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Problem 5. The probability of at least one hit with two shots is 0.96. Find the probability of four hits with five shots. | Solution. By the Bernoulli formula (19), we have $p_{5}(4)=C_{5}^{4} \cdot p^{4} \cdot q^{1}$. Here, the probability $p$ of hitting the target with one shot is unknown. To find it, we use the condition of the problem.
Let's denote the events:
$A_{1}$ - hitting the target with the first shot;
$A_{2}$ - hitting the ta... | 0.4096 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,411 |
Problem 6. The probability of a shooter hitting the target is 0.5. Find the probability that the target will be hit from 5 to 7 times inclusive out of 8 shots. | Solution. By formula (20) we have:
$$
\begin{aligned}
& p_{8}(5 \leqslant k \leqslant 7)=p_{8}(5)+p_{8}(6)+p_{8}(7)= \\
& \quad=C_{8}^{5} \cdot p^{5} \cdot q^{3}+C_{8}^{6} \cdot p^{6} \cdot q^{2}+C_{8}^{7} \cdot p^{7} \cdot q=\frac{23}{64} \approx 0.3534
\end{aligned}
$$
Answer. $0.3534$. | 0.3534 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,412 |
Task 7. Among some products, the share of top-grade products is $31 \%$. What is the most probable number of top-grade products in a randomly selected batch of 75 products? | Solution. It is known that $p=0.31; q=1-p=0.69; n=75$. By formula (22) we have: $75 \cdot 0.31-0.69 \leqslant k_{0} \leqslant 75 \cdot 0.31+0.31 ; 22.56 \leqslant$ $\leqslant k_{0} \leqslant 23.56 ;$ since $k_{0}$ must be an integer, it follows that $k_{0}=23$.
Answer. 23.
## Exercises
1. Two equally skilled chess p... | 23 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,413 |
Task 1. A radio device contains 1000 elements. The probability of failure of any element within a certain time $T$ is $p=0.004$. Find the probability that exactly 5 elements will fail during time $T$. | Solution. We have $n=1000, p=0.004, q=1-p=0.996, \lambda=$ $=pn=4$. We need to calculate $p_{1000}(5)$. The Poisson formula gives:
$$
p_{1000}(5)=\frac{4^{5} \cdot e^{-4}}{5!}=\frac{1024 \cdot 0.0183}{120}=0.1562
$$
Answer. 0.1562.
Problem 2. The average number of taxi orders per hour is three. Find the probabilitie... | 0.1512 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,414 |
Task 3. The average number of airplanes arriving at the airport per minute is 4. Find the probabilities that in 3 minutes a) two airplanes will arrive, b) fewer than two, c) no fewer than two airplanes will arrive. | Solution. We are dealing with the simplest flow, for which $\lambda=4$, so by formula (24) we have: $p_{t}(k)=\frac{(4 t)^{k} \cdot e^{-4 t}}{k!}$.
a) $t=3, k=2, p_{3}(2)=\frac{(4 \cdot 3)^{2} \cdot e^{-12}}{2!}=0.0005$.
b) The event "fewer than two airplanes arrived" is the sum of the events "no airplane arrived" an... | )0.0005;b)0.0001;)0.9999 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,415 |
Task 4. Among the rye seeds, there is $0.4\%$ weed seeds. What is the probability of finding 5 weed seeds when randomly selecting 5000 seeds? | Solution. Using the Poisson formula (23), we find for $\lambda=n p=20$.
$$
p_{5000}(5)=\frac{20^{5} \cdot e^{-20}}{5!}=0.000055
$$
Answer. 0.000055.
## Exercises
1. The probability that a phone call will be interrupted due to technical reasons is 0.005. What is the probability that out of 1000 phone calls, exactly ... | 0.000055 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,416 |
Task 1. The germination rate of the given variety is $90 \%$. Determine the probability that out of 400 planted seeds, 350 will germinate. | Solution. Given: $n=400, p=0.9, q=0.1, k=350$. We need to calculate $p_{400}(350)$. Note that it is not easy to numerically implement the formulas of Bernoulli and Poisson. We will use the Laplace formula. First, we calculate $\sqrt{n p q}=\sqrt{400 \cdot 0.9 \cdot 0.1}=6$, then $k-n p=350-400 \cdot 0.9=-10$, next $x=-... | 0.2848 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,417 |
Problem 5. The probability that an item has not passed the inspection, $p=0.2$. Find the probability that among 400 randomly selected items, between 70 and 100 items will not be inspected. | Solution. The answer to the question is given by formula (26) of the integral Laplace theorem (Theorem 10). We have: \( n=400 ; k_{1}=70 ; k_{2}=100 ; p=0.2 \); \( q=0.8 \). Therefore, \( x_{1}=-1.25 ; x_{2}=2.5 \) and \( p_{400}(70,100)=\Phi\left(x_{2}\right)-\Phi\left(x_{1}\right)= \Phi(2.5)+\Phi(1.25) \approx 0.4938... | 0.8882 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,418 |
Task 1. The probability of event $A$ occurring in each of 484 independent trials is 0.75. Find the probability that the relative frequency of event $A$ deviates from its probability in absolute value by no more than 0.03. | Solution. Given: $n=484 ; p=0.75 ; q=0.25 ; \varepsilon=0.03$. By formula (27) we have:
$$
\begin{aligned}
& P\left(\left|\frac{m}{n}-0.75\right|<0.03\right) \approx 2 \Phi\left(0.03 \sqrt{\frac{484}{0.75 \cdot 0.25}}\right)=2 \Phi(1.524)=0.872 \\
& \text { Answer. } 0.872
\end{aligned}
$$ | 0.872 | Other | math-word-problem | Yes | Yes | olympiads | false | 31,419 |
Task 2. The quality control department checks 900 items for standard compliance. The probability that an item is standard is 0.9. Find, with a probability of 0.95, the boundaries within which the number $m$ of standard items among those checked will be contained. | Solution. Given: $n=900 ; p=0.9 ; q=0.1 ; P=2 \Phi\left(\varepsilon \sqrt{\frac{n}{p q}}\right)=$ $=0.95$. First, let's determine the deviation of the relative frequency from the probability of the event itself. The maximum value of this deviation $\varepsilon$ can be calculated from the relation $2 \Phi\left(\varepsil... | 793\leqslant\leqslant827 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,420 |
Task 3. The probability that a part is standard is 0.8. Find the minimum number of parts that should be selected so that with a probability of at least 0.96, one can assert that the relative frequency of the appearance of standard parts among the selected ones deviates in absolute value from the probability of 0.8 by n... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,421 | |
Task 1. A discrete random variable $X$ takes possible values $1,2,3,4$ with probabilities respectively 0.25, 0.35, 0.3, and $p$. Find $p$, construct the distribution table, and plot the distribution polygon. | Solution. The number $p$ is found from the condition $0.25+0.35+0.3+p=1$. We get $p=0.1$. The polygon is shown in Fig. 7.2, and the table is as follows:
| $X$ | 1 | 2 | 3 | 4 | $\Sigma$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $p$ | 0.25 | 0.35 | 0.3 | 0.1 | 1 |
=C_{5}^{0} \cdot 0.6^{0} \cdot 0.4^{5}=0.01024 \\
p(X=1)=C_{5}^{1} \cdot 0.6 \cdot 0.4^{4}=0.0768 \\
p(X=2)=C_{5}^{2} \cdot 0.6^{2} \cdot 0.4^{3}=0.2304 \\
... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,423 |
Problem 4. A basketball player throws the ball at the hoop. The probability of hitting the hoop with one throw is 0.6. Formulate the probability distribution law for the number of throws, if the basketball player stops throwing as soon as he hits the hoop. | Solution. The number of throws $X$ until the first hit in the ring is a random variable that can take possible values $1,2,3, \ldots$.
$X=1$ means that the basketball player hit on the first throw. The probability of this value is 0.6, i.e., $p(X=1)=0.6$. $X=2$ means that the basketball player made two throws, which m... | p(X=k)=0.4^{k-1}\cdot0.6 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,424 |
Problem 5. In an urn, there are 9 balls, of which 5 are white and 4 are black. Four balls are randomly drawn from the urn. Let $X$ be the number of white balls among the drawn ones. Construct the distribution law of the discrete random variable $X$.
untranslated text:
Задача 5. В урне находятся 9 шаров, из них 5 белы... | Solution. A similar problem in a more general form is considered in Chapter 2, Section VI (see problem 6): $N=9, M=5, n=4, k=0,1,2,3,4$.
Since $C_{9}^{4}=\frac{9 \cdot 8 \cdot 7 \cdot 6}{4 \cdot 3 \cdot 2 \cdot 1}=126$, the denominators of all 5 fractions are equal to 126. The results will be rounded to three decimal ... | \begin{pmatrix}x&p\\\hline0&0.008\\1&0.159\\2&0.476\\3&0.317\\4&0.040\\\end{pmatrix} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,425 |
Task 1. Calculate the expected value of a random variable $X$, distributed according to the binomial law with parameters $p=0.6$ and $n=5$. | Solution. Let's form the distribution series of $X$:
$$
p(X=k)=p_{5}(k)=C_{5}^{k} \cdot(0.6)^{k} \cdot(0.4)^{5-k}, \quad k=0,1,2,3,4,5
$$
We will place the values of $x_{k}$ and $p_{k}$ in a table (p. 305), to which we will add another row for the products $x_{k} \cdot p_{k}$. The sum of the numbers in this row const... | Ek(X) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,426 |
Task 1. The random variable $X$ is given by the probability density function $f(x)=a x^{2}$ for $x \in(0 ; 1)$ and $f(x)=0$ for $x \notin(0 ; 1)$. Determine the parameter $a$ and the distribution function $F(x)$. Also, find the probability that in four trials, $X$ will exactly three times take values in the interval $(... | Solution. The value of the parameter $a$ is determined from the condition $\int_{-\infty}^{+\infty} f(x) d x=1$. We have $\int_{0}^{1} a x^{2} d x=\left.\frac{a x^{3}}{3}\right|_{0} ^{1}=\frac{a}{3}=1$. We considered that $f(x)=0$ for $x \notin(0 ; 1)$. Hence, $a=3$. Thus, we have $f(x)=3 x^{2}, x \in(0 ; 1)$ and $f(x)... | 0.1536 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,428 |
Task 2. The random variable $X$ is given by the cumulative distribution function
$$
F(x)=\left[\begin{array}{ll}
0 & \text { for } x \leqslant 1 \\
a \ln x & \text { for } 1 < x \leqslant e \\
1 & \text { for } x > e
\end{array}\right.
$$
It is required: a) find the value of the parameter $a$; b) find the probability... | Solution. a) The function $F(x)$ is continuous on the entire line (see property 5) of point $2^{\circ}$), in particular, it is continuous at the value $x=e$. Therefore, $\lim _{x \rightarrow e-0} F(x)=\lim _{x \rightarrow e+0} F(x)=1$, and then $\lim _{x \rightarrow e} a \ln x=a=1$. Thus,
$$
F(x)=\left[\begin{array}{l... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,429 |
Task 1. For a random variable $X$, given by the distribution function $F(x)=0$ for $x<1$, $F(x)=a(x-1)^2$ for $1 \leq x \leq 4$, and $F(x)=1$ for $x>4$, determine: the value of the parameter $a$, the density function $f(x)$, the initial and central moments of the first four orders, as well as the skewness and kurtosis.... | Solution. We have $1=F(4)=a(x-2)^{2}=4 a$. From this, $a=0.25$.
Next, $f(x)=F^{\prime}(x)=0.5(x-2), x \in(2,4)$. Thus (see Fig. 7.12 and 7.13)
Fig. 7.12
=\frac{10}{3},D(X)=\frac{2}{9},\sigma(X)=\frac{\sqrt{2}}{3}\approx0.47,A(X)\approx-0.56,Ek(X)=-0.6 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,430 |
Problem 3. The quantity $X$ is normally distributed with parameters $a=2$ and $\sigma=3$. Calculate the probabilities that: a) $1 \leqslant X \leqslant 2 ;$ b) $X \geqslant 2$; c) $|X-2| \leqslant 3 ;$ d) $X \leqslant-1$; e) $|X-2| \leqslant 9$. | Solution. We will use equality 2) from point $3^{\circ}$. Let's recall that $\Phi(-x)=-\Phi(x)$, and the values of $\Phi(x)$ are found in table 1 of the appendix.
a) Here $\alpha=1$ and $\beta=2$. Therefore, $p(1 \leqslant X \leqslant 2)=\Phi\left(\frac{2-2}{3}\right)-\Phi\left(\frac{1-2}{3}\right)=\Phi(0)-\Phi\left(-... | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,433 | |
Task 1. In the production of costumes, the defect rate is $2 \%$. Calculate the probability that when inspecting a batch of 800 costumes, the deviation from the established defect rate will be less than $1 \%$. | Solution. It is necessary to evaluate the probability $p\left(\left|\frac{m}{800}-0.02\right| \leqslant\right.$ $\leqslant 0.01)$, where $m$ is the number of defective suits out of 800, and $\frac{m}{800}$ is the proportion of defective suits. According to Bernoulli's theorem, the required probability is estimated from... | 0.755 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,434 |
Problem 4. Given a sequence of random variables $X_{1}, X_{2}$, $\ldots, X_{n}$, where the discrete random variable $X_{n}$ can take three values: $-n \alpha, 0$ and $n \alpha$ ( $\alpha$ - a constant) with probabilities respectively $\frac{1}{2 n^{2}}, 1-\frac{1}{n^{2}}$, $\frac{1}{2 n^{2}}$. Is the law of large numbe... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,435 | |
Task 1. Given a statistical series - the number of days missed due to illness by employees of a laboratory.
| Number of days | 0 | 2 | 3 | 4 | 5 | 7 | 10 | Total |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Number of employees | 7 | 3 | 5 | 2 | 5 | 6 | 2 | 30 |
Determine the average n... | Solution. Let's determine the sample mean of size $n=30(k=7)$ using formula (1):
$$
\bar{x}_{3}=\frac{1}{30}(7 \cdot 0+3 \cdot 2+5 \cdot 3+2 \cdot 4+5 \cdot 5+6 \cdot 7+2 \cdot 10)=\frac{116}{30}=3.87
$$
We will calculate the variance and standard deviation using formula (2):
$$
\begin{aligned}
\overline{x_{\mathrm{... | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,436 |
Problem 3. The random variable $X$ - the number of weed seeds in a grain sample, is distributed according to the Poisson law. In the table
| $x_{i}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | $\Sigma$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $m_{i}$ | 405 | 366 | 175 | 40 | 8 | 5 | 2 | 1000 |
... | Solution. The Poisson law has the form $p_{m}\left(x_{\imath}\right)=\frac{\lambda^{x_{2}} e^{-\lambda}}{x_{\imath}!}$, where $m$ is the number of trials conducted in one experiment, $x_{\imath}$ is the number of occurrences of the event in the $i$-th experiment. For the Poisson law, the mathematical expectation coinci... | \lambda=0.9 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,437 |
Problem 4. The operating time of a radio device element follows the law $f(x)=\lambda e^{-\lambda x}$. In the table
| $x_{2}$ | 2.5 | 7.5 | 12.5 | 17.5 | 22.5 | 27.5 | $\Sigma$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $n_{2}$ | 133 | 45 | 15 | 4 | 2 | 1 | 200 |
the empirical distribution ... | Solution. As is known, for the exponential law, the mathematical expectation $M(X)$ is equal to $\frac{1}{\lambda}$. Therefore, the point estimate of $\lambda$ is $\frac{1}{\overline{\bar{x}}_{\mathrm{B}}}$. Let's calculate $\bar{x}_{\mathrm{B}}:$
$$
\bar{x}_{\mathrm{B}}=\frac{1}{200}(133 \cdot 2.5+45 \cdot 7.5+15 \cd... | \lambda=0.2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,438 |
Problem 5. Given a variational series of feature $X$:
| $x$ | 1 | 3 | 5 | 7 | 9 | $\Sigma$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $n_{2}$ | 10 | 40 | 20 | 20 | 10 | 100 |
Find the initial and central moments of the first four orders of feature $X$, as well as determine the skewness and kurtosis... | Solution. Calculations are performed using the formulas of point $7^{\circ}$ for $M_{k}$ and point $8^{\circ}$ for $m_{k}, h=1$.
1) $M_{\mathrm{I}}=\frac{1}{100}(1 \cdot 10+3 \cdot 40+5 \cdot 20+7 \cdot 20+9 \cdot 10)=4.6$
2) $M_{2}=\frac{1}{100}(1 \cdot 10+9 \cdot 40+25 \cdot 20+49 \cdot 20+81 \cdot 10)=26.6$;
3) $M_... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,439 |
Task 1. Construct the theoretical distribution law of the general population of the random variable \( X \), given in the form of a sample of size \( n=100 \) variants using a variational series
\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline x_{2} & 2 & 5 & 8 & 11 & 14 & 17 & 20 & \Sigma \\
\hline n_{2} & 9 & 13 & 20 & ... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,440 | |
Task 2. The technical control department inspected $n=100$ batches of $N=10$ items each, and as a result, obtained a distribution table of the random variable $X$ - the number of non-standard items in one batch:
| $x_{\imath}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | ... | Solution. We will construct a distribution polygon (Fig. 8.7). The Poisson law has the form:
$$
p(k)=\lambda^{k} \cdot \frac{e^{-\lambda}}{k!} \quad(\lambda=p \cdot n)
$$
We will apply the rule (see above).
1) We have $\bar{x}_{\text {v }}=\frac{1}{200}(116 \cdot 0+56 \cdot 1+$ $+22 \cdot 2+4 \cdot 3+2 \cdot 4)=0.6$... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,441 |
Task 1. Given the interval distribution:
| Intervals | $(0 ; 5)$ | $(5,10)$ | $(10 ; 15)$ | $(15 ; 20)$ | $(20 ; 25)$ | $(25 ; 30)$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Midpoints | 2.5 | 7.5 | 12.5 | 17.5 | 22.5 | 27.5 |
| Frequencies | 133 | 45 | 15 | 4 | 2 | 1 |
It is required, at the signi... | Solution. 1) The empirical sample mean $\widetilde{x}_{B}$ is calculated using the centers of the intervals:
$\widetilde{x}_{\mathrm{B}}=\frac{1}{200}(133 \cdot 2.5+45 \cdot 7.5+15 \cdot 12.5+4 \cdot 17.5+2 \cdot 22.5+$ $+1 \cdot 27.5)=5$
We take $\lambda=\frac{1}{\widetilde{x}_{\mathrm{B}}}=\frac{1}{5}=0.2$. Thus, t... | Thehypothesisisconfirmed | Other | math-word-problem | Yes | Yes | olympiads | false | 31,442 |
Task 1 Find the sample regression equation of the straight line for $Y$ on $X$ based on the data in the correlation table, and calculate the residual variance:
| | $X$ | | | | | |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $Y$ | 20 | 25 | 30 | 35 | 40 | |
| 16 | 4 | 6 | | | | 10 |
| 26 | | ... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,443 | |
Example. Find the decomposition of the vector $\vec{x}=\{3,-1,2\}$ in terms of the vectors $\vec{p}=\{2,0,1\}, \vec{q}=\{1,-1,1\}$ and $\vec{r}=\{1,-1,-2\}$. | ## Solution.
1. The desired decomposition of vector $\vec{x}$ has the form
$$
\vec{x}=\alpha \vec{p}+\beta \vec{q}+\gamma \vec{r}
$$
2. This vector equation with respect to $\alpha, \beta$, and $\gamma$ is equivalent to a system of three linear equations with three unknowns
$$
\left\{\begin{aligned}
2 \alpha+\beta+... | {-1,0,3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,444 |
Example. Are the vectors $\vec{p}=4 \vec{a}-3 \vec{b}, \vec{q}=9 \vec{b}-12 \vec{a}$ collinear, where $\vec{a}=\{-1,2,8\}$ and $\vec{b}=\{3,7,-1\} ?$ | Solution.
1. We find the coordinates of vectors $\vec{p}$ and $\vec{q}$, using the fact that when vectors are added, their coordinates are added, and when multiplied by a number, the coordinates are multiplied by that number:
$$
\vec{p}=\{-13,-13,35\}, \quad \vec{q}=\{39,39,-105\}
$$
2. Since
$$
\frac{-13}{39}=\fra... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,445 |
Example. Given points $A(-2,4,-6), B(0,2,-4)$ and $C(-6,8,-10)$. Find the cosine of the angle between vectors $\overline{A B}$ and $\overline{A C}$. | Solution.
1. We find the coordinates of the vectors $\overline{A B}=\{2,-2,2\}$ and $\overline{A C}=\{-4,4,-4\}$.
2. Using the formulas for the length of a vector and the scalar product of vectors, we have
$$
\begin{gathered}
|\overline{A B}|=\sqrt{2^{2}+(-2)^{2}+2^{2}}=2 \sqrt{3}, \quad|\overline{A C}|=\sqrt{(-4)^{2... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,446 |
Example. Calculate the area of the parallelogram constructed on the vectors $\vec{a}=3 \vec{p}+2 \vec{q}$ and $\vec{b}=2 \vec{p}-\vec{q}$, given that $|\vec{p}|=4,|\vec{q}|=3$ and the angle between the vectors $\vec{p}$ and $\vec{q}$ is $3 \pi / 4$. | ## Solution.
1. We calculate $[\vec{a}, \vec{b}]$ using the properties of the vector product
$$
[\vec{a}, \vec{b}]=[3 \vec{p}+2 \vec{q}, 2 \vec{p}-\vec{q}]=6[\vec{p}, \vec{p}]-3[\vec{p}, \vec{q}]+4[\vec{q}, \vec{p}]-2[\vec{q}, \vec{q}]=-7[\vec{p}, \vec{q}]
$$
2. We calculate the magnitude of the vector product
$$
|... | 42\sqrt{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,447 |
Example. Are the vectors $\vec{a}=\{7,4,6\}, \vec{b}=\{2,1,1\}$ and $\vec{c}=\{19,11,17\}$ coplanar? | ## Solution.
1. We calculate the mixed product of vectors:
$$
(\vec{a}, \vec{b}, \vec{c})=\left|\begin{array}{rrr}
7 & 4 & 6 \\
2 & 1 & 1 \\
19 & 11 & 17
\end{array}\right|=0
$$
2. Since $(\vec{a}, \vec{b}, \vec{c})=0$, the vectors $\vec{a}, \vec{b}$, and $\vec{c}$ are coplanar.
Answer. The vectors $\vec{a}, \vec{b... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,448 |
Example. Calculate the volume of the tetrahedron with vertices $A_{1}(2,3,1)$, $A_{2}(4,1,-2)$, $A_{3}(6,3,7)$, and $A_{4}(-5,-4,8)$, and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$. | ## Solution.
1. From vertex $A_{1}$, we draw vectors $\overline{A_{1} A_{2}}=\{2,-2,-3\}, \overline{A_{1} A_{3}}=$ $=\{4,0,6\}$ and $\overline{A_{1} A_{4}}=\{-7,-7,7\}$.
2. We calculate the mixed product:
$$
\begin{aligned}
& \left(\overline{A_{1} A_{2}}, \overline{A_{1} A_{3}}, \overline{A_{1} A_{4}}\right)=\left|\b... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 31,449 |
Example. Find the distance from the point $M_{0}(1,-1,2)$ to the plane passing through the points $M_{1}(1,5,-7), M_{2}(-3,6,3), M_{3}(-2,7,3)$. | ## Solution.
1. We find the coordinates of the vectors:
$$
\overline{M_{1} M_{2}}=\{-4,1,10\}, \quad \overline{M_{1} M_{3}}=\{-3,2,10\}, \quad \overline{M_{1} M_{0}}=\{0,-6,9\}
$$
and the normal vector of the plane:
$$
\vec{n}=\left[\overline{M_{1} M_{2}}, \overline{M_{1} M_{3}}\right]=\left|\begin{array}{ccc}
\vec... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 31,450 |
Example. Write the equation of the plane passing through the point $M_{0}(2,5,-3)$ and perpendicular to the vector $\overline{M_{1} M_{2}}$, where points $M_{1}$ and $M_{2}$ have coordinates $(7,8,-1)$ and $(9,7,4)$. | Solution.
1. As the normal vector of the plane $\vec{n}$, we choose the vector $\overline{M_{1} M_{2}}=\{2,-1,5\}$.
2. We form the equation of the plane (1) with the normal vector $\vec{n}=\{2,-1,5\}$, passing through the point $M_{0}(2,5,-3)$:
$$
2(x-2)-1(y-5)+5(z+3)=0
$$
Answer. The equation of the plane is $2 x-y... | 2x+2y+z | Geometry | math-word-problem | Yes | Yes | olympiads | false | 31,451 |
Example. Find the angle between the planes
$$
x+2 y-2 z-7=0, \quad x+y-35=0
$$ | Solution. The dihedral angle between the planes is equal to the angle between their normal vectors $\vec{n}_{1}=\{1,2,-2\}$ and $\vec{n}_{2}=\{1,1,0\}$. Therefore, the angle $\varphi$ between the planes is determined by the equality
$$
\cos \varphi=\frac{\left(\vec{n}_{1}, \vec{n}_{2}\right)}{\left|\vec{n}_{1}\right| ... | \varphi=\pi/4 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 31,452 |
Example. Write the canonical equations of the line given as the line of intersection of two planes (by general equations)
$$
\left\{\begin{array}{l}
2 x+3 y+z-8=0 \\
x-2 y-2 z+1=0
\end{array}\right.
$$ | Solution.
1. Let's check that the vectors $\vec{n}_{1}=\{2,3,1\}$ and $\vec{n}_{2}=\{1,-2,-2\}$ are non-collinear (see problem 1.2). We have
$$
\frac{2}{1} \neq \frac{3}{-2}
$$
The vectors $\vec{n}_{1}=\{2,3,1\}$ and $\vec{n}_{2}=\{1,-2,-2\}$ are non-collinear, as their coordinates are not proportional. Therefore, t... | \frac{x-3}{-4}=\frac{y}{5}=\frac{z-2}{-7} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 31,453 |
Example. Find the point of intersection of the line
$$
\frac{x-1}{2}=\frac{y+1}{0}=\frac{z}{-1}
$$
and the plane
$$
2 x-3 y+z-8=0
$$ | Solution.
1. We have
$$
(\vec{a}, \vec{n})=2 \cdot 2+0 \cdot(-3)+(-1) \cdot 1=3 \neq 0
$$
Therefore, the direction vector of the line and the normal vector of the plane are not orthogonal, i.e., the line and the plane intersect at a unique point.
2. Let
$$
\frac{x-1}{2}=\frac{y+1}{0}=\frac{z}{-1}=t .
$$
Then the ... | (3,-1,-1) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,454 |
Example. Find the coordinates of the projection $P^{\prime}$ of the point $P(1,2,-1)$ onto the plane $3 x-y+2 z-4=0$. | SOLUTION.
1. We compose the equations of the line passing through point $P$ and perpendicular to the given plane. For this, we take the normal vector of the plane as the direction vector of the line: $\vec{a}=\vec{n}=$
$=\{3,-1,2\}$. Then the canonical equations of the line are
$$
\frac{x-1}{3}=\frac{y-2}{-1}=\frac{z... | (7,0,1) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 31,455 |
Example. Find the coordinates of the point $Q$, symmetric to the point $P(2,-1,2)$ with respect to the line
$$
\frac{x-1}{1}=\frac{y}{0}=\frac{z+1}{-2}
$$ | Solution.
1. Let's find the projection of point $P$ onto the given line, i.e., point $P^{\prime}$. For this:
a) we will form the equation of the plane passing through point $P$ and perpendicular to the given line. As the normal vector $\vec{n}$ of this plane, we can take the direction vector of the given line: $\vec{... | (-2,1,0) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 31,456 |
Example. Solve the system of equations
$$
\left\{\begin{array}{r}
x_{1}+2 x_{2}+x_{3}=4 \\
3 x_{1}-5 x_{2}+3 x_{3}=1 \\
2 x_{1}+7 x_{2}-x_{3}=8
\end{array}\right.
$$
using Cramer's rule. | SOLUTION.
1. We calculate the determinant of the system matrix, expanding it along the first row:
$$
\Delta=\left|\begin{array}{rrr}
1 & 2 & 1 \\
3 & -5 & 3 \\
2 & 7 & -1
\end{array}\right|=1 \cdot(-16)-2 \cdot(-9)+1 \cdot 31=33
$$
Since it is not equal to zero, the system of equations has a unique solution.
2. We ... | x_{1}=1,x_{2}=1,x_{3}=1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,457 |
Example. Given a third-order square matrix
$$
C=\left(\begin{array}{rrr}
1 & 2 & 1 \\
3 & -5 & 3 \\
2 & 7 & -1
\end{array}\right)
$$
Determine the existence and find the inverse matrix $C^{-1}$. | Solution.
1. We compute the determinant of matrix $\operatorname{det} C$:
$$
\operatorname{det} C=\left|\begin{array}{rrr}
1 & 2 & 1 \\
3 & -5 & 3 \\
2 & 7 & -1
\end{array}\right|=1 \cdot(-16)-2 \cdot(-9)+1 \cdot 31=33
$$
Since $\operatorname{det} C \neq 0$, the matrix $C$ has an inverse.
2. We form the matrix of c... | C^{-1}=\frac{1}{33}\cdot(\begin{pmatrix}-16&9&11\\9&-3&0\\31&-3&-11\end{pmatrix}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,458 |
Example. Does the set of positive numbers $X=\mathbb{R}_{+}$ form a linear space, in which the operations of "addition" and "multiplication by a number" are defined as follows: $\forall a, b \in \mathbb{R}_{+}$ and $\forall \alpha \in \mathbb{R}$
$$
a \oplus b=a \cdot b, \quad \alpha \odot a=a^{\alpha} ?
$$ | ## Solution.
1. The operations introduced in this way are closed in the given set, since if $a, b \in \mathbb{R}_{+}$ and $\alpha \in \mathbb{R}$, then
$$
a \oplus b=a \cdot b>0, \quad \alpha \odot a=a^{\alpha}>0
$$
i.e., $a \oplus b \in \mathbb{R}_{+}$ and $\alpha \odot a \in \mathbb{R}_{+}$.
2. The zero element s... | proof | Algebra | proof | Yes | Yes | olympiads | false | 31,459 |
Example 1. Find the dimension $d$ of the solution space, its basis (fundamental system of solutions), and the general solution of the homogeneous system of linear equations
\[
\left\{\begin{array}{r}
x_{2}+2 x_{3}-3 x_{4}=0 \\
2 x_{1}-x_{2}+3 x_{3}+4 x_{5}=0 \\
2 x_{1}+5 x_{3}-3 x_{4}+4 x_{5}=0
\end{array}\right.
\] | ## Solution.
1. We write down the matrix of the system and transform it to the reduced form using elementary row operations:
$$
\left(\begin{array}{rrrrr}
0 & 1 & 2 & -3 & 0 \\
2 & -1 & 3 & 0 & 4 \\
2 & 0 & 5 & -3 & 4
\end{array}\right) \sim\left(\begin{array}{rrrrr}
1 & 0 & 5 / 2 & -3 / 2 & 2 \\
0 & 1 & 2 & -3 & 0 \... | C_{1}(\begin{pmatrix}-5/2\\-2\\1\\0\\0\end{pmatrix})+C_{2}(\begin{pmatrix}3/2\\3\\0\\1\\0\end{pmatrix})+C_{3}(\begin{pmatrix}-2\\0\\0\\0\\1\end{pmatrix}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,460 |
EXAMPLE 2. Find the general solution of the non-homogeneous system of linear equations
$$
\left\{\begin{aligned}
x_{2}+2 x_{3}-3 x_{4} & =-1 \\
2 x_{1}-x_{2}+3 x_{3}+4 x_{5} & =5 \\
2 x_{1}+5 x_{3}-3 x_{4}+4 x_{5} & =4
\end{aligned}\right.
$$ | Solution.
1. We write the augmented matrix of the system and use elementary row transformations to reduce the matrix $A_{\text {ext. }}$ to its reduced form:
$$
\left(\begin{array}{rrrrr|r}
0 & 1 & 2 & -3 & 0 & -1 \\
2 & -1 & 3 & 0 & 4 & 5 \\
2 & 0 & 5 & -3 & 4 & 4
\end{array}\right) \sim\left(\begin{array}{rrrrr|r}
... | X_{\text{0n.}}=(\begin{pmatrix}2\\-1\\0\\0\\0\end{pmatrix})+C_{1}(\begin{pmatrix}-5/2\\-2\\1\\0\\0\end{pmatrix})+C_{2}(\begin{pmatrix}3/2\\3\\0\\1\\0\end | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,461 |
Example. Let in some basis of the linear space $X_{3}$ a vector $x=\left\{x_{1}, x_{2}, x_{3}\right\}$ be given. Is the operator $\widehat{A}: X_{3} \mapsto X_{3}$ such that
$$
\widehat{A} x=\left\{x_{1}-x_{2}, 2 x_{1}+x_{3}, 3 x_{1}\right\} ?
$$ | Solution. Let $x=\{x_{1}, x_{2}, x_{3}\}$ and $y=\{y_{1}, y_{2}, y_{3}\}$ be arbitrary vectors in the space $X_{3}$. Then $x+y=\{x_{1}+y_{1}, x_{2}+y_{2}, x_{3}+y_{3}\}$ and $\alpha x=\{\alpha x_{1}, \alpha x_{2}, \alpha x_{3}\}$.
*) Verify the fundamental systems of solutions of homogeneous systems of equations and p... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,462 |
Example. Prove linearity, find the matrix (in the basis $\vec{i}, \vec{j}, \vec{k}$), image, kernel, rank, and nullity of the projection operator of the space of geometric vectors $V_{3}$ onto the plane $X O Y$. | Solution.
1. We will prove the linearity of the projection operator by definition. Let in the basis $\vec{i}, \vec{j}, \vec{k}$ we have an arbitrary vector $\vec{x}=\left\{x_{1}, x_{2}, x_{3}\right\}$. Then its image (projection) is $\widehat{P} \vec{x}=\left\{x_{1}, x_{2}, 0\right\}$.
According to the rules of opera... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,463 |
Example. In a certain basis of a three-dimensional linear space $X_{3}$, the mappings are given by
$$
\begin{aligned}
x & \mapsto \widehat{A} x=\left\{x_{1}+x_{2}-x_{3}, x_{2}+x_{3}, x_{3}\right\} \\
x & \mapsto \widehat{B} x=\left\{x_{2}+2 x_{3},-x_{1}, x_{2}\right\}
\end{aligned}
$$
where $x=\left\{x_{1}, x_{2}, x_... | ## Solution.
1. We construct the matrices of operators $\widehat{A}$ and $\widehat{B}$:
$$
A=\left(\begin{array}{rrr}
1 & 1 & -1 \\
0 & 1 & 1 \\
0 & 0 & 1
\end{array}\right) \quad \text { and } \quad B=\left(\begin{array}{rrr}
0 & 1 & 2 \\
-1 & 0 & 0 \\
0 & 1 & 0
\end{array}\right)
$$
2. According to the rules of ma... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,464 |
Example. The vector $\mathbf{x}$ in the basis $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ has coordinates $\{1,2,3\}$. Find the coordinates of the vector $\mathbf{x}$ in the basis $\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \mathbf{e}_{3}^{\prime}$, where
$$
\begin{aligned}
& \mathbf{e}_{1}^{\prime}=\mathb... | Solution.
1. Find the transition matrix
$$
C=\left(\begin{array}{ccc}
1 & 0 & -1 \\
0 & 1 & -1 \\
2 & 1 & -2
\end{array}\right)
$$
2. Find the inverse matrix $C^{-1}$ using the Gaussian method:
$$
2. Find the inverse matrix $C^{-1}$ using the Gaussian method:
 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,466 |
Example. Find the eigenvalues and eigenvectors of the operator $\widehat{A}: X_{3} \mapsto X_{3}$, given in some basis by the matrix
$$
A=\left(\begin{array}{rrr}
3 & 0 & 0 \\
1 & 2 & -1 \\
1 & -1 & 2
\end{array}\right)
$$ | Solution.
1. We form the characteristic equation:
$$
\left|\begin{array}{ccc}
3-\lambda & 0 & 0 \\
1 & 2-\lambda & -1 \\
1 & -1 & 2-\lambda
\end{array}\right|=0 \Longleftrightarrow(3-\lambda)\left(\lambda^{2}-4 \lambda+3\right)=0
$$
Therefore, $\lambda_{1,2}=3, \lambda_{3}=1$.
2. For the eigenvalue $\lambda_{1,2}=3... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,467 |
Example. Using the definition of the limit of a sequence, prove that
$$
\lim _{n \rightarrow \infty} \frac{2 n^{3}}{n^{3}-2}=2
$$ | SOLUTION.
1. By definition, the number 2 is the limit of the numerical sequence $\left\{\frac{2 n^{3}}{n^{3}-2}\right\}$ if
$$
\forall \varepsilon>0 \quad \exists N(\varepsilon): n>N(\varepsilon) \Longrightarrow\left|\frac{2 n^{3}}{n^{3}-2}-2\right|<\varepsilon
$$
We have:
$$
\left|\frac{2 n^{3}}{n^{3}-2}-2\right|=... | proof | Calculus | proof | Yes | Yes | olympiads | false | 31,468 |
Example. Compute the limit
$$
\lim _{n \rightarrow \infty} \frac{(2 n+1)^{2}-(n+1)^{2}}{n^{2}+n+1}
$$ | Solution. Here $(2 n+1)^{2}-(n+1)^{2}=3 n^{2}+2 n-$ is a polynomial of the second degree (an infinitely large sequence of order $n^{2}$) and $n^{2}+n+1$ is a polynomial of the second degree (an infinitely large sequence of order $n^{2}$).
1. Factor out $n^{2}$ in the numerator, we get
$$
(2 n+1)^{2}-(n+1)^{2}=n^{2}\l... | 3 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,469 |
Example. Compute the limit
$$
\lim _{n \rightarrow \infty} \frac{n \sqrt[6]{n}+\sqrt[5]{32 n^{10}+1}}{(n+\sqrt[4]{n}) \sqrt[3]{n^{3}-1}}
$$ | Solution. The numerator $n \sqrt[6]{n}+\sqrt[5]{32 n^{10}+1}$ is an infinitely large sequence of order $n^{2}$, and the denominator $(n+\sqrt[4]{n}) \sqrt[3]{n^{3}-1}$ is an infinitely large sequence of order $n^{2}$.
1. Factor out $n^{2}$ in the numerator, we get
$$
n \sqrt[6]{n}+\sqrt[5]{32 n^{10}+1}=n^{2}\left(\fr... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,470 |
Example. Compute the limit
$$
\lim _{n \rightarrow \infty}\left(\frac{4 n^{2}+4 n-1}{4 n^{2}+2 n+3}\right)^{1-2 n}
$$ | Solution.
1. As $n \rightarrow \infty$, the expression under the limit sign represents a power, the base of which tends to one:
$$
\lim _{n \rightarrow \infty}\left(\frac{4 n^{2}+4 n-1}{4 n^{2}+2 n+3}\right)=1
$$
and the exponent tends to minus infinity:
$$
\lim _{n \rightarrow \infty}(1-2 n)=-\infty
$$
We transfo... | e^{-1} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,471 |
Example. Using the definition, prove that the function $f(x)=5 x^{2}+5$ is continuous at the point $a=8$. | Solution.
1. We calculate $f(8)=325$.
The function $f(x)$ is called continuous at the point $x=8$ if
$$
\forall \varepsilon>0 \quad \exists \delta(\varepsilon)>0: \quad|x-8|<\delta(\varepsilon) \Longrightarrow|f(x)-325|<\varepsilon.
$$
For any $\varepsilon>0$, we need to find $\delta(\varepsilon)>0$ such that for a... | proof | Calculus | proof | Yes | Yes | olympiads | false | 31,473 |
Example. Compute the limit
$$
\lim _{x \rightarrow 3} \frac{x^{3}-4 x^{2}-3 x+18}{x^{3}-5 x^{2}+3 x+9}
$$ | Solution.
1. The expression under the limit sign (a rational fraction) is the ratio of two infinitesimal functions as $x \rightarrow 3$.
Factorize the numerator and the denominator:
$$
\frac{x^{3}-4 x^{2}-3 x+18}{x^{3}-5 x^{2}+3 x+9}=\frac{(x-3)^{2}(x+2)}{(x-3)^{2}(x+1)}
$$
2. Since in the definition of the limit o... | \frac{5}{4} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,474 |
## Example. Compute the limit
$$
\lim _{x \rightarrow 0} \frac{2 x \sin x}{1-\cos x}
$$ | Solution. The expression under the limit sign is the ratio of two infinitesimals at the point $x=0$, since
$$
\lim _{x \rightarrow 0}(2 x \sin x)=0, \quad \lim _{x \rightarrow 0}(1-\cos x)=0
$$
The infinitesimals in the numerator and denominator are replaced by equivalent ones:
$$
\begin{array}{ll}
2 x \sin x \sim 2... | 4 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,475 |
Example. Calculate the limit
$$
\lim _{x \rightarrow \pi} \frac{\cos 3 x-\cos x}{\tan^{2} 2 x}
$$ | Solution.
1. Since
$$
\lim _{x \rightarrow \pi}[\cos 3 x-\cos x]=0, \quad \lim _{x \rightarrow \pi} \operatorname{tg}^{2} 2 x=0
$$
the expression under the limit sign is a ratio of two infinitesimal functions as $x \rightarrow \pi$. We need to replace these infinitesimal functions with equivalent ones. For this, we ... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,476 |
Example. Compute the limit
$$
\lim _{x \rightarrow 0}\left(\frac{1+x^{2} 2^{x}}{1+x^{2} 5^{x}}\right)^{1 / \sin ^{3} x}
$$ | Solution. When $x \rightarrow 0$, the expression under the limit sign represents a power, the base of which tends to one:
$$
\lim _{x \rightarrow 0} \frac{1+x^{2} 2^{x}}{1+x^{2} 5^{x}}=1
$$
and the exponent tends to infinity:
$$
\lim _{x \rightarrow 0} \frac{1}{\sin ^{3} x}=\infty
$$
1. Transform the expression und... | \frac{2}{5} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,477 |
Example. Calculate the limit of the function
$$
\lim _{x \rightarrow 1}\left(\frac{2 x-1}{x}\right)^{\ln (3+2 x) / \ln (2-x)}
$$ | Solution. As $x \rightarrow 1$, the expression under the limit sign represents a power, the base of which tends to one:
$$
\lim _{x \rightarrow 1} \frac{2 x-1}{x}=1
$$
and the exponent tends to infinity:
$$
\lim _{x \rightarrow 1} \frac{\ln (3+2 x)}{\ln (2-x)}=\infty
$$
1. To use the table of equivalent infinitesim... | \frac{1}{5} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,478 |
Example. Compute the limit
$$
\lim _{x \rightarrow 0} \sqrt[3]{x\left(2+\sin \frac{1}{x}\right)+8 \cos x}
$$ | Solution.
1. Since the function $y=\sqrt[3]{x}$ is continuous for all $x$, by passing to the limit under the sign of a continuous function, we get
$$
\lim _{x \rightarrow 0} \sqrt[3]{x\left(2+\sin \frac{1}{x}\right)+8 \cos x}=\sqrt[3]{\lim _{x \rightarrow 0}\left[x\left(2+\sin \frac{1}{x}\right)+8 \cos x\right]}
$$
... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,479 |
Example. According to the definition, find the derivative of the function
$$
f(x)=\left[\begin{array}{ll}
1-\cos \left(x \sin \frac{1}{x}\right), & x \neq 0 \\
0, & x=0
\end{array}\right.
$$
at the point $x=0$. | Solution.
1. By definition
$$
f^{\prime}(0)=\lim _{x \rightarrow 0} \frac{f(x)-f(0)}{x}=\lim _{x \rightarrow 0} \frac{1-\cos (x \sin (1 / x))-0}{x}
$$
2. Since $\sin (1 / x)$ is a bounded function and $x$ is an infinitesimal function as $x \rightarrow 0$, by the theorem on the product of an infinitesimal function an... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,480 |
Example. Find the derivative of the function
$$
y=\frac{3 x^{6}+4 x^{4}-x^{2}-2}{15 \sqrt{1+x^{2}}}
$$ | ## Solution.
1. The function $y(x)$ has the form
$$
\frac{1}{15} \frac{u}{v}
$$
where $u(x)=3 x^{6}+4 x^{4}-x^{2}-2$ and $v(x)=\sqrt{1+x^{2}}$. Using the formula for the derivative of a quotient, we get
$$
\begin{aligned}
& y^{\prime}= \\
& =\frac{1}{15} \frac{\left(3 x^{6}+4 x^{4}-x^{2}-2\right)^{\prime} \sqrt{1+x... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,481 |
Example. Find the equations of the tangent and normal to the curve
$$
y=6 \sqrt[3]{x}-\frac{16}{3} \sqrt[4]{x}
$$
at the point with abscissa $a=1$. | Solution.
1. Find $f(1)=2 / 3$.
2. Find the derivative $f^{\prime}(1)=2 / 3$. Since $f^{\prime}(1) \neq 0$ and $f^{\prime}(1) \neq \infty$, we will use equations (1) and (2).
3. Substituting the found values $f(a)=2 / 3$ and $f^{\prime}(a)=2 / 3$ into (1) and (2), we get the equations of the tangent and normal:
$$
y=... | 2x-30,9x+6y-13=0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,482 |
Example. Approximate the value of the function $y=\sqrt{x^{2}+5}$ at the point $x=1.97$ using the differential. | ## Solution.
1. The point closest to 1.97 where it is easy to calculate the values of $f(a)$ and $f'(a)$ is the point $a=2$.
2. We calculate:
$$
\begin{gathered}
\Delta x=x-a=1.97-2=-0.03 \\
f(a)=f(2)=3, \quad f'(x)=\frac{x}{\sqrt{x^{2}+5}}, \quad f'(a)=y'(2)=\frac{2}{3}
\end{gathered}
$$
3. According to formula (1)... | 2.98 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,483 |
Example. Find the derivative of the function $y=x^{e^{x}} x^{9}$.
---
Translation:
Example. Find the derivative of the function $y=x^{e^{x}} x^{9}$. | Solution.
1. The logarithm of the given function is
$$
\ln y=\ln \left(x^{e^{x}} x^{9}\right)=e^{x} \ln x+9 \ln x
$$
2. Differentiating both sides of this equation, we get
$$
\frac{y^{\prime}}{y}=e^{x} \ln x+e^{x} \frac{1}{x}+\frac{9}{x}
$$
Therefore,
$$
y^{\prime}=y\left(e^{x} \ln x+\frac{e^{x}+9}{x}\right)
$$
... | y^{\}=x^{e^{x}}x^{9}(e^{x}\cdot\lnx+\frac{e^{x}+9}{x}) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,484 |
Example. Find the derivative $y_{x}^{\prime}$, if
\[
\left\{\begin{array}{l}
x=\ln \left(t+\sqrt{1+t^{2}}\right) \\
y=\sqrt{1+t^{2}}-\ln \frac{1+\sqrt{1+t^{2}}}{t}
\end{array}\right.
\] | Solution. We calculate:
$$
\begin{aligned}
& \frac{d x}{d t}=\frac{1}{t+\sqrt{1+t^{2}}}\left(1+\frac{t}{\sqrt{1+t^{2}}}\right)=\frac{1}{\sqrt{1+t^{2}}} \\
& \frac{d y}{d t}=\frac{t}{\sqrt{1+t^{2}}}-\frac{t}{1+\sqrt{1+t^{2}}} \frac{\frac{t^{2}}{\sqrt{1+t^{2}}}-\left(1+\sqrt{1+t^{2}}\right)}{t^{2}}=\frac{\sqrt{t^{2}+1}}... | {\begin{pmatrix}\ln(+\sqrt{1+^{2}}),\\y^{\}=\frac{1+^{2}}{}\end{pmatrix}.}\] | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,485 |
Example. Find the equations of the tangent and normal to the curve
\[
\left\{\begin{array}{l}
x=2 e^{t} \\
y=e^{-t}
\end{array}\right.
\]
at the point \( A \), corresponding to the parameter value \( t=0 \). | SOLUTION.
1. We calculate the coordinates of point $A: a=2, y(a)=1$.
2. We find the derivative $y^{\prime}$ at point $A$:
$$
f^{\prime}(0)=\left.2 e^{t}\right|_{t=0}=2, \quad g^{\prime}(0)=-\left.e^{-t}\right|_{t=0}=-1 \Rightarrow y^{\prime}(0)=\frac{g^{\prime}(0)}{f^{\prime}(0)}=-\frac{1}{2}
$$
Since $f^{\prime}(0)... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,486 |
Example. Find the $n$-th order derivative of the function $y=3^{2 x+5}$.
The above text has been translated into English, preserving the original text's line breaks and format. | Solution.
1. Let's find sequentially
$$
\begin{aligned}
& y^{\prime}(x)=\left(3^{2 x+1}\right)^{\prime}=3^{2 x+1}(\ln 3) 2 \\
& y^{\prime \prime}(x)=y^{\prime}(x)^{\prime}=\left(3^{2 x+1}(\ln 3) 2\right)^{\prime}=3^{2 x+1}(\ln 3)^{2} 2^{2} \\
& y^{\prime \prime \prime}(x)=y^{\prime \prime}(x)^{\prime}=\left(3^{2 x+1}... | y^{(n)}(x)=3^{2x+1}(2\ln3)^{n} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,487 |
Example. Find the fourth-order derivative of the function
$$
y=\left(x^{3}+2\right) e^{4 x+3}
$$ | ## Solution.
1. We apply Leibniz's formula (1). In this case,
$$
n=4, \quad u(x)=x^{3}+2, \quad v(x)=e^{4 x+3}
$$
We have
$$
\begin{gathered}
u^{\prime}(x)=3 x^{2}, \quad u^{\prime \prime}(x)=6 x, \quad u^{\prime \prime \prime}(x)=6, \quad u^{(4)}(x)=0 \\
v^{\prime}(x)=4 e^{4 x+3}, \quad v^{\prime \prime}(x)=4^{2} ... | y^{(4)}=32e^{4x+3}(8x^{3}+24x^{2}+18x+19) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,488 |
Example. Find the second-order derivative of the function given parametrically:
$$
\left\{\begin{array}{l}
x=\ln t \\
y=\operatorname{arctg} t
\end{array}\right.
$$ | ## Solution.
1. We compute
$$
\frac{d x}{d t}=\frac{1}{t}, \quad \frac{d y}{d t}=\frac{1}{1+t^{2}}
$$
and substitute these values into formula (1):
$$
\left\{\begin{array}{l}
x=\ln t \\
y^{\prime}=\frac{t}{1+t^{2}}
\end{array}\right.
$$
Differentiating $y^{\prime}$ with respect to $x$ as a composite function of $x... | y^{\\}=\frac{(1-^{2})}{(1+^{2})^{2}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,489 |
Example. Find the maximum and minimum values of the function
$$
y=\frac{10 x+10}{x^{2}+2 x+2}
$$
on the interval $[-1,2]$. | Solution.
1. The given function is continuous on the interval $[-1,2]$, as it is the ratio of continuous functions with a denominator not equal to zero $\left(x^{2}+2 x+2>0\right)$.
2. We calculate the derivative of the given function:
$$
y^{\prime}=-10 \frac{x^{2}+2 x}{\left(x^{2}+2 x+2\right)^{2}}
$$
3. The critic... | =0,M=5 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,491 |
Example. Investigate the function
$$
y=\sin ^{2}(x-1)-x^{2}+2 x
$$
in the neighborhood of the point $a=1$ using higher-order derivatives. | ## SOLUTION.
1. We compute the derivatives of the given function at the point $a=1$:
$$
\begin{array}{ll}
y^{\prime}=\sin (2 x-2)-2 x+2, & y^{\prime}(1)=0 \\
y^{\prime \prime}=2 \cos (2 x-2)-2, & y^{\prime \prime \prime}(1)=0 \\
y^{\prime \prime \prime}=-4 \sin (2 x-2), & y^{\prime \prime \prime}(1)=0 \\
y^{\prime \p... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,492 |
Example. Find the partial derivatives up to the second order inclusive of the function $z=x^{y} \quad(x>0)$. | Solution.
1. To find the partial derivative with respect to $x$, we fix $y$ and differentiate the function $z=x^{y}$ as a function of one variable $x$. Using the formula for the derivative of a power function $\left(x^{\alpha}\right)^{\prime}=\alpha x^{\alpha-1}$, we get
$$
z_{x}^{\prime}=y x^{y-1}
$$
To find the pa... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,493 |
Example. Find the gradient of the function
$$
u=x^{2}-\operatorname{arctg}(y+z)
$$
at the point $M(2,1,1)$. | SOLUTION.
1. We find the partial derivatives of the function $u=x^{2}-\operatorname{arctg}(y+z)$:
$$
\frac{\partial f}{\partial x}=2 x, \quad \frac{\partial f}{\partial y}=-\frac{1}{1+(y+z)^{2}}, \quad \frac{\partial f}{\partial z}=-\frac{1}{1+(y+z)^{2}}
$$
2. We compute the partial derivatives of the function $u=x^... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,494 |
Example. Find the derivative of the function
$$
u=x^{2}-\operatorname{arctg}(y+z)
$$
at the point $A(2,1,1)$ in the direction of the point $B(2,4,-3)$. | Solution.
1. Since the function $u=x^{2}-\operatorname{arctg}(y+z)$ is differentiable at the point $A(2,1,1)$, the derivative of this function in any direction $\vec{l}$ exists at this point, which is determined by formula (1).
2. We find the coordinates of the vector $\vec{l}$. In this case,
$$
\vec{l}=\overline{A B... | \frac{1}{25} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,495 |
Example. Find the derivatives $z_{x}^{\prime}$ and $z_{y}^{\prime}$ of the function $z=u / v$, where $u=x^{y}$ and $v=\sqrt{x y}$. | Solution.
1. We compute the partial derivatives
$$
\begin{gathered}
\frac{\partial z}{\partial u}=\frac{1}{v}, \quad \frac{\partial z}{\partial v}=-\frac{u}{v^{2}} \\
\frac{\partial u}{\partial x}=y x^{y-1}, \quad \frac{\partial u}{\partial y}=x^{y} \ln x, \quad \frac{\partial v}{\partial x}=\frac{\sqrt{y}}{2 \sqrt{x... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,496 |
Example. Find the derivative of the function $y=y(x)$, given implicitly by the equation
$$
\ln \sqrt{x^{2}+y^{2}}=\operatorname{arctg} \frac{y}{x}
$$ | SOLUTION.
1. In this case, $F(x, y)=\ln \sqrt{x^{2}+y^{2}}-\operatorname{arctg} \frac{y}{x}$. We compute its partial derivatives:
$$
F_{x}^{\prime}=\frac{x}{x^{2}+y^{2}}-\frac{1}{1+(y / x)^{2}}\left(-\frac{y}{x^{2}}\right)=\frac{x+y}{x^{2}+y^{2}}
$$
$$
F_{y}^{\prime}=\frac{y}{x^{2}+y^{2}}-\frac{1}{1+(y / x)^{2}}\lef... | y^{\}=\frac{x_{0}+y_{0}}{x_{0}-y_{0}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,497 |
Example. Find the equations of the tangent plane and normal to the surface given by the equation
$$
z=x y
$$
at the point $M(1,1)$. | Solution. Let's write the equation of the surface as $xy - z = 0$, i.e., $F = xy - z$.
Coordinates of point $M: x_0 = 1$ and $y_0 = 1$. The coordinate $z_0$ is determined from the condition that point $M$ belongs to the given surface, i.e., $F(1, 1, z_0) = 0$. We get $z_0 = 1$.
1. We find the partial derivatives $F_x... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,498 |
Example. Find the stationary points of the function
$$
z=x^{3}+y^{3}-3 x y
$$
and investigate their nature. | Solution.
1. We compute the partial derivatives
$$
z_{x}^{\prime}=3 x^{2}-3 y, \quad z_{y}=3 y^{2}-3 x
$$
2. To find the stationary points of the function, we solve the system of two equations with two unknowns
$$
\left\{\begin{array}{l}
3 x^{2}-3 y=0 \\
3 y^{2}-3 x=0
\end{array}\right.
$$
We obtain two solutions:... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,499 |
Example. Find the indefinite integral
$$
\int \operatorname{ctg} x \ln \sin x d x
$$ | ## Solution.
1. Represent the integrand as a product of two functions $F(x) g(x)$, where $g(x)$ has an obvious antiderivative $G(x)$, and $F(x)$ is a function of this antiderivative, i.e., $F(x)=u(G(x))$.
In this case,
$$
F(x)=\frac{\ln \sin x}{\sin x}, \quad g(x)=\cos x, \quad G(x)=\sin x, \quad F(x)=\frac{\ln G}{G... | \frac{\ln^{2}\sinx}{2}+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,500 |
Example. Find the indefinite integral
$$
\int \frac{x d x}{\cos ^{2} x}
$$ | Solution.
1. Represent the integrand as a product of two functions $F(x) g(x)$, where $g(x)$ has an obvious antiderivative $G(x)$, and $F(x)$ is a differentiable function, with its derivative $f(x)=F'(x)$ being a simpler function than $F(x)$.
## In this case
$$
F(x)=x, \quad g(x)=\frac{1}{\cos ^{2} x}, \quad G(x)=\t... | x\tanx+\ln|\cosx|+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,501 |
Example. Find the indefinite integral
$$
\int \frac{2 x^{3}-40 x-8}{x(x+4)(x-2)} d x
$$ | SOLUTION.
1. The integrand is an improper rational fraction since $n=m=3$. We separate the integer part:
$$
\frac{2 x^{3}-40 x-8}{x(x+4)(x-2)}=2-\frac{4 x^{2}+24 x+8}{x(x+4)(x-2)}
$$
2. Since the denominator of the last fraction has three distinct real roots $x=0, x=-4$, and $x=2$, its decomposition into partial fra... | 2x+\ln\frac{|x||x+4|}{(x-2)^{6}}+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,502 |
Example. Find the indefinite integral
$$
\int \frac{x^{3}+6 x^{2}+13 x+6}{(x-2)(x+2)^{3}} d x
$$ | Solution.
1. The integrand is a proper rational fraction.
2. Decompose it into partial fractions. Since the denominator has two real roots: $r_{1}=2$ of multiplicity one and $r_{2}=-2$ of multiplicity three, the partial fraction decomposition has the form
$$
\frac{x^{3}+6 x^{2}+13 x+6}{(x-2)(x+2)^{3}}=\frac{A_{11}}{x... | \ln|x-2|-\frac{1}{2(x+2)^{2}}+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,503 |
Example. Find the indefinite integral
$$
\int \frac{2 x^{3}+3 x^{2}+3 x+2}{\left(x^{2}+x+1\right)\left(x^{2}+1\right)} d x
$$ | ## Solution.
1. The integrand is a proper rational fraction.
2. Decompose it into partial fractions. The denominator has two pairs of complex conjugate roots: $r_{1,2}=-1 / 2 \pm i \sqrt{3} / 2$ and $r_{3,4}= \pm i$. Therefore, the partial fraction decomposition has the form
$$
\frac{2 x^{3}+3 x^{2}+3 x+2}{\left(x^{2... | \frac{1}{2}\ln(x^{2}+x+1)(x^{2}+1)+\frac{1}{\sqrt{3}}\operatorname{arctg}\frac{2x+1}{\sqrt{3}}+\operatorname{arctg}x+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,504 |
Example 1. Find the indefinite integral
$$
\int \frac{\sin x}{2+\sin x} d x
$$ | ## Solution.
1. Let's make the substitution $t=\operatorname{tg}(x / 2)$.
Substituting into the integrand,
$$
\sin x=\frac{2 t}{1+t^{2}}, \quad d x=\frac{2}{1+t^{2}} d t
$$
we get
$$
\frac{\sin x}{2+\sin x} d x=\frac{\frac{2 t}{1+t^{2}}}{2+\frac{2 t}{1+t^{2}}} \frac{2}{1+t^{2}} d t=\frac{2 t}{\left(t^{2}+t+1\right... | \frac{x-\frac{4}{\sqrt{3}}\operatorname{arctg}\frac{2\operatorname{tg}(x}{2)+1}{\sqrt{3}}+C} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,505 |
Example 2. Find the indefinite integral
$$
\int \frac{3 \operatorname{tg}^{2} x-1}{\operatorname{tg}^{2} x+5} d x
$$ | Solution.
1. Since the integrand has the form $R(\operatorname{tg} x)$, we make the substitution $\operatorname{tg} x=t$.
Substituting into the integrand
$$
\operatorname{tg} x=t, \quad d x=\frac{d t}{1+t^{2}}
$$
we get
$$
\frac{3 \operatorname{tg}^{2} x-1}{\operatorname{tg}^{2} x+5} d x=\frac{3 t^{2}-1}{t^{2}+5} ... | -x+\frac{4}{\sqrt{5}}\operatorname{arctg}(\frac{\operatorname{tg}x}{\sqrt{5}})+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,506 |
Example. Find the indefinite integral
$$
\int \sin ^{4} 3 x \cos ^{4} 3 x d x
$$ | SOLUTION. Applying the formulas for reducing the degree, we have $\int \sin ^{4} 3 x \cos ^{4} 3 x d x=2^{-4} \int(2 \sin 3 x \cos 3 x)^{4} d x=$
$$
\begin{aligned}
& =2^{-4} \int \sin ^{4} 6 x d x=2^{-6} \int(1-\cos 12 x)^{2} d x= \\
& =2^{-6} \int d x-2^{-5} \int \cos 12 x d x+2^{-6} \int \cos ^{2} 12 x d x= \\
& =2... | \frac{3}{2^{7}}x-\frac{1}{3\cdot2^{7}}\sin12x+\frac{1}{3\cdot2^{10}}\sin24x+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,507 |
Example. Find the indefinite integral
$$
\int \frac{4 \sqrt{2-x}-\sqrt{2+x}}{(\sqrt{x+2}+4 \sqrt{2-x})(x+2)^{2}} d x
$$ | ## Solution.
1. To make a substitution leading to an integral of a rational function, it is necessary to transform the integrand so that it contains roots of any degree, but from the same expression of the form $\frac{a x+b}{c x+d}$.
Let's transform the integrand, highlighting $\sqrt{\frac{2-x}{2+x}}$:
$$
\int \frac... | \frac{1}{4}\frac{2-x}{2+x}-\frac{1}{4}\sqrt{\frac{2-x}{2+x}}+\frac{1}{16}\ln(4\sqrt{\frac{2-x}{2+x}}+1)+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,508 |
Example. Find the indefinite integral
$$
\int \frac{x^{2}}{\sqrt{9-x^{2}}} d x
$$ | Solution.
1. To get rid of the radical, we will use the substitution $x=3 \sin t$. Then $d x=3 \cos t$ and $\sqrt{9-x^{2}}=3 \cos t$.
2. We will make a variable substitution in the indefinite integral:
$$
\int \frac{x^{2}}{\sqrt{9-x^{2}}} d x=\int \frac{9 \sin ^{2} t 3 \cos t}{\sqrt{9-9 \sin ^{2} t}} d t=9 \int \sin ... | \int\frac{x^{2}}{\sqrt{9-x^{2}}}=\frac{9}{2}\arcsin\frac{x}{3}-\frac{x}{2}\sqrt{9-x^{2}}+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,509 |
Example. Find the indefinite integral
$$
\int \frac{\sqrt[5]{1+\sqrt[3]{x}}}{x \sqrt[5]{x^{2}}} d x
$$ | Solution. Rewrite the integral as
$$
\int \frac{\sqrt[5]{1+\sqrt[3]{x}}}{x \sqrt[5]{x^{2}}} d x=\int x^{-7 / 5}\left(1+x^{1 / 3}\right)^{1 / 5} d x
$$
The integrand has the form $x^{m}\left(1+x^{n}\right)^{p}$ for
$$
m=-\frac{7}{5}, \quad n=\frac{1}{3}, \quad p=\frac{1}{5}, \quad \frac{m+1}{n}+p=-1
$$
Therefore, th... | -\frac{5}{2}(\frac{1+\sqrt[3]{x}}{\sqrt[3]{x}})^{6/5}+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,510 |
Example. Evaluate the definite integral
$$
\int_{0}^{1} \frac{x d x}{x^{4}+1}
$$ | Solution.
1. Let's represent the integrand as a product of two functions $F(x) g(x)$, where $g(x)$ has an obvious antiderivative $G(x)$, and $F(x)$ is a function of this antiderivative, i.e., $F(x)=u(G(x))$.
In this case,
$$
F(x)=\frac{1}{2} \frac{1}{1+\left(x^{2}\right)^{2}}, \quad g(x)=2 x, \quad G(x)=x^{2}, \quad... | \frac{\pi}{8} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,511 |
Example. Calculate the definite integral
$$
\int_{1}^{2} x \ln ^{2} x d x
$$ | ## Solution.
1. Let's represent the integrand as a product of two functions $F(x) g(x)$, where $g(x)$ has an obvious antiderivative $G(x)$, and $F(x)$ is a differentiable function, with its derivative $f(x)=F'(x)$ being a simpler function than $F(x)$.
In this case,
$$
F(x)=\ln^2 x, \quad g(x)=x, \quad G(x)=\frac{x^2... | 2\ln^22-2\ln2+\frac{3}{4} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,512 |
Example 1. Calculate the definite integral
$$
\int_{0}^{\pi / 2} \frac{\sin x}{2+\sin x} d x
$$ | Solution.
1. Since the function $\operatorname{tg}(x / 2)$ is defined on $[0, \pi / 2]$, we make the substitution
$$
t=\operatorname{tg} \frac{x}{2}
$$
Substituting into the integrand
$$
\sin x=\frac{2 t}{1+t^{2}}, \quad d x=\frac{2}{1+t^{2}} d t
$$
we get
$$
\frac{\sin x}{2+\sin x} d x=\frac{\frac{2 t}{1+t^{2}}}... | \frac{\pi}{2}-\frac{2\pi}{3\sqrt{3}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,513 |
EXAMPLE 2. Calculate the definite integral
$$
\int_{0}^{\arccos (1 / \sqrt{6})} \frac{3 \tan^{2} x-1}{\tan^{2} x+5} d x
$$ | Solution.
1. Since the integrand has the form $R(\operatorname{tg} x)$ and the function $\operatorname{tg} x$ is defined on $[0, \arccos (1 / \sqrt{6})]$, we make the substitution $\operatorname{tg} x=t$.
Substituting into the integrand,
$$
\operatorname{tg} x=t, \quad d x=\frac{d t}{1+t^{2}}
$$
we obtain a rationa... | \frac{\pi}{\sqrt{5}}-\operatorname{arctg}\sqrt{5} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,514 |
Example. Calculate the definite integral
$$
\int_{0}^{2 \pi} \sin ^{4} 3 x \cos ^{4} 3 x d x
$$ | Solution. Applying the formulas for reducing the degree, we have $\int_{0}^{2 \pi} \sin ^{4} 3 x \cos ^{4} 3 x d x=2^{-4} \int_{0}^{2 \pi}(2 \sin 3 x \cos 3 x)^{4} d x=$
$$
\begin{aligned}
&= 2^{-4} \int_{0}^{2 \pi} \sin ^{4} 6 x d x=2^{-6} \int_{0}^{2 \pi}(1-\cos 12 x)^{2} d x= \\
&= 2^{-6} \int_{0}^{2 \pi} d x-2^{-5... | \frac{3\pi}{64} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,515 |
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