toloka/u-math huggingface dataset:
Subject: differential_calc

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

Problem: If $x = \sin(\theta)$, find $\cot(\arcsin(x))$.

Answer: $\cot(\arcsin(x))$ = $\frac{\sqrt{1-x^2}}{x}$ (Enter your solution as an expression using only the variable $x$.)
Subject: multivariable_calculus

Problem: The force of gravity $\vec{F}$ acting on an object is given by $\vec{F} = m \cdot \vec{g}$, where $m$ is the mass of the object (expressed in kilograms) and $\vec{g}$ is acceleration resulting from gravity, with $\left\lVert\vec{g}\right\rVert = 9.8$ N/kg. A $2$-kg disco ball hangs by a chain from the ceiling of a room.

1. Find the force of gravity $\vec{F}$ acting on the disco ball and find its magnitude.
2. Find the force of tension $\vec{T}$ in the chain and its magnitude.

Express answers using standard unit vectors.

Answer: 1. $\vec{F} = $-19.6\cdot\vec{k}$$; $\left\lVert\vec{F}\right\rVert = $19.6$$
2. $\vec{T} = $-19.6\cdot\vec{k}$$; $\left\lVert\vec{T}\right\rVert = $19.6$$
Subject: sequences_series

Problem: Evaluate the integral:
$$
\int_{0}^{\frac{ 1 }{ 2 }} \sqrt[5]{1+x^3} \, dx
$$
with accuracy $\frac{ 1 }{ 100 }$ using power series expansion.

Answer: The final answer: $0.503$
Subject: differential_calc

Problem: Compute the limit:
$$
\lim_{x \to 0} \left( \frac{ x-1 }{ 2 \cdot x^2 } + \frac{ 1 }{ x \cdot \left( e^{2 \cdot x} - 1 \right) } \right)
$$

Answer: The final answer to the problem is: $\frac{1}{6}$
Subject: algebra

Problem: For what value of $m$ does the inequality $m \cdot x + 4 \le 2 \cdot x + 5 \cdot (x - 1)$ have no real solutions?

Answer: $m$ = $7$
Subject: sequences_series

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

Answer: The final answer: $\frac{1}{180}$
Subject: algebra

Problem: Rewrite the quadratic expression $3 + 2 \cdot z - 5 \cdot z^2$ by completing the square.

Answer: $3 + 2 \cdot z - 5 \cdot z^2$ = $-5\cdot\left(z-\frac{1}{5}\right)^2+\frac{16}{5}$
Subject: differential_calc

Problem: Sketch the curve:  
$$
y = \sqrt{\frac{ 216-x^3 }{ 4 \cdot x }}
$$
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): $(0,6]$
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: $(0,6]$
7. Intervals where the function is concave up: $\left(0,3\cdot\sqrt[3]{2}\right)$
8. Intervals where the function is concave down: $\left(3\cdot\sqrt[3]{2},6\right)$
9. Points of inflection: $P\left(3\cdot\sqrt[3]{2},3.2734\right)$
Subject: integral_calc

Problem: Find the integral:
$$
\int \frac{ \arcsin(4 \cdot x) }{ \sqrt{4 \cdot x+1} } \, dx
$$

Answer: Answer is: $\frac{1}{2}\cdot\sqrt{4\cdot x+1}\cdot\arcsin(4\cdot x)-\left(C-\sqrt{1-4\cdot x}\right)$
Subject: precalculus_review

Problem: $P = \left(x,-\frac{ \sqrt{15} }{ 4 }\right)$, $x > 0$ 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: $\frac{1}{4}$
2. The values of the six trigonometric functions are:
    * $\sin\left(\theta\right)$ = $-\frac{\sqrt{15}}{4}$
    * $\cos\left(\theta\right)$ = $\frac{1}{4}$
    * $\tan\left(\theta\right)$ = $-\sqrt{15}$
    * $\csc\left(\theta\right)$ = $-\frac{4\cdot\sqrt{15}}{15}$
    * $\sec\left(\theta\right)$ = $4$
    * $\cot\left(\theta\right)$ = $-\frac{\sqrt{15}}{15}$
Subject: differential_calc

Problem: Find all values of the constant $c$ such that the limit 
$$\lim_{x \to -\infty} \left(\frac{ 3^{c \cdot x}+4 }{ 3^{4 \cdot x}+4 }\right)$$
exists. Enter the range for the constant $c$ as an interval of the real line.

Answer: The final answer: $(-\infty,\infty)$
Subject: precalculus_review

Problem: Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{0}^{2 \cdot \pi} \left(\sin(x) \cdot \sin(2 \cdot x) \cdot \sin(3 \cdot x)\right) \, dx
$$

Answer: $\int_{0}^{2 \cdot \pi} \left(\sin(x) \cdot \sin(2 \cdot x) \cdot \sin(3 \cdot x)\right) \, dx$ = $0$
Subject: multivariable_calculus

Problem: $E$ is located inside the sphere $x^2+y^2+z^2=1$, above the xy-plane, and inside the circular cone $z=\sqrt{x^2+y^2}$. Find the volume of $E$.

Answer: Volume = $\frac{2\cdot\pi-\pi\cdot\sqrt{2}}{3}$
Subject: precalculus_review

Problem: Find expressions for $\cosh(x) + \sinh(x)$ and $\cosh(x) - \sinh(x)$.

Answer: The final answer:

$\cosh(x) + \sinh(x)$: $e^x$

$\cosh(x) - \sinh(x)$: $e^{-x}$
Subject: precalculus_review

Problem: Find points on a coordinate plane that satisfy the following equation:
$$
x^4 + y^4 + 2 \cdot x^2 \cdot y^2 - 9 \cdot x^2 - 10 \cdot y^2 - 4 \cdot x + 29 = 0
$$

Answer: The final answer: $(2,-1)$, $(2,1)$
Subject: algebra

Problem: Divide the rational expressions:
$$
\frac{ q^2-9 }{ q^2+6 \cdot q+9 } \div \frac{ q^2-2 \cdot q-3 }{ q^2+2 \cdot q-3 }
$$

Answer: The final answer: $\frac{q-1}{q+1}$
Subject: sequences_series

Problem: Find a “reasonable” upper-bound on the error in approximating $f(x) = (x-1) \cdot \ln(x-1)$ by its 3rd order Taylor polynomial $P_{3}(x)$ about $a=2$ valid for all values of $x$ such that $|x-2| \le 0.3$.

Answer: The final answer: $\frac{2}{(0.7)^3}\cdot\frac{(0.3)^4}{4!}$
Subject: precalculus_review

Problem: Evaluate the sum $S = \sin(a) + \sin(2 \cdot a) + \sin(3 \cdot a) + \ldots + \sin(n \cdot a)$, where $a \ne \pi \cdot k$.

Answer: The final answer: $S=\frac{\sin\left(\frac{(n+1)\cdot a}{2}\right)\cdot\sin\left(\frac{n\cdot a}{2}\right)}{\sin\left(\frac{a}{2}\right)}$
Subject: differential_calc

Problem: Find the derivative of the function $y = \frac{ 2 \cdot \csc(x) - 7 \cdot \sin(x) }{ 4 \cdot \left(\cos(x)\right)^5 } - \frac{ 3 }{ 5 } \cdot \cot(2 \cdot x)$.

Answer: $y'$ = $\frac{6}{5\cdot\left(\sin(2\cdot x)\right)^2}+\frac{28\cdot\left(\cos(x)\right)^6-25\cdot\left(\cos(x)\right)^4-2\cdot\left(\cos(x)\right)^6\cdot\left(\csc(x)\right)^2}{4\cdot\left(\cos(x)\right)^{10}}$
Subject: integral_calc

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

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

Problem: Solve the integral:
$$
\int 2 \cdot \tan(-10 \cdot x)^4 \, dx
$$

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

Problem: A falling object travels a distance given by the formula $d = 5 \cdot t + 16 \cdot t^2$ (ft), where $t$ is measured in seconds. How long will it take for the object to travel $74$ ft?

Answer: The final answer: $t=2$
Subject: algebra

Problem: A town's population has been decreasing at a constant rate. In 2010, the population was $5900$. By 2012, the population had dropped to $4700$. Assuming this trend continues, predict the population in 2016.

Answer: The population in 2016 is predicted to be $2300$
Subject: algebra

Problem: Find the formula for an exponential function that passes through the two points: $P(-2,6)$ and $P(3,1)$.

Answer: $f(x)$ = $6^{\frac{3}{5}}\cdot6^{-\frac{x}{5}}$
Subject: integral_calc

Problem: Find $I=\int \frac{ 5 }{ 1+\sqrt{(x+1)^2+1} } \, dx$.

Answer: The final answer: $I=5\cdot\ln\left(x+1+\sqrt{x^2+2\cdot x+2}\right)+\frac{10}{x+2+\sqrt{x^2+2\cdot x+2}}+C$
Subject: multivariable_calculus

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

Answer: $I$ = $\frac{73}{30}-\frac{52\cdot\sqrt{2}}{35}$
Subject: integral_calc

Problem: Calculate the integral:
$$
\int \frac{ M \cdot x + N }{ \left( x^2 + p \cdot x + q \right)^m } \, dx
$$
where $M = 4$, $N = 5$, $p = 2$, $q = 9$, and $m = 2$.

Answer: $\int \frac{ M \cdot x + N }{ \left( x^2 + p \cdot x + q \right)^m } \, dx$ = $C+\frac{x+1}{128+16\cdot(x+1)^2}+\frac{\sqrt{2}}{64}\cdot\arctan\left(\frac{1}{2\cdot\sqrt{2}}\cdot(x+1)\right)-\frac{2}{8+(x+1)^2}$
Subject: differential_calc

Problem: Make full curve sketching of $y = \sqrt[3]{5 \cdot x^2 - \frac{ x^3 }{ 2 }}$. 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) $(-1\cdot\infty,\infty)$
2. Vertical asymptotes None
3. Horizontal asymptotes None
4. Slant asymptotes $y=-\frac{1}{\sqrt[3]{2}}\cdot x+\frac{10}{3\cdot\sqrt[3]{2}}$
5. Intervals where the function is increasing $\left(0,\frac{20}{3}\right)$
6. Intervals where the function is decreasing $(-\infty,0)$, $(0,10)$
7. Intervals where the function is concave up $(10,\infty)$
8. Intervals where the function is concave down $\left(\frac{20}{3},\infty\right)$, $(-\infty,0)$
9. Points of inflection $P(10,0)$
Subject: integral_calc

Problem: Find the mass of an oversized hockey puck of radius 2 in. with density function $\rho(x) = x^3 - 2 \cdot x + 5$ that is centered at the origin.

Answer: $m$ = $\frac{332\cdot\pi}{15}$
Subject: multivariable_calculus

Problem: The position function for an object in three dimensions is given by the equation $\vec{r}(t) = t \cdot \cos(t) \cdot \vec{i} + t \cdot \sin(t) \cdot \vec{j} + 3 \cdot t \cdot \vec{k}$. Find the tangential and normal components of acceleration when $t = 1.5$. Round your answer to two decimal digits.

Answer: $a_{T}(1.5)$ = $0.43$ meters per second squared

$a_{N}(1.5)$ = $2.46$ meters per second squared
Subject: sequences_series

Problem: Find the Fourier series of the function $f(x) = \frac{ 1 }{ 3 } \cdot x$ in the interval $[-4,4]$.

Answer: The Fourier series is: $\sum_{n=1}^\infty\left(\frac{\frac{8}{3}\cdot(-1)^{n+1}}{\pi\cdot n}\cdot\sin\left(\frac{\pi\cdot n\cdot x}{4}\right)\right)$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \frac{ 7 }{ \sin(-2 \cdot x)^3 \cdot \cos(2 \cdot x)^2 } \, dx
$$

Answer: $\int \frac{ 7 }{ \sin(-2 \cdot x)^3 \cdot \cos(2 \cdot x)^2 } \, dx$ = $-7\cdot\left(C+\frac{3}{8}\cdot\ln\left(\left|\cos(2\cdot x)-1\right|\right)+\frac{1}{2\cdot\cos(2\cdot x)}+\frac{1}{8\cdot\left(1+\cos(2\cdot x)\right)}+\frac{1}{8\cdot\left(\cos(2\cdot x)-1\right)}-\frac{3}{8}\cdot\ln\left(\left|1+\cos(2\cdot x)\right|\right)\right)$
Subject: sequences_series

Problem: Expand the function $f(x) = \ln(1 + 4 \cdot x)$ given on the interval $[0,1]$ in powers of $x$ using the Maclaurin formula. Estimate the error allowed with the retention of the first ten members.

Submit as your final answer:
1. the resulting expansion of the function (the first ten terms)
2. the estimate of the absolute value of the remainder term (in the form of an inequality for $\left|R_{10}(x)\right|$)

Answer: 1. $4\cdot x-\frac{(4\cdot x)^2}{2}+\frac{(4\cdot x)^3}{3}-\frac{(4\cdot x)^4}{4}+\frac{(4\cdot x)^5}{5}-\frac{(4\cdot x)^6}{6}+\frac{(4\cdot x)^7}{7}-\frac{(4\cdot x)^8}{8}+\frac{(4\cdot x)^9}{9}$
2. $\left|R_{10}(x)\right|<\frac{4^{10}}{10}$
Subject: integral_calc

Problem: Compute the integral using the Substitution Rule:
$$
\int \frac{ x^2+3 }{ \sqrt{(2 \cdot x-5)^3} } \, dx
$$

Answer: The final answer: $C+\frac{60\cdot x+(2\cdot x-5)^2-261}{12\cdot\sqrt{2\cdot x-5}}$
Subject: sequences_series

Problem: Use partial fractions to find the power series of the function $\frac{ 5 }{ \left(x^2+4\right) \cdot \left(x^2-1\right) }$.

Answer: $\frac{ 5 }{ \left(x^2+4\right) \cdot \left(x^2-1\right) }$ = $\sum_{n=0}^\infty\left(\left(-1+(-1)^{n+1}\cdot\frac{1}{2^{2\cdot n+2}}\right)\cdot x^{2\cdot n}\right)$
Subject: precalculus_review

Problem: Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{0}^{\frac{ \pi }{ 2 }} \sqrt{1-\cos(2 \cdot x)} \, dx
$$

Answer: $\int_{0}^{\frac{ \pi }{ 2 }} \sqrt{1-\cos(2 \cdot x)} \, dx$ = $\sqrt{2}$
Subject: integral_calc

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

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

Problem: Find the solutions of $(x+2)^4 + (x+5)^4 = 17$.

Answer: The final answer: $x_1=-4$, $x_2=-3$
Subject: sequences_series

Problem: Find the sum of the series $\sum_{n=0}^\infty \left(\frac{ (-1)^n }{ (2 \cdot n+1)! }\right)$ with an estimate error of $0.01$.

Answer: The final answer: $\frac{101}{120}$
Subject: algebra

Problem: Use the vertex $P(h,k) = P(-3,-2)$ and a point $P(x,y) = P(-1,3)$ on the graph of $f(x)$ to find the general form of the quadratic function.

Answer: The final answer: $f(x)=\frac{1}{4}\cdot\left(5\cdot x^2+30\cdot x+37\right)$
Subject: sequences_series

Problem: Compute the integral $\int_{0}^1{e^{-x^2} \, dx}$ with an estimated error of $0.0001$ using its series expansion.

Answer: The final answer: $0.74683603$
Subject: differential_calc

Problem: For the function $f(x) = x^{11} - 6 \cdot x^{10}$, determine:

1. Intervals where:
    1. $f$ is increasing
    2. $f$ is decreasing
    3. $f$ is concave up
    4. $f$ is concave down

2. find:
    1. local minima
    2. local maxima
    3. the inflection points of $f$

Answer: 1. Intervals where:
    1. $f$ is increasing: $(-\infty,0)$, $\left(\frac{60}{11},\infty\right)$
    2. $f$ is decreasing: $\left(0,\frac{60}{11}\right)$
    3. $f$ is concave up: $\left(\frac{54}{11},\infty\right)$
    4. $f$ is concave down: $(-\infty,0)$, $\left(0,\frac{54}{11}\right)$

2. find:
    1. local minima: $\frac{60}{11}$
    2. local maxima: $0$
    3. the inflection points of $f$: $P\left(\frac{54}{11},-\frac{2529990231179046912}{285311670611}\right)$
Subject: multivariable_calculus

Problem: Consider points $P(3,7,-2)$ and $Q(1,1,-3)$. Determine the angle between vectors $\vec{OP}$ and $\vec{OQ}$. Express the answer in radians, rounded to two decimal places.

Answer: $\theta$ = $0.91$
Subject: multivariable_calculus

Problem: Determine the equation of the parabola with a focus at $(-1,4)$ and a directrix of $x=5$.

Answer: The equation is: $(y-4)^2=-12\cdot(x-2)$
Subject: algebra

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

$$
\frac{ x-1 }{ x+1 } - \frac{ 2 \cdot x + 3 }{ 2 \cdot x + 1 }
$$

Answer: The final answer: $\frac{-6\cdot x-4}{(x+1)\cdot(2\cdot x+1)}$
Subject: multivariable_calculus

Problem: Find the distance from $\left\langle 2,3,1 \right\rangle$ to the line through the origin parallel to $\left\langle 5,1,4 \right\rangle$.

Answer: The final answer: $\sqrt{\frac{299}{42}}$
Subject: algebra

Problem: What are the intercepts and asymptotes of the function $P(t) = \frac{ 20 }{ 1+4 \cdot e^{-0.5 \cdot t} }$? If there are no asymptotes or intercepts, then write $\text{None}$.

Answer: x-intercept: None
y-intercept: $P(0,4)$

asymptotes: $y=20$, $y=0$
Subject: multivariable_calculus

Problem: Evaluate the double integral:
$$
\int \int \rho^2 \, d \varphi \, d \rho
$$
where the integration area $Q$ is bounded by the curve $\rho = a \cdot \sin(2 \cdot \varphi)$.

Answer: $\int \int \rho^2 \, d \varphi \, d \rho$ = $\frac{8}{9}\cdot a^3$
Subject: sequences_series

Problem: Let $f(x) = \ln\left(2 \cdot x^2 + 1\right)$.

1. Find the 4th order Taylor polynomial $P_{4}(x)$ of $f(x)$ about $a=0$.
2. Compute $\left|f(1) - P_{4}(1)\right|$.
3. Compute $\left|f(0.1) - P_{4}(0.1)\right|$.

Answer: 1. $2\cdot x^2-2\cdot x^4$
2. $1.09861229$
3. $2.62729\cdot10^{-6}$
Subject: differential_calc

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

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

Problem: Find the Fourier series expansion of the function $f(x) = \frac{ x }{ 2 }$ with the period $4$ on the interval $[-2,2]$.

Answer: The Fourier series is: $f(x)=\sum_{n=1}^\infty\left(\frac{(-1)^{n+1}\cdot2}{\pi\cdot n}\cdot\sin\left(\frac{\pi\cdot n\cdot x}{2}\right)\right)$
Subject: differential_calc

Problem: Find the derivative of the function: $y = -4 \cdot x^{\sqrt{5 \cdot x}}$

Answer: $\frac{ d y }{d x}$ = $-\left(\frac{4\cdot\sqrt{5}}{\sqrt{x}}+\frac{2\cdot\sqrt{5}\cdot\ln(x)}{\sqrt{x}}\right)\cdot x^{\sqrt{5}\cdot\sqrt{x}}$
Subject: sequences_series

Problem: Write the Taylor series for the function $f(x) = x \cdot \sin(2 \cdot x)$ at the point $x = \pi$ up to the third term (zero or non-zero).

Answer: The final answer: $0+2\cdot\pi\cdot(x-\pi)+\frac{4}{2}\cdot(x-\pi)^2$
Subject: algebra

Problem: A substance has a half-life of $2.045$ minutes. If the initial amount of the substance was $132.8$ grams, find the exponential decay model formula. How many half-lives will have passed before the substance decays to $8.3$ grams?

Answer: Model formula: $A=132.8\cdot e^{\frac{\ln(0.5)}{2.045}\cdot t}$
Half-lives count: $4$
Subject: algebra

Problem: A town has an initial population of $53\ 000$. It grows at a constant rate of $2100$ per year for $5$ years. The linear function that models the town’s population $P$ as a function of the year is: $P(t) = 53\ 000 + 2100 \cdot t$, where $t$ is the number of years since the model began. Find the $x$- and $y$-intercepts.

Answer: $x$-intercept: $P\left(-\frac{530}{21},0\right)$
$y$-intercept: $P(0,53000)$
Subject: differential_calc

Problem: Find the derivative of the function $y = \arcsin\left(\sqrt{1-x^2}\right)$.

Answer: $y'$ = $-\frac{x}{|x|\cdot\sqrt{1-x^2}}$
Subject: integral_calc

Problem: Solve the integral:
$$
\int 5 \cdot \cos(3 \cdot x)^6 \, dx
$$

Answer: $\int 5 \cdot \cos(3 \cdot x)^6 \, dx$ = $\frac{5}{8}\cdot\left(\frac{5}{2}\cdot x+\frac{2\cdot\sin(6\cdot x)}{3}+\frac{\sin(12\cdot x)}{8}-\frac{\sin(6\cdot x)^3}{18}\right)+C$
Subject: precalculus_review

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

Answer: The final answer: $-\frac{1}{2}\le y\le\frac{1}{2}$
Subject: differential_calc

Problem: Consider a limousine that gets $m(v) = \frac{ 120-2 \cdot v }{ 5 }$ mi/gal at speed $v$, the chauffer costs $\$15$/h, and gas is $\$3.5$/gal. Find the cheapest driving speed.

Answer: approximately $34.02$ mph.
Subject: precalculus_review

Problem: Find the degree, $y$-intercept, and zeros for the polynomial function $f(x) = 2 \cdot x^2 + 9 \cdot x - 5$.

1. Degree
2. $y$-intercept
3. Zeros

Answer: 1. Degree: $2$
2. $y$-intercept: $-5$, $\frac{1}{2}$
3. Zeros: $-5$, $\frac{1}{2}$
Subject: integral_calc

Problem: Evaluate $\int_{0}^\pi \sin(5 \cdot x) \cdot \sqrt{\cos(5 \cdot x)} \, dx$ using substitution.

Answer: The final answer: $\frac{2+2\cdot\sqrt{-1}}{15}$
Subject: precalculus_review

Problem: Given two functions $f(x) = \sqrt{x^2 - 1}$ and $g(x) = \sqrt{3 - x}$:

1. Compute $f\left(g(x)\right)$
2. Compute $\frac{ f(x) }{ g(x) }$ and find the domain of the new function.

Answer: 1. The new function $f\left(g(x)\right)$ is $f\left(g(x)\right)=\sqrt{2-x}$
2. The function $\frac{ f(x) }{ g(x) }$ is $\frac{\sqrt{x^2-1}}{\sqrt{3-x}}$, and the domain for the function is $1\le x<3 \lor x\le-1$
Subject: multivariable_calculus

Problem: Find the center of mass of the region $\rho(x,y) = x \cdot y$ on the circle with radius $1$ in the first quadrant only.

Answer: Center of mass: $P\left(\frac{8}{15},\frac{8}{15}\right)$
Subject: precalculus_review

Problem: Find the domain of the function $f(x) = \sqrt{\cos\left(\sin(x)\right)} + \arcsin\left(\frac{ 1 + x^2 }{ 2 \cdot x }\right)$.

Answer: The final answer: $x=-1 \lor x=1$
Subject: algebra

Problem: A town has an initial population of $80\ 000$. It grows at a constant rate of $2200$ per year for $5$ years. The linear function that models the town’s population $P$ as a function of the year is: $P(t) = 80\ 000 + 2200 \cdot t$, where $t$ is the number of years since the model began. When will the population reach $120\ 000$?

Answer: $t$: $\frac{200}{11}$
Subject: integral_calc

Problem: Find the arc length of the parametric curve given by $x(t) = \frac{ 1 }{ 6 } \cdot t^3$, $y(t) = \frac{ 1 }{ 9 } \cdot t^3$ on $[1,3]$.

Answer: The final answer: $\frac{1573}{90}$
Subject: algebra

Problem: Calculate the average rate of change of the function $f(x) = 20 \cdot x - 2 \cdot x^2$ near $a=3$ over the intervals $[a,a+h]$ for $h=0.1$, $0.01$, and $0.001$. Use these values to estimate the instantaneous rate of change.

Answer: 1. The average rate of change for $f$ near $a$ with $h=0.1$ is: $7.8$
2. The average rate of change for $f$ near $a$ with $h=0.01$ is: $7.98$
3. The average rate of change for $f$ near $a$ with $h=0.001$ is: $7.998$
4. An estimate of the instantaneous rate of change is: $8$