \ln \left(2 \pi \sigma_{0}^{2}\... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,138 |
43. (On the statistics of the normal distribution $\mathscr{N}\left(a, \sigma^{2}\right)$: $a=a_{0}, \sigma^{2}$ is unknown.) If $a$ is known $\left(a=a_{0}\right)$, then it is natural to take not the value $s^{2}(x)=\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}$, but the value
$$
s_{0}^{2}(x)=\frac{1}{n} \... | Solution. (a) The random vector $\zeta=\left(\frac{\xi_{i}-a_{0}}{\sigma}, 1 \leqslant i \leqslant n\right)$ consists of independent Gaussian variables, each having a distribution $\mathscr{N}(0,1)$. Therefore,
$$
\frac{n s_{0}^{2}(\xi)}{\sigma^{2}}=\|\zeta\|^{2} \sim \chi_{n}^{2}
$$
We apply the formula for calculat... | \gamma=\alpha/2 | Algebra | proof | Yes | Yes | olympiads | false | 34,139 |
44. (On the statistics of the normal distribution $\mathscr{N}\left(a, \sigma^{2}\right)$: $a$ and $\sigma^{2}$ are unknown.)
(a) Show that in the considered case, unbiased estimators for $a$ and $\sigma^{2}$ are, for $n>1$,
$$
\bar{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i}, \quad s_{1}^{2}(x) \equiv \frac{n}{n-1} s^{2}(x)... | Solution. (a) Obviously,
$$
\mathrm{E} \bar{\xi}=\frac{1}{n} \sum_{i=1}^{n} \mathrm{E} \xi_{i}=\mathrm{E} \xi_{1}=a
$$
Moreover, $(n-1) \frac{s_{1}^{2}(\xi)}{\sigma^{2}} \sim \chi_{n-1}^{2}$ due to problem II.13.40, therefore
$$
\mathrm{E}(n-1) \frac{s_{1}^{2}(\xi)}{\sigma^{2}}=n-1
$$
which is equivalent to the unb... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,140 |
45. Let $X$ be a random vector in $\mathbb{R}^{n}$ with a non-degenerate covariance matrix $\mathrm{D} X$. Consider the following extremal problem:
$$
\mathrm{E} \ln f(X ; a, \Sigma) \rightarrow \sup _{a, \Sigma}
$$
where the supremum is taken over all vectors $a$ from some set $A \subseteq \mathbb{R}^{n}$ and all po... | Solution. Letting $Z=Z(a, \Sigma)=\Sigma^{-1 / 2}(X-a)$, we obtain $\mathrm{E} \ln f(X ; a, \Sigma)=-\frac{n}{2} \ln 2 \pi+\ln \operatorname{det} \Sigma^{-1}-\frac{1}{2} \mathrm{E} Z^{\top} Z=$
\[
\begin{aligned}
& =-\frac{n}{2} \ln 2 \pi+\ln \operatorname{det} \Sigma^{-1}-\frac{1}{2} \operatorname{tr}\left[\mathrm{E}... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,141 |
46. Let $\xi \sim \mathscr{N}(0,1)$. Show that
$$
\mathrm{E}|\xi|^{\alpha}=\frac{2^{\alpha / 2}}{\sqrt{\pi}} \Gamma\left(\frac{\alpha+1}{2}\right), \quad \alpha>-1
$$
where $\Gamma(s)=\int_{\mathbb{R}_{+}} x^{s-1} e^{-t} d t$ is the Euler gamma function. In particular,
$$
\mathrm{E} \xi^{2 n}=(2 n-1)!!=1 \cdot 3 \cd... | Solution. We write $\mathrm{E}|\xi|^{\alpha}$ using the density of the magnitude $|\xi|$:
$$
\begin{array}{rl}
\mathrm{E}|\xi|^{\alpha}=\sqrt{\frac{2}{\pi}} \int_{\mathbb{R}_{+}} x^{\alpha} e^{-x^{2} / 2} d x=\sqrt{\frac{2}{\pi}} \int_{\mathbb{R}_{+}}(2 z)^{(\alpha-1) / 2} e^{-z} & d z \\
& =\frac{2^{\alpha / 2}}{\sqr... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,142 |
47. Let $X$ and $Y$ be independent random variables with distribution $\mathscr{N}(0,1)$. Define
$$
T=\frac{X^{2}+Y^{2}}{2}, \quad g=\frac{X^{2}}{X^{2}+Y^{2}}
$$
Using problem II.8.11, find the joint distribution of $(T, g)$. | Solution. Due to problem II.8.11, the quantities $T$ and $g$ are independent, so it is sufficient to compute the distribution functions of these quantities. The quantity $T$ has an exponential distribution $\mathrm{P}(T>t)=e^{-t}, t>0$. The quantity $g$ has an arcsin distribution with density $\frac{1}{\pi \sqrt{x(1-x)... | T\simExponential(1),\quad\simArcsin(\frac{1}{\pi\sqrt{x(1-x)}},x\in(0,1)) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 34,143 |
48. Let $B=\left(B_{t}\right)_{t \geqslant 0}$ be a Brownian motion process and
$$
T_{a}=\inf \left\{t \geqslant 0: B_{t}=a\right\}
$$
- the first hitting time of level $a>0$. (We set $T_{a}=\infty$ if $\{\cdot\}=\varnothing$.
Using the reflection principle for Brownian motion
$$
\mathrm{P}\left(\sup _{s \leqslant ... | Solution. We should use the fact that
$$
\mathrm{P}\left(T_{a} \leqslant t\right)=2 \mathrm{P}\left(B_{t} \geqslant a\right)=2-2 \Phi\left(\frac{a}{\sqrt{t}}\right)
$$ | p_{}()=\frac{}{\sqrt{2\pi^{3}}}e^{-\frac{^{2}}{2}} | Calculus | proof | Yes | Yes | olympiads | false | 34,144 |
49. Let $T=T_{1}$, where $T_{a}$ is defined in problem II.13.48. Show that
$$
T \stackrel{d}{=} N^{-2},
$$
where $N \sim \mathscr{N}(0,1)$ is a standard Gaussian random variable. Show also that the Laplace transform is given by
$$
\mathrm{E} e^{-\frac{\lambda^{2} T}{2}}=\mathrm{E} e^{-\frac{\lambda^{2}}{2 N^{2}}}=e^... | Solution. Reasoning as in the solution of II.13.48, we find that $\mathrm{P}\left(T_{1} \leqslant t\right)=2 \mathrm{P}\left(B_{t} \geqslant 1\right)=2 \mathrm{P}(\sqrt{t} N \geqslant 1)=\mathrm{P}\left(t N^{2} \geqslant 1\right)=\mathrm{P}\left(N^{-2} \leqslant t\right)$. Hence,
$$
\begin{aligned}
\mathrm{E} e^{-\fra... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,145 |
50. (See [75].) Let $X=\left(X_{1}, \ldots, X_{2 n}\right), n \geqslant 1,$ be a Gaussian vector with zero mean and covariance matrix $C=\left(c_{i j}\right)_{i, j=1}^{2 n}$. Prove that
$$
\mathrm{E} X_{1} X_{2} \ldots X_{2 n}=\frac{1}{n!} \sum_{\sigma} c_{\sigma_{1} \sigma_{2}} \ldots c_{\sigma_{2 n-1} \sigma_{2 n}}
... | Solution. The characteristic function of the random variable $X$ has the form
$$
\varphi(t)=\mathrm{E} e^{i(t, X)}=\exp \left\{-\frac{1}{2} \sum_{j, k=1}^{2 n} c_{j k} t_{j} t_{k}\right\}, \quad t=\left(t_{1}, \ldots, t_{2 n}\right) \in \mathbb{R}^{2 n}
$$
In this case,
$$
\mathrm{E} X_{1} X_{2} \ldots X_{2 n}=(-1)^... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 34,146 |
51. Let $X$ and $Y$ be independent $\mathscr{N}\left(0, \sigma^{2}\right)$-distributed random variables.
(a) Show that
$$
\left(\frac{X^{2}-Y^{2}}{\sqrt{X^{2}+Y^{2}}}, \frac{2 X Y}{\sqrt{X^{2}+Y^{2}}}\right) \stackrel{d}{=}(X, Y) .
$$
Also verify that if $\mathrm{E}|X|0$ the equality
$$
C-\frac{a}{C} \stackrel{d}{=... | Solution. (a) If $X$ and $Y$ are Gaussian variables with zero mean, then the desired relation follows from Problem II.8.11. Conversely, in polar coordinates $X=R \cos \Theta, Y=R \sin \Theta$, where $R \geqslant 0, \Theta \in[0,2 \pi)$ a.s., property $(*)$ can be rewritten as
$$
\begin{gathered}
(R \cos 2 \Theta, R \s... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,147 |
52. (a) Starting from the fact that complex-valued functions of the form \( g(t) \overline{g(s)} \) are non-negative definite, show that the function
\[
R_{H}(s, t)=\frac{1}{2}\left[|t|^{2 H}+|s|^{2 H}-|t-s|^{2 H}\right], \quad s, t \in \mathbb{R}
\]
is also non-negative definite for \( H \in (0,1] \) and that there ... | Solution. (a) Let $H \in(0,1)$. It is easy to verify that any complex-valued function $g(s) \overline{g(t)}, s, t \in \mathbb{R}$, is positive definite. Therefore, for any non-negative function $f$, the function $\int_{\mathbb{R}} g(s, x) \overline{g(t, x)} f(x) d x$ is also positive definite (assuming all integrals co... | proof | Other | proof | Yes | Yes | olympiads | false | 34,148 |
53. Using the fact that the function
$$
R_{1}(s, t)=\frac{|t|+|s|}{2}-\frac{|s-t|}{2}=s \wedge t, \quad s, t \in \mathbb{R}
$$
is non-negatively defined, prove the non-negative definiteness of the function
$$
R_{n}(s, t)=\frac{|t|+|s|}{2}-\frac{|s-t|}{2}, \quad s, t \in \mathbb{R}^{n}
$$
Establish a similar result ... | Solution. Let $X_{1}, \ldots, X_{n}$ be independent and identically distributed (i.i.d.) Gaussian random variables with zero mean and $\mathrm{E}\left|X_{1}\right|=1$. Then $\mathbf{E}\left|t^{\top} X\right|=|t|$ for $X=\left(X_{1}, \ldots, X_{n}\right)$, and thus,
$$
R_{n}(s, t)=\mathrm{E} R_{1}\left(s^{\top} X, t^{\... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,149 |
54. Let $X_{1}, \ldots, X_{n}$ be i.i.d. random variables, $\mathrm{E} X_{1}=0, \mathrm{E} X_{1}^{2}=1$. Suppose also that for some constants $a_{1}, \ldots, a_{n}$, different from 0, the quantities
$$
\sum_{k=1}^{n} X_{k} \quad \text { and } \quad \sum_{k=1}^{n} a_{k} X_{k}
$$
are independent. Prove that $X_{1} \sim... | Solution. For simplicity, we will assume that $\sum_{k} a_{k}^{2}=1 ;$ here and in the following, the index of summation $k$ ranges from 1 to $n$. Also note that
$$
\sum_{k} a_{k}=\mathrm{E} \sum_{k} X_{k} \cdot \sum_{k} a_{k} X_{k}=\mathrm{E} \sum_{k} X_{k} \cdot \mathrm{E} \sum_{k} a_{k} X_{k}=0
$$
Let $\varphi=\va... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,150 |
55. The application of the Student's distribution in statistics is primarily based on the fact that the statistics
$$
\bar{X}=\frac{1}{n} \sum_{k=1}^{n} X_{k} \quad \text { and } \quad \sum_{k=1}^{n}\left(X_{k}-\bar{X}\right)^{2}
$$
are independent when $X_{1}, \ldots, X_{n}$ are independent and identically distribut... | Solution. By analogy with the solution of problem II.13.54, consider the equality
$$
\mathrm{E} e^{i t \sum_{k=1}^{n} X_{k}} \sum_{k=1}^{n}\left(X_{k}-\bar{X}\right)^{2}=\mathrm{E} e^{i t \sum_{k=1}^{n} X_{k}} \cdot \mathrm{E} \sum_{k=1}^{n}\left(X_{k}-\bar{X}\right)^{2}=(n-1) \mathrm{E} e^{i t \sum_{k=1}^{n} X_{k}}
$... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,151 |
56. (See [96].) Let $X_{1}, \ldots, X_{n}$ be independent random variables. Suppose also that for some distinct constants $a_{1}, \ldots, a_{n}$, the quantities
$$
\sum_{k=1}^{n} X_{k} \quad \text { and } \quad \sum_{k=1}^{n} a_{k} X_{k}
$$
are independent. Prove the Darmois-Skitovich theorem, which states that all $... | Solution. (a) Let $\varphi_{k}=\varphi_{k}(t)$ be the characteristic function of the random variable $X_{k}$; here and in the following, $k$ ranges from 1 to $n$. Then, by the condition, for all $s, t \in \mathbb{R}$ we have
$$
\prod_{k} \varphi_{k}\left(s+a_{k} t\right)=\mathrm{E} e^{i s \sum_{k} X_{k}+i t \sum_{k} a... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,152 |
59. Let $(X, Y)$ be a two-dimensional Gaussian vector with $\mathrm{E} X=\mathrm{E} Y=0$, $\mathrm{D} X=\mathrm{D} Y=1$ and correlation coefficient $\rho=\mathrm{E} X Y$. Show that the correlation coefficient $\rho_{\Phi(X) \Phi(Y)}$ of the quantities $\Phi(X)$ and $\Phi(Y)$ is given by the formula
$$
\rho_{\Phi(X) \P... | Solution. The quantities $\Phi(X)$ and $\Phi(Y)$ have a uniform distribution on $[0,1]$, since $\mathrm{P}(\Phi(X) \leqslant x)=\Phi\left(\Phi^{-1}(x)\right)=x, x \in(0,1)$. Therefore, $\mathrm{E} \Phi(X)=1 / 2, \mathrm{D} \Phi(X)=1 / 12$ and
$$
\rho_{\Phi(X) \Phi(Y)}=\frac{\mathrm{E} \Phi(X) \Phi(Y)-1 / 4}{1 / 12}
$$... | \rho_{\Phi(X)\Phi(Y)}=\frac{6}{\pi}\arcsin\frac{\rho}{2} | Algebra | proof | Yes | Yes | olympiads | false | 34,155 |
60. Let $(X, Y, Z)$ be a Gaussian vector with $\mathrm{E} X=\mathrm{E} Y=\mathrm{E} Z=0$, $\mathrm{D} X=\mathrm{D} Y=\mathrm{D} Z=1$ and correlation coefficients $\rho_{X Y}, \rho_{X Z}, \rho_{Y Z}$. In addition to problem II.13.31(b), show that
$$
\mathrm{P}(X>0, Y>0, Z>0)=\frac{1}{8}+\frac{1}{4 \pi}\left(\arcsin \rh... | Solution. Let $A=\{X \leqslant 0\}, B=\{Y \leqslant 0\}, C=\{Z \leqslant 0\}$. Then if $p=\mathrm{P}(\bar{A} \cap \bar{B} \cap \bar{C})$, by the inclusion-exclusion formula from problem II.8.60 we have
$$
\begin{aligned}
1-p= & \mathrm{P}(A \cup B \cup C)= \\
& =[\mathrm{P}(A)+\mathrm{P}(B)+\mathrm{P}(C)]-[\mathrm{P}(... | notfound | Algebra | proof | Yes | Yes | olympiads | false | 34,156 |
61. Prove that the Laplace transform $E e^{-\lambda \mathscr{R}^{2}}, \lambda>0$, of the square $\mathscr{R}^{2}$ of the "range" $\mathscr{R}$ of the Brownian bridge $B^{\circ}=\left(B_{t}^{\circ}\right)_{0 \leqslant t \leqslant 1}$, i.e., the random variable
$$
\mathscr{R}=\max _{0 \leqslant t \leqslant 1} B_{t}^{\ci... | Solution. The statement follows from Problem II.8.80 and the subsequent remark, which, in particular, guarantees the equality $\mathscr{R}^{2} \stackrel{d}{=} Y_{2}$. | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,157 |
62. Let $\xi \sim \mathscr{N}(0,1)$. Define the Hermite polynomials by the formula
$$
\mathrm{H}_{n}(x)=(-1)^{n} \frac{D^{n} \varphi(x)}{\varphi(x)}, \quad n \geqslant 1
$$
where $D^{n}$ is the $n$-th order derivative ($n \geqslant 0$), and $\varphi=\varphi(x)$ is the density of $\xi$.
(a) Show that
$$
\begin{gathe... | Solution. (a) The generating functions of the sequences $\mathrm{E}(x+ i \xi)^{n}$ and $\mathrm{H}_{n}(x)$ coincide, so $\mathrm{H}_{n}(x)=\mathrm{E}(x+i \xi)^{n}$ for all $n$. Indeed,
$$
\sum_{n=0}^{\infty} \mathrm{E}(x+i \xi)^{n} \frac{t^{n}}{n!}=\mathrm{E} e^{t x+i t \xi}=e^{t x-t^{2} / 2}=\frac{\varphi(x-t)}{\varp... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,158 |
2. (See [13], [39].) Assuming that $\int_{\mathbb{R}} x p(x) d x=0$ and $\int_{\mathbb{R}} x^{2} p(x) d x = \sigma^{2} > 0$, provide a formal derivation of another widely used expansion - the Edgeworth expansion:
$$
\sigma p(\sigma x)=\varphi(x)+\varphi(x) \sum_{n \geqslant 1} \frac{1}{\sigma^{n}} \sum_{n} \mathrm{H}_... | Solution. 1. (a) According to problem II.13.62(c), the system $\left(\mathrm{H}_{n}\right)_{n \geqslant 1}$ is orthogonal and dense everywhere in the Hilbert space of measurable functions with the scalar product $(f, g)=\mathrm{E} f(\xi) g(\xi)$, where $\xi \sim \mathscr{N}(0,1)$. Thus, the specified expansion in terms... | notfound | Calculus | proof | Yes | Yes | olympiads | false | 34,159 |
64. (See [45].) Let $(\xi, \eta)$ be a two-dimensional Gaussian vector with $\mathrm{E} \xi = \mathrm{E} \eta = 0, \mathrm{D} \xi = \mathrm{D} \eta = 1$ and correlation coefficient $\rho$. Establish the equality
$$
\mathrm{P}(\xi > a, \eta > b) - \mathrm{P}(\xi > a) \mathrm{P}(\eta > b) = \int_{0}^{\rho} \phi(a, b, r)... | Solution. We will assume that $\rho \geqslant 0$, otherwise we replace $\eta$ with $-\eta$ and use the relation
$$
\operatorname{cov}(I(\xi>a), I(\eta>b))=-\operatorname{cov}(I(\xi>a), I(-\eta>-b))
$$
We will compute $\mathrm{P}(\xi>a, \eta>b)$ using Hermite polynomials. We have
$$
\begin{gathered}
I(\xi>a)=\sum_{n}... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,160 |
65. Let $(\xi, \eta)$ be a two-dimensional Gaussian vector with $\mathrm{E} \xi=\mathrm{E} \eta=0$, $\mathrm{D} \xi=\mathrm{D} \eta=1$ and correlation coefficient $\rho$. Show that for any Borel functions $f$ and $g$ satisfying the condition $\mathrm{E} f(\xi)^{2}$, $\mathrm{E} g(\eta)^{2}1$, there exists a constant $C... | Solution. We will assume that $\rho>0$ (otherwise, we replace $\eta$ with $-\eta$), and also that $\mathrm{E} f(\xi)=\mathrm{E} g(\eta)=0$. First, let's address the first part of the problem. By problem II.13.62(c), for some constants $a_{n}$ and $b_{n}$, the following expansions hold in the space $L^{2}$:
$$
f(\xi)=\... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 34,161 |
66. (Viskov.) Let $\xi, \eta \sim \mathscr{N}(0,1)$ be independent Gaussian variables. Show that for any entire function $f=f(z), z \in \mathbb{C}$, satisfying the condition $\mathrm{E}|f(x+\xi+i \eta)|<\infty$, the following "averaging" property holds:
$$
f(x)=\mathrm{E} f(x+\xi+i \eta) .
$$ | Solution. It is sufficient to prove the desired equality for the case when $f(z)$ is a polynomial $\sum_{k=0}^{n} a_{k} z^{k}$. According to problem II.13.62(a), we have
$$
\begin{aligned}
& \sum_{k=0}^{n} a_{k} x^{k}=\sum_{k=0}^{n} a_{k} \mathrm{EH}_{k}(x+\xi)=\sum_{k=0}^{n} a_{k} \mathrm{EE}\left[(x+\xi+i \eta)^{k} ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,162 |
67. Let $\xi, \eta$ be independent random variables. Prove Cramér's theorem.
The variable $\xi+\eta$ has a Gaussian distribution if and only if $\xi$ and $\eta$ are Gaussian variables.
To derive this statement, use a variant of Hadamard's theorem.
If $f(z)=\sum_{n \geqslant 0} a_{n} z^{n}$ for all $z \in \mathbb{C}$... | Solution. If $\xi$ and $\eta$ are independent Gaussian variables, then $(\xi, \eta)$ is a Gaussian vector, which means that $\xi+\eta$ has a Gaussian distribution.
Conversely, let $\xi+\eta \sim \mathscr{N}(0,1)$. Then, due to the symmetry of the law $\mathscr{N}(0,1)$, we have
$$
\mathrm{E} e^{|z| |\xi+\eta|} \leqsl... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,163 |
68. Let's call a function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$ almost differentiable if for some measurable function $g: \mathbb{R}^{d} \rightarrow \mathbb{R}$, the following equality holds:
$$
f(b)-f(a)=\int_{0}^{1}(b-a)^{\top} g(a+x(b-a)) d x, \quad a, b \in \mathbb{R}^{d}
$$
In this case, we will write $\nab... | Solution. Let $\varphi=\varphi(x)$ be the density of the random variable $\xi$. Then
$$
\varphi(x)=\int_{x}^{\infty} z \varphi(z) d z=-\int_{-\infty}^{x} z \varphi(z) d z
$$
By Fubini's theorem
$$
\begin{aligned}
\mathrm{E} f^{\prime}(\xi) & =\int_{\mathbb{R}} f^{\prime}(x) \varphi(x) d x= \\
& =\int_{0}^{\infty} f^... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,164 |
70. (See [103].) Let the function $f: \mathbb{R}^{d} \rightarrow(0, \infty)$ be smooth up to the second order, and for all $x \in \mathbb{R}^{d}$, the following equality holds:
$$
\nabla^{2} h(x)=\nabla^{\top} \nabla h(x)=\sum_{i=1}^{d} h_{x_{i} x_{i}}^{\prime \prime}(x) \leqslant 0, \quad h=\sqrt{f}
$$
Also,
$$
\ma... | Solution. We have
$\mathrm{E}\|\xi+\nabla \ln f(\xi)-\mathrm{E} \xi\|^{2}=$
$$
=\mathrm{E}\|\xi-\mathrm{E} \xi\|^{2}+2 \mathrm{E}(\xi-\mathrm{E} \xi)^{\top} \nabla \ln f(\xi)+\mathrm{E}\|\nabla \ln f(\xi)\|^{2}
$$
and by problem II.13.68 we obtain
$$
\mathrm{E}(\xi-\mathrm{E} \xi)^{\top} \nabla \ln f(\xi)=\mathrm{E... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,166 |
71. (See [4].) Consider the following Heisenberg-Weyl relation
$$
\mathrm{pq}-\mathrm{qp}=1
$$
where $\mathbf{p}, \mathbf{q}$ are elements of some linear space $S$ with an associative multiplication and a unit element 1.
Remark. This relation was studied by C. Graves in [68] long before H. Weyl and W. Heisenberg, wh... | Solution. (a) It is sufficient to prove the desired relation for polynomials of the form $P(x)=x^{n}, n \geqslant 1$ (the case $n=0$ is trivial). For $n>1$ we have
$$
\begin{aligned}
\mathbf{p} \mathbf{q}^{n}=(\mathbf{p q}) \mathbf{q}^{n-1}=(\mathbf{q} \mathbf{p}+\mathbf{1}) \mathbf{q}^{n-1}=\mathbf{q}(\mathbf{p q}) \... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,167 |
72. Let $X \sim \mathscr{N}(0,1)$. Prove the Poincaré inequality $\left[\mathrm{E} f^{\prime}(X)\right]^{2} \leqslant \mathrm{D} f(X) \leqslant \mathrm{E}\left[f^{\prime}(X)\right]^{2} \quad$ for all smooth functions $f$ satisfying the condition $\mathrm{E}[f(X)]^{2}<\infty$. Refine this inequality by showing that $\ma... | Solution. The lower bound follows from the fact that $\mathrm{E} f^{\prime}(X)=\mathrm{E} X f(X)$ and
$$
|\mathrm{E} X f(X)|^{2}=|\mathrm{E} X[f(X)-\mathrm{E} f(X)]|^{2} \leqslant \mathrm{E} X^{2} \mathrm{D} f(X)=\mathrm{D} f(X)
$$
The upper bound is proved as follows. Let $Y$ be an independent copy of the variable $... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 34,168 |
73. Show that the Poincaré inequality from problem II. 13.72 provides a complete characterization of the distribution $\mathscr{N}(0,1)$ in the sense that if for some random variable $X, \mathrm{E} X=0, \mathrm{E} X^{2}=1$, we have
$\mathrm{D} f(X) \leqslant \mathrm{E}\left[f^{\prime}(X)\right]^{2}$ for all smooth fun... | Solution. Let $X$ satisfy condition (*). Then for any smooth function $g, \mathrm{E}[g(X)]^{2}<\infty$, we have $\mathrm{D}[X+z g(X)] \leqslant \mathrm{E}\left[1+z g^{\prime}(X)\right]^{2}$ for all $z \in \mathbb{R}$. The latter is equivalent to $a z^{2}+2 b z \geqslant 0$ for each $z \in \mathbb{R}$, where
$$
\begin{... | X\sim\mathscr{N}(0,1) | Inequalities | proof | Yes | Yes | olympiads | false | 34,169 |
75. (See [49].) Let $\left\{X_{1}, X_{2}, \ldots\right\}$ be a uniformly integrable family of random variables. Show that (as $n \rightarrow \infty$)
$$
X_{n} \xrightarrow{d} \mathscr{N}(0,1) \Leftrightarrow \mathrm{E}\left[X_{n} e^{i t X_{n}}-i t e^{i t X_{n}}\right] \rightarrow 0 \text { for all } t \in \mathbb{R} .... | Solution. The family of random variables $\left\{X_{n}\right\}_{n \geqslant 1}$ is tight, since
$$
\sup _{n} \mathrm{P}\left(\varepsilon\left|X_{n}\right|>1\right) \leqslant \sup _{n} \varepsilon \mathrm{E}\left|X_{n}\right| I\left(\varepsilon\left|X_{n}\right|>1\right) \rightarrow 0, \quad \varepsilon \downarrow 0 .
... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,171 |
76. Let the functions $R=R(s, t)$ and $K=K(s, t), s, t \in T$, be non-negatively defined. Show that the same property holds for the functions $a R + b K$ when $a, b > 0$ and $R K$, as well as for the function $C=C(u, v)$, $u, v \in T^{2}$, defined by the formula $C(u, v) = R(u) K(v)$.
From this, derive that for any an... | Solution. It is sufficient to consider the case of finite $T$. Let $\left(X_{t}\right)$ and $\left(Y_{t}\right), t \in T,$ be independent Gaussian systems,
$$
\mathrm{E} X_{t}=\mathrm{E} Y_{t}=0, \quad R(s, t)=\mathrm{E} X_{s} X_{t}, \quad K(s, t)=\mathrm{E} Y_{s} Y_{t}.
$$
Then the covariance functions of the system... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,172 |
77. Let $R=R(s, t)$ be a symmetric non-negative definite function on $\mathbb{R}^{2}$, and $X$ be a random variable such that $\mathrm{E} \sqrt{R(X, X)}<\infty$. Prove that $0 \leqslant \mathrm{E} R(X, Y)<\infty$, when $Y$ is an independent copy of the variable $X$. | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,173 | |
5. Determine which of the given functions is even, odd, or neither even nor odd:
$$
\begin{aligned}
& \text { 1) } f(x)=\frac{x^{2}}{\sin 2 x} ; \quad \text { 2) } \varphi(x)=4-2 x^{4}+\sin ^{2} x \\
& \text { 3) } u(x)=x^{3}+2 x-1 ; \quad \text { 4) } y(x)=\frac{1+a^{k x}}{1-a^{k x}}
\end{aligned}
$$ | Solution. To determine whether some function $Q(x)$ is even or odd, it is necessary to find $Q(-x)$.
By substituting $x$ with $-x$, we get:
$$
\text { 1) } f(-x)=\frac{(-x)^{2}}{\sin 2(-x)}=\frac{x^{2}}{-\sin 2 x}=-\frac{x^{2}}{\sin 2 x}
$$
i.e., $f(-x)=-f(x)$, hence the function $f(x)$ is odd;
2) $\varphi(-x)=4-2(-... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 34,175 |
12. Find the domain of each of the following functions:
1) $y=\sqrt{1-x^{2}}$
2) $u=\frac{x-1}{x^{2}-5 x+6}+\sqrt[3]{2 x+1}$
3) $v=\arccos \frac{1-2 x}{3}$
4) $p=\frac{x}{\sin x}$
5) $q=\log _{2}\left(x^{2}-9\right)$. | Solution. 1) Since the argument $x$ is under a radical of even degree, the function $y$ will have real values only for those values of $x$ for which the radicand is non-negative, i.e., $1-x^{2} \geqslant 0$. Solving this inequality, we get
$$
x^{2} \leqslant 1 ; \quad|x| \leqslant 1 ; \quad-1 \leqslant x \leqslant 1
$... | \begin{pmatrix}1)[-1;1]\\2)(-\infty,2)\cup(2,3)\cup(3,+\infty)\\3)[-1;2]\\4)(-\infty,0)\cup(0,\pi)\cup(\pi,2\pi)\cup\cdots\\5)(-\infty,-3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 34,176 |
14. Plot the graphs of the functions:
1) $y=x^{2}-2 x-1$ on the interval $[-2 ; 4]$;
2) $y=-\frac{4 x}{x^{2}+1} \quad$ on the interval $|-5 ; 5|$;
3) $y=7 x^{2}-100 \sqrt{1+x^{2}}$ on the interval $|x| \leq 7$;
4) $y=x^{2}-4|x-1|+1$ on the interval $[-6 ; 5]$;
5) $y=\frac{16}{x^{2}}-1$ between the points of intersectio... | Solution. 1) The problem statement indicates that the independent variable $x$ can only take values within the interval $[-2 ; 4]$. Considering this, we will create the following table, using only integer values of $x$ for simplicity and calculating the corresponding values of $y$ from the given equation:
| $x$ | $y$ ... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 34,177 |
16. Find the approximate values of the roots of the function $y=0.3 x^{3}-2 x^{2}-0.2 x+0.5$ by plotting its graph on the interval $[-1 ; 3]$. | Solution. The roots of the function, i.e., the values of the argument that make it zero, can be found as the abscissas of the points of intersection of the function's graph with the x-axis, since at these points $y=0$.
By creating a table of numerical values of the variables $x$ and $y$, we will plot the graph of the ... | x_{1}\approx-0.4;x_{2}\approx0.5;x_{3}\approx2.6 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 34,178 |
21. Construct the graph of the function $y=\sqrt{\bar{x}}$ on the interval $[0 ; 9]$ using points, and then, based on this graph, construct the graph of the function $y=$ $=2 \sqrt{-3(x+1.5)}-1.2$ through sequential deformations and shifts of it. | Solution. We will compile a table of corresponding values of the variables $x$ and $y$ for the function $y=\sqrt{x}$ and plot its graph (Fig. 14).
| $\boldsymbol{x}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $y$ | 0 | 1.0 | 1.4... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 34,179 |
22. Based on the graph of the function $y=\sin x$, by its deformations and shifts construct the graph of the function $y=-3 \sin (2 x+8)$. | Solution. Replacing in expression (1) the symbol of an arbitrary function $f$ with the symbol of a trigonometric function $\sin$, we get
$$
y=A \sin k(x-a)+b
$$
Transforming the given function:
$$
y=-3 \sin (2 x+8)=-3 \sin 2(x+4)
$$
and comparing it with expression (2), we determine the following parameter values: ... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 34,180 |
27. Assuming $n=0,1,2,3, \ldots$, construct a table of values for the variables
$$
x=1+0.1^{n} ; \quad y=-0.1^{-n}, \quad z=(-0.1)^{n}, \quad u=(-1)^{n}+0.1^{n}
$$
and determine the nature of their change as $n$ increases without bound, i.e., as $n \rightarrow+\infty$. | Solution. Calculating the values of the given variables at the specified values of $n$, we obtain the following table:
| $n$ | $0 ;$ | $1 ;$ | $2 ;$ | $3 ;$ | $4 ;$ | $5 ;$ | $\ldots ; n \rightarrow+\infty$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $x$ | $2 ;$ | 1.1 ; | 1.01 ; | 1.001 ; | 1... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 34,181 |
28. Prove that $\lim _{n \rightarrow+\infty} a^{n}= \begin{cases}0, & \text { if } 0<a<1 \\ 1, & \text { if } a=1 \\ +\infty, & \text { if } a>1 .\end{cases}$ | Solution. 1) Let the constant a be a proper positive fraction $0n_{0}$, the values of the function $a^{n}$ will be less than any given positive number $\varepsilon$.
Assuming $a^{n_{0}}\frac{\lg \varepsilon}{\lg a}$ (the inequality sign changes because when $0n_{0}$ will be less than $\varepsilon$, no matter how small... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,182 |
29. Prove that:
1) $\lim _{x \rightarrow \infty} \frac{2 x+3}{3 x}=\frac{2}{3}$
2) $\lim _{x \rightarrow 3}(2 x+1)=7$ | Solution. 1) Let's form the difference $\frac{2 x+3}{3 x}-\frac{2}{3}=\frac{1}{x}$. As $x \rightarrow \infty$, this difference is an infinitesimal quantity, being the reciprocal of an infinitely large quantity. If the variable $\frac{2 x+3}{3 x}$ differs from the constant $\frac{2}{3}$ by an infinitesimal quantity, the... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,183 |
30. Find the limits of the function $y=\frac{5}{2-x}$: 1) as $x \rightarrow 2-0$ and 2) as $x \rightarrow 2+0$. Explain the solution using tables. | Solution. 1) If $x$ tends to 2 from the left, remaining less than 2, then $2-x$ will be a positive infinitesimal,
and $\frac{5}{2-x}$ will be a positive infinitely large, i.e., if $x \rightarrow 2-0$, then $(2-x) \rightarrow+0$, and $\frac{5}{2-x} \rightarrow+\infty$, so $\lim _{x \rightarrow 2-0} \frac{5}{2-x}+\infty$... | \lim_{xarrow2-0}\frac{5}{2-x}=+\infty\lim_{xarrow2+0}\frac{5}{2-x}=-\infty | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,184 |
31. Find the limits of the function $y=2^{\frac{1}{x}}$ as $x$ approaches zero: 1) from the left, 2) from the right, and 3) in any manner. | Solution. 1) If the variable $x$ tends to zero from the left, remaining negative, i.e., if $x$ is a negative infinitesimal, then $\frac{1}{x}$ will be a negative infinitely large quantity. Thus, $\lim _{x \rightarrow-0} 2^{\frac{1}{x}}=\lim \left(\frac{1}{2}\right)^{-\frac{1}{x}}=\left(\frac{1}{2}\right)^{+\infty}=0$, ... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,185 |
38. Find the limits of the following functions:
1) $f(x)=2 x-3-\frac{1}{x}$ as $x \rightarrow 1$:
2) $y=\frac{x^{3}-3 x^{2}+2 x-5}{x^{2}+2}$ as $x \rightarrow-1$;
3) $y=x \sin \frac{1}{x}$ as $x \rightarrow 0$. | Solution. Using the indicated theorems, we sequentially find:
1) $\lim _{x \rightarrow 1}\left(2 x-3-\frac{1}{x}\right)=\lim 2 \cdot \lim x-\lim 3-\frac{\lim 1}{\lim x}=$ $=2 \cdot 1-3-\frac{1}{1}=-2$
2) $\lim _{x \rightarrow-1} \frac{x^{3}-3 x^{2}+2 x-5}{x^{2}+2}=\frac{(\lim x)^{3}-3(\lim x)^{2}+2 \lim x-5}{(\lim x)^... | -2,-\frac{11}{3},0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,186 |
39. For $n \rightarrow+\infty$ find the limits of the following functions:
1) $S_{1}(n)=\frac{1}{n}+\frac{2}{n}+\frac{3}{n}+\ldots+\frac{n-1}{n}$;
2) $S_{2}(n)=\frac{1}{n^{2}}+\frac{2}{n^{2}}+\frac{3}{n^{2}}+\ldots+\frac{n-1}{n^{2}}$;
3) $S_{3}(n)=\frac{1}{n^{3}}+\frac{2}{n^{3}}+\frac{3}{n^{3}}+\ldots+\frac{n-1}{n^{3}}... | Solution. Each of these functions represents the sum of $n-1$ terms of an arithmetic progression. The common difference of the first progression is $\frac{1}{n}$, the second is $\frac{1}{n^{2}}$, and the third is $-\frac{1}{n^{3}}$.
By performing the addition and taking the limit, we find:
1) $S_{1}=\frac{n-1}{2}\lef... | S_1arrow+\infty,\quadS_2arrow\frac{1}{2},\quadS_3arrow0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 34,187 |
40. Prove that $\lim _{n \rightarrow+\infty} \frac{x^{n}}{n!}=0$ for any value of $x$. ${ }^{*}$ | Solution. No matter what the value of $x$ is, there will always be two consecutive positive integers $k$ and $k+1$ such that $|x|$ is between them, i.e., $k<|x|<k+1$.
From this, we obtain the obvious inequality:
$$
\left|\frac{x^{n}}{n!}\right|=\left|\frac{x^{k}}{k!} \cdot \frac{x}{k+1} \cdot \frac{x}{k+2} \cdot \fra... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,188 |
47. Find the limit of the function:
1) $f(x)=x^{3}-5 x^{2}+2 x+4$ as $x \rightarrow-3$;
2) $\varphi(t)=t \sqrt{t^{2}-20}-\lg \left(t+\sqrt{t^{2}-20}\right)$ as $t \rightarrow 6$. | Solution. The given function is elementary, it is defined at the limit point, so we find the limit of the function as its particular value at the limit point:
1) $\lim _{x \rightarrow-3} f(x)=f(-3)=(-3)^{3}-5 \cdot(-3)^{2}+2 \cdot(-3)+4=-74$;
2) $\lim \varphi(t)=\varphi(6)=6 \sqrt{6^{2}-20}-\lg \left(6+\sqrt{6^{2}-20}... | 23 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,189 |
53. 54) $\lim _{x \rightarrow 0} \frac{1-\sqrt{x+1}}{x}$;
3) $\lim _{x \rightarrow 0} \frac{\tan x}{1-\sqrt{1+\tan x}}$
4) $\lim _{x \rightarrow 1} \frac{2-\sqrt{x}}{3-\sqrt{2 x+1}}$
5) $\lim _{x \rightarrow 1} \frac{1-\sqrt{x}}{1-\sqrt[3]{x}}$. | Solution. First, we find that the given function represents the ratio of two infinitesimally small quantities (case $\frac{0}{0}$) when the argument is changed. We then transform the fraction to eliminate the factor that tends to zero:
1) We eliminate the irrationality in the numerator by multiplying the numerator and... | -\frac{1}{2},\frac{3}{4},-2,\frac{3}{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,190 |
54. 55) $\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}$; 2) $\lim _{x \rightarrow 0} \frac{x^{2}}{1-\cos x}$;
3) $\lim _{x \rightarrow 1} \frac{\cos \frac{\pi x}{2}}{1-x}$; 4) $\lim _{x \rightarrow-2} \frac{x^{2}-4}{\operatorname{arctg}(x+2)}$. | Solution. We establish that the given function is not defined at the limit point, and that with the given change in the argument, it represents the ratio of two infinitesimally small quantities (the case $\frac{0}{0}$). After this, we subject the function to transformations in order to use the 1st remarkable limit:
$$... | 3,2,\frac{\pi}{2},-4 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,191 |
71. 72) $\lim _{x \rightarrow \infty} \frac{3 x^{2}-1}{5 x^{2}+2 x}$
2) $\lim _{n \rightarrow-\infty} \frac{n}{\sqrt{n^{2}+1}}$
3) $\lim _{n \rightarrow+\infty} \frac{1+7^{n+2}}{3-7^{n}}$
4) $\lim _{n \rightarrow+\infty} \frac{2+4+6+\ldots+2 n}{1+3+5+\ldots+(2 n+1)}$
5) $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\tan 2... | Solution. Having made sure that the case is $\frac{\infty}{\infty}$, we subject the function to transformations.
1) Dividing the numerator and denominator of the fraction by $x^{2}$ (the highest power of $x$ here), we find
$$
\lim _{x \rightarrow \infty} \frac{3 x^{2}-1}{5 x^{2}+2 x}=\lim \frac{3-\frac{1}{x^{2}}}{5+\... | \frac{3}{5},-1,-49,1,-\frac{1}{2},3 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,192 |
88. 1) $\lim _{n \rightarrow \infty}\left(1+\frac{a}{n}\right)^{n}$
2) $\lim _{x \rightarrow 0} \sqrt[x]{1-2 x}$
3) $\lim _{t \rightarrow \infty}\left(\frac{t-3}{t+2}\right)^{2 t+1}$
4) $\lim (\tan x)^{\tan 2 x}$.
$x \rightarrow \frac{\pi}{4}$ | Solution. First, we ensure that with the specified change in the argument, the function represents a power where the base tends to one and the exponent tends to infinity (case $1^{*}$). Then, we transform the function to use the 2nd remarkable limit.
1) Assuming $n=a x$, we get $x \rightarrow \infty$ when $n \rightarr... | e^{},e^{-2},e^{-10},e^{-1} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,195 |
113. If $x \rightarrow 0$, which of the following infinitesimals:
1) $10 x$;
2) $x^{3}$; 3) $\sqrt{3 x}$
3) $\tan \frac{x}{5}$;
4) $\log (1+x)$ have a higher order than $x$, a lower order than $x$, and the same order as $x$? | Solution. We find the limit of the ratio of each given infinitesimal to the infinitesimal $x$:
1) $\lim _{x \rightarrow 0} \frac{10 x}{x}=10$.
Therefore, $10 x$ is an infinitesimal of the same order as $x$;
2) $\lim _{x \rightarrow 0} \frac{x^{3}}{x}=\lim x^{2}=0$
$x^{3}$ is an infinitesimal of a higher order than $... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,196 |
114. Prove that as $x \rightarrow 0$:
1) $\sin a x \approx a x$; 2) $\tan a x \approx a x$; 3) $\arcsin a x \approx a x$;
2) $\arctan a x \approx a x$
3) $\sqrt{1+x}-1 \approx \frac{1}{2} x$ | Solution. To prove the equivalence of two infinitesimals, we need to find the limit of their ratio. If this limit turns out to be equal to one, then the infinitesimals are equivalent.
1) \(\lim _{x \rightarrow 0} \frac{\sin a x}{a x}=\lim _{a x \rightarrow 0} \frac{\sin a x}{a x}=1\).
2) \(\lim _{x \rightarrow 0} \fra... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,197 |
115. Using the fact that when finding the limit of the ratio of two infinitesimals, they can be replaced by equivalent infinitesimals (property II), find the following limits:
1) $\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 3 x}$
2) $\lim _{x \rightarrow 0} \frac{\tan^{2} 2 x}{\sin ^{2} \frac{x}{3}}$;
3) $\lim _{x \ri... | Solution. Using the fact that $\sin \alpha \approx \tan \alpha \approx \arcsin \alpha \approx \arctan x \approx \alpha$ as $\alpha \rightarrow 0$, which follows from the solution of problem 114, and applying the property of equivalent infinitesimals, we get:
1) $\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 3 x}=\lim \... | \frac{4}{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,198 |
4. Show that the elementary functions: 1) $y=2 x^{2}-1$; 2) $v=\operatorname{cosec} x$ are continuous throughout their domain of definition. | Solution. We will find the domain of the function and then, based on the definition of continuity, ensure that the function will be continuous in the same domain.
1) The domain of the function \( y \) is the entire number line. Next, we will give the argument \( x \) an arbitrary increment \( \Delta x \) and, substitu... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,199 |
124. By computing the limit $\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$, find the derivatives of the following functions:
1) $y=3 x^{2}-4 x$
2) $y=\frac{1}{x}$
3) $y=\sqrt{x}$
4) $y=\cos 3 x$. | Solution: Guided by the specified general rule for directly finding the derivative, we sequentially find:
1) For the function $y=3 x^{2}-4 x$:
I) $y+\Delta y=3(x+\Delta x)^{2}-4(x+\Delta x)=3 x^{2}+6 x \Delta x+$
$$
+3 \Delta x^{2}-4 x-4 \Delta x
$$
II) $\Delta y=\left(3 x^{2}+6 x \Delta x+3 \Delta x^{2}-4 x-4 \Delt... | 6x-4,-\frac{1}{x^2},\frac{1}{2\sqrt{x}},-3\sin3x | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,202 |
126. Using differentiation formulas, find the derivatives of the following functions:
1) $y=x^{2}-5 x+4$
2) $y=\sqrt{x}+\frac{5}{\sqrt[3]{x}}-\frac{1}{x^{2}}+\frac{1}{3 x^{3}}$;
3) $z=x^{5}\left(2-\frac{x}{3}+3 x^{2}\right)$;
4) $f(x)=\frac{x^{2}}{x^{2}+1}$;
5) $\varphi(t)=\frac{10}{a \sin t-b \cos t}$;
6) $R(a)=\frac{... | Solution. 1) $y^{\prime}=\left(x^{2}-5 x+4\right)^{\prime}=\left(x^{2}\right)^{\prime}-(5 x)^{\prime}+(4)^{\prime}$ (by formula 2);
$$
y^{\prime}=2 x-5 \cdot 1+0=2 x-5 \text { (by formulas } 5 \text {, 3a and } 1 \text {). }
$$
2) Introducing fractional and negative exponents, we transform the given function:
$$
y=x... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,203 |
127. Find the derivative of the given function and then calculate its particular value at the specified argument value:
1) $F(x)=\frac{(1-\sqrt{\bar{x}})^{2}}{x}, \quad x=0.01$
2) $z=\frac{\cos t}{1-\sin t}, \quad t=\frac{\pi}{6}$
3) $y=\frac{a+b}{3-2 x}+\frac{5 x^{4}-1}{a-b}, x=0$. | Solution. 1) First, we expand the brackets and perform the division, then differentiate:
$$
\begin{gathered}
F(x)=\frac{1-2 V x+x}{x}=\frac{1}{x}-\frac{2}{V x}+1=x^{-1}-2 x^{-\frac{1}{2}}+1 \\
F^{\prime}(x)=-x^{-2}-2\left(-\frac{1}{2}\right) x^{-\frac{3}{2}}=-\frac{1}{x^{2}}+\frac{1}{\sqrt{x^{3}}}
\end{gathered}
$$
S... | -9000,2,\frac{2}{9}(+b) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,204 |
143. 144) \( y=(1+5x)^{3} \); 2) \( y=\sin 5x \); 3) \( y=\cos^{2} x \);
4) \( y=\sin x^{2} \);
5) \( y=\sqrt[3]{2+x^{4}} \). | Solution. 1) Assuming $y=u^{3}$, where $u=1+5 x$, and applying the rule of differentiation of a composite function, we get:
$$
\frac{d y}{d u}=3 u^{2} ; \quad \frac{d u}{d \bar{x}}=5 ; \quad \frac{d y}{d x}=\frac{d y}{d u} \cdot \frac{d u}{d x}=3 u^{2} \cdot 5=15(1+5 x)^{2}
$$
It is easy to verify the correctness of ... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,205 |
144. 145) \( z=\left(3 a x-x^{2}\right)^{k} ; z^{\prime} \) ?
2) \( \beta=2 \sqrt{\sin \frac{\alpha}{3}} ; \frac{d \beta}{d \alpha} \) ?
3) \( s=\left(\frac{t}{2 t+1}\right)^{10} \), compute \( s^{\prime}(-1) \).
4) \( r=\sin ^{3} 2 \varphi-\cos ^{3} 2 \varphi \), compute \( r^{\prime}\left(\frac{\pi}{8}\right) \). | Solution. 1) Applying formulas 5 and 2, we find
$$
z^{\prime}=k\left(3 a x-x^{2}\right)^{k-1} \cdot\left(3 a x-x^{2}\right)^{\prime}=k(3 a-2 x)\left(3 a x-x^{2}\right)^{k-1}
$$
2) Using formulas 5 and 6:
$$
\begin{gathered}
\beta^{\prime}=2 \cdot \frac{1}{2}\left(\sin \frac{\alpha}{3}\right)^{-\frac{1}{2}} \cdot\lef... | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,206 | |
158. Find the derivatives of the following functions:
1) $y=x^{3} 3^{x}$.
2) $f(x)=\sqrt[x]{3}+\frac{1}{25^{5 x}}+6^{1 / x}$; calculate $f^{\prime}(1)$.
3) $y=\ln \cos 3 x$
4) $r=a^{4} b^{\prime \prime} c^{\prime \prime}+\lg (5 p)-4 \lg \sqrt{\varphi}$.
5) $y=\ln \frac{a^{2}-x^{2}}{a^{2}+x^{2}}$.
6) $y=\ln \sqrt{\frac{... | Solution. 1) Differentiate as a product and by formulas 5 and 10b:
$$
y^{\prime}=\left(x^{3}\right)^{\prime} 3^{x}+x^{3}\left(3^{x}\right)^{\prime}=3 x^{2} 3^{x}+x^{3} 3^{x} \ln 3=x^{2} 3^{x}(3+x \ln 3)
$$
2) Introduce fractional and negative exponents, then differentiate as a sum and by formula 10:
$$
\begin{aligne... | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,207 | |
170. Find the derivatives of the following functions:
1) $y=5 \arcsin k x+3 \arccos k x$; 2) $y=\arcsin \frac{a}{x}-\operatorname{arcctg} \frac{x}{a}$;
2) $r=\operatorname{arctg} \frac{m}{\varphi}+\operatorname{arcctg}(m \operatorname{ctg} \varphi)$; $r^{\prime}(0)$ ?, $r^{\prime}(\pi)$ ? | Solution. 1) Using formulas 12 and 13, we find
$$
\begin{gathered}
y^{\prime}=5 \frac{(k x)^{\prime}}{\sqrt{1-(k x)^{2}}}+3\left[-\frac{(k x)^{\prime}}{\sqrt{1-(k x)^{2}}}\right]=\frac{5 k}{\sqrt{1-k^{2} x^{2}}}- \\
-\frac{3 k}{\sqrt{1-k^{2} x^{2}}}=\frac{2 k}{\sqrt{1-k^{2} x^{2}}}
\end{gathered}
$$
2) Using formulas... | \begin{aligned}1)&\quady^{\}=\frac{2k}{\sqrt{1-k^{2}x^{2}}}\\2)&\quady^{\}=\frac{}{^{2}+x^{2}}-\frac{}{|x|\sqrt{x^{2}-^{2}}}\\3)&\quadr^{\}(0)=0,\ | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,208 |
181. 182) $y=-\frac{\cos x}{\sin ^{2} x}+\ln \operatorname{tg} \frac{x}{2} ; y^{\prime}$ ? 2) $y=\arcsin (\cos x) ; y^{\prime}$ ?
3) $r=\varphi^{2} \arccos \frac{2}{\varphi}-2 \sqrt{\varphi^{2}-4}$; compute $r^{\prime}(2)$ and $r^{\prime}(-2)$.
4)* $y=\left|1-x^{2}\right| ;$ find $y^{\prime}\left(\frac{1}{2}\right), y... | Solution. 1) Sequentially applying formulas 2, 4, 7, 5, 6, 11, and 14, we get
\[
\begin{aligned}
& y' = -\frac{(\cos x)' \sin^2 x - \cos x (\sin^2 x)'}{\sin^4 x} + \frac{(\tan \frac{x}{2})'}{\tan \frac{x}{2}} = \\
& = \frac{\sin^3 x + 2 \sin x \cos^2 x}{\sin^4 x} + \frac{\sec^2 \frac{x}{2}}{2 \tan \frac{x}{2}} = \frac... | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,210 | |
203. Find the derivatives of the following functions:
1) $y=x^{x}$;
2) $r=(\cos \alpha)^{\sin 2 \alpha}$;
3) $s=\frac{2 t}{\sqrt{1-t^{2}}}$;
4) $R=(x-1) \sqrt[3]{(x+1)^{2}(x-2)}$. | Solution. Applying logarithmic differentiation, we sequentially find:
1) a) $\ln y = x \ln x$;
b) $\frac{y'}{y} = (x)' \ln x + x (\ln x)' = \ln x + x \cdot \frac{1}{x} = \ln x + 1$;
c) $y' = y(1 + \ln x) = x^x (1 + \ln x)$.
2) a) $\ln r = \sin 2\alpha \ln \cos \alpha$;
b) $\frac{r'}{r} = (\sin 2\alpha)' \ln \cos \... | \begin{aligned}1)&\quady'=x^x(1+\lnx)\\2)&\quadr'=2(\cos2\alpha\ln\cos\alpha-\sin^2\alpha)(\cos\alpha)^{\sin2\alpha}\\3)&\quad'=\frac{2}{\sqrt{(1-^2)^3}}\\4)&\quad | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,211 |
232.
$$
\text { 1) }\left\{\begin{array}{l}
x=k \sin t+\sin k t \\
y=k \cos t+\cos k t ;\left(\frac{d y}{d x}\right)_{t=0} ?
\end{array}\right.
$$
What is the geometric meaning of the result?
2) $\left\{\begin{array}{l}x=\alpha^{2}+2 \alpha \\ y=\ln (\alpha+1) ; \frac{d^{2} y}{d x^{2}} ?\end{array}\right.$
3) $\left... | Solution. 1) We find the derivatives of $x$ and $y$ with respect to the parameter $t$:
$$
\frac{d x}{d t}=k \cos t+k \cos k t ; \quad \frac{d y}{d t}=-k \sin t-k \sin k t
$$
The desired derivative of $y$ with respect to $x$ is found as the ratio of the derivatives of $y$ and $x$ with respect to $t$:
$\frac{d y}{d x}... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,214 |
239. Form the equations of the tangent and normal:
1) to the parabola $y=x^{2}-4 x$ at the point where $x=1$;
2) to the circle $x^{2}+y^{2}-2 x+4 y-3=0$ at the points of its intersection with the $O x$ axis;
3) to the cycloid $x=t-\sin t, y=1-\cos t$ at the point where $t=\frac{\pi}{2}$;
4) * to the curve $y=\left|x^{3... | Solution. 1) Substituting the given abscissa of the point of tangency \(x=1\) into the equation of the parabola, we find its ordinate \(y=-3\).
To determine the slope of the tangent \(y_{0}^{\prime}\), we find the derivative of \(y\) with respect to \(x\) from the equation of the parabola and compute its value at the ... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,215 |
240. Find the angles at which the following lines intersect:
1) the line $x+y-4=0$ and the parabola $2 y=8-x^{2}$;
2) the ellipse $x^{2}+4 y^{2}=4$ and the parabola $4 y=4-5 x^{2}$;
3) the sine curve $y=\sin x$ and the cosine curve $y=\cos x$. | Solution. 1) Solving the equations of the parabola and the line together, we find that they intersect at two points: $A(0 ; 4)$ and $B(2 ; 2)$, see Fig. 38.
Next, we find the derivative of $y$ with respect to $x$ from the equation of the parabola: $2 y^{\prime}=-2 x, y^{\prime}=-x$ and determine the slopes of the tang... | 45,18.5,92,0,70.5,109.5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 34,216 |
241. At which points of the curve $x=t-1, y=t^{3}-12 t+1$ is the tangent parallel to: 1) the $Ox$ axis; 2) the line $9 x+y+3=0$? | Solution. Here we use the condition of parallelism of lines, which consists in the equality of their angular coefficients.
Let's find the derivative of $y$ with respect to $x$ from the equations of the curve:
$$
y^{\prime}=\frac{d y}{d t}: \frac{d x}{d t}=\frac{3 t^{2}-12}{1}=3 t^{2}-12
$$
This derivative represents... | (1,-15),(-3,17),(0,-10),(-2,12) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,217 |
262. A point moves along the cubic parabola $12 y=x^{3}$. Which of its coordinates changes faster? | Solution. Considering $y$ as a composite function of time $t$ in the parabola equation and differentiating it with respect to $t$, we get
$$
12 \frac{d y}{d t}=3 x^{2} \frac{d x}{d t}
$$
From this, we find the ratio of the rates of change of the ordinate and the abscissa:
$$
\frac{d y}{d t}: \frac{d x}{d t}=\frac{x^... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,218 |
263. A reservoir in the form of a hemisphere with an internal radius of $R(m)$ is being filled with water at a rate of $Q(l)$ per second. Determine the rate of increase of the water level in the reservoir at the moment when it is equal to $0.5 R$. | Solution. Let $h$ be the water level in $m$ and $v$ its volume in $\mu^{3}$. We will find the dependence between the variables $h$ and $v$ using the formula for the volume of a spherical cap:
$$
v=\pi h^{2}\left(R-\frac{h}{3}\right)
$$
Differentiating this equality with respect to time $t$, we will find the dependenc... | \frac{0.004Q}{3\piR^{2}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,219 |
264. The speed of a body in rectilinear motion is proportional to the square root of the distance traveled (as, for example,
in free fall). Prove that this motion occurs under the action of a constant force. | Solution. According to Newton's law, the force $F$ causing the motion is proportional to the acceleration
$$
F=k \frac{d^{2} s}{d t^{2}}
$$
According to the condition $\frac{d s}{d t}=\lambda \sqrt{s}$. Differentiating this equality, we find
$$
\frac{d^{2} s}{d t^{2}}=\frac{\lambda}{2 \sqrt{s}} \cdot \frac{d s}{d t}... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,220 |
265. A point performs a straight-line oscillatory motion according to the law $x=A \sin \omega t$. Determine the velocity and acceleration of the motion at the moment of time $t=\frac{2 \pi}{\omega}$. Show that the acceleration of the motion is proportional to the displacement $x$. | Solution. Let's find the velocity $v$ and the acceleration of the motion at any moment of time $t$:
$$
v=\frac{d x}{d t}=A \omega \cos \omega t ; \quad \omega=\frac{d^{2} x}{d t^{2}}=-A \omega^{2} \sin \omega t
$$
At $t=\frac{2 \pi}{\omega}, v=A \omega, \tau=0$.
By comparing the expressions for the acceleration and ... | -\omega^{2}x | Calculus | proof | Yes | Yes | olympiads | false | 34,221 |
271. Find the differentials of the functions:
1) $y=x^{3}-3^{x}$;
2) $F(\varphi)=\cos \frac{\varphi}{3}+\sin \frac{3}{\varphi}$
3) $z=\ln \left(1+e^{10 x}\right)+\operatorname{arcctg} e^{5 x} ;$ calculate $\left.d z\right|_{x=0 ; d x=0,1}$ | Solution. To find the derivative of the given function and, by multiplying it by the differential of the independent variable, we obtain[^8]the desired differential of the given function:
1) $d y=y^{\prime} d x=\left(x^{3}-3^{x}\right)^{\prime} d x=\left(3 x^{2}-3^{x} \ln 3\right) d x$;
2) $d F(\varphi)=d\left(\cos \fr... | 0.25 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,222 |
272. Compute the approximate value: 1) $\sqrt[4]{17} ; 2$ ) $\operatorname{arc} \operatorname{tg} 0.98$ ; 3) $\sin 29^{\circ}$. | Solution. If it is required to calculate $f\left(x_{1}\right)$ and it is easier to calculate $f\left(x_{0}\right)$ and $f^{\prime}\left(x_{0}\right)$, then for a sufficiently small absolute value of the difference $x_{1}-x_{0}=d x$, the increment of the function can be replaced by its differential $f\left(x_{1}\right)-... | \sqrt[4]{17}\approx2.031,\operatorname{arctg}0.98\approx0.7754,\sin29\approx0.4848 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 34,223 |
286. Find the equations of the tangent line and the normal plane to the curve:
1) $x=t^{3}, y=t^{2}, z=t$ at the point where $t=-1$;
2) $x=y^{2}, y=z^{2}$ at the point where $z=2$. | Solution. 1) Determine the coordinates of the point of tangency: $x=-1, y=1, z=-1$ (substituting $t=-1$ into the given equations). Find the derivatives of $x, y$, and $z$ with respect to $t$ and calculate their values at the point of tangency: $\dot{x}=3 t^{2}, \dot{y}=2 t, \dot{z}=1 ; \dot{x}(-1)=3, \dot{y}(-1)=-2$, $... | 3x-2y+z+6=032x+4y+z-530=0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,224 |
287. Find the equation of the tangent to the helical line $y=a \cos t$, $y=a \sin t, z=b t$ at the point where $t=t_{0}$, and the angle it forms with the $O z$ axis. | Solution. Denoting the coordinates of the point of tangency ( $x_{0}, y_{0}, z_{0}$ ) and using the general equations (1), we obtain the following equations of the tangent:
$$
\frac{x-x_{0}}{-a \sin t_{0}}=\frac{y-y_{0}}{a \cos t_{0}}=\frac{z-z_{0}}{b}
$$
From here, the directional cosine of the angle formed by the t... | \frac{b}{\sqrt{^{2}+b^{2}}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,225 |
292. Given the equation of motion of a point, determine (name) the line that represents its trajectory and find the velocity and acceleration of this point:
1) $\bar{r}=(3 t-2) \bar{i}-4 t j ; \quad$ 2) $\bar{r}=2 \cos t \cdot \bar{i}+\sin t \cdot \bar{k} ;$
2) $\bar{r}=\left(2 t^{2}-3\right) \bar{i}-3 t^{2} j+\left(4 ... | Solution. 1) The trajectory of the point is the hodograph of its radius vector $r\{3 t-2 ;-4 t\}$, i.e., the line defined by the parametric equations $x=3 t-2, y=-4 t$. By eliminating the parameter (time) $t$ from these equations, we obtain the straight line $4 x+3 y+8=0$, located in the $x O y$ plane.
The velocity $\... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,226 |
298. Approximate the functions: 1) $x^{m}$ and 2) $\ln x$ by polynomials of degree $n$ with respect to the binomial $x-1$ and estimate the error. Then, setting $x-1=t$, obtain the expansions of the functions in powers of $t$. | Solution. To approximate the given function $f(x)$ by a polynomial with respect to the binomial $x-1$, one should write the Taylor polynomial for it, setting $a=1$. The error arising from replacing the given function with the Taylor polynomial is determined by the magnitude of the remainder term $R_{n}$ of the Taylor f... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,229 |
315. Find the limits:
1) $\lim x \operatorname{ctg} 2 x$
2) $\lim _{x \rightarrow+0} \sqrt[3]{x} \ln x$
3) $\lim (\operatorname{tg} \varphi-\sec \varphi)$; $\varphi \rightarrow \frac{\pi}{2}$
4) $\lim _{x \rightarrow 1}\left(\frac{1}{\ln x}-\frac{x}{x-1}\right)$;
5) $\lim _{t \rightarrow 0}\left(\frac{1}{\sin t}-\frac{... | Solution. By establishing that the case is $0 \cdot \infty$ or $\infty - \infty$, we transform the function into a fraction where both the numerator and the denominator simultaneously tend to zero or infinity, then apply L'Hôpital's rule:
1) $\lim _{x \rightarrow 0} x \operatorname{ctg} 2 x=\lim \frac{x}{\operatorname... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,232 |
350. Find the greatest and least values of each of the following functions:
1) $u=x^{3}-3 x^{2}-9 x+35$ on the interval $[-4 ; 4]$;
2) $p=x^{2} \ln x$ on the interval $[1, e]$
3) $r=2 \sin x+\sin 2 x$ on the interval $\left[0 ; \frac{3}{2} \pi\right]$;
4) $y=\operatorname{arctg} x^{2}$. | Solution. According to the practical rule:
1) I. Find the critical points of the function lying within the interval $[-4 ; 4]$, and calculate its values at these points: $u' = 3x^2 - 6x - 9 ; u' = 0$ at points $x = -1$ and $x = 3$. These points lie within the interval $[-4 ; 4]$ and are critical. There are no other cr... | 40,-41,0,e^2 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,236 |
357. Find the dimensions of a cylindrical closed tank with a given volume $v$ and the smallest total surface area. | Solution. Denoting the radius and height of the cylinder by $r$ and $h$, and its total surface area by $s$, we get
$$
s=2 \pi r h + 2 \pi r^{2}
$$
Here, the variables $r$ and $h$ are not independent but are related by the equation $v=\pi r^{2} h$, since according to the condition, the cylinder must have a given volum... | r=\sqrt[3]{\frac{v}{2\pi}},=2\sqrt[3]{\frac{v}{2\pi}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,238 |
359. Choose a place to build a bridge across the river so that the length of the road between two points located on opposite sides of the river is the shortest. | Solution. Let's make a schematic plan of the terrain near the objects specified in the condition (Fig. 58). Distances $a, b, c$ and $h$ are constant according to the condition of the problem. If the bridge is built in the place indicated in the plan, then the length of the road between points $A$ and $B$
$$
l=A C+h+D ... | \frac{}{+b} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 34,240 |
371. Determine the direction of convexity and the points of inflection of the curves:
1) $y=3 x^{5}-5 x^{4}+4$
2) $y=3-\sqrt[5]{(x+2)^{2}}$
3) $y=4 \sqrt{(x-1)^{5}}+20 \sqrt{(x-1)^{3}}$;
4) $y=\frac{1}{(x+1)^{3}}$;
5)* $y=2-\left|x^{5}-1\right|$. | Solution. To find the inflection points of the curve, we follow the given rule.
1) I. We find the points $x$ where $y''==0$ or does not exist, and the curve is continuous and lies within the domain of the curve:
$$
y' = 15x^4 - 20x^3; \quad y'' = 60x^3 - 60x^2 = 60x^2(x-1)
$$
$y'' = 0$ at points $x=0$ and $x=1$. The... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,241 |
387. Investigate the functions and plot their graphs:
1) $y=\frac{4 x^{3}-x^{4}}{5}$;
2) $y=\frac{1-x^{3}}{x^{2}}$;
3) $y=\sqrt[3]{(x+1)^{2}}-\sqrt[3]{(x-1)^{2}}$
4) $y=\sin ^{4} x+\cos ^{4} x$
5) $y=x^{2} \sqrt[x]{e}$
6) $y=x+2 \operatorname{arcctg} x$
7) $^{*} y=\left|e^{x}-1\right|$ | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,243 | |
400. Separate the real roots of the following equations: 1) $x^{2}-\cos x=0$; 2) $2 x^{3}+x+1=0$; 3) $x-\operatorname{ctg} x=0$.[^16] | Solution. To separate the real roots of the given equation, i.e., to enclose each of them within a special small interval, we will use the graphical method.
1) Transform the given equation to the form $x^{2}=\cos x$ and plot the curves $y=x^{2}$ and $y=\cos x$ on the same coordinate axes and with the same scale unit (... | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,244 | |
402. Calculate with an accuracy of 0.000001 the real root of the equation $2-x-\lg x=0$. | Solution. To isolate the root,
Fig. 80. Transform the equation to the form $\lg x=2-x$ and plot the curves $y=\lg x$ and $y=2-x$ (Fig. 80). From the graph, we determine that the desired roo... | 1.755581 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 34,245 |
411. Find the curvature of the curve: 1) $x=t^{2}, y=2 t^{3}$ at the point where $t=1$; 2) $y=\cos 2 x$ at the point where $x=\frac{\pi}{2}$. | Solution. 1) We find the derivatives $\dot{x}=2 t, \ddot{x}=2, \dot{y}=6 t^{2}$, $\ddot{y}=12 t$, and compute their values at the point where $t=1$:
$$
\dot{x}=2, \ddot{x}=2, \dot{y}=6, \ddot{y}=12
$$
and, substituting into formula (1), we get
$$
K=\frac{|\ddot{x} \ddot{y}-\ddot{y} x|}{\left(\dot{x}^{2}+\dot{y}^{2}\... | 4 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,246 |
412. Determine the radii of curvature at the vertices of the ellipse $x=a \cos t, y=b \sin t$. | Solution. Let's find the derivatives $\dot{x}=-a \sin t, \ddot{x}=-a \cos t$, $\dot{y}=b \cos t, \ddot{y}=-b \sin t$ and determine the radius of curvature of the ellipse at any of its points:
$$
R(t)=\frac{1}{K(t)}=\frac{\left(\dot{x}^{2}+\dot{y}^{2}\right)^{\frac{3}{2}}}{|\dddot{x} \ddot{y}-\dot{y}| \dot{x} \mid}=\fr... | R(0)=R(\pi)=\frac{b^{2}}{},\quadR(\frac{\pi}{2})=R(\frac{3\pi}{2})=\frac{^{2}}{b} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,247 |
413. Find the coordinates of the center of curvature and construct the curve and the circle of curvature for the curve:
1) $y=4 x-x^{2}$ at its vertex;
2) $x=t-\sin t, y=1-\cos t$ at the point where $t=\frac{\pi}{2}$. | Solution. 1) The given equation defines a parabola, the axis of which is parallel to the $O y$ axis. We find its vertex as the point where the tangent is parallel to the $O x$ axis, i.e., where $y^{\prime}=0$:
$$
y^{\prime}=4-2 x ; y^{\prime}=0 \text { when } x=2 ; y(2)=4
$$
Next, using formulas (2), we find the coor... | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,248 | |
414. At which points of the parabola $y=\sqrt{2} x^{2}$ is the radius of curvature equal to one? | Solution. We find the derivatives $y^{\prime}=2 \sqrt{2} x, y^{\prime \prime}=2 \sqrt{2}$ and by formula (1) the radius of curvature of the parabola at any point with abscissa $x$:
$$
R(x)=\frac{\left(1+8 x^{2}\right)^{\frac{3}{2}}}{2 \sqrt{2}}
$$
Setting $R(x)=1$, we get the abscissas of the desired points $2 \sqrt{... | \\frac{1}{2\sqrt{2}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,249 |
415. At what point does the curve $y=e^{x}$ have the greatest curvature? | Solution. We find the derivatives $y^{\prime}=y^{\prime \prime}=e^{x}$ and the curvature of the given curve at any point:
$$
K(x)=\frac{e^{x}}{\left(1+e^{2 x}\right)^{\frac{3}{3}}}
$$
Next, we find the maximum value of the function $K(x)$, which is defined and continuous over the entire number line:
$$
K^{\prime}(x)... | (-\frac{\ln2}{2};\frac{\sqrt{2}}{2}) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,250 |
416. Find the equation of the evolute of the curve and plot the curve and its evolute:
1) $x^{2}=2(1-y)$
2) $x=a \cos t, y=b \sin t$. | Solution. 1) From the given parabola equation, we find the derivatives: \( y' = -x, y'' = -1 \) and using formulas (2), we find the coordinates of any point on its evolute:
\[
\begin{aligned}
& X = x - \frac{1 + (y')^2}{y''} y' = x - \frac{1 + x^2}{-1}(-x); \\
& Y = y + \frac{1 + (y')^2}{y''} = 1 - \frac{x^2}{2} + \fr... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,251 |
430. Find the following integrals and verify the results by differentiation:
1) $\int \frac{d x}{x^{3}}$; 2) $\int \frac{d x}{\sqrt{2-x^{2}}}$; 3) $\int 3^{t} 5^{t} d t$; 4) $\int \sqrt{y+1} d y$; 5) $\int \frac{d x}{2 x^{2}-6}$. | Solution: 1) $\int \frac{d x}{x^{3}}=\int x^{-3} d x=\frac{x^{-2}}{-2}+C=C-\frac{1}{2 x^{2}}$, by formula 1, where $u=x, a=-3$.
Verification. We find the differential of the obtained function and ensure that it equals the integrand:
$$
d\left(C-\frac{1}{2 x^{2}}\right)=-\frac{1}{2}\left(x^{-2}\right)^{\prime} d x=x^{... | 1)C-\frac{1}{2x^{2}}\\2)\arcsin\frac{x}{\sqrt{2}}+C\\3)\frac{15^{}}{\ln15}+C\\4)\frac{2}{3}\sqrt{(y+1)^{3}}+C\\5)\frac{1}{4\sqrt{3}}\ln|\frac{x-\sqrt{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,252 |
431. Find the integrals:
1) $\int \frac{d x}{\sqrt[3]{5 x}}$
2) $\int \frac{d t}{\sqrt{3-4 t^{2}}}$
3) $\int \cos 3 \varphi d \varphi$
4) $\int e^{-\frac{x}{2}} d x$
5) $\int \sin (a x+b) d x$
6) $\int \frac{1}{5 x+4} d x$ | Solution.
1) $\int \frac{d x}{\sqrt[3]{5 x}}=\frac{1}{\sqrt[3]{5}} \int x^{-\frac{1}{3}} d x=\frac{1}{\sqrt[3]{5}} \cdot \frac{3}{2} x^{\frac{2}{3}}+C=$ $=\frac{3}{2 \sqrt[3]{5} 5} \sqrt[3]{x^{2}}+C$, according to property III and formula 1, when $u=x, a=-\frac{1}{3}$.
2) $\int \frac{d t}{\sqrt{3-4 t^{2}}}=\frac{1}{2}... | \begin{aligned}1)&\frac{3}{2\sqrt[3]{5}5}\sqrt[3]{x^{2}}+C\\2)&\frac{1}{2}\arcsin\frac{2}{\sqrt{3}}+C\\3)&\frac{1}{3}\sin3\varphi+C\\4)&-2e^{-\frac{x}{2}}+ | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,253 |
432. Find the integrals
1) $\int(3-2 x)^{2} d x$
2) $\int \sec ^{2}(m-n x) d x$
3) $\int \operatorname{tg} \varphi d \varphi$. | Solution. 1) We multiply and divide by -2, introduce the factor -2 under the integral sign according to property III, and replace $-2 dx$ with $d(3-2x)$, which is the same, to get:
$\int(3-2 x)^{7} d x=-\frac{1}{2} \int(3-2 x)^{7}(-2 d x)=-\frac{1}{2} \int(3-2 x)^{7} d(3-2 x)=$
$$
=-\frac{1}{2} \cdot \frac{(3-2 x)^{8... | \begin{aligned}1)&-\frac{1}{16}(3-2x)^8\\2)&-\frac{1}{n}\operatorname{tg}(-nx)\\3)&-\ln|\cos\varphi|\end{aligned} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,254 |
449. Find the integrals:
1) $\int\left(3 x^{2}-2 x+5\right) d x$;
2) $\int \frac{2 x^{2}+x-1}{x^{3}} d x$;
3) $\int\left(1+e^{x}\right)^{2} d x$
4) $\int \frac{2 x+3}{x^{2}-5} d x$;
5) $\int \frac{x^{2}}{x^{2}+1} d x$;
6) $\int \operatorname{tg}^{2} \varphi d \varphi$. | Solution. 1) Integrating each term separately, we get:
$$
\begin{aligned}
& \int\left(3 x^{2}-2 x+5\right) d x=\int 3 x^{2} d x-\int 2 x d x+\int 5 d x=3 \int x^{2} d x- \\
& -2 \int x d x+5 \int d x=3 \cdot \frac{x^{3}}{3}-2 \cdot \frac{x^{2}}{2}+5 x+C=x^{3}-x^{2}+5 x+C
\end{aligned}
$$
by formula 1.
2) We decompos... | \begin{aligned}1)&\quadx^{3}-x^{2}+5x+C\\2)&\quad2\ln|x|-\frac{1}{x}+\frac{1}{2x^{2}}+C\\3)&\quadx+2e^{x}+\frac{1}{2}e^{2x}+C\\4)&\quad\ln|x^{2}-5\ | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,255 |
458. Find the integrals:
1) $\int \frac{2 x d x}{x^{4}+3}$
2) $\int \frac{\sin x d x}{\sqrt{1+2 \cos x}}$
3) $\int \frac{x d x}{\sqrt[3]{x^{2}+a}}$;
4) $\int \frac{\sqrt{1+\ln x}}{x} d x$
5) $\int \frac{d y}{\sqrt{e^{y}+1}}$
6) $\int \frac{d t}{\sqrt{\left(1-t^{2}\right)^{3}}}$. | Solution. 1) Let $x^{2}=t$; differentiate $2 x d x=d t$, substitute into the integrand, find the new integral, and return to the original variable $x$:
$$
\int \frac{2 x d x}{x^{4}+3}=\int \frac{d t}{t^{2}+3}=\frac{1}{\sqrt{3}} \operatorname{arctg} \frac{t}{\sqrt{3}}+C=\frac{1}{\sqrt{3}} \operatorname{arctg} \frac{x^{... | \begin{aligned}1)&\quad\frac{1}{\sqrt{3}}\operatorname{arctg}\frac{x^{2}}{\sqrt{3}}+C\\2)&\quadC-\sqrt{1+2\cosx}\\3)&\quad\frac{3}{4}\sqrt[3]{(x^{2}+)^{2}}+C\\4) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,256 |
475. Find the integrals:
1) $\int x \cos x d x$
2) $\int \frac{\ln x}{x^{3}} d x$
3) $\int x \operatorname{arctg} x d x$
4) $\int \arcsin x d x$
5) $\int x^{2} e^{3 x} d x$
6) $\int e^{-x} \cos \frac{x}{2} d x$ | Solution. 1) Let $u=x, d v=\cos x d x$, then we find: $d u=d x$, $v=\int \cos x d x=\sin x$. Substituting into formula (*), we get $\int x \cos x d x=x \sin x-\int \sin x d x=x \sin x+\cos x+C$.
2) Let $u=\ln x, d v=\frac{d x}{x^{3}}$, then $d u=\frac{d x}{x}, v=\int \frac{d x}{x^{3}}=$ $=\int x^{-3} d x=-\frac{1}{2 x... | \begin{aligned}1)&\quad\intx\cosx=x\sinx+\cosx+C\\2)&\quad\int\frac{\lnx}{x^{3}}=C-\frac{1+2\lnx}{4x^{2}}\\3)&\quad\intx\operatorname{arctg}x=C-\frac{x}{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,257 |
488. Find the integrals:
1) $\int \frac{d x}{x^{2}+4 x+8}$
2) $\int \frac{7-8 x}{2 x^{3}-3 x+1} d x$;
3) $\int \frac{3 x-2}{x^{2}+6 x+9} d x$;
4) $\int \frac{6 x^{3}-7 x^{2}+3 x-1}{2 x-3 x^{2}} d x$. | Solution. 1) By completing the square of the quadratic trinomial $x^{2}+4 x+8=(x+2)^{2}+4$, writing $d(x+2)$ instead of $d x$ and integrating, we get
$$
\int \frac{d x}{x^{2}+4 x+8}=\int \frac{d(x+2)}{(x+2)^{2}+4}=\frac{1}{2} \operatorname{arctg} \frac{x+2}{2}+C,
$$
using formula 8, with $u=x+2, a=2$.
2) Completing ... | \begin{aligned}1)&\quad\frac{1}{2}\operatorname{arctg}\frac{x+2}{2}+C,\\2)&\quad\ln|\frac{x-1}{x-0.5}|-2\ln|x^{2}-1.5x+0.5|+C,\\3)&\quad3\ | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,258 |
489. Find the integrals:
1) $\int \frac{d x}{\sqrt{x^{2}-4 x-3}}$
2) $\int \frac{(3 x-5) d x}{\sqrt{9+6 x-3 x^{2}}}$ | Solution. 1) By completing the square of the trinomial $x^{2} - 4x - 3 = (x-2)^{2} - 7$, writing $d(x-2)$ instead of $dx$, and integrating, we find
$$
\int \frac{dx}{\sqrt{x^{2} - 4x - 3}} = \int \frac{d(x-2)}{\sqrt{(x-2)^{2} - 7}} = \ln \left| x-2 + \sqrt{(x-2)^{2} - 7} \right| + C
$$
(according to formula 11, with ... | C-\sqrt{9+6x-3x^{2}}-\frac{2}{\sqrt{3}}\arcsin\frac{x-1}{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,259 |
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