toloka/u-math huggingface dataset:
Subject: multivariable_calculus

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

Answer: The final answer: $L=\frac{486}{5}$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ 10 }{ \sin(4 \cdot x)^6 } \, dx
$$

Answer: $\int \frac{ 10 }{ \sin(4 \cdot x)^6 } \, dx$ = $C-\frac{1}{2}\cdot\left(\cot(4\cdot x)\right)^5-\frac{5}{2}\cdot\cot(4\cdot x)-\frac{5}{3}\cdot\left(\cot(4\cdot x)\right)^3$
Subject: precalculus_review

Problem: Find the zeros of the function $f(x) = \left(1 - \tan(x)\right) \cdot \left(1 + \sin(2 \cdot x)\right) - \tan(x) + \cos(2 \cdot x) - 1$.

Answer: The final answer: $x_1=-\frac{\pi}{4}+\pi\cdot n$, $x_2=\arctan\left(\frac{1}{2}\right)+\pi\cdot n$
Subject: sequences_series

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

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

Problem: Evaluate the telescoping series or state whether the series diverges:

$$
\sum_{n=1}^\infty \left(\sin(n)-\sin(n+1)\right)
$$

Answer: $S_{k}$ -> None as $k$ -> $\infty$.
Subject: multivariable_calculus

Problem: Express the region $D$ in polar coordinates. $D$ = $\left\{(x,y)|x^2+y^2 \le 4 \cdot x\right\}$  ($a \le \theta \le b$, $a \ge 0$, $b \ge 0$).

Answer: 1. The interval for $r$ is $\left[0,4\cdot\cos\left(\theta\right)\right]$;
2. The interval for $\theta$ is $\left[\frac{3}{2}\cdot\pi,\frac{5\cdot\pi}{2}\right]$.
Subject: precalculus_review

Problem: Simplify the expression $\frac{ 1+\tan\left(\alpha\right)^2 }{ 1+\cot\left(\alpha\right)^2 }$ by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

Answer: $\frac{ 1+\tan\left(\alpha\right)^2 }{ 1+\cot\left(\alpha\right)^2 }$ = $\tan\left(\alpha\right)^2$
Subject: integral_calc

Problem: Find the integral:
$$
\int \frac{ 1 }{ \sqrt[3]{\left(\sin(x)\right)^{11} \cdot \cos(x)} } \, dx
$$

Answer: $\int \frac{ 1 }{ \sqrt[3]{\left(\sin(x)\right)^{11} \cdot \cos(x)} } \, dx$ = $-\frac{3\cdot\left(1+4\cdot\left(\tan(x)\right)^2\right)}{8\cdot\left(\tan(x)\right)^2\cdot\sqrt[3]{\left(\tan(x)\right)^2}}+C$
Subject: multivariable_calculus

Problem: Find the tangential and normal components of acceleration for $\vec{r}(t) = \left\langle a \cdot \cos\left(\omega \cdot t\right), b \cdot \sin\left(\omega \cdot t\right) \right\rangle$ at $t=0$.

Answer: The final answer:

1. The tangential component of acceleration, $a_{t}$, is: $0$
2. The normal component of acceleration, $a_{N}$, is: $\omega^2\cdot|a|$
Subject: integral_calc

Problem: Consider the Karun-$3$ dam in Iran. Its shape can be approximated as an isosceles triangle with height $205$ m and width $388$ m. Assume the current depth of the water is $180$ m. The density of water is $1000$ kg/$m^3$. Find the total force on the wall of the dam.

Answer: The total force : $18028940487.8049$
Subject: precalculus_review

Problem: Calculate $\cos\left(\frac{ \pi }{ 12 }\right)$.

Answer: The final answer: $\cos\left(\frac{\pi}{12}\right)=\frac{\sqrt{2}+\sqrt{6}}{4}$
Subject: differential_calc

Problem: Sketch the curve:  
$y = 3 \cdot x \cdot \sqrt{2-x^2}$.

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): $\left[-1\cdot2^{2^{-1}},2^{2^{-1}}\right]$
2. Vertical asymptotes: None
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: $(-1,1)$
6. Intervals where the function is decreasing: $\left(-2^{2^{-1}},-1\right)$, $\left(1,2^{2^{-1}}\right)$
7. Intervals where the function is concave up: $\left(-2^{2^{-1}},0\right)$
8. Intervals where the function is concave down: $\left(0,2^{2^{-1}}\right)$
9. Points of inflection: $P(0,0)$
Subject: sequences_series

Problem: Find the sum of the series $\sum_{n=1}^\infty \frac{ x^{4 \cdot n-3} }{ 4 \cdot n-3 }$. (Use differentiation of the 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: differential_calc

Problem: For the function $y = \sqrt[3]{2-x} + \frac{ 1 }{ 9 } \cdot x^2$ determine the intervals, where the function is concave up and concave down and points of inﬂection. Submit as your final answer:

1. Interval(s) where the function is concave up
2. Interval(s) where the function is concave down
3. Point(s) of inﬂection

Answer: 1. Interval(s) where the function is concave up: $(-\infty,1)$, $(2,\infty)$
2. Interval(s) where the function is concave down: $(1,2)$
3. Point(s) of inﬂection: $P\left(1,\frac{10}{9}\right)$, $P\left(2,\frac{4}{9}\right)$
Subject: sequences_series

Problem: Determine the Taylor series for $f(x) = \frac{ 2 \cdot x - 1 }{ x^2 - 3 \cdot x + 2 }$, centered at $x_{0} = 4$. Write out the sum of the first four non-zero terms, followed by dots.

Answer: The final answer: $\frac{7}{6}+\left(\frac{1}{3^2}-\frac{3}{2^2}\right)\cdot(x-4)-\left(\frac{1}{3^3}-\frac{3}{2^3}\right)\cdot(x-4)^2+\left(\frac{1}{3^4}-\frac{3}{2^4}\right)\cdot(x-4)^3+\cdots$
Subject: integral_calc

Problem: Compute the length of the arc $y = 2 \cdot \ln(3 \cdot x)$ between the points $x = \sqrt{5}$ and $x = 2 \cdot \sqrt{3}$.

Answer: Arc Length: $1+\ln\left(\frac{5}{3}\right)$
Subject: precalculus_review

Problem: Solve $2 \cdot \left(\arcsin(x)\right)^2 - \arcsin(x) - 6 = 0$.

Answer: The final answer: $x=-\sin(1.5)$
Subject: algebra

Problem: Let $f(x) = \frac{ 1 }{ x }$. Find the number $b$ such that the average rate of change of $f$ on the interval $(2,b)$ is $-\frac{ 1 }{ 10 }$.

Answer: The final answer: $b=5$
Subject: multivariable_calculus

Problem: Use two circular permutations of the variables $x$, $y$, and $z$ to write new integral whose value equal the value of the original integral. A circular permutation of $x$, $y$, and $z$ is the arrangement of the numbers in one of the following orders: $y$, $z$, and $x$ or $z$, $x$, and $y$:
$$
\int_{0}^1 \int_{1}^3 \int_{2}^4 \left(x^2 \cdot z^2 + 1\right) \, dx \, dy \, dz
$$

Answer: 1. $I$ with $d z \cdot d x \cdot d y$ is $\frac{148}{9}$
2. $I$ with $d y \cdot d z \cdot d x$ is $\frac{148}{9}$
Subject: algebra

Problem: Solve the following equations:

1. $-15c = -75$
2. $-8 + r = 27$
3. $19 + m = 3$
4. $-\frac{w}{8} = 1$
5. $p - 4.8 = 11.2$
6. $\frac{g}{3.4} = 2.2$
7. $3.6m = 25.2$
8. $\frac{2}{5}t = \frac{8}{25}$
9. $5\frac{5}{6} + m = 3\frac{5}{12}$

Answer: The solutions to the given equations are:
1. $c=5$
2. $r=35$
3. $m=-16$
4. $w=-8$
5. $p=16$
6. $g=7.48$
7. $m=7$
8. $t=\frac{ 4 }{ 5 }$
9. $m=-\frac{ 29 }{ 12 }$
Subject: multivariable_calculus

Problem: The volume of a right circular cylinder is given by $V(x,y) = \pi \cdot x^2 \cdot y$ where $x$ is the radius of the cylinder and $y$ is the cylinder height. Suppose $x$ and $y$ are functions of $t$ given by $x = \frac{ 1 }{ 2 } \cdot t$ and $y = \frac{ 1 }{ 3 } \cdot t$ so that $x$ and $y$ are both increasing with time. How fast is the volume increasing when $x=2$ and $y=5$?

Answer: The volume is increasing at a rate of $\frac{34\cdot\pi}{3}$
Subject: differential_calc

Problem: For the function $y = \frac{ 2 \cdot x + 3 }{ 4 \cdot x + 5 }$ find the derivative $y^{(n)}$.

Answer: The General Form of the Derivative of $y = \frac{ 2 \cdot x + 3 }{ 4 \cdot x + 5 }$: $y^{(n)}=\frac{1}{2}\cdot(-1)^n\cdot\left(n!\right)\cdot4^n\cdot(4\cdot x+5)^{-(n+1)}$
Subject: differential_calc

Problem: Find the $x$-coordinate for each critical point of $f(x) = e^{-x} \cdot \sqrt{x}$

Answer: $x$ = $\frac{1}{2}$, $0$
Subject: sequences_series

Problem: Find the 3rd order Taylor polynomial $P_{3}(x)$ for the function $f(x) = \arctan(x)$ in powers of $x-1$ and give the Lagrange form of the remainder.

Answer: $P_{3}(x)$ = $\frac{ \pi }{ 4 }+\frac{ 1 }{ 2 } \cdot (x-1)-\frac{ 1 }{ 4 } \cdot (x-1)^2+\frac{ 1 }{ 12 } \cdot (x-1)^3$

$R_{3}(x)$ = $\frac{ -\frac{ 48 \cdot c^3 }{ \left(1+c^2\right)^4 }+\frac{ 24 \cdot c }{ \left(1+c^2\right)^3 } }{ 4! } \cdot (x-1)^4$
Subject: differential_calc

Problem: Find $\frac{ d y }{d x}$ if $y = \frac{ 5 \cdot x^2 - 3 \cdot x }{ \left( 3 \cdot x^7 + 2 \cdot x^6 \right)^4 }$.

Answer: $\frac{ d y }{d x}$ = $\frac{-390\cdot x^2+23\cdot x+138}{x^{24}\cdot(3\cdot x+2)^5}$
Subject: sequences_series

Problem: Find $\lim_{n \to \infty} \left(x_{n}\right)$, where $x_{n} = \left(\frac{ 2 \cdot n^3 }{ 2 \cdot n^2+3 }+\frac{ 1-5 \cdot n^2 }{ 5 \cdot n+1 }\right)$ is the general term of a sequence.

Answer: The final answer: $L=\frac{1}{5}$
Subject: precalculus_review

Problem: Is the statement True or False?

The following linear programming problem is infeasible.

Maximize $f = 4 \cdot x_{1} - x_{2}$ subject to

$x_{1} - x_{2} \le 3$

$-x_{1} + 3 \cdot x_{2} \le -5$

$x_{1} \ge 0$, $x_{2} \ge 0$

Answer: The final answer: $T\cdot r\cdot u\cdot e$
Subject: precalculus_review

Problem: Solve the exponential equation exactly: $4^{x+1} - 32 = 0$

Answer: $x$ = $\frac{3}{2}$
Subject: differential_calc

Problem: Sketch the curve:

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

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

Problem: Make full curve sketching of $f(x) = \frac{ 3 \cdot x^3 }{ 2 \cdot x^2 - 3 }$.

Submit as your final answer:

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

Answer: 1. The domain (in interval notation) $\left(-1\cdot\infty,-1\cdot3^{2^{-1}}\cdot2^{-1\cdot2^{-1}}\right)\cup\left(-1\cdot3^{2^{-1}}\cdot2^{-1\cdot2^{-1}},3^{2^{-1}}\cdot2^{-1\cdot2^{-1}}\right)\cup\left(3^{2^{-1}}\cdot2^{-1\cdot2^{-1}},\infty\right)$
2. Vertical asymptote(s) $x=\frac{\sqrt{3}}{\sqrt{2}}$, $x=-\frac{\sqrt{3}}{\sqrt{2}}$
3. Horizontal asymptote(s) None
4. Slant asymptote(s) $y=\frac{3}{2}\cdot x$
5. Interval(s) where the function is increasing $\left(-\infty,-\frac{3}{\sqrt{2}}\right)$, $\left(\frac{3}{\sqrt{2}},\infty\right)$
6. Interval(s) where the function is decreasing $\left(-\frac{3}{\sqrt{2}},-\frac{\sqrt{3}}{\sqrt{2}}\right)$, $\left(-\frac{\sqrt{3}}{\sqrt{2}},0\right)$, $\left(0,\frac{\sqrt{3}}{\sqrt{2}}\right)$, $\left(\frac{\sqrt{3}}{\sqrt{2}},\frac{3}{\sqrt{2}}\right)$
7. Interval(s) where the function is concave up $\left(-\frac{\sqrt{3}}{\sqrt{2}},0\right)$, $\left(\frac{\sqrt{3}}{\sqrt{2}},\infty\right)$
8. Interval(s) where the function is concave down $\left(-\infty,-\frac{\sqrt{3}}{\sqrt{2}}\right)$, $\left(0,\frac{\sqrt{3}}{\sqrt{2}}\right)$
9. Point(s) of inflection $P(0,0)$
Subject: integral_calc

Problem: Find the area of the surface formed by rotating the arc of the circle $x^2 + y^2 = 1$ between the points $(1,0)$ and $(0,1)$ in the first quadrant, around the line $x + y = 1$.

Answer: The final answer: $\frac{4\cdot\pi-\pi^2}{\sqrt{2}}$
Subject: precalculus_review

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

Answer: $\int \sin(x)^2 \, dx + \int \cos(x)^2 \, dx$ = $x+C$
Subject: differential_calc

Problem: Calculate the derivative of the function $r = 9 \cdot \ln\left(\sqrt[3]{\frac{ 1+\tan\left(\frac{ \varphi }{ 3 }\right) }{ 1-\tan\left(\frac{ \varphi }{ 3 }\right) }}\right)$.

Answer: $r'$ = $2\cdot\sec\left(\frac{2\cdot\varphi}{3}\right)$
Subject: sequences_series

Problem: Find the radius of convergence and sum of the series:
$$
\frac{ 1 }{ 3 }+\frac{ x }{ 1 \cdot 4 }+\frac{ x^2 }{ 1 \cdot 2 \cdot 5 }+\cdots+\frac{ x^n }{ \left(n!\right) \cdot (n+3) }+\cdots
$$

Answer: 1. Radius of convergence: $R=\infty$
2. Sum: $f(x)=\begin{cases}\frac{x\cdot\left(2\cdot e^x-2\cdot x\cdot e^x\right)+e^x\cdot x^3}{x^4}-\frac{2}{x^3},&x\ne0\\\frac{1}{3},&x=0\end{cases}$
Subject: integral_calc

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

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

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

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 $a=-\frac{1}{2}$
2. The radius of convergence is $R=1$
Subject: sequences_series

Problem: Express the series $\sum_{k=0}^\infty\left(x^k-x^{2 \cdot k+1}\right)$ in terms of elementary functions and find the radius of convergence of the sum.

Answer: 1. $\sum_{k=0}^\infty\left(x^k-x^{2 \cdot k+1}\right)$ = $\frac{1}{1-x^2}$
2. radius of convergence $R$ = $1$
Subject: multivariable_calculus

Problem: Determine a definite integral that represents the area enclosed by the inner loop of $r = 3 - 4 \cdot \cos(\theta)$.

Answer: The final answer: $\int_0^{\arccos\left(\frac{3}{4}\right)}\left(3-4\cdot\cos\left(\theta\right)\right)^2d\theta$
Subject: multivariable_calculus

Problem: The position of a particle at time $t$ is given by $\vec{r}(t) = \left\langle \frac{ t^2 }{ 2 },\frac{ 2 \cdot \sqrt{2} }{ 3 } \cdot t^{\frac{ 3 }{ 2 }},t \right\rangle$.

1. Find the velocity vector of the particle.
2. Find the acceleration vector of the particle.
3. Find the distance traveled by the particle between the points $P(0,0,0)$ and $P\left(\frac{ 1 }{ 2 },\frac{ 2 \cdot \sqrt{2} }{ 3 },1\right)$.

Answer: 1. $\left\langlet,\sqrt{2}\cdot\sqrt{t},1\right\rangle$
2. $\left\langle1,\frac{1}{\sqrt{2}\cdot\sqrt{t}},0\right\rangle$
3. $\frac{3}{2}$
Subject: sequences_series

Problem: Expand the function $y = \ln\left(x + \sqrt{1 + x^2}\right)$ in a power series.

Answer: The final answer: $x-\frac{1}{2}\cdot\frac{x^3}{3}+\frac{1\cdot3}{4\cdot2}\cdot\frac{x^5}{5}-\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\cdot\frac{x^7}{7}+\cdots+\frac{(2\cdot n-1)!!}{(2\cdot n)!!}\cdot\frac{x^{2\cdot n+1}}{2\cdot n+1}+\cdots$
Subject: sequences_series

Problem: Compute $\sqrt[4]{80}$ with accuracy $0.0001$.

Answer: The final answer: $2.9907$
Subject: precalculus_review

Problem: Use properties of the natural logarithm to write the expression $\ln\left(x^4 \cdot y\right)$ as an expression of $\ln(x)$ and $\ln(y)$.

Answer: $\ln\left(x^4 \cdot y\right)$ = $4\cdot\ln(x)+\ln(y)$
Subject: integral_calc

Problem: Find the moment of inertia of the figure bounded by the arc of the semicircle $x^2 + y^2 = 9$, $y > 0$ relative to the x-axis.

Answer: The moment of inertia is: $\frac{243}{8}\cdot\pi$
Subject: multivariable_calculus

Problem: Find the maximum volume of the parallelepiped inscribed in the half-sphere of radius $R$.

Answer: $V$ = $\frac{4}{3\cdot\sqrt{3}}\cdot R^3$
Subject: algebra

Problem: Determine all rational values of $x$ at which $y = \sqrt{x^2 + x + 3}$ is a rational number.

Answer: The final answer: $x=\frac{q^2-3}{(1-2\cdot q)}$
Subject: differential_calc

Problem: Find the second derivative $\frac{d ^2y}{ d x^2}$ of the function $x = \left(4 \cdot \sin(t)\right)^3$, $y = 2 \cdot \sin(2 \cdot t)$.

Answer: $\frac{d ^2y}{ d x^2}$ = $\frac{\left(2304\cdot\left(\sin(t)\right)^3-1536\cdot\sin(t)\right)\cdot\cos(2\cdot t)-1536\cdot\left(\sin(t)\right)^2\cdot\cos(t)\cdot\sin(2\cdot t)}{7077888\cdot\left(\cos(t)\right)^3\cdot\left(\sin(t)\right)^6}$
Subject: multivariable_calculus

Problem: Evaluate the area of the figure bounded by $\rho = a \cdot \cos(2 \cdot \varphi)$.

Answer: The area is $\frac{\pi}{2}\cdot a^2$
Subject: algebra

Problem: 1. On a map, the scale is 1.5 inches : 20 miles. If two cities are 115 miles apart, how far apart are they on the map?

2. A model of a dinosaur built to a scale of 3 cm : 1 m is 24 centimeters tall. How tall was the actual dinosaur?

3. What are the dimensions of a building whose dimensions on a blueprint are 1.8 cm by 2.4 cm? The blueprint’s scale is 1 cm : 15 feet.

4. The distance between two cities is 180 miles. How far does that measure on a map with a scale of 1 inch : 25 miles?

Answer: 1. $8.625$ inches

2. $8$ m

3. $27$ feet by $36$ feet

4. $7.2$ inches
Subject: multivariable_calculus

Problem: Compute the area of the region bounded by the graph of $\left(x^2+y^2\right)^2=2 \cdot y^3$. (Hint: use the polar coordinates).

Answer: $S$ = $\frac{5}{4}\cdot\pi$
Subject: algebra

Problem: Find an exponential function that passes through the points:

| $x$ | $1$ | $2$ | $3$ | $4$ |
| --- | --- | --- | --- | --- |
| $f(x)$ | $10$ | $20$ | $40$ | $80$ |

Answer: $f(x)$ = $5\cdot2^x$
Subject: differential_calc

Problem: Find the local extrema of the function $f(x) = 3 \cdot \left(x^2\right)^{\frac{ 1 }{ 3 }} - 4 \cdot x^2$ using the First Derivative Test.

Answer: Local maxima: $x=-\frac{1}{2\cdot\sqrt[4]{4}}$, $x=\frac{1}{2\cdot\sqrt[4]{4}}$

Local minima: $x=0$
Subject: precalculus_review

Problem: Solve the exponential equation exactly: $3^{\frac{ x }{ 14 }} = \frac{ 1 }{ 10 }$

Answer: $x$ = $-\frac{14}{\log_{10}(3)}$
Subject: multivariable_calculus

Problem: Use the method of Lagrange multipliers to find the maximum and minimum values of the function $f(x,y,z) = y \cdot z + x \cdot y$ subject to the constraints $x \cdot y = 1$ and $y^2 + z^2 = 1$.

Answer: A minimum of $f(x,y,z)$ is $\frac{1}{2}$

A maximum of $f(x,y,z)$ is $\frac{3}{2}$
Subject: differential_calc

Problem: Sketch the curve:
$$
y = 25 \cdot x^2 \cdot e^{\frac{ 1 }{ 5 \cdot x }}
$$
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: $\left(\frac{1}{10},\infty\right)$
6. Intervals where the function is decreasing: $(-\infty,0)$, $\left(0,\frac{1}{10}\right)$
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: algebra

Problem: 1 cup = 8 fluid oz
1 pint = 2 cups = 16 fl oz
1 quart = 2 pints = 4 cups = 32 fluid oz
1 gal = 4 quarts = 8 pints = 16 cups = 128 fl oz

1 lb = 16 oz
1 ton = 2,000 lbs

1 kg = 1,000 g
1 metric ton = 1,000 kg

1 m = 100 cm
1 km = 1,000 m

1. Miles has a bag of peanuts that weighs 64 oz. How many lbs of peanuts does he have?

2. Michelle has 8 pints of milk. How many quarts of milk does she have?

3. Peter needs 8 cups of water for his drink mix. How many pints is this?

4. What is 1 metric ton and 500 kg multiplied by 4?

5. What is 50 lb and 20 oz divided by 5?

Answer: 1. $4$
2. $4$
3. $4$
4. $6,000$
5. $10.25$
Subject: multivariable_calculus

Problem: Evaluate the integral by interchanging the order of integration:
$$
\int_{1}^{27} \int_{1}^2 \left(\sqrt[3]{x} + \sqrt[3]{y}\right) \, dy \, dx
$$

Answer: $\int_{1}^{27} \int_{1}^2 \left(\sqrt[3]{x} + \sqrt[3]{y}\right) \, dy \, dx$ = $39\cdot\sqrt[3]{2}+\frac{81}{2}$
Subject: differential_calc

Problem: For the function $y = \ln\left(x^2 - \frac{ 3 }{ 2 }\right) - 4 \cdot x$ determine the intervals, where the function is increasing and decreasing.

Submit as your final answer:

1. Interval(s) where the function is increasing
2. Interval(s) where the function is decreasing

Answer: 1. Interval(s) where the function is increasing: $\left(\sqrt{\frac{3}{2}},\frac{3}{2}\right)$
2. Interval(s) where the function is decreasing: $\left(-\infty,-\sqrt{\frac{3}{2}}\right)$, $\left(\frac{3}{2},\infty\right)$
Subject: precalculus_review

Problem: Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{0}^\pi \sin\left(\frac{ 1 }{ 2 } \cdot x\right) \cdot \cos\left(\frac{ 1 }{ 4 } \cdot x\right) \, dx
$$

Answer: $\int_{0}^\pi \sin\left(\frac{ 1 }{ 2 } \cdot x\right) \cdot \cos\left(\frac{ 1 }{ 4 } \cdot x\right) \, dx$ = $\frac{8-2\cdot\sqrt{2}}{3}$
Subject: multivariable_calculus

Problem: Calculate the second-order partial derivatives. (Treat $A$,$B$,$C$,$D$ as constants.)

$f(x,y,z) = \arctan(x \cdot y \cdot z)$.

Answer: $f_{xx}(x,y,z)$ = $\frac{-2\cdot x\cdot y^3\cdot z^3}{\left(1+x^2\cdot y^2\cdot z^2\right)^2}$
$f_{xy}(x,y,z)$ = $f_{yx}(x,y,z)$ = $\frac{z-x^2\cdot y^2\cdot z^3}{\left(1+x^2\cdot y^2\cdot z^2\right)^2}$

$f_{yy}(x,y,z)$ = $\frac{-2\cdot x^3\cdot y\cdot z^3}{\left(1+x^2\cdot y^2\cdot z^2\right)^2}$

$f_{yz}(x,y,z)$ = $f_{zy}(x,y,z)$ = $\frac{x-x^3\cdot y^2\cdot z^2}{\left(1+x^2\cdot y^2\cdot z^2\right)^2}$

$f_{zz}(x,y,z)$ = $\frac{-2\cdot x^3\cdot y^3\cdot z}{\left(1+x^2\cdot y^2\cdot z^2\right)^2}$

$f_{xz}(x,y,z)$ = $f_{zx}(x,y,z)$ = $\frac{y-x^2\cdot y^3\cdot z^2}{\left(1+x^2\cdot y^2\cdot z^2\right)^2}$
Subject: differential_calc

Problem: For the function $f(x) = x^3 + x^4$, determine:

1. Intervals where $f$ is increasing or decreasing,
2. Local minima and maxima of $f$,
3. Intervals where $f$ is concave up and concave down, and
4. The inflection points of $f$.

Answer: 1. Increasing over $\left(-\frac{3}{4},0\right)\cup(0,\infty)$; decreasing over $\left(-\infty,-\frac{3}{4}\right)$
2. Local maxima at None; local minima at $x=-\frac{3}{4}$
3. Concave up for $x<-\frac{1}{2}$, $x>0$; concave down for $-\frac{1}{2}<x<0$
4. Inflection points at $P\left(-\frac{1}{2},-\frac{1}{16}\right)$
Subject: integral_calc

Problem: Find the area of the figure enclosed between the curves $y = 3 \cdot x^2$, $y = \frac{ x^2 }{ 6 }$, and $y = 2$.

Answer: Area: $\frac{48-8\cdot\sqrt{2}}{3\cdot\sqrt{3}}$
Subject: sequences_series

Problem: Compute $\sqrt[5]{1000}$ with accuracy $10^{-5}$.

Answer: The final answer: $3.981072$
Subject: differential_calc

Problem: Given the curve $y = 2 \cdot x - \arccos\left(\frac{ 1 }{ x }\right)$, find all

1. horizontal asymptotes,
2. vertical asymptotes,
3. slant asymptotes.

Answer: 1. horizontal asymptotes: None
2. vertical asymptotes: None
3. slant asymptotes: $y=2\cdot x-\frac{\pi}{2}$
Subject: algebra

Problem: Find the area of a triangle bounded by the x-axis, the line $f(x) = 12 - \frac{ 1 }{ 3 } \cdot x$, and the line perpendicular to $f(x)$ that passes through the origin.

Answer: The area of the triangle is $\frac{972}{5}$
Subject: precalculus_review

Problem: Use properties of the natural logarithm to write the expression $\ln\left(\frac{ 6 }{ \sqrt{e^3} }\right)$ as an expression of $\ln(6)$.

Answer: $\ln\left(\frac{ 6 }{ \sqrt{e^3} }\right)$ = $\ln(6)-\frac{3}{2}$
Subject: sequences_series

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

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