toloka/u-math huggingface dataset:
Subject: multivariable_calculus

Problem: Determine the coordinates, if any, for which $f(x,y) = 6 \cdot x^2 - 3 \cdot x^2 \cdot y + y^3 + 12$ 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: multivariable_calculus

Problem: Evaluate the integral by choosing the order of integration:
$$
\int_{0}^1 \int_{1}^2 \left(\frac{ y }{ x+y^2 }\right) \, dy \, dx
$$

Answer: $\int_{0}^1 \int_{1}^2 \left(\frac{ y }{ x+y^2 }\right) \, dy \, dx$ = $\ln\left(\frac{25\cdot\sqrt{5}}{32}\right)$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \frac{ 1 }{ \sin(x)^7 \cdot \cos(x) } \, dx
$$

Answer: $\int \frac{ 1 }{ \sin(x)^7 \cdot \cos(x) } \, dx$ = $C+\ln\left(\left|\tan(x)\right|\right)-\frac{3}{2\cdot\left(\tan(x)\right)^2}-\frac{3}{4\cdot\left(\tan(x)\right)^4}-\frac{1}{6\cdot\left(\tan(x)\right)^6}$
Subject: sequences_series

Problem: Find the Fourier series of the periodic function $f(x) = x^2$ in the interval $-2 \cdot \pi \leq x < 2 \cdot \pi$ if $f(x) = f(x + 4 \cdot \pi)$.

Answer: The Fourier series is: $\frac{4\cdot\pi^2}{3}+\sum_{n=1}^\infty\left(\frac{16\cdot(-1)^n}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)$
Subject: multivariable_calculus

Problem: If $z = x \cdot y \cdot e^{\frac{ x }{ y }}$, $x = r \cdot \cos\left(\theta\right)$, $y = r \cdot \sin\left(\theta\right)$, find $\frac{ d z }{d r}$ and $\frac{ d z }{d \theta}$ when $r = 2$ and $\theta = \frac{ \pi }{ 6 }$.

Answer: The final answer:

$\frac{ d z }{d r}$: $\sqrt{3}\cdot e^{\left(\sqrt{3}\right)}$

$\frac{ d z }{d \theta}$: $2\cdot e^{\left(\sqrt{3}\right)}-4\cdot\sqrt{3}\cdot e^{\left(\sqrt{3}\right)}$
Subject: sequences_series

Problem: Consider the function $y = \left| \cos\left( \frac{ x }{ 8 } \right) \right|$.

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 $\frac{2}{\pi}-\frac{4}{\pi}\cdot\sum_{n=1}^\infty\left(\frac{(-1)^n}{\left(4\cdot n^2-1\right)}\cdot\cos\left(\frac{n\cdot x}{4}\right)\right)$
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: differential_calc

Problem: For the function $r = \arctan\left(\frac{ m }{ \varphi }\right) + \arccot\left(m \cdot \cot(\varphi)\right)$, find the derivative $r'(0)$ and $r'(2 \cdot \pi)$. Submit as your final answer:

1. $r'(0)$
2. $r'(2 \cdot \pi)$

Answer: 1. $0$
2. $\frac{1}{m}-\frac{m}{m^2+4\cdot\pi^2}$
Subject: algebra

Problem: Rewrite the quadratic expression $x^2 + \frac{ 2 }{ 3 } \cdot x - \frac{ 1 }{ 3 }$ by completing the square.

Answer: $x^2 + \frac{ 2 }{ 3 } \cdot x - \frac{ 1 }{ 3 }$ = $\left(x+\frac{1}{3}\right)^2-\frac{4}{9}$
Subject: sequences_series

Problem: Using the Taylor formula, decompose the function $f(x) = \ln(1+4 \cdot x)$ 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. $\ln(1+4 \cdot x)$ = $4\cdot x-\frac{(4\cdot x)^2}{2}+\frac{(4\cdot x)^3}{3}-\frac{(4\cdot x)^4}{4}+\frac{(4\cdot x)^5}{5}-\frac{(4\cdot x)^6}{6}+\frac{(4\cdot x)^7}{7}-\frac{(4\cdot x)^8}{8}+\frac{(4\cdot x)^9}{9}$
2. Accuracy is not more than $\frac{4^{10}}{10}$
Subject: differential_calc

Problem: Compute the limit:
$$
\lim_{x \to 0}\left(\frac{ x-x \cdot \cos(x) }{ x-\sin(x) }\right)
$$

Answer: $\lim_{x \to 0}\left(\frac{ x-x \cdot \cos(x) }{ x-\sin(x) }\right)$ = $3$
Subject: differential_calc

Problem: Compute the derivative of the function $y = \sqrt{\frac{ x^5 \cdot \left(2 \cdot x^6+3\right) }{ \sqrt[3]{1-2 \cdot x} }}$ by taking the natural log of both sides of the equation.

Answer: Derivative: $y'=\frac{-128\cdot x^7+66\cdot x^6-84\cdot x+45}{-24\cdot x^8+12\cdot x^7-36\cdot x^2+18\cdot x}\cdot\sqrt{\frac{x^5\cdot\left(2\cdot x^6+3\right)}{\sqrt[3]{1-2\cdot x}}}$
Subject: algebra

Problem: 1. On a map, the scale is 1/2 inch : 25 miles. What is the actual distance between two cities that are 3 inches apart?

2. What are the dimensions of a 15 foot by 10 foot room on a blueprint with a scale of 1.5 inches : 2 feet?

3. The distance between Newark, NJ and San Francisco, CA is 2,888 miles. How far would that measure on a map with a scale of 3 cm to 1,000 miles?

4. If the scale for a model train is 1 inch for every 10 feet of a real train, how big is the model for an 83-foot train?

Answer: 1. $150$ miles

2. $11.25$ feet by $7.5$ feet

3. $8.664$ cm

4. $8.3$ inches
Subject: sequences_series

Problem: Find the radius of convergence $R$ and interval of convergence for the power series $\sum_{n=0}^\infty a_{n} \cdot x^n$:

$$
\sum_{n=1}^\infty (-1)^n \cdot \frac{ x^n }{ \ln(2 \cdot n) }
$$

Answer: * $R$ = $1$
* $I$ = $(-1,1]$
Subject: differential_calc

Problem: Sketch the curve:

$$
y = 16 \cdot x^2 \cdot e^{\frac{ 1 }{ 4 \cdot x }}
$$

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

Problem: Evaluate $L=\lim_{P(x,y) \to P\left(1,\frac{ 1 }{ 2 }\right)}\left(f(x,y)\right)$ given $f(x,y) = \frac{ x^2 - 2 \cdot x^3 \cdot y - 2 \cdot x \cdot y^3 + y^2 }{ 1 + x + y - 2 \cdot x \cdot y - 2 \cdot x^2 \cdot y - 2 \cdot x \cdot y^2 }$.

Answer: The final answer: $L=\frac{1}{2}$
Subject: precalculus_review

Problem: Form the compositions $f\left(g(x)\right)$ and $g\left(f(x)\right)$ if $f(x) = \sin(3 \cdot x)$ and $g(x) = \frac{ 1 }{ \sqrt{2-x^2} }$.

1. Find $f\left(g(x)\right)$ and $g\left(f(x)\right)$.
2. Find the domain and range of $f\left(g(x)\right)$ and $g\left(f(x)\right)$.

Answer: 1. $f\left(g(x)\right)$ = $\sin\left(\frac{3}{\sqrt{2-x^2}}\right)$  
$g\left(f(x)\right)$ = $\frac{1}{\sqrt{2-\sin(3\cdot x)^2}}$  

2. Domain of $f\left(g(x)\right)$ is $\left(-\sqrt{2},\sqrt{2}\right)$  
Range of $f\left(g(x)\right)$ is $[-1,1]$  

Domain of $g\left(f(x)\right)$ is $(-\infty,\infty)$  
Range of $g\left(f(x)\right)$ is $\left[\frac{1}{\sqrt{2}},1\right]$
Subject: differential_calc

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

Problem: Solve the following equation:

$$
x^2 - x + 1 = \frac{ 1 }{ 2 } + \sqrt{x - \frac{ 3 }{ 4 }}
$$

Answer: The final answer: $x=1$
Subject: integral_calc

Problem: Consider the function $f(x) = x^2$ on $[-1,1]$ and the partition $\left\{-1, -\frac{ 1 }{ 2 }, \frac{ 1 }{ 4 }, 1\right\}$. Find the upper and lower sums.

Answer: The upper sum is: $\frac{23}{16}$
The lower sum is: $\frac{11}{64}$
Subject: multivariable_calculus

Problem: Evaluate $\int\int\int_{E}{(x+2 \cdot y \cdot z) \, dV}$, where $E = \left\{(x,y,z) | 0 \le x \le 1, 0 \le y \le x, 0 \le z \le 5-x-y \right\}$.

Answer: $I$ = $\frac{439}{120}$
Subject: sequences_series

Problem: Calculate $e$ with an estimated error of $0.001$, using the series expansion.

Answer: The final answer: $2.7181$
Subject: integral_calc

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

Answer: $-2 \cdot \int x^{-4} \cdot \left(4+x^2\right)^{\frac{ 1 }{ 2 }} \, dx$ = $C+\frac{1}{6}\cdot\left(\frac{4}{x^2}+1\right)\cdot\sqrt{\frac{4}{x^2}+1}$
Subject: sequences_series

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

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

Problem: Compute $\lim_{x \to 0}\left(\frac{ 2 \cdot \cos(x)+4 }{ 5 \cdot x^3 \cdot \sin(x) }-\frac{ 6 }{ 5 \cdot x^4 }\right)$. Use the expansion of the function in the Taylor series.

Answer: The final answer: $\frac{1}{150}$
Subject: precalculus_review

Problem: Evaluate $\cos\left(2 \cdot \arctan(-7)\right)$.

Answer: The final answer: $\cos(a)=-\frac{24}{25}$
Subject: precalculus_review

Problem: Find the equivalent of $\sin\left(\arctan(x)\right)$ in terms of an algebraic expression.

Answer: The final answer: $\frac{x}{\sqrt{1+x^2}}$
Subject: multivariable_calculus

Problem: Differentiate $y = \sqrt{\frac{ (x-1) \cdot (x-2) }{ (x-3) \cdot (x-4) }}$.

Answer: $\frac{ d y }{d x}$ = $-\left(\frac{2\cdot x^2-10\cdot x+11}{(x-1)^{\frac{1}{2}}\cdot(x-2)^{\frac{1}{2}}\cdot(x-3)^{\frac{3}{2}}\cdot(x-4)^{\frac{3}{2}}}\right)$
Subject: differential_calc

Problem: For the function $y = 3 \cdot x \cdot \sqrt[3]{x} - 6 \cdot \sqrt[3]{x^2} + 24 \cdot \sqrt[3]{x} - 8 \cdot x$, specify the points where local maxima and minima of $y$ 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: $P(1,13)$
2. The point(s) where local minima occur: $P(-1,-19)$, $P(8,8)$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \left(\frac{ x+4 }{ x-4 } \right)^{\frac{ 3 }{ 2 }} \, dx
$$

Answer: $\int \left(\frac{ x+4 }{ x-4 } \right)^{\frac{ 3 }{ 2 }} \, dx$ = $C+\sqrt{\frac{x+4}{x-4}}\cdot(x-20)-12\cdot\ln\left(\left|\frac{\sqrt{x-4}-\sqrt{x+4}}{\sqrt{x-4}+\sqrt{x+4}}\right|\right)$
Subject: precalculus_review

Problem: Solve the differential equation:
$$
\frac{ d y }{d \theta} = \sin\left(\pi \cdot \theta\right)^4
$$

Answer: $y(\theta)$ = $\frac{\sin\left(4\cdot\pi\cdot\theta\right)}{32\cdot\pi}+\frac{3\cdot\theta}{8}-\frac{\sin\left(2\cdot\pi\cdot\theta\right)}{4\cdot\pi}+C$
Subject: algebra

Problem: A doctor injects a patient with $13$ milligrams of radioactive dye that decays exponentially. After $12$ minutes, there are $4.75$ milligrams of dye remaining in the patient's system. Which is an appropriate model for this situation?

Answer: Model formula: $f(t)=13\cdot e^{\frac{\ln\left(\frac{4.75}{13}\right)}{12}\cdot t}$
Subject: multivariable_calculus

Problem: Find the first derivative $y_{x}'$ of the function:
$$
x = \arcsin\left(\frac{ 2 \cdot t }{ \sqrt{4+4 \cdot t^2} }\right), \quad y = \arccos\left(\frac{ 2 }{ \sqrt{4+4 \cdot t^2} }\right), \quad t \ge 0
$$

Answer: $y_{x}'$ = $1$
Subject: algebra

Problem: Find all real solutions of the following equation:

$$
\frac{ x }{ 2 x+6 }-\frac{ 2 }{ 5 x+5 }=\frac{ 5 x^2-3 x-7 }{ 10 x^2+40 x+30 }
$$

Answer: The real solution(s) to the given equation are: $x=\frac{5}{4}$
Subject: algebra

Problem: Solve the following equations:

1. $8.6 = 6j + 4j$
2. $12z - (4z + 6) = 82$
3. $5.4d - 2.3d + 3(d - 4) = 16.67$
4. $2.6f - 1.3(3f - 4) = 6.5$
5. $-5.3m + (-3.9m) - 17 = -94.28$
6. $6(3.5y + 4.2) - 2.75y = 134.7$

Answer: The solutions to the given equations are:
1. $j=0.86$
2. $z=11$
3. $d=\frac{ 47 }{ 10 }$
4. $f=-1$
5. $m=\frac{ 42 }{ 5 }$
6. $y=6$
Subject: multivariable_calculus

Problem: Compute the second order derivative $\frac{d ^2y}{ d x^2}$ for the parametrically defined function $x = 2 \cdot e^{3 \cdot t} \cdot \cos(t)$, $y = e^{3 \cdot t} \cdot \sin(t)$.

Answer: $\frac{d ^2y}{ d x^2}$ = $\frac{5}{2\cdot e^{3\cdot t}\cdot\left(3\cdot\cos(t)-\sin(t)\right)^3}$
Subject: multivariable_calculus

Problem: Find the volume of the prism with vertices $A(0,0,0)$, $B(4,0,0)$, $C(4,6,0)$, $D(0,6,0)$, $E(0,0,1)$, and $F(4,0,1)$.

Answer: $V$ = $12$
Subject: multivariable_calculus

Problem: Find the point of intersection of the following surfaces:

1. $\frac{ 1 }{ 2 } \cdot y = x - z - 5$
2. $4 \cdot x^2 + 9 \cdot y^2 - 72 \cdot z = 0$

Answer: The final answer: $x=9$, $y=-2$, $z=5$
Subject: sequences_series

Problem: Consider the power series $\sum_{k=0}^\infty \frac{ (2 \cdot x-1)^k }{ 3^k \cdot \sqrt{k+1} }$.

1. Find the center of convergence of the series.
2. Find the radius of convergence of the series.

Answer: 1. The center of convergence is $\frac{1}{2}$.
2. The radius of convergence is $\frac{3}{2}$.
Subject: multivariable_calculus

Problem: Find the points at which the following polar curve $r = 4 \cdot \cos(\theta)$ has a horizontal or vertical tangent line.

Answer: The final answer:

1. Horizontal tangents at: $P\left(2\cdot\sqrt{2},\frac{\pi}{4}\right)$, $P\left(-2\cdot\sqrt{2},\frac{3\cdot\pi}{4}\right)$
2. Vertical tangents at: $P\left(0,\frac{\pi}{2}\right)$, $P(4,0)$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \sin(x)^4 \cdot \cos(x)^6 \, dx
$$

Answer: $\int \sin(x)^4 \cdot \cos(x)^6 \, dx$ = $C+\frac{1}{320}\cdot\left(\sin(2\cdot x)\right)^5+\frac{1}{128}\cdot\left(\frac{3\cdot x}{2}-\frac{\sin(4\cdot x)}{2}+\frac{\sin(8\cdot x)}{16}\right)$
Subject: precalculus_review

Problem: A biologist observes that a certain bacterial colony triples every 4 hours. Approximately how many hours does it take for it to double in size? (Give your answer to two decimal places.)

Answer: The final answer: $2.52$
Subject: multivariable_calculus

Problem: Compute the second order derivative $\frac{d ^2y}{ d x^2}$ for the parametrically defined function
$$
x = 4 \cdot e^t \cdot \cos(3 \cdot t), \quad y = e^t \cdot \sin(3 \cdot t)
$$

Answer: $\frac{d ^2y}{ d x^2}$ = $\frac{15}{8\cdot e^t\cdot\left(\cos(3\cdot t)-3\cdot\sin(3\cdot t)\right)^3}$
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 $x^{\frac{ 1 }{ 2 }}+y^{\frac{ 1 }{ 2 }}=6^{\frac{ 1 }{ 2 }}$.

Answer: Volume = $\pi\cdot\frac{72}{5}$
Subject: precalculus_review

Problem: Solve $3 \cdot \sin(x) + 4 \cdot \cos(x) = 5$.

Answer: The final answer: $x=-\arcsin\left(\frac{4}{5}\right)+\frac{\pi}{2}+2\cdot\pi\cdot n$, $x=-\arccos\left(\frac{3}{5}\right)+\frac{\pi}{2}+2\cdot\pi\cdot n$
Subject: integral_calc

Problem: Calculate the integral:
$$
\int_{-\sqrt{5}}^{\sqrt{5}} \frac{ 4 \cdot x^7+6 \cdot x^6-12 \cdot x^5-14 \cdot x^3-24 \cdot x^2+2 \cdot x+4 }{ x^2+2 } \, dx
$$

Answer: $\int_{-\sqrt{5}}^{\sqrt{5}} \frac{ 4 \cdot x^7+6 \cdot x^6-12 \cdot x^5-14 \cdot x^3-24 \cdot x^2+2 \cdot x+4 }{ x^2+2 } \, dx$ = $20\cdot\sqrt{5}+4\cdot\sqrt{2}\cdot\arctan\left(\frac{\sqrt{5}}{\sqrt{2}}\right)$
Subject: multivariable_calculus

Problem: Find a rectangular equation which is equivalent to the following parametric equations:

1. $x^2 = t^3 - 3 \cdot t^2 + 3 \cdot t - 1$
2. $y^2 = t^3 + 6 \cdot t^2 + 12 \cdot t + 8$

Answer: The final answer: $\sqrt[3]{x^2}-\sqrt[3]{y^2}=-3$
Subject: algebra

Problem: Solve the following equations:

1. Five less than twice a number $x$ is nineteen. What is the number $x$?
2. Four times the sum of a number $x$ and −3 is 28.
3. Three times twenty-one more than a number $x$ is twelve. What is the number?
4. Find the width $w$ of a rectangle if its length is 5 more than the width and its perimeter is 90 cm.

Answer: The solutions to the given equations are:

1. $x=12$
2. $x=10$
3. $x=-17$
4. $w=20$
Subject: differential_calc

Problem: $f(x) = x + \sin(2 \cdot x)$ over $x = \left[-\frac{ \pi }{ 2 },\frac{ \pi }{ 2 }\right]$. Determine:
1. intervals where $f$ is increasing
2. intervals where $f$ is decreasing
3. local minima of $f$
4. local maxima of $f$
5. intervals where $f$ is concave up
6. intervals where $f$ is concave down
7. the inflection points of $f$

Answer: 1. intervals where $f$ is increasing: $\left(-\frac{\pi}{3},\frac{\pi}{3}\right)$
2. intervals where $f$ is decreasing: $\left(-\frac{\pi}{2},-\frac{\pi}{3}\right)$, $\left(\frac{\pi}{3},\frac{\pi}{2}\right)$
3. local minima of $f$: $-\frac{\pi}{3}$
4. local maxima of $f$: $\frac{\pi}{3}$
5. intervals where $f$ is concave up: $\left(-\frac{\pi}{2},0\right)$
6. intervals where $f$ is concave down: $\left(0,\frac{\pi}{2}\right)$
7. the inflection points of $f$: $0$
Subject: algebra

Problem: Solve the following equations:

1. $13x + 6 = 6$
2. $\frac{x}{-4} + 11 = 5$
3. $-4.5x + 12.3 = -23.7$
4. $\frac{x}{5} + 4 = 4.3$
5. $-\frac{x}{3} + (-7.2) = -2.1$
6. $5.4x - 8.3 = 14.38$
7. $\frac{x}{3} - 14 = -8$

Answer: The solutions to the given equations are:
1. $x=0$
2. $x=24$
3. $x=8$
4. $x=\frac{ 3 }{ 2 }$
5. $x=\frac{ -153 }{ 10 }$
6. $x=\frac{ 21 }{ 5 }$
7. $x=18$
Subject: integral_calc

Problem: Compute the integral:
$$
-10 \cdot \int \frac{ \cos(5 \cdot x)^4 }{ \sin(5 \cdot x)^3 } \, dx
$$

Answer: $-10 \cdot \int \frac{ \cos(5 \cdot x)^4 }{ \sin(5 \cdot x)^3 } \, dx$ = $C+3\cdot\cos(5\cdot x)+\frac{\left(\cos(5\cdot x)\right)^3}{1-\left(\cos(5\cdot x)\right)^2}-\frac{3}{2}\cdot\ln\left(\frac{1+\cos(5\cdot x)}{1-\cos(5\cdot x)}\right)$
Subject: algebra

Problem: Divide the rational expressions:
$$
\frac{ q^2-36 }{ q^2+12 \cdot q+36 } \div \frac{ q^2-4 \cdot q-12 }{ q^2+4 \cdot q-12 }
$$

Answer: The final answer: $\frac{q-2}{q+2}$
Subject: precalculus_review

Problem: Find all values of $t$ that satisfy the following equation:
$$
\log_{a}(t) + \log_{2 \cdot a}(t) = \frac{ \ln\left(2 \cdot a^2\right) }{ \ln(2 \cdot a) }
$$
for $a > 0$ and $a \ne 1$.

Answer: $t$ = $a$
Subject: differential_calc

Problem: Find the maximum and minimum values of the function $y = 4 \cdot \sin(x) + \sin(4 \cdot x)$ in the closed interval $[0, \pi]$.

Answer: Maximum Value: $\frac{5\cdot\sqrt{2}\cdot\sqrt{5+\sqrt{5}}}{4}$
Minimum Value: $0$
Subject: algebra

Problem: Divide the rational expressions:
$$
\frac{ q^2-16 }{ q^2+8 \cdot q+16 } \div \frac{ q^2+q-20 }{ q^2-q-20 }
$$

Answer: The final answer: $\frac{q-5}{q+5}$
Subject: differential_calc

Problem: Sketch the curve:  

$y = \sqrt{\frac{ 27 - x^3 }{ 2 \cdot x }}$.

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): $(0,3]$
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: $(0,3]$
7. Intervals where the function is concave up: $\left(0,\frac{3}{\sqrt[3]{4}}\right)$
8. Intervals where the function is concave down: $\left(\frac{3}{\sqrt[3]{4}},3\right)$
9. Points of inflection: $P\left(\frac{3}{\sqrt[3]{4}},2.3146\right)$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \frac{ \cos(x)^3 }{ \sin(x)^9 } \, dx
$$

Answer: $\int \frac{ \cos(x)^3 }{ \sin(x)^9 } \, dx$ = $C-\left(\frac{1}{3}\cdot\left(\cot(x)\right)^6+\frac{1}{4}\cdot\left(\cot(x)\right)^4+\frac{1}{8}\cdot\left(\cot(x)\right)^8\right)$
Subject: algebra

Problem: An epidemiological study of the spread of a certain influenza strain that hit a small school population found that the total number of students, $P$, who contracted the flu $t$ days after it broke out is given by the model $P = t^2 - 14t + 200$, where $1 \leq t \leq 6$. Find the day that 160 students had the flu. Recall that the restriction on $t$ is at most 6.

Answer: The final answer: $4$
Subject: algebra

Problem: Using the Rational Zero Theorem, list all possible rational zeros of the following polynomial:

$p(x) = 2 \cdot x^3 + 3 \cdot x^2 - 8 \cdot x + 5$

Answer: Possible rational zeros are $1$, $-1$, $5$, $-5$, $\frac{1}{2}$, $-\frac{1}{2}$, $\frac{5}{2}$, $-\frac{5}{2}$
Subject: algebra

Problem: You sold 4 more than three times as many newspapers this week as last week. If you sold 112 newspapers altogether, how many did you sell this week?

Answer: The number of newspapers sold this week is: $85$
Subject: sequences_series

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

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

Problem: Identify all points of removable discontinuity (singularity) of the function $f(x) = \frac{ x^2 - 16 }{ x - 4 }$.

Answer: $f(x)$ has removable discontinuities at $x=4$
Subject: algebra

Problem: Find the equations of the following objects:

1. The sphere centered at $P(1,3,5)$ through $P(0,1,7)$. Also find its radius.
2. The points equidistant from $P(0,0,0)$ and $P(0,1,3)$.
3. The cylinder with radius $10$ and central axis line $y=3$, $z=5$.

Answer: 1. Equation: $(x-1)^2+(y-3)^2+(z-5)^2=9$ Radius: $3$
2. Equation: $y+3\cdot z=5$
3. Equation: $(y-3)^2+(z-5)^2=10^2$
Subject: multivariable_calculus

Problem: Find the volume of the solid whose boundaries are given in the rectangular coordinates:
$$
\sqrt{x^2+y^2} \le z \le \sqrt{16-x^2-y^2}, \quad x \ge 0, \quad y \ge 0
$$

Answer: Volume: $\frac{64\cdot\pi-32\cdot\pi\cdot\sqrt{2}}{6}$
Subject: multivariable_calculus

Problem: Evaluate $L=\lim_{(x,y) \to (2,2)}\left(\frac{ 2 \cdot x^2+2 \cdot x^2 \cdot y+2 \cdot x-2 \cdot x \cdot y-2 \cdot x \cdot y^2-2 \cdot y }{ 3 \cdot x^2-6 \cdot x \cdot y-3 \cdot x^2 \cdot y+3 \cdot x \cdot y^2+3 \cdot y^2 }\right)$

Answer: The final answer: $L=-\frac{7}{6}$
Subject: differential_calc

Problem: Consider the differential equation $\frac{ d y }{d x} = \frac{ 4+y }{ x }$. Find the particular solution $y=f(x)$ to the given differential equation with the initial condition $f(3) = -3$.

Answer: $y$ = $\frac{|x|}{3}-4$
Subject: algebra

Problem: Given the rational function $f(x) = \frac{ 3 \cdot x - 4 }{ x^3 - 16 \cdot x }$, find:

1. the domain (in interval notation),
2. vertical asymptotes (in the form $x=a$),
3. horizontal asymptotes (in the form $y=c$).

Answer: 1. The domain in interval notation is $(-\infty,-4)\cup(-4,0)\cup(0,4)\cup(4,\infty)$
2. Vertical asymptotes of $f(x)$: $x=4$, $x=0$, $x=-4$
3. Horizontal asymptotes of $f(x)$: $y=0$
Subject: multivariable_calculus

Problem: Determine a definite integral that represents the region enclosed by one petal of $r = \cos(3 \cdot \theta)$.

Answer: The final answer: $\int_0^{\frac{\pi}{6}}\cos\left(3\cdot\theta\right)^2d\theta$
Subject: integral_calc

Problem: Calculate the integral:
$$
\int \frac{ \sqrt[5]{x}+\sqrt[5]{x^4}+x \cdot \sqrt[5]{x} }{ x \cdot \left(1+\sqrt[5]{x^2}\right) } \, dx
$$

Answer: $\int \frac{ \sqrt[5]{x}+\sqrt[5]{x^4}+x \cdot \sqrt[5]{x} }{ x \cdot \left(1+\sqrt[5]{x^2}\right) } \, dx$ = $C+5\cdot\arctan\left(\sqrt[5]{x}\right)+\frac{5}{4}\cdot\sqrt[5]{x}^4$