toloka/u-math huggingface dataset:
Subject: differential_calc

Problem: Make full curve sketching of $f(x) = \frac{ 3 \cdot x^3 }{ 3 \cdot x^2 - 4 }$.

Submit as your final answer:

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

Answer: 1. The domain (in interval notation) $\left(-1\cdot\infty,-2\cdot3^{-1\cdot2^{-1}}\right)\cup\left(-2\cdot3^{-1\cdot2^{-1}},2\cdot3^{-1\cdot2^{-1}}\right)\cup\left(2\cdot3^{-1\cdot2^{-1}},\infty\right)$
2. Vertical asymptote(s) $x=\frac{2}{\sqrt{3}}$, $x=-\frac{2}{\sqrt{3}}$
3. Horizontal asymptote(s) None
4. Slant asymptote(s) $y=x$
5. Interval(s) where the function is increasing $(-\infty,-2)$, $(2,\infty)$
6. Interval(s) where the function is decreasing $\left(-2,-\frac{2}{\sqrt{3}}\right)$, $\left(-\frac{2}{\sqrt{3}},0\right)$, $\left(0,\frac{2}{\sqrt{3}}\right)$, $\left(\frac{2}{\sqrt{3}},2\right)$
7. Interval(s) where the function is concave up $\left(-\frac{2}{\sqrt{3}},0\right)$, $\left(\frac{2}{\sqrt{3}},\infty\right)$
8. Interval(s) where the function is concave down $\left(-\infty,-\frac{2}{\sqrt{3}}\right)$, $\left(0,\frac{2}{\sqrt{3}}\right)$
9. Point(s) of inflection $P(0,0)$
Subject: algebra

Problem: Find the degree and leading coefficient for the given polynomial $f(x) = x^3 \cdot (4 \cdot x - 3)^2$.

Answer: The degree of the polynomial: $5$
The leading coefficient of the polynomial: $16$
Subject: differential_calc

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

$$
\lim_{x \to 0} \left( \frac{ 1 }{ x^2 } - \cot(x)^2 \right)
$$

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

Problem: Evaluate $\lim_{x \to 0}\left(\left(\frac{ \sin(2 \cdot x) }{ 2 \cdot x }\right)^{\frac{ 1 }{ 4 \cdot x^2 }}\right)$.

Answer: $\lim_{x \to 0}\left(\left(\frac{ \sin(2 \cdot x) }{ 2 \cdot x }\right)^{\frac{ 1 }{ 4 \cdot x^2 }}\right)$ = $e^{-\frac{1}{6}}$
Subject: sequences_series

Problem: Using the fact that:
$$
e^x = \sum_{n=0}^\infty \left(\frac{ x^n }{ n! }\right)
$$
Evaluate:
$$
\sum_{n=0}^\infty \left(\frac{ 3 \cdot n }{ n! } \cdot 7^{1-3 \cdot n}\right)
$$

Answer: The final answer: $\frac{3}{49}\cdot e^{\frac{1}{343}}$
Subject: integral_calc

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

Answer: Answer is: $\frac{1}{32}\cdot x-\frac{1}{32}\cdot\frac{1}{8}\cdot\sin(8\cdot x)-\frac{1}{8}\cdot\frac{1}{4}\cdot\frac{1}{3}\cdot\sin(4\cdot x)^3+\frac{1}{128}\cdot x-\frac{1}{128}\cdot\frac{1}{16}\cdot\sin(16\cdot x)+C$
Subject: precalculus_review

Problem: Simplify the following expression:

$$
E = 2 \cdot \left( \left( \sin(x) \right)^6 + \left( \cos(x) \right)^6 \right) - 3 \cdot \left( \left( \sin(x) \right)^4 + \left( \cos(x) \right)^4 \right)
$$

Answer: The final answer: $E=-1$
Subject: precalculus_review

Problem: For both $y=a^x$ and $y=\log_{a}(x)$ where $a>0$ and $a \ne 1$, find the $P(x,y)$ coordinate where the slope of the functions are $1$. Write your answers only in terms of natural logarithms.

Answer: Coordinates of $a^x$: $x$ = $-\frac{\ln\left(\ln(a)\right)}{\ln(a)}$ , $y$ = $y=\frac{1}{\ln(a)}$  

Coordinates of $\log_{a}(x)$: $x$ = $x=\frac{1}{\ln(a)}$ , $y$ = $-\frac{\ln\left(\ln(a)\right)}{\ln(a)}$
Subject: differential_calc

Problem: Find the first derivative of the function: $y = (x + 11)^5 \cdot (3 \cdot x - 7)^4 \cdot (x - 12) \cdot (x + 4)$.

Answer: $y'$ = $\left(\frac{5}{x+11}+\frac{12}{3\cdot x-7}+\frac{1}{x-12}+\frac{1}{x+4}\right)\cdot(x+11)^5\cdot(3\cdot x-7)^4\cdot(x-12)\cdot(x+4)$
Subject: multivariable_calculus

Problem: Find the directional derivative of $f(x,y,z) = x^2 + y \cdot z$ at $P(1,-3,2)$ in the direction of increasing $t$ along the path

$\vec{r}(t) = t^2 \cdot \vec{i} + 3 \cdot t \cdot \vec{j} + \left(1-t^3\right) \cdot \vec{k}$.

Answer: $f_{u}(P)$ = $\frac{11}{\sqrt{22}}$
Subject: algebra

Problem: Find the solution to the following inequality and express it in interval notation:
$$-4 (x-5) (x+4) (x-1) \le 0$$

Answer: The solution set to the inequality is: $\left[-4,\ 1\right]\cup\left[5,\ \infty\right)$
Subject: integral_calc

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

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

Problem: Find the inverse function of $f(x) = \frac{ 2 \cdot x + 3 }{ 5 \cdot x + 4 }$.

Answer: $f^{-1}(x)$ = $\frac{3-4\cdot x}{5\cdot x-2}$
Subject: precalculus_review

Problem: Find zeros of $f(x) = \sqrt{(x+3)^2} + \sqrt{(x-2)^2} + \sqrt{(2 \cdot x-8)^2} - 9$

Answer: The final answer: $x\in[2,4]$
Subject: integral_calc

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

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

Problem: Compute the limit:
$$
\lim_{x \to 5} \left( \frac{ 3 \cdot x }{ x-5 }-\frac{ 3 }{ \ln\left(\frac{ x }{ 5 }\right) } \right)
$$

Answer: $\lim_{x \to 5} \left( \frac{ 3 \cdot x }{ x-5 }-\frac{ 3 }{ \ln\left(\frac{ x }{ 5 }\right) } \right)$ = $\frac{3}{2}$
Subject: precalculus_review

Problem: Find the domain of the function $f(x) = \frac{ 1 }{ \sqrt{ |x| - x } }$.

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

Problem: Solve the integral:
$$
\int 3 \cdot \cot(-7 \cdot x)^6 \, dx
$$

Answer: $\int 3 \cdot \cot(-7 \cdot x)^6 \, dx$ = $C-\frac{3}{7}\cdot\left(\frac{1}{5}\cdot\left(\cot(7\cdot x)\right)^5+\cot(7\cdot x)-\frac{1}{3}\cdot\left(\cot(7\cdot x)\right)^3-\arctan\left(\cot(7\cdot x)\right)\right)$
Subject: multivariable_calculus

Problem: Evaluate the triple integral of the function $f(x,y,z) = z$ over the solid $B$ bounded by the half-sphere $x^2+y^2+z^2=16$ with $z \ge 0$ and below by the cone $2 \cdot z^2 = x^2+y^2$.

Answer: $\int\int\int_{B}{f(x,y,z) d V}$ = $\frac{128\cdot\pi}{3}$
Subject: multivariable_calculus

Problem: Find a normal vector and a tangent vector for $2 \cdot x^3 - x^2 \cdot y^2 = 3 \cdot x - y - 7$ at point $P : (1,-2)$

Answer: Normal vector: $\vec{N}=\vec{i}-\vec{j}$
Tangent vector: $\vec{T}=\vec{i}+\vec{j}$
Subject: differential_calc

Problem: $f(x) = \frac{ 1 }{ 4 } \cdot \sqrt{x} + \frac{ 1 }{ x }$, $x > 0$. Determine:
1. Intervals where $f$ is increasing
2. Intervals where $f$ is decreasing
3. Local minima of $f$
4. Local maxima of $f$
5. Intervals where $f$ is concave up
6. Intervals where $f$ is concave down
7. The inflection points of $f$

Answer: 1. Intervals where $f$ is increasing: $(4,\infty)$
2. Intervals where $f$ is decreasing: $(0,4)$
3. Local minima of $f$: $4$
4. Local maxima of $f$: None
5. Intervals where $f$ is concave up: $\left(0,8\cdot\sqrt[3]{2}\right)$
6. Intervals where $f$ is concave down: $\left(8\cdot\sqrt[3]{2},\infty\right)$
7. The inflection points of $f$: $8\cdot\sqrt[3]{2}$
Subject: multivariable_calculus

Problem: Find the average value of the function $f(x,y) = \arctan(x \cdot y)$ over the region $R = [0,1] \times [0,1]$.

Answer: $f_{ave}$ = $\frac{\pi-\ln(4)}{4}-\frac{\pi^2}{48}$
Subject: differential_calc

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

Answer: 1. Extrema points: $P\left(\frac{3}{2},\frac{1}{32}\right)$, $P\left(-1,\frac{4}{3}\right)$, $P(0,2)$
2. The largest value: $\frac{254}{3}$
3. The smallest value: $\frac{1}{32}$
Subject: algebra

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

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

Answer: The rational function satisfying the given conditions is $f(x)=\frac{15\cdot(x-1)\cdot(x+4)}{2\cdot(x+3)\cdot(x-2)}$
Subject: integral_calc

Problem: Solve the integral:
$$
\int \cot(x)^4 \, dx
$$

Answer: $\int \cot(x)^4 \, dx$ = $C+\cot(x)-\frac{1}{3}\cdot\left(\cot(x)\right)^3-\arctan\left(\cot(x)\right)$
Subject: multivariable_calculus

Problem: Use the method of Lagrange multipliers to maximize $U(x,y) = 8 \cdot x^{\frac{ 4 }{ 5 }} \cdot y^{\frac{ 1 }{ 5 }}$ subject to the constraint $4 \cdot x + 2 \cdot y = 12$.

Answer: Answer: maximum $16.715$ at $P(2.4,1.2)$
Subject: algebra

Problem: Note that $1 \in (-3,7)$. Think of the solution of the inequality $|x-1|<\delta$, where $\delta$ (delta, a Greek letter) is a constant.

Find the greatest $\delta$ so that the solution is completely on the interval $(-3,7)$.

[Hint: How far you can go within $(-3,7)$, starting from $1$?]

Answer: The final answer: $4$
Subject: integral_calc

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

Answer: $\int \left(\frac{ x+3 }{ x-3 }\right)^{\frac{ 3 }{ 2 }} \, dx$ = $C+\sqrt{\frac{x+3}{x-3}}\cdot(x-15)-9\cdot\ln\left(\left|\frac{\sqrt{x-3}-\sqrt{x+3}}{\sqrt{x-3}+\sqrt{x+3}}\right|\right)$
Subject: multivariable_calculus

Problem: $E$ is located above the xy-plane, below $z=1$, outside the one-sheeted hyperboloid $x^2+y^2-z^2=1$, and inside the cylinder $x^2+y^2=2$. Find the volume of $E$.

Answer: Volume = $\frac{2\cdot\pi}{3}$
Subject: multivariable_calculus

Problem: Develop the Cartesian equation for the following parametric equations:

$x = 3 \cdot t + \frac{ 3 }{ 4 \cdot t }$

$y = 5 \cdot t - \frac{ 5 }{ 4 \cdot t }$

Answer: The final answer: $\frac{x^2}{9}-\frac{y^2}{25}=1$
Subject: multivariable_calculus

Problem: Evaluate the integral by choosing the order of integration:
$$
\int_{1}^e \int_{1}^e \left( \frac{ x \cdot \ln(y) }{ \sqrt{y} } + \frac{ y \cdot \ln(x) }{ \sqrt{x} } \right) \, dy \, dx
$$

Answer: $\int_{1}^e \int_{1}^e \left( \frac{ x \cdot \ln(y) }{ \sqrt{y} } + \frac{ y \cdot \ln(x) }{ \sqrt{x} } \right) \, dy \, dx$ = $2\cdot\sqrt{e}+4\cdot e^2-4-2\cdot e^2\cdot\sqrt{e}$
Subject: differential_calc

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

Problem: Compute the length of the arc $y = 3 \cdot \ln(2 \cdot x)$ between the points $x = \sqrt{7}$ and $x = 4$.

Answer: Arc Length: $1+\frac{3}{2}\cdot\ln\left(\frac{7}{4}\right)$
Subject: sequences_series

Problem: Find $L=\lim_{n \to \infty}\left(x_{n}\right)$, where $x_{n} = \frac{ \sqrt{n^2+1} + \sqrt{n} }{ \sqrt[4]{n^3+n} - \sqrt{n} }$ is the general term of a sequence.

Answer: The final answer: $L=\infty$
Subject: precalculus_review

Problem: Find points on a coordinate plane that satisfy the equation $x \cdot (x-2) + y \cdot (y+4) + 5 = 0$.

Answer: The final answer: $(1,-2)$
Subject: precalculus_review

Problem: Find points on a coordinate plane that satisfy the equation $y = \sqrt{\ln\left(\cos(x)\right)}$.

Answer: The final answer: $(2\cdot k\cdot\pi,0)$
Subject: differential_calc

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

Answer: $y^{(5)}$ = $\frac{1}{7776}\cdot e^{\frac{x}{3}}\cdot\left(122\cdot\sin\left(\frac{x}{2}\right)-597\cdot\cos\left(\frac{x}{2}\right)\right)$
Subject: algebra

Problem: Identify all points of removable discontinuity (singularity) of the function $f(x) = \frac{ x^2 - 4 }{ x - 2 }$.

Answer: $f(x)$ has removable discontinuities at $x=2$
Subject: differential_calc

Problem: The cost for printing a book can be given by the equation $C(x) = 1000 + 12 \cdot x + \frac{ 1 }{ 2 } \cdot x^{\frac{ 2 }{ 3 }}$. Use Newton’s method to find the break-even point if the printer sells each book for $\$20$.

Answer: $127$ books
Subject: multivariable_calculus

Problem: Use the method of Lagrange multipliers to find the maximum and minimum values of $f(x,y,z) = x^2 + y^2 + z^2$ subject to the constraint $x^4 + y^4 + z^4 = 1$. If there is no maximum or minimum, then write $\text{None}$ in the answer.

Answer: Maximum: $\sqrt{3}$

Minimum: $1$
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 $\$33$, they would lose $7000$ 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: $33$
Subject: differential_calc

Problem: Find the derivative of $f(x) = \frac{ \left(\left(\tan(x)\right)^2-1\right) \cdot \left(\left(\tan(x)\right)^4+10 \cdot \left(\tan(x)\right)^2+1\right) }{ 3 \cdot \left(\tan(x)\right)^3 }$

Answer: The final answer: $f'(x)=\left(\tan(x)\right)^4+4\cdot\left(\tan(x)\right)^2+\frac{4}{\left(\tan(x)\right)^2}+\frac{1}{\left(\tan(x)\right)^4}+6$
Subject: sequences_series

Problem: Decompose the function $f(x) = \frac{ 1 }{ 2 } \cdot (\pi-x)$ into a trigonometric series on the interval $[0,2 \cdot \pi]$.

Answer: The final answer: $f(x)=\sin(x)+\frac{1}{2}\cdot\sin(2\cdot x)+\frac{1}{3}\cdot\sin(3\cdot x)+\cdots+\frac{1}{n}\cdot\sin(n\cdot x)+\cdots$
Subject: integral_calc

Problem: Compute the integral:
$$
\int_{0}^1 \frac{ \sqrt{x}+1 }{ \sqrt[3]{x}+1 } \, dx
$$

Answer: $\int_{0}^1 \frac{ \sqrt{x}+1 }{ \sqrt[3]{x}+1 } \, dx$ = $3\cdot\ln(2)+\frac{3\cdot\pi}{2}-\frac{409}{70}$
Subject: multivariable_calculus

Problem: Evaluate the integral by choosing the order of integration:
$$
\int_{0}^1 \int_{0}^{\frac{ 1 }{ 2 }} \left(\arcsin(x) + \arcsin(y)\right) \, dy \, dx
$$

Answer: $\int_{0}^1 \int_{0}^{\frac{ 1 }{ 2 }} \left(\arcsin(x) + \arcsin(y)\right) \, dy \, dx$ = $\frac{\pi}{12}+\frac{\sqrt{3}}{2}+\frac{\arcsin(1)}{2}-\frac{3}{2}$
Subject: multivariable_calculus

Problem: Compute the partial derivatives of the implicit function $z(x,y)$, given by the equation $-x-6 \cdot y+z=3 \cdot \cos(-x-6 \cdot y+z)$.

Submit as your final answer:

1. $\frac{\partial z}{\partial x}$;

2. $\frac{\partial z}{\partial y}$.

Answer: 1. $1$

2. $6$
Subject: sequences_series

Problem: Find the Fourier integral of the function $q(x) = \begin{cases} 0, & x < 0 \\ \pi \cdot x, & 0 \le x \le 1 \\ 0, & x > 1 \end{cases}$

Answer: $q(x) = $\int_0^\infty\left(\frac{\left(\alpha\cdot\sin\left(\alpha\right)+\cos\left(\alpha\right)-1\right)\cdot\cos\left(\alpha\cdot x\right)+\left(\sin\left(\alpha\right)-\alpha\cdot\cos\left(\alpha\right)\right)\cdot\sin\left(\alpha\cdot x\right)}{\alpha^2}\right)d\alpha$
Subject: sequences_series

Problem: Find the Maclaurin series for the function: $f(x) = \cos(x) - x \cdot \sin(x)$.

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

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

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

Problem: A phone company has a monthly cellular plan, where a customer pays a flat monthly fee and then a certain amount of money per minute used on the phone. If a customer uses 410 minutes, the monthly cost will be $71.5. If the customer uses 720 minutes, the monthly cost will be $118.1.

1. Find a linear equation for the monthly cost of the cell plan as a function of $x$, the number of monthly minutes used.
2. Interpret the slope and y-intercept of the equation.
3. Use your equation to find the total monthly cost if 687 minutes are used.

Answer: 1. The monthly cost function is $C(x)=0.15\cdot x+10$
2. The flat monthly fee is $10$ and the fee for each additional minute used is $0.15$
3. The total monthly cost if 687 minutes are used is $113.05$
Subject: precalculus_review

Problem: A car is racing along a circular track with a diameter of 1 mile. A trainer standing in the center of the circle marks his progress every 5 seconds. After 5 seconds, the trainer has to turn $55^o$ to keep up with the car. How fast is the car traveling?

Answer: Velocity of car is $345.57519189$ mph.
Subject: precalculus_review

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

$$
\left|\left(x^2+4 \cdot x+9\right)+(2 \cdot x-3)\right|=\left|x^2+4 \cdot x+9\right|+|2 \cdot x-3|
$$

Answer: The final answer: $x\ge\frac{3}{2}$
Subject: precalculus_review

Problem: Find the zeros of $\tan(x) + \tan\left(\frac{ \pi }{ 4 } + x\right) = -2$.

Answer: The final answer: $x_{1}=-\frac{ \pi }{ 3 }+\pi \cdot n$, $x_{2}=\frac{ \pi }{ 3 }+n \cdot \pi$
Subject: precalculus_review

Problem: Calculate $E = \left(\sin\left(\frac{ \pi }{ 8 }\right)\right)^4 + \left(\sin\left(\frac{ 3 \cdot \pi }{ 8 }\right)\right)^4 + \left(\sin\left(\frac{ 5 \cdot \pi }{ 8 }\right)\right)^4 + \left(\sin\left(\frac{ 7 \cdot \pi }{ 8 }\right)\right)^4$

Answer: The final answer: $E=\frac{3}{2}$
Subject: multivariable_calculus

Problem: Compute the partial derivatives of the implicit function $z(x,y)$, given by the equation
$$
-10 \cdot x-9 \cdot y+8 \cdot z=4 \cdot \cos(-10 \cdot x-9 \cdot y+8 \cdot z)
$$

Submit as your final answer:

1. $\frac{\partial z}{\partial x}$;

2. $\frac{\partial z}{\partial y}$.

Answer: 1. $\frac{5}{4}$

2. $\frac{9}{8}$
Subject: multivariable_calculus

Problem: For the following exercise, line $L$ is given.

1. Find point $P$ that belongs to the line and direction vector $\vec{v}$ of the line. Express $\vec{v}$ in component form.
2. Find the distance from the origin to line $L$.

Line $L$: $-x = y + 1$, $z = 2$

Answer: 1. $P$: $P(0,-1,2)$ ; $\vec{v}$= $\left\langle-1,1,0\right\rangle$
2. $d$= $\frac{3}{\sqrt{2}}$
Subject: integral_calc

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

Answer: $\int x^{-6} \cdot \left(1+x^2\right)^{\frac{ 1 }{ 2 }} \, dx$ = $C+\frac{1}{3}\cdot\left(\frac{1}{x^2}+1\right)\cdot\sqrt{\frac{1}{x^2}+1}-\frac{1}{5}\cdot\left(\frac{1}{x^2}+1\right)^2\cdot\sqrt{\frac{1}{x^2}+1}$
Subject: precalculus_review

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

$p(x) = x^3 - 3 \cdot x^2 - 10 \cdot x + 24$

Answer: The real zeros are $2$, $-3$, $4$
Subject: sequences_series

Problem: Find the radius of convergence and the interval of convergence for the series:
$$
\sum_{n=0}^\infty \left(\frac{ x^n }{ n^n }\right)
$$

Answer: $R$ = $\infty$
$I$ = $(-\infty,\infty)$
Subject: precalculus_review

Problem: Find the period of $f(x) = \left| \cos(x) \right|$

Answer: The final answer: $T=\pi$
Subject: algebra

Problem: Rewrite the quadratic expression $x^2 - 11 \cdot x - 20$ by completing the square.

Answer: $x^2 - 11 \cdot x - 20$ = $(x-5.5)^2-50.25$
Subject: algebra

Problem: Use the fact that the vertex of the graph of the quadratic function is $(2,-3)$ and the graph opens up to find the domain and range of the function.

Answer: The domain is $(-\infty,\infty)$ and the range is $[-3,\infty)$
Subject: integral_calc

Problem: Compute the area of the figure bounded by curves $y = 2 \cdot x^2$, $y = 1 + x^2$, lines $x = 3$, $x = -2$, and the $x$-axis.

Answer: Area = $\frac{46}{3}$
Subject: precalculus_review

Problem: An alternating current for outlets in a home has voltage given by the function $V(t) = 150 \cdot \cos(368 \cdot t)$, where $V$ is the voltage in volts at time $t$ in seconds.

1. Find the period of the function.
2. Determine the number of periods that occur when $1$ sec. has passed.

Answer: 1. The period of the function is $\frac{\pi}{184}$
2. The number of periods that occur when $1$ sec. has passed is $58.56901906$
Subject: precalculus_review

Problem: Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{-\pi}^\pi \left(\cos(3 \cdot x)\right)^2 \, dx
$$

Answer: $\int_{-\pi}^\pi \left(\cos(3 \cdot x)\right)^2 \, dx$ = $\pi$
Subject: integral_calc

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

Answer: $\int \frac{ -4 }{ 3+\sin(4 \cdot x)+\cos(4 \cdot x) } \, dx$ = $C-\frac{2}{\sqrt{7}}\cdot\arctan\left(\frac{2}{\sqrt{7}}\cdot\left(\frac{1}{2}+\tan(2\cdot x)\right)\right)$
Subject: differential_calc

Problem: Make full curve sketching of $y = \arcsin\left(\frac{ 1-x^2 }{ 1+x^2 }\right)$. 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 $(-\infty,0)$
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: multivariable_calculus

Problem: Find the point on the surface $f(x,y) = x^2 + y^2 + 10$ nearest the plane $x + 2 \cdot y - z = 0$. Identify the point on the plane.

Answer: Answer: $P\left(\frac{47}{24},\frac{47}{12},\frac{235}{24}\right)$
Subject: differential_calc

Problem: Sketch the curve:  
$$
y = \frac{ x^3 }{ 4 \cdot (x+3)^2 }
$$
Provide the following:

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): $(-1\cdot\infty,-3)\cup(-3,\infty)$
2. Vertical asymptotes: $x=-3$
3. Horizontal asymptotes: None
4. Slant asymptotes: $y=\frac{x}{4}-\frac{3}{2}$
5. Intervals where the function is increasing: $(-\infty,-9)$, $(-3,0)$, $(0,\infty)$
6. Intervals where the function is decreasing: $(-9,-3)$
7. Intervals where the function is concave up: $(-\infty,-9)$, $(-3,0)$, $(0,\infty)$
8. Intervals where the function is concave down: $(-3,0)$, $(-\infty,-3)$
9. Points of inflection: $P(0,0)$