toloka/u-math huggingface dataset:
Subject: multivariable_calculus

Problem: A small appliances company makes toaster ovens and pizza cookers, and have noticed their customer base's purchasing habits give them the price-demand equations given below, where $p$ is the price and $x$ is the quantity of toaster ovens, $q$ is the price and $y$ is the quantity of pizza cookers. The Cost function is $C(x,y) = 800 + 10 \cdot x + 16 \cdot y$.

$p = 50 - 5 \cdot x + y$ and $q = 90 + x - 3 \cdot y$

1. Construct the 2-product Revenue function, $R(x,y)$
2. Find $R(6,10)$
3. Construct the 2-product Profit function, $P(x,y)$
4. Find $P(6,10)$

Answer: 1. $R(x,y)$ = $R(x,y)=50\cdot x+90\cdot y+2\cdot x\cdot y-3\cdot y^2-5\cdot x^2$
2. $R(6,10)$ = $840$
3. $P(x,y)$ = $P(x,y)=40\cdot x+74\cdot y+2\cdot x\cdot y-800-3\cdot y^2-5\cdot x^2$
4. $P(6,10)$ = $-180$
Subject: integral_calc

Problem: Compute the integral:
$$
\int_{0}^3 \frac{ 1 }{ x^2 + 2 \cdot x - 8 } \, dx
$$

Answer: $\int_{0}^3 \frac{ 1 }{ x^2 + 2 \cdot x - 8 } \, dx$ = $\infty$
Subject: sequences_series

Problem: Find the Taylor series of $f'(x)$ about $a=0$ if $f(x) = \frac{ x - \ln(1 + x) }{ x^2 }$. Use sigma notation in the final answer.

Answer: The final answer: $\sum_{k=0}^\infty\left((-1)^{k+1}\cdot\frac{(k+1)}{(k+3)}\cdot x^k\right)$
Subject: differential_calc

Problem: Find the maximum and minimum values of the function $r = 3 \cdot \sin(x) + \sin(3 \cdot x)$ in the closed interval $\left[0, \frac{ 3 }{ 2 } \cdot \pi\right]$.

Answer: Maximum Value: $2\cdot\sqrt{2}$
Minimum Value: $-2\cdot\sqrt{2}$
Subject: multivariable_calculus

Problem: Find $L=\lim_{(x,y) \to (2,3)}\left(\frac{ x^2-y^2+10 \cdot y-25 }{ x^2-y^2-10 \cdot x+25 }\right)$.

Answer: The final answer: $L=-\frac{2}{3}$
Subject: multivariable_calculus

Problem: Find the mass of the solid $Q=\left\{(x,y,z) | 1 \le x^2+z^2 \le 25, y \le 1-x^2-z^2 \right\}$ whose density is $\rho(x,y,z) = k$, where $k > 0$.

Answer: $m$ = $288\cdot\pi\cdot k$
Subject: multivariable_calculus

Problem: Find the directional derivative using the limit definition only:
$$
f(x,y) = y^2 \cdot \cos(2 \cdot x)
$$
at point $P\left(\frac{ \pi }{ 3 },2\right)$ in the direction of $\vec{u}=\left\langle \cos\left(\frac{ \pi }{ 4 }\right),\sin\left(\frac{ \pi }{ 4 }\right) \right\rangle$.

Answer: Directional derivative: $-\frac{2+4\cdot\sqrt{3}}{\sqrt{2}}$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ 4 \cdot x+\sqrt{4 \cdot x-5} }{ 5 \cdot \sqrt[4]{4 \cdot x-5}+\sqrt[4]{(4 \cdot x-5)^3} } \, dx
$$

Answer: $\int \frac{ 4 \cdot x+\sqrt{4 \cdot x-5} }{ 5 \cdot \sqrt[4]{4 \cdot x-5}+\sqrt[4]{(4 \cdot x-5)^3} } \, dx$ = $C+25\cdot\sqrt[4]{4\cdot x-5}+\frac{1}{5}\cdot\sqrt[4]{4\cdot x-5}^5-\frac{4}{3}\cdot\sqrt[4]{4\cdot x-5}^3-\frac{125}{\sqrt{5}}\cdot\arctan\left(\frac{1}{\sqrt{5}}\cdot\sqrt[4]{4\cdot x-5}\right)$
Subject: algebra

Problem: A basic cellular package costs $\$30$/mo. for $60$ min of calling, with an additional charge of $\$0.4$/min beyond that time. The cost formula would be: $C = 30 + 0.4 \cdot (x - 60)$. If you have to keep your bill lower than $\$50$, what is the maximum calling minutes you can use?

Answer: Maximum calling minutes: $110$
Subject: multivariable_calculus

Problem: Evaluate the triple integral $\int \int \int f(x,y,z) \, dx \, dy \, dz$ over the solid $f(x,y,z) = e^{\sqrt{x^2+y^2}}$, $1 \le x^2+y^2 \le 4$, $y \le 0$, $x \le y \cdot \sqrt{3}$, $2 \le z \le 3$.

Answer: $\int \int \int f(x,y,z) \, dx \, dy \, dz$ = $\frac{\pi\cdot e^2}{6}$
Subject: precalculus_review

Problem: Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{0}^{4 \cdot \pi} \left(\cos\left(\frac{ x }{ 2 }\right) \cdot \sin\left(\frac{ x }{ 2 }\right)\right) \, dx
$$

Answer: $\int_{0}^{4 \cdot \pi} \left(\cos\left(\frac{ x }{ 2 }\right) \cdot \sin\left(\frac{ x }{ 2 }\right)\right) \, dx$ = $0$
Subject: multivariable_calculus

Problem: Find the volume of the solid that lies under the double cone $z^2 = 4 \cdot x^2 + 4 \cdot y^2$, inside the cylinder $x^2 + y^2 = x$, and above the plane $z = 0$.

Answer: The volume is $\frac{8}{9}$
Subject: sequences_series

Problem: Find the Fourier series of the function $u = \left| \sin\left( \frac{ x }{ 2 } \right) \right|$ in the interval $[-2 \cdot \pi,2 \cdot \pi]$.

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

Problem: Find the expansion in the series of the integral $\int_{0}^x{\frac{ \arctan\left(x^2\right) }{ 4 \cdot x } d x}$, using the expansion of the integrand in the series. Find the radius of convergence of the series.

Answer: 1. $\int_{0}^x{\frac{ \arctan\left(x^2\right) }{ 4 \cdot x } d x}$ = $\sum_{n=1}^\infty\left((-1)^{n+1}\cdot\frac{x^{2\cdot(2\cdot n-1)}}{4\cdot2\cdot(2\cdot n-1)^2}\right)$
2. $R$ = $1$
Subject: precalculus_review

Problem: Calculate $E = \frac{ \cos(70) \cdot \cos(10) + \cos(80) \cdot \cos(20) }{ \cos(68) \cdot \cos(8) + \cos(82) \cdot \cos(22) }$.

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

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

Answer: $\int \frac{ -9 \cdot \sqrt[3]{x} }{ 9 \cdot \sqrt[3]{x^2} + 3 \cdot \sqrt{x} } \, dx$ = $-\left(C+\frac{1}{3}\cdot\sqrt[6]{x}^2+\frac{2}{27}\cdot\ln\left(\frac{1}{3}\cdot\left|1+3\cdot\sqrt[6]{x}\right|\right)+\frac{3}{2}\cdot\sqrt[6]{x}^4-\frac{2}{3}\cdot\sqrt[6]{x}^3-\frac{2}{9}\cdot\sqrt[6]{x}\right)$
Subject: algebra

Problem: Find all real solutions of the following equation:

$$
\frac{ 2 }{ x+7 }-\frac{ 3 }{ x-3 }=1
$$

Answer: The real solution(s) to the given equation are: $x=-3$, $x=-2$
Subject: multivariable_calculus

Problem: Determine a definite integral that represents the region common to $r=2$ and $r=4 \cdot \cos\left(\theta\right)$.

Answer: The final answer: $4\cdot\int_0^{\frac{\pi}{3}}1d\theta+16\cdot\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\cos\left(\theta\right)^2d\theta$
Subject: integral_calc

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

Answer: $\int \frac{ 6 }{ \sin(3 \cdot x)^6 } \, dx$ = $-\frac{2\cdot\cos(3\cdot x)}{5\cdot\sin(3\cdot x)^5}+\frac{24}{5}\cdot\left(-\frac{\cos(3\cdot x)}{9\cdot\sin(3\cdot x)^3}-\frac{2}{9}\cdot\cot(3\cdot x)\right)+C$
Subject: multivariable_calculus

Problem: Find the curvature for the vector function: $\vec{r}(t) = \left\langle 2 \cdot \sin(t), -4 \cdot t, 2 \cdot \cos(t) \right\rangle$.

Answer: The final answer: $\frac{1}{10}$
Subject: algebra

Problem: Solve the inequality $|-6 x-6| \le 1$.

Express your answer using interval notation.

Answer: Solution using interval notation: $\left[-\frac{ 7 }{ 6 },\ -\frac{ 5 }{ 6 }\right]$

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

Problem: Integrate the approximation $\sin(t) \approx t - \frac{ t^3 }{ 6 } + \frac{ t^5 }{ 120 } - \frac{ t^7 }{ 5040 } + \cdots$ evaluated at $\pi \cdot t$ to approximate $\int_{0}^1{\frac{ \sin(\pi \cdot t) }{ \pi \cdot t } d t}$

Answer: $\int_{0}^1{\frac{ \sin(\pi \cdot t) }{ \pi \cdot t } d t}$ ≈ $0.58678687$
Subject: sequences_series

Problem: Find the Fourier integral of the function $q(x) = \begin{cases} 0, & x < 0 \\ 2 \cdot \pi \cdot x, & 0 \le x \le 1 \\ 0, & x > 1 \end{cases}$

Answer: $q(x) = $\int_0^\infty\left(\frac{2\cdot\left(\alpha\cdot\sin\left(\alpha\right)+\cos\left(\alpha\right)-1\right)\cdot\cos\left(\alpha\cdot x\right)+2\cdot\left(\sin\left(\alpha\right)-\alpha\cdot\cos\left(\alpha\right)\right)\cdot\sin\left(\alpha\cdot x\right)}{\alpha^2}\right)d\alpha$
Subject: differential_calc

Problem: Find the second derivative $\frac{d ^2y}{ d x^2}$ of the function $x = \left(2 \cdot \sin(3 \cdot t)\right)^2$, $y = \sin(2 \cdot t)$.

Answer: $\frac{d ^2y}{ d x^2}$ = $\frac{\left(288\cdot\left(\sin(3\cdot t)\right)^2-144\right)\cdot\cos(2\cdot t)-96\cdot\cos(3\cdot t)\cdot\sin(2\cdot t)\cdot\sin(3\cdot t)}{13824\cdot\left(\cos(3\cdot t)\right)^3\cdot\left(\sin(3\cdot t)\right)^3}$
Subject: integral_calc

Problem: Calculate the integral:
$$
I = \int 4 \cdot \cos\left(3 \cdot \ln(2 \cdot x)\right) \, dx
$$

Answer: The final answer: $\frac{1}{10}\cdot\left(C+4\cdot x\cdot\cos\left(3\cdot\ln(2\cdot x)\right)+12\cdot x\cdot\sin\left(3\cdot\ln(2\cdot x)\right)\right)$
Subject: sequences_series

Problem: Find the sum of the series $\sum_{n=1}^\infty \frac{ 7 }{ 2 \cdot n \cdot 3^n }$. (Use the formula $\int_{3}^\infty \frac{ 7 }{ 2 \cdot x^{n+1} } \, dx = \frac{ 7 }{ 2 \cdot n \cdot 3^n }$)

Answer: The sum of the series is $-\frac{7\cdot\ln\left(\frac{2}{3}\right)}{2}$
Subject: precalculus_review

Problem: A vehicle with a 20-gallon tank gets 15 mpg. The number of miles $N$ that can be driven depends on the amount of gas $x$ in the tank.

1. Write a formula that models this situation.
2. Determine the number of miles the vehicle can travel on
    * a full tank of gas and
    * $\frac{ 3 }{ 4 }$ of a tank of gas.
3. Determine the 
    * domain and 
    * range of the function.
4. Determine how many times the driver had to stop for gas if she has driven a total of 578 mi.

Answer: 1. The formula for the function is $N(x)$ = $15\cdot x$
2. The number of miles the vehicle can travel on
    * full tank of gas is $300$
    * $\frac{ 3 }{ 4 }$ of a tank of gas is $225$
3. Domain and range (in interval notation)
    * domain is $[0,20]$
    * range is $[0,300]$
4. Number of times that the driver had to stop = $1$
Subject: sequences_series

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

Answer: $R$ = $\infty$
Subject: multivariable_calculus

Problem: Consider points $A(1,1)$, $B(2,-7)$, and $C(6,3)$.

1. Determine vectors $\vec{BA}$ and $\vec{BC}$. Express the answer in component form.
2. Determine the measure of angle $B$ in triangle $ABC$. Express the answer in radians, rounded to two decimal places.

Answer: 1. $\vec{BA}$ = $\left\langle-1,8\right\rangle$ , $\vec{BC}$ = $\left\langle4,10\right\rangle$
2. $B$ = $0.50$
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 $P=-t^2+13 \cdot t+130$, where $1 \le t \le 6$. Find the day that $160$ students had the flu. Recall that the restriction on $t$ is at most $6$.

Answer: The final answer: $3$
Subject: multivariable_calculus

Problem: Evaluate $\int\int\int_{E}{(2 \cdot x+5 \cdot y+7 \cdot z) d V}$, where $E$ is the region defined by:
$$
E = \left\{(x,y,z) | 0 \le x \le 1, 0 \le y \le -x+1, 1 \le z \le 2\right\}
$$

Answer: $I$ = $\frac{77}{12}$
Subject: multivariable_calculus

Problem: Let $Q$ be the solid situated outside the sphere $x^2+y^2+z^2=z$ and inside the upper hemisphere $x^2+y^2+z^2=R^2$, where $R>1$. If the density of the solid is $\rho(x,y,z) = \frac{ 1 }{ \sqrt{x^2+y^2+z^2} }$, find $R$ such that the mass of the solid is $\frac{ 7 \cdot \pi }{ 2 }$.

Answer: $R$ = $\frac{\sqrt{138}}{6}$
Subject: differential_calc

Problem: Find the minimum value of $y = \frac{ \left(\cos(x)\right)^2 - 4 \cdot \cos(x) + 5 }{ 3 - 2 \cdot \cos(x) }$.

Answer: The final answer: $\frac{1+\sqrt{5}}{2}$
Subject: sequences_series

Problem: Compute the first 5 nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of $f(x) = e^{\sin(x)}$.

Answer: $f(x)$ = $1+x+\frac{x^2}{2}-\frac{x^4}{8}-\frac{x^5}{15}+\cdots$
Subject: differential_calc

Problem: Compute the limits:

1. $\lim_{x \to \infty}\left(x \cdot \left(\sqrt{x^2+1}-x\right)\right)$
2. $\lim_{x \to -\infty}\left(x \cdot \left(\sqrt{x^2+1}-x\right)\right)$

Answer: 1. $\frac{1}{2}$
2. $-\infty$
Subject: integral_calc

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

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

Problem: Find the extrema of a function $y = \frac{ 3 \cdot x^4 }{ 4 } - \frac{ 4 \cdot x^3 }{ 3 } - \frac{ 3 \cdot x^2 }{ 2 } + 2$. Then determine the largest and smallest value of the function when $-2 \le x \le 4$.

Answer: 1. Extrema points: $P\left(\frac{4-2\cdot\sqrt{13}}{6},1.8363\right)$, $P(0,2)$, $P\left(\frac{4+2\cdot\sqrt{13}}{6},-2.7931\right)$
2. The largest value: $\frac{254}{3}$
3. The smallest value: $-2.7931$
Subject: sequences_series

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

Answer: The final answer: $\frac{1}{90}$
Subject: algebra

Problem: Rewrite the quadratic expression $x^2 - 9 \cdot x - 22$ by completing the square.

Answer: $x^2 - 9 \cdot x - 22$ = $(x-4.5)^2-42.25$
Subject: precalculus_review

Problem: Find the degree, $y$-intercept, and zeros for the polynomial function $f(x) = x^3 + 2 \cdot x^2 - 2 \cdot x$.

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

Answer: 1. Degree: $3$
2. $y$-intercept: $0$, $\sqrt{3}-1$, $-1-\sqrt{3}$
3. Zeros: $0$, $\sqrt{3}-1$, $-1-\sqrt{3}$
Subject: sequences_series

Problem: Find the Fourier series of the periodic function $f(x) = \frac{ x^2 }{ 3 }$ in the interval $-4 \cdot \pi \le x < 4 \cdot \pi$ if $f(x) = f(x + 8 \cdot \pi)$.

Answer: The Fourier series is: $\frac{16\cdot\pi^2}{9}+\sum_{n=1}^\infty\left(\frac{64\cdot(-1)^n}{3\cdot n^2}\cdot\cos\left(\frac{n\cdot x}{4}\right)\right)$
Subject: algebra

Problem: Rewrite the quadratic function $h(x) = 2 \cdot x^2 + 12 \cdot x - 4$ in standard form and give the vertex.

### Forms of quadratic functions

A quadratic function is a polynomial function of degree two. The graph of a quadratic function is a parabola.

The general form of a quadratic function is $f(x) = a \cdot x^2 + b \cdot x + c$ where $a$, $b$, and $c$ are real numbers and $a \ne 0$.

The standard form of a quadratic function is $f(x) = a(x - h)^2 + k$ where $a \ne 0$.

The vertex $P(h, k)$ is located at
$k = f(h)$ and $h = -\frac{b}{2 \cdot a}$

Answer: The final answer: The standard form of the quadratic function: $2\cdot(x+3)^2-22$ Vertex $(h,k)$ = $P(-3,-22)$
Subject: multivariable_calculus

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

Answer: The final answer: $L=\frac{a^2+2\cdot b^2}{a^2-2\cdot b^2}$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ x-\sqrt[3]{4 \cdot x^2}-\sqrt[6]{2 \cdot x} }{ x \cdot \left(4+\sqrt[3]{2 \cdot x}\right) } \, dx
$$

Answer: $\int \frac{ x-\sqrt[3]{4 \cdot x^2}-\sqrt[6]{2 \cdot x} }{ x \cdot \left(4+\sqrt[3]{2 \cdot x}\right) } \, dx$ = $C+36\cdot\ln\left(\left|4+\sqrt[3]{2}\cdot\sqrt[3]{x}\right|\right)+\frac{\left(3\cdot2^{\frac{2}{3}}\right)}{4}\cdot x^{\frac{2}{3}}-3\cdot\arctan\left(\frac{1}{2^{\frac{5}{6}}}\cdot\sqrt[6]{x}\right)-9\cdot\sqrt[3]{2}\cdot\sqrt[3]{x}$
Subject: multivariable_calculus

Problem: A person is standing 8 feet from the nearest wall in a whispering gallery. If that person is at one focus and the other focus is 80 feet away, what is the length and the height at the center of the gallery?

Answer: Length is $96$ feet and height is approximately $26.53$ feet.
Subject: sequences_series

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

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

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

Problem: Using the scoring system in the game of Jeopardy (you earn points for correct responses and lose points for incorrect responses), what would be the final score after the following 5 responses:

1. Player 1 answers a 200-point question correct, a 50-point question wrong, a 250-point question correct, a 50-point question wrong, and a 150-point question correct.
2. Player 2 answers a 200-point question wrong, a 100-point question correct, a 100-point question wrong, a 100-point question wrong, and a 50-point question correct.
3. Player 3 answers a 150-point question correct, a 100-point question wrong, a 50-point question wrong, a 100-point question correct, and a 250-point question correct.

Answer: 1. The final score for Player 1 is $500$
2. The final score for Player 2 is $-250$
3. The final score for Player 3 is $350$
Subject: multivariable_calculus

Problem: Determine a definite integral that represents the region common to $r = 3 \cdot \cos\left(\theta\right)$ and $r = 3 \cdot \sin\left(\theta\right)$.

Answer: The final answer: $9\cdot\int_0^{\frac{\pi}{4}}\sin\left(\theta\right)^2d\theta$
Subject: algebra

Problem: Solve the following equations:

1. $4 a - 15 = -23$
2. $44 = 5 x - 6$
3. $-3.7 x + 5.6 = -5.87$
4. $\frac{ x }{ 5 } + 3.5 = 4.16$
5. $-\frac{ x }{ 4 } + (-9.8) = -5.6$
6. $3.7 x - 2.2 = 16.3$
7. $\frac{ x }{ 6 } - 3.7 = 11$

Answer: The solutions to the given equations are:
1. $a=-2$
2. $x=10$
3. $x=\frac{ 31 }{ 10 }$
4. $x=\frac{ 33 }{ 10 }$
5. $x=-\frac{ 84 }{ 5 }$
6. $x=5$
7. $x=\frac{ 441 }{ 5 }$
Subject: differential_calc

Problem: Evaluate $\lim_{x \to \frac{ \pi }{ 2 }}\left(\frac{ \tan(3 \cdot x) }{ \tan(5 \cdot x) }\right)$ using L'Hopital's Rule.

Answer: $\lim_{x \to \frac{ \pi }{ 2 }}\left(\frac{ \tan(3 \cdot x) }{ \tan(5 \cdot x) }\right)$ = $\frac{5}{3}$
Subject: sequences_series

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

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. $x^2-\frac{x^4}{2}$
2. $0.1931$
3. $0.0003\cdot10^{-3}$
Subject: differential_calc

Problem: Differentiate $y = b \cdot \tan\left(\sqrt{1-x}\right)$.

Answer: The final answer: $\frac{dy}{dx}=\frac{b\left(\sec\left(\sqrt{1-x}\right)\right)^2}{2\cdot\sqrt{1-x}}$
Subject: precalculus_review

Problem: Use the double-angle formulas to evaluate the integral:
$$
\int \sin(x)^2 \cdot \cos(x)^2 \, dx
$$

Answer: $\int \sin(x)^2 \cdot \cos(x)^2 \, dx$ = $\frac{1}{8}\cdot x-\frac{1}{32}\cdot\sin(4\cdot x)+C$
Subject: differential_calc

Problem: Compute the derivative of the implicit function $e^{\frac{ y }{ 2 }} = x^{\frac{ x }{ 3 } - \frac{ y }{ 4 }}$.

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

Problem: Solve the following series via differentiation of the geometric series:

1. $\sum_{n=1}^\infty\left(\frac{ n }{ 6^{n-1} }\right)$
2. $\sum_{n=2}^\infty\left(\frac{ n }{ 3^{n-1} }\right)$
3. $\sum_{n=3}^\infty\left(\frac{ n }{ 2^{n+1} }\right)$
4. $\sum_{n=2}^\infty\left(\frac{ n+1 }{ 5^n }\right)$
5. $\sum_{n=2}^\infty\left(\frac{ n \cdot (n-1) }{ 4^n }\right)$
6. $\sum_{n=2}^\infty\left(\frac{ n^2 }{ 4^n }\right)$

Answer: 1. $\frac{36}{25}$
2. $\frac{5}{4}$
3. $\frac{1}{2}$
4. $\frac{13}{80}$
5. $\frac{8}{27}$
6. $\frac{53}{108}$
Subject: multivariable_calculus

Problem: Determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbit of the comet or planet. Distance is given in astronomical units (AU):

1. Halley’s Comet: length of major axis = $35.88$, eccentricity = $0.967$.

Answer: 1. $r$ = $\frac{1.16450334}{1+0.967\cdot\cos(t)}$
Subject: precalculus_review

Problem: Find the zeros of the function $f(x) = x^3 - (3 + \sqrt{3}) \cdot x + 3$.

Answer: The final answer: $x_1=\sqrt{3}$, $x_2=\frac{-\sqrt{3}-\sqrt{3+4\cdot\sqrt{3}}}{2}$, $x_3=\frac{-\sqrt{3}+\sqrt{3+4\cdot\sqrt{3}}}{2}$
Subject: multivariable_calculus

Problem: Evaluate the triple integral $\int_{0}^2 \int_{4}^6 \int_{3 \cdot z}^{3 \cdot z+2} (5-4 \cdot y) \, dx \, dz \, dy$ by using the transformation $u=x-3 \cdot z$, $v=4 \cdot y$, and $w=z$.

Answer: $I$ = $8$
Subject: precalculus_review

Problem: Solve the trigonometric equation $6 \cdot \cos(x)^2 - 3 = 0$ on the interval $[-2 \cdot \pi, 2 \cdot \pi]$ exactly.

Answer: $x$ = $\frac{\pi}{4}$, $-\frac{\pi}{4}$, $\frac{3}{4}\cdot\pi$, $-\frac{3}{4}\cdot\pi$, $\frac{5}{4}\cdot\pi$, $-\frac{5}{4}\cdot\pi$, $\frac{7}{4}\cdot\pi$, $-\frac{7}{4}\cdot\pi$
Subject: integral_calc

Problem: Calculate $I=\int_{\pi}^{\frac{ 5 }{ 4 } \cdot \pi}{\frac{ \sin(2 \cdot x) }{ \left(\cos(x)\right)^4+\left(\sin(x)\right)^4 } \, dx}$

Answer: The final answer: $I=\frac{\pi}{4}$
Subject: multivariable_calculus

Problem: The period $T$ of a simple pendulum with small oscillations is calculated from the formula $T = 2 \cdot \pi \cdot \sqrt{\frac{ L }{ g }}$, where $L$ is the length of the pendulum and $g$ is the acceleration resulting from gravity. Suppose that $L$ and $g$ have errors of, at most, $0.5\%$ and $0.1\%$ respectively. Use differentials to approximate the maximum percentage error in the calculated value of $T$.

Answer: The maximum percentage error is: $0.003$
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).

$2 \cdot x^2 - 8 \cdot x - 5 = 0$

Answer: $x$ = $\sqrt{\frac{13}{2}}+2$, $-\sqrt{\frac{13}{2}}+2$
Subject: differential_calc

Problem: Compute the derivative of the function $f(x) = \sqrt[x]{3} + \frac{ 1 }{ 6^{5 \cdot x} } + 4^{\sqrt{x}}$ at $x = 1$.

Answer: $f'(1)$ = $-0.5244$
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 $a = 4$.

Answer: The curvature is: $\frac{1}{16\cdot\left|\sin\left(\frac{t}{2}\right)\right|}$
Subject: sequences_series

Problem: Determine the Taylor series for $y = \left(\cos(2 \cdot x)\right)^2$, centered at $x_{0} = 0$. Write out the sum of the first three non-zero terms, followed by dots.

Answer: The final answer: $1-\frac{2^3}{2!}\cdot x^2+\frac{2^7}{4!}\cdot x^4+\cdots$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \frac{ 1 }{ \sin(8 \cdot x)^5 } \, dx
$$

Answer: The final answer: $C+\frac{1}{128}\cdot\left(2\cdot\left(\tan(4\cdot x)\right)^2+6\cdot\ln\left(\left|\tan(4\cdot x)\right|\right)+\frac{1}{4}\cdot\left(\tan(4\cdot x)\right)^4-\frac{2}{\left(\tan(4\cdot x)\right)^2}-\frac{1}{4\cdot\left(\tan(4\cdot x)\right)^4}\right)$
Subject: precalculus_review

Problem: Find the domain of the function $g(x) = \ln(2 \cdot x - 3) + \sqrt{1 - x}$. Give your answer using interval notation.

Answer: The final answer: $\varnothing$
Subject: algebra

Problem: Rewrite the quadratic expression $2 \cdot x^2 - 10 \cdot x - 15$ by completing the square.

Answer: $2 \cdot x^2 - 10 \cdot x - 15$ = $2\cdot(x-2.5)^2-27.5$
Subject: integral_calc

Problem: Leaves of deciduous trees fall in an exponential rate during autumn. A large maple has 10,000 leaves and the wind starts blowing. If there are 8,000 leaves left after 3 hours, how many will be left after 4 hours?

Answer: There will be $7426$ leaves after 4 hours.
Subject: algebra

Problem: The fox population in a certain region has an annual growth rate of 9 percent. In the year 2012, there were 23 900 fox counted in the area. Create a function representing the population. 
What is the fox population to be in the year 2020? Round your answer to the nearest whole number.

Answer: The final answer: $47622$
Subject: precalculus_review

Problem: Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{0}^\pi \left(\sin(3 \cdot x) \cdot \sin(5 \cdot x)\right) \, dx
$$

Answer: $\int_{0}^\pi \left(\sin(3 \cdot x) \cdot \sin(5 \cdot x)\right) \, dx$ = $0$
Subject: sequences_series

Problem: The $k$ th term of the given series has a factor $x^k$. Find the range of $x$ for which the ratio test implies that the series converges:

$$\sum_{k=1}^\infty \left(\frac{ x^k }{ k^2 }\right)$$

Answer: $|x|$ < $1$