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Example 3.21. Find the points of discontinuity of the function
$$
z=\frac{x y+1}{x^{2}-y}
$$ | Solution. The function will lose its meaning if the denominator becomes zero. But $x^{2}-y=0$ or $y=x^{2}$ is the equation of a parabola. Therefore, the given function has a line of discontinuity, which is the parabola $y=x^{2}$. | x^{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,322 |
Example 3.25. Find the gradient of the function $z=x^{2}-x y+y^{3}$ at the point $A(1, -1)$ and the derivative in the direction of the vector $\bar{a}=3 \bar{i}-4 \bar{j}$. | Solution. The partial derivatives are:
$$
\frac{\partial z}{\partial x}=2 x-y, \quad \frac{\partial z}{\partial y}=-x+3 y^{2}
$$
By formula (3.12):
$$
\begin{aligned}
\left.\operatorname{grad} z\right|_{A}= & \frac{\partial z}{\partial x} \bar{l}+\frac{\partial z}{\partial y} \bar{j} \quad=(2 x-y)_{A} \cdot \bar{l}+... | \frac{1}{5} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,323 |
Example 3.28. Compute the double integral $\iint_{D}(x+2 y) d x d y$, where the region of integration is bounded by the parabolas $y=x-x^{2}$, $y=1-x^{2}$, and the $O y$ axis (Fig. 48).
Fig... | Solution. The parabolas intersect at point $A(1 ; 0)$. The region of integration $D$ is regular in the direction of the $O y$ axis and is defined by the inequalities
$$
\begin{aligned}
0 & \leq x \leq 1 \\
x-x^{2} & \leq y \leq 1-x^{2}
\end{aligned}
$$
Therefore,
$$
\iint_{D}(x+2 y) d x d y=\int_{0}^{1} d x \int_{x-... | \frac{2}{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,325 |
Example 3.29. Express the double integral of the function $f(x, y)$ over the region $D$, bounded by the parabola $y=3-x^{2}$ and the line $y=-1$, as iterated integrals with different orders of integration. | Solution. Solving the system of equations $\begin{aligned} & y=3-x^{2} \\ & y=-1\end{aligned}$, we find the points of intersection of the parabola and the line. These points are $A(-2, -1)$ and
$B(2, -1)$. The vertex of the parabola is at point $C(0, 3)$; the $O y$ axis is the axis of symmetry of the parabola (Fig. 49)... | \iint_{(D)}f(x,y)\int_{-2}^{2}\int_{-1}^{3-x^{2}}f(x,y)=\int_{-1}^{3}\int_{-\sqrt{3-y}}^{\sqrt{3-y}}f(x,y) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,326 |
Example 3.31. Compute the volume of the body bounded by the surfaces $z=0, x=0, y=x, z=1-y \sqrt{y}$. | Solution. We will construct a drawing showing the body bounded by the specified surfaces (Fig. 51a). The volume of the body \( V = \iint_{D} z \, dx \, dy \), where \( z = 1 - y \sqrt{y} \) is the equation of the surface bounding the body from above, and \( D \) is the region of integration, representing a triangle in ... | \frac{3}{14} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,327 |
Example 4.1. Verify that the function $y=\cos x$ is a solution to the equation $y^{\prime \prime}+y=0$. | Solution. We have
$$
y^{\prime}=-\sin x, \quad y^{\prime \prime}=-\cos x
$$
Substituting the expressions for $y^{\prime \prime}$ and $y$ into the given equation, we get:
$$
y^{\prime \prime}+y=-\cos x+\cos x=0,
$$
i.e., indeed, the function $y=\cos x$ is a solution to the given differential equation. | proof | Calculus | proof | Yes | Yes | olympiads | false | 30,328 |
Example 4.2. Show that the function $y$, defined by the equation $x^{2}-y^{2}=4$, is an integral of the differential equation $y^{\prime}=\frac{x}{y}$. | Solution. Differentiating both sides of the equality with respect to the variable $x$, we get:
$$
2 x-2 y y^{\prime}=0
$$
from which $y^{\prime}=\frac{x}{y}$.
## 4.2. Equations with Separable Variables
A differential equation with separable variables has the form:
$$
M_{1}(x) N_{1}(y) d x+M_{2}(x) N_{2}(y) d y=0
$... | proof | Calculus | proof | Yes | Yes | olympiads | false | 30,329 |
Example 4.3. Find the general integral of the equation
$$
\cos ^{2} y \operatorname{ctg} x d x+\sin ^{2} x \operatorname{tg} y d y=0
$$ | Solution. Let's separate the variables in the given equation by dividing both sides by the expression $\cos ^{2} y \cdot \sin ^{2} x$:
$$
\frac{\operatorname{ctg} x}{\sin ^{2} x} d x+\frac{\operatorname{tg} y}{\cos ^{2} y} d y=0
$$
Integrating both sides of the equation, we get
$$
\int \frac{\operatorname{ctg} x}{\s... | \operatorname{tg}^{2}y-\operatorname{ctg}^{2}x=C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,330 |
Example 4.4. Find the general solution of the first-order differential equation $x y^{\prime}-\frac{y}{\ln x}=0$. | Solution. Express the derivative $y^{\prime}$ from the equation $y^{\prime}=\frac{y}{x \ln x}$. The right-hand side can be factored as $f(x, y)=\phi(x) \cdot \psi(y)$, hence this is a separable differential equation. Separate the variables $\frac{d y}{d x}=\frac{y}{x \ln x}$ or $\frac{d y}{y}=\frac{d x}{x \ln x}$. Now ... | \lnx | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,331 |
Example 4.5. Find a particular solution of the equation $\left(1+e^{x}\right) y y^{\prime}=\mathrm{e}^{x}$, satisfying the initial condition $\left.y\right|_{x=0}=1$. | Solution. We have $\left(1+e^{x}\right) y \frac{d y}{d x}=e^{x}$. Separating variables, we get: $y d y=\frac{e^{x} d x}{1+e^{x}}$. Integrating, we find the general integral:
$$
\frac{y^{2}}{2}=\ln \left(1+e^{x}\right)+C
$$
Now let's find $C$. Setting $x=0, y=1$, we get
$$
\frac{1}{2}=\ln \left(1+e^{0}\right)+C \Righ... | \sqrt{1+\ln\frac{1+e^{x^{2}}}{2}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,332 |
Example 4.6. Find a particular integral of the equation
$$
y^{\prime} \sin ^{2} x \ln y+y=0,
$$
satisfying the initial conditions $y \left(\frac{\pi}{4}\right)=1$. | Solution. Let's find the general integral of the given equation. For this, we will separate the variables:
$$
\sin ^{2} x \ln y d y+y d x=0
$$
or
$$
\frac{\ln y}{y} d y=-\frac{d x}{\sin ^{2} x} .
$$
Integrating, we get
$$
\frac{\ln ^{2} y}{2}=\operatorname{ctg} x+C
$$
This is the general integral of the given equ... | \frac{\ln^{2}y}{2}=\operatorname{ctg}x-1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,333 |
Example 4.7. Find the general integral of the equation
$$
\left(x^{2}+y^{2}\right) d x-x y d y=0
$$ | Solution. Let's solve the equation with respect to the derivative $\frac{d y}{d x}$:
$$
y^{\prime}=\frac{x^{2}+y^{2}}{x y} \text {. }
$$
By dividing the numerator and the denominator of the right-hand side of the equation by $x^{2}$, we get:
$$
y^{\prime}=\frac{1+\frac{y^{2}}{x^{2}}}{\frac{y}{x}}
$$
i.e., $y^{\prim... | Ce^{\frac{y^{2}}{2x^{2}}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,334 |
Example 4.8. Consider the following example:
$$
\frac{d y}{d x}=\frac{x^{2}+y^{2}}{2 x y}
$$ | Solution. By making the substitution $y=x u$, we reduce the equation to the form:
$$
x \frac{d u}{d x}+u=\frac{1+u^{2}}{2 u}
$$
From this,
$$
x \frac{d u}{d x}=\frac{1+u^{2}}{2 u}-u=\frac{1-u^{2}}{2 u}
$$
and therefore
$$
\begin{gathered}
\frac{d x}{x}=\frac{2 u d u}{1-u^{2}} ; \ln |x|=-\ln \left|1-u^{2}\right|+\l... | Ce^{\frac{\phi}{k}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,335 |
Example 4.9. Find the general solution of the equation
$$
y^{\prime}-y \tan x=\sin x
$$ | Solution. Let $y=u v$, then $y^{\prime}=u^{\prime} v+u v^{\prime}$ and the given equation takes the form
$$
u^{\prime} v+u v^{\prime}-u v \operatorname{tg} x=\sin x \text {, }
$$
or
$$
u^{\prime} v+u\left(v^{\prime}-v \operatorname{tg} x\right)=\sin x \text {. }
$$
Solving the equation $U^{\prime}-U \operatorname{t... | \frac{\sin^{2}x}{2}+C\frac{1}{\cosx} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,336 |
Example 4.10. Solve the differential equation:
$$
y^{\prime}+y \cdot \tan x=\cos ^{2} x
$$ | Solution. Assuming $y=u v$, we transform this equation into the form
$$
v\left[u^{\prime}+u \operatorname{tg} x\right]+u u^{\prime}=\cos ^{2} x
$$
Setting the square bracket to zero:
$$
u^{\prime}+u \operatorname{tg} x=0
$$
we get that
$$
\frac{d u}{u}=-\operatorname{tg} x d x ; \quad \ln u=\ln \cos x, \quad u=\co... | \cosx(\sinx+C) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,337 |
Example 4.14. Find the general solution of the equation
$$
y^{\prime \prime}=1-x^{2}
$$ | Solution. Let $\frac{d y}{d x}=z$, then the given equation can be written as:
$$
\frac{d z}{d x}=1-x^{2}, \text { or } d z=\left(1-x^{2}\right) d x
$$
After integration, we find
$$
z=x-\frac{x^{3}}{3}+C_{1}
$$
Hence,
$$
\frac{d y}{d x}=x-\frac{x^{3}}{3}+C_{1}, \text { or } \quad d y=x-\frac{x^{3}}{3}+C_{1} d x .
$... | \frac{x^{2}}{2}-\frac{x^{4}}{12}+C_{1}x+C_{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,338 |
Example 4.15. Find the particular solution of the equation
$$
y^{\prime \prime}=\sin x-1,
$$
satisfying the given initial conditions $y(0)=-1$, $y^{\prime}(0)=1$. | Solution. We have $\frac{d y}{d x}=z$, hence
$$
\begin{gathered}
\frac{d z}{d x}=\sin x-1, d z=(\sin x-1) d x \\
z=-\cos x-x+C_{1}
\end{gathered}
$$
Therefore,
$$
y^{\prime}=-\cos x-x+C_{1} .
$$
Using the initial condition $y^{\prime}(0)=1$, we get
$$
1=-\cos 0+C_{1},
$$
from which $C_{1}=2$. Thus,
$$
\frac{d y}... | -\sinx-\frac{x^{2}}{2}+2x-1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,339 |
Example 4.16. Find the general solution of the equation
$$
y^{\prime \prime}-5 y^{\prime}-6 y=0
$$ | Solution. Let's write down the characteristic equation; for this, we replace the function $y$ and its derivatives $y^{\prime}$ and $y^{\prime \prime}$ with the corresponding powers of $r: r^{0}=1, r u r^{2}$. Then we get
$$
r^{2}-5 r-6=0
$$
from which $r_{1}=-1, r_{2}=6$. Since the roots of the characteristic equatio... | C_{1}e^{-x}+C_{2}e^{6x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,340 |
Example 4.17. Find the general solution of the equation
$$
y^{\prime \prime}-4 y^{\prime}+4 y=0
$$ | Solution. We form the characteristic equation
$$
r^{2}-4 r+4=0
$$
from which we find $r_{1,2}=2$. The characteristic equation has equal real roots, so according to formula (4), the general solution is written as follows:
$$
y=e^{2 x}\left(C_{1}+C_{2} x\right)
$$ | e^{2x}(C_{1}+C_{2}x) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,341 |
Example 4.18. Find the general solution of the equation
$$
y^{\prime \prime}+9 y=0
$$ | Solution. This equation corresponds to the characteristic equation
$$
r^{2}+9=0
$$
which has two imaginary conjugate roots $r_{1,2}= \pm 3 i$. Using formula (5) with $\alpha=0$ and $\beta=3$, we obtain the general solution
$$
y=C_{1} \cos 3 x+C_{2} \sin 3 x
$$ | C_{1}\cos3x+C_{2}\sin3x | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,342 |
Example 4.19. Find the general solution of the equation
$$
y^{\prime \prime}+6 y^{\prime}+25 y=0 \text {. }
$$ | Solution. The characteristic equation
$$
r^{2}+6 r+25=0
$$
has two complex conjugate roots $r_{1,2}=-3 \pm 4 i$. Using formula (5) with $\alpha=-3$ and $\beta=4$, we obtain the general solution
$$
y=e^{-3 x}\left(C_{1} \cos 4 x+C_{2} \sin 4 x\right)
$$ | e^{-3x}(C_{1}\cos4x+C_{2}\sin4x) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,343 |
Example 4.20. Find the particular solution of the equation $y^{\prime \prime}-3 y^{\prime}+2 y=0$, satisfying the given initial conditions $y(0)=1$, $y^{\prime}(0)=-1$. | Solution. Let's write down the characteristic equation
$$
r^{2}-3 r+2=0
$$
its roots are $r_{1}=1, r_{2}=2$. Therefore, the general solution has the form:
$$
y=C_{1} e^{x}+C_{2} e^{2 x}
$$
Next, using the initial conditions, we determine the values of the constants $C_{1}$ and $C_{2}$. For this, we substitute the g... | 3e^{x}-2e^{2x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,344 |
Example 4.21. Find the general solution of the equation
$$
y^{\prime \prime}+4 y^{\prime}+3 y=\left(8 x^{2}+84 x\right) e^{x}
$$ | ## Solution:
1. Find the general solution $\bar{y}$ of the corresponding homogeneous equation
$$
y^{\prime \prime}+4 y^{\prime}+3 y=0
$$
Solving the characteristic equation
$$
r^{2}+4 r+3=0
$$
we obtain the roots $r_{1}=-3, r_{2}=-1$. Therefore,
$$
\bar{y}=C_{1} e^{-3 x}+C_{2} e^{-x}
$$
2. Proceed to finding a p... | C_{1}e^{-3x}+C_{2}e^{-x}+(x^{2}+9x-7)e^{x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,345 |
Example 4.22. Find the general solution of the equation
$$
y^{\prime \prime}+6 y^{\prime}+9 y=14 e^{-3 x} .
$$ | Solution. 1. Find $\bar{y}$.
The characteristic equation $r^{2}+6 r+9=0$, has roots $r_{1}=r_{2}=-3$. Therefore,
$$
\bar{y}=\left(C_{1}+C_{2} x\right) e^{-3 x}
$$
2. Now find $y^{*}$. Here the right-hand side has the form (3): $n=0$, $P_{0}=14, k=-3$. Since $k=-3$ is a double root of the characteristic equation, the... | (C_{1}+C_{2}x)e^{-3x}+7x^{2}e^{-3x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,346 |
Example 4.23. Find the general solution of the equation
$$
y^{\prime \prime}-4 y^{\prime}+5 y=2 \cos x+6 \sin x
$$ | Solution. 1. Find $\bar{y}$. The characteristic equation
$$
r^{2}-4 r+5=0
$$
has roots $r_{1,2}=2 \pm i$. Therefore,
$$
\bar{y}=e^{2 x}\left(C_{1} \cos x+C_{2} \sin x\right)
$$
2. We will now find $y^{*}$. Here the right-hand side $f(x)$ has the form (4): $a=2, b=6, \lambda= \pm i$. The numbers $\pm i$ are not root... | e^{2x}(C_{1}\cosx+C_{2}\sinx)+\cosx+\frac{1}{2}\sinx | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,347 |
Example 4.24. Find the general solution of the equation
$$
y^{\prime \prime}+4 y=12 \cos 2 x
$$ | Solution. 1. First, let's find $\bar{y}$. The characteristic equation $r^{2}+4=0$ has roots $r_{1,2}= \pm 2 i$. Therefore,
$$
\bar{y}=C_{1} \cos 2 x+C_{2} \sin 2 x
$$
2. Now, we proceed to finding $y^{*}$. Here, the right-hand side $f(x)$ has the form (4): $a=12, b=0, \lambda= \pm 2 i$. Since the numbers $\pm 2 i$ ar... | C_{1}\cos2x+C_{2}\sin2x+3x\sin2x | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,348 |
Example 4.25. Find a particular solution of the equation
$$
y^{\prime \prime}+2 y^{\prime}-8 y=(12 x+20) e^{2 x}
$$
satisfying the initial conditions $y(0)=0, y^{\prime}(0)=1$. | Solution. 1. The characteristic equation $r^{2}+2 r-8=0$ has roots $r_{1}=-4, r_{2}=2$. Therefore,
$$
\bar{y}=C_{1} e^{-4 x}+C_{2} e^{2 x}
$$
2. The right-hand side of the given equation has the form (3): $n=1$, $P_{1}(x)=12 x+20, k=2$. Since $k=2$ is a single root of the characteristic equation, we seek the particul... | \frac{1}{3}e^{-4x}-\frac{1}{3}e^{2x}+(x^{2}+3x)e^{2x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,349 |
Example 5.1. Write the first five terms of the sequence if its $n$-th term $a_{n}$ is given by:
1) $\frac{1}{4 n-1}$
2) $(-1)^{n} \cdot \frac{1}{4 n-1}$
3) $\frac{2^{n}+1}{2^{n}}$
4) $\frac{2+(-1)^{n-1}}{n}$. | Solution. Substitute $1,2, \ldots, 5$ for $n$:
1) $\frac{1}{4 n-1}: \frac{1}{3}, \frac{1}{7}, \frac{1}{11}, \frac{1}{15}, \frac{1}{19}$
2) $(-1)^{n} \cdot \frac{1}{4 n-1}:-\frac{1}{3}, \frac{1}{7},-\frac{1}{11}, \frac{1}{15},-\frac{1}{19}$;
3) $\frac{2^{n}+1}{2^{n}}: \frac{3}{2}, \frac{5}{4}, \frac{9}{8}, \frac{17}{16... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,350 |
Example 5.2. Using the definition directly, show that the series converges, and find its sum.
\[
\sum_{n=1}^{\infty} \frac{1}{n(n+1)}=\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\ldots+\frac{1}{n(n+1)}+\ldots
\] | Solution. By the definition of the partial sum of a series, we have:
$$
\begin{aligned}
& S_{1}=a_{1}=\frac{1}{2} \\
& S_{2}=a_{1}+a_{2}=\frac{1}{2}+\frac{1}{6}=\frac{2}{3} \\
& S_{3}=a_{1}+a_{2}+a_{3}=\frac{2}{3}+\frac{1}{12}=\frac{3}{4} \\
& S_{4}=a_{1}+a_{2}+a_{3}+a_{4}=\frac{3}{4}+\frac{1}{20}=\frac{4}{5}
\end{ali... | 1 | Algebra | proof | Yes | Yes | olympiads | false | 30,351 |
Example 5.3. Using the necessary condition for convergence, show that the series
$$
1+\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\ldots+\frac{n}{n+1}+\ldots
$$
diverges. | Solution. Let's find
$$
\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} \frac{n}{n+1}=1
$$
Thus, the limit of the general term of the series as $n \rightarrow \infty$ is not zero, i.e., the necessary condition for convergence is not satisfied. This means that the given series diverges. | proof | Calculus | proof | Yes | Yes | olympiads | false | 30,352 |
Example 5.4. Investigate the convergence of the series
$$
\frac{1}{5 \cdot 2}+\frac{1}{5 \cdot 2^{2}}+\frac{1}{5 \cdot 2^{3}}+\ldots+\frac{1}{5 \cdot 2^{n}}+\ldots
$$ | Solution. Let's compare the given series with the series
$$
\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\ldots+\frac{1}{2^{n}}+\ldots
$$
The series (*) converges because its terms form a decreasing geometric progression with a common ratio $q=\frac{1}{2}$. Moreover, each term $a_{n}=\frac{1}{5 \cdot 2^{n}}$ of the gi... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,353 |
Example 5.5. Investigate the convergence of the series
$$
1+\frac{1}{\sqrt[3]{2}}+\frac{1}{\sqrt[3]{3}}+\ldots+\frac{1}{\sqrt[3]{n}}+\ldots
$$ | Solution. Let's compare the given series with the harmonic series
$$
1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}+\ldots
$$
Each term \(a_{n}=\frac{1}{\sqrt[3]{n}}\) of the given series, starting from the second, is greater than the corresponding term \(b_{n}=\frac{1}{n}\) of the harmonic series. Since the harmonic s... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,354 |
Example 5.6. Investigate the convergence of the series
$$
1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\ldots+\frac{1}{n^{2}}+\ldots
$$ | Solution. Each term of the series
$$
\frac{1}{2^{2}}+\frac{1}{3^{2}}+\ldots+\frac{1}{n^{2}}+\ldots
$$
is less than the corresponding term of the series
$$
\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\ldots+\frac{1}{n(n+1)}+\ldots
$$
As shown in problem 5.2, the latter series converges. Therefore, the series (*) also co... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,355 |
Example 5.7. Using the D'Alembert's criterion, determine the convergence of the series
$$
\frac{1}{3}+\frac{2}{3^{2}}+\frac{3}{3^{3}}+\ldots+\frac{n}{3^{n}}+\ldots
$$ | Solution. To use the D'Alembert's criterion, we need to know the ( $n+1$ )-th term of the series. It is obtained by substituting $n+1$ for $n$ in the expression for the general term of the series $a_{n}=\frac{n}{3^{n}}$: $a_{n+1}=\frac{n+1}{3^{n+1}}$. Now we find the limit of the ratio of the ( $n+1$ )-th term to the $... | \frac{1}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,356 |
Example 5.8. Using the D'Alembert's criterion, investigate the convergence of the series
$$
\frac{1}{10}+\frac{1 \cdot 2}{10^{2}}+\frac{1 \cdot 2 \cdot 3}{10^{3}}+\ldots+\frac{n!}{10^{n}}+\ldots
$$ | Solution. Knowing $a_{n}=\frac{n!}{10^{n}}$, let's find the $(n+1)$-th term of the series: $a_{n+1}=\frac{(n+1)!}{10^{n+1}}$. We will compute
$$
\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=\lim _{n \rightarrow \infty} \frac{(n+1)!}{10^{n+1}}: \frac{n!}{10^{n}}=\lim _{n \rightarrow \infty} \frac{(n+1)!10^{n}}{n!... | \infty | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,357 |
Example 5.9. Based on the D'Alembert's criterion, investigate the convergence of the series
$$
3+\frac{3^{2}}{2^{2}}+\frac{3^{3}}{3^{3}}+\frac{3^{4}}{4^{4}}+\ldots+\frac{3^{n}}{n^{n}}+\ldots
$$ | Solution. Knowing the $n$-th term of the series $a_{n}=\frac{3^{n}}{n^{n}}$, we write the $(n+1)$-th term: $a_{n+1}=\frac{3^{n+1}}{(n+1)^{n+1}} \cdot$ From this,
\[
\begin{gathered}
\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=\lim _{n \rightarrow \infty} \frac{3^{n+1}}{(n+1)^{n+1}}: \frac{3^{n}}{n^{n}}=\lim _{n... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,358 |
Example 5.10. Using the Leibniz criterion, investigate the convergence of the alternating series
$$
1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+(-1)^{n-1} \frac{1}{n}+\ldots
$$ | Solution. Since the terms of the given series decrease monotonically in absolute value:
$$
1>\frac{1}{2}>\frac{1}{3}>\frac{1}{4}>\ldots
$$
and the general term approaches zero as $n \rightarrow \infty$:
$$
\lim _{n \rightarrow \infty} a_{n}=0
$$
then by the Leibniz criterion, the series converges. | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,359 |
Example 5.11. Estimate the error made when replacing the sum of the series
$$
1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+(-1)^{n-1} \frac{1}{n}+\ldots
$$
with the sum of its first four terms. | Solution. The given alternating series converges (see problem 5.10). The error $\Delta S_{4}$, obtained by replacing the sum $S$ of this series with the sum of its first four terms, is less than the absolute value of the fifth term: $\Delta S_{4}<0.2$.
## 5.4. Absolute and Conditional Convergence of an Alternating Ser... | 0.2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,360 |
Example 5.12. Investigate the convergence of the series
$$
1-\frac{1}{2!}+\frac{1}{3!}-\frac{1}{4!}+\ldots+(-1)^{n-1} \frac{1}{n!}+\ldots
$$ | Solution. Consider the series composed of the absolute values of the terms of the given series:
$$
1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\ldots+\frac{1}{n!}+\ldots
$$
By the D'Alembert's ratio test, this series converges, since
$$
l=\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=\lim _{n \rightarrow \infty} \... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,361 |
Example 5.13. Investigate the convergence of the series
$$
\frac{\sin 1}{1^{2}}+\frac{\sin 2}{2^{2}}+\frac{\sin 3}{3^{2}}+\ldots+\frac{\sin n}{n^{2}}+\ldots
$$ | Solution. The series under consideration is alternating, as it contains both positive terms $\frac{\sin 1}{1^{2}}, \frac{\sin 2}{2^{2}}, \frac{\sin 3}{3^{2}}, \frac{\sin 7}{7^{2}}, \ldots$, and negative terms $\frac{\sin 4}{4^{2}}, \frac{\sin 5}{5^{2}}, \frac{\sin 6}{6^{2}}, \ldots$.
The series
$$
\frac{|\sin 1|}{1^{... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,362 |
Example 5.14. Investigate the convergence of the series
$$
1-\frac{1}{\sqrt[3]{2}}+\frac{1}{\sqrt[3]{3}}-\ldots+(-1)^{n-1} \cdot \frac{1}{\sqrt[3]{n}}+\ldots
$$ | Solution. The series
$$
1+\frac{1}{\sqrt[3]{2}}+\frac{1}{\sqrt[3]{3}}+\ldots+\frac{1}{\sqrt[3]{n}}+\ldots
$$
composed of the absolute values of the terms of the given series diverges (see problem 5.5). Therefore, the series (*) is not absolutely convergent. It remains to determine whether it converges (conditionally)... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,363 |
Example 5.15. Investigate the convergence of the series
$$
\sin \frac{\pi}{3}+\sin \frac{2 \pi}{3}+\sin \frac{3 \pi}{3}+\ldots+\sin \frac{n \pi}{3}+\ldots
$$ | Solution. The given alternating series diverges because the necessary condition for convergence is not satisfied: $\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} \sin \frac{n \pi}{3}$ does not exist.
## 5.5. Power Series
A power series is a series of the form
$$
a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{n... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,364 |
Example 5.16. Investigate the convergence of the series
$$
\frac{1}{2} x+\frac{2}{2^{2}} x^{2}+\frac{3}{2^{3}} x^{3}+\frac{4}{2^{4}} x^{4}+\ldots+\frac{n}{2^{n}} x^{n}+\ldots
$$
at the points $x=1, x=3, x=-2$. | Solution. When $x=1$, the given series turns into a numerical series
$$
\frac{1}{2}+\frac{2}{2^{2}}+\frac{3}{2^{3}}+\ldots+\frac{n}{2^{n}}+\ldots
$$
By the D'Alembert's ratio test, this series converges, since
$$
l=\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=\lim _{n \rightarrow \infty} \frac{n+1}{2^{n+1}}: \... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,365 |
Example 5.17. Find the interval of convergence of the power series
$$
1-\frac{x}{2 \cdot 2}+\frac{x^{2}}{3 \cdot 2^{2}}-\frac{x^{3}}{4 \cdot 2^{3}}+\ldots+(-1)^{n} \frac{x^{n}}{(n+1) 2^{n}}+\ldots
$$ | Solution. Let's form a series of the absolute values of the terms of the given series:
$$
1+\frac{|x|}{2 \cdot 2}+\frac{|x|^{2}}{3 \cdot 2^{2}}+\frac{|x|^{3}}{4 \cdot 2^{3}}+\ldots+\frac{|x|^{n}}{(n+1) 2^{n}}+\ldots
$$
According to the D'Alembert's ratio test, the resulting positive series converges (absolutely) for ... | -2<x\leq2 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,366 |
Example 5.18. Find the region of convergence of the power series
$$
x+2!x^{2}+3!x^{3}+\ldots+n!x^{n}+\ldots
$$ | Solution. Here $u_{n}=n!|x|^{n}, u_{n+1}=(n+1)!|x|^{n+1}$.
From this,
$$
l=\lim _{n \rightarrow \infty} \frac{u_{n+1}}{u_{n}}=\lim _{n \rightarrow \infty} \frac{(n+1)!|x|^{n+1}}{n!|x|^{n}}=|x| \lim _{n \rightarrow \infty}(n+1)
$$
Thus,
$$
l=\begin{array}{ccc}
\infty & \text { when } & x \neq 0, \\
0 & \text { when ... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,367 |
Example 5.19. Find the interval of convergence of the power series
$$
1+\frac{3}{1!} x+\frac{3^{2}}{2!} x^{2}+\frac{3^{3}}{3!} x^{3}+\ldots+\frac{3^{n}}{n!} x^{n}+\ldots
$$ | Solution. We have $u_{n}=\frac{3^{n}}{n!}|x|^{n}, u_{n+1}=\frac{3^{n+1}}{(n+1)!}|x|^{n+1}$.
From this,
$$
l=\lim _{n \rightarrow \infty} \frac{u_{n+1}}{u_{n}}=\lim _{n \rightarrow \infty} \frac{3^{n+1} \cdot n!|x|^{n+1}}{(n+1)!3^{n}|x|^{n}}=|x| \lim _{n \rightarrow \infty} \frac{3}{n+1}=0<1
$$
Therefore, for any fin... | (-\infty,\infty) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,368 |
Example 5.20. Expand the function $f(x)=e^{3 x}$ into a Taylor series. | Solution. Let's compute the values of the given function and its successive derivatives at $x=0$:
$$
\begin{array}{rr}
f(x)=e^{3 x}, & f(0)=1, \\
f^{\prime}(x)=3 e^{3 x}, & f^{\prime}(0)=3 \\
f^{\prime \prime}(x)=3^{2} e^{3 x}, & f^{\prime \prime}(0)=3^{2} \\
f^{\prime \prime \prime}(x)=3^{3} e^{3 x}, & f^{\prime \pri... | e^{3x}=1+\frac{3}{1!}x+\frac{3^{2}}{2!}x^{2}+\frac{3^{3}}{3!}x^{3}+\ldots+\frac{3^{n}}{n!}x^{n}+\ldots | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,369 |
Example 5.21. Expand the function $f(x)=\ln (1-2 x)$ into a Taylor series. | Solution. Replacing $x$ with $-2x$ in the expansion (7), we get:
$$
\ln (1-2 x)=(-2 x)-\frac{(-2 x)^{2}}{2}+\frac{(-2 x)^{3}}{3}-\ldots+(-1)^{n-1} \frac{(-2 x)^{n}}{n}+\ldots,
$$
or
$$
\ln (1-2 x)=-2 x-\frac{2^{2}}{2} x^{2}-\frac{2^{3}}{3} x^{3}-\ldots-\frac{2^{n}}{n} x^{n}-\ldots
$$
The expansion (7) is valid in t... | -\frac{1}{2}\leqx<\frac{1}{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,370 |
Example 5.22. Expand the function $f(x)=\cos ^{2} x$ into a Taylor series. | Solution. By a well-known trigonometric formula, we have:
$$
\cos ^{2} x=\frac{1+\cos 2 x}{2}
$$
We expand the function $\cos 2 x$ into a Taylor series, replacing $x$ with $2 x$ in expansion (4):
$$
\cos 2 x=1-\frac{(2 x)^{2}}{2!}+\frac{(2 x)^{4}}{4!}-\ldots+(-1)^{n} \frac{(2 x)^{n}}{(2 n)!}+\ldots
$$
or
$$
\cos 2... | 1-\frac{2}{2!}x^{2}+\frac{2^{3}}{4!}x^{4}-\ldots+(-1)^{n}\frac{2^{2n-1}}{(2n)!}x^{2n}+\ldots | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,371 |
Example 5.23. Using differentiation and integration, find the Taylor series expansion for the given function $f(x)=\operatorname{arc} \operatorname{tg} x$ and specify the intervals in which these expansions are valid. | Solution. Let's write the expression of the given function in the form of an integral:
$$
\operatorname{arctg} x=\int_{0}^{x} \frac{d t}{1+t^{2}}
$$
Expand the integrand function $f(t)=\frac{1}{1+t^{2}}$ into a Taylor series. For this, replace $x$ with $-t^{2}$ in expansion (6):
$$
f(t)=\frac{1}{1-\left(-t^{2}\right... | \operatorname{arctg}x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\ldots+(-1)^{n}\frac{x^{2n+1}}{2n+1}+\ldots | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,372 |
Example 5.24. Calculate $\sin 18^{\circ}$, using the first two terms of series (3), and estimate the resulting error. | Solution. Since the Taylor series expansion of $\sin x$ is valid for any $x$, in particular, for $x=\frac{\pi}{10}$ we have
$$
\sin \frac{\pi}{10}=\frac{\pi}{10}-\frac{\frac{\pi}{10}^{3}}{3!}+\frac{\frac{\pi}{10}^{5}}{5!}-\ldots
$$
The obtained series is alternating. By limiting ourselves to the first two terms of th... | 0.3091 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,373 |
Example 5.25. Calculate $e^{2}$ with an accuracy of 0.01. | Solution. Using expansion (2), for $x=2$ we get
$$
e^{2}=1+2+\frac{2^{2}}{2!}+\frac{2^{3}}{3!}+\ldots+\frac{2^{n}}{n!}+\ldots
$$
It remains to determine how many terms of this series need to be taken to obtain the value of $e^{2}$ with the required accuracy. Let the desired number of terms be $k$. This means that the... | 7.38 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,374 |
Example 5.26. Compute $\int_{0}^{1} \frac{\sin x}{x} d x$ with an accuracy of 0.01. | Solution. This definite integral can only be calculated approximately. For this, we expand the integrand function into a Taylor series:
$$
\frac{\sin x}{x}=1-\frac{x^{2}}{3!}+\frac{x^{4}}{5!}-\frac{x^{6}}{7!}+\ldots
$$
Hence,
$$
\begin{aligned}
& \int_{0}^{1} \frac{\sin x}{x} d x=\int_{0}^{1} 1-\frac{x^{2}}{3!}+\fra... | 0.94 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,375 |
4.4.1. By first dividing the integration interval into 10 equal parts and then into 20 parts, find the approximate integrals $J_{10}$ and $J_{20}$. Determine the accuracy using the difference $\varepsilon=\left|J_{10}-J_{20}\right|$.
$$
J=\int_{-1}^{9} \sqrt{x^{2}+4} d x
$$
a) using the trapezoidal rule;
b) using Si... | Solution. We have the integrand function $y=\sqrt{x^{2}+4}$. Let's construct an auxiliary table.
| $x_{i}$ | $x_{i}^{2}$ | $x_{i}^{2}+4$ | When divided into 10 parts $y_{i}=\sqrt{x_{i}^{2}+4}$ | When divided into 20 parts $y_{i}=\sqrt{x_{i}^{2}+4}$ |
| :---: | :---: | :---: | :---: | :---: |
| -1 | 1 | 5 | $y_{0}=2.23... | 0.08926 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,376 |
5.2.1. Derive the equation of the tangent plane and the equation of the normal to the surface $4 z=x y-2 x-4 y+8$ at the point $M(-4, -2, 8)$. | Solution. Let's check if point $M$ belongs to the surface:
$$
4 \cdot 8=(-4) \cdot(-2)-2 \cdot(-4)-4(-2)+8,32=32
$$
thus, point $M$ belongs to the surface.
The equation of the tangent plane is:
$$
z-z_{M}=\frac{\partial z}{\partial x}_{M}\left(x-x_{M}\right)+\frac{\partial z}{\partial y}_{M}\left(y-y_{M}\right)
$$
... | x+2y+0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,377 |
5.2.2. Find the gradient and the directional derivative in the direction $\bar{a}=4 \bar{i}-2 \bar{j}$ of the function $z=\ln \left(4 x^{2}+2 y^{2}\right)$ at the point $A(-2 ; 4)$. | Solution. The gradient of the function $z=f(x, y)$ is:
$$
\overline{\operatorname{grad}} z=\frac{\partial z}{\partial x} \bar{l}+\frac{\partial z}{\partial y} \cdot \bar{j}
$$
Let's find the partial derivatives:
$$
\begin{gathered}
\frac{\partial z}{\partial x}=\left(\ln \left(4 x^{2}+2 y^{2}\right)\right)_{x}=\frac... | -\frac{\sqrt{5}}{15} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,378 |
5.2.3. Find the maximum and minimum values of the function
$$
z=4 x^{2}+y^{2}-16 x-4 y+20
$$
in the closed domain $D$, defined by the inequalities:
$$
x \geq 0, \quad x-2 y \leq 0, \quad x+y-6 \leq 0 .
$$ | ## Solution.
a) Let's find the partial derivatives and set them to zero (necessary conditions for extremum):
$$
\begin{gathered}
\frac{\partial z}{\partial x}=\left(4 x^{2}+y^{2}-16 x-4 y+20\right)_{x}^{\prime}=8 x-16 \\
\frac{\partial z}{\partial x}=0 ; 8 x-16=0 ; x_{0}=2 \\
\frac{\partial z}{\partial y}=\left(4 x^{... | z_{\text{}}=0,z_{\text{max}}=32 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,379 |
6.2.1. Find the integral $\iiint_{V} x d x d y d z$, if the body $V$ is bounded by the planes: $x=0, y=0, x+4 y-2 z=0$ and $x+y+z-6=0$. | Solution. The body V is bounded by the coordinate planes $x \mathrm{Oz}$ and $y \mathrm{Oz}$, and from below and above by the planes: $z=\frac{1}{2}(x+4 y)$ and $z=6-x-y$. Let's find the line of intersection of these planes:
$$
\begin{aligned}
& z=\frac{1}{2}(x+4 y) \\
& z=6-x-y
\end{aligned}
$$
or
$$
\frac{1}{2}(x+... | 8 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,380 |
6.2.2. Find the volume of the body bounded by the surfaces: $2 z=x^{2}+y^{2} ; z=2 ; x=0 ; y=2 x$. | Solution. The body is bounded from below by the surface $z=\frac{1}{2}\left(x^{2}+y^{2}\right)$, and from above by $-z=2$. On the plane $x O y$, we construct the region $D$ (Fig. 63).
Fig. ... | 1.704\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,381 |
6.3.2. Compute the line integral
$$
J=\int_{C}(2 x+4 z) d x+(2 y+2 z) d y-12 z d z
$$
along one turn of the helical line $C$ :
$$
\begin{aligned}
& x=2 \cos 4 t, \\
& y=2 \sin 4 t, \\
& z=6 t, \quad 0 \leq t \leq 2 \pi
\end{aligned}
$$ | Solution.
$$
\begin{gathered}
J=\int_{0}^{2 \pi}((4 \cos 4 t+24 t)(-2 \cdot 4 \sin 4 t) d t+ \\
+(4 \sin 4 t+12 t) \cdot(2 \cdot 4 \cos 4 t) d t-12 \cdot 6 \cdot t \cdot 6 d t)= \\
=16 \int_{0}^{2 \pi}(-2 \sin 4 t \cos 4 t-12 t \sin 4 t+2 \sin 4 t \cdot \cos 4 t+6 t \cos 4 t-27 t) d t= \\
=16 \int_{0}^{2 \pi}(-12 t \s... | 96\pi(1-9\pi) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,382 |
7.2.1. Verify that the field $\bar{F}(x, y, z)=3 z \bar{\iota}+5 z \bar{j}+(3 x+5 y) \bar{k}$ is potential and find its potential. | Solution. A necessary and sufficient condition for the potentiality of the field $\bar{F}(x, y, z)$ is the equality to zero of the rotor (vortex) of this field: $\operatorname{rot} \bar{F}=0$. We find the rotor vector of the field $\bar{F}=P \bar{\iota}+Q \bar{j}+R \bar{k}=3 z \bar{\imath}+5 z \bar{j}+(3 x+5 y) \bar{k}... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,383 |
9.3.1. Expand the function $f(x)$ into a Fourier series in the given interval:
$$
f(x)=(x-4)^{2} \text { in the interval }(0 ; 4) .
$$ | Solution. We will extend (continue) the given function to the segment $[-4 ; 0]$ in an even manner and expand it into a Fourier series, considering it defined on the segment $[-4; 4]$, where it will be even (Fig. 67). Then its Fourier series coefficients will be:
$$
\frac{a_{0}}{2}+\sum_{m=1}^{\infty} a_{m} \cdot \cos... | \frac{16}{3}+\frac{64}{\pi^{2}}\cdot\sum_{=1}^{\infty}\frac{1}{^{2}}\cdot\cos\frac{\pix}{4} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,385 |
11.2.1. Solve the differential equation using the operational method: $x^{\prime \prime}-x^{\prime}-20 \cdot x=\mathrm{e}^{-t}, x(0)=0, x^{\prime}(0)=4$. | Solution. We transition from originals to images of functions.
Let $x(t) \risingdotseq X(p)=x$,
$$
\begin{aligned}
& x^{\prime}(t) \risingdotseq p \cdot X(p)-x_{0}=p \cdot X-0=p \cdot X \\
& x^{\prime \prime}(t) \risingdotseq p^{2} \cdot X(p)-p \cdot x_{0}-x_{0}^{\prime}=p^{2} \cdot X-4 .
\end{aligned}
$$
Then, taki... | x()=-\frac{1}{18}\cdote^{-}+\frac{25}{54}\cdote^{5}-\frac{11}{27}\cdote^{-4} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,386 |
Example 6.1. A ball is randomly drawn from an urn containing 4 white, 9 black, and 7 red balls. What is the probability of drawing a white ball? | Solution. Here, the elementary outcome is the extraction of any ball from the urn. The number of all such outcomes is equal to the number of balls in the urn, i.e., \( n=20 \). The number of outcomes favorable to the appearance of a white ball (event \( A \)), obviously, is equal to the number of white balls in the urn... | \frac{1}{5} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 30,387 |
Example 6.2. A die is rolled twice. What is the probability that the sum of the points rolled will be eight? | Solution. Let $A_{ij}$ be the event that $i$ points fall on the first throw, and $j$ points fall on the second throw. Then the 36 events
$$
\begin{array}{cccc}
A_{11}, & A_{12}, & \ldots, & A_{16} \\
A_{21}, & A_{22}, & \ldots, & A_{26} \\
\ldots, & \ldots, & \ldots, & \ldots, \\
A_{61}, & A_{62}, & \ldots, & A_{66}
\... | \frac{5}{36} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 30,388 |
Example 6.3. A square is inscribed in a circle. What is the probability that a point randomly placed in the circle will be inside the square? | Solution. The area of the circle $S=\pi r^{2}$, the area of the square $S_{0}=2 r^{2}$, where $r$ is the radius of the circle (Fig. 69). From here, using formula (2), we find the required probability:
$$
p=\frac{S_{0}}{S}=\frac{2 r^{2}}{\pi r^{2}}=\frac{2}{\pi}
$$
. The
. | ## Solution.
The first method. Consider the opposite event $\bar{B}$, which means the failure of both machines during time $t$. Obviously, the event $\bar{B}$ is the conjunction of events $\bar{A}_{1}$ and $\bar{A}_{2}$ - the failures of the first and second machines, i.e., $\bar{B}=\bar{A}_{1} \bar{A}_{2}$. Since eve... | 0.98 | Other | math-word-problem | Yes | Yes | olympiads | false | 30,393 |
Example 6.8. When the voltage increases, a break in the electrical circuit may occur due to the failure of one of three elements connected in series; the probabilities of failure of the elements are 0.2; 0.3; 0.4, respectively. Determine the probability that the circuit will not break. | Solution. Let events $A_{1}, A_{2}, A_{3}$ denote the failure of the first, second, and third elements, respectively. Their probabilities are given as: $P\left(A_{1}\right)=0.2$; $P\left(A_{2}\right)=0.3$; $P\left(A_{3}\right)=0.4$. Then the probabilities of the opposite events $\bar{A}_{1}, \bar{A}_{2}, \bar{A}_{3}$ (... | 0.336 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 30,394 |
Example 6.9. In an urn, there are 6 black, 5 red, and 4 white balls. Three balls are drawn sequentially. Find the probability that the first ball will be black, the second - red, and the third - white. | Solution. Consider the following events: $A$ - the first ball drawn is black, $B$ - the second ball is red, $C$ - the third ball is white. Let $D$ be the event that the balls are drawn in the sequence: black, red, white. Obviously, $D=A \cdot B \cdot C$.
By formula (4) we have:
$$
P(D)=P(A) \cdot P(B / A) \cdot P(C /... | \frac{4}{91} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 30,395 |
Example 6.10. A factory produces a certain type of product; each product has a defect with a probability of $p_{1}=0.1$. Each product is inspected by one inspector; he detects an existing defect with a probability of $p_{2}=0.8$, and if the defect is not detected, the product is released into the finished product. In a... | Solution. Consider the following events:
$B_{1}$ - the item has a defect;
$B_{2}$ - the inspector will detect the existing defect;
$B_{3}$ - the inspector will reject an item that does not have a defect.
According to the problem, $\quad P\left(B_{1}\right)=p_{1}=0.1 ; \quad P\left(B_{2}\right)=p_{2}=0.8 ;$ $P_{\bar... | 0.27,0.02,0.35 | Other | math-word-problem | Yes | Yes | olympiads | false | 30,396 |
Example 6.11. There are three urns with balls. The first urn contains 4 white and 5 black balls, the second - 5 white and 4 black balls, the third - 6 white balls. Someone randomly selects one of the urns and draws a ball from it. Find the probability that: a) the ball will be white; b) the white ball is drawn from the... | ## Solution:
a) Let $A$ be the event that a white ball is drawn. Consider three hypotheses:
$$
\begin{aligned}
& H_{1} \text { - the first urn is chosen; } \\
& \mathrm{H}_{2} \text { - the second urn, } \text {; } \\
& H_{3} \text { - the third urn, . . }
\end{aligned}
$$
Since the urn from which the ball is drawn ... | \frac{5}{18} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 30,397 |
Example 6.12. Two signals $A$ and $B$ are transmitted over a communication line with probabilities 0.72 and 0.28, respectively. Due to interference, $\frac{1}{6}$ of the $A$-signals are distorted and received as $B$-signals, and $\frac{1}{7}$ of the transmitted $B$-signals are received as $A$-signals.
a) Determine the... | ## Solution:
a) Let event $A$ be the appearance of an $A$-signal at the receiving point. Introduce hypotheses: $H_{A}$ - signal $A$ is transmitted, $H_{B}$ - signal $B$ is transmitted. According to the problem, $P\left(H_{A}\right)=0.72 ; P\left(H_{B}\right)=0.28$.
The probability that an $A$-signal is received given... | \frac{0.6}{0.64} | Other | math-word-problem | Yes | Yes | olympiads | false | 30,398 |
Example 6.14. There are three batches of parts, each containing 30 parts. The number of standard parts in the first, second, and third batches is $20, 15, 10$, respectively. A part is randomly drawn from a randomly selected batch and turns out to be standard. Then, a second part is randomly drawn from the same batch an... | Solution. Let $A$ be the event that a standard part was drawn in each of the two trials.
We can make three assumptions (hypotheses): $H_{1}$ - parts were drawn from the first batch; $H_{2}$ - parts were drawn from the second batch; $H_{3}$ - parts were drawn from the third batch.
Since the parts were drawn from a ran... | \frac{9}{68} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 30,400 |
Example 6.15. Among the parts processed by a worker, on average $4 \%$ are non-standard. Find the probability that among 30 parts taken for testing, two parts will be non-standard. What is the most probable number of non-standard parts in the sample of 30 parts and what is its probability? | Solution. Here, the experiment involves checking each of the 30 parts for quality. Event $A$ - the appearance of a non-standard part; its probability $P=0.04$, then $q=0.96$. From this, using the Bernoulli formula, we find:
$$
P_{30}(2)=C_{30}^{2}(0.04)^{2}(0.96)^{28} \approx 0.202
$$
The most probable number of non-... | 0.202,1,0.305 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 30,401 |
Example 6.16. The probability of hitting the target with one shot is 0.8. Find the probability that in a series of four shots there will be: a) at least one hit; b) no less than three hits; c) no more than one hit. | Solution. Here $n=4, p=0.8, q=0.2$. a) Let's find the probability of the opposite event - in a series of four shots, there is not a single hit on the target:
$$
P_{4}(0)=C_{4}^{0} p^{0} q^{4}=0.2^{4}=0.0016
$$
From this, we find the probability of at least one hit on the target:
$$
P_{4}(k \geq 1)=1-0.0016=0.9984.
$... | 0.9984,0.8192,0.2576 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 30,402 |
Example 6.17. The probability of manufacturing a part of the highest grade on a given machine is 0.4. Find the probability that half of 26 randomly selected parts will be of the highest grade. | Solution. It is required to find the probability $P_{13,26}$. Knowing the Laplace theorem, the necessary calculations are performed according to the following scheme:
$$
\begin{aligned}
& p=0.4 ; \quad n p=26 \cdot 0.4=10.4 \\
& q=1-0.4=0.6 ; \quad n p q=10.4 \cdot 0.6=6.24 \\
& n=26 ; \quad \sqrt{n p q}=\sqrt{6.24}=2... | 0.093 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 30,403 |
Example 6.18. When a technological process is established, a factory produces on average $70 \%$ of first-grade products. What is the probability that in a batch of 1000 items, the number of first-grade items is between 652 and 760? | Solution. The number of independent trials $n=1000$ and the probability of the event occurring in a single trial $p=0.7$ are known. It is required to find the probability that the number of occurrences of the event is between $a=652$ and $b=760$. The required probability is found using formula (5)
$$
P(a<m<b) \approx ... | 0.99905 | Other | math-word-problem | Yes | Yes | olympiads | false | 30,404 |
Example 6.19. Buffon tossed a coin 4040 times, with heads appearing 2048 times. Can the obtained deviation in the number of heads from 2020 be considered random, or is it due to a systematic cause? | Solution. The discrepancy between Buffon's empirical frequency and the theoretical one can be considered random if the probability that the deviation of the number of heads from 2020 is equal to or greater than Buffon's in absolute value when a coin is tossed 4040 times is sufficiently large. Let $m$ be the number of h... | 0.3783 | Other | math-word-problem | Yes | Yes | olympiads | false | 30,405 |
Example 6.20. 600 corn seeds are planted with a probability of 0.9 for each seed to germinate. Find the boundary of the absolute value of the deviation of the frequency of germinated seeds from the probability $p=0.9$, if this boundary is to be guaranteed with a probability $P=0.995$. | Solution. We know that if $n$ is the number of independent trials and $p$ is the probability of the event occurring in a single trial, then for any $\varepsilon>0$, the approximate equality holds:
$$
P\left|\frac{m}{n}-p\right|<\varepsilon \approx \Phi \varepsilon \sqrt{\frac{n}{p q}}
$$
where $q=1-p$. In our case, $... | 0.034 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,406 |
Example 6.21. On average, $85 \%$ of the items coming off the conveyor are of the first grade. How many items need to be taken so that with a probability of 0.997, the deviation of the frequency of first-grade items from 0.85 in absolute value does not exceed 0.01? | Solution. Here $p=0.85, q=1-0.85=0.15, \varepsilon=0.01, P=0.997$, $n-$? Since in the equation $P\left|\frac{m}{n}-p\right|<\varepsilon \approx \Phi \varepsilon \sqrt{\frac{n}{p q}}$ the probability $P$ on the left is known, we first solve the equation $\Phi(t)=P$. Let $t_{P}$ be the root of this equation. Then $\varep... | 11475 | Other | math-word-problem | Yes | Yes | olympiads | false | 30,407 |
Example 6.23. The probability of placing an order in a tailor shop is $p<1.0$. A persistent customer visits all the tailor shops in the city until he succeeds. What is the probability that he will succeed no earlier than on the third attempt, if on average the number of attempts is 5. | Solution. The random variable is distributed according to the geometric law. Therefore, $M(X)=5=\frac{1}{p}$, from which $p=0.2$. Let's find $P(1)=p \cdot q^{1-1}=0.2$ and $P(2)=p \cdot q^{2-1}=0.2 \cdot 0.8=0.16$ — the probabilities that the client will achieve success on the first and second attempts, respectively.
... | 0.64 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,409 |
Example 6.24. Five televisions have been put on subscription service. It is known that for a group of five televisions, the expected number of failures per year is one. If the televisions have the same probability of working without failure, what is the probability that at least one repair will be needed within a year? | Solution. The random variable is distributed according to the binomial law, since the trials are independent and the probability $P(A)=$ const. Since the expected number of failed televisions is one, the expected number of good ones is:
$$
M\left(X_{A}\right)=n-M\left(X_{\bar{A}}\right)=5-1=4 .
$$
The parameter $P$ i... | 0.67 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 30,410 |
Example 6.25. A clerk accepts on average two applications per hour. It is assumed that the flow of applications is a simple one. What is the probability of receiving four applications in four hours? | Solution. For the simplest event flow, we can use the Poisson distribution, which will have the form:
$$
P_{n}(m)=\frac{(\lambda t)^{m}}{m!} e^{-\lambda t},
$$
where, according to the condition, $\lambda=2 ; t=4$ and thus $\lambda t=8$, and $m$ is the number of requests.
The probability of the occurrence of at least... | 0.9576 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 30,411 |
Example 6.26. A master, who performs repairs at home, can appear at any time from 10 to 18 hours. The client, having waited until 14 o'clock, left for one hour. What is the probability that the master (his arrival is mandatory) will not find him at home? | Solution. From the analysis of the problem statement, we accept the uniform distribution law for the random arrival time of the master. To determine the boundaries of the time interval, we note that since the master did not arrive by 14:00, he will definitely arrive by 18:00. Therefore, \(a=14\) and \(b=18\), and the c... | \frac{1}{4} | Other | math-word-problem | Yes | Yes | olympiads | false | 30,412 |
Example 6.28. A radio system, having 1000 elements (with a failure rate $\lambda_{i}=10^{-6}$ failures/hour), has passed testing and has been accepted by the customer. It is required to determine the probability of failure-free operation of the system in the interval $t_{1}<\left(t=t_{1}+\Delta t\right)<t_{2}$, where $... | Solution. The failure intensity of a system consisting of $n=1000$ non-redundant elements, each with $\lambda_{i}=10^{-6}$ failures/hour, is $\lambda_{s}=n \cdot \lambda_{i}=10^{-3}$ failures/hour. Given these initial conditions, the probability of system failure over a time $t=1000$ hours is:
$$
P(X>t)=e^{-\lambda t}... | 0.37 | Other | math-word-problem | Yes | Yes | olympiads | false | 30,413 |
Example 6.29. A refrigerator has a constant failure rate equal to $\lambda=10^{-5}$ failures/hour. What is the probability that it will fail after the warranty period $\tau=20000$ hours? | Solution. Given the constant failure intensity, the exponential distribution law can be adopted. Then the probability of failure in the interval $(0<20000)=0.819$.
The expected value of a continuous random variable distributed according to the exponential law is equal to the value inverse to the distribution parameter... | Other | math-word-problem | Yes | Yes | olympiads | false | 30,414 | |
Example 6.30. A random variable $X$ is normally distributed. The mathematical expectation and standard deviation of this variable are 2 and 5, respectively. Find the probability that in the experiment, the RV $X$ will take a value belonging to the interval $(1 ; 4)$. | Solution. We will use the formula:
$$
P(\alpha<X<\beta)=\Phi \frac{\beta-a}{\sigma}-\Phi \frac{\alpha-a}{\sigma}
$$
According to the condition: $\alpha=1, \beta=4, a=2, \sigma=5$, therefore,
$$
P(1<X<4)=\Phi \frac{4-2}{5}-\Phi \frac{1-2}{5}=\Phi(0.4)-\Phi(-0.2)
$$
Since the Laplace function is odd, then
$$
\Phi(-0... | 0.2347 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,415 |
Example 6.31. Three continuous random variables have different distributions: a) uniform; b) exponential; c) normal. For all three distributions, $M\left(X_{i}\right)=\sigma\left(X_{i}\right)=4$.
Find for all distribution laws the probability that in the result of the trial, the RV $X_{i}$ will take a value contained ... | ## Solution.
a) Uniform distribution.
The differential function of the uniform distribution of the random variable $X_{1}$ is:
$$
\begin{aligned}
& 0 \quad \text { for } \quad xb \text {. }
\end{aligned}
$$
The parameters $a$ and $b$ are found from the condition that for a uniform distribution
$$
\begin{aligned}
&... | 0.428,0.2367,0.3785 | Other | math-word-problem | Yes | Yes | olympiads | false | 30,416 |
Example 6.35. Find the confidence interval for estimating the mathematical expectation $a$ of a normal distribution with a reliability of 0.95, knowing the sample mean $\bar{X}_{\mathrm{B}}=10.43$, the sample size $n=100$, and the standard deviation $\sigma=5$. | Solution. We will use the formula
$$
\bar{x}_{\mathrm{B}}-t \cdot \frac{\sigma}{\sqrt{n}}<a<\bar{x}_{\mathrm{B}}+t \cdot \frac{\sigma}{\sqrt{n}}
$$
Let's find $t$. From the relation $2 \Phi(t)=0.95$ we get: $\Phi(t)=0.475$. By the table, we find $t=1.96$. Let's find the accuracy of the estimate
$$
\delta=\frac{t \si... | (9.45<11.41) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,418 |
Example 6.36. Given the sample values $x_{i}$ and corresponding frequencies $n_{i}$ of a quantitative characteristic $X$:
| $x_{i}$ | 10 | 15 | 20 | 25 | 30 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $n_{i}$ | 6 | 16 | 50 | 24 | 4 |
Find the sample mean, variance, and standard deviation using the method of... | Solution. We will form a calculation table for which:
1) we will write the variants $x_{i}$ in the first column;
2) we will write the frequencies $n_{i}$ in the second column;
3) as the false zero $C$, we will choose the variant 20 (this variant has the highest frequency); in the cell of the third column, which belong... | \bar{x}_{\mathrm{B}}=20.2,\sigma_{\mathrm{B}}=4.47 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,419 |
Example 6.37. At a significance level of $\alpha=0.05$, test the hypothesis of a normal distribution of the general population, given the empirical and theoretical frequencies:
| $n_{i}$ | 6 | 12 | 16 | 40 | 13 | 8 | 5 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $n_{i}^{\prime}$ | 4 | 11 | 15 | 43 | ... | Solution. The number of different variants $m$ is 7, so the number of degrees of freedom of the $\chi^{2}$ distribution is $7-3=4$. According to the table of critical points of the $\chi^{2}$ distribution, at the significance level $\alpha=0.05$ and 4 degrees of freedom, we find $\chi_{\text {crit. }}^{2}=9.5$. We will... | \chi_{\text{obs.}}^{2}<\chi_{\text{crit.}}^{2} | Other | math-word-problem | Yes | Yes | olympiads | false | 30,420 |
Example 6.38. A statistical distribution of a sample is given:
| $x_{i}$ | 1.6 | 3.0 | 4.4 | 5.8 | 7.2 | 6.6 | 10.0 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $n_{i}$ | 3 | 7 | 15 | 35 | 22 | 13 | 5 | | ## Solution.
1. We will find the sample mean, variance, and standard deviation using the method of products. To do this, we will create Table 1.
Table 1
| $x_{i}$ | $n_{i}$ | $u_{i}$ | $n_{i} u_{i}$ | $n_{i} u_{i}^{2}$ | $n_{i}\left(u_{i}+1\right)^{2}$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| 1.6 | 3 | ... | \ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,421 |
12.1.1. In a box, there are 6 identical pairs of black gloves and 4 identical pairs of beige gloves. Find the probability that two randomly drawn gloves form a pair. | Solution. Consider the event $A$ - two randomly drawn gloves form a pair; and hypotheses: $B_{1}$ - a pair of black gloves is drawn, $B_{2}$ - a pair of beige gloves is drawn, $B_{3}$ - the drawn gloves do not form a pair.
The probability of hypothesis $B_{1}$ by the multiplication theorem is the product of the probab... | \frac{47}{95} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 30,422 |
12.1.2. In an urn, there are 3 white balls and 5 black balls. Three balls are drawn at random one by one, and after each draw, the ball is returned to the urn. Find the probability that among the drawn balls there will be:
a) exactly two white balls, b) at least two white balls. | Solution. We have a scheme with replacement, i.e., the composition of balls does not change each time:
a) when drawing three balls, two of them should be white, and one should be black. In this case, the black ball can be either the first, the second, or the third. Applying the theorems of addition and multiplication ... | \frac{162}{512} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 30,423 |
12.1.3. An urn contains 6 white and 5 black balls. Three balls are randomly drawn sequentially without replacement. Find the probability that the third ball drawn will be white. | Solution. If the third ball must be white, then the first two balls can be white, or white and black, or black and white, or black, i.e., there are four groups of non-
mutually exclusive events. Applying the multiplication theorem of probabilities to them, we get:
$$
\begin{aligned}
P & =P_{16} \cdot P_{2 \sigma} \cdo... | \frac{6}{11} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 30,424 |
12.2.1. The distribution law of a discrete random variable $X$ is as follows:
| $x_{i}$ | -2 | -1 | 0 | $m$ | $m+n$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $P_{i}$ | 0.2 | 0.1 | 0.2 | $P_{4}$ | $P_{5}$ |
Find the probabilities $P_{4}, P_{5}$ and the variance $D(x)$, if the mathematical expectation $M(x)... | Solution. From the condition $\sum P_{i}=1$ we have $P_{4}+P_{5}+0.2+0.1+0.2=1$ or $P_{4}+P_{5}=0.5$.
From the definition of mathematical expectation $M(x)=\sum x_{i} \cdot p_{i}$ we get:
$$
-0.4-0.1+m \cdot P_{4}+(m+n) P_{5}=-0.5+0.5 m+0.1 n
$$
or
$$
m \cdot P_{4}+(m+n) P_{5}=0.5 m+0.1 n
$$
or
$$
m\left(P_{4}+P_... | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,425 | |
13.1.2. Calculate the sample mean $\bar{X}$, sample variance $D_{x}$, and sample standard deviation $\sigma_{x}$. | Solution. To calculate $\bar{X}, D_{x}$, and $\sigma_{x}$, we will use the method of products. We introduce conditional variants
$$
u_{1}=\frac{\left(x_{i}-c_{x}\right)}{h_{x}}
$$
where $c_{x}$ is the value of $x_{i}$ corresponding to the highest frequency, $c_{x}=3.8$, and $h_{x}$ is the step of the sample, $h_{x}=1... | \bar{X}=3.58,D_{x}=2.949,\sigma_{x}=1.72 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,427 |
13.1.3. By the $\chi^{2}$ criterion, test the hypothesis about the normal distribution of the general population at the significance level $\alpha=0.05$. | Solution. We will test the hypothesis about the normal distribution of the general population using the $\chi^{2}$ (Pearson) criterion at $\alpha=0.05$.
The basis of the criterion is the comparison of frequencies $m_{i}$ and theoretical frequencies $m_{i}^{T}$, calculated under the assumption of a normal distribution ... | \bar{X}=3.584;D_{x}=2.949;\sigma_{x}=1.72; | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,428 |
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