toloka/u-math huggingface dataset:
Subject: precalculus_review

Problem: $P = \left(\frac{ 7 }{ 25 },y\right)$, $y>0$ is a point on the unit circle.

1. Find the (exact) missing coordinate value of the point.
2. Find the values of the six trigonometric functions for the angle $\theta$ with a terminal side that passes through point $P$.

Rationalize denominators.

Answer: 1. The (exact) missing coordinate value of the point is: $\frac{24}{25}$
2. The values of the six trigonometric functions are:
    * $\sin\left(\theta\right)$ = $\frac{24}{25}$
    * $\cos\left(\theta\right)$ = $\frac{7}{25}$
    * $\tan\left(\theta\right)$ = $\frac{24}{7}$
    * $\csc\left(\theta\right)$ = $\frac{25}{24}$
    * $\sec\left(\theta\right)$ = $\frac{25}{7}$
    * $\cot\left(\theta\right)$ = $\frac{7}{24}$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \sqrt{\frac{ -16 \cdot \sin(-10 \cdot x) }{ 25 \cdot \cos(-10 \cdot x)^9 }} \, dx
$$

Answer: $\int \sqrt{\frac{ -16 \cdot \sin(-10 \cdot x) }{ 25 \cdot \cos(-10 \cdot x)^9 }} \, dx$ = $C+\frac{2}{25}\cdot\left(\frac{2}{3}\cdot\left(\tan(10\cdot x)\right)^{\frac{3}{2}}+\frac{2}{7}\cdot\left(\tan(10\cdot x)\right)^{\frac{7}{2}}\right)$
Subject: differential_calc

Problem: Find the derivative of $f(x) = \frac{ 1 }{ 15 } \cdot \left(\cos(x)\right)^3 \cdot \left(\left(\cos(x)\right)^2-5\right)$.

Answer: The final answer: $f'(x)=-\frac{\left(\cos(x)\right)^2}{15}\cdot\left(3\cdot\sin(x)\cdot\left(\left(\cos(x)\right)^2-5\right)+\cos(x)\cdot\sin(2\cdot x)\right)$
Subject: sequences_series

Problem: Compute $\sqrt[4]{90}$ with accuracy $0.0001$.

Answer: The final answer: $3.0801$
Subject: algebra

Problem: Solve the following compound (double) inequality and write your final answer in interval notation:

$-4 < 3 \cdot x + 2 \le 18$

Answer: The final answer: $\left(-2,\frac{16}{3}\right]$
Subject: differential_calc

Problem: Find the local minimum and local maximum values of the function $f(x) = \frac{ x^4 }{ 4 } - \frac{ 11 }{ 3 } \cdot x^3 + 15 \cdot x^2 + 17$.

Answer: The point(s) where the function has a local minimum: $P(0,17)$, $P(6,89)$
The point(s) where the function has a local maximum: $P\left(5,\frac{1079}{12}\right)$
Subject: sequences_series

Problem: Compute the first 3 nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of $f(x) = \ln(\cos(x))$.

Answer: $f(x)$ = $-\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^6}{45}+\cdots$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ 8 }{ 7 \cdot x^2 \cdot \sqrt{5 \cdot x^2-2 \cdot x+1} } \, dx
$$

Answer: Answer is: $\frac{8}{7}\cdot\left(C-\sqrt{5+\frac{1}{x}^2-\frac{1\cdot2}{x}}-\ln\left(\left|\frac{1}{x}+\sqrt{5+\frac{1}{x}^2-\frac{1\cdot2}{x}}-1\right|\right)\right)$
Subject: algebra

Problem: Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeros of the following polynomial:

$p(x) = x^3 - 2 \cdot x^2 + x - 1$

Answer: The number of positive zeros: $1$, $3$
The number of negative zeros: $0$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ \sqrt{25+x^2} }{ 5 \cdot x } \, dx
$$

Answer: $\int \frac{ \sqrt{25+x^2} }{ 5 \cdot x } \, dx$ = $C+\frac{1}{2}\cdot\ln\left(\left|\frac{\sqrt{25+x^2}-5}{5+\sqrt{25+x^2}}\right|\right)+\frac{1}{5}\cdot\sqrt{25+x^2}$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \sin\left(\frac{ x }{ 2 }\right)^5 \, dx
$$

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

Problem: Evaluate the integral $\int\int\int_{E}{(x+y) \, dV}$, where $E$ is the region defined by:
$$
E = \left\{ (x,y,z) \, | \, 0 \le x \le \sqrt{1-y^2}, \, 0 \le y \le 1, \, 0 \le z \le 1-x \right\}
$$

Answer: $I$ = $\frac{1}{48}\cdot(26-3\cdot\pi)$
Subject: multivariable_calculus

Problem: Use the second derivative test to identify any critical points of the function $f(x,y) = 7 \cdot x^2 \cdot y + 9 \cdot x \cdot y^2$, and determine whether each critical point is a maximum, minimum, saddle point, or none of these.

Answer: Maximum: None
Minimum: None

Saddle point: None

The second derivative test is inconclusive at: $P(0,0)$
Subject: differential_calc

Problem: Find any local extrema for $s = \frac{ \pi }{ 2 } - \left| \arctan\left( x^2 - 1 \right) \right|$.

Submit as your final answer:

1. The point(s), where the function has local maximum(s);
2. The point(s), where the function has local minimum(s).

Answer: 1. Local Maximum(s) $P\left(-1,\frac{\pi}{2}\right)$, $P\left(1,\frac{\pi}{2}\right)$
2. Local Minimum(s) $P\left(0,\frac{\pi}{4}\right)$
Subject: sequences_series

Problem: Evaluate the sum of the series $\sum_{n=1}^\infty\left(\frac{ 1 }{ n \cdot (n+1) \cdot (n+2) }\right)$.

(Hint: $\frac{ 1 }{ n \cdot (n+1) \cdot (n+2) }=\frac{ 1 }{ 2 \cdot n }-\frac{ 1 }{ n+1 }+\frac{ 1 }{ 2 \cdot (n+2) }$)

Answer: $\sum_{n=1}^\infty\left(\frac{ 1 }{ n \cdot (n+1) \cdot (n+2) }\right)$ = $\frac{1}{4}$
Subject: multivariable_calculus

Problem: Determine the coordinates, if any, for which $f(x,y) = 24 \cdot x + 12 \cdot y - 8 \cdot x \cdot y - 4 \cdot x^2 - 6 \cdot y^2$ 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 $P(6,-3)$ with the value(s) $54$
3. The function $f(x,y)$ has a Saddle Point(s) at None
Subject: sequences_series

Problem: Use the substitution $(b+x)^r = (b+a)^r \cdot \left(1+\frac{ x-a }{ b+a }\right)^r$ in the binomial expansion to find the Taylor series of the function $\sqrt{2 \cdot x-x^2}$ with the center $a=1$.

Answer: $\sqrt{2 \cdot x-x^2}$ = $\sum_{n=0}^\infty\left((-1)^n\cdot C_{\frac{1}{2}}^n\cdot(x-1)^{2\cdot n}\right)$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \tan(x)^4 \, dx
$$

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

Problem: Find the Fourier series of the function $u = \left| \sin(x) \right|$ in the interval $[-\pi, \pi]$.

Answer: The Fourier series is: $\frac{2}{\pi}-\frac{4}{\pi}\cdot\left(\frac{\cos(2\cdot x)}{1\cdot3}+\frac{\cos(4\cdot x)}{3\cdot5}+\frac{\cos(6\cdot x)}{5\cdot7}+\cdots\right)$
Subject: precalculus_review

Problem: The cost to remove a toxin from a lake is modeled by the function $C(p) = \frac{ 75 \cdot p }{ 85-p }$, where $C$ is the cost (in thousands of dollars) and $p$ is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than $85$ ppb.

1. Find the cost to remove $25$ ppb, $40$ ppb, and $50$ ppb of the toxin from the lake.
2. Find the inverse function.
3. Use part b. to determine how much toxin is removed for $\$50000$.

Answer: 1. The cost to remove $25$ ppb is: $31.25$ thousand dollars
The cost to remove $40$ ppb is: $66.66666667$ thousand dollars
The cost to remove $50$ ppb is: $107.14285714$ thousand dollars
2. $p(C)=\frac{85\cdot C}{75+C}$
3. $34$ ppb
Subject: multivariable_calculus

Problem: Calculate the area of the surface formed by rotating the astroid $x^{\frac{ 2 }{ 3 }} + y^{\frac{ 2 }{ 3 }} = 2^{\frac{ 2 }{ 3 }}$ about the x-axis.

Answer: The final answer: $\frac{48\cdot\pi}{5}$
Subject: algebra

Problem: Use the properties of logarithms to expand the logarithm $\ln\left(y \cdot \sqrt{\frac{ y }{ 1-y }}\right)$ as much as possible. Rewrite the expression as a sum, difference, or product of logs.

Answer: The final answer: $\frac{3}{2}\cdot\ln(y)-\frac{1}{2}\cdot\ln(1-y)$
Subject: algebra

Problem: Multiply the rational expressions and express the product in simplest form:

$$
\frac{ 2 \cdot d^2 + d - 45 }{ d^2 + 7 \cdot d + 10 } \cdot \frac{ 4 \cdot d^2 + 7 \cdot d - 2 }{ 4 \cdot d^2 + 31 \cdot d - 8 }
$$

Answer: The final answer: $\frac{(2\cdot d-9)}{(d+8)}$
Subject: multivariable_calculus

Problem: Evaluate $\int\int_{D}{\left(\int_{0}^{4 \cdot x^2+4 \cdot y^2}{y \, dz}\right) \, dA}$, where $D = \left\{(x,y) | x^2+y^2 \le 4, y \ge 1, x \ge 0\right\}$ is the projection of $E$ onto the $x \cdot y$-plane.

Answer: $I$ = $\frac{274}{15}$
Subject: sequences_series

Problem: Consider the function $y = \left| \cos\left( \frac{ x }{ 4 } \right) \right|$.

1. Find the Fourier series of the function.
2. Using this decomposition, calculate the sum of the series $\sum_{n=1}^\infty \frac{ (-1)^n }{ 4 \cdot n^2 - 1 }$.
3. Using this decomposition, calculate the sum of the series $\sum_{n=1}^\infty \frac{ 1 }{ 4 \cdot n^2 - 1 }$.

Answer: 1. The Fourier series is $\frac{2}{\pi}+\sum_{n=1}^\infty\left(\frac{4\cdot(-1)^n}{\pi\cdot\left(1-4\cdot n^2\right)}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)$
2. The sum of the series $\sum_{n=1}^\infty \frac{ (-1)^n }{ 4 \cdot n^2 - 1 }$ is $\frac{(2-\pi)}{4}$
3. The sum of the series $\sum_{n=1}^\infty \frac{ 1 }{ 4 \cdot n^2 - 1 }$ is $\frac{1}{2}$
Subject: precalculus_review

Problem: Solve $\cos(2 \cdot t) - 5 \cdot \sin(t) - 3 = 0$.

Answer: The final answer: $t=(-1)^{n+1}\cdot\frac{\pi}{6}+n\cdot\pi$
Subject: multivariable_calculus

Problem: Find the center of mass of the region $\rho(x,y,z) = z$ on the inverted cone with radius $2$ and height $2$.

Answer: Center of mass: $P\left(0,0,\frac{8}{5}\right)$
Subject: algebra

Problem: Solve the following equations:

1. $-10 c = -80$
2. $n - (-6) = 12$
3. $-82 + x = -20$
4. $- \frac{ r }{ 2 } = 5$
5. $r - 3.4 = 7.1$
6. $\frac{ g }{ 2.5 } = 1.8$
7. $4.8 m = 43.2$
8. $\frac{ 3 }{ 4 } t = \frac{ 9 }{ 20 }$
9. $3\frac{ 2 }{ 3 } + m = 5\frac{ 1 }{ 6 }$

Answer: The solutions to the given equations are:
1. $c=8$
2. $n=6$
3. $x=62$
4. $r=-10$
5. $r=10.5$
6. $g=\frac{ 9 }{ 2 }$
7. $m=9$
8. $t=\frac{3}{5}$
9. $m=\frac{3}{2}$
Subject: integral_calc

Problem: The region bounded by the arc of the curve $y = \sqrt{2} \cdot \sin(2 \cdot x)$, $0 \le x \le \frac{ \pi }{ 2 }$, is revolved around the x-axis. Compute the surface area of this solid of revolution.

Answer: Surface Area: $\frac{\pi}{4}\cdot\left(12\cdot\sqrt{2}+\ln\left(17+12\cdot\sqrt{2}\right)\right)$
Subject: sequences_series

Problem: Given that $\frac{ 1 }{ 1-x } = \sum_{n=0}^\infty x^n$, use term-by-term differentiation or integration to find a power series for the function $f(x) = \frac{ 2 \cdot x }{ \left(1-x^2\right)^2 }$ centered at $x=0$.

Answer: $\frac{ 2 \cdot x }{ \left(1-x^2\right)^2 }$ = $\sum_{n=0}^\infty\left(2\cdot(n+1)\cdot x^{2\cdot n+1}\right)$
Subject: algebra

Problem: Solve the following inequality: $|-12 x-6| \ge 2$.

Express your answer in the interval form.

Answer: Solution in the interval form: $\left(-\infty,\ -\frac{ 2 }{ 3 }\right] \cup \left[-\frac{ 1 }{ 3 },\ \infty\right)$

*Note: enter an interval or union of intervals. If there is no solution, leave empty or enter "none".*
Subject: differential_calc

Problem: Analyze the curve
$$
y = \frac{ 2 \cdot x^3 }{ x^2 - 4 }
$$
and submit as your final answer the following:

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): $(-\infty,-2)\cup(-2,2)\cup(2,\infty)$
2. Vertical asymptotes (leave blank if none): $x=-2$, $x=2$
3. Horizontal asymptotes (leave blank if none): None
4. Slant asymptotes (leave blank if none): $y=2\cdot x$
5. Intervals where the function is increasing (leave blank if none): $\left(-\infty,-2\cdot\sqrt{3}\right)\cup\left(2\cdot\sqrt{3},\infty\right)$
6. Intervals where the function is decreasing (leave blank if none): $\left(-2\cdot\sqrt{3},-2\right)\cup(-2,2)\cup\left(2,2\cdot\sqrt{3}\right)$
7. Intervals where the function is concave up (leave blank if none): $(-2,0)\cup(2,\infty)$
8. Intervals where the function is concave down (leave blank if none): $(-\infty,-2)\cup(0,2)$
9. Points of inflection (leave blank if none): $x=0$
Subject: differential_calc

Problem: Sketch the curve:  
$$
y = \frac{ x^3 }{ 3 \cdot (x+2)^2 }
$$
Provide the following:

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): $(-1\cdot\infty,-2)\cup(-2,\infty)$
2. Vertical asymptotes: $x=-2$
3. Horizontal asymptotes: None
4. Slant asymptotes: $y=\frac{x}{3}-\frac{4}{3}$
5. Intervals where the function is increasing: $(-\infty,-6)$, $(0,\infty)$, $(-2,0)$
6. Intervals where the function is decreasing: $(-6,-2)$
7. Intervals where the function is concave up: $(0,\infty)$
8. Intervals where the function is concave down: $(-2,0)$, $(-\infty,-2)$
9. Points of inflection: $P(0,0)$
Subject: precalculus_review

Problem: Solve the inequality:

$2 \le |x+5| \le 4$

Answer: Write your final answer as intervals: $[-9,-7]\cup[-3,-1]$
Subject: sequences_series

Problem: Find the Taylor polynomial $P_{4}$ for the function $f(x) = \sec(x)$.

Answer: $P_{4}$ = $1+\frac{1}{2}\cdot x^2+\frac{5}{24}\cdot x^4$
Subject: multivariable_calculus

Problem: Determine the equation of the ellipse using the information given: Endpoints of major axis at $(0,2)$, $(0,-2)$ and foci located at $(3,0)$, $(-3,0)$.

Answer: The equation is: $\frac{x^2}{13}+\frac{y^2}{4}=1$
Subject: differential_calc

Problem: Find the equation(s) of the tangent line(s) to the curve $y = 3 \cdot x^4 - 9 \cdot x^2 + 5$, that also passes through the point $\left(0,\frac{ 29 }{ 4 }\right)$.

Answer: Tangent Line(s): $4\cdot y-24\cdot\sqrt{2}\cdot x-29=0$, $4\cdot y+24\cdot\sqrt{2}\cdot x-29=0$
Subject: precalculus_review

Problem: If a culture of bacteria doubles in $3$ hours, how many hours does it take to multiply by $10$? Use $y=y_{0} \cdot e^{k \cdot t}$, where $y_{0}$ is the beginning amount, $y$ is the ending amount, $k$ is the growth or decay rate, and $t$ is time. Leave your answer in exact form.

Answer: It takes $\frac{ 3 \cdot \ln(10) }{ \ln(2) }$ hours.
Subject: differential_calc

Problem: Sketch the curve:  

$y = \sqrt{\frac{ 64-x^3 }{ 3 \cdot x }}$.

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): $(0,4]$
2. Vertical asymptotes: $x=0$
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: None
6. Intervals where the function is decreasing: $(0,4]$
7. Intervals where the function is concave up: $\left(0,2\cdot\sqrt[3]{2}\right)$
8. Intervals where the function is concave down: $\left(2\cdot\sqrt[3]{2},4\right)$
9. Points of inflection: $P\left(2\cdot\sqrt[3]{2},2.5198\right)$
Subject: sequences_series

Problem: Find $\lim_{x \to 0}\frac{ 2 \cdot \left(\tan(x)-\sin(x)\right)-x^3 }{ x^5 }$, using the expansion of the function in the Maclaurin series.

Answer: $\lim_{x \to 0}\frac{ 2 \cdot \left(\tan(x)-\sin(x)\right)-x^3 }{ x^5 }$ = $\frac{1}{4}$
Subject: sequences_series

Problem: Find the 5th order Taylor polynomial $P_{5}$ (in powers of $x$) for the function $f(x) = e^x \cdot \sin(x)$.

Answer: $P_{5}(x)$ = $x+x^2+\frac{1}{3}\cdot x^3-\frac{1}{30}\cdot x^5$
Subject: differential_calc

Problem: For the function $f(x) = -\frac{ 3 }{ 2 } \cdot x^4 + 2 \cdot x^3 + 3 \cdot x^2 - 6 \cdot x + \frac{ 1 }{ 2 }$, specify the points where local maxima and minima of $f(x)$ occur.

1. The point(s) where local maxima occur
2. The point(s) where local minima occur

Answer: 1. The point(s) where local maxima occur: $P(-1,6)$
2. The point(s) where local minima occur: None
Subject: multivariable_calculus

Problem: Apply the gradient descent algorithm to the function $g(x,y) = \left(x^2-1\right) \cdot \left(x^2-3 \cdot x+1\right)+y^2$ with step size $\frac{ 1 }{ 5 }$ and initial guess $p_{0} = \left\langle -1,0 \right\rangle$ for three steps (so steps $p_{1}$, $p_{2}$, and $p_{3}$).

Answer: | $i$ | $1$ | $2$ | $3$ |
| --- | --- | --- | --- |
| $p_{i}$ | $\left\langle1,0\right\rangle$ | $\left\langle\frac{7}{5},0\right\rangle$ | $\left\langle\frac{1333}{625},0\right\rangle$ |
| $g\left(p_{i}\right)$ | $0$ | $-\frac{744}{625}$ | $-\frac{460\ 046\ 957\ 304}{152\ 587\ 890\ 625}$ |
Subject: algebra

Problem: Construct a polynomial $p(x)$ of least degree possible such that its zeros are $-2$, $\frac{ 1 }{ 2 }$ (each with multiplicity $2$), and $p(-3) = 5$.

Answer: $p(x)$ = $\frac{20}{49}\cdot(x+2)^2\cdot\left(x-\frac{1}{2}\right)^2$
Subject: differential_calc

Problem: Find $y'$ and $y''$ for $x^2 + 6 \cdot x \cdot y - 2 \cdot y^2 = 3$.

Answer: $y'$ = $\frac{x+3\cdot y}{2\cdot y-3\cdot x}$
$y''$ = $\frac{11\cdot\left(x^2+6\cdot x\cdot y-2\cdot y^2\right)}{(3\cdot x-2\cdot y)^3}$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ -1 }{ \sqrt[3]{\tan\left(\frac{ x }{ 2 }\right)} } \, dx
$$

Answer: $\int \frac{ -1 }{ \sqrt[3]{\tan\left(\frac{ x }{ 2 }\right)} } \, dx$ = $C+\frac{1}{2}\cdot\ln\left(\left|1+\sqrt[3]{\tan\left(\frac{1}{2}\cdot x\right)}\cdot\tan\left(\frac{1}{2}\cdot x\right)-\sqrt[3]{\tan\left(\frac{x}{2}\right)^2}\right|\right)-\sqrt{3}\cdot\arctan\left(\frac{1}{\sqrt{3}}\cdot\left(2\cdot\sqrt[3]{\tan\left(\frac{x}{2}\right)^2}-1\right)\right)-\ln\left(1+\sqrt[3]{\tan\left(\frac{x}{2}\right)^2}\right)$
Subject: sequences_series

Problem: Find the sum of the series $\sum_{n=1}^\infty \frac{ x^{4 \cdot n-1} }{ 4 \cdot n-1 }$ using the integration of series.

Answer: The sum of the series is $\frac{1}{4}\cdot\ln\left(\frac{|x+1|}{|x-1|}\right)-\frac{1}{2}\cdot\arctan(x)$
Subject: precalculus_review

Problem: Find the center of symmetry of the curve of $f(x) = \frac{ x^2 + 2 \cdot x - 1 }{ 1 - x }$.

Answer: The final answer: $(1,-4)$
Subject: integral_calc

Problem: Evaluate the integral:
$$
I = \int \left(x^3 + 3\right) \cdot \cos(2 \cdot x) \, dx
$$

Answer: The final answer: $\frac{1}{256}\cdot\left(384\cdot\sin(2\cdot x)+128\cdot x^3\cdot\sin(2\cdot x)+192\cdot x^2\cdot\cos(2\cdot x)-96\cdot\cos(2\cdot x)-256\cdot C-192\cdot x\cdot\sin(2\cdot x)\right)$
Subject: differential_calc

Problem: Given the function $y = \frac{ x^3 + x }{ x^4 - x^2 + 1 }$, identify the maxima and/or minima by using the Second Derivative Test.

Answer: Maxima: $P(1,2)$
Minima: $P(-1,-2)$
Subject: algebra

Problem: The radius of the right circular cylinder is $\frac{ 1 }{ 3 }$ meter greater than the height. The volume is $\frac{ 98 \cdot \pi }{ 9 }$ cubic meters. Find the dimensions.

Answer: $r$: $\frac{7}{3}$ $h$: $2$
Subject: precalculus_review

Problem: Solve the following inequality:

$$
\left| |x+3| + |x-2| \right| \le 5
$$

By using rules of the absolute value function.

Answer: The final answer: $[-3,2]$
Subject: integral_calc

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

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

Problem: Transform $z = -3 + i$ to a trigonometric form.

Answer: The final answer: $z=\sqrt{10}\cdot\left(\cos\left(\arccos\left(-\frac{3}{\sqrt{10}}\right)\right)+i\cdot\sin\left(\arccos\left(-\frac{3}{\sqrt{10}}\right)\right)\right)$
Subject: algebra

Problem: Solve the following equations:

1. $\frac{ 3 }{ 4 } (8 x+12)=21$
2. $\frac{ 1 }{ 2 } (8 r+10)=17$
3. $\frac{ 5 x }{ 8 }+9=4$
4. $\frac{ 5 }{ 9 } x-\frac{ 4 }{ 3 }=\frac{ 7 }{ 9 }$
5. $2\frac{2}{3} p+3 \left(\frac{ 2 }{ 3 } p-8\right)=-10$
6. $-5\frac{1}{4} u+2 \left(\frac{ 3 }{ 8 } u-9\right)=\frac{ 5 }{ 2 }$

Answer: The solutions to the given equations are:
1. $x=2$
2. $r=3$
3. $x=-8$
4. $x=\frac{ 19 }{ 5 }$
5. $p=3$
6. $u=-\frac{ 41 }{ 9 }$
Subject: sequences_series

Problem: Expand function $y = \sin\left(\frac{ \pi \cdot x }{ 4 }\right)$ in Taylor series at $x = 2$, given that $\cos(x) = \sum_{n=0}^\infty \left((-1)^n \cdot \frac{ x^{2 \cdot n} }{ (2 \cdot n)! }\right)$.

Answer: The final answer: $\sum_{n=0}^\infty\left((-1)^n\cdot\frac{\left(\frac{\pi}{4}\cdot(x-2)\right)^{2\cdot n}}{(2\cdot n)!}\right)$
Subject: precalculus_review

Problem: $P = \left(x, \frac{ \sqrt{7} }{ 3 }\right)$, $x<0$ is a point on the unit circle.

1. Find the (exact) missing coordinate value of the point.
2. Find the values of the six trigonometric functions for the angle $\theta$ with a terminal side that passes through point $P$.

Rationalize denominators.

Answer: 1. The (exact) missing coordinate value of the point is: $-\frac{\sqrt{2}}{3}$
2. The values of the six trigonometric functions is:
    * $\sin\left(\theta\right)$ = $\frac{\sqrt{7}}{3}$
    * $\cos\left(\theta\right)$ = $-\frac{\sqrt{2}}{3}$
    * $\tan\left(\theta\right)$ = $-\frac{\sqrt{14}}{2}$
    * $\csc\left(\theta\right)$ = $\frac{3\cdot\sqrt{7}}{7}$
    * $\sec\left(\theta\right)$ = $-\frac{3\cdot\sqrt{2}}{2}$
    * $\cot\left(\theta\right)$ = $-\frac{\sqrt{14}}{7}$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \frac{ \sin(x)^5 }{ \cos(x)^4 } \, dx
$$

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

Problem: Divide rational expressions and simplify it:

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

Answer: The final answer: $\frac{3}{2}\cdot x-3$
Subject: precalculus_review

Problem: Simplify $E = \left(\sin(2 \cdot a)\right)^3 \cdot \left(\cos(6 \cdot a)\right) + \left(\cos(2 \cdot a)\right)^3 \cdot \left(\sin(6 \cdot a)\right)$.

Answer: The final answer: $E=\frac{3}{4}\cdot\sin(8\cdot a)$
Subject: integral_calc

Problem: Compute the integral:
$$
8 \cdot \int \cot(-4 \cdot x)^5 \cdot \csc(4 \cdot x)^4 \, dx
$$

Answer: $8 \cdot \int \cot(-4 \cdot x)^5 \cdot \csc(4 \cdot x)^4 \, dx$ = $C+\frac{1}{3}\cdot\left(\cot(4\cdot x)\right)^6+\frac{1}{4}\cdot\left(\cot(4\cdot x)\right)^8$
Subject: multivariable_calculus

Problem: Find vector $\vec{c} \times \left(\vec{a}+3 \cdot \vec{b}\right)$ where
$$
\vec{a}=\begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 5 & 0 & 9 \\ 0 & 1 & 0 \end{vmatrix}, \quad
\vec{b}=\begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 0 & -1 & 1 \\ 7 & 1 & -1 \end{vmatrix}, \quad \text{and} \quad
\vec{c}=\vec{i}-\vec{k}
$$

Answer: $\vec{c} \times \left(\vec{a}+3 \cdot \vec{b}\right)$ = $21\cdot\vec{i}-17\cdot\vec{j}+21\cdot\vec{k}$
Subject: precalculus_review

Problem: Solve $\left(\sin(x)\right)^2 + \left(\cos(3 \cdot x)\right)^2 = 1$.

Answer: The final answer: $x=\frac{n\cdot\pi}{4}$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ -1 }{ x^2 \cdot \left(3+x^3\right)^{\frac{ 5 }{ 3 }} } \, dx
$$

Answer: $\int \frac{ -1 }{ x^2 \cdot \left(3+x^3\right)^{\frac{ 5 }{ 3 }} } \, dx$ = $C+\frac{1}{9}\cdot\sqrt[3]{1+\frac{3}{x^3}}+\frac{1}{18\cdot\left(1+\frac{3}{x^3}\right)^{\frac{2}{3}}}$
Subject: multivariable_calculus

Problem: Change the order of integration by integrating first with respect to $z$, then $x$, then $y$:
$$
\int_{0}^1 \int_{1}^2 \int_{2}^3 \left( x^2 + \ln(y) + z \right) \, dx \, dy \, dz
$$

Answer: $I$ = $2\cdot\ln(2)+\frac{35}{6}$
Subject: algebra

Problem: Rewrite the quadratic expression $x^2 + \frac{ 2 }{ 5 } \cdot x - \frac{ 1 }{ 5 }$ by completing the square.

Answer: $x^2 + \frac{ 2 }{ 5 } \cdot x - \frac{ 1 }{ 5 }$ = $\left(x+\frac{1}{5}\right)^2-\frac{6}{25}$