toloka/u-math huggingface dataset:
Subject: precalculus_review

Problem: Use the Rational Zero Theorem to find all real zeros of the following polynomial:

$p(x) = 9 \cdot x^3 + 6 \cdot x^2 - 29 \cdot x - 10$

Answer: The real zeros are $-2$, $-\frac{1}{3}$, $\frac{5}{3}$
Subject: integral_calc

Problem: Compute the integral:
$$
3 \cdot \int \frac{ \cos(3 \cdot x)^4 }{ \sin(3 \cdot x)^3 } \, dx
$$

Answer: $3 \cdot \int \frac{ \cos(3 \cdot x)^4 }{ \sin(3 \cdot x)^3 } \, dx$ = $C+\frac{3}{4}\cdot\ln\left(\frac{1+\cos(3\cdot x)}{1-\cos(3\cdot x)}\right)-\frac{\left(\cos(3\cdot x)\right)^3}{2-2\cdot\left(\cos(3\cdot x)\right)^2}-\frac{3}{2}\cdot\cos(3\cdot x)$
Subject: sequences_series

Problem: Evaluate $\sum_{n=2}^\infty\frac{ n \cdot (n-1) }{ 2^n }$ as $f''\left(\frac{ 1 }{ 2 }\right)$ where $f(x) = \sum_{n=0}^\infty x^n$.

Answer: $\sum_{n=2}^\infty\frac{ n \cdot (n-1) }{ 2^n }$ = $4$
Subject: precalculus_review

Problem: Find the zeros of $f(x) = (2 - x)^4 + (2 \cdot x - 1)^4 - (x + 1)^4$.

Answer: The final answer: $x_1=2 \land x_2=\frac{1}{2}$
Subject: sequences_series

Problem: Using the Taylor formula, decompose the function $f(x) = \ln\left(1+\frac{ x }{ 5 }\right)$ in powers of the variable $x$ on the segment $[0,1]$. Use the first nine terms.

Then estimate the accuracy obtained by dropping an additional term after the first nine terms.

Answer: 1. $\ln\left(1+\frac{ x }{ 5 }\right)$ = $\frac{x}{5}-\frac{x^2}{5^2\cdot2}+\frac{x^3}{5^3\cdot3}-\frac{x^4}{5^4\cdot4}+\frac{x^5}{5^5\cdot5}-\frac{x^6}{5^6\cdot6}+\frac{x^7}{5^7\cdot7}-\frac{x^8}{5^8\cdot8}+\frac{x^9}{5^9\cdot9}$
2. Accuracy is not more than $\frac{1}{10\cdot5^{10}}$
Subject: algebra

Problem: A local band sells out for their concert. They sell all 1175 tickets for a total purse of $28,112.5. The tickets were priced at $20 for student tickets, $22.5 for children, and $29 for adult tickets. If the band sold twice as many adult as children tickets, how many of each type was sold?

Answer: Student tickets: $500$ Children tickets: $225$  
Adult tickets: $450$
Subject: algebra

Problem: Luna is going bowling. Shoe rental costs $3 and the lane costs $2 per game. If Luna paid a total of $17, how many games did she bowl?

Answer: Luna bowled $7$ games.
Subject: sequences_series

Problem: Find the radius of convergence of the series:
$$
\sum_{n=1}^\infty \left(\frac{ \left((2 \cdot n)!\right) \cdot x^n }{ n^{2 \cdot n} }\right)
$$

Answer: $R$ = $\frac{e^2}{4}$
Subject: multivariable_calculus

Problem: Evaluate $L=\lim_{P(x,y) \to P(m+n,m-n)}\left(\frac{ x^2-m \cdot x-x \cdot y+m \cdot y-2 \cdot n \cdot x+2 \cdot m \cdot n }{ x \cdot y-n \cdot x-y^2-n \cdot y+2 \cdot n^2 }\right)$, given $m-2 \cdot n=7 \cdot n$

Answer: The final answer: $L=\frac{1}{7}$
Subject: algebra

Problem: Use the Rational Zero Theorem to find the real solution(s) for the equation: $8 \cdot x^4 + 26 \cdot x^3 + 39 \cdot x^2 + 26 \cdot x + 6 = 0$

Answer: By the Rational Zero Theorem, possible solutions occur at values $-\frac{1}{2}$, $-\frac{3}{4}$, $1$, $-1$, $2$, $-2$, $3$, $-3$, $\frac{ 1 }{ 2 }$, $\frac{ 3 }{ 2 }$, $-\frac{ 3 }{ 2 }$, $\frac{ 1 }{ 4 }$, $-\frac{ 1 }{ 4 }$, $\frac{ 3 }{ 4 }$, $\frac{ 1 }{ 8 }$, $-\frac{ 1 }{ 8 }$, $\frac{ 3 }{ 8 }$, $-\frac{ 3 }{ 8 }$, $6$, $-6$. (Be sure to submit each possibility only once; hint: there are 20 unique possibilities by RZT).
$x$ = $-\frac{1}{2}$, $-\frac{3}{4}$, $1$, $-1$, $2$, $-2$, $3$, $-3$, $\frac{ 1 }{ 2 }$, $\frac{ 3 }{ 2 }$, $-\frac{ 3 }{ 2 }$, $\frac{ 1 }{ 4 }$, $-\frac{ 1 }{ 4 }$, $\frac{ 3 }{ 4 }$, $\frac{ 1 }{ 8 }$, $-\frac{ 1 }{ 8 }$, $\frac{ 3 }{ 8 }$, $-\frac{ 3 }{ 8 }$, $6$, $-6$
Subject: sequences_series

Problem: Compute $\int_{0}^{\frac{ 1 }{ 5 }} e^{-2 \cdot x^2} \, dx$ with accuracy $0.00001$.

Answer: The final answer: $0.1948$
Subject: integral_calc

Problem: Find the integral:
$$
\int \frac{ 4 \cdot x^2+25 \cdot x+7 }{ \sqrt{x^2+8 \cdot x} } \, dx
$$

Answer: $\int \frac{ 4 \cdot x^2+25 \cdot x+7 }{ \sqrt{x^2+8 \cdot x} } \, dx$ = $(2\cdot x+1)\cdot\sqrt{x^2+8\cdot x}+3\cdot\ln\left(\left|x+4+\sqrt{x^2+8\cdot x}\right|\right)+C$
Subject: precalculus_review

Problem: Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{\frac{ -\pi }{ 3 }}^{\frac{ \pi }{ 3 }} \sqrt{\left(\sec(x)\right)^2-1} \, dx
$$

Answer: $\int_{\frac{ -\pi }{ 3 }}^{\frac{ \pi }{ 3 }} \sqrt{\left(\sec(x)\right)^2-1} \, dx$ = $0$
Subject: sequences_series

Problem: Find a “reasonable” upper-bound on the error in approximating $f(x) = x^7$ by its 3rd order Taylor polynomial $P_{3}(x)$ about $a = -1$ valid for all values of $x$ such that $|x + 1| \leq 0.1$.

Answer: The final answer: $840\cdot(1.1)^3\cdot\frac{(0.1)^4}{4!}$
Subject: integral_calc

Problem: Compute the volume of the solid formed by rotating about the x-axis the area bounded by the axes and the parabola $x^{\frac{ 1 }{ 2 }}+y^{\frac{ 1 }{ 2 }}=5^{\frac{ 1 }{ 2 }}$.

Answer: Volume = $\pi\cdot\frac{25}{3}$
Subject: differential_calc

Problem: Make full curve sketching of $y = \arcsin\left(\frac{ 1-3 \cdot x^2 }{ 1+3 \cdot x^2 }\right)$. Submit as your final answer:

1. The domain (in interval notation)
2. Vertical asymptotes
3. Horizontal asymptotes
4. Slant asymptotes
5. Intervals where the function is increasing
6. Intervals where the function is decreasing
7. Intervals where the function is concave up
8. Intervals where the function is concave down
9. Points of inflection

Answer: 1. The domain (in interval notation) $(-1\cdot\infty,\infty)$
2. Vertical asymptotes None
3. Horizontal asymptotes $y=-\frac{\pi}{2}$
4. Slant asymptotes None
5. Intervals where the function is increasing $(-\infty,0)$, $(0,\infty)$
6. Intervals where the function is decreasing $(-\infty,0)$, $(0,\infty)$
7. Intervals where the function is concave up $(-\infty,0)$
8. Intervals where the function is concave down None
9. Points of inflection None
Subject: multivariable_calculus

Problem: Determine the coordinates, if any, for which $f(x,y) = x^3 - 3 \cdot x \cdot y^2 + 6 \cdot y^2 - 8$ has 

1. a Relative Minimum(s)
2. a Relative Maximum(s)
3. a Saddle Point(s)

If a Relative Minimum or Maximum, find the Minimum or Maximum value. If none, enter None.

Answer: 1. The function $f(x,y)$ has Relative Minimum(s) at None with the value(s) None
2. The function $f(x,y)$ has Relative Maximum(s) at None with the value(s) None
3. The function $f(x,y)$ has a Saddle Point(s) at $P(2,-2)$, $P(2,2)$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \frac{ 20 \cdot \cos(-10 \cdot x)^3 }{ 21 \cdot \sin(-10 \cdot x)^7 } \, dx
$$

Answer: $\int \frac{ 20 \cdot \cos(-10 \cdot x)^3 }{ 21 \cdot \sin(-10 \cdot x)^7 } \, dx$ = $C+\frac{1}{21}\cdot\left(\frac{1}{2}\cdot\left(\cot(10\cdot x)\right)^4+\frac{1}{3}\cdot\left(\cot(10\cdot x)\right)^6\right)$
Subject: algebra

Problem: The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a local newspaper currently has $84\ 000$ subscribers at a quarterly charge of $\$30$. Market research has suggested that if the owners raise the price to $\$32$, they would lose $5000$ subscribers. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue?

Answer: The final answer: $31.8$
Subject: sequences_series

Problem: Find the Taylor polynomial $P_{5}$ for the function $f(x) = x \cdot \cos\left(x^2\right)$.

Answer: $P_{5}$ = $x-\frac{1}{2}\cdot x^5$
Subject: integral_calc

Problem: Solve the integral:
$$
-\int \frac{ 1 }{ \cos(x)^3 \cdot \sin(x)^2 } \, dx
$$

Answer: $-\int \frac{ 1 }{ \cos(x)^3 \cdot \sin(x)^2 } \, dx$ = $C+\frac{\sin(x)}{2\cdot\left(\left(\sin(x)\right)^2-1\right)}+\frac{3}{4}\cdot\ln\left(\left|\sin(x)-1\right|\right)+\frac{1}{\sin(x)}-\frac{3}{4}\cdot\ln\left(\left|1+\sin(x)\right|\right)$
Subject: sequences_series

Problem: Find the Fourier series of the function $\psi(x) = e^{-2 \cdot x}$ in the interval $(-2 \cdot \pi, 2 \cdot \pi)$.

Answer: The Fourier series is: $e^{-2\cdot x}=\frac{\left(e^{4\cdot\pi}-e^{-4\cdot\pi}\right)}{\pi}\cdot\left(\frac{1}{8}+\sum_{n=1}^\infty\frac{(-1)^n\cdot\left(4\cdot\cos(n\cdot x)+n\cdot\sin(n\cdot x)\right)}{16+n^2}\right)$
Subject: precalculus_review

Problem: Find zeros of $f(x) = \sqrt{2 \cdot x - 1} - \sqrt{x - 1} - 5$

Answer: The final answer: $x=145$
Subject: algebra

Problem: Write an expression for a rational function with the given characteristics:

1. vertical asymptotes $x=-6$ and $x=6$,
2. x-intercepts at $P(1,0)$ and $P(-1,0)$,
3. y-intercept at $P(0,2)$.

Answer: The rational function satisfying the given conditions is $f(x)=\frac{72\cdot(x-1)\cdot(x+1)}{(x+6)\cdot(x-6)}$
Subject: integral_calc

Problem: Find the area of the figure enclosed between the curves $y = 4 \cdot x^2$, $y = \frac{ x^2 }{ 3 }$, and $y = 2$.

Answer: Area: $\frac{\left(8\cdot\sqrt{3}-4\right)\cdot\sqrt{2}}{3}$
Subject: multivariable_calculus

Problem: Find the measure of the angle $\theta$ between the three-dimensional vectors $\vec{a}$ and $\vec{b}$, expressed in radians rounded to two decimal places, if it is not possible to express it exactly.

Given:
$\vec{a} = 3 \cdot \vec{i} - \vec{j} - 2 \cdot \vec{k}$
$\vec{b} = \vec{v} - \vec{w}$, where $\vec{v} = 2 \cdot \vec{i} + \vec{j} + 4 \cdot \vec{k}$ and $\vec{w} = 6 \cdot \vec{i} + \vec{j} + 2 \cdot \vec{k}$

Answer: $\theta$ = $2.84$
Subject: integral_calc

Problem: Evaluate the integral:
$$
I = \int 3 \cdot x \cdot \ln\left(4 + \frac{ 1 }{ x } \right) \, dx
$$

Answer: The final answer: $\left(\frac{3}{2}\cdot x^2\cdot\ln(4\cdot x+1)-\frac{3\cdot x^2}{4}+\frac{3\cdot x}{8}-\frac{3}{32}\cdot\ln\left(x+\frac{1}{4}\right)\right)-\left(\frac{3}{2}\cdot x^2\cdot\ln(x)-\left(C+\frac{3}{4}\cdot x^2\right)\right)$
Subject: sequences_series

Problem: A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms.

Let $a_{n} = f(n) - 2 \cdot f(n+1) + f(n+2)$, in which $f(n)$ -> 0 as $n$ -> $\infty$. Find $\sum_{n=1}^\infty\left(a_{n}\right)$.

Answer: $\sum_{n=1}^\infty\left(a_{n}\right)$ = $f(1)-f(2)$
Subject: precalculus_review

Problem: Solve $z^2 - (2 \cdot i + 1) \cdot z - (3 - i) = 0$.

Answer: The final answer: $z_1=2+i \lor z_1=-1+i$
Subject: multivariable_calculus

Problem: Calculate the double integral:
$$
\int\int_{R}{\left(x \cdot y \cdot \sqrt{x^2+y^2}\right) d A}
$$
where $R=\left\{(x,y)|0 \le x \le 1,0 \le y \le 2\right\}$.

Answer: The final answer: $\frac{25\cdot\sqrt{5}-33}{15}$
Subject: sequences_series

Problem: Find the Fourier series of the function $\psi(x) = e^{-x}$ in the interval $(-\pi,\pi)$.

Answer: The Fourier series is: $e^{-x}=\frac{e^\pi-e^{-\pi}}{2\cdot\pi}\cdot\left(\frac{1}{2}+\sum_{n=1}^\infty\left(\frac{(-1)^n}{1+n^2}\cdot\left(\cos(n\cdot x)+n\cdot\sin(n\cdot x)\right)\right)\right)$
Subject: sequences_series

Problem: Express the series $\sum_{n=1}^\infty \left(\frac{ 1 }{ (x-3)^{2 \cdot n-1} }-\frac{ 1 }{ (x-2)^{2 \cdot n-1} }\right)$ as a rational function.

Answer: $\sum_{n=1}^\infty \left(\frac{ 1 }{ (x-3)^{2 \cdot n-1} }-\frac{ 1 }{ (x-2)^{2 \cdot n-1} }\right)$ = $\frac{x-3}{(x-3)^2-1}-\frac{x-2}{(x-2)^2-1}$
Subject: multivariable_calculus

Problem: Find the equations of the common tangent lines to the following ellipses:

1. $rac{ x^2 }{ 6 } + y^2 = 1$  
2. $rac{ x^2 }{ 4 } + rac{ y^2 }{ 9 } = 1$

Answer: The equations are: $2\cdot x+y-5=0 \lor 2\cdot x+y+5=0 \lor 2\cdot x-y-5=0 \lor 2\cdot x-y+5=0$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \sin(5 \cdot x)^6 \cdot \cos(5 \cdot x)^2 \, dx
$$

Answer: Answer is: $\frac{1}{32}\cdot x-\frac{1}{32}\cdot\frac{1}{20}\cdot\sin(20\cdot x)-\frac{1}{8}\cdot\frac{1}{10}\cdot\frac{1}{3}\cdot\sin(10\cdot x)^3+\frac{1}{128}\cdot x-\frac{1}{128}\cdot\frac{1}{40}\cdot\sin(40\cdot x)+C$
Subject: precalculus_review

Problem: Recall the following double-angle formulas:

1. $\sin\left(2 \cdot \theta\right) = 2 \cdot \sin\left(\theta\right) \cdot \cos\left(\theta\right)$
2. $\cos\left(2 \cdot \theta\right) = 2 \cdot \cos\left(\theta\right)^2 - 1$
3. $\cos\left(2 \cdot \theta\right) = 1 - 2 \cdot \sin\left(\theta\right)^2$

From the formulae, find appropriate expressions for the following, in terms of $\sin\left(2 \cdot \theta\right)$ or $\cos\left(2 \cdot \theta\right)$:  
[Note: your end answer may contain only one of them.]

1. $\sin\left(\theta\right)^2$
2. $\cos\left(\theta\right)^2$

Answer: Part (a): $\sin\left(\theta\right)^2=\frac{1}{2}\cdot\left(1-\cos\left(2\cdot\theta\right)\right)$ Part (b): $\cos\left(\theta\right)^2=\frac{1}{2}\cdot\left(1+\cos\left(2\cdot\theta\right)\right)$
Subject: sequences_series

Problem: Compute the first 4 nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of $f(x) = e^x \cdot \cos(x)$.

Answer: $f(x)$ = $1+x-\frac{x^3}{3}-\frac{x^4}{6}+\cdots$
Subject: differential_calc

Problem: Find the local minimum and local maximum values of the function $f(x) = \frac{ 3 }{ 4 } \cdot x^4 - 10 \cdot x^3 + 24 \cdot x^2 - 4$.

Answer: The point(s) where the function has a local minimum: $P(0,-4)$, $P(8,-516)$
The point(s) where the function has a local maximum: $P(2,24)$
Subject: precalculus_review

Problem: Find zeros of $f(x) = \sqrt{x-3} + \sqrt{x+2} - \sqrt{2 \cdot x-5} - 2$

Answer: The final answer: $x=7$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ x^3-2 \cdot x^2+x }{ 3+2 \cdot x-x^2 } \, dx
$$

Answer: $\int \frac{ x^3-2 \cdot x^2+x }{ 3+2 \cdot x-x^2 } \, dx$ = $-\frac{1}{2}\cdot x^2-2\cdot\ln\left(\left|x^2-2\cdot x-3\right|\right)-\ln\left(\left|\frac{x-3}{x+1}\right|\right)+C$
Subject: multivariable_calculus

Problem: Evaluate $L=\lim_{(x,y) \to (-1,1)}\left(\frac{ x \cdot y \cdot (x+y)-5 \cdot y \cdot (y+5)+5 \cdot x \cdot (x-5) }{ x^2+y^2+2 \cdot x \cdot y+5 \cdot x+5 \cdot y }\right)$.

Answer: The final answer: $L=-\frac{36}{5}$
Subject: precalculus_review

Problem: A rental car company rents cars for a flat fee of $\$20$ and an hourly charge of $\$10.25$. Therefore, the total cost $C$ to rent a car is a function of the hours $t$ the car is rented plus the flat fee.

1. Write the formula for the function that models this situation.
2. Find the total cost to rent a car for 2 days and 7 hours.
3. Determine how long the car was rented if the bill is $\$432.73$.

Answer: 1. The formula for the function is $C(t)$ = $10.25\cdot t+20$
2. The total cost to rent a car is $ $583.75$
3. The time of car rental in hours is $40.26634146$
Subject: differential_calc

Problem: A box is to be made with the following properties:

1. The length of the base, $l$, is twice the length of a width $w$.
2. The cost of material to be used for the lateral faces and the top of the box is three times the cost of the material to be used for the lower base.

Find the dimensions of the box in terms of its fixed volume $V$ such that the cost of the used material is the minimum.

Answer: The final answer: $w=\frac{1}{2}\cdot\sqrt[3]{\frac{9\cdot V}{2}}$, $l=\sqrt[3]{\frac{9\cdot V}{2}}$, $y=\frac{2}{3}\cdot\sqrt[3]{\frac{4\cdot V}{3}}$
Subject: algebra

Problem: Solve the quadratic equation by completing the square. Show each step. (Give your answer either exactly or rounded to two decimal places).

$2 + z = 6 \cdot z^2$

Answer: $z$ = $\frac{2}{3}$, $-\frac{1}{2}$
Subject: differential_calc

Problem: Find the derivative of the 27th order $y^{(27)}$ for a function $y = 2 \cdot x^2 \cdot \sin(x)$.

Answer: $y^{(27)}$ = $1404\cdot\cos(x)-2\cdot x^2\cdot\cos(x)-108\cdot x\cdot\sin(x)$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \frac{ 1 }{ \sin(x)^5 } \, dx
$$

Answer: The final answer: $C+\frac{1}{16}\cdot\left(2\cdot\left(\tan\left(\frac{x}{2}\right)\right)^2+6\cdot\ln\left(\left|\tan\left(\frac{x}{2}\right)\right|\right)+\frac{1}{4}\cdot\left(\tan\left(\frac{x}{2}\right)\right)^4-\frac{2}{\left(\tan\left(\frac{x}{2}\right)\right)^2}-\frac{1}{4\cdot\left(\tan\left(\frac{x}{2}\right)\right)^4}\right)$
Subject: multivariable_calculus

Problem: The solid $Q$ has the mass given by the triple integral:
$$
\int_{0}^1 \left( \int_{0}^{\frac{ \pi }{ 2 }} \left( \int_{0}^{r^2} \left( r^4 + r \right) dz \right) d \theta \right) dr
$$
1. Find the density of the solid in rectangular coordinates.
2. Find the moment $M_{xy}$ about the xy-plane.

Answer: 1. $\rho(x,y,z)$ = $1+\left(x^2+y^2\right)^{\frac{3}{2}}$
2. $M_{xy}$ = $\frac{5\cdot\pi}{72}$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \frac{ \sqrt{9 \cdot x+4} }{ -3 \cdot x^2 } \, dx
$$

Answer: $\int \frac{ \sqrt{9 \cdot x+4} }{ -3 \cdot x^2 } \, dx$ = $C+\frac{\sqrt{9\cdot x+4}}{3\cdot x}-\frac{3}{4}\cdot\ln\left(\frac{\left|\sqrt{9\cdot x+4}-2\right|}{2+\sqrt{9\cdot x+4}}\right)$
Subject: algebra

Problem: Evaluate the expression $\frac{ (2+i) \cdot (4-2 \cdot i) }{ (1+i) }$ and write the result as a simplified complex number.

Answer: $\frac{ (2+i) \cdot (4-2 \cdot i) }{ (1+i) }$ = $5-5\cdoti$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ 2 \cdot x+1 }{ (x-1) \cdot \sqrt{x^2-4 \cdot x+2} } \, dx
$$

Answer: $\int \frac{ 2 \cdot x+1 }{ (x-1) \cdot \sqrt{x^2-4 \cdot x+2} } \, dx$ = $2\cdot\ln\left(\left|x-2+\sqrt{x^2-4\cdot x+2}\right|\right)-3\cdot\arcsin\left(\frac{x}{(x-1)\cdot\sqrt{2}}\right)+C$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ -1 }{ 3 \cdot \sin\left(\frac{ x }{ 3 }\right)^6 } \, dx
$$

Answer: $\int \frac{ -1 }{ 3 \cdot \sin\left(\frac{ x }{ 3 }\right)^6 } \, dx$ = $\frac{\cos\left(\frac{x}{3}\right)}{5\cdot\sin\left(\frac{x}{3}\right)^5}-\frac{4}{15}\cdot\left(-\frac{\cos\left(\frac{x}{3}\right)}{\sin\left(\frac{x}{3}\right)^3}-2\cdot\cot\left(\frac{x}{3}\right)\right)+C$
Subject: multivariable_calculus

Problem: Find the equation of a quadratic curve, which passes through the origin, and is tangent to $4 \cdot x + 3 \cdot y + 2 = 0$ at $(1,-2)$ and to $x - y - 1 = 0$ at $(0,-1)$.

Answer: The final answer: $6\cdot x^2-y^2+3\cdot x\cdot y+2\cdot x-y=0$
Subject: multivariable_calculus

Problem: Find the first derivative $y_{x}'$ of the function:
$$
x = \arcsin\left(\frac{ t }{ \sqrt{2+2 \cdot t^2} }\right), \quad y = \arccos\left(\frac{ 1 }{ \sqrt{2+2 \cdot t^2} }\right), \quad t \ge 0
$$

Answer: $y_{x}'$ = $\frac{t\cdot\sqrt{t^2+2}}{\sqrt{2\cdot t^2+1}}$
Subject: precalculus_review

Problem: 1. Find the inverse function of $f(x) = (x-1)^2$, for $x \le 1$.
2. Find the domain of the inverse function.
3. Find the range of the inverse function.

Answer: 1. The inverse function is: $1-\sqrt{x}$
2. The domain of the inverse function is: $x\ge0$
3. The range of the inverse function is: $y\le1$
Subject: multivariable_calculus

Problem: Find the area of the region $R$ bounded by circles $x^2+y^2=1$ and $x^2+y^2=4$ and the lines $y=x$ and $y=\sqrt{3} \cdot x$.

Answer: The final answer: $\frac{\pi}{8}$
Subject: algebra

Problem: Solve the following equations:

1. $\frac{ 3 }{ 4 } (x+9) = 15$
2. $\frac{ 2 }{ 3 } (m-8) = 4$
3. $\frac{ 1 }{ 8 } (b+2) = \frac{ 3 }{ 8 }$
4. $\frac{ 1 }{ 7 } (x-4) = -\frac{ 1 }{ 2 }$
5. $\frac{ 2 }{ 5 } (-3-2 x) = \frac{ 18 }{ 5 }$
6. $\frac{ 4 }{ 5 } (j+5) = \frac{ 2 }{ 5 }$

Answer: The solutions to the given equations are:  
1. $x=11$
2. $m=14$
3. $b=1$
4. $x=\frac{ 1 }{ 2 }$
5. $x=-6$
6. $j=\frac{ -9 }{ 2 }$
Subject: multivariable_calculus

Problem: Consider a rectangular box of width $w$, length $l$, and height $h$ without a lid. If the box is to be made from $12$ sq. cm of cardboard, find the dimensions that maximize the volume of the box and the maximum possible volume itself.

Answer: Optimal dimensions:

$w$: $2$

$l$: $2$

$h$: $1$

Maximum volume: $4$
Subject: precalculus_review

Problem: State the domain and range of the function: $f(x) = \ln(x-5)$

1. Domain
2. Range

Answer: 1. Domain: $(5,\infty)$
2. Range: $(-\infty,\infty)$
Subject: differential_calc

Problem: Make full curve sketching of $y = \sqrt[3]{9 \cdot x^2 - x^3}$. Submit as your final answer:

1. The domain (in interval notation)
2. Vertical asymptotes
3. Horizontal asymptotes
4. Slant asymptotes
5. Intervals where the function is increasing
6. Intervals where the function is decreasing
7. Intervals where the function is concave up
8. Intervals where the function is concave down
9. Points of inflection

Answer: 1. The domain (in interval notation) $(-1\cdot\infty,\infty)$
2. Vertical asymptotes None
3. Horizontal asymptotes None
4. Slant asymptotes $y=-x+3$
5. Intervals where the function is increasing $(0,6)$
6. Intervals where the function is decreasing $(-\infty,0)$, $(6,\infty)$
7. Intervals where the function is concave up $(9,\infty)$
8. Intervals where the function is concave down $(-\infty,0)$, $(0,9)$
9. Points of inflection $P(9,0)$
Subject: algebra

Problem: Recall the formula for calculating the magnitude of an earthquake, $M=\frac{ 2 }{ 3 } \cdot \log_{10}\left(\frac{ S }{ S_{0} }\right)$. One earthquake has magnitude $3.9$ on the MMS scale. If a second earthquake has $750$ times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth.

Answer: Magnitude of the second quake: $5.82$
Subject: algebra

Problem: Find the solution to the following inequality and express it in interval notation:
$$
5 (x-7) (x+3) (x-2) < 0
$$

Answer: The solution set to the inequality is: $\left(-\infty,\ -3\right)\cup\left(2,\ 7\right)$
Subject: integral_calc

Problem: Compute the area of the figure bounded by curves $y = 8 \cdot x^2$, $y = 4 + 4 \cdot x^2$, lines $x = 3$, $x = -2$, and the $x$-axis.

Answer: Area = $\frac{184}{3}$
Subject: precalculus_review

Problem: Find points on a coordinate plane that satisfy the following equation:

$$
10 \cdot x^2 + 29 \cdot y^2 + 34 \cdot x \cdot y + 8 \cdot x + 14 \cdot y + 2 = 0
$$

Answer: The final answer: $(3,-2)$
Subject: sequences_series

Problem: Find the radius of convergence and sum of the series:
$$
\frac{ 3 }{ 2 }+\frac{ 3 \cdot x }{ 1 \cdot 3 }+\frac{ 3 \cdot x^2 }{ 1 \cdot 2 \cdot 4 }+\cdots+\frac{ 3 \cdot x^n }{ \left(n!\right) \cdot (n+2) }+\cdots
$$

Answer: 1. Radius of convergence: $R=\infty$
2. Sum: $f(x)=\begin{cases}\frac{3}{x^2}+\frac{3\cdot x\cdot e^x-3\cdot e^x}{x^2},&x\ne0\\\frac{3}{2},&x=0\end{cases}$
Subject: precalculus_review

Problem: 1. Find the inverse function of $f(x) = \frac{ 1 }{ x+2 }$
2. Find the domain of the inverse function
3. Find the range of the inverse function

Answer: 1. The inverse function is: $\frac{1}{x}-2$
2. The domain of the inverse function is: $(-\infty,0)\cup(0,\infty)$
3. The range of the inverse function is: $(-\infty,-2)\cup(-2,\infty)$
Subject: multivariable_calculus

Problem: Evaluate $I=\int_{0}^4 \int_{0}^x \int_{0}^{x+y} \left(3 \cdot e^x \cdot (y+2 \cdot z)\right) dz dy dx$.

Answer: The final answer: $I=19\cdot\left(17\cdot e^4+3\right)$
Subject: algebra

Problem: Divide rational expressions and simplify it:

$$
\frac{ x^2-9 }{ 3 } \div \frac{ x-3 }{ 4 }
$$

Answer: The final answer: $\frac{4}{3}\cdot x+4$
Subject: differential_calc

Problem: For the curve $x = a \cdot \left(t - \sin(t)\right)$, $y = a \cdot \left(1 - \cos(t)\right)$, determine the curvature. Use $a = 12$.

Answer: The curvature is: $\frac{1}{48\cdot\left|\sin\left(\frac{t}{2}\right)\right|}$
Subject: multivariable_calculus

Problem: Let $z = x^2 + 3 \cdot x \cdot y - y^2$. Find the exact change $\Delta z = f\left(x+\Delta x,y+\Delta y\right) - f(x,y)$ in the function and find the approximate change $d z$ in the function as $x$ changes from $2$ to $2.05$ and $y$ changes from $3$ to $2.96$.

Answer: The final answer:

1. The exact change $\Delta z$ is $0.6449$
2. The approximate change $d z$ is $0.65$
Subject: algebra

Problem: Add and subtract the rational expressions, and then simplify:

$$
\frac{ x-4 }{ x+4 } - \frac{ 2 \cdot x+6 }{ 2 \cdot x+1 }
$$

Answer: The final answer: $\frac{-21\cdot x-28}{2\cdot x^2+9\cdot x+4}$
Subject: precalculus_review

Problem: Simplify the expression $\frac{ \cos(t) }{ \sin(t) }+\frac{ \sin(t) }{ 1+\cos(t) }$ by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

Answer: $\frac{ \cos(t) }{ \sin(t) }+\frac{ \sin(t) }{ 1+\cos(t) }$ = $\csc(t)$