toloka/u-math huggingface dataset:
Subject: algebra

Problem: The population $P$ of a koi pond over $x$ months is modeled by the function $P(x) = \frac{ 68 }{ 1 + 16 \cdot e^{-0.28 \cdot x} }$. How many months will it take before there are $20$ koi in the pond?

Answer: $x$: $\frac{\ln(0.15)}{-0.28}$
Subject: sequences_series

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

Answer: The Fourier series is: $\frac{\pi}{2}-\frac{\sin(x)}{1}-\frac{\sin(2\cdot x)}{2}-\frac{\sin(3\cdot x)}{3}-\cdots$
Subject: algebra

Problem: Find the area of a triangle bounded by the x-axis, the line $f(x) = 12 - \frac{ 1 }{ 5 } \cdot x$, and the line perpendicular to $f(x)$ that passes through the origin.

Answer: The area of the triangle is $\frac{4500}{13}$
Subject: differential_calc

Problem: Find the derivative of $y = \sin(2 \cdot x) \cdot \cos(3 \cdot x) - \frac{ \ln(x-1) }{ \ln(x+1) } + c$

Answer: The final answer: $y'=2\cdot\cos(5\cdot x)-\sin(3\cdot x)\cdot\sin(2\cdot x)-\frac{(x+1)\cdot\ln(x+1)-(x-1)\cdot\ln(x-1)}{(x-1)\cdot(x+1)\cdot\left(\ln(x+1)\right)^2}$
Subject: differential_calc

Problem: Compute the limit using L'Hopital's Rule:

$$
\lim_{x \to \infty} \left(x - x^2 \cdot \ln\left(1 + \frac{ 1 }{ x }\right)\right)
$$

Answer: The final answer: $\frac{1}{2}$
Subject: differential_calc

Problem: Find all values of the constant $c$ such that the limit 
$$
\lim_{x \to -\infty} \left(\frac{ 3^{c \cdot x}+1 }{ 3^{2 \cdot x}+1 }\right)
$$
exists. Enter the range for the constant $c$ as an interval of the real line.

Answer: The final answer: $[0,\infty)$
Subject: integral_calc

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

Answer: $\int \frac{ 1 }{ (x+4) \cdot \sqrt{x^2+2 \cdot x+5} } \, dx$ = $C+\frac{1}{\sqrt{13}}\cdot\ln\left(\sqrt{13}-4-x-\sqrt{x^2+2\cdot x+5}\right)-\frac{1}{\sqrt{13}}\cdot\ln\left(4+\sqrt{13}+x+\sqrt{x^2+2\cdot x+5}\right)$
Subject: precalculus_review

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

$p(x) = x^3 - 8 \cdot x^2 + 17 \cdot x - 10$

Answer: The real zeros are $1$, $2$, $5$
Subject: integral_calc

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

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

Problem: Compute the integral:
$$
\int \sqrt[3]{x \cdot \left(8-x^2\right)} \, dx
$$

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

Problem: Solve $|3 \cdot x - 11| = 4 \cdot x - 3$.

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

Problem: Find points on the curve at which the tangent line is horizontal or vertical:
$$
x = \frac{ 3 \cdot t }{ 1+t^3 }, \quad y = \frac{ 3 \cdot t^2 }{ 1+t^3 }
$$

Answer: 1. Horizontal: $P(0,0)$, $P\left(\sqrt[3]{2},\sqrt[3]{4}\right)$
2. Vertical: $P\left(\sqrt[3]{4},\sqrt[3]{2}\right)$
Subject: integral_calc

Problem: Solve the integral:
$$
\int 22 \cdot \cot(-11 \cdot x)^5 \, dx
$$

Answer: $\int 22 \cdot \cot(-11 \cdot x)^5 \, dx$ = $C+\frac{1}{2}\cdot\left(\cot(11\cdot x)\right)^4+\ln\left(1+\left(\cot(11\cdot x)\right)^2\right)-\left(\cot(11\cdot x)\right)^2$
Subject: differential_calc

Problem: Calculate the derivative of the function $r = 2 \cdot \ln\left(\sqrt[6]{\frac{ 1+\tan(3 \cdot \varphi) }{ 1-\tan(3 \cdot \varphi) }}\right)$.

Answer: $r'$ = $2\cdot\sec(6\cdot\varphi)$
Subject: multivariable_calculus

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

Answer: Moment of Inertia: $\frac{256}{15}\cdot a^3$
Subject: precalculus_review

Problem: Suppose a function $f(x)$ is periodic with the period $T$. What is the period of the function $f(a \cdot x + b)$, when $a > 0$?

Answer: The final answer: $\frac{T}{a}$
Subject: algebra

Problem: Solve the equation for $x$:
$$
\frac{ 3 }{ \log_{2}(10) }-\log_{10}(x-9)=\log_{10}(44)
$$

Answer: $x$ = $\frac{101}{11}$
Subject: sequences_series

Problem: Suppose that $\sum_{n=0}^\infty\left(a_{n} \cdot x^n\right)$ converges to a function $y$ such that $y'' - y' + y = 0$ where $y(0) = 1$ and $y'(0) = 0$. Compute $a_{0}$, ..., $a_{5}$.

Answer: The final answer:

$a_{0}$: $1$

$a_{1}$: $0$

$a_{2}$: $\frac{1}{2}$

$a_{3}$: $\frac{1}{6}$

$a_{4}$: $0$

$a_{5}$: $-\frac{1}{120}$
Subject: precalculus_review

Problem: Solve the logarithmic equation exactly, if possible: $\ln(x) + \ln(x-2) = \ln(4)$

If the solution does not exist, enter undefined.

Answer: $x$ = $1+\sqrt{5}$
Subject: algebra

Problem: Given the rational function $f(x) = \frac{ x^2-1 }{ x^3+9 \cdot x^2+14 \cdot x }$, find

1. the domain (in interval notation),
2. vertical asymptotes (in the form $x=a$),
3. horizontal asymptotes (in the form $y=c$).

Answer: 1. The domain in interval notation is $(-\infty,-7)\cup(-7,-2)\cup(-2,0)\cup(0,\infty)$
2. Vertical asymptotes of $f(x)$: $x=-7$, $x=0$, $x=-2$
3. Horizontal asymptotes of $f(x)$: $y=0$
Subject: sequences_series

Problem: Evaluate $\int_{0}^{\frac{ \pi }{ 2 }}{\sin\left(\theta\right)^4 d \theta}$ and $\int_{0}^{\frac{ \pi }{ 2 }}{\sin\left(\theta\right)^2 d \theta}$ in the approximation $T=4 \cdot \sqrt{\frac{ L }{ g }} \cdot \int_{0}^{\frac{ \pi }{ 2 }}{\left(1+\frac{ 1 }{ 2 } \cdot k^2 \cdot \sin\left(\theta\right)^2+\frac{ 3 }{ 8 } \cdot k^4 \cdot \sin\left(\theta\right)^4\right) d \theta}$ to obtain an improved estimate for $T$.

Answer: $T$: $2\cdot\pi\cdot\sqrt{\frac{L}{g}}\cdot\left(1+\frac{k^2}{4}+\frac{9\cdot k^4}{64}\right)$
Subject: integral_calc

Problem: Evaluate the integral:
$$
I = \int 3 \cdot \ln\left(\sqrt{2-x}+\sqrt{2+x}\right) \, dx
$$

Answer: The final answer: $3\cdot x\cdot\ln\left(\sqrt{2-x}+\sqrt{2+x}\right)-\frac{3}{2}\cdot\left((C+x)-2\cdot\arcsin\left(\frac{x}{2}\right)\right)$
Subject: algebra

Problem: Use the vertex $P(h,k) = P(-4,-5)$ and a point $P(x,y) = P(-2,7)$ on the graph of $f(x)$ to find the general form of the quadratic function.

Answer: The final answer: $f(x)=3\cdot x^2+24\cdot x+43$
Subject: multivariable_calculus

Problem: Consider points $A(3,-1,2)$, $B(2,1,5)$, and $C(1,-2,-2)$.

1. Find the area of parallelogram $ABCD$ with adjacent sides $\vec{AB}$ and $\vec{AC}$.
2. Find the area of triangle $ABC$.
3. Find the distance from point $A$ to line $BC$.

Answer: 1. $A$ = $5\cdot\sqrt{6}$
2. $A$ = $\frac{5\cdot\sqrt{6}}{2}$
3. $d$ = $\frac{5\cdot\sqrt{6}}{\sqrt{59}}$
Subject: integral_calc

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

Answer: $\int \left(\frac{ x+2 }{ x-2 } \right)^{\frac{ 3 }{ 2 }} \, dx$ = $C+\sqrt{\frac{x+2}{x-2}}\cdot(x-10)-6\cdot\ln\left(\left|\frac{\sqrt{x-2}-\sqrt{x+2}}{\sqrt{x-2}+\sqrt{x+2}}\right|\right)$
Subject: integral_calc

Problem: Use integration by substitution and/or by parts to compute the integral:
$$
\int x \cdot \ln(5+x) \, dx
$$

Answer: The final answer: $D+5\cdot(x+5)+\left(\frac{1}{2}\cdot(x+5)^2-5\cdot(x+5)\right)\cdot\ln(x+5)-\frac{1}{4}\cdot(x+5)^2$
Subject: precalculus_review

Problem: Find the period of the function $f(x) = \left(\sin(x)\right)^4 + \left(\cos(x)\right)^4$.

Answer: The final answer: $T=\frac{\pi}{2}$
Subject: multivariable_calculus

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

Answer: Moment of Inertia: $\frac{256}{15}\cdot a^3$
Subject: multivariable_calculus

Problem: For the function $f(x,y) = 2 \cdot x \cdot y \cdot e^{-x^2-y^2}$, use the second derivative test to identify any critical points and determine whether each critical point is a local minimum, local maximum, saddle point, or none of these.

Answer: $f$ has a local minimum at $P\left(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$, $P\left(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)$ .

$f$ has a local maximum at $P\left(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)$, $P\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$ .

$f$ has a saddle point at $P(0,0)$ .

The second derivative test is inconclusive at None .
Subject: precalculus_review

Problem: Identify the set of real numbers for which $x \cdot (x-1) \cdot (x-4) \le 0$.

Answer: The final answer: $(-\infty,0]\cup[1,4]$
Subject: multivariable_calculus

Problem: Calculate the average lengths of all vertical chords of the parabola $\frac{ x^2 }{ a^2 } - \frac{ y^2 }{ b^2 } = 1$ over the interval $a \le x \le 2 \cdot a$.

Answer: The final answer: $\mu=b\left(2\cdot\sqrt{3}-\ln\left(2+\sqrt{3}\right)\right)$
Subject: multivariable_calculus

Problem: Given $\vec{r}(t) = \left\langle e^t \cdot \cos(t), e^t \cdot \sin(t), e^t \right\rangle$. Find the tangential and normal components of acceleration.

Answer: The final answer:

1. The tangential component of acceleration, $a_{t}$, is: $\sqrt{3}\cdot e^t$
2. The normal component of acceleration, $a_{N}$, is: $\sqrt{2}\cdot e^t$
Subject: sequences_series

Problem: Expand the function $f(x) = e^x$ in terms of powers of $x + 1$.

Answer: The final answer: $e^x=\frac{1}{e}+(x+1)\cdot\frac{1}{e}+\frac{(x+1)^2}{2!}\cdot\frac{1}{e}+\frac{(x+1)^3}{3!}\cdot\frac{1}{e}+\frac{(x+1)^4}{4!}\cdot e^{\xi}$
Subject: precalculus_review

Problem: If $y=100$ at $t=4$ and $y=10$ at $t=8$, when does $y=1$? Use $y=y_{0} \cdot e^{k \cdot t}$, where $y_{0}$ is the beginning amount, $y$ is the ending amount, $k$ is the growth or decay rate, and $t$ is time.

Answer: $y$ = 1 at $t$ = $12$
Subject: precalculus_review

Problem: Simplify the radical expression:
$$
\frac{ 4 \cdot \sqrt{2 \cdot n} }{ \sqrt{16 \cdot n^4} }
$$

Answer: The final answer: $\frac{\sqrt{2}}{n^{\frac{3}{2}}}$
Subject: differential_calc

Problem: Differentiate the function
$$
f(x) = \frac{ 5 \cdot x^3-\sqrt[3]{x}+6 \cdot x+2 }{ \sqrt{x} }
$$

Answer: $\frac{ d }{d x}\left(\frac{ 5 \cdot x^3-\sqrt[3]{x}+6 \cdot x+2 }{ \sqrt{x} }\right)$: $\frac{18\cdot x\cdot\sqrt[6]{x}+75\cdot x^3\cdot\sqrt[6]{x}+\sqrt{x}-6\cdot\sqrt[6]{x}}{6\cdot x\cdot x^{\frac{2}{3}}}$
Subject: algebra

Problem: Rewrite the quadratic expression $2 \cdot x^2 - 6 \cdot x - 9$ by completing the square.

Answer: $2 \cdot x^2 - 6 \cdot x - 9$ = $2\cdot(x-1.5)^2-13.5$
Subject: sequences_series

Problem: Compute the first $6$ nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of $f(x) = \sin(x) \cdot \cos\left(\frac{ \pi }{ 4 }\right) + \cos(x) \cdot \sin\left(\frac{ \pi }{ 4 }\right)$.

Answer: $f(x)$ = $\frac{1}{34560\cdot\sqrt{2}}\cdot\left(288\cdot x^5+1440\cdot x^4-5760\cdot x^3-17280\cdot x^2+34560\cdot x+34560\right)+\cdots$
Subject: precalculus_review

Problem: $P = \left(x, \frac{ \sqrt{7} }{ 3 }\right)$, $x<0$ is a point on the unit circle.

1. Find the (exact) missing coordinate value of the point.
2. Find the values of the three trigonometric functions $\sin\left(\theta\right)$, $\cos\left(\theta\right)$, $\tan\left(\theta\right)$ for the angle $\theta$ with a terminal side that passes through point $P$.

Rationalize denominators.

Answer: 1. The (exact) missing coordinate value of the point is: $-\frac{\sqrt{2}}{3}$
2. The values of the three trigonometric functions are:
    * $\sin\left(\theta\right) = $\frac{\sqrt{7}}{3}$$
    * $\cos\left(\theta\right) = $-\frac{\sqrt{2}}{3}$$
    * $\tan\left(\theta\right) = $-\frac{\sqrt{14}}{2}$$
Subject: algebra

Problem: Find the dimensions of the box whose length is twice as long as the width, whose height is $2$ inches greater than the width, and whose volume is $192$ cubic inches.

Answer: The dimensions of the box are $4$, $6$, $8$
Subject: differential_calc

Problem: Given $y = \frac{ 2 }{ 5 } \cdot x^6 + \frac{ 6 }{ 5 } \cdot x^5 + x^4 + 2$, find where the function is 

1. concave up
2. concave down
3. point(s) of inflection

Answer: 1. Concave up: $(-\infty,-1)$, $(-1,0)$, $(0,\infty)$
2. Concave down: None
3. Point(s) of Inflection: None
Subject: differential_calc

Problem: Sketch the curve:

$$
y = 9 \cdot x^2 \cdot e^{\frac{ 1 }{ 3 \cdot x }}
$$

Submit as your final answer:

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

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

Problem: Calculate the derivative of the function $r = \ln\left(\sqrt[4]{\frac{ 1+\tan(\varphi) }{ 1-\tan(\varphi) }}\right)$.

Answer: $r'$ = $\frac{1}{2}\cdot\sec(2\cdot\varphi)$
Subject: algebra

Problem: The radius of the right circular cylinder is $\frac{ 1 }{ 3 }$ meter greater than the height. The volume is $\frac{ 74 \cdot \pi }{ 9 }$ cubic meters. Find the dimensions.

Answer: $r$: $\frac{1+5.40740373}{3}$ $h$: $1.80246791$
Subject: integral_calc

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

Answer: $\int \frac{ 3 }{ 3+\sin(2 \cdot x)+\cos(2 \cdot x) } \, dx$ = $C+\frac{3}{\sqrt{7}}\cdot\arctan\left(\frac{1}{\sqrt{7}}\cdot\left(1+2\cdot\tan(x)\right)\right)$
Subject: multivariable_calculus

Problem: Use the second derivative test to identify any critical points of the function $f(x,y) = x^2 \cdot y^2$ and determine whether each critical point is a maximum, minimum, saddle point, or none of these.

Answer: Maximum: None
Minimum: $P(0,0)$

Saddle point: None

The second derivative test is inconclusive at: None
Subject: multivariable_calculus

Problem: Use the second derivative test to identify any critical points of the function $f(x,y) = 8 \cdot x \cdot y \cdot (x+y) + 7$, and determine whether each critical point is a maximum, minimum, saddle point, or none of these.

Answer: Maximum: None
Minimum: None

Saddle point: None

The second derivative test is inconclusive at: $P(0,0)$
Subject: sequences_series

Problem: Find the power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left(n \cdot x^n\right)$ and $g(x) = \sum_{n=1}^\infty \left(n \cdot x^n\right)$.

Answer: $f(x) \cdot g(x)$ = $\sum_{n=2}^\infty\left(\frac{1}{6}\cdot n\cdot\left(n^2-1\right)\cdot x^n\right)$
Subject: multivariable_calculus

Problem: Find the equation of the tangent line to the curve: $r = 3 + \cos(2 \cdot t)$, $t = \frac{ 3 \cdot \pi }{ 4 }$.

Answer: $y$ = $\frac{1}{5}\cdot\left(x+\frac{3}{\sqrt{2}}\right)+\frac{3}{\sqrt{2}}$
Subject: multivariable_calculus

Problem: Find and classify all critical points of the function $f(x,y) = x \cdot y \cdot (1-7 \cdot x-9 \cdot y)$.

Answer: Points of local minima: None

Points of local maxima: $P\left(\frac{1}{21},\frac{1}{27}\right)$

Saddle points: $P(0,0)$, $P\left(\frac{1}{7},0\right)$, $P\left(0,\frac{1}{9}\right)$
Subject: integral_calc

Problem: Compute the integral:
$$
\int x \cdot \arctan(2 \cdot x)^2 \, dx
$$

Answer: $\int x \cdot \arctan(2 \cdot x)^2 \, dx$ = $\frac{1}{16}\cdot\left(2\cdot\left(\arctan(2\cdot x)\right)^2+2\cdot\ln\left(4\cdot x^2+1\right)+8\cdot x^2\cdot\left(\arctan(2\cdot x)\right)^2-8\cdot x\cdot\arctan(2\cdot x)\right)+C$
Subject: algebra

Problem: When hired at a new job selling electronics, you are given two pay options:

Option A: Base salary of $20,000$ USD a year with a commission of $12\%$ of your sales.

Option B: Base salary of $26,000$ USD a year with a commission of $3\%$ of your sales.

How much electronics would you need to sell for Option A to produce a larger income? Give your answer either exactly or rounded to two decimal places.

Answer: The final answer: $66666.67$
Subject: precalculus_review

Problem: Use the double-angle formulas to evaluate the integral:
$$
\int_{0}^\pi \sin(x)^4 \, dx
$$

Answer: $\int_{0}^\pi \sin(x)^4 \, dx$ = $\frac{3\cdot\pi}{8}$
Subject: integral_calc

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

Answer: $3 \cdot \int \cos(2 \cdot x)^6 \, dx$ = $\frac{3}{8}\cdot\sin(4\cdot x)+\frac{9}{128}\cdot\sin(8\cdot x)+\frac{15}{16}\cdot x-\frac{1}{32}\cdot\left(\sin(4\cdot x)\right)^3+C$
Subject: precalculus_review

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

Answer: $\int \sin(x)^2 \cdot \cos(2 \cdot x)^2 \, dx$ = $\frac{x}{4}-\frac{3}{16}\cdot\sin(2\cdot x)+\frac{1}{16}\cdot\sin(4\cdot x)-\frac{1}{48}\cdot\sin(6\cdot x)+C$
Subject: multivariable_calculus

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

$f(x,y,z) = \sin\left(x+z^y\right)$.

Answer: $f_{xx}(x,y,z)$ = $-\sin\left(x+z^y\right)$
$f_{xy}(x,y,z)$ = $f_{yx}(x,y,z)$ = $-z^y\cdot\ln(z)\cdot\sin\left(x+z^y\right)$

$f_{yy}(x,y,z)$ = $-z^y\cdot\left(\ln(z)\right)^2\cdot\left(-\cos\left(x+z^y\right)+z^y\cdot\sin\left(x+z^y\right)\right)$

$f_{yz}(x,y,z)$ = $f_{zy}(x,y,z)$ = $z^{-1+y}\cdot\left(\cos\left(x+z^y\right)\cdot\left(1+y\cdot\ln(z)\right)-y\cdot z^y\cdot\ln(z)\cdot\sin\left(x+z^y\right)\right)$

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

$f_{xz}(x,y,z)$ = $f_{zx}(x,y,z)$ = $-y\cdot z^{-1+y}\cdot\sin\left(x+z^y\right)$
Subject: multivariable_calculus

Problem: Use the method of Lagrange multipliers to find the maximum and minimum values of the function $f(x,y) = x^2 - y^2$ with the constraint $x + 6 \cdot y = 4$.

Answer: Minimum value of the function $f(x,y)$ is $-\frac{16}{35}$

Maximum value of the function $f(x,y)$ is None
Subject: differential_calc

Problem: Find $\frac{d^3}{dx^3}f(x)$, given $f(x) = \ln\left(\frac{ x+7 }{ x-7 }\right)$.

Answer: The final answer: $\frac{d^3}{dx^3}f(x)=-\frac{84\cdot x^2+1372}{\left(x^2-49\right)^3}$
Subject: integral_calc

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

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

Problem: Calculate the double integral $\int\int_{R}{\left(x^2+y^2\right) d A}$, where $R$ is the parallelogram with the sides $y=x$, $y=x+2$, $y=2$, and $y=6$.

Answer: The final answer: $224$
Subject: multivariable_calculus

Problem: Find the curvature for the vector function: $\vec{r}(t) = \left\langle \sqrt{2} \cdot e^t, \sqrt{2} \cdot e^{-t}, 2 \cdot t \right\rangle$.

Answer: The final answer: $\frac{2\cdot e^{2\cdot t}}{2+2\cdot e^{4\cdot t}+4\cdot e^{2\cdot t}}$
Subject: precalculus_review

Problem: Multiply the rational expressions and express the product in simplest form:

$$
\frac{ x^2-x-6 }{ 2 \cdot x^2+x-6 } \cdot \frac{ 2 \cdot x^2+7 \cdot x-15 }{ x^2-9 }
$$

Answer: The final answer: $\frac{x+5}{x+3}$
Subject: integral_calc

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

Answer: $\int \frac{ 2 \cdot x+\sqrt{2 \cdot x-3} }{ 3 \cdot \sqrt[4]{2 \cdot x-3}+\sqrt[4]{(2 \cdot x-3)^3} } \, dx$ = $C+2\cdot\left(9\cdot\sqrt[4]{2\cdot x-3}+\frac{1}{5}\cdot\sqrt[4]{2\cdot x-3}^5-\frac{2}{3}\cdot\sqrt[4]{2\cdot x-3}^3-\frac{27}{\sqrt{3}}\cdot\arctan\left(\frac{1}{\sqrt{3}}\cdot\sqrt[4]{2\cdot x-3}\right)\right)$
Subject: sequences_series

Problem: Expand the function $f(x) = \ln\left(1+\frac{ x }{ 5 }\right)$ 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. $\frac{x}{5}-\frac{x^2}{5^2\cdot2}+\frac{x^3}{5^3\cdot3}-\frac{x^4}{5^4\cdot4}+\frac{x^5}{5^5\cdot5}-\frac{x^6}{5^6\cdot6}+\frac{x^7}{5^7\cdot7}-\frac{x^8}{5^8\cdot8}+\frac{x^9}{5^9\cdot9}$
2. $\left|R_{10}(x)\right|<\frac{1}{5^{10}\cdot10}$
Subject: differential_calc

Problem: Find the extrema of a function $y = \frac{ x^4 }{ 4 } - \frac{ 2 \cdot x^3 }{ 3 } - \frac{ x^2 }{ 2 } + 2$. Then determine the largest and smallest value of the function when $-2 \le x \le 4$.

Answer: 1. Extrema points: $P\left(\frac{2-2\cdot\sqrt{2}}{2},1.969\right)$, $P(0,2)$, $P\left(\frac{2+2\cdot\sqrt{2}}{2},-1.8023\right)$
2. The largest value: $\frac{46}{3}$
3. The smallest value: $-1.8023$
Subject: differential_calc

Problem: Given $y = \frac{ 1 }{ \sqrt{a \cdot x^2 + b \cdot x + c} }$, evaluate $D = y \cdot y'' - 3 \cdot \left(y' \right)^2 + y^4$.

Answer: The final answer: $D=\frac{1-a}{\left(a\cdot x^2+b\cdot x+c\right)^2}$
Subject: multivariable_calculus

Problem: The distances of all the points of a curve from two fixed points $M$ and $N$ with coordinates $(c,0)$ and $(-c,0)$ are equal to $16$. Find the equation of the curve.

Answer: The final answer: $\frac{x^2}{64}+\frac{y^2}{\left(64-c^2\right)}=1$
Subject: precalculus_review

Problem: A professor asks her class to report the amount of time $t$ they spent writing two assignments. Most students report that it takes them about $45$ minutes to type a four-page assignment and about $90$ minutes to type a nine-page assignment.

1. Find the linear function $y = N(t)$ that models this situation, where $N$ is the number of pages typed and $t$ is the time in minutes.

2. Use part a. to determine how many pages can be typed in $120$ minutes (round to the nearest integer).

3. Use part a. to determine how long it takes to type a $20$-page assignment.

Answer: 1. $N(t)$ = $\frac{1}{9}\cdot t-1$
2. $12$
3. $189$
Subject: integral_calc

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

Answer: $-\int \cos(6 \cdot x)^6 \, dx$ = $C+\frac{1}{288}\cdot\left(\sin(12\cdot x)\right)^3-\frac{1}{24}\cdot\sin(12\cdot x)-\frac{1}{128}\cdot\sin(24\cdot x)-\frac{5}{16}\cdot x$