toloka/u-math huggingface dataset:
Subject: sequences_series

Problem: Find the power series representation for the function $f(x) = \frac{ 8 \cdot x + 2 }{ 2 \cdot x^2 - 3 \cdot x + 1 }$. Determine the radius of convergence and the interval of convergence for that series.

Answer: 1. The series : $\sum_{n=0}^\infty\left(12\cdot(2\cdot x)^n-10\cdot x^n\right)$
2. $R$ = $\frac{1}{2}$
3. $I$ = $\left(-\frac{1}{2},\frac{1}{2}\right)$
Subject: algebra

Problem: Note that $a^3-b^3=(a-b) \cdot \left(a^2+a \cdot b+b^2\right)$. Using this factorization principle, factor the cubic polynomial $64 \cdot x^3-125$.

Answer: The final answer: $(4\cdot x-5)\cdot\left(16\cdot x^2+20\cdot x+25\right)$
Subject: precalculus_review

Problem: Solve $0 < \left| \frac{ 5 \cdot x - 6 }{ 7 } \right| < 3$.

Answer: The final answer: $\left(-3,\frac{6}{5}\right)\cup\left(\frac{6}{5},\frac{27}{5}\right)$
Subject: multivariable_calculus

Problem: Evaluate the integral $\int \int \int f(x,y,z) \, dx \, dy \, dz$ over the solid $f(x,y,z) = x^2 + y^2$, $0 \le x^2 + y^2 \le 4$, $y \ge 0$, $0 \le z \le 3 - x$.

Answer: $\int \int \int f(x,y,z) \, dx \, dy \, dz$ = $12\cdot\pi$
Subject: precalculus_review

Problem: Find zeros of the absolute function $f(x) = \left| x^2 - 4 \right| + |x| + 2 \cdot x - 2$.

Answer: The final answer: $x_1=-3$, $x_2=-1$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \frac{ -8 \cdot \cos(-4 \cdot x)^3 }{ 5 \cdot \sin(-4 \cdot x)^9 } \, dx
$$

Answer: $\int \frac{ -8 \cdot \cos(-4 \cdot x)^3 }{ 5 \cdot \sin(-4 \cdot x)^9 } \, dx$ = $C-\frac{2}{5}\cdot\left(\frac{1}{3}\cdot\left(\cot(4\cdot x)\right)^6+\frac{1}{4}\cdot\left(\cot(4\cdot x)\right)^4+\frac{1}{8}\cdot\left(\cot(4\cdot x)\right)^8\right)$
Subject: algebra

Problem: Find the zeros of the polynomial function with complex roots: $f(x) = 2 \cdot x^3 - 2 \cdot x^2 + 5 \cdot x - 5$.

Answer: $x$ = $1$, $i\cdot\sqrt{\frac{5}{2}}$, $-i\cdot\sqrt{\frac{5}{2}}$
Subject: differential_calc

Problem: Calculate the integral:
$$
\int \frac{ 1 }{ \sqrt[4]{(x-1)^3 \cdot (x+2)^5} } \, dx
$$

Answer: $\int \frac{ 1 }{ \sqrt[4]{(x-1)^3 \cdot (x+2)^5} } \, dx$ = $\frac{4}{3}\cdot\sqrt[4]{\frac{x-1}{x+2}}+C$
Subject: sequences_series

Problem: For a sequence with the general term $x_{n} = \frac{ 3 \cdot n - 5 }{ 9 \cdot n + 4 }$, it is known that $\lim_{n \to \infty} \left(x_{n}\right) = \frac{ 1 }{ 3 }$.

Find the number of points $x_{n}$ lying outside the open interval $\left(\frac{ 1 }{ 3 } - \frac{ 1 }{ 1000 }, \frac{ 1 }{ 3 } + \frac{ 1 }{ 1000 }\right)$.

Answer: The final answer: $703$
Subject: algebra

Problem: Find two consecutive even numbers whose product is $624$.

Use the method of factoring, the square root principle, or the Quadratic Formula.

Answer: The final answer: $24$, $26$, $-24$, $-26$
Subject: multivariable_calculus

Problem: Evaluate the integral $\int_{0}^1 \int_{x^3}^{x^2} (x+y) \, dy \, dx$ by reversing the order of integration.

Answer: The integral corresponding to the reversed order of integration: $\int_0^1\int_{\sqrt{y}}^{\sqrt[3]{y}}(x+y)dxdy$

The value of the given integral: $\frac{11}{140}$
Subject: sequences_series

Problem: Find the Fourier series of the function $f(x) = \frac{ -1 }{ 2 } \cdot x$ in the interval $[-2,2]$.

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

Problem: Find the moment of inertia of one arch of the cycloid $x = 3 \cdot a \cdot \left(\frac{ t }{ 2 } - \sin\left(\frac{ t }{ 2 }\right)\right)$, $y = 3 \cdot a \cdot \left(1 - \cos\left(\frac{ t }{ 2 }\right)\right)$ relative to the x-axis.

Answer: Moment of Inertia: $\frac{1152}{5}\cdot a^3$
Subject: precalculus_review

Problem: Find zeros of $f(x) = 2 \cdot x^4 - 13 \cdot x^3 + 24 \cdot x^2 - 13 \cdot x + 2$.

Answer: The final answer: $x_1=2+\sqrt{3}$, $x_2=2-\sqrt{3}$, $x_3=2$, $x_4=\frac{1}{2}$
Subject: sequences_series

Problem: Let $P_{n}(x)$ be the n-th order Taylor polynomial of $f(x) = \cos(x)$ about $a = 0$. Find the minimum $n$ such that the approximation $f(x) \approx P_{n}(x)$ is accurate to within $0.00001$ for every value of $x$ satisfying $|x| \le 0.1$.

Answer: The final answer: $n=3$
Subject: integral_calc

Problem: Compute the volume of the solid formed by rotating about the x-axis the area bounded by the axes and the parabola $x^{\frac{ 1 }{ 2 }}+y^{\frac{ 1 }{ 2 }}=3^{\frac{ 1 }{ 2 }}$.

Answer: Volume = $\pi\cdot\frac{9}{5}$
Subject: differential_calc

Problem: Solve to four decimal places using Newton's method: $x^3 + (x + 1)^3 = 10^3$. Choose any initial guess $x_0$ that is not the exact root.

Answer: $x$ ≈ $x=7.4055$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ 3 }{ 4 \cdot x^2 \cdot \sqrt{5 \cdot x^2-2 \cdot x+1} } \, dx
$$

Answer: Answer is: $\frac{3}{4}\cdot\left(C-\sqrt{5+\frac{1}{x}^2-\frac{1\cdot2}{x}}-\ln\left(\left|\frac{1}{x}+\sqrt{5+\frac{1}{x}^2-\frac{1\cdot2}{x}}-1\right|\right)\right)$
Subject: differential_calc

Problem: Use implicit differentiation to find an equation of the tangent line to the curve 
$x^4 - x^3 \cdot y^2 - y^3 \cdot x + 5 \cdot y^4 = 17$ at the point 
$P(-2,-1)$.

Answer: The line tangent to the curve $x^4 - x^3 \cdot y^2 - y^3 \cdot x + 5 \cdot y^4 = 17$ at the point $(-2,-1)$ is: $y+1=-\frac{43}{30}\cdot(x+2)$
Subject: integral_calc

Problem: Calculate the integral:
$$
\int_{-\sqrt{2}}^{\sqrt{2}} \frac{ 23 \cdot x^7+7 \cdot x^6-130 \cdot x^5-72 \cdot x^3-112 \cdot x^2+4 \cdot x+7 }{ x^2+4 } \, dx
$$

Answer: $\int_{-\sqrt{2}}^{\sqrt{2}} \frac{ 23 \cdot x^7+7 \cdot x^6-130 \cdot x^5-72 \cdot x^3-112 \cdot x^2+4 \cdot x+7 }{ x^2+4 } \, dx$ = $7\cdot\arctan\left(\frac{1}{\sqrt{2}}\right)-\frac{392\cdot\sqrt{2}}{15}$
Subject: multivariable_calculus

Problem: Find the first derivative $y_{x}'$ of the function:
$$
x = \arcsin\left(\frac{ t }{ \sqrt{1+t^2} }\right), \quad y = \arccos\left(\frac{ 1 }{ \sqrt{1+t^2} }\right), \quad t \ge 0
$$

Answer: $y_{x}'$ = $1$
Subject: integral_calc

Problem: Compute the integral:
$$
3 \cdot \int x^{-8} \cdot \left(9+x^2\right)^{\frac{ 1 }{ 2 }} \, dx
$$

Answer: $3 \cdot \int x^{-8} \cdot \left(9+x^2\right)^{\frac{ 1 }{ 2 }} \, dx$ = $C+\frac{2}{1215}\cdot\left(1+\frac{9}{x^2}\right)^2\cdot\sqrt{1+\frac{9}{x^2}}-\frac{1}{729}\cdot\left(1+\frac{9}{x^2}\right)\cdot\sqrt{1+\frac{9}{x^2}}-\frac{1}{1701}\cdot\left(1+\frac{9}{x^2}\right)^3\cdot\sqrt{1+\frac{9}{x^2}}$
Subject: algebra

Problem: Perform the indicated operation and express the result as a simplified complex number:

$$
\frac{ 4+5 \cdot i }{ 4-5 \cdot i }
$$

Answer: The final answer: $-\frac{9}{41}+\frac{40}{41}\cdoti$
Subject: sequences_series

Problem: For which $p$ does the series $\sum_{n=1}^\infty \left(\frac{ p^{n^2} }{ 2^n }\right)$ converge?

Answer: Converges for : $p\le1$ (leave empty, if the series diverges for any $p$).
Subject: precalculus_review

Problem: A rental car company rents cars for a flat fee of $\$20$ and an hourly charge of $\$10.25$. Therefore, the total cost $C$ to rent a car is a function of the hours $t$ the car is rented plus the flat fee.

1. Write the formula for the function that models this situation.
2. Find the total cost to rent a car for 2 days and 7 hours.
3. Determine how long the car was rented if the bill is $\$432.73$. (Round to 2 decimal places)

Answer: 1. The formula for the function is $C(t)$ = $10.25\cdot t+20$
2. The total cost to rent a car is $ $583.75$
3. The time of car rental in hours is $40.27$
Subject: differential_calc

Problem: For the function $r = \frac{ 1 }{ 2 } \cdot \arctan\left(\frac{ 3 }{ \beta }\right) + \arccot\left(3 \cdot \cot\left(\beta\right)\right)$, find the derivative $r'(0)$ and $r'(2 \cdot \pi)$. Submit as your final answer:

1. $r'(0)$
2. $r'(2 \cdot \pi)$

Answer: 1. $r'(0)$ = $\frac{1}{6}$
2. $r'(2 \cdot \pi)$ = $\frac{9+8\cdot\pi^2}{54+24\cdot\pi^2}$
Subject: multivariable_calculus

Problem: Find the tangential and normal components of acceleration if $\vec{r}(t) = \left\langle 6 \cdot t, 3 \cdot t^2, 2 \cdot t^3 \right\rangle$

Answer: $a_{T}$ = $\frac{12\cdot t^3+6\cdot t}{\sqrt{t^4+t^2+1}}$ ; $a_{N}$ = $\frac{6\cdot\sqrt{t^4+4\cdot t^2+1}}{\sqrt{t^4+t^2+1}}$
Subject: differential_calc

Problem: Find $\frac{ d y }{d x}$, given $y=\tan(2 \cdot v)$ and $v=\arctan(2 \cdot x-1)$.

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

Problem: A sky diver jumps from a reasonable height above the ground. The air resistance she experiences is proportional to her velocity, and the constant of proportionality is $0.24$. It can be shown that the downward velocity of the sky diver at time $t$ is given by
$$
v(t) = 180 \cdot \left(1 - e^{-0.24 \cdot t}\right)
$$
where $t$ is measured in seconds and $v(t)$ is measured in feet per second.

1. Find the initial velocity of the sky diver.
2. Find the velocity after $4$ seconds (round your answer to one decimal place).
3. The maximum velocity of a falling object with wind resistance is called its terminal velocity. Find the terminal velocity of this sky diver. (round your answer to the nearest whole number).

Answer: 1. The initial velocity of the sky diver is $0$.
2. The velocity after $4$ seconds is $111.1$.
3. The terminal velocity of the sky diver is $180$.
Subject: algebra

Problem: Determine the point(s) where the line $y = m \cdot x$ intersects the circle $x^2 + y^2 = 4$.

Answer: The final answer: $P\left(\sqrt{\frac{4}{1+m^2}},\sqrt{\frac{4\cdot m^2}{1+m^2}}\right)$, $P\left(-\sqrt{\frac{4}{1+m^2}},-\sqrt{\frac{4\cdot m^2}{1+m^2}}\right)$
Subject: multivariable_calculus

Problem: Let $z = e^{1 - x \cdot y}$, $x = t^{\frac{ 1 }{ 3 }}$, $y = t^3$. Find $\frac{ d z }{ d t }$.

Answer: $\frac{ d z }{ d t }$ = $\frac{-(10\cdot e)}{3}\cdot e^{-t^3\cdot\sqrt[3]{t}}\cdot t^2\cdot\sqrt[3]{t}$
Subject: algebra

Problem: Consider the equation $25 + 8x + x^2 + y^2 - 10y = 0$.

1. Complete the square to write the equation in the standard form of a circle.
2. Identify the center and the radius of the circle.

*Note: The standard form of a circle is $(x-a)^2 + (y-b)^2 = r^2$.*

Answer: 1. Standard form: $(x+4)^2+(y-5)^2=16$
2. Center $(a,b) = $ $P\left(-4,\ 5\right)$
3. Radius $r = $ $4$
Subject: algebra

Problem: To convert from $x$ degrees Celsius to $y$ degrees Fahrenheit, we use the formula $f(x) = \frac{ 9 }{ 5 } \cdot x + 32$. Find the inverse function, if it exists. If it doesn't exist, write $\text{None}$.

Answer: Inverse function: $g(x)=\frac{5}{9}\cdot(x-32)$
Subject: integral_calc

Problem: The chain of length $200$ rises up, winding on the winch. Compute the work of the weight force when lifting the chain, neglecting the size of the winch, if the running meter of the chain weighs $50$ kg.

Answer: $W$ = $-1000000$
Subject: algebra

Problem: Rewrite the quadratic expression $2 + z - 6 \cdot z^2$ by completing the square.

Answer: $2 + z - 6 \cdot z^2$ = $-6\cdot\left(z-\frac{1}{12}\right)^2+\frac{49}{24}$
Subject: sequences_series

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

Answer: The final answer: $\frac{1}{60}$
Subject: integral_calc

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

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

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

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

Problem: Compute the limit:
$$
\lim_{x \to 3} \left( \frac{ x }{ x-3 } - \frac{ 1 }{ \ln\left( \frac{ x }{ 3 } \right) } \right)
$$

Answer: $\lim_{x \to 3} \left( \frac{ x }{ x-3 } - \frac{ 1 }{ \ln\left( \frac{ x }{ 3 } \right) } \right)$ = $\frac{1}{2}$
Subject: differential_calc

Problem: Compute the derivative of the function $y = \sqrt[4]{\frac{ x^5 \cdot \left(5 \cdot x^2+1\right) }{ \sqrt[3]{2+3 \cdot x} }}$ by taking the natural log of both sides of the equation.

Answer: Derivative: $y'=\frac{50\cdot x^3+35\cdot x^2+14\cdot x+10}{30\cdot x^4+20\cdot x^3+12\cdot x^2+8\cdot x}\cdot\sqrt[4]{\frac{x^5\cdot\left(5\cdot x^2+1\right)}{\sqrt[3]{2+3\cdot x}}}$
Subject: sequences_series

Problem: Compute $\lim_{x \to 0}\left(\frac{ 3 \cdot \cos(x)+6 }{ 4 \cdot x^3 \cdot \sin(x) }-\frac{ 9 }{ 4 \cdot x^4 }\right)$. Use the expansion of the function in the Taylor series.

Answer: The final answer: $\frac{1}{80}$
Subject: precalculus_review

Problem: Solve $2 \cdot z = |z| + 2 \cdot i$.

Answer: The final answer: $z=\frac{1}{\sqrt{3}}+i$
Subject: multivariable_calculus

Problem: Find $f(x,y) = e^{x^2} + \sqrt{y}$ at $P(0.1,9.1)$. Find the linear approximation for this function.

Answer: The final answer:

1. $f\left(x_{0},y_{0}\right)$: $4.02667079$

2. $L(x,y)$: $4+\frac{1}{6}\cdot(y-9)$
Subject: differential_calc

Problem: Compute the derivative of the implicit function:
$$
2 \cdot y^4 - \frac{ 3 \cdot x + 3 \cdot y }{ 2 \cdot x - 2 \cdot y } = 0
$$

Answer: $y'$: $y'=-\frac{y^2}{2\cdot\left(x^2-y^2\right)-x\cdot y}$
Subject: integral_calc

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

Answer: $\int \frac{ 2 \cdot x+\sqrt{x-2} }{ \sqrt[4]{x-2}+\sqrt[4]{(x-2)^3} } \, dx$ = $C+20\cdot\sqrt[4]{x-2}+\frac{8\cdot1}{5}\cdot\sqrt[4]{x-2}^5-20\cdot\arctan\left(\sqrt[4]{x-2}\right)-\frac{1\cdot4}{3}\cdot\sqrt[4]{x-2}^3$
Subject: precalculus_review

Problem: Solve $\sqrt{3 \cdot x^2+5 \cdot x+8}-\sqrt{3 \cdot x^2+5 \cdot x+1}=1$.

Answer: The final answer: $x=-\frac{8}{3} \lor x=1$
Subject: algebra

Problem: Find the zeros of the polynomial function with complex roots: $f(x) = 2 \cdot x^3 + 4 \cdot x^2 + 3 \cdot x + 6$.

Answer: $x$ = $-i\cdot\sqrt{\frac{3}{2}}$, $i\cdot\sqrt{\frac{3}{2}}$, $-2$
Subject: differential_calc

Problem: For the function $y = \frac{ x }{ x^2 - 1 }$ find the derivative $y^{(n)}$.

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

Problem: Compute the integral:
$$
\int \frac{ 4 \cdot x-\sqrt[3]{36 \cdot x^2}-\sqrt[6]{6 \cdot x} }{ x \cdot \left(1+\sqrt[3]{6 \cdot x}\right) } \, dx
$$

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

Problem: Given $y = 3 \cdot x^5 + 10 \cdot x^4 - 20$, find where the function is 
1. concave up,
2. concave down, and
3. point(s) of inflection.

Answer: 1. Concave up: $(-2,0)$, $(0,\infty)$
2. Concave down: $(-\infty,-2)$
3. Point(s) of Inflection: $P(-2,44)$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ x^3 }{ \sqrt{x^2+x+1} } \, dx
$$

Answer: $\int \frac{ x^3 }{ \sqrt{x^2+x+1} } \, dx$ = $\left(\frac{1}{3}\cdot x^2-\frac{5}{12}\cdot x-\frac{1}{24}\right)\cdot\sqrt{x^2+x+1}+\frac{7}{16}\cdot\ln\left(\left|x+\frac{1}{2}+\sqrt{x^2+x+1}\right|\right)+C$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ -3 }{ e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} } \, dx
$$

Answer: $\int \frac{ -3 }{ e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} } \, dx$ = $C-\frac{1}{4}\cdot\left(\frac{3}{e^{4\cdot x}+\sqrt{1+e^{8\cdot x}}}+3\cdot\ln\left(\frac{e^{4\cdot x}+\sqrt{1+e^{8\cdot x}}-1}{1+e^{4\cdot x}+\sqrt{1+e^{8\cdot x}}}\right)\right)$
Subject: sequences_series

Problem: Since $1 + x^3 + x^6 + \cdots + x^{3 \cdot n} + \cdots = \frac{ 1 }{ 1 - x^3 }$, when $|x| < 1$, then find the sum of the series:
$$
\sum_{n=1}^\infty \left(3 \cdot n \cdot x^{3 \cdot n-1}\right)
$$

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

Problem: Find the derivative of $y = x \cdot \sin(x) + 2 \cdot x \cdot \cos(x) - 2 \cdot \sin(x) + \ln(\sin(x)) + c^2$.

Answer: The final answer: $y'=\frac{\left(\sin(x)\right)^2+x\cdot\sin(x)\cdot\cos(x)-2\cdot x\cdot\left(\sin(x)\right)^2-\cos(x)}{\sin(x)}$
Subject: algebra

Problem: Evaluate the expression $\frac{ (3+i)^2 }{ (1+2 \cdot i)^2 }$ and write the result as a simplified complex number.

Answer: $\frac{ (3+i)^2 }{ (1+2 \cdot i)^2 }$ = $-2\cdoti$
Subject: precalculus_review

Problem: Use properties of logarithms to write the expression as a sum, difference, and/or product of logarithms:

$$
\log_{4}\left(\frac{ \sqrt[3]{x \cdot y} }{ 64 }\right)
$$

Answer: $\log_{4}\left(\frac{ \sqrt[3]{x \cdot y} }{ 64 }\right)$ = $-3+\frac{1}{3}\cdot\log_4(x)+\frac{1}{3}\cdot\log_4(y)$
Subject: precalculus_review

Problem: Find the domain of the function $f(x) = \arccos\left(\frac{ 3 }{ 4+2 \cdot \sin(x) }\right)$

Answer: The final answer: $-\frac{\pi}{6}+2\cdot k\cdot\pi\le x\le\frac{7\cdot\pi}{6}+2\cdot k\cdot\pi$
Subject: sequences_series

Problem: Given that $\frac{ 1 }{ 1-x } = \sum_{n=0}^\infty x^n$, use term-by-term differentiation or integration to find a power series for the function $f(x) = \ln\left(1+x^2\right)$ centered at $x=0$.

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

Problem: Find the derivative of the function: $y = -3 \cdot x^{\sqrt[3]{2 \cdot x}}$

Answer: $\frac{ d y }{d x}$ = $-\left(\frac{3\cdot\sqrt[3]{2}}{x^{\frac{2}{3}}}+\frac{\sqrt[3]{2}\cdot\ln(x)}{x^{\frac{2}{3}}}\right)\cdot x^{\sqrt[3]{2}\cdot\sqrt[3]{x}}$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ 1 }{ \left(\cos(2 \cdot x)\right)^3 } \, dx
$$

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

Problem: Compute the volume of the solid formed by rotating about the x-axis the area bounded by the axes and the parabola $x^{\frac{ 1 }{ 2 }}+y^{\frac{ 1 }{ 2 }}=2^{\frac{ 1 }{ 2 }}$.

Answer: Volume = $\frac{8\cdot\pi}{15}$
Subject: precalculus_review

Problem: Find all values of $x$ that satisfy the following equation:

$$
\left|\left(x^4-4\right)-\left(x^2+2\right)\right|=\left|x^4-4\right|-\left|x^2+2\right|
$$

Answer: The final answer: $|x|\ge\sqrt{3}$
Subject: multivariable_calculus

Problem: Use the method of Lagrange multipliers to find the maximum and minimum values of the function $f(x,y,z) = x + y + z$ subject to the constraint $\frac{ 1 }{ x }+\frac{ 1 }{ y }+\frac{ 1 }{ z }=1$.

Answer: Minimum: $1$

Maximum: $9$
Subject: differential_calc

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

Problem: Solve the following equation for $x$:

$3 \cdot x - (x - 4) = 10 + 5 \cdot (2 \cdot x + 2)$

Answer: $x$ = $-2$
Subject: multivariable_calculus

Problem: The integral has been converted to polar coordinates. Verify that the identity is true and choose the easiest way to evaluate the integral, in rectangular or polar coordinates:
$$
\int_{0}^1 \int_{x^2}^x \frac{ y }{ \sqrt{x^2+y^2} } \, dy \, dx = \int_{0}^{\frac{ \pi }{ 4 }} \int_{0}^{\sec(\varphi) \cdot \tan(\varphi)} r \cdot \sin(\varphi) \, dr \, d\varphi
$$

Answer: $I$ = $\frac{1}{6}\cdot\left(2-\sqrt{2}\right)$
Subject: differential_calc

Problem: The position function of a ball dropped from the top of a $200$-meter tall building is given by $s(t) = 200 - 4.9 \cdot t^2$, where position $s$ is measured in meters and time $t$ is measured in seconds. Compute the average velocity of the ball over the given time intervals. Round your answer to eight significant digits:

1. $[4.99,5]$
2. $[5,5.01]$
3. $[4.999,5]$
4. $[5,5.001]$

Answer: 1. $-48.951000$ m/sec.
2. $-49.049000$ m/sec.
3. $-48.995100$ m/sec.
4. $-49.004900$ m/sec.
Subject: multivariable_calculus

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

$$
f(x,y) = \frac{ A \cdot x + B \cdot y }{ C \cdot x + D \cdot y }
$$

Answer: $f_{xx}(x,y)$ = $\frac{2\cdot C\cdot(B\cdot C-A\cdot D)\cdot y}{(C\cdot x+D\cdot y)^3}$
$f_{xy}(x,y)$ = $f_{yx}(x,y)$ = $\frac{-(B\cdot C-A\cdot D)\cdot(C\cdot x-D\cdot y)}{(C\cdot x+D\cdot y)^3}$
$f_{yy}(x,y)$ = $-2\cdot D\cdot\frac{B\cdot C\cdot x-A\cdot D\cdot x}{(C\cdot x+D\cdot y)^3}$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ 3 \cdot x+\sqrt[3]{9 \cdot x^2}+\sqrt[6]{3 \cdot x} }{ x \cdot \left(4+\sqrt[3]{3 \cdot x}\right) } \, dx
$$

Answer: $\int \frac{ 3 \cdot x+\sqrt[3]{9 \cdot x^2}+\sqrt[6]{3 \cdot x} }{ x \cdot \left(4+\sqrt[3]{3 \cdot x}\right) } \, dx$ = $C+3\cdot\arctan\left(\frac{\sqrt[6]{3}}{2}\cdot\sqrt[6]{x}\right)+36\cdot\ln\left(\left|4+\sqrt[3]{3}\cdot\sqrt[3]{x}\right|\right)+\frac{\left(3\cdot3^{\frac{2}{3}}\right)}{2}\cdot x^{\frac{2}{3}}-9\cdot\sqrt[3]{3}\cdot\sqrt[3]{x}$
Subject: multivariable_calculus

Problem: The integral has been converted to polar coordinates. Verify that the identity is true and choose the easiest way to evaluate the integral, in rectangular or polar coordinates.
$$
\int_{0}^1 \int_{x^2}^x \frac{ 1 }{ \sqrt{x^2+y^2} } \, dy \, dx = \int_{0}^{\frac{ \pi }{ 4 }} \int_{0}^{\tan(\varphi) \cdot \sec(\varphi)} 1 \, dr \, d \varphi
$$

Answer: $I$ = $\sqrt{2}-1$
Subject: differential_calc

Problem: Consider a wire 4 ft long cut into two pieces. One piece forms a circle with radius $r$ and the other forms a square of side $x$. Choose $x$ to maximize the sum of their areas.

Answer: $x$ = $0$
Subject: algebra

Problem: Solve the following equations:

1. $\frac{ 2 }{ 3 } (6 x+18)=20$
2. $\frac{ 6 }{ 5 } (10 z+15)=-6$
3. $\frac{ 3 z }{ 8 }-4=5$
4. $\frac{ 1 }{ 2 } x+\frac{ 3 }{ 4 }=-\frac{ 7 }{ 8 }$
5. $3\frac{1}{4} p+4 \left(\frac{ 3 }{ 4 } p-12\right)=-79.25$
6. $3\frac{1}{2} w-2 \left(\frac{ 5 }{ 6 } w+2\right)=-\frac{ 1 }{ 3 }$

Answer: The solutions to the given equations are:
1. $x=2$
2. $z=-2$
3. $z=24$
4. $x=-\frac{ 13 }{ 4 }$
5. $p=-5$
6. $w=2$
Subject: precalculus_review

Problem: Use the Rational Zero Theorem to find all real zeros of the following polynomial:

$p(x) = 2 \cdot x^3 - 5 \cdot x^2 + 9 \cdot x - 9$

Answer: The real zeros are $\frac{3}{2}$
Subject: precalculus_review

Problem: 1. Find the inverse function of $f(x) = \sqrt[3]{x-4}$
2. Find the domain of the inverse function
3. Find the range of the inverse function

Answer: 1. The inverse function is: $x^3+4$
2. The domain of the inverse function is: $(-\infty,\infty)$
3. The range of the inverse function is: $(-\infty,\infty)$
Subject: precalculus_review

Problem: Find zeros of $f(x) = \sin(x) + \sin(2 \cdot x) + 2 \cdot \sin(x) \cdot \sin(2 \cdot x) - 2 \cdot \cos(x) - \cos(2 \cdot x)$.

Answer: The final answer: $x_1=-\frac{\pi}{2}+2\cdot\pi\cdot n$, $x_2=-\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n$, $x_3=\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n$, $x_4=(-1)^n\cdot\frac{\pi}{6}+\pi\cdot n$