toloka/u-math huggingface dataset:
Subject: integral_calc

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

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

Problem: Let $u(x,y,z) = x \cdot y + y \cdot z + x \cdot z$, where $x = r \cdot \cos(\theta)$, $y = r \cdot \sin(\theta)$, and $z = r \cdot \theta$.

Evaluate $\frac{\partial w}{\partial r}(r,\theta)$ and $\frac{\partial w}{\partial \theta}(r,\theta)$ for $r = 1$, $\theta = \frac{ 3 \cdot \pi }{ 2 }$, where $w(r,\theta) = u(x(r,\theta),y(r,\theta),z(r,\theta))$.

Answer: The final answer:

$\frac{\partial w}{\partial r}(1,\frac{ 3 \cdot \pi }{ 2 })$ is equal to $-3\cdot\pi$ .

$\frac{\partial w}{\partial \theta}(1,\frac{ 3 \cdot \pi }{ 2 })$ is equal to $\frac{3\cdot\pi}{2}-2$ .
Subject: sequences_series

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

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

Problem: Find the Fourier series of the function $\psi(x) = e^{-x}$ in the interval $(-2 \cdot \pi, 2 \cdot \pi)$.

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

Problem: If it takes $6$ years for an exponentially growing quantity to triple in size, then by what percentage does it increase every year? Round your answer to two decimal places, and omit the % sign.

Answer: The final answer: $20.09$
Subject: precalculus_review

Problem: 1. Find the inverse function of $f(x) = x^2 - 4$, for $x \ge 0$.
2. Find the domain of the inverse function.
3. Find the range of the inverse function.

Answer: 1. The inverse function is: $\sqrt{x+4}$
2. The domain of the inverse function is: $x\ge-4$
3. The range of the inverse function is: $y\ge0$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \frac{ 4 \cdot \cos(6 \cdot x)^3 }{ 9 \cdot \sin(6 \cdot x)^9 } \, dx
$$

Answer: $\int \frac{ 4 \cdot \cos(6 \cdot x)^3 }{ 9 \cdot \sin(6 \cdot x)^9 } \, dx$ = $C-\frac{2}{27}\cdot\left(\frac{1}{3}\cdot\left(\cot(6\cdot x)\right)^6+\frac{1}{4}\cdot\left(\cot(6\cdot x)\right)^4+\frac{1}{8}\cdot\left(\cot(6\cdot x)\right)^8\right)$
Subject: integral_calc

Problem: $\int \frac{ 7+2 \cdot x-4 \cdot x^2 }{ 2 \cdot x^2+x-3 } \, dx$

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

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

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

Problem: Find all the solutions of the equation $1 + \left(\sin(x) - \cos(x)\right) \cdot \sin\left(\frac{ \pi }{ 4 }\right) = 2 \cdot \left(\cos\left(\frac{ 5 }{ 2 } \cdot x\right)\right)^2$ which satisfy the condition $\sin(6 \cdot x) < 0$.

Answer: The final answer: $x=\frac{5\cdot\pi}{16}+\pi\cdot k$
Subject: sequences_series

Problem: Using the series expansion for the function $(1+x)^m$, calculate approximately $\sqrt[3]{7}$ with an accuracy of 0.0001.

Answer: The final answer: $1.9129$
Subject: algebra

Problem: Use the values of $f(x)$ listed in the table below to solve the equation $f^{-1}(x) = 5$ for $x$.

| $x$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| $f(x)$ | $8$ | $0$ | $7$ | $4$ | $2$ | $6$ | $5$ | $3$ | $9$ |

To find the solution, we need to find the value of $x$ such that $f(x) = 5$.

Answer: The solution is $x$ = $5$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \sin(3 \cdot x)^6 \cdot \cos(3 \cdot x)^2 \, dx
$$

Answer: Answer is: $\frac{1}{32}\cdot x-\frac{1}{32}\cdot\frac{1}{12}\cdot\sin(12\cdot x)-\frac{1}{8}\cdot\frac{1}{6}\cdot\frac{1}{3}\cdot\sin(6\cdot x)^3+\frac{1}{128}\cdot x-\frac{1}{128}\cdot\frac{1}{24}\cdot\sin(24\cdot x)+C$
Subject: precalculus_review

Problem: Simplify the expression $\tan(x)^2 + \sin(x) \cdot \csc(x)$ 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: $\tan(x)^2 + \sin(x) \cdot \csc(x)$ = $\sec(x)^2$
Subject: sequences_series

Problem: Suppose that $\lim_{n \to \infty}\left(\left|\frac{ a_{n+1} }{ a_{n} }\right|\right)=p$. For which values of $p$ must $\sum_{n=1}^\infty\left(2^n \cdot a_{n}\right)$ converge?

Answer: $|p|$ < $\frac{1}{2}$
Subject: multivariable_calculus

Problem: Find the equation of an ellipse with the following conditions:

1. The ellipse is tangent to the y-axis at $(0,3)$.
2. The ellipse intersects the x-axis at $(3,0)$ and $(7,0)$.
3. The axes of the ellipse are parallel to the cartesian axes.

Answer: The final answer: $\frac{(x-5)^2}{25}+\frac{7\cdot(y-3)^2}{75}=1$
Subject: differential_calc

Problem: For the function $y = (4 - x)^3 \cdot (x + 1)^2$ specify the points where local maxima and minima of $y$ occur.

1. The point(s) where local maxima occur
2. The point(s) where local minima occur

Answer: 1. The point(s) where local maxima occur: $P(1,108)$
2. The point(s) where local minima occur: $P(-1,0)$
Subject: differential_calc

Problem: Compute the limit:
$$
\lim_{x \to 4}\left(\frac{ x }{ x-4 }-\frac{ 1 }{ \ln\left(\frac{ x }{ 4 }\right) }\right)
$$

Answer: $\lim_{x \to 4}\left(\frac{ x }{ x-4 }-\frac{ 1 }{ \ln\left(\frac{ x }{ 4 }\right) }\right)$ = $\frac{1}{2}$
Subject: integral_calc

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

Answer: $\int \frac{ -2 }{ e^{3 \cdot x} + \sqrt{1 + e^{6 \cdot x}} } \, dx$ = $C-\frac{1}{3}\cdot\left(\frac{2}{e^{3\cdot x}+\sqrt{1+e^{6\cdot x}}}+2\cdot\ln\left(\frac{e^{3\cdot x}+\sqrt{1+e^{6\cdot x}}-1}{1+e^{3\cdot x}+\sqrt{1+e^{6\cdot x}}}\right)\right)$
Subject: multivariable_calculus

Problem: Evaluate the iterated integral:
$$
\int_{e}^{e^2} \int_{\ln(u)}^2 \left(v + \ln(u)\right) \, dv \, du
$$

Answer: $\int_{e}^{e^2} \int_{\ln(u)}^2 \left(v + \ln(u)\right) \, dv \, du$ = $e^2-\frac{e}{2}$
Subject: precalculus_review

Problem: Find the product $C=\left(-\frac{ 2 }{ 3 }-\frac{ 1 }{ 2 } \cdot i\right) \cdot \left(\frac{ 1 }{ 5 }+\frac{ 4 }{ 5 } \cdot i\right)$.

Answer: The final answer: $C=\frac{4}{15}-\frac{19}{30}\cdot i$
Subject: sequences_series

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

Answer: The final answer: $x-2\cdot x^2+3\cdot x^3-4\cdot x^4+\cdots$
Subject: algebra

Problem: You inherit one hundred thousand dollars. You invest it all in three accounts for one year. The first account pays $4\%$ compounded annually, the second account pays $3\%$ compounded annually, and the third account pays $2\%$ compounded annually. After one year, you earn $\$3650$ in interest. If you invest five times the money in the account that pays $4\%$ compared to $3\%$, how much did you invest in each account?

Answer: The first account investment: $75000$
The second account investment: $15000$
The third account investment: $10000$
Subject: differential_calc

Problem: Given $y = 3 \cdot x^5 + 20 \cdot x^4 + 40 \cdot x^3 + 100$ find where the function is 
1. concave up,
2. concave down, and
3. point(s) of inflection.

Answer: 1. Concave up: $(0,\infty)$
2. Concave down: $(-\infty,-2)$, $(-2,0)$
3. Point(s) of Inflection: $P(0,100)$
Subject: integral_calc

Problem: Calculate the integral:
$$
\int_{-\sqrt{2}}^{\sqrt{2}} \frac{ 2 \cdot x^7+3 \cdot x^6-10 \cdot x^5-7 \cdot x^3-12 \cdot x^2+x+1 }{ x^2+2 } \, dx
$$

Answer: $\int_{-\sqrt{2}}^{\sqrt{2}} \frac{ 2 \cdot x^7+3 \cdot x^6-10 \cdot x^5-7 \cdot x^3-12 \cdot x^2+x+1 }{ x^2+2 } \, dx$ = $\frac{5\cdot\pi-64}{10\cdot\sqrt{2}}$
Subject: differential_calc

Problem: Make full curve sketching of $y = \sqrt[3]{x^2 - \frac{ x^3 }{ 8 }}$. 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 $y=-\frac{1}{2}\cdot x+\frac{4}{3}$
5. Intervals where the function is increasing $\left(0,\frac{16}{3}\right)$
6. Intervals where the function is decreasing $(-\infty,0)$, $\left(\frac{16}{3},8\right)$, $(8,\infty)$
7. Intervals where the function is concave up $(8,\infty)$
8. Intervals where the function is concave down $(-\infty,0)$, $(0,8)$
9. Points of inflection $P(8,0)$
Subject: differential_calc

Problem: Consider a bank investment: The initial investment is $\$10000$. After $25$ years, the investment has tripled to $\$30000$. Use Newton’s method to determine the interest rate if the interest was compounded annually.

Answer: $4.492$%
Subject: precalculus_review

Problem: Simplify the expression $\sin(x) \cdot \left(\csc(x)-\sin(x)\right)$ 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: $\sin(x) \cdot \left(\csc(x)-\sin(x)\right)$ = $\cos(x)^2$
Subject: sequences_series

Problem: Find the Fourier series of the function $\psi(x) = 2 \cdot e^{-2 \cdot x}$ in the interval $(-\pi, \pi)$.

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

Problem: Evaluate the limit:
$$
\lim_{x \to 0}\left(\frac{ \ln\left(1+x+x^2\right)+\ln\left(1-x+x^2\right) }{ x^2 }\right)
$$

Answer: $\lim_{x \to 0}\left(\frac{ \ln\left(1+x+x^2\right)+\ln\left(1-x+x^2\right) }{ x^2 }\right)$ = $1$
Subject: algebra

Problem: Use the Rational Zero Theorem to find the real solution(s) of the equation: $x^4 + 2 \cdot x^3 - 4 \cdot x^2 - 10 \cdot x - 5 = 0$

Answer: By the Rational Zero Theorem, possible rational zeros of the function are: $1$, $-1$, $5$, $-5$ $x$ = $\sqrt{5}$, $-\sqrt{5}$, $-1$
Subject: sequences_series

Problem: Find the interval of convergence of the series $\sum_{n=1}^\infty \frac{ 3 \cdot x^n }{ n \cdot 7^n }$. (Use interval notation)

Answer: The final answer: $[-7,7)$
Subject: algebra

Problem: Solve the following equations:

1. $8 s - (8 + 6 s) = 20$
2. $34 = 2 x + 8 (x + 3)$
3. $3 (x + 9) = 60$
4. $2 (m - 8) = 12$
5. $35 = 22 x - 12 x + 5$
6. $6 (b + 8) = 54$
7. $99 = 33 x + 3 (3 x + 5)$

Answer: The solutions to the given equations are:
1. $s=14$
2. $x=1$
3. $x=11$
4. $m=14$
5. $x=3$
6. $b=1$
7. $x=2$
Subject: multivariable_calculus

Problem: Evaluate the triple integral $\int_{0}^1{\int_{1}^2{\int_{z}^{z+1}{(y+1) d x} d y} d z}$ by using the transformation $u = x - z$, $v = 3 \cdot y$, and $w = \frac{ z }{ 2 }$.

Answer: $I$ = $\frac{5}{2}$
Subject: sequences_series

Problem: Consider the function $y = \left| \cos(2 \cdot x) \right|$.

1. Find the Fourier series of the function.
2. Using this decomposition, calculate the sum of the series $\sum_{n=1}^\infty\frac{ (-1)^n }{ 4 \cdot n^2-1 }$.
3. Using this decomposition, calculate the sum of the series $\sum_{n=1}^\infty\frac{ 1 }{ 4 \cdot n^2-1 }$.

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

Problem: Evaluate $\int\int\int_{E}{\left(x^3+y^3+z^3\right) d V}$, where $E$ is the region defined by:
$$
E = \left\{(x,y,z) | 0 \le x \le 2, 0 \le y \le 2 \cdot x, 0 \le z \le 4-x-y\right\}
$$

Answer: $I$ = $\frac{112}{5}$
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 $\sqrt{x^2+2}$ with the center $a=0$.

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

Problem: Find the solution to the following inequality and express it in interval notation:
$$8 (x-9) (x+5) (x-3) > 0$$

Answer: The solution set to the inequality is $\left(-5,\ 3\right)\cup\left(9,\ \infty\right)$
Subject: differential_calc

Problem: For what values of $a$, $b$, and $c$ does the curve $y = a \cdot x^4 + b \cdot x^3 + c \cdot x^2 + e \cdot x + f$ have points of inflection?

Answer: The final answer: $3\cdot b^2-8\cdot a\cdot c>0$
Subject: precalculus_review

Problem: Simplify the expression $\left(1+\tan\left(\theta\right)\right)^2-2 \cdot \tan\left(\theta\right)$ 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: $\left(1+\tan\left(\theta\right)\right)^2-2 \cdot \tan\left(\theta\right)$ = $\sec\left(\theta\right)^2$
Subject: integral_calc

Problem: Compute the integral:
$$
\int \frac{ -12 }{ \sin(6 \cdot x)^6 } \, dx
$$

Answer: $\int \frac{ -12 }{ \sin(6 \cdot x)^6 } \, dx$ = $C+2\cdot\cot(6\cdot x)+\frac{2}{5}\cdot\left(\cot(6\cdot x)\right)^5+\frac{4}{3}\cdot\left(\cot(6\cdot x)\right)^3$
Subject: precalculus_review

Problem: Calculate $E = \frac{ 1 }{ \sin(10) } - \frac{ \sqrt{3} }{ \cos(10) }$.

Answer: The final answer: $E=4$
Subject: multivariable_calculus

Problem: Find the equations of the planes below:

1. The plane through $P(2,2,1)$ perpendicular to the vector $\left\langle 1,0,-1 \right\rangle$
2. The plane containing the line $\left\langle x,y,z \right\rangle = t \cdot \left\langle 1,2,3 \right\rangle$ and through the point $P(2,1,-1)$
3. The plane containing the points $P(2,0,3)$, $P(0,4,1)$, $P(-1,3,3)$

Answer: 1. $x-z=1$
2. $-5\cdot x+7\cdot y-3\cdot z=0$
3. $x+y+z=5$
Subject: algebra

Problem: Use the Remainder Theorem to find the remainder when dividing $5 \cdot x^5 - 4 \cdot x^4 + 3 \cdot x^3 - 2 \cdot x^2 + x - 1$ by $x + 6$.

Answer: The remainder is $-44791$
Subject: differential_calc

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

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

Problem: Evaluate $\int\int\int_{E}{z \, dV}$, where $E$ is the region defined by:
$$
E = \left\{ (x,y,z) \mid -y \le x \le y, \, 0 \le y \le 1, \, 0 \le z \le 1-x^4-y^4 \right\}
$$

Answer: $I$ = $\frac{113}{450}$
Subject: integral_calc

Problem: Calculate the integral:
$$
\int \frac{ 3 \cdot x + 4 }{ \left( x^2 + 1 \cdot x + 7 \right)^2 } \, dx
$$

Answer: $\int \frac{ 3 \cdot x + 4 }{ \left( x^2 + 1 \cdot x + 7 \right)^2 } \, dx$ = $\frac{\frac{5}{27}\cdot x-\frac{38}{27}}{\frac{27}{4}+\left(x+\frac{1}{2}\right)^2}+\frac{30\cdot\sqrt{3}}{729}\cdot\arctan\left(\sqrt{\frac{4}{27}}\cdot\left(x+\frac{1}{2}\right)\right)+C$
Subject: algebra

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

$p(x) = 2 \cdot x^3 - x^2 + 7 \cdot x - 1$

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

Problem: Solve the trigonometric equation $\sec(x)^2 - 2 \cdot \sec(x) + 1 = 0$ on the interval $[-2 \cdot \pi, 2 \cdot \pi]$.

Answer: $x$ = $-2\cdot\pi$, $2\cdot\pi$, $0$
Subject: algebra

Problem: Write an expression for a rational function with the given characteristics:

1. vertical asymptotes $x=-5$ and $x=5$,
2. x-intercepts at $P(2,0)$ and $P(-1,0)$,
3. y-intercept at $P(0,4)$.

Answer: The rational function satisfying the given conditions is $f(x)=\frac{50\cdot(x-2)\cdot(x+1)}{(x+5)\cdot(x-5)}$
Subject: differential_calc

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

Problem: For the function $y = x \cdot \sqrt[3]{2 \cdot x - \frac{ 4 }{ 3 }}$ 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: $\left(-\infty,\frac{2}{3}\right)$, $(1,\infty)$
2. Interval(s) where the function is concave down: $\left(\frac{2}{3},1\right)$
3. Point(s) of inﬂection: $P\left(\frac{2}{3},0\right)$, $P\left(1,\sqrt[3]{\frac{2}{3}}\right)$
Subject: precalculus_review

Problem: Solve $\left(\sin(x)\right)^{10} + \left(\cos(x)\right)^{10} = \frac{ 29 }{ 16 } \cdot \left(\cos(2 \cdot x)\right)^4$.

Answer: The final answer: $x=\frac{\pi}{8}+\frac{\pi\cdot k}{4}$
Subject: differential_calc

Problem: Where is the parabola $y = x^2$ closest to the point $(2,0)$?

Answer: The final answer: $P(0.8351,0.6974)$
Subject: multivariable_calculus

Problem: Use Lagrange multipliers to find the maximum volume of a rectangular box that can be inscribed in the ellipsoid $\frac{ x^2 }{ 9 } + \frac{ y^2 }{ 25 } + \frac{ z^2 }{ 4 } = 1$.

Answer: The final answer: $\frac{80}{\sqrt{3}}$
Subject: sequences_series

Problem: Give the first six terms of the sequence and then give the $n$th term.

1. $a_{1} = 1$
2. $a_{2} = 3$

$a_{n+1} = 3 \cdot a_{n} - 2 \cdot n - 1$ for $n \ge 2$

Answer: $a_{1}$ = $1$ , $a_{2}$ = $3$ , $a_{3}$ = $4$ , $a_{4}$ = $5$ , $a_{5}$ = $6$ , $a_{6}$ = $7$ $a_{n}$ = $n+1$ for $n \ge 2$
Subject: differential_calc

Problem: Given $g(x) = \frac{ 1 }{ 3 } \cdot (a+b) \cdot x^3 + \frac{ 1 }{ 2 } \cdot (a+b+c) \cdot x^2 - (a+b+c+d) \cdot x + a \cdot b \cdot c \cdot d$, simplify the derivative of $g(x)$ if $x^2 + x = a + b$.

Answer: The final answer: $g'(x)=(a+b)^2+c\cdot x-(a+b+c+d)$
Subject: differential_calc

Problem: For the function $y = (3 - x)^3 \cdot (x + 2)^2$ specify the points where local maxima and minima of $y$ occur.

1. The point(s) where local maxima occur
2. The point(s) where local minima occur

Answer: 1. The point(s) where local maxima occur: $P(0,108)$
2. The point(s) where local minima occur: $P(-2,0)$
Subject: integral_calc

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

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

Problem: Solve the integral:
$$
2 \cdot \int \sin(-2 \cdot x)^5 \cdot \cos(2 \cdot x)^2 \, dx
$$

Answer: $2 \cdot \int \sin(-2 \cdot x)^5 \cdot \cos(2 \cdot x)^2 \, dx$ = $C+\frac{1}{3}\cdot\left(\cos(2\cdot x)\right)^3+\frac{1}{7}\cdot\left(\cos(2\cdot x)\right)^7-\frac{2}{5}\cdot\left(\cos(2\cdot x)\right)^5$
Subject: sequences_series

Problem: Find the Fourier expansion of this function:
$$
f(x) = \begin{cases} -\frac{ \pi }{ 4 }, & -\pi \le x < 0 \\ \frac{ \pi }{ 4 }, & 0 \le x \le \pi \end{cases}
$$
at $(-\pi, \pi)$.

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

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

Answer: The final answer: $\sum_{k=1}^\infty\left((-1)^k\cdot\frac{(2\cdot k-1)\cdot x^{2\cdot k-2}}{(2\cdot k+1)!}\right)$
Subject: sequences_series

Problem: Given that $\frac{ 1 }{ 1-x } = \sum_{n=0}^\infty \left(x^n\right)$ with convergence in $(-1,1)$, find the power series for the function with the given center $a$, and identify its interval of convergence:

1. $f(x) = \frac{ x^2 }{ 5-4 \cdot x+x^2 }$; $a=2$.

Answer: 1. $f(x)$ = $\sum_{n=0}^\infty\left(x^2\cdot\left(-(x-2)^2\right)^n\right)$
2. $I$ = $(1,3)$
Subject: differential_calc

Problem: A rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by $h(t) = 600 + 78.4 \cdot t - 4.9 \cdot t^2$, where $t$ is measured in seconds. Compute the average velocity of the rocket over the given time intervals. Round your answer to eight significant digits.

1. $[9,9.01]$
2. $[8.99,9]$
3. $[9,9.001]$
4. $[8.999,9]$

Answer: 1. $-9.8490000$ m/sec.
2. $-9.7510000$ m/sec.
3. $-9.8049000$ m/sec.
4. $-9.7951000$ m/sec.
Subject: algebra

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

$i^{15}$

Answer: The final answer: $-i$
Subject: differential_calc

Problem: Find any local extrema for $s = \left| \arctan\left(4 \cdot x^2 - 12 \cdot x + 8\right) \right|$.

Submit as your final answer:
1. The point(s), where the function has local maximum(s);
2. The point(s), where the function has local minimum(s).

Answer: 1. Local Maximum(s) $P\left(\frac{3}{2},\frac{\pi}{4}\right)$;
2. Local Minimum(s) $P(1,0)$, $P(2,0)$.
Subject: precalculus_review

Problem: Calculate the derivative $\frac{ d }{d x}\left(\log_{x}(a)\right)$ for $x > 0$, $a > 0$, $x \ne 1$, $a \ne 1$.

Answer: $\frac{ d }{d x}\left(\log_{x}(a)\right)$ = $\frac{-\ln(a)}{x\cdot\left(\ln(x)\right)^2}$
Subject: sequences_series

Problem: Consider the power series $\sum_{k=0}^\infty \frac{ (1-3 \cdot x)^k }{ 4^k \cdot \sqrt{k+1} }$.

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 $\frac{1}{3}$.
2. The radius of convergence is $\frac{4}{3}$.
Subject: algebra

Problem: Solve the following inequality: $-5 x^2 + 10 x + 15 \le 0$.

Express your answer in the interval form.

Answer: Solution in the interval form: $\left(-\infty,\ -1\right] \cup \left[3,\ \infty\right)$

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

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

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

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

Answer: Inverse function: $g(x)=\frac{9}{5}\cdot y+32$