uuid stringlengths 36 36 | subject stringclasses 6
values | has_image bool 2
classes | image stringclasses 160
values | problem_statement stringlengths 32 784 | golden_answer stringlengths 7 1.13k |
|---|---|---|---|---|---|
b750bd34-f50c-493d-b569-d3bc6ab886f1 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAAB1HElEQVR4nO3deViVdf7/8ddhR1ZRELTUygQzNbXSciHLTLPNmrTJspopS1uNabOxmsx2ahzT0RrnO03p5JQ11VRqppKZS6Kh5b7hAioCIhzWA/fvD3/nHg6LIuJ9c+D5uK5zec59Pofz9hbwfp3P5jAMwxAAAAAASyQnJys5OVmSlJSUpKSkJJsrspaP3QUAAAAAaD4IIAAAAAAsQwABAAAAYBkCCAAAAADLEEAAAAAAWIYAAgAAAMAyBB... | Let $Q$ be the region bounded by the graph of $x=\frac{ 2 }{ 1-y }$, the line $x=-1$, and the line $y=\frac{ 5 }{ 4 }$, as shown in the figure above.
Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when $Q$ is revolved about the line $x=-1$. | $V$ = $\int_{\frac{5}{4}}^3\left(\pi\cdot\left(\frac{2}{1-y}+1\right)^2\right)dy$ |
b75b8718-0c2d-4333-9d47-a9a7a16878ae | precalculus_review | false | null | 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 $\sin(6 \cdot x) < 0$. | The final answer: $x=\frac{5\cdot\pi}{16}+\pi\cdot k$ |
b775a64a-639f-497b-b064-ba7db40ab3ca | sequences_series | false | null | Using the series expansion for the function $(1+x)^m$, calculate approximately $\sqrt[3]{7}$ with an accuracy of 0.0001. | The final answer: $1.9129$ |
b78a35bf-86eb-4ed0-8899-09518687eaeb | algebra | false | null | 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 t... | The solution is $x$ = $5$ |
b78b640e-2077-4261-8d0d-6af62d82047e | integral_calc | false | null | Compute the integral:
$$
\int \sin(3 \cdot x)^6 \cdot \cos(3 \cdot x)^2 \, dx
$$ | Answer is: $\frac{1}{32}\cdot x-\frac{1}{32}\cdot\frac{1}{12}\cdot\sin(12\cdot x)-\frac{1}{8}\cdot\frac{1}{6}\cdot\frac{1}{3}\cdot\sin(6\cdot x)^3+\frac{1}{128}\cdot x-\frac{1}{128}\cdot\frac{1}{24}\cdot\sin(24\cdot x)+C$ |
b7a0b462-28e2-47c0-be0c-1c233c09fe99 | precalculus_review | false | null | 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. | $\tan(x)^2 + \sin(x) \cdot \csc(x)$ = $\sec(x)^2$ |
b7a66b26-e0ab-4533-a404-db2f9afeb3b0 | sequences_series | false | null | 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? | $|p|$ < $\frac{1}{2}$ |
b800fdc0-a9c8-47b9-b98f-9c4ec5811709 | multivariable_calculus | false | null | 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. | The final answer: $\frac{(x-5)^2}{25}+\frac{7\cdot(y-3)^2}{75}=1$ |
b82a9cc4-bafb-4dff-b474-ea2287346a3d | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAT4AAABICAYAAABvNJlYAAAuNUlEQVR4nO2dd3gcxf3GP7vX1YtVrW5ZcpdcsI2NO9jYwaaZXpJQQgkBQgshkISEUAIJhIQfoQUSSEJJAGOMe++23JvcJVmWZKt3Xdv9/bG7dyonWeUkO7Dv88iW7nZnvjs78863zYwgy7KMDh06dHyHIJ5vAXTo0KGjr6ETnw4dOr5z0IlPhw4d3znoxKdDh47vHHTi06FDx3cOOvHp0KHjOwfj+Rbg2wxZlpFlGQQBURDOtzg6dOhQoR... | Find values of $a$ and $b$, making function continuous and differentiable at $x=1$.
The graph is shown. | The $a$ and $b$ values are: $a=-4$, $b=2$ |
b84018c9-fd9f-4ef5-917a-9ed1849998ea | differential_calc | false | null | 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 | 1. The point(s) where local maxima occur: $P(1,108)$
2. The point(s) where local minima occur: $P(-1,0)$ |
b887d056-ce73-4520-a35b-d59f1198f8a3 | differential_calc | false | null | Compute the limit:
$$
\lim_{x \to 4}\left(\frac{ x }{ x-4 }-\frac{ 1 }{ \ln\left(\frac{ x }{ 4 }\right) }\right)
$$ | $\lim_{x \to 4}\left(\frac{ x }{ x-4 }-\frac{ 1 }{ \ln\left(\frac{ x }{ 4 }\right) }\right)$ = $\frac{1}{2}$ |
b8a1a55a-f36b-4033-9813-55d603b2edd9 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ -2 }{ e^{3 \cdot x} + \sqrt{1 + e^{6 \cdot x}} } \, dx
$$ | $\int \frac{ -2 }{ e^{3 \cdot x} + \sqrt{1 + e^{6 \cdot x}} } \, dx$ = $C-\frac{1}{3}\cdot\left(\frac{2}{e^{3\cdot x}+\sqrt{1+e^{6\cdot x}}}+2\cdot\ln\left(\frac{e^{3\cdot x}+\sqrt{1+e^{6\cdot x}}-1}{1+e^{3\cdot x}+\sqrt{1+e^{6\cdot x}}}\right)\right)$ |
b8e659f6-36d7-4648-9a18-b9a397fa9c13 | multivariable_calculus | false | null | Evaluate the iterated integral:
$$
\int_{e}^{e^2} \int_{\ln(u)}^2 \left(v + \ln(u)\right) \, dv \, du
$$ | $\int_{e}^{e^2} \int_{\ln(u)}^2 \left(v + \ln(u)\right) \, dv \, du$ = $e^2-\frac{e}{2}$ |
b9549494-f051-4d1b-891b-874e88c55728 | precalculus_review | false | null | Find the product $C=\left(-\frac{ 2 }{ 3 }-\frac{ 1 }{ 2 } \cdot i\right) \cdot \left(\frac{ 1 }{ 5 }+\frac{ 4 }{ 5 } \cdot i\right)$. | The final answer: $C=\frac{4}{15}-\frac{19}{30}\cdot i$ |
b95c92b9-de56-4e8a-8b1a-c7cd1d375bf5 | sequences_series | false | null | 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. | The final answer: $x-2\cdot x^2+3\cdot x^3-4\cdot x^4+\cdots$ |
b9ac8e90-d4b9-4750-906c-54109d1396d7 | algebra | false | null | 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 t... | The first account investment: $75000$
The second account investment: $15000$
The third account investment: $10000$ |
b9ba91e6-fd9d-4131-9c7d-ba92a0947e30 | differential_calc | false | null | 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. | 1. Concave up: $(0,\infty)$
2. Concave down: $(-\infty,-2)$, $(-2,0)$
3. Point(s) of Inflection: $P(0,100)$ |
b9cdefb2-0696-45ba-a3d0-d4804c70d4f0 | integral_calc | false | null | Calculate the integral:
$$
\int_{-\sqrt{2}}^{\sqrt{2}} \frac{ 2 \cdot x^7+3 \cdot x^6-10 \cdot x^5-7 \cdot x^3-12 \cdot x^2+x+1 }{ x^2+2 } \, dx
$$ | $\int_{-\sqrt{2}}^{\sqrt{2}} \frac{ 2 \cdot x^7+3 \cdot x^6-10 \cdot x^5-7 \cdot x^3-12 \cdot x^2+x+1 }{ x^2+2 } \, dx$ = $\frac{5\cdot\pi-64}{10\cdot\sqrt{2}}$ |
b9fdcdf9-caa8-482e-b135-c2073d4e0c25 | differential_calc | false | null | 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 ... | 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(... |
ba021e0e-5fd6-428c-91e4-fcd26bec895f | differential_calc | false | null | 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. | $4.492$% |
ba6d988a-5f01-4dfe-a6c1-f85fb7f22f01 | precalculus_review | false | null | 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. | $\sin(x) \cdot \left(\csc(x)-\sin(x)\right)$ = $\cos(x)^2$ |
baea3a26-3788-4b32-b19e-931a1c4bed1d | sequences_series | false | null | Find the Fourier series of the function $\psi(x) = 2 \cdot e^{-2 \cdot x}$ in the interval $(-\pi, \pi)$. | The Fourier series is: $2\cdot e^{-2\cdot x}=\frac{\left(e^{2\cdot\pi}-e^{-2\cdot\pi}\right)}{\pi}\cdot\left(\frac{1}{2}+\sum_{n=1}^\infty\left(\frac{2\cdot(-1)^n\cdot\left(2\cdot\cos(n\cdot x)+n\cdot\sin(n\cdot x)\right)}{4+n^2}\right)\right)$ |
bba192e9-a029-4393-a68e-e42c6444ae52 | differential_calc | false | null | Evaluate the limit:
$$
\lim_{x \to 0}\left(\frac{ \ln\left(1+x+x^2\right)+\ln\left(1-x+x^2\right) }{ x^2 }\right)
$$ | $\lim_{x \to 0}\left(\frac{ \ln\left(1+x+x^2\right)+\ln\left(1-x+x^2\right) }{ x^2 }\right)$ = $1$ |
bbe52c80-dc6a-4752-8fa2-e540fdd12adb | algebra | false | null | Use the Rational Zero Theorem to find the real solution(s) of the equation: $x^4 + 2 \cdot x^3 - 4 \cdot x^2 - 10 \cdot x - 5 = 0$ | By the Rational Zero Theorem, possible rational zeros of the function are: $1$, $-1$, $5$, $-5$ $x$ = $\sqrt{5}$, $-\sqrt{5}$, $-1$ |
bc1290d0-681b-468e-b494-3a277399c84f | sequences_series | false | null | Find the interval of convergence of the series $\sum_{n=1}^\infty \frac{ 3 \cdot x^n }{ n \cdot 7^n }$. (Use interval notation) | The final answer: $[-7,7)$ |
bc24c3db-73a9-4f4d-84c9-58dcebd518f9 | algebra | false | null | 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. $99 = 33 x + 3 (3 x + 5)$ | 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. $x=2$ |
bc3c0d6d-ed63-46d2-9537-1d1c3ad95d1c | multivariable_calculus | false | null | 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 $w = \frac{ z }{ 2 }$. | $I$ = $\frac{5}{2}$ |
bc5b6d16-0655-4241-850e-6c54cc736d93 | sequences_series | false | null | 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^... | 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... |
bcc5b742-cc7b-42d6-98eb-9db904ab8078 | multivariable_calculus | false | null | Evaluate $\int\int\int_{E}{\left(x^3+y^3+z^3\right) d V}$, where $E$ is the region defined by:
$$
E = \left\{(x,y,z) | 0 \le x \le 2, 0 \le y \le 2 \cdot x, 0 \le z \le 4-x-y\right\}
$$ | $I$ = $\frac{112}{5}$ |
be0450d9-d002-494c-b831-25492d662296 | sequences_series | false | null | 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 $a=0$. | $\sqrt{x^2+2}$ = $\sum_{n=0}^\infty\left(2^{\frac{1}{2}-n}\cdot C_{\frac{1}{2}}^n\cdot x^{2\cdot n}\right)$ |
be5b328d-cd61-45c8-97bf-93d2e3f5c525 | algebra | false | null | Find the solution to the following inequality and express it in interval notation:
$$8 (x-9) (x+5) (x-3) > 0$$ | The solution set to the inequality is $\left(-5,\ 3\right)\cup\left(9,\ \infty\right)$ |
be642fd8-0fc8-450f-867d-eea4c7765a27 | differential_calc | false | null | 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? | The final answer: $3\cdot b^2-8\cdot a\cdot c>0$ |
be8194d4-6693-48d6-a382-70e25ddc9b86 | precalculus_review | false | null | 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. | $\left(1+\tan\left(\theta\right)\right)^2-2 \cdot \tan\left(\theta\right)$ = $\sec\left(\theta\right)^2$ |
bea9777f-c66e-48e1-b32b-f2795b3f8c4f | integral_calc | false | null | Compute the integral:
$$
\int \frac{ -12 }{ \sin(6 \cdot x)^6 } \, dx
$$ | $\int \frac{ -12 }{ \sin(6 \cdot x)^6 } \, dx$ = $C+2\cdot\cot(6\cdot x)+\frac{2}{5}\cdot\left(\cot(6\cdot x)\right)^5+\frac{4}{3}\cdot\left(\cot(6\cdot x)\right)^3$ |
bf49a8b7-ff89-4c97-86c5-94549ff3ce80 | sequences_series | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAeQAAACrCAIAAAA4tXijAABGi0lEQVR4nO2dd1wU1/f378xsYaU3pQgoYEOqYm9INHZjicYaTYwtGiMmahJrLLHE2GLUWGKPsRCNFaKIJSJ2UeyCFOl9YfuU54/zc579LogI28D7/sMX7s7OnL175zPnnnvuuQTHcQiDwWAw5g1pagMwGAwG83awWGMwGEwtAIs1BoPB1AKwWGMwGEwtAIt1HQFPFGMwdRss1nWB58+fx8XFYb3GYOowBL7DaztyuTwgIKCwsPDGjRtNmj... | Find the Fourier series expansion of the function $f(x) = \begin{cases} -x, & -\pi < x \le 0 \\ \frac{ x^2 }{ \pi }, & 0 < x \le \pi \end{cases}$ with the period $2 \cdot \pi$ on the interval $[-\pi,\pi]$. | The Fourier series is: $f(x)=\frac{5\cdot\pi}{12}+\sum_{n=1}^\infty\left(\frac{(-1)^n\cdot3-1}{\pi\cdot n^2}\cdot\cos(n\cdot x)+\left(\frac{2}{\pi^2\cdot n^3}\cdot\left((-1)^n-1\right)\right)\cdot\sin(n\cdot x)\right)$ |
bf6ce9fb-0a58-4dab-8385-8e9e64099565 | precalculus_review | false | null | Calculate $E = \frac{ 1 }{ \sin(10) } - \frac{ \sqrt{3} }{ \cos(10) }$. | The final answer: $E=4$ |
bf760c7e-8f0f-4e8f-8add-9a1541343d1a | multivariable_calculus | false | null | 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(... | 1. $x-z=1$
2. $-5\cdot x+7\cdot y-3\cdot z=0$
3. $x+y+z=5$ |
bfd92140-56a4-4cd9-a0d4-c99aef34eb24 | algebra | false | null | 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 $x + 6$. | The remainder is $-44791$ |
bff02121-9696-4fd4-9951-072d213b75f0 | differential_calc | false | null | Evaluate $\lim_{x \to 0^{+}} \left( \left( \frac{ \tan(x) }{ x } \right)^{\frac{ 1 }{ x^2 }} \right)$ using l'Hospital's Rule. | $\lim_{x \to 0^{+}} \left( \left( \frac{ \tan(x) }{ x } \right)^{\frac{ 1 }{ x^2 }} \right)$ = $e^{\frac{1}{3}}$ |
c1776c95-1585-4f95-8a20-535567d00fae | multivariable_calculus | false | null | Evaluate $\int\int\int_{E}{z \, dV}$, where $E$ is the region defined by:
$$
E = \left\{ (x,y,z) \mid -y \le x \le y, \, 0 \le y \le 1, \, 0 \le z \le 1-x^4-y^4 \right\}
$$ | $I$ = $\frac{113}{450}$ |
c1c7b5c8-ba37-4871-9cc7-79a575e299a1 | integral_calc | false | null | Calculate the integral:
$$
\int \frac{ 3 \cdot x + 4 }{ \left( x^2 + 1 \cdot x + 7 \right)^2 } \, dx
$$ | $\int \frac{ 3 \cdot x + 4 }{ \left( x^2 + 1 \cdot x + 7 \right)^2 } \, dx$ = $\frac{\frac{5}{27}\cdot x-\frac{38}{27}}{\frac{27}{4}+\left(x+\frac{1}{2}\right)^2}+\frac{30\cdot\sqrt{3}}{729}\cdot\arctan\left(\sqrt{\frac{4}{27}}\cdot\left(x+\frac{1}{2}\right)\right)+C$ |
c1fc8501-1c68-469f-a8b9-22b27702d1c8 | algebra | false | null | Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeros of the following polynomial:
$p(x) = 2 \cdot x^3 - x^2 + 7 \cdot x - 1$ | The number of positive zeros: $1$, $3$
The number of negative zeros: $0$ |
c221a7ed-a23f-41f3-89c0-2b6faf6228de | precalculus_review | false | null | Solve the trigonometric equation $\sec(x)^2 - 2 \cdot \sec(x) + 1 = 0$ on the interval $[-2 \cdot \pi, 2 \cdot \pi]$. | $x$ = $-2\cdot\pi$, $2\cdot\pi$, $0$ |
c251e922-18a0-40d3-a506-3a4d6f32a485 | algebra | false | null | 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 $P(0,4)$. | The rational function satisfying the given conditions is $f(x)=\frac{50\cdot(x-2)\cdot(x+1)}{(x+5)\cdot(x-5)}$ |
c2a16bfe-a68f-49e2-a46b-b76210e6995a | differential_calc | false | null | 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 decreasi... | 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 ... |
c2c64c79-64e8-4c01-8d35-17ac590004cd | differential_calc | false | null | 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 inflectio... | 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: $P\left(\frac{2}{3},0\right)$, $P\left(1,\sqrt[3]{\frac{2}{3}}\right)$ |
c2d80e7a-26ff-46c4-a9ac-1d3f772e77ab | precalculus_review | false | null | Solve $\left(\sin(x)\right)^{10} + \left(\cos(x)\right)^{10} = \frac{ 29 }{ 16 } \cdot \left(\cos(2 \cdot x)\right)^4$. | The final answer: $x=\frac{\pi}{8}+\frac{\pi\cdot k}{4}$ |
c3062353-e56a-4397-bc22-1ea7e9e7fc72 | differential_calc | false | null | Where is the parabola $y = x^2$ closest to the point $(2,0)$? | The final answer: $P(0.8351,0.6974)$ |
c37c3f43-24ad-4906-a400-c14ace455f14 | multivariable_calculus | false | null | Use Lagrange multipliers to find the maximum volume of a rectangular box that can be inscribed in the ellipsoid $\frac{ x^2 }{ 9 } + \frac{ y^2 }{ 25 } + \frac{ z^2 }{ 4 } = 1$. | The final answer: $\frac{80}{\sqrt{3}}$ |
c38d8b60-f7e8-433d-ac1c-3d3adc457a9b | sequences_series | false | null | 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 $n \ge 2$ | $a_{1}$ = $1$ , $a_{2}$ = $3$ , $a_{3}$ = $4$ , $a_{4}$ = $5$ , $a_{5}$ = $6$ , $a_{6}$ = $7$ $a_{n}$ = $n+1$ for $n \ge 2$ |
c3b0d303-3a1c-431d-9b8e-84e4d9d8a53e | differential_calc | false | null | 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 $x^2 + x = a + b$. | The final answer: $g'(x)=(a+b)^2+c\cdot x-(a+b+c+d)$ |
c47c3a9b-3f0f-4752-b421-b9a8a197b1e9 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAcAAAADrCAIAAABxW9FlAACEKElEQVR4nO1dd5wV1fX/3ntn5r1tLAs2pINioaOioHSxxKjE8lOjYG+JscQkdqUZTSzRRI3GjsYkdqPGgoIUxUQ6dnpREGnLlvdm5t57fn+c92Zndykp7LrgfD/vs/vevHnT5zvnnvM95wgiQkPCWiul5PdEZK1VSllrhRBCCCIiIimlMYaIHMfh+YMg8DwPgO/7qVSKiIQQDbqdCRIkSPCfQjQ0gTLCMHRdV2vtOA5PiTgxYlV+D6AOVz... | A projectile is shot in the air from ground level with an initial velocity of $500$ m/sec at an angle of $60$ deg with the horizontal. At what time is the maximum range of the projectile attained? The graph is shown here: | $t$ = $88.37$ |
c487e6c3-04f0-44a1-9e8f-4512dccbeacf | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARMAAAD+CAIAAACWZnYHAADq7UlEQVR4nOy92Y9d15UfvM9w57q3qm7NE2dSHCSRFCVbA23Jsi2r1e0x7U7Qje5GggAZkIcg/0YegiAPAbqD7nQSJOnu2PEgdyxLsmTNE0VxJkUWySrWPN+57j3T9/DL+n3rnlsDLTXw4QN0HgpVp87ZZ++199p7jb9lRVFkvri+uL64fsvL/v+6A19cX1z/v7y+4Jwvri+uz3J9wTlfXF9cn+X6gnO+uL64Psv1Bed8cX1xfZbrC8754v... | The graph below is the derivative of a function, $f$, whose domain is the set of all real numbers and is continuous everywhere. Determine the x values for the relative extrema for $f$. | There is a local maximum at x-value(s): $x=-3$, $x=4$
There is a local minimum at x-value(s): $x=-1$ |
c57070ee-de68-4104-beda-984585448856 | differential_calc | false | null | 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 | 1. The point(s) where local maxima occur: $P(0,108)$
2. The point(s) where local minima occur: $P(-2,0)$ |
c57bcca2-8fe0-433f-98b9-bd31a9a66e49 | integral_calc | false | null | Solve the integral:
$$
\int \frac{ 4 }{ \cos(-3 \cdot x)^3 \cdot \sin(-3 \cdot x)^2 } \, dx
$$ | $\int \frac{ 4 }{ \cos(-3 \cdot x)^3 \cdot \sin(-3 \cdot x)^2 } \, dx$ = $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\cd... |
c600dd96-8ec6-4564-8cfc-0dcecb9d1084 | integral_calc | false | null | Solve the integral:
$$
2 \cdot \int \sin(-2 \cdot x)^5 \cdot \cos(2 \cdot x)^2 \, dx
$$ | $2 \cdot \int \sin(-2 \cdot x)^5 \cdot \cos(2 \cdot x)^2 \, dx$ = $C+\frac{1}{3}\cdot\left(\cos(2\cdot x)\right)^3+\frac{1}{7}\cdot\left(\cos(2\cdot x)\right)^7-\frac{2}{5}\cdot\left(\cos(2\cdot x)\right)^5$ |
c6544023-d8e0-45df-9798-832448bac8aa | sequences_series | false | null | 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 $(-\pi, \pi)$. | The Fourier series is: $\sum_{n=1}^\infty\left(\frac{\sin\left((2\cdot n-1)\cdot x\right)}{2\cdot n-1}\right)$ |
c65eeba7-8616-499c-97b1-31abc7253b2d | sequences_series | false | null | 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. | The final answer: $\sum_{k=1}^\infty\left((-1)^k\cdot\frac{(2\cdot k-1)\cdot x^{2\cdot k-2}}{(2\cdot k+1)!}\right)$ |
c6e74489-488e-4d9a-8ad5-45abc3c647e9 | sequences_series | false | null | 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. $f(x) = \frac{ x^2 }{ 5-4 \cdot x+x^2 }$; $a=2$. | 1. $f(x)$ = $\sum_{n=0}^\infty\left(x^2\cdot\left(-(x-2)^2\right)^n\right)$
2. $I$ = $(1,3)$ |
c724aa02-d543-4850-97d9-517c1f1eeb74 | differential_calc | false | null | 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. ... | 1. $-9.8490000$ m/sec.
2. $-9.7510000$ m/sec.
3. $-9.8049000$ m/sec.
4. $-9.7951000$ m/sec. |
c7fcefca-8951-42f5-99b4-83026cbb571f | algebra | false | null | Perform the indicated operation and express the result as a simplified complex number:
$i^{15}$ | The final answer: $-i$ |
c80d8185-edc4-4a44-9e58-dcab9d666178 | differential_calc | false | null | 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). | 1. Local Maximum(s) $P\left(\frac{3}{2},\frac{\pi}{4}\right)$;
2. Local Minimum(s) $P(1,0)$, $P(2,0)$. |
c890e92c-0ebd-42a5-a3d3-fce40464ed16 | precalculus_review | false | null | Calculate the derivative $\frac{ d }{d x}\left(\log_{x}(a)\right)$ for $x > 0$, $a > 0$, $x \ne 1$, $a \ne 1$. | $\frac{ d }{d x}\left(\log_{x}(a)\right)$ = $\frac{-\ln(a)}{x\cdot\left(\ln(x)\right)^2}$ |
c89d463c-5220-451b-80db-0200a72cbdcb | sequences_series | false | null | 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. | 1. The center of convergence is $\frac{1}{3}$.
2. The radius of convergence is $\frac{4}{3}$. |
c89f34ab-0533-4173-aba0-76f2b2800224 | algebra | false | null | Solve the following inequality: $-5 x^2 + 10 x + 15 \le 0$.
Express your answer in the interval form. | 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".* |
c8c08145-397a-4331-8934-956d0e39af2d | sequences_series | false | null | 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. | The final answer: $\left(1-\frac{3}{2}\right)+\left(1-\frac{3}{2}\cdot\frac{1}{2}\right)\cdot x+\left(1-\frac{3}{2}\cdot\frac{1}{2^2}\right)\cdot x^2+\left(1-\frac{3}{2}\cdot\frac{1}{2^3}\right)\cdot x^3+\cdots$ |
c8dc16e5-519c-4ebe-9432-d1f13b782008 | algebra | false | null | 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 $\text{None}$. | Inverse function: $g(x)=\frac{9}{5}\cdot y+32$ |
c9256ca7-b498-433f-975c-11aa311ca307 | precalculus_review | false | null | Use the Rational Zero Theorem to find all real zeros of the following polynomial:
$p(x) = 9 \cdot x^3 + 6 \cdot x^2 - 29 \cdot x - 10$ | The real zeros are $-2$, $-\frac{1}{3}$, $\frac{5}{3}$ |
c92615c8-ce81-46c3-8693-7b13461c9371 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXcAAAFACAIAAADI3au8AADTvElEQVR4nOyddVwU29vAZ7tg6e6UBhFQQBAQEERMLKyL3V67u7v12t1iICACgkiLIC0gKd3LLrG97x/nc+ed3y4s2Hjd718wO3P2zOzMM895EiUQCKDfBIFAgEKhfvUsfhKSk5XwnwH9qycgQYKE/zgSKSNBgoQfi0TKSJAg4ccikTISJEj4sUikjAQJEn4sEikjQYKEH4tEykiQIOHHIpEyEiRI+LFIpIwECRJ+LBIpI0GChB+LRMpIkC... | Find the surface area bounded by the curves $q = a \cdot \cos(\varphi)$, $p = b \cdot \cos(\varphi)$, $b > a > 0$. | $S$ = $\frac{\pi}{4}\cdot\left(b^2-a^2\right)$ |
c9286655-d7f7-45da-a36c-6736332eca0d | integral_calc | false | null | Compute the integral:
$$
3 \cdot \int \frac{ \cos(3 \cdot x)^4 }{ \sin(3 \cdot x)^3 } \, dx
$$ | $3 \cdot \int \frac{ \cos(3 \cdot x)^4 }{ \sin(3 \cdot x)^3 } \, dx$ = $C+\frac{3}{4}\cdot\ln\left(\frac{1+\cos(3\cdot x)}{1-\cos(3\cdot x)}\right)-\frac{\left(\cos(3\cdot x)\right)^3}{2-2\cdot\left(\cos(3\cdot x)\right)^2}-\frac{3}{2}\cdot\cos(3\cdot x)$ |
c93e2665-a4ed-40fb-85e2-4c28ad0dced4 | sequences_series | false | null | Evaluate $\sum_{n=2}^\infty\frac{ n \cdot (n-1) }{ 2^n }$ as $f''\left(\frac{ 1 }{ 2 }\right)$ where $f(x) = \sum_{n=0}^\infty x^n$. | $\sum_{n=2}^\infty\frac{ n \cdot (n-1) }{ 2^n }$ = $4$ |
c9dd4213-29b4-46b3-86dd-b34b06ed004a | precalculus_review | false | null | Find the zeros of $f(x) = (2 - x)^4 + (2 \cdot x - 1)^4 - (x + 1)^4$. | The final answer: $x_1=2 \land x_2=\frac{1}{2}$ |
c9e88bba-f8b2-49e6-8c20-2bb176663e89 | sequences_series | false | null | Using the Taylor formula, decompose the function $f(x) = \ln\left(1+\frac{ x }{ 5 }\right)$ in powers of the variable $x$ on the segment $[0,1]$. Use the first nine terms.
Then estimate the accuracy obtained by dropping an additional term after the first nine terms. | 1. $\ln\left(1+\frac{ x }{ 5 }\right)$ = $\frac{x}{5}-\frac{x^2}{5^2\cdot2}+\frac{x^3}{5^3\cdot3}-\frac{x^4}{5^4\cdot4}+\frac{x^5}{5^5\cdot5}-\frac{x^6}{5^6\cdot6}+\frac{x^7}{5^7\cdot7}-\frac{x^8}{5^8\cdot8}+\frac{x^9}{5^9\cdot9}$
2. Accuracy is not more than $\frac{1}{10\cdot5^{10}}$ |
c9fab8b1-59aa-4e6f-9428-ef2aa6679b28 | algebra | false | null | A local band sells out for their concert. They sell all 1175 tickets for a total purse of $28,112.5. The tickets were priced at $20 for student tickets, $22.5 for children, and $29 for adult tickets. If the band sold twice as many adult as children tickets, how many of each type was sold? | Student tickets: $500$ Children tickets: $225$
Adult tickets: $450$ |
ca5a60b8-627d-46bb-af40-542e361e629d | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQUAAAEGCAIAAADjYBdoAABKO0lEQVR4nO2dZ1wU19fH753ZvsDSBSkKClYQMBpL/rHEEqypJiaxoUFUihQBiYpSNIlKUUqCJXajxscSxRITUy3Yu0YsCEhnF5btO3OfF6NIaILO7CLO94Uf3J2558zs/Ob2cyBCCLCwsAAAAMCM7QALSxuC1QMLyzNYPbCwPIPVAwvLM1g9sLA8g2NsB55AkuSjR48ePXrUtWvXjh07Uh/m5t4tL6/o0KGDi4tL3YNlMhmEUCKRGMNTlv... | Use the following figure to find the indicated derivatives, if they exist (enter 'undefined' if the derivative doesn't exist):
Let $h(x) = f(x) + g(x)$. Find:
1. $h'(1)$
2. $h'(3)$
3. $h'(4)$ | 1. $h'(1)$ = $0$
2. $h'(3)$ = None
3. $h'(4)$ = $1$ |
ca5b0525-4e4c-4698-8367-146ba9ca18e3 | algebra | false | null | Luna is going bowling. Shoe rental costs $3 and the lane costs $2 per game. If Luna paid a total of $17, how many games did she bowl? | Luna bowled $7$ games. |
ca5ffe7c-f495-43dc-a653-de477cabc185 | sequences_series | false | null | Find the radius of convergence of the series:
$$
\sum_{n=1}^\infty \left(\frac{ \left((2 \cdot n)!\right) \cdot x^n }{ n^{2 \cdot n} }\right)
$$ | $R$ = $\frac{e^2}{4}$ |
ca6b1413-d76d-4992-89ed-aaad88dc94ef | multivariable_calculus | false | null | Evaluate $L=\lim_{P(x,y) \to P(m+n,m-n)}\left(\frac{ x^2-m \cdot x-x \cdot y+m \cdot y-2 \cdot n \cdot x+2 \cdot m \cdot n }{ x \cdot y-n \cdot x-y^2-n \cdot y+2 \cdot n^2 }\right)$, given $m-2 \cdot n=7 \cdot n$ | The final answer: $L=\frac{1}{7}$ |
cac53af9-3149-4ea9-b0c9-13b6a9d9a18f | algebra | false | null | Use the Rational Zero Theorem to find the real solution(s) for the equation: $8 \cdot x^4 + 26 \cdot x^3 + 39 \cdot x^2 + 26 \cdot x + 6 = 0$ | By the Rational Zero Theorem, possible solutions occur at values $-\frac{1}{2}$, $-\frac{3}{4}$, $1$, $-1$, $2$, $-2$, $3$, $-3$, $\frac{ 1 }{ 2 }$, $\frac{ 3 }{ 2 }$, $-\frac{ 3 }{ 2 }$, $\frac{ 1 }{ 4 }$, $-\frac{ 1 }{ 4 }$, $\frac{ 3 }{ 4 }$, $\frac{ 1 }{ 8 }$, $-\frac{ 1 }{ 8 }$, $\frac{ 3 }{ 8 }$, $-\frac{ 3 }{ 8 ... |
cb4f082e-28c9-4e3d-bc92-12d111f5c953 | sequences_series | false | null | Compute $\int_{0}^{\frac{ 1 }{ 5 }} e^{-2 \cdot x^2} \, dx$ with accuracy $0.00001$. | The final answer: $0.1948$ |
cb72c058-1dd3-47a6-bf34-25ab75f7a436 | integral_calc | false | null | Find the integral:
$$
\int \frac{ 4 \cdot x^2+25 \cdot x+7 }{ \sqrt{x^2+8 \cdot x} } \, dx
$$ | $\int \frac{ 4 \cdot x^2+25 \cdot x+7 }{ \sqrt{x^2+8 \cdot x} } \, dx$ = $(2\cdot x+1)\cdot\sqrt{x^2+8\cdot x}+3\cdot\ln\left(\left|x+4+\sqrt{x^2+8\cdot x}\right|\right)+C$ |
cbb4b10b-ecb3-497b-8a66-81c54a1d265a | precalculus_review | false | null | Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{\frac{ -\pi }{ 3 }}^{\frac{ \pi }{ 3 }} \sqrt{\left(\sec(x)\right)^2-1} \, dx
$$ | $\int_{\frac{ -\pi }{ 3 }}^{\frac{ \pi }{ 3 }} \sqrt{\left(\sec(x)\right)^2-1} \, dx$ = $0$ |
cc16099c-195d-4735-9d83-0b18f8f27a27 | sequences_series | false | null | Find a “reasonable” upper-bound on the error in approximating $f(x) = x^7$ by its 3rd order Taylor polynomial $P_{3}(x)$ about $a = -1$ valid for all values of $x$ such that $|x + 1| \leq 0.1$. | The final answer: $840\cdot(1.1)^3\cdot\frac{(0.1)^4}{4!}$ |
cc733108-05a4-4478-9f32-0eea85157535 | integral_calc | false | null | Compute the volume of the solid formed by rotating about the x-axis the area bounded by the axes and the parabola $x^{\frac{ 1 }{ 2 }}+y^{\frac{ 1 }{ 2 }}=5^{\frac{ 1 }{ 2 }}$. | Volume = $\pi\cdot\frac{25}{3}$ |
cc73f597-71ca-49bd-8bc4-62c5b0c48753 | differential_calc | false | null | Make full curve sketching of $y = \arcsin\left(\frac{ 1-3 \cdot x^2 }{ 1+3 \cdot x^2 }\right)$. 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 decreasi... | 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 ... |
cc752972-ff6b-4b03-8a40-722afdd8bb50 | multivariable_calculus | false | null | Determine the coordinates, if any, for which $f(x,y) = x^3 - 3 \cdot x \cdot y^2 + 6 \cdot y^2 - 8$ has
1. a Relative Minimum(s)
2. a Relative Maximum(s)
3. a Saddle Point(s)
If a Relative Minimum or Maximum, find the Minimum or Maximum value. If none, enter None. | 1. The function $f(x,y)$ has Relative Minimum(s) at None with the value(s) None
2. The function $f(x,y)$ has Relative Maximum(s) at None with the value(s) None
3. The function $f(x,y)$ has a Saddle Point(s) at $P(2,-2)$, $P(2,2)$ |
ccfa5962-71e5-4955-bc2d-c91098f2cd13 | integral_calc | false | null | Solve the integral:
$$
\int \frac{ 20 \cdot \cos(-10 \cdot x)^3 }{ 21 \cdot \sin(-10 \cdot x)^7 } \, dx
$$ | $\int \frac{ 20 \cdot \cos(-10 \cdot x)^3 }{ 21 \cdot \sin(-10 \cdot x)^7 } \, dx$ = $C+\frac{1}{21}\cdot\left(\frac{1}{2}\cdot\left(\cot(10\cdot x)\right)^4+\frac{1}{3}\cdot\left(\cot(10\cdot x)\right)^6\right)$ |
cd1538cc-ca04-4235-819e-b0d2b3542eb2 | algebra | false | null | 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 $\$32$, they... | The final answer: $31.8$ |
cdfb4e3a-81b3-4b5e-84f0-0e48d5898fe4 | sequences_series | false | null | Find the Taylor polynomial $P_{5}$ for the function $f(x) = x \cdot \cos\left(x^2\right)$. | $P_{5}$ = $x-\frac{1}{2}\cdot x^5$ |
ce47631d-a24b-4d1b-8f75-aeaf64f9da34 | integral_calc | false | null | Solve the integral:
$$
-\int \frac{ 1 }{ \cos(x)^3 \cdot \sin(x)^2 } \, dx
$$ | $-\int \frac{ 1 }{ \cos(x)^3 \cdot \sin(x)^2 } \, dx$ = $C+\frac{\sin(x)}{2\cdot\left(\left(\sin(x)\right)^2-1\right)}+\frac{3}{4}\cdot\ln\left(\left|\sin(x)-1\right|\right)+\frac{1}{\sin(x)}-\frac{3}{4}\cdot\ln\left(\left|1+\sin(x)\right|\right)$ |
ce5fb045-4688-4338-8225-f0a5c9b38ee2 | sequences_series | false | null | Find the Fourier series of the function $\psi(x) = e^{-2 \cdot x}$ in the interval $(-2 \cdot \pi, 2 \cdot \pi)$. | The Fourier series is: $e^{-2\cdot x}=\frac{\left(e^{4\cdot\pi}-e^{-4\cdot\pi}\right)}{\pi}\cdot\left(\frac{1}{8}+\sum_{n=1}^\infty\frac{(-1)^n\cdot\left(4\cdot\cos(n\cdot x)+n\cdot\sin(n\cdot x)\right)}{16+n^2}\right)$ |
ce91d092-7d61-4244-85a6-f8d040bb6264 | precalculus_review | false | null | Find zeros of $f(x) = \sqrt{2 \cdot x - 1} - \sqrt{x - 1} - 5$ | The final answer: $x=145$ |
ce97eb96-8d97-4889-97b8-deb2bb3023a4 | algebra | false | null | Write an expression for a rational function with the given characteristics:
1. vertical asymptotes $x=-6$ and $x=6$,
2. x-intercepts at $P(1,0)$ and $P(-1,0)$,
3. y-intercept at $P(0,2)$. | The rational function satisfying the given conditions is $f(x)=\frac{72\cdot(x-1)\cdot(x+1)}{(x+6)\cdot(x-6)}$ |
ceae34e5-3a64-44a0-95c9-a2e5f16fe8ae | integral_calc | false | null | Find the area of the figure enclosed between the curves $y = 4 \cdot x^2$, $y = \frac{ x^2 }{ 3 }$, and $y = 2$. | Area: $\frac{\left(8\cdot\sqrt{3}-4\right)\cdot\sqrt{2}}{3}$ |
cf789d84-24bd-45fa-89da-772ce496d1d0 | multivariable_calculus | false | null | Find the measure of the angle $\theta$ between the three-dimensional vectors $\vec{a}$ and $\vec{b}$, expressed in radians rounded to two decimal places, if it is not possible to express it exactly.
Given:
$\vec{a} = 3 \cdot \vec{i} - \vec{j} - 2 \cdot \vec{k}$
$\vec{b} = \vec{v} - \vec{w}$, where $\vec{v} = 2 \cdot \... | $\theta$ = $2.84$ |
cfeeda2e-b0df-47fc-89b0-3f8cb8938309 | integral_calc | false | null | Evaluate the integral:
$$
I = \int 3 \cdot x \cdot \ln\left(4 + \frac{ 1 }{ x } \right) \, dx
$$ | The final answer: $\left(\frac{3}{2}\cdot x^2\cdot\ln(4\cdot x+1)-\frac{3\cdot x^2}{4}+\frac{3\cdot x}{8}-\frac{3}{32}\cdot\ln\left(x+\frac{1}{4}\right)\right)-\left(\frac{3}{2}\cdot x^2\cdot\ln(x)-\left(C+\frac{3}{4}\cdot x^2\right)\right)$ |
cfef8ba8-ebc2-4ce0-a35f-c900e2bf8c99 | sequences_series | false | null | A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms.
Let $a_{n} = f(n) - 2 \cdot f(n+1) + f(n+2)$, in which $f(n)$ -> 0 as $n$ -> $\infty$. Find $\sum_{n=1}^\infty\left(a_{n}\right)$. | $\sum_{n=1}^\infty\left(a_{n}\right)$ = $f(1)-f(2)$ |
d0331d53-4efd-46ce-a9d3-d10570408bd7 | precalculus_review | false | null | Solve $z^2 - (2 \cdot i + 1) \cdot z - (3 - i) = 0$. | The final answer: $z_1=2+i \lor z_1=-1+i$ |
d0e2146e-63e1-4b1d-8f2c-6c0c62ab256f | multivariable_calculus | false | null | Calculate the double integral:
$$
\int\int_{R}{\left(x \cdot y \cdot \sqrt{x^2+y^2}\right) d A}
$$
where $R=\left\{(x,y)|0 \le x \le 1,0 \le y \le 2\right\}$. | The final answer: $\frac{25\cdot\sqrt{5}-33}{15}$ |
d1263456-584b-4e04-be1c-20da3101099e | sequences_series | false | null | Find the Fourier series of the function $\psi(x) = e^{-x}$ in the interval $(-\pi,\pi)$. | The Fourier series is: $e^{-x}=\frac{e^\pi-e^{-\pi}}{2\cdot\pi}\cdot\left(\frac{1}{2}+\sum_{n=1}^\infty\left(\frac{(-1)^n}{1+n^2}\cdot\left(\cos(n\cdot x)+n\cdot\sin(n\cdot x)\right)\right)\right)$ |
Subsets and Splits
Differential Calc Problems
Retrieves specific math problems related to differential calculus, providing basic filtering but limited analytical value beyond finding relevant entries.