toloka/u-math huggingface dataset:
Subject: precalculus_review

Problem: A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by $r(t) = 6 - \frac{ 5 }{ t^2 + 1 }$, where $t$ is time measured in hours since a circle of a $1$-cm radius of the bacterium was put into the culture.

1. Express the area of the bacteria, $A(t)$, as a function of time.
2. Find the exact and approximate area of the bacterial culture in $3$ hours.
3. Express the circumference of the bacteria, $C(t)$, as a function of time.
4. Find the exact and approximate circumference of the bacteria in $3$ hours.

Answer: 1. $A(t)$ = $\pi\cdot\left(6-\frac{5}{t^2+1}\right)^2$ square centimeters
2. The exact area of the bacterial culture in $3$ hours = $\frac{121}{4}\cdot\pi$ square centimeters. The approximate area of the bacterial culture in $3$ hours = $95.03317777$ square centimeters.
3. $C(t)$ = $2\cdot\pi\cdot\left(6-\frac{5}{t^2+1}\right)$ centimeters
4. The exact circumference of the bacteria in $3$ hours = $11\cdot\pi$ centimeters. The approximate circumference of the bacteria in $3$ hours = $34.55751919$ centimeters.
Subject: algebra

Problem: Admission into an amusement park for $4$ children and $2$ adults is $\$116.90$. For $6$ children and $3$ adults, the admission is $\$175.35$. Assuming a different price for children and adults, what is the price of the child’s ticket and the price of the adult ticket? If there is no solution or infinitely many solutions, write $\text{None}$.

Answer: Price of the child’s ticket: None Price of the adult ticket: None
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ x^3 }{ \sqrt{4 \cdot x^2+4 \cdot x+5} } \, dx
$$

Answer: $\int \frac{ x^3 }{ \sqrt{4 \cdot x^2+4 \cdot x+5} } \, dx$ = $\left(\frac{1}{12}\cdot x^2-\frac{5}{48}\cdot x-\frac{5}{96}\right)\cdot\sqrt{4\cdot x^2+4\cdot x+5}+\frac{5}{16}\cdot\ln\left(x+\frac{1}{2}+\sqrt{1+\left(x+\frac{1}{2}\right)^2}\right)+C$
Subject: sequences_series

Problem: Expand the function $f(x) = \ln(1 + 3 \cdot x)$ 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. $3\cdot x-\frac{(3\cdot x)^2}{2}+\frac{(3\cdot x)^3}{3}-\frac{(3\cdot x)^4}{4}+\frac{(3\cdot x)^5}{5}-\frac{(3\cdot x)^6}{6}+\frac{(3\cdot x)^7}{7}-\frac{(3\cdot x)^8}{8}+\frac{(3\cdot x)^9}{9}$
2. $\left|R_{10}(x)\right|<\frac{3^{10}}{10}$
Subject: differential_calc

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

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): $\left[-1\cdot3^{2^{-1}},3^{2^{-1}}\right]$
2. Vertical asymptotes: None
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: $\left(-\sqrt{\frac{3}{2}},\sqrt{\frac{3}{2}}\right)$
6. Intervals where the function is decreasing: $\left(-3^{2^{-1}},-\sqrt{\frac{3}{2}}\right)$, $\left(\sqrt{\frac{3}{2}},3^{2^{-1}}\right)$
7. Intervals where the function is concave up: $\left(-3^{2^{-1}},0\right)$
8. Intervals where the function is concave down: $\left(0,3^{2^{-1}}\right)$
9. Points of inflection: $P(0,0)$
Subject: algebra

Problem: What is the domain of the function $f(x) = \ln\left(\frac{ x+2 }{ x-4 }\right)$?

Answer: Domain: $(-\infty,-2)\cup(4,\infty)$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ 1 }{ (x-3) \cdot \sqrt{10 \cdot x-24-x^2} } \, dx
$$

Answer: $\int \frac{ 1 }{ (x-3) \cdot \sqrt{10 \cdot x-24-x^2} } \, dx$ = $-\frac{2}{\sqrt{3}}\cdot\arctan\left(\frac{\sqrt{-x^2+10\cdot x-24}}{\sqrt{3}\cdot x-4\cdot\sqrt{3}}\right)+C$
Subject: algebra

Problem: Using properties of radicals and the absolute value, simplify the expression $\sqrt{(x-y)^2} + \sqrt{(x+y)^2}$, where $x < y < 0$.

Answer: The final answer: $-2\cdot x$
Subject: sequences_series

Problem: Find the sum of the series $\sum_{n=1}^\infty \frac{ 10 }{ n \cdot 9^n }$. (Use the formula $\int_{9}^\infty \frac{ 10 }{ x^{n+1} } \, dx = \frac{ 10 }{ n \cdot 9^n }$)

Answer: The sum of the series is $-10\cdot\ln\left(\frac{8}{9}\right)$
Subject: sequences_series

Problem: Find the Fourier series of the function $\varphi(x) = \frac{ x }{ 2 }$ in the interval $(0, 4 \cdot \pi)$.

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

Problem: The function $s(t) = \frac{ t }{ 1+t^2 }$ represents the position of a particle traveling along a horizontal line.

1. Find the velocity and acceleration functions.
2. Determine the time intervals when the object is slowing down or speeding up.

Answer: 1. The velocity function $v(t)$ = $\frac{1-t^2}{\left(1+t^2\right)^2}$ and acceleration function $a(t)$ = $\frac{2\cdot t\cdot\left(t^2-3\right)}{\left(1+t^2\right)^3}$
2. The time intervals when the object speeds up: $\left(1,\sqrt{3}\right)$ and slows down: $(0,1)$, $\left(\sqrt{3},\infty\right)$
Subject: differential_calc

Problem: Compute the derivative $y^{(5)}$ of the function $y = 2 \cdot e^{3 \cdot x} \cdot \sin(2 \cdot x)$.

Answer: $y^{(5)}$ = $2\cdot e^{3\cdot x}\cdot\left(122\cdot\cos(2\cdot x)-597\cdot\sin(2\cdot x)\right)$
Subject: sequences_series

Problem: Compute $\sqrt[3]{130}$ with an estimate error of $0.0001$ using series expansion.

Answer: The final answer: $5.06577778$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ 2 \cdot x-1 }{ \sqrt{8 \cdot x^2+4 \cdot x-10} } \, dx
$$

Answer: $\int \frac{ 2 \cdot x-1 }{ \sqrt{8 \cdot x^2+4 \cdot x-10} } \, dx$ = $\frac{1}{32\cdot\sqrt{2}}\cdot\left(16\cdot\sqrt{4\cdot x^2+2\cdot x-5}-24\cdot\ln\left(\left|2+4\cdot\sqrt{4\cdot x^2+2\cdot x-5}+8\cdot x\right|\right)\right)+C$
Subject: multivariable_calculus

Problem: Find the area of the region bounded by the parametric curves and the indicated values of the parameter:
$$
x = a \cdot \sin(2 \cdot t), \quad y = b \cdot \sin(t), \quad 0 \le t < 2 \cdot \pi
$$
(the "hourglass").

Answer: Area = $\frac{8}{3}\cdot a\cdot b$
Subject: algebra

Problem: In 1991, the moose population in a park was measured to be $4360$. By 1999, the population was measured again to be $5880$. Assume the population continues to change linearly.

1. Find a formula for the moose population, $P$, since 1991.
2. What does your model predict the moose population to be in 2003?

Answer: 1. Formula for the moose population, $P$, since 1991: $P(x)=190\cdot x+4360$
2. The moose population in 2003 is expected to be: $6640$
Subject: sequences_series

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

Answer: $f(x)$ = $x+\frac{2\cdot x^3}{6}+\frac{16\cdot x^5}{120}+\cdots$
Subject: precalculus_review

Problem: What are the vertical asymptotes of the function $y = \frac{ x+11 }{ x^3-x^2-4 \cdot x+4 }$?

Answer: The final answer: $x=1$, $x=2$, $x=-2$
Subject: algebra

Problem: Rewrite the quadratic expression $x^2 - 8 \cdot x + 12$ by completing the square.

Answer: $x^2 - 8 \cdot x + 12$ = $(x-4)^2-4$
Subject: multivariable_calculus

Problem: Consider the solid $Q=\left\{(x,y,z)|0 \le x \le 1,0 \le y \le 2,0 \le z \le 3\right\}$ with the density function $\rho(x,y,z)=x+y+1$.

1. Find the mass of $Q$.
2. Find the moments $M_{xy}$, $M_{xz}$, and $M_{yz}$ about the xy-plane, xz-plane, and yz-plane, respectively.
3. Find the center of mass of $Q$.

Answer: 1. $m$ = $15$
2. $M_{xy}$ = $\frac{45}{2}$, $M_{xz}$ = $17$, $M_{yz}$ = $8$
3. $x_{0}$ = $\frac{8}{15}$, $y_{0}$ = $\frac{17}{15}$, $z_{0}$ = $\frac{3}{2}$
Subject: sequences_series

Problem: Find the Taylor series of the given function $f(x) = \frac{ 1 }{ (x-1)^3 }$ centered at the indicated point: $a=0$.

Answer: $\frac{ 1 }{ (x-1)^3 }$ = $\sum_{n=0}^\infty\left(-\frac{(n+2)\cdot(n+1)\cdot x^n}{2}\right)$
Subject: integral_calc

Problem: A square with side $a$ rotates about a straight line passing through its vertex and forming an angle $\varphi$ with its diagonal $\frac{ \pi }{ 4 }<\varphi<\frac{ \pi }{ 2 }$. Find
1. the volume of the solid of revolution and
2. its surface area.

Answer: 1. Volume of a solid of revolution = $a^3\cdot\pi\cdot\sqrt{2}\cdot\sin(\varphi)$
2. Surface area = $4\cdot a^2\cdot\pi\cdot\sqrt{2}\cdot\sin(\varphi)$
Subject: precalculus_review

Problem: 1. Find the inverse function of $f(x) = \sqrt{x-1}$. Format: plain expression $(e \cdot t \cdot c)$; not in the form $f(x)^{-1}=(e \cdot t \cdot c)$

Answer: 1. the inverse function is : $x^2+1$
2. the domain of the inverse function is: $[0,\infty)$
3. the range of the inverse function is: $[1,\infty)$
Subject: multivariable_calculus

Problem: Evaluate the integral:
$$
\int_{0}^1 \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_{-\sqrt{1-x^2-y^2}}^{\sqrt{1-x^2-y^2}} 1 \, dz \, dy \, dx
$$

Answer: $\int_{0}^1 \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_{-\sqrt{1-x^2-y^2}}^{\sqrt{1-x^2-y^2}} 1 \, dz \, dy \, dx$ = $\frac{2\cdot\pi}{3}$
Subject: precalculus_review

Problem: Solve the equation:
$$
\tan\left(\frac{ \pi }{ 4 }+x\right)+\tan\left(\frac{ \pi }{ 4 }-x\right)=2
$$

Answer: The final answer: $x=\pi\cdot n$
Subject: sequences_series

Problem: Suppose that we have the series: $1-\frac{ 1 }{ 2^2 }+\frac{ 1 }{ 3^2 }-\frac{ 1 }{ 4^2 }+\cdots$ . Find the smallest $n$ - number of members of the series such that the accuracy of the partial sum is equal to:

1. $0.01$
2. $0.001$

Answer: 1. The smallest $n$ for an accuracy of $0.01$ is $n=10$
2. The smallest $n$ for an accuracy of $0.001$ is $n=31$
Subject: multivariable_calculus

Problem: Electrical power $P$ is given by $P = \frac{ V^2 }{ R }$, where $V$ is the voltage and $R$ is the resistance. Approximate the maximum percentage error in calculating power if $120$ V is applied to a $2000$-ohm resistor and the possible percent errors in measuring $V$ and $R$ are $3$% and $4$% respectively.

Answer: The maximum percentage error is: $0.1$
Subject: integral_calc

Problem: A tank initially holds 100 gallons of soapy water, containing 5 oz of soap. At $t = 0$, more soap solution is poured in containing 10 oz of soap/gallon at a rate of 1 gal/min. Allowing for the even dispersal of the soap, the new solution leaves the tank at the same rate. Find the amount of soap in the water after 5 minutes.

Answer: $y=53.5267$ oz of soap
Subject: multivariable_calculus

Problem: $E$ is located inside the cylinder $x^2+y^2=1$ and between the circular paraboloids $z=1-x^2-y^2$ and $z=x^2+y^2$. Find the volume of $E$.

Answer: Volume = $\frac{\pi}{4}$
Subject: integral_calc

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

Answer: $3 \cdot \int \cos(3 \cdot x)^6 \, dx$ = $C+\frac{1}{4}\cdot\sin(6\cdot x)+\frac{3}{64}\cdot\sin(12\cdot x)+\frac{15}{16}\cdot x-\frac{1}{48}\cdot\left(\sin(6\cdot x)\right)^3$
Subject: integral_calc

Problem: Find the antiderivative of $-\frac{ 1 }{ x \cdot \sqrt{1-x^2} }$.

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

Problem: Find the expansion in the series of the integral $\int_{-1}^x{\frac{ \arctan(x+1) }{ 2 } d x}$, using the expansion of the integrand in the series. Find the radius of convergence of the series.

Answer: 1. $\int_{-1}^x{\frac{ \arctan(x+1) }{ 2 } d x}$ = $\sum_{n=1}^\infty\left((-1)^{n+1}\cdot\frac{(x+1)^{2\cdot n}}{2\cdot2\cdot n\cdot(2\cdot n-1)}\right)$
2. The radius of convergence of the series is $R$ = $1$
Subject: differential_calc

Problem: For the curve $x = 10 \left(t - \sin(t)\right)$, $y = 10 \left(1 - \cos(t)\right)$, determine the curvature. Use $a=10$.

Answer: The curvature is: $\frac{1}{40\cdot\left|\sin\left(\frac{t}{2}\right)\right|}$
Subject: algebra

Problem: The equation $N(t) = \frac{ 500 }{ 1+49 \cdot e^{-0.7 \cdot t} }$ models the number of people in a town who have heard a rumor after $t$ days. To the nearest whole number, how many people will have heard the rumor after $3$ days?

Answer: The final answer: $71$
Subject: differential_calc

Problem: Given positive numbers $a$, $b$, $c$, and $d$, find the minimum value of

$$
S = \frac{ a }{ b+c+d }+\frac{ b }{ a+c+d }+\frac{ c }{ a+b+d }+\frac{ d }{ a+b+c }+\frac{ b+c+d }{ a }+\frac{ a+c+d }{ b }+\frac{ a+b+d }{ c }+\frac{ a+b+c }{ d }
$$

Answer: The final answer: $\frac{40}{3}$
Subject: multivariable_calculus

Problem: The electrical resistance $R$ produced by wiring resistors $R_{1}$ and $R_{2}$ in parallel can be calculated from the formula $\frac{ 1 }{ R }=\frac{ 1 }{ R_{1} }+\frac{ 1 }{ R_{2} }$. If $R_{1}$ and $R_{2}$ are measured to be $7$ ohm and $6$ ohm respectively, and if these measurements are accurate to within $0.05$ ohm, estimate the maximum possible error in computing $R$.

Answer: Maximum possible error: $0.02514793$
Subject: differential_calc

Problem: Make full curve sketching of $f(x) = 2 \cdot x^4 - 9 \cdot x^2 + 1$. 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 None
5. Intervals where the function is increasing $\left(-\frac{3}{2},0\right)$, $\left(\frac{3}{2},\infty\right)$
6. Intervals where the function is decreasing $\left(-\infty,-\frac{3}{2}\right)$, $\left(0,\frac{3}{2}\right)$
7. Intervals where the function is concave up $\left(-\infty,-\frac{\sqrt{3}}{2}\right)$, $\left(\frac{\sqrt{3}}{2},\infty\right)$
8. Intervals where the function is concave down $\left(-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}\right)$
9. Points of inflection $P\left(-\frac{\sqrt{3}}{2},-\frac{37}{8}\right)$, $P\left(\frac{\sqrt{3}}{2},-\frac{37}{8}\right)$
Subject: sequences_series

Problem: Compute $\sqrt[3]{130}$ with accuracy $0.0001$.

Answer: The final answer: $5.0658$
Subject: precalculus_review

Problem: Solve the equation $2 \cdot \left(\cos\left(\theta\right)\right)^2-\cos\left(\theta\right)=1$ for $0 \le \theta < \pi$.

Answer: $\theta_{1}$= $0$
$\theta_{2}$= $\frac{2\cdot\pi}{3}$
Subject: sequences_series

Problem: Find the interval of convergence and the sum of the series:
$$
2 + \frac{ 1 }{ 2 } \cdot x \cdot (1-x) + \frac{ 1 }{ 2 } \cdot x^2 \cdot (1-x) + \cdots + \frac{ 1 }{ 2 } \cdot x^{n-1} \cdot (1-x) + \cdots
$$

Answer: 1. Interval of convergence: $(-1,1]$
2. Sum of the series: $S(x)=\begin{cases}2+\frac{1}{2}\cdot x,&-1<x<1\\2,&x=1\end{cases}$
Subject: algebra

Problem: Find the formula for an exponential function that passes through the two points: $P(3,1)$ and $P(5,4)$.

Answer: $f(x)$ = $\frac{1}{8}\cdot2^x$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ x }{ \left(x^2-4 \cdot x+8\right)^2 } \, dx
$$

Answer: $\int \frac{ x }{ \left(x^2-4 \cdot x+8\right)^2 } \, dx$ = $C+\frac{x-2}{2\cdot\left(8+2\cdot(x-2)^2\right)}+\frac{1}{8}\cdot\arctan\left(\frac{1}{2}\cdot(x-2)\right)-\frac{1}{2\cdot\left(x^2-4\cdot x+8\right)}$
Subject: algebra

Problem: A statue is to be placed in the center of a park. The area of the base of the statue is $4 \cdot x^2 + 12 \cdot x + 9$ square meters. Factor the area to find the lengths of the sides of the statue.

Answer: The final answer: the length is $2\cdot x+3$, the width is $2\cdot x+3$
Subject: algebra

Problem: Find the unit rate for the following:

1. $72.00 in 12 hours

2. 8 slices per 2 people

3. 60 votes per 10 hours

4. 1500 km in 3 hours

5. 45 bags in 5 minutes

Answer: 1. $6$ per hour

2. $4$ per person

3. $6$ per hour

4. $500$ per hour

5. $9$ per minute
Subject: integral_calc

Problem: Solve the integral:
$$
\int \frac{ 6 }{ \cos(-4 \cdot x)^7 \cdot \sin(4 \cdot x) } \, dx
$$

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

Problem: Given $f\left(\frac{ y }{ x }\right) = \frac{ \sqrt{x^2+y^2} }{ x }$ and $x > 0$, find $f(x)$.

Answer: The final answer: $f(x)=\sqrt{1+x^2}$
Subject: differential_calc

Problem: Given the curve $y = x \cdot \arctan(x)$, find all 

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

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

Problem: Solve $\left(\sin(4 \cdot x)\right)^2 + \left(\cos(x)\right)^2 = 2 \cdot \sin(4 \cdot x) \cdot \left(\cos(x)\right)^4$.

Answer: The final answer: $x=\frac{\pi}{2}+n\cdot\pi$
Subject: algebra

Problem: A stone is dropped from a $196$-foot platform. Use the formula $h=-16 \cdot t^2 + v_{0} \cdot t + 196$ to determine how many seconds it will take for the stone to hit the ground.

Hint: Since the stone is dropped, $v_{0}=0$.

Answer: The final answer: $\frac{7}{2}$
Subject: sequences_series

Problem: Using the Taylor formula, decompose the function $f(x) = \ln\left(1+\frac{ x }{ 2 }\right)$ 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. $\ln\left(1+\frac{ x }{ 2 }\right)$ = $\frac{x}{2}-\frac{x^2}{2^2\cdot2}+\frac{x^3}{2^3\cdot3}-\frac{x^4}{2^4\cdot4}+\frac{x^5}{2^5\cdot5}-\frac{x^6}{2^6\cdot6}+\frac{x^7}{2^7\cdot7}-\frac{x^8}{2^8\cdot8}+\frac{x^9}{2^9\cdot9}$
2. Accuracy is not more than $\frac{1}{10\cdot2^{10}}$
Subject: differential_calc

Problem: Evaluate $\lim_{x \to 0^{+}} \left( \left( \frac{ \tan(2 \cdot x) }{ 2 \cdot x } \right)^{\frac{ 1 }{ 3 \cdot x^2 }} \right)$ using L'Hopital's Rule.

Answer: $\lim_{x \to 0^{+}} \left( \left( \frac{ \tan(2 \cdot x) }{ 2 \cdot x } \right)^{\frac{ 1 }{ 3 \cdot x^2 }} \right)$ = $e^{\frac{4}{9}}$
Subject: algebra

Problem: Use synthetic division to find the quotient and remainder. Ensure the equation is in the form required by synthetic division:

$$
\frac{ 4 \cdot x^3-33 }{ x-2 }
$$

Hint: Divide the dividend and divisor by the coefficient of the linear term in the divisor. Solve on a paper, if it is more convenient for you.

Answer: Quotient: $4\cdot x^2+8\cdot x+16$ Remainder: $-1$
Subject: sequences_series

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

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

Problem: Sketch the curve:  

$y = \sqrt{\frac{ 8-x^3 }{ x }}$.

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): $(0,2]$
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: $(0,2]$
7. Intervals where the function is concave up: $\left(0,\sqrt[3]{2}\right)$
8. Intervals where the function is concave down: $\left(\sqrt[3]{2},2\right)$
9. Points of inflection: $P\left(\sqrt[3]{2},2.1822\right)$
Subject: differential_calc

Problem: Find the derivative of the function: $f\left(\alpha\right) = \ln\left(\sqrt{\frac{ 3-\sin\left(\alpha\right) }{ 4+2 \cdot \sin\left(\alpha\right) }}\right)$ at $\alpha = \frac{ \pi }{ 4 }$.

Answer: $f'\left(\frac{ \pi }{ 4 }\right)$ = $-0.2848$
Subject: multivariable_calculus

Problem: The solid $Q$ has the mass given by the triple integral:
$$
\int_{0}^{\frac{ \pi }{ 4 }} \left( \int_{0}^{2 \cdot \sec\left(\theta\right)} \left( \int_{0}^1 \left( r^3 \cdot \cos\left(\theta\right) \cdot \sin\left(\theta\right) + 2 \cdot r \right) dz \right) dr \right) d \theta
$$
1. Find the density of the solid in rectangular coordinates.
2. Find the moment $M_{xz}$ about the xz-plane.

Answer: 1. $\rho(x,y,z)$ = $x\cdot y+2$
2. $M_{xz}$ = $\frac{24}{5}$
Subject: multivariable_calculus

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

Answer: $\frac{d ^2y}{ d x^2}$ = $\frac{30}{e^t\cdot\left(\cos(2\cdot t)-2\cdot\sin(2\cdot t)\right)^3}$
Subject: algebra

Problem: Solve the following equations:

1. $7x - 2 = 26$
2. $2m - 6 = 48$
3. $-6h - 6 = 30$
4. $5x + 20 = -20$
5. $3 = -3y - 15$
6. $-24 = 14y - 5$
7. $7r - 5 = 10$
8. $9 = 16y + 51$

Answer: The solutions to the given equations are:
1. $x=4$
2. $m=27$
3. $h=-6$
4. $x=-8$
5. $y=-6$
6. $y=-\frac{ 19 }{ 14 }$
7. $r=\frac{ 15 }{ 7 }$
8. $y=-\frac{ 21 }{ 8 }$
Subject: multivariable_calculus

Problem: Determine the concavity of the curve $x = 2 \cdot t + \ln(t)$, $y = 2 \cdot t - \ln(t)$.

Answer: * Concave up on $(0,\infty)$, $\left(-\infty,-\frac{1}{2}\right)$
* Concave down on $\left(-\frac{1}{2},0\right)$
Subject: algebra

Problem: Evaluate the function at the indicated values $f(-3)$, $f(2)$, $f(-a)$, $-f(a)$, and $f(a+h)$.

$f(x) = |x-1| - |x+1|$

Answer: 1. $f(-3)$ = $2$
2. $f(2)$ = $-2$
3. $f(-a)$ = $|a+1|-|-a+1|$
4. $-f(a)$ = $-|a-1|+|a+1|$
5. $f(a+h)$ = $|a+h-1|-|a+h+1|$
Subject: integral_calc

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

Answer: $\int \frac{ \sin(x)^4 }{ \cos(x) } \, dx$ = $C-\frac{1}{2}\cdot\ln\left(\left|\frac{1-\sin(x)}{1+\sin(x)}\right|\right)-\frac{1}{3}\cdot\left(\sin(x)\right)^3-\sin(x)$
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 $P = t^2 - 12t + 180$, 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: $2$
Subject: algebra

Problem: Use the Factor Theorem to find all real zeros of the polynomial $p(x) = 2 \cdot x^3 - 9 \cdot x^2 + 13 \cdot x - 6$ given that $x - 1$ is a factor of $p(x)$.

Answer: The real zeros are $1$, $2$, $\frac{3}{2}$
Subject: sequences_series

Problem: Find the Fourier expansion of this function: $f(x) = x$ at $(-\pi, \pi)$.

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

Problem: Find the inverse of the function $f(x) = \log_{a}\left(x + \sqrt{x^2 + 1}\right)$, where $a > 1$.

Answer: The final answer: $\frac{a^{2\cdot x}-1}{2\cdot a^x}$
Subject: sequences_series

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

Answer: The final answer: $2\cdot\pi\cdot(x-\pi)+2\cdot(x-\pi)^2$
Subject: algebra

Problem: Find the solution to the following inequality and express it in interval notation:
$$-4 (x-6) (x+3) (x-2) > 0$$

Answer: The solution set to the inequality is: $\left(-\infty,\ -3\right)\cup\left(2,\ 6\right)$