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329. Find the value of the arbitrary function $f(x)=\sin ^{4} x-$ $-\cos ^{4} x$ at $x=\pi / 12$. | Solution. First, we transform the given function:
$$
f(x)=\left(\sin ^{2} x+\cos ^{2} x\right)\left(\sin ^{2} x-\cos ^{2} x\right)=-\cos 2 x
$$
(since $\sin ^{2} \alpha+\cos ^{2} \alpha=1, \cos ^{2} \alpha-\sin ^{2} \alpha=\cos 2 \alpha$). Using the rule for differentiating a composite function, we get
$$
f^{\prime}... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
374. Find the slope of the tangent line drawn to the curve $y=x^{3}$ at the point $C(-2;-8)$. | Solution. Let's find the derivative of the function $y=x^{3}$ at the point $x=-2$:
$$
y^{\prime}=\left(x^{3}\right)^{\prime}=3 x^{2} ; \quad y_{x=-2}^{\prime}=3(-2)^{2}=12
$$
Thus, the slo... | 12 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
410. The height of a body thrown vertically upwards changes with time according to the law $H=200 t-4.9 t^{2}$. Find the velocity of the body at the end of the 10th second. How many seconds will the body fly upwards and what is the greatest height it will reach? | Solution. The velocity of the body is determined by the expression $v=\frac{d H}{d t}=$ $=200-9.8 t$; at $t=10$ we have $v=102($ m $/ \mathrm{s})$.
At the moment when the body reaches its maximum height, its velocity is zero. Therefore, for this moment $\frac{d H}{d t}=200-$ $-9.8 t=0$, from which $t=\frac{200}{9.8} \... | 102 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
413. A body with a mass of 8 kg moves in a straight line according to the law $s=$ $=2 t^{2}+3 t-1$. Find the kinetic energy of the body $\left(m v^{2} / 2\right)$ 3 seconds after the start of the motion. | Solution. Let's find the velocity of the body at any moment of time $t$:
$$
v=\frac{d s}{d t}=4 t+3
$$
Calculate the velocity of the body at the moment $t=3$:
$$
v_{t=3}=4 \cdot 3+3=15(\mathrm{M} / \mathrm{c})
$$
Determine the kinetic energy of the body at the moment $t=3$:
$$
\frac{m v^{2}}{2}=\frac{8 \cdot 15^{2... | 900 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
434. A material point moves according to the law $s=2 t^{3}-6 t^{2}+$ $+4 t$. Find its acceleration at the end of the 3rd second. | Solution. We find $v=s^{\prime}=6 t^{2}-12 t+4, a=s^{\prime \prime}=12 t-12$, from which at $t=3$ we get $a=12 \cdot 3-12=24\left(\mathrm{m} / \mathrm{c}^{2}\right)$. | 24 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
435. At time $t$, the body is at a distance of $s=\frac{1}{4} t^{4}+4 t^{3}+16 t^{2}$ km from the starting point. Find its acceleration after 2 hours. | Solution. We find $v=s^{\prime}=t^{3}+12 t^{2}+32 t, a=v^{\prime}=s^{\prime \prime}=3 t^{2}+$ $+24 t+32$. For $t=2$ we have $a=3 \cdot 4+24 \cdot 2+32=92$ (km $\left./ \mathrm{u}^{2}\right)$. | 92 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
448. The sides $a$ and $b$ of a rectangle change according to the laws $a=(2 t+1)$ cm, $b=(3 t+2)$ cm. At what rate is its area $S$ changing at the moment $t=4$ s? | Solution. We find $S=a b=(2 t+1)(3 t+2)=6 t^{2}+7 t+2, \quad v=$ $=S_{t}^{\prime}=\left(6 t^{2}+7 t+2\right)^{\prime}=12 t+7$. For $t=4$ we get $v_{t=4}=12 \cdot 4+7=$ $=55(\mathrm{~cm} / \mathrm{s})$. | 55 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
455. The amount of electricity flowing through a conductor, starting from the moment of time $t=0$, is given by the formula $Q=$ $=3 t^{2}-3 t+4$. Find the current strength at the end of the 6th second. | Solution. The current is the derivative of the quantity of electricity with respect to time: therefore, we need to find the derivative of the function $Q=3 t^{2}-3 t+4$ and calculate its value at $t=6 \mathrm{c}$. We have $I=Q^{\prime}=$ $=6 t-3$, from which at $t=6$ we get $I=33$ (A). | 33 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
461. The law of temperature change $T$ of a body depending on time $t$ is given by the equation $T=0.2 t^{2}$. At what rate is this body heating at the moment of time $10 \mathrm{c}$? | Solution. The heating rate of a body is the derivative of temperature $T$ with respect to time $t$:
$$
\frac{d T}{d t}=\left(0.2 t^{2}\right)^{\prime}=0.4 t
$$
Determine the heating rate of the body at $t=10$:
$$
\left(\frac{d T}{d t}\right)_{t=10}=0.4 \cdot 10=4(\text{ deg } / \mathrm{s})
$$ | 4 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
564. $y=x^{2}+2$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
564. $y=x^{2}+2$. | Solution. $1^{0}$. Find the derivative: $y^{\prime}=\left(x^{2}+2\right)^{\prime}=2 x$.
$2^{0}$. Set it to zero; $2 x=0$, from which $x=0$ - the critical point.
$3^{\circ}$. Determine the sign of the derivative at the value $x0$, for example at $x=1$: $y_{x=1}^{\prime}=2 \cdot 1=2$. Since the derivative changes sign ... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
582. $y=4x-x^{2}$. | Solution. $1^{0}$. Find the first derivative: $y^{\prime}=4-2 x$. $2^{0}$. Solving the equation $4-2 x=0$, we get the critical point $x=2$. $3^{0}$. Calculate the second derivative: $y^{\prime \prime}=-2$.
$4^{0}$. Since $y^{\prime \prime}$ is negative at any point, then $y^{\prime \prime}(2)=-2<0$. This means that th... | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
585. $y=x^{6}$. | Solution. $1^{0} . y^{\prime}=6 x^{5}$.
$2^{0} .6 x^{5}=0$, i.e. $x=0$.
$3^{0} \cdot y^{\prime \prime}=30 x^{4}$.
$4^{0}$. Since $y^{\prime \prime}(0)=0$, the method of investigation using the second derivative is not applicable.
We will investigate the function for an extremum using the first derivative. If $x<0$,... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
614. There is a square sheet of tin, the side of which $a=$ $=60 \mathrm{~cm}$. By cutting out equal squares from all its corners and folding up the remaining part, a box (without a lid) needs to be made. What should be the dimensions of the squares to be cut out so that the box has the maximum volume? | Solution. According to the condition, the side of the square $a=60$. Let the side of the squares cut out from the corners be denoted by $x$. The bottom of the box is a square with side $a-2 x$, and the height of the box is equal to the side $x$ of the cut-out square. Therefore, the volume of the box can be expressed by... | 10 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
616. An irrigation channel has the shape of an isosceles trapezoid, the lateral sides of which are equal to the smaller base. At what angle of inclination of the lateral sides is the cross-sectional area of the channel the largest? | Solution. Let the smaller base of the trapezoid be denoted by $a$, the angle of inclination of the lateral sides by $\alpha$, and the area of the section by $S$ (Fig. 124). According to the condition, $|A B|=|A D|=|B C|=a$.
=2 x+4$. Find the increment of any of its antiderivatives when $x$ changes from -2 to 0. | Solution. Let's find the antiderivative of the given function:
$$
F=\int(2 x+4) d x=x^{2}+4 x+C
$$
Consider, for example, the antiderivatives $F_{1}=x^{2}+4 x, F_{2}=x^{2}+4 x+2$, $F_{3}=x^{2}+4 x-1$, and compute the increment of each of them on the interval $[-2,0]: \quad F_{1}(0)-F_{1}(-2)=0-(-4)=4 ; \quad F_{2}(0)... | 4 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
244. Find $\int_{1}^{2}\left(5 x^{4}+2 \dot{x}-8\right) d x$. | Solution. $\int_{1}^{2}\left(5 x^{4}+2 x-8\right) d x=\int_{1}^{2} 5 x^{4} d x+\int_{1}^{2} 2 x d x-\int_{1}^{2} 8 d x=\left.x^{5}\right|_{1} ^{2}+$ $+x^{2}-\left.8 x\right|_{1} ^{2}=\left(2^{5}-1^{5}\right)+\left(2^{2}-1^{2}\right)-8(2-1)=26$. | 26 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
245. Find $\int_{\pi / 2}^{\pi} \frac{2 \sin x d x}{(1-\cos x)^{2}}$. | Solution. Let's use the substitution $u=1-\cos x$, from which $d u=$ $=\sin x d x$. Then we will find the new limits of integration; substituting into the equation $u=1-\cos x$ the values $x_{1}=\pi / 2$ and $x_{2}=\pi$, we will respectively obtain $u_{1}=1-\cos (\pi / 2)=1$ and $u_{2}=1-\cos \pi=2$. The solution is wr... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
246. $\int 8 x^{3} d x$ | Solution. $\int_{1}^{3} 8 x^{3} d x=\left.8 \cdot \frac{x^{4}}{4}\right|_{1} ^{3}=\left.2 x^{4}\right|_{1} ^{3}=2\left(3^{4}-1^{4}\right)=160$. | 160 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
294. $\int_{1}^{e} \frac{3 \ln ^{2} x d x}{x}$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
294. $\int_{1}^{e} \frac{3 \ln ^{2} x \, dx}{x}$. | Solution. $\int_{1}^{e} \frac{3 \ln ^{2} x d x}{x}=\left|\begin{array}{l}\ln x=t, t_{1}=\ln 1=0, \\ \frac{d x}{x}=d t ; t_{2}=\ln e=1\end{array}\right|=3 \int_{0}^{1} t^{2} d t=\left.t^{3}\right|_{0} ^{1}=1$. | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
307. $y^{2}=9 x, x=16, x=25$ and $y=0$ (Fig. 150). | Solution. For any $x \in [16,25]$, the function $y=\sqrt{9 x}$ takes positive values; therefore, to calculate the area of the given curvilinear trapezoid, we should use formula (1):
$$
S=\int_{16}^{25} \sqrt{9} x d x=\int_{16}^{25} 3 x^{1 / 2} d x=\left.3 \frac{x^{3 / 2}}{3 / 2}\right|_{16} ^{25}=\left.2 x \sqrt{x}\ri... | 122 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
320. $y=4x-x^2$, $y=0$ and $x=5$. | The parabola $y=4 x-x^{2}$ intersects the x-axis at points $x=0$ and $x=4$. The figure whose area we need to find is marked in color in Fig. 167. Let $S_{1}$ and $S_{2}$ be the areas of the parts of this figure corresponding to the segments $[0,4]$ and $[4, 5]$, and let $S$ be the required area; then $S=S_{1}+S_{2}$. U... | 13 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
324. $y=\frac{1}{2} x^{3}, x=-2, x=4$ and $y=0$. | Solution. The graph of the function $y=\frac{1}{2} x^{3}$ lies below the $O x$ axis on the interval $[-2,0]$, and above the $O x$ axis on the interval $[0,4]$ (Fig. 171). Therefore,
$S=\left|\int_{-2}^{0} \frac{1}{2} x^{3} d x\right|+\int_{0}^{4} \frac{1}{2} x^{3} d x=\left|\frac{1}{8} x^{4}\right|_{-2}^{0}+\left.\fra... | 34 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
358. Calculate the integral $\int_{1}^{9} \sqrt{6 x-5} d x$ using the Newton-Leibniz formula and approximate formulas for rectangles and trapezoids, dividing the interval of integration into 8 equal parts. Estimate the error of the results. | Solution. According to the Newton-Leibniz formula, we find
$$
I=\int_{1}^{9} \sqrt{6 x-5} d x=\frac{1}{6} \int_{1}^{9}(6 x-5)^{1 / 2} d(6 x-5)=\left.\frac{1}{9}(6 x-5)^{3 / 2}\right|_{1} ^{9}=\frac{7^{3}-1}{9}=38
$$
Since the interval $[1,9]$ is divided into 8 equal parts, then $\Delta x=1$. We find the values $y_{i}... | 38 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
362. Calculate the integral $\int_{1}^{4} x^{2} d x$ using Simpson's formula. | Solution. Divide the interval of integration into 10 equal parts. Then $(b-a) / 3 n=3 / 30=1 / 10=0.1$. Substituting into the integrand function $y=x^{2}$ the values of the argument $x_{0}=1, x_{1}=1.3 ; x_{2}=1.6, \ldots$, $x_{10}=4$, we find the corresponding values of the ordinates: $y_{0}=1 ; y_{1}=1.69$; $y_{2}=2.... | 21 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
363. The velocity of a material point is given by the formula $v=\left(4 t^{3}-2 t+1\right)$ m/s. Find the distance traveled by the point in the first 4 s from the start of the motion. | Solution. According to formula (1), we have
$$
s=\int_{0}^{4}\left(4 t^{3}-2 t+1\right) d t=\left.\left(t^{4}-t^{2}+t\right)\right|_{0} ^{4}=256-16+4=244(\text { m })
$$
Thus, in 4 s, the point has traveled 244 m. | 244 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
370. The velocity of a body is given by the equation $v=$ $=\left(12 t-3 t^{2}\right) \mathrm{m} / \mathrm{s}$. Determine the distance traveled by the body from the start of the motion until it stops. | Solution. The speed of the body is zero at the beginning of its motion and at the moment of stopping. To find the moment of stopping, we set the speed to zero and solve the equation with respect to $t$; we get $12 t-3 t^{2}=0 ; 3 t(4-t)=0 ; t_{1}=0, t_{2}=4$. Therefore,
$$
s=\int_{0}^{4}\left(12 t-3 t^{2}\right) d t=\... | 32 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
380. Two bodies simultaneously started linear motion from a certain point in the same direction with speeds $v_{1}=$ $=\left(6 t^{2}+4 t\right) \mathrm{m} /$ s and $v_{2}=4 t \mathrm{m} / \mathrm{s}$. After how many seconds will the distance between them be 250 m? | Solution. Let $t_{1}$ be the moment of the meeting. Then
$$
s_{1}=\int_{0}^{t_{1}}\left(6 t^{2}+4 t\right) d t=\left.\left(2 t^{3}+2 t^{2}\right)\right|_{0} ^{t_{1}}=2 t_{1}^{3}+2 t_{1}^{2} ; \quad s_{2}=\int_{0}^{t_{1}} 4 t d t=2 t_{1}^{2}
$$
Since $s_{1}-s_{2}=250$, we get the equation $2 t_{1}^{3}+2 t_{1}^{2}-2 t_... | 5 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
392. A rectangular plate is immersed in water, positioned vertically. Its horizontal side is 1 m, the vertical side is $2 \mathrm{m}$. The top side is at a depth of 0.5 m. Determine the force of water pressure on the plate. | Solution. Here $y=1, \quad a=0.5, \quad b=2+0.5=2.5$ (m), $\gamma=$ $=1000 \mathrm{kr} / \mathrm{m}^{3}$. Therefore,
$$
\begin{gathered}
P=9810 \int_{a}^{b} x y d x=9810 \int_{a}^{b} x d x=\left.9810 \frac{x^{2}}{2}\right|_{0.5} ^{2.5}=9810 \frac{2.5^{2}-0.5^{2}}{2}= \\
=29430 \text { (N) }
\end{gathered}
$$ | 29430 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
400. Calculate the force of water pressure on a dam that has the shape of a trapezoid, where the upper base, coinciding with the water surface, is $10 \mathrm{~m}$ long, the lower base is $20 \mathrm{~m}$, and the height is $3 \mathrm{m}$. | Solution. Using formula (6), we find
$$
P=9810 \frac{(10+2 \cdot 20) 3^{2}}{6}=9810 \cdot \frac{450}{6}=735750(\mathrm{H})
$$ | 735750 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
403. Determine the force of water pressure on a vertical parabolic segment, the base of which is $4 \mathrm{m}$ and is located on the water surface, while the vertex is at a depth of $4 \mathrm{m}$ (Fig. 204). | Solution. We have $|B A|=2 x=4$ (m). Point $A$ in the chosen coordinate system has coordinates $(2 ; 4)$. The equation of the parabola relative to this system is $y = a x^{2}$ or $4=a \cdot 2^{2}$, from which $a=1$, i.e., $y=x^{2}$.
Consider an elementary area $d S$ at a distance $y$ from the origin. The length of thi... | 167424 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
65. The speed of a body coming out of a state of rest is equal to $5 t^{2} \mathrm{m} / \mathrm{s}$ after $t$ seconds. Determine the distance the body will travel in 3 s (see problem 20). | Solution. Using the condition, we form the differential equation: $\frac{d s}{d t}=5 t^{2}$, since velocity $v=\frac{d s}{d t}$.
Let's find the general solution of this equation:
$$
d s=5 t^{2} d t ; \int d s=5 \int t^{2} d t ; s=\frac{5}{3} t^{3}+C
$$
Now, let's find the particular solution of this equation. The in... | 45 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
192. A metal ball, the temperature of which at the beginning of the experiment was $12^{\circ} \mathrm{C}$, is cooled by a stream of water having a temperature of $0^{\circ}$. After 8 minutes, the ball cooled down to $9^{\circ}$. Assuming the cooling rate is proportional to the difference between the temperature of the... | Solution. $1^{\circ}$. Let $T$ denote the temperature of the sphere, and $t-$ the time elapsed since the start of the experiment. Then the cooling rate of the sphere is the derivative $\frac{d T}{d t}\left(T^{\prime}\right)$. According to the condition, $T^{\prime}=k(T-0)=k T$, where $k$ is the proportionality coeffici... | 15 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
195. In a motorboat with an initial speed of $v_{0}=$ $=5 \mathrm{m} / \mathrm{s}$, the engine was turned off. During movement, the boat experiences water resistance, the force of which is proportional to the square of the boat's speed, with a proportionality coefficient of $m / 50$, where $m$ is the mass of the boat. ... | Solution. $1^{0}$. According to the condition, we can take the function as the path $s$, and the argument as time $t$. Using Newton's second law $F=m a$, we obtain the equation
$$
m s_{(t)}^{\prime \prime}=-\frac{m}{50}\left(s_{(t)}^{\prime}\right)^{2}, \text { or } s_{(t)}^{\prime \prime} \doteq-\frac{1}{50}\left(s_{... | 10 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
18. How many two-digit numbers can be formed from the five digits $1,2,3,4,5$ if no digit is repeated? | Solution. Since two-digit numbers differ from each other either by the digits themselves or by their order, the desired number is equal to the number of arrangements of two out of five elements: \(A_{5}^{2}=5 \cdot 4=20\). Therefore, 20 different two-digit numbers can be formed.
When finding the number of arrangements... | 20 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
19. Calculate in factorial form $A_{6}^{3}$. | Solution. $A_{6}^{3}=\frac{6!}{(6-3)!}=6 \cdot 5 \cdot 4=120$.
20-25. Calculate in any way: | 120 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
30. Calculate: a) $C_{8}^{3}$; b) $C_{10}^{8}$. | Solution. a) Applying formula (6) for $m=8, n=3$, we find
$$
C_{8}^{3}=-\frac{8!}{(8-3)!3!}=-\frac{8!}{5!3!}=\frac{5!6 \cdot 7 \cdot 8}{5!3 \cdot 2 \cdot 1}=56
$$
b) $C_{10}^{\beta}=\frac{10!}{(10-8)!\cdot 8!}=\frac{10!}{2!\cdot 8!}=\frac{8!9 \cdot 10}{1 \cdot 2 \cdot 8!}=45$.
Let's note the main property of the num... | 56 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
129. The probability of an event occurring in each of the independent trials is 0.8. How many trials need to be conducted to expect with a probability of 0.9 that the event will occur at least 75 times? | Solution. According to the condition, $p=0.8 ; q=0.2 ; k_{i}=75 ; k_{2}=n$; $P_{n}=(75, n)=0.9$.
Let's use the integral theorem of Laplace:
$$
P_{n}\left(k_{1} ; n\right)=\Phi\left(x^{\prime \prime}\right)-\Phi\left(x^{\prime}\right)=\Phi\left[\frac{k_{2}-n p}{\sqrt{n p q}}\right]-\Phi\left[\frac{k_{1}-n p}{\sqrt{n p... | 100 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
136. How many times should a die be thrown so that the probability of the inequality
$$
|m / n-1 / 6| \leqslant 0.01
$$
is not less than the probability of the opposite inequality, where $m$ is the number of times one dot appears in $n$ throws of a die? | Solution. We will use the formula
$$
P\left(\left|\frac{m}{n}-p\right|<\varepsilon\right)=1-2 \Phi\left(\varepsilon \sqrt{\frac{n}{p q}}\right)
$$
According to the condition, the inequality should hold
$$
2 \Phi\left(\varepsilon \sqrt{\frac{n}{p q}}\right) \geqslant 1-2 \Phi\left(\varepsilon \sqrt{\frac{n}{p q}}\rig... | 632 | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
145. Each of the 15 elements of a certain device is tested. The probability that an element will withstand the test is 0.9. Find the most probable number of elements that will withstand the test. | Solution. According to the condition, $n=15, p=0.9, q=0.1$. Let's find the most probable number $k_{0}$ from the double inequality
$$
n p-q<k_{0}<n p+p
$$
Substituting the data from the problem, we get
$$
15 \cdot 0.9-0.1 \leqslant k_{0}<15 \cdot 0.9+0.9, \text{ or } 13.5 \leqslant k_{0}<14.4 \text{. }
$$
Since $k_... | 14 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
150. Two shooters shoot at a target. The probability of a miss with one shot for the first shooter is 0.2, and for the second shooter it is 0.4. Find the most probable number of volleys in which there will be no hits on the target, if the shooters will make 25 volleys. | Solution. Misses by the shooters are independent events, so the multiplication theorem of probabilities of independent events applies. The probability that both shooters will miss in one volley, $p=0.2 \cdot 0.4=0.08$.
Since the product $n p=25 \cdot 0.08=2$ is an integer, the most probable number of volleys in which ... | 2 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
172. After the student answers the questions in the examination ticket, the examiner asks the student additional questions. The teacher stops asking additional questions as soon as the student shows ignorance of the given question. The probability that the student will answer any additional question is 0.9. Required: a... | Solution. a) The discrete random variable $X$ - the number of additional questions asked - has the following possible values: $x_{1}=1, x_{2}=2, x_{3}=3, \ldots, x_{k}=k, \ldots$ Let's find the probabilities of these possible values.
The variable $X$ will take the possible value $x_{i}=1$ (the examiner will ask only o... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
183. The probability of winning with one lottery ticket is $p=0.01$. How many tickets need to be bought to win at least one of them with a probability $P$, not less than 0.95? | Solution. The probability of winning is small, and the number of tickets that need to be purchased is obviously large, so the number of winning tickets is approximately Poisson distributed.
It is clear that the events "none of the purchased tickets are winning" and "at least one ticket is winning" are opposite. Theref... | 300 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
189. Find the mathematical expectation of the random variable $Z$, if the mathematical expectations of $X$ and $Y$ are known:
a) $Z=X+2 Y, M(X)=5, M(Y)=3 ;$ b) $Z=3 X+4 Y$, $M(X)=2, M(Y)=6$ | Solution. a) Using the properties of mathematical expectation (the expectation of a sum is equal to the sum of the expectations of the terms; a constant factor can be factored out of the expectation), we get
$$
\begin{aligned}
M(Z)=M(X+2 Y)= & M(X)+M(2 Y)=M(X)+2 M(Y)= \\
& =5+2 \cdot 3=11 .
\end{aligned}
$$ | 11 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
208. Random variables $X$ and $Y$ are independent. Find the variance of the random variable $Z=3X+2Y$, given that $D(X)=5, D(Y)=6$. | Solution. Since the variables $X$ and $Y$ are independent, the variables $3X$ and $2Y$ are also independent. Using the properties of variance (the variance of the sum of independent random variables is equal to the sum of the variances of the addends; a constant factor can be factored out of the variance sign, squaring... | 69 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
277. The random variable $X$ in the interval (-c, c) is given by the probability density function $f(x)=1 /\left(\pi \sqrt{c^{2}-x^{2}}\right)$; outside this interval, $f(x)=0$. Find the mathematical expectation of the variable $X$. | Solution. We use the formula $M(X)=\int_{a} x f(x) \mathrm{d} x$. Substituting $a=-c, b=c, f(x)=1 /\left(\pi \sqrt{c^{2}-x^{2}}\right)$, we get
$$
M(X)=\frac{1}{\pi} \int_{-c}^{c} \frac{x d x}{\sqrt{c^{2}-x^{2}}}
$$
Considering that the integrand is an odd function and the limits of integration are symmetric with res... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
280. Find the mathematical expectation of the random variable $X$, given by the distribution function
$$
F(x)=\left\{\begin{array}{ccr}
0 & \text { for } & x \leqslant 0 \\
x / 4 & \text { for } & 0 < x \leqslant 4 \\
1 & \text { for } & x > 4
\end{array}\right.
$$ | Solution. Find the density function of the random variable $\boldsymbol{X}$:
$$
f(x)=F^{\prime}(x)=\left\{\begin{array}{ccc}
0 & \text { for } & x4 .
\end{array}\right.
$$
Find the required expected value:
$$
M(X)=\int_{0}^{4} x f \cdot(x) \mathrm{d} x=\int_{0}^{4} x \cdot(1 / 4) \mathrm{d} x=2
$$ | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
334. A machine manufactures balls. A ball is considered suitable if the deviation $X$ of the ball's diameter from the design size in absolute value is less than 0.7 mm. Assuming that the random variable $X$ is normally distributed with a standard deviation $\sigma=0.4$ mm, find how many suitable balls on average will b... | Solution. Since $X$ is the deviation (of the diameter of the ball from the design size), then $M(X)=a=0$.
Let's use the formula $P(|X|<\delta)=2 \Phi(\delta / \sigma)$. Substituting $\delta=0.7, \sigma=0.4$, we get
$$
P(|X|<0.7)=2 \Phi\left(\frac{0.7}{0.4}\right)=2 \Phi(1.75)=2 \cdot 0.4599=0.92
$$
Thus, the probabi... | 92 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
453. Find the sample mean for the given distribution of sample size $n=10$ :
| $x_{i}$ | 1250 | 1270 | 1280 |
| :--- | :---: | :---: | :---: |
| $n_{i}$ | 2 | 5 | 3 | | Solution. The initial variants are large numbers, so we will switch to conditional variants $u_{i}=x_{i}-1270$. As a result, we will get the distribution of conditional variants:
$$
\begin{array}{rrrr}
u_{i} & -20 & 0 & 10 \\
n_{i} & 2 & 5 & 3
\end{array}
$$
We will find the required sample mean:
$$
\bar{x}_{\mathrm... | 1269 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
506. Find the minimum sample size for which with a reliability of 0.975, the accuracy of estimating the mean $a$ of the population using the sample mean is $\delta=0.3$, given that the standard deviation $\sigma=1.2$ of the normally distributed population is known. | Solution. We will use the formula that defines the accuracy of estimating the mathematical expectation of the general population by the sample mean: $\delta=t \sigma / \sqrt{n}$. From this,
$$
n=t^{2} \sigma^{2} / \delta^{2}
$$
According to the condition, $\gamma=0.975$; therefore, $\Phi(t)=0.975 / 2=0.4875$. From th... | 81 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
576. a) From a sample of size $n$, extracted from a normal general population with a known standard deviation $\sigma$, the sample mean $\bar{x}$ is found. At a significance level $\alpha$, it is required: 1) to find the critical region if the null hypothesis $H_{0}$: $a=a_{0}$ about the equality of the general mean $a... | Solution. 1) We use the formula
$$
1-\beta=0.5-\Phi\left(u_{\mathrm{kp}}-\lambda\right) .
$$
By rule 2, we find the critical point of the right-sided critical region $u_{\text {kp}}=1.65$.
We calculate $\lambda$, considering that, according to the condition, $a_{1}=3, a_{0}=2, n=16$, $\sigma=4:$
$$
\lambda=\left(a_... | 58 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
647. Why, when checking the hypothesis of exponential distribution of the population by the Pearson criterion, is the number of degrees of freedom determined by the equation $k=s-2$, where $s$ is the number of sample intervals? | Solution. When using the Pearson criterion, the number of degrees of freedom is $k=s-1-r$, where $r$ is the number of parameters estimated from the sample. The exponential distribution is defined by one parameter $\lambda$. Since this parameter is estimated from the sample, $r=1$, and therefore, the number of degrees o... | -2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
657. Why, when testing the hypothesis of a uniform distribution of the population $X$ using the Pearson's criterion, is the number of degrees of freedom determined by the equation $k=s-3$, where $s$ is the number of intervals in the sample? | Solution. When using the Pearson criterion, the number of degrees of freedom $k=s-1-r$, where $r$ is the number of parameters estimated from the sample. The uniform distribution is defined by two parameters $a$ and $b$. Since these two parameters are estimated from the sample, $r=2$, and therefore, the number of degree... | -3 | Other | math-word-problem | Yes | Yes | olympiads | false |
732. In a single-channel queuing system with rejections, a Poisson flow of requests arrives.
The time between the arrival of two consecutive requests is distributed according to the law \( f(\tau)=0.8 \mathrm{e}^{-0.8 \tau} \); the service time for requests is random and distributed according to the law \( f_{1}(t)=1.... | The solution is as follows. The time between the arrival of two consecutive requests is distributed according to the law $f(\tau)=0.8 \mathrm{e}^{-0.8 \tau}$, so the values $\tau_{i}$ will be generated using the formula
$$
\tau_{i}=-(1 / 0.8) \ln r_{i}=1.25\left(-\ln r_{i}\right)
$$
Random numbers $r_{i}$ are taken f... | 66 | Other | math-word-problem | Yes | Yes | olympiads | false |
910. A stationary random function $X(t)$ with mean $m_{x}=5$ is input to a linear stationary dynamic system described by the equation $Y^{\prime}(t)+2 Y(t)=5 X^{\prime}(t)+$ $+6 X(t)$. Find the mean of the random function $Y(t)$ at the output of the system in the steady state (after the transient process has decayed). | Solution. Let's equate the mathematical expectations of the left and right parts of the given differential equation:
$$
\begin{aligned}
M\left[Y^{\prime}(t)+2 Y(t)\right]= & M\left[5 X^{\prime}(t)+6 X(t)\right], \text{ or } M\left[Y^{\prime}(t)\right]+2 m_{y}= \\
& =5 M\left[X^{\prime}(t)\right]+6 m_{x} .
\end{aligned... | 15 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Find $\lim _{x \rightarrow 2}\left(4 x^{2}-6 x+3\right)$. | Solution. Since the limit of the algebraic sum of variables is equal to the same algebraic sum of the limits of these variables (formula (1.42)), a constant multiplier can be factored out of the limit sign (formula (1.44)), the limit of an integer positive power is equal to the same power of the limit (formula (1.45)),... | 7 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 3. Find $\lim _{x \rightarrow 4} \frac{x^{2}-6 x+8}{x-4}$. | Solution. When $x=4$, both the numerator and the denominator of the given function become zero. This results in an indeterminate form $\frac{0}{0}$, which needs to be resolved. We will transform the given function by factoring the numerator using the formula
$$
x^{2}+p x+q=\left(x-x_{1}\right)\left(x-x_{2}\right)
$$
... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 6. Find $\lim _{x \rightarrow+\infty} \frac{6 x^{2}+5 x+4}{3 x^{2}+7 x-2}$. | Solution. As $x \rightarrow+\infty$, the numerator and denominator increase without bound (we get an indeterminate form of $\frac{\infty}{\infty}$). To find the limit, we transform the given fraction by dividing its numerator and denominator by $x^{2}$, i.e., by the highest power of $x$. Using the properties of limits,... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 7. Find $\lim _{x \rightarrow+\infty} \frac{7 x^{2}+6 x-3}{9 x^{3}+8 x^{2}-2}$. | Solution. By dividing the numerator and the denominator of the fraction by $x^{3}$, i.e., by the highest degree, we get
$$
\begin{aligned}
\lim _{x \rightarrow+\infty} \frac{7 x^{2}+6 x-3}{9 x^{3}+8 x^{2}-2}= & \lim _{x \rightarrow+\infty} \frac{\frac{7}{x}+\frac{6}{x^{2}}-\frac{3}{x^{3}}}{9+\frac{8}{x}-\frac{2}{x^{3}... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 9. Find $\lim _{x \rightarrow 3} \frac{x^{2}-9}{\sqrt{x+1}-2}$. | Solution. When $x=3$, both the numerator and the denominator of the fraction become zero. The denominator contains an irrational expression $\sqrt{x+1}$. To eliminate the irrationality in the denominator, we multiply the numerator and the denominator by $(\sqrt{x+1}+2)$. We get
$$
\begin{gathered}
\lim _{x \rightarrow... | 24 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 11. Find $\lim _{n \rightarrow \infty} \frac{1+3+5+\ldots+(2 n-1)}{1+2+3+\ldots+n}$. | Solution. The numerator and denominator of the fraction are the sums of $n$ terms of the corresponding arithmetic progressions. Finding these sums using the known formula, we get
\[
\begin{gathered}
\lim _{n \rightarrow \infty} \frac{1+3+5+\ldots+(2 n-1)}{1+2+3+\ldots+n}=\lim _{n \rightarrow \infty} \frac{\frac{1+(2 n... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 14. Find $\lim _{n \rightarrow \infty} \frac{12 n+5}{\sqrt[3]{27 n^{3}+6 n^{2}+8}}$. | Solution. As $n \rightarrow \infty$, the numerator and denominator also tend to infinity, resulting in an indeterminate form $\frac{\infty}{\infty}$. To find the limit, we divide the numerator and denominator by $\boldsymbol{n}$ and bring $\boldsymbol{n}$ under the root sign:
$$
\lim _{n \rightarrow \infty} \frac{12 n... | 4 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Find $\lim _{x \rightarrow 0} \frac{\ln (1+x)}{x}$. | Solution. Since
$$
\frac{\ln (1+x)}{x}=\frac{1}{x} \ln (1+x)=\ln (1+x)^{\frac{1}{x}}
$$
then based on formula (1.58) we find
$$
\lim _{x \rightarrow 0} \frac{\ln (1+x)}{x}=\lim _{x \rightarrow 0}\left[\ln (1+x)^{\frac{1}{x}}\right]=\ln \left[\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x}}\right]=\ln e=1
$$
Therefore,
$... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 6. Find $\lim _{x \rightarrow 0} \frac{\tan x}{x}$. | Solution. Taking into account that $\operatorname{tg} x=\frac{\sin x}{\cos x}$ and $\lim _{x \rightarrow 0} \cos x=\cos 0=1$, based on the properties of limits (1.43) and (1.46) and formula (1.59), we obtain
$$
\begin{aligned}
\lim _{x \rightarrow 0} \frac{\operatorname{tg} x}{x} & =\lim _{x \rightarrow 0}\left(\frac{... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 9. Find $\lim _{x \rightarrow 0} \frac{\sin x}{\sqrt{x+9}-3}$. | Solution. When $x=0$, both the numerator and the denominator become zero. The denominator contains an irrationality. We will eliminate the irrationality and use formula (1.59)
$$
\begin{gathered}
\lim _{x \rightarrow 0} \frac{\sin x}{\sqrt{x+9}-3}=\lim _{x \rightarrow 0} \frac{\sin x(\sqrt{x+9}+3)}{(\sqrt{x+9}-3)(\sqr... | 6 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 10. Find $\lim _{x \rightarrow 0}\left(\frac{\sin 3 x}{x}\right)^{x+2}$. | Solution. This is a limit of the form (1.60), where $\varphi(x)=\frac{\sin 3 x}{x}, \psi(x)=x+2$.
From (1.67)
$$
\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}=3, \lim _{x \rightarrow 0}(x+2)=2
$$
According to formula (1.62), we get
$$
\lim _{x \rightarrow 0}\left(\frac{\sin 3 x}{x}\right)^{x+2}=3^{2}=9
$$ | 9 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 11. Find $\lim _{x \rightarrow \infty}\left(\frac{2 x-1}{3 x+4}\right)^{x^{2}}$. | Solution. This is also a limit of the form $(1.60)$, where $\varphi(x)=\frac{2 x-1}{3 x+4}, \psi(x)=x^{2}$
Since
$$
\lim _{x \rightarrow \infty} \frac{2 x-1}{3 x+4}=\lim _{x \rightarrow \infty} \frac{2-\frac{1}{x}}{3+\frac{4}{x}}=\frac{2}{3}, \lim _{x \rightarrow \infty} x^{2}=\infty
$$
then according to formula (1.... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 13. Find $\lim _{x \rightarrow 0}(\cos x)^{\frac{1}{x}}$. | Solution. By adding and subtracting 1 from $\cos x$ and applying the corresponding formula, we get
$$
\begin{aligned}
& \lim _{x \rightarrow 0}(\cos x)^{\frac{1}{x}}=\lim _{x \rightarrow 0}[1-(1-\cos x)]^{\frac{1}{x}}=\lim _{x \rightarrow 0}\left(1-2 \sin ^{2} \frac{x}{2}\right)^{\frac{1}{x}}= \\
& =\lim _{x \rightarr... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Find $\lim _{x \rightarrow+\infty}\left(\sqrt{x^{2}+6 x+5}-x\right)$. | Solution. When $x \rightarrow+\infty$, the given function represents the difference of two infinitely large quantities taking positive values (case $\infty-\infty$). By multiplying and dividing the given function by $\left(\sqrt{x^{2}+6 x+5}+x\right)$, we get
$$
\begin{gathered}
\lim _{x \rightarrow+\infty}\left(\sqrt... | 3 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 4. Find $\lim _{x \rightarrow 0} x \operatorname{ctg} \frac{x}{3}$. | Solution. When $x \rightarrow 0$, we get an indeterminate form of $0 \cdot \infty$. Rewriting the given function in another form and applying formula (1.59), we find
$$
\lim _{x \rightarrow 0} x \operatorname{ctg} \frac{x}{3}=\lim _{x \rightarrow 0} x \frac{\cos \frac{x}{3}}{\sin \frac{x}{3}}=\lim _{x \rightarrow 0} \... | 3 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 6. Find $\lim _{x \rightarrow+\infty} x\left(\operatorname{arctg} x-\frac{\pi}{2}\right)$. | Solution. Let $\operatorname{arctg} x=\alpha$, then $x=\operatorname{tg} \alpha$, if $x \rightarrow+\infty$, then $\alpha \rightarrow \frac{\pi}{2}$.
Consequently,
$$
\begin{gathered}
\lim _{x \rightarrow+\infty} x\left(\operatorname{arctg} x-\frac{\pi}{2}\right)=\lim _{\alpha \rightarrow \frac{\pi}{2}} \operatorname... | -1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 6. Find $\lim _{x \rightarrow 3} \frac{\ln \left(x^{2}-5 x+7\right)}{x-3}$. | Solution. When $x \rightarrow 3$, we get an indeterminate form of $\frac{0}{0}$, since $x^{2}-5 x+7 \rightarrow 1$ and $\ln \left(x^{2}-5 x+7\right) \rightarrow 0$. The expression $x^{2}-5 x+7$ can be represented as:
$$
x^{2}-5 x+7=1+\left(x^{2}-5 x+6\right)=1+z
$$
where $\left(x^{2}-5 x+6\right)=z \rightarrow 0$ as ... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 3. Calculate the value of the derivative of the implicit function $x y^{2}=4$ at the point $M(1,2)$. | Solution. First, let's find the derivative:
$$
x^{\prime} y^{2}+x 2 y y^{\prime}=0, y^{\prime}=-\frac{y}{2 x}
$$
Substituting the values \(x=1\), \(y=2\) into the right-hand side of the last equation, we get
$$
y^{\prime}=-\frac{2}{2 \cdot 1}=-1
$$ | -1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Find the third-order derivative of the function $y=x^{2}+3 x+2$. | Solution. Differentiating successively, we obtain:
$$
\begin{gathered}
y^{\prime}=\left(x^{2}+3 x+2\right)^{\prime}=2 x+3 ; y^{\prime \prime}=(2 x+3)^{\prime}=2 \\
y^{\prime \prime \prime}=(2)^{\prime}=0
\end{gathered}
$$ | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 4. Find the derivative $y_{x}^{\prime}$ of the function $x=e^{t} \cos t$; $y=e^{t} \sin t$ at $t=0$. | Solution. The functions $x$ and $y$ have the following derivatives with respect to $t$:
$$
\begin{aligned}
& x_{i}^{\prime}=e^{t} \cos t-e^{t} \sin t=e^{t}(\cos t-\sin t) \\
& y_{t}^{\prime}=e^{t} \sin t+e^{t} \cos t=e^{t}(\sin t+\cos t)
\end{aligned}
$$
therefore
$$
y_{x}^{\prime}=\frac{\sin t+\cos t}{\cos t-\sin t... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 3. Find $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan x}{\tan 3 x}$. | Solution. Using the L'Hôpital-Bernoulli rule, we get
$$
\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan x}{\tan 3 x}=\lim _{x \rightarrow \frac{\pi}{2}} \frac{\frac{1}{\cos ^{2} x}}{\frac{3}{\cos ^{2} 3 x}}=\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos ^{2} 3 x}{3 \cos ^{2} x}
$$
The limit of the ratio of the first d... | 3 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 6. Find $\lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\sin x}\right)$. | Solution. When $x \rightarrow \mathbf{0}$, we get an indeterminate form of $\infty-\infty$. We will resolve this indeterminacy by converting it to an indeterminate form of $\frac{0}{0}$ and applying L'Hôpital-Bernoulli's rule,
$$
\begin{gathered}
\lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\sin x}\right)=\lim _{... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 7. Find $\lim _{x \rightarrow 0}[(x-\sin x) \ln x]$. | Solution. Here we have an indeterminate form $0 \cdot \infty$. The given function can be represented as
$$
(x-\sin x) \ln x=\frac{\ln x}{\frac{1}{x-\sin x}}
$$
The resulting indeterminate form $\frac{\infty}{\infty}$ is resolved using L'Hôpital's rule:
$$
\lim _{x \rightarrow 0} \frac{\ln x}{\frac{1}{x-\sin x}}=\lim... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 8. Find $\lim _{x \rightarrow 1}(x-1)^{\ln x}$. | Solution. When $x \rightarrow 1$, we have the indeterminate form $0^{0}$. We will use the identity
$$
[f(x)]^{\varphi(x)}=e^{\varphi(x) \ln f(x)}
$$
which in this case will be
$$
(x-1)^{\ln x}=e^{\ln x \cdot \ln (x-1)}
$$
We have
$$
\lim _{x \rightarrow 1}(x-1)^{\ln x}=\lim _{x \rightarrow 1} e^{\ln x \cdot \ln (x... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 9. Find $\lim _{x \rightarrow 0}\left[\frac{\sin x}{x}\right]^{\frac{1}{x}}$. | Solution. Since $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1, \lim _{x \rightarrow 0} \frac{1}{x}=\infty$, we have an indeterminate form of $1^{\infty}$ here.
Taking into account identity (A) (see example 8), using L'Hôpital-Bernoulli's rule, we find
$$
\lim _{x \rightarrow 0}\left[\frac{\sin x}{x}\right]^{\frac{1}{x}... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 11. Find $\lim _{x \rightarrow \infty} \frac{x-\sin x}{x+\sin x}$. | Solution. Apply L'Hôpital's rule:
$$
\lim _{x \rightarrow \infty} \frac{x-\sin x}{x+\sin x}=\lim _{x \rightarrow \infty} \frac{1-\cos x}{1+\cos x}
$$
In the right-hand side of the last equality, the limit does not exist, so L'Hôpital's rule is not applicable here.
The specified limit can be found directly:
$$
\lim ... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Calculate the definite integral $\int_{1}^{4} x^{2} d x$. | Solution. By formula (5.3) we have
$$
\int_{1}^{4} x^{2} d x=\left.\frac{x^{3}}{3}\right|_{1} ^{4}=\frac{4^{3}}{3}-\frac{1^{3}}{3}=\frac{64}{3}-\frac{1}{3}=21
$$
Remark According to the Newton-Leibniz theorem, we can take any antiderivative of the integrand. In this case, instead of $\frac{x^{3}}{3}$, we could have t... | 21 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Calculate the integral $\int_{0}^{\frac{\pi}{2}} \cos x d x$. | Solution.
$$
\int_{0}^{\frac{\pi}{2}} \cos x d x=\left.\sin x\right|_{0} ^{\frac{\pi}{2}}=\sin \frac{\pi}{2}-\sin 0=1-0=1
$$ | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 3. Calculate $\int_{4}^{9}\left(\frac{2 x}{5}+\frac{1}{2 \sqrt{x}}\right) d x$. | Solution. Based on properties 5 and 6 of the definite integral and formula (5.3), we obtain
$$
\begin{aligned}
& \int_{4}^{9}\left(\frac{2 x}{5}+\frac{1}{2 \sqrt{x}}\right) d x=\frac{2}{5} \int_{4}^{9} x d x+\int_{4}^{9} \frac{1}{2 \sqrt{x}} d x=\left.\frac{2}{5} \cdot \frac{x^{2}}{2}\right|_{4} ^{9}+ \\
& +\left.\sqr... | 14 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 6. Calculate $\int_{1}^{e} \ln x d x$. | Solution. We apply the integration by parts formula (5.5).
Letting $u=\ln x, d v=d x$, we determine $d u=\frac{1}{x} d x, v=x$.
Therefore,
$$
\begin{gathered}
\int_{1}^{e} \ln x d x=\left.x \ln x\right|_{1} ^{e}-\int_{1}^{e} \frac{1}{x} x d x=\left.x \ln x\right|_{1} ^{e}-\int_{1}^{e} d x=\left.x \ln x\right|_{1} ^{... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 8. Determine the area bounded by the arc of the cosine curve from $x=-\frac{\pi}{2}$ to $x=\frac{\pi}{2}$ and the $O x$ axis. | Solution. Based on the geometric meaning of the definite integral, we conclude that the desired area is expressed by the integral
$$
S=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x d x
$$
Evaluating this integral, we get
$$
S=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x d x=\left.\sin x\right|_{-\frac{\pi}{2}} ^{\fra... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Determine the area bounded by the lines $x y=6$, $x=1, x=e, y=0$ (Fig. 5.5).
Fig. 5.5
 into formula (5.13), we find:
$$
S=\int_{1}^{e} \frac{6}{x} d x=6 \int_{1}^{e} \frac{d x}{x}=\left.6 \ln x\rig... | 6 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 6. Calculate the area bounded by the parabola $x=8 y-y^{2}-7$ and the $O y$ axis. | Solution. To determine the limits of integration, we find the points of intersection of the curve with the $O y$ axis.
Solving the system of equations $x=8 y-y^{2}-7, x=0$, we get $\quad y_{1}=1, \quad y_{2}=7$
(Fig. 5.10). By formula (5.16) we find
$$
\begin{aligned}
& S=\int_{1}^{7}\left(8 y-y^{2}-7\right) d y=8 \... | 36 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Find the length of the arc of the astroid $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$. What is the length of the astroid when $a=1, a=\frac{2}{3}?$ | Solution. Since the astroid
Fig. 5.13 is symmetric with respect to the coordinate axes (Fig. 5.13), it is sufficient to compute the length of the arc $AB$ and multiply the result by 4.
Diff... | 4 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Find $\lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} \frac{\sin x y}{y}$. | Solution. At the point $M(2,0)$, the function $z=\frac{\sin x y}{y}$ is undefined.
By multiplying and dividing the given function by $x \neq 0$, we get
$$
\frac{\sin x y}{y}=\frac{x \sin x y}{x y}=x \frac{\sin x y}{x y}
$$
Taking the limit in the last equality, we obtain
$$
\lim _{\substack{x \rightarrow 2 \\ y \ri... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Given the function
$$
f(x, y)=x \sin \frac{1}{y}+y \sin \frac{1}{x} \quad\left(x^{2}+y^{2} \neq 0\right), \quad f(0, y)=0, \quad f(x, 0)=0
$$
find $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} f(x, y)$. | Solution. Let $\varepsilon>0$, then for $|x|<\frac{\varepsilon}{2},|y|<\frac{\varepsilon}{2}$ we get $\rho=$
$$
=\sqrt{\left(\frac{\varepsilon}{2}\right)^{2}+\left(\frac{\varepsilon}{2}\right)^{2}}=\frac{\sqrt{2}}{2} \varepsilon . \text { We form the difference } f(x, y)-0 \text { and estimate it: }
$$
$$
|f(x, y)-0|... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 3. Find $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x^{2} y}{x^{2}+y^{2}}$. | Solution. The given function can be represented as:
$$
f(x, y)=\frac{x^{2} y}{x^{2}+y^{2}}=\frac{x y}{x^{2}+y^{2}} x .
$$
Since
$$
\left|\frac{x y}{x^{2}+y^{2}}\right| \leq \frac{1}{2}
$$
(this can be derived from the inequality $(x-y)^{2} \geq 0$ : $x^{2}-2 x y+y^{2} \geq 0$, $\left.x^{2}+y^{2} \geq 2 x y, \frac{1... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Investigate the function for extremum
$$
f(x, y)=x^{3}+y^{3}+9 x y
$$ | Solution. We find the first and second partial derivatives:
$$
\begin{gathered}
f_{x}^{\prime}(x, y)=3 x^{2}+9 y ; f_{y}^{\prime}(x, y)=3 y^{2}+9 x \\
f_{x x}^{\prime \prime}(x, y)=6 x ; f_{x y}^{\prime \prime}(x, y)=9 ; f_{y y}^{\prime \prime}(x, y)=6 y
\end{gathered}
$$
Setting the first derivatives to zero, we obt... | 27 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Find the maximum and minimum values of the function $z=f(x, y)=x^{3}+y^{3}+6 x y \quad$ in the rectangle with vertices $A(-3,-3), B(-3,2), C(1,2), D(1,-3)$. | Solution. We take the partial derivatives of the given function:
$$
f_{x}^{\prime}(x, y)=3 x^{2}+6 y ; f_{y}^{\prime}(x, y)=3 y^{2}+6 x
$$
From the system of equations
$$
\left.\left.\begin{array}{c}
3 x^{2}+6 y=0 \\
3 y^{2}+6 x=0
\end{array}\right\} \text { or } \begin{array}{r}
x^{2}+2 y=0 \\
y^{2}+2 x=0
\end{arra... | -55 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Into how many parts should the interval of integration be divided to calculate $\int_{2}^{7} \frac{d x}{\sqrt{x+2}}$ with an accuracy of 0.1? | Solution. The absolute error in calculating a definite integral using the rectangle method is determined by inequality (13.4). In the problem, the condition $\left|R_{n}(f)\right| \leq \varepsilon$ is set, where $\varepsilon=0.1$. The inequality $\left|R_{n}(f)\right| \leq \varepsilon$ will be satisfied if $\frac{(b-a)... | 8 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Construct a table of differences of various orders for the following values of $x$ and $f(x)$:
\[
\begin{gathered}
x_{0}=-1, x_{1}=-2, x_{2}=1, x_{3}=2, x_{4}=3 \\
y_{0}=0, y_{1}=7, y_{2}=30, y_{3}=-16, y_{4}=-45
\end{gathered}
\] | Solution. Using formulas (15.4), we find the first differences:
$$
\begin{gathered}
\Delta y_{0}=y_{1}-y_{0}=7-0=7 ; \Delta y_{1}=y_{2}-y_{1}=30-7=23 \\
\Delta y_{2}=y_{3}-y_{2}=-16-30=-46 ; \Delta y_{3}=y_{4}-y_{3}=-45-(-16)=-29
\end{gathered}
$$
In accordance with formulas (15.5), we calculate the second-order diff... | 171 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Construct a table of divided differences of various orders for the following values of $x$ and $y=f(x)$:
\[
\begin{gathered}
x_{0}=-3, x_{1}=-2, x_{2}=-1, x_{3}=1, x_{4}=2 \\
y_{0}=-9, y_{1}=-16, y_{2}=-3, y_{3}=11, y_{4}=36
\end{gathered}
\] | Solution. According to the definitions, we find the divided differences of the first order:
$$
\begin{gathered}
f\left(x_{1}, x_{0}\right)=\frac{y_{1}-y_{0}}{x_{1}-x_{0}}=\frac{-16-(-9)}{-2-(-3)}=-7 ; f\left(x_{2}, x_{1}\right)=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{-3-(-16)}{-1-(-2)}=13 \\
f\left(x_{3}, x_{2}\right)=\... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 2. How many elementary outcomes favor the event "the same number of points fell on both dice" when two gaming dice are rolled? | Solution. This event is favored by 6 elementary outcomes (see Table 1.1$):(1 ; 1),(2 ; 2),(3 ; 3),(4 ; 4),(5 ; 5),(6 ; 6)$. | 6 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Example 1. In how many different ways can three people be selected for three different positions from ten candidates? | Solution. We will use formula (1.3.3). For $n=10, m=3$ we get
$$
A_{10}^{3}=10 \cdot 9 \cdot 8=720
$$ | 720 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Example 3. In how many ways can three people be selected for three identical positions from ten candidates? | Solution. According to formula (1.3.4), we find
$$
C_{10}^{3}=\frac{10!}{3!\cdot(10-3)!}=\frac{10!}{3!\cdot 7!}=\frac{10 \cdot 9 \cdot 8}{1 \cdot 2 \cdot 3}=120
$$ | 120 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Example 4. How many different six-digit numbers can be written using the digits $1 ; 1 ; 1 ; 2 ; 2 ; 2$? | Solution. Here we need to find the number of permutations with repetitions, which is determined by formula (1.3.7). For $k=2, n_{1}=3, n_{2}=3, n=6$ this formula gives us
$$
P_{6}(3 ; 3)=\frac{6!}{3!\cdot 3!}=\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6}{1 \cdot 2 \cdot 3 \cdot 1 \cdot 2 \cdot 3}=20
$$ | 20 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
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