toloka/u-math huggingface dataset:
Subject: algebra

Problem: Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeros of the following polynomial:

$p(x) = x^3 - 6 \cdot x^2 + 2 \cdot x - 3$

Answer: The number of positive zeros: $1$, $3$
The number of negative zeros: $0$
Subject: precalculus_review

Problem: A restaurant owner wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $\$440$ for $20$ shirts and $\$1000$ for $100$ shirts.

1. Find the equation $C=f(x)$ that describes the total cost as a function of the number of shirts.
2. Determine how many shirts he must sell to break even if he sells the shirts for $\$10$ each.

Answer: 1. $f(x)$ = $7\cdot x+300$
2. $100$ shirts.
Subject: multivariable_calculus

Problem: Find the moment of inertia of one arch of the cycloid $x = a^2 \cdot \left(\frac{ t }{ 4 } - \sin\left(\frac{ t }{ 4 }\right)\right)$, $y = a^2 \cdot \left(1 - \cos\left(\frac{ t }{ 4 }\right)\right)$ relative to the x-axis.

Answer: Moment of Inertia: $\frac{2}{15}\cdot\left(64-43\cdot\sqrt{2}\right)\cdot a^6$
Subject: algebra

Problem: The radius and height of the right circular cylinder differ by two meters. The height is greater than the radius, and the volume is $15.125 \cdot \pi$ cubic meters. Find the dimensions of the cylinder.

Answer: $r$: $3.95546093-2$ $h$: $3.95546093$
Subject: sequences_series

Problem: Find the sum of the series $\sum_{k=2}^\infty \left(\frac{ 3^{k-1} }{ 4^{3 \cdot k+1} }\right)$.

Answer: The final answer: $\frac{3}{15616}$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ 6 \cdot x^3-7 \cdot x^2+3 \cdot x-1 }{ 2 \cdot x-3 \cdot x^2 } \, dx
$$

Answer: Answer is: $-x^2+x-\frac{1}{3}\cdot\ln\left(\left|x-\frac{2}{3}\right|\right)+\frac{1}{2}\cdot\ln\left(\left|1-\frac{2}{3\cdot x}\right|\right)+C$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ x+2 }{ \sqrt{6+10 \cdot x+25 \cdot x^2} } \, dx
$$

Answer: $\int \frac{ x+2 }{ \sqrt{6+10 \cdot x+25 \cdot x^2} } \, dx$ = $\frac{1}{25}\cdot\sqrt{6+10\cdot x+25\cdot x^2}+\frac{9}{25}\cdot\ln\left(1+5\cdot x+\sqrt{1+(5\cdot x+1)^2}\right)+C$
Subject: sequences_series

Problem: Determine the Taylor series for the function $f(x) = rac{ 1-x }{ (1+2 \cdot x)^3 }$, centered at $x = 0$.

Use this decomposition to find the sum of the series $\sum_{n=0}^\inftyrac{ (-1)^n \cdot (n+1) \cdot (3 \cdot n+4) }{ 2^{n+2} }$.  

1. Submit the Taylor series in summation notation.
2. Submit the sum of the series.

Answer: 1. The Taylor series in summation notation is $\sum_{n=0}^\infty\left(\frac{(-1)^{n+2}\cdot\left((n+1)!\right)\cdot2^{n-2}\cdot(3\cdot n+4)}{n!}\cdot x^n\right)$
2. The sum of the series is $\frac{2}{9}$
Subject: differential_calc

Problem: For the function $y = 2 \cdot x + \ln\left(\frac{ 1 }{ \frac{ x^2 }{ 2 } - 1 }\right)$, 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: $\left(-\infty,-\sqrt{2}\right)$, $(2,\infty)$
2. Interval(s) where the function is decreasing: $\left(\sqrt{2},2\right)$
Subject: precalculus_review

Problem: Divide $C = \frac{ (3-i)^2 }{ (3+i)^2 }$.

Answer: The final answer: $C=\frac{7}{25}-\frac{24}{25}\cdot i$
Subject: algebra

Problem: If an investor invests $\$23\ 000$ into two bonds, the first one that pays $4\%$ in simple interest, and the second one paying $2\%$ simple interest, and the investor earns $\$710$ annual interest, how much was invested in each account?

Answer: Add both values to your final answer here: $x=12500$, $y=10500$
Subject: algebra

Problem: Solve the following equations:

1. $-t + (5t - 7) = -5$
2. $21 - 3(2 - w) = -12$
3. $9 = 8b - (2b - 3)$
4. $4.5r - 2r + 3(r - 1) = 10.75$
5. $1.2(x - 8) + 2.4(x + 1) = 7.2$
6. $4.9m + (-3.2m) - 13 = -2.63$
7. $4(2.25w + 3.1) - 2.75w = 44.9$

Answer: The solutions to the given equations are:
1. $t=\frac{ 1 }{ 2 }$
2. $w=-9$
3. $b=1$
4. $r=\frac{ 5 }{ 2 }$
5. $x=4$
6. $m=\frac{ 61 }{ 10 }$
7. $w=\frac{ 26 }{ 5 }$
Subject: sequences_series

Problem: Calculate $\sqrt[3]{30}$ with an estimate error of $0.001$, using series expansion.

Answer: The final answer: $\frac{755}{243}$
Subject: multivariable_calculus

Problem: Find $I=\int_{0}^\pi \int_{0}^{4 \cdot \sin\left(\theta\right)} \frac{ 4 }{ \sqrt{16-r^2} } \cdot r \, d r \, d \theta$.

Answer: The final answer: $I=32\cdot\left(\frac{\pi}{2}-1\right)$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ \sqrt{1+x^2} }{ x } \, dx
$$

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

Problem: Newton’s Law of Cooling states that the temperature $T$ of an object at any time $t$ can be described by the equation $T = T_{s} + (T_{0} - T_{s}) \cdot e^{-k \cdot t}$, where $T_{s}$ is the temperature of the surrounding environment, $T_{0}$ is the initial temperature of the object, and $k$ is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time $t$ such that $t$ is equal to a single logarithm.

Answer: $t$ = $\ln\left(\left(\frac{T-T_s}{T_0-T_s}\right)^{-\frac{1}{k}}\right)$
Subject: precalculus_review

Problem: Solve $x \cdot (x+1) \cdot (x+2) \cdot (x+3) = 0.5625$.

Answer: The final answer: $x=-\frac{3}{2} \lor x=\frac{-3+\sqrt{10}}{2} \lor x=\frac{-3-\sqrt{10}}{2}$
Subject: integral_calc

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

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

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

Problem: Solve the integral:
$$
\int 3 \cdot \sin(2 \cdot x)^4 \cdot \cos(2 \cdot x)^2 \, dx
$$

Answer: $\int 3 \cdot \sin(2 \cdot x)^4 \cdot \cos(2 \cdot x)^2 \, dx$ = $\frac{3}{16}\cdot\left(x-\frac{\sin(8\cdot x)}{8}-\frac{\sin(4\cdot x)}{8}+\frac{\sin(12\cdot x)}{24}\right)+C$
Subject: differential_calc

Problem: Sketch the curve:  
$y = 4 \cdot x \cdot \sqrt{5-x^2}$.

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): $\left[-1\cdot5^{2^{-1}},5^{2^{-1}}\right]$
2. Vertical asymptotes: None
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: $\left(-\frac{\sqrt{5}}{\sqrt{2}},\frac{\sqrt{5}}{\sqrt{2}}\right)$
6. Intervals where the function is decreasing: $\left(-\sqrt{5},-\frac{\sqrt{5}}{\sqrt{2}}\right)$, $\left(\frac{\sqrt{5}}{\sqrt{2}},\sqrt{5}\right)$
7. Intervals where the function is concave up: $\left(-\sqrt{5},0\right)$
8. Intervals where the function is concave down: $\left(0,\sqrt{5}\right)$
9. Points of inflection: $P(0,0)$
Subject: sequences_series

Problem: Compute $\sqrt[5]{240}$ with an accuracy of $0.0001$.

Answer: The final answer: $2.9926$
Subject: differential_calc

Problem: Find the derivative of the function $y = \frac{ 3 \cdot \csc(x) - 2 \cdot \sin(x) }{ 5 \cdot \left(\cos(x)\right)^5 } - \frac{ 16 }{ 5 } \cdot \cot(2 \cdot x)$.

Answer: $y'$ = $\frac{1}{\left(\sin(x)\right)^2\cdot\left(\cos(x)\right)^6}$
Subject: algebra

Problem: Subtract the rational expression, then simplify:

$$
\frac{ 12 }{ 2 \cdot q }-\frac{ 6 }{ 3 \cdot p }
$$

Answer: The final answer: $\frac{6\cdot p-2\cdot q}{p\cdot q}$
Subject: precalculus_review

Problem: If $\tan(\theta) = \sqrt{x^2 - 1}$, find $\sec(\theta) + \tan(\theta)$.

Answer: $\sec(\theta) + \tan(\theta)$ = $x+\sqrt{x^2-1}$ (Enter your solution as an expression using only the variable $x$.)
Subject: algebra

Problem: The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a local newspaper currently has $84\ 000$ subscribers at a quarterly charge of $\$30$. Market research has suggested that if the owners raise the price to $\$34$, they would lose $9000$ subscribers. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue?

Answer: The final answer: $\frac{101}{3}$
Subject: algebra

Problem: Solve the following equations:

1. $6 (m+4) - 2m = -8$
2. $44 = 4 (8+h)$
3. $0.75 (8t - 4) = -2$
4. $3 (5-t) - 4t = 18$
5. $2 (y-5) = 16$
6. $0.1 (h+20) = 3$
7. $7y + 3 (8-y) = -12$

Answer: The solutions to the given equations are:
1. $m=-8$
2. $h=3$
3. $t=0.166666666666667$
4. $t=-\frac{ 3 }{ 7 }$
5. $y=13$
6. $h=10$
7. $y=-9$
Subject: precalculus_review

Problem: Find the value $k$ that satisfies the following integral equation:
$$
\int_{1}^k \log_{2}(x) \, dx = 2 - \frac{ 1 }{ \ln(2) }
$$

Answer: $k$ = $2$
Subject: sequences_series

Problem: Find the radius of convergence of the series:
$$
\sum_{k=1}^\infty \left(\frac{ \left(k!\right)^3 \cdot x^k }{ (2 \cdot k)! }\right)
$$

Answer: $R$ = $0$
Subject: precalculus_review

Problem: Solve the following absolute value equation:

$x^2 - 2 \cdot |x| - 3 = 0$

Answer: The final answer: $x=-3 \lor x=3$
Subject: multivariable_calculus

Problem: The surface of a large cup is formed by revolving the graph of the function $y = 0.25 \cdot x^{1.6}$ from $x = 0$ to $x = 5$ about the $y$-axis (measured in centimeters). Find the curvature $\kappa$ of the generating curve as a function of $x$.

Answer: $\kappa$ = $\frac{30}{x^{\frac{2}{5}}\cdot\left(25+4\cdot x^{\frac{6}{5}}\right)^{\frac{3}{2}}}$
Subject: multivariable_calculus

Problem: Evaluate the integral:
$$
\int \int \int_{R} 3 \cdot y \, dV
$$
where $R$ is bounded by:
1. $0 \le x \le 1$
2. $0 \le y \le x$
3. $0 \le z \le \sqrt{9-y^2}$

Answer: $\int \int \int_{R} 3 \cdot y \, dV$ = $\frac{216-86\cdot\sqrt{2}-243\cdot\arcsin\left(\frac{1}{3}\right)}{8}$
Subject: sequences_series

Problem: Use the substitution $(b+x)^r = (b+a)^r \cdot \left(1 + \frac{ x-a }{ b+a }\right)^r$ in the binomial expansion to find the Taylor series of the function $\sqrt{x+2}$ with the center $a=1$.

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

Problem: Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeros of the following polynomial:

$f(x) = 10 \cdot x^4 - 21 \cdot x^2 + 11$

Answer: The possible number of positive zeros: $2$, $0$
The possible number of negative zeros: $2$, $0$
Subject: sequences_series

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

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

Problem: The function $y=f(x)$ in parametric form is:

$x = t^5 - 5 \cdot t^3 - 20 \cdot t + 7$   

$y = 4 \cdot t^3 - 3 \cdot t^2 - 18 \cdot t + 3$ on the interval $\left(-2<t<2\right)$.

Find extreme values of this function. Submit as your final answer:

1. The point(s) where the function has a maximum or maxima
2. The point(s) where the function has a minimum or minima

Answer: 1. The point(s) where the function has a maximum: $P(31,14)$
2. The point(s) where the function has a minimum: $P\left(-\frac{1033}{32},-\frac{69}{4}\right)$
Subject: differential_calc

Problem: The following polynomial functions are given:

$f(x) = x^5 + x^4 + x + 11$

$g(x) = x^3 + x^2 - 10$

What is the value of $\frac{ f'(2) }{ g'(-2) }$?

Answer: The final answer: $\frac{f'(2)}{g'(-2)}=\frac{123}{8}$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ x + \sqrt[3]{x^2} + \sqrt[6]{x} }{ x \cdot \left(1 + \sqrt[3]{x}\right) } \, dx
$$

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

Problem: For the function $z = \ln\left(e^x + e^t\right)$, find:

1. $\frac{ d z }{d t}$
2. $\frac{ d z }{d t}$ when $x = t^3$

Answer: 1. $\frac{ d z }{d t}$: $\frac{e^t}{e^t+e^x}$
2. $\frac{ d z }{d t}$, when $x = t^3$: $\frac{3\cdot e^{t^3}\cdot t^2+e^t}{e^t+e^x}$
Subject: multivariable_calculus

Problem: The following vectors are given:

$u = 2 \cdot \mathbf{i} - 3 \cdot \mathbf{j} + 2 \cdot \mathbf{k}$

$v = 3 \cdot \mathbf{i} + 2 \cdot \mathbf{j} - 3 \cdot \mathbf{k}$

Find the following component:

Project of $u$ onto $v$

Answer: The final answer: $w=\frac{9}{11}\cdot i+\frac{6}{11}\cdot j-\frac{9}{11}\cdot k$
Subject: multivariable_calculus

Problem: Find the equation of a circle which passes through $(-3,1)$ and its center is on the same line with the centers of the following circles:

1. $(x-5)^2 + y^2 = 5$
2. $x^2 + (y-10)^2 = 130$

Answer: The final answer: $(x-2)^2+(y-6)^2=50$
Subject: differential_calc

Problem: Sketch the curve:  

$y = 3 \cdot x^2 - x^4 - 2$.

Submit as your final answer:

1. The domain (in interval notation)
2. Vertical asymptotes (Leave blank if there are no vertical asymptotes)
3. Horizontal asymptotes (Leave blank if there are no horizontal asymptotes)
4. Slant asymptotes (Leave blank if there are no slant asymptotes)
5. Intervals where the function is increasing (Leave blank if there are no such intervals)
6. Intervals where the function is decreasing (Leave blank if there are no such intervals)
7. Intervals where the function is concave up (Leave blank if there are no such intervals)
8. Intervals where the function is concave down (Leave blank if there are no such intervals)
9. Points of inflection (Leave blank if there are no 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: $\left(-\infty,-\frac{\sqrt{3}}{\sqrt{2}}\right)$, $\left(0,\frac{\sqrt{3}}{\sqrt{2}}\right)$
6. Intervals where the function is decreasing: $\left(-\frac{\sqrt{3}}{\sqrt{2}},0\right)$, $\left(\frac{\sqrt{3}}{\sqrt{2}},\infty\right)$
7. Intervals where the function is concave up: $\left(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$
8. Intervals where the function is concave down: $\left(-\infty,-\frac{1}{\sqrt{2}}\right)$, $\left(\frac{1}{\sqrt{2}},\infty\right)$
9. Points of inflection: $P\left(-\frac{1}{\sqrt{2}},-\frac{3}{4}\right)$, $P\left(\frac{1}{\sqrt{2}},-\frac{3}{4}\right)$
Subject: differential_calc

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

Problem: Evaluate the integral:
$$
I = \int 2 \cdot \ln\left(\sqrt{2-x}+\sqrt{2+x}\right) \, dx
$$

Answer: The final answer: $2\cdot x\cdot\ln\left(\sqrt{2-x}+\sqrt{2+x}\right)-\left((C+x)-2\cdot\arcsin\left(\frac{x}{2}\right)\right)$
Subject: precalculus_review

Problem: Solve $\frac{ 1 }{ x \cdot (x+2) }-\frac{ 1 }{ (x+1)^2 }=\frac{ 1 }{ 12 }$

Answer: The final answer: $x=-3 \lor x=1$
Subject: algebra

Problem: A formula for the normal systolic blood pressure for a man age $A$, measured in mmHg, is given as $P = 0.006 \cdot A^2 - 0.02 \cdot A + 120$. Find the age to the nearest year of a man whose normal blood pressure measures $125$ mmHg.

Answer: The final answer: $31$
Subject: integral_calc

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

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

Problem: Use the table of integrals to evaluate the integral $\int{\left(\sin(y)\right)^2 \cdot \left(\cos(y)\right)^3 \, dy}$.

Use this link to access the table of integrals: [Table of Integrals](https://openstax.org/books/calculus-volume-2/pages/a-table-of-integrals)

Answer: 1. Submit the formula used: $\int{\left(\sin(u)\right)^n \cdot \left(\cos(u)\right)^m d u}=-\frac{ \left(\sin(u)\right)^{n-1} \cdot \left(\cos(u)\right)^{m+1} }{ n+m }+\frac{ n-1 }{ n+m } \cdot \int{\left(\sin(u)\right)^{n-2} \cdot \left(\cos(u)\right)^m d u}$, $\int{\left(\cos(u)\right)^3 d u}=\frac{ 1 }{ 3 } \cdot \left(2+\left(\cos(u)\right)^2\right) \cdot \sin(u)+c$ (For example: to evaluate $\int{(x+3)^2 \, dx}$ you would use and submit the formula $\int{u^n \, du}=\frac{ u^{n+1} }{ n+1 }+C$).
2. $\int{\left(\sin(y)\right)^2 \cdot \left(\cos(y)\right)^3 \, dy}$ = $-\frac{\sin(y)\cdot\left(\cos(y)\right)^4}{5}+\frac{1}{5}\cdot\frac{1}{3}\cdot\left(2+\left(\cos(y)\right)^2\right)\cdot\sin(y)+c$
Subject: differential_calc

Problem: Find the first derivative of the function:
$$
y = \left(3 \cdot a^2 - 2 \cdot a \cdot b \cdot x + \frac{ 5 }{ 3 } \cdot b^2 \cdot x^2\right) \cdot \sqrt[3]{\left(\frac{ a }{ 3 } + \frac{ b }{ 3 } \cdot x\right)^2}
$$

Answer: The first derivative is: $\frac{40\cdot b^3\cdot x^2}{9\cdot3^{\frac{2}{3}}\cdot\sqrt[3]{a+b\cdot x}}$
Subject: sequences_series

Problem: Find the Taylor series of the given function $f(x) = \frac{ 1 }{ (x-1)^2 }$ centered at the indicated point: $a=0$ . (Hint: Differentiate $\frac{ 1 }{ 1-x }$.)

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

Problem: A biologist observes that a certain bacterial colony triples every 4 hours and after 12 hours occupies 1 square centimeter.

1. How much area was occupied by the colony when first observed?
2. What is the doubling time for the colony?

Answer: 1. $\frac{1}{27}$ cm². (Enter an exact solution as a fraction.)
2. $2.52$ hours. (Enter an approximate solution rounded to 2 decimal places.)
Subject: sequences_series

Problem: Find a “reasonable” upper-bound on the error in approximating $f(x) = x \cdot \ln(x)$ by its 3rd order Taylor polynomial $P_{3}(x)$ at $a = 1$ valid for all values of $x$ such that $|x - 1| \leq 0.7$.

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

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

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

Problem: Compute the limit:
$$
\lim_{x \to 0}\left(\frac{ -\sin(x) }{ x }\right)^{\frac{ 1 }{ 4 \cdot x^2 }}
$$

Answer: $\lim_{x \to 0}\left(\frac{ -\sin(x) }{ x }\right)^{\frac{ 1 }{ 4 \cdot x^2 }}$ = $e^{-\frac{1}{24}}$
Subject: precalculus_review

Problem: Solve the following system of equations:

1. $x + y = \frac{ 2 \cdot \pi }{ 3 }$
2. $\frac{ \sin(x) }{ \sin(y) } = 2$

Answer: The final answer: $x=\frac{\pi}{2}+\pi\cdot k$, $y=\frac{\pi}{6}-\pi\cdot k$
Subject: differential_calc

Problem: Find the maximum and minimum values of the function $r = \frac{ 1 }{ 3 } \cdot \sin(x) + \sin\left(\frac{ x }{ 3 }\right)$ in the closed interval $\left[0, \frac{ 3 }{ 2 } \cdot \pi\right]$.

Answer: Maximum Value: $\frac{2\cdot\sqrt{2}}{3}$
Minimum Value: $0$
Subject: algebra

Problem: For what values of $x$ and $y$ does the following expression attain the minimum possible value?

$E = 5 \cdot x^2 + 9 \cdot y^2 - 12 \cdot x \cdot y - 6 \cdot x + 14$

Answer: The final answer: $x=3$, $y=2$
Subject: precalculus_review

Problem: Evaluate $\tan\left(\arcsin\left(\frac{ 3 }{ 5 }\right)+\arccos\left(\frac{ 5 }{ 13 }\right)\right)$.

Answer: The final answer: $\tan\left(\arcsin\left(\frac{3}{5}\right)+\arccos\left(\frac{5}{13}\right)\right)=-\frac{63}{16}$
Subject: precalculus_review

Problem: Vertices of a triangle are at the points $A(-1,3)$, $B(3,0)$, and $C(8,12)$. Calculate the area of the triangle.

Answer: The final answer: $A=\frac{63}{2}$
Subject: precalculus_review

Problem: Find zeros of $f(x) = \sqrt[3]{2 \cdot x - 3} + \sqrt[3]{3 \cdot x + 2} - 3$

Answer: The final answer: $x=2$
Subject: precalculus_review

Problem: State the domain and range of the function $g\left(f(x)\right)$, if $g=\ln(x-5)$ and $f=x^2+2 \cdot x-3$.

1. Domain
2. Range

Answer: 1. Domain (with union): $(-\infty,-4)\cup(2,\infty)$
2. Range: $(-\infty,\infty)$
Subject: algebra

Problem: Find the area of a triangle bounded by the x-axis, the line $f(x) = 20 - \frac{ 1 }{ 3 } \cdot x$, and the line perpendicular to $f(x)$ that passes through the origin.

Answer: The area of the triangle is $540$
Subject: sequences_series

Problem: Determine the Taylor series for the function $f(x) = rac{ 2+x }{ (1-x)^2 }$, centered at $x = 0$.

Use this decomposition to find the sum of the series $\sum_{n=0}^\inftyrac{ 3 \cdot n+2 }{ 3^n }$.  

1. Submit the Taylor series in summation notation.
2. Submit the sum of the series.

Answer: 1. The Taylor series in summation notation is $\sum_{n=0}^\infty\left((3\cdot n+2)\cdot x^n\right)$
2. The sum of the series is $\frac{21}{4}$
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. A cake recipe calls for 12 cups of sugar at a bakery. How many pints is this?
2. Sandy has 1 quart and 1 pint of soda. How many cups can she divide this amount into?
3. Sonya drove 25 km to her friend’s house. What is this distance in meters?
4. What is 10 km and 90 m divided by 5?
5. What is 100 km and 60 m multiplied by 10?

Answer: 1. $6$
2. $6$
3. $25,000$
4. $2,018$
5. $1,000,600$