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 |
|---|---|---|---|---|---|
15ae9179-a0f2-4848-9b6e-0a1f50f70491 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAaoAAAEbCAIAAAAEe1dxAAAKMWlDQ1BJQ0MgUHJvZmlsZQAAeJydlndUU9kWh8+9N71QkhCKlNBraFICSA29SJEuKjEJEErAkAAiNkRUcERRkaYIMijggKNDkbEiioUBUbHrBBlE1HFwFBuWSWStGd+8ee/Nm98f935rn73P3Wfvfda6AJD8gwXCTFgJgAyhWBTh58WIjYtnYAcBDPAAA2wA4HCzs0IW+EYCmQJ82IxsmRP4F726DiD5+yrTP4zBAP+flLlZIjEAUJiM5/... | Use the graph in figure, which shows the profit, $y$, in thousands of dollars, of a company in a given year, $t$, where $t$ represents the number of years since $1980$:
Find the slope. | Slope: $30$ |
15c84a6a-1fd0-4fb0-977a-daae876ccb9b | sequences_series | false | null | Find the Fourier series of the function $u = \left| \sin\left( \frac{ x }{ 2 } \right) \right|$ in the interval $[-2 \cdot \pi,2 \cdot \pi]$. | The Fourier series is: $\frac{2}{\pi}+\frac{4}{\pi}\cdot\sum_{k=1}^\infty\left(\frac{1}{\left(1-4\cdot k^2\right)}\cdot\cos(k\cdot x)\right)$ |
15dd94cf-dae0-43c2-837b-d48e826a391b | sequences_series | false | null | Find the expansion in the series of the integral $\int_{0}^x{\frac{ \arctan\left(x^2\right) }{ 4 \cdot x } d x}$, using the expansion of the integrand in the series. Find the radius of convergence of the series. | 1. $\int_{0}^x{\frac{ \arctan\left(x^2\right) }{ 4 \cdot x } d x}$ = $\sum_{n=1}^\infty\left((-1)^{n+1}\cdot\frac{x^{2\cdot(2\cdot n-1)}}{4\cdot2\cdot(2\cdot n-1)^2}\right)$
2. $R$ = $1$ |
15ddf157-e25a-4c30-aa8b-dcc9d5b8abb5 | precalculus_review | false | null | Calculate $E = \frac{ \cos(70) \cdot \cos(10) + \cos(80) \cdot \cos(20) }{ \cos(68) \cdot \cos(8) + \cos(82) \cdot \cos(22) }$. | The final answer: $E=1$ |
15e0826d-2594-436a-a05c-e9872b5222e4 | integral_calc | false | null | Solve the integral:
$$
\int \frac{ -9 \cdot \sqrt[3]{x} }{ 9 \cdot \sqrt[3]{x^2} + 3 \cdot \sqrt{x} } \, dx
$$ | $\int \frac{ -9 \cdot \sqrt[3]{x} }{ 9 \cdot \sqrt[3]{x^2} + 3 \cdot \sqrt{x} } \, dx$ = $-\left(C+\frac{1}{3}\cdot\sqrt[6]{x}^2+\frac{2}{27}\cdot\ln\left(\frac{1}{3}\cdot\left|1+3\cdot\sqrt[6]{x}\right|\right)+\frac{3}{2}\cdot\sqrt[6]{x}^4-\frac{2}{3}\cdot\sqrt[6]{x}^3-\frac{2}{9}\cdot\sqrt[6]{x}\right)$ |
16393e13-072d-439a-b975-4363298c4da4 | algebra | false | null | Find all real solutions of the following equation:
$$
\frac{ 2 }{ x+7 }-\frac{ 3 }{ x-3 }=1
$$ | The real solution(s) to the given equation are: $x=-3$, $x=-2$ |
16c8afab-e3d5-432e-8e76-b8d3674e86cd | multivariable_calculus | false | null | Determine a definite integral that represents the region common to $r=2$ and $r=4 \cdot \cos\left(\theta\right)$. | The final answer: $4\cdot\int_0^{\frac{\pi}{3}}1d\theta+16\cdot\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\cos\left(\theta\right)^2d\theta$ |
16e404c3-c9da-465c-a97d-0b7a6e1e5ef9 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ 6 }{ \sin(3 \cdot x)^6 } \, dx
$$ | $\int \frac{ 6 }{ \sin(3 \cdot x)^6 } \, dx$ = $-\frac{2\cdot\cos(3\cdot x)}{5\cdot\sin(3\cdot x)^5}+\frac{24}{5}\cdot\left(-\frac{\cos(3\cdot x)}{9\cdot\sin(3\cdot x)^3}-\frac{2}{9}\cdot\cot(3\cdot x)\right)+C$ |
171de471-6ade-4285-bf3c-033a5f936e58 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAATcAAAEjCAIAAADYB9HoAABBh0lEQVR4nO2deWBU1fn3n+fOPlkm+76QhCUkshMWEQQRLQpqFde6VKkWl1pbbK3a+qqgv5/1dbdatdW+ahVRXJFFEQmbLLITEpYkZN+Tyezrfd4/zp3JGFQCmWRmbs7HVpOZmzvn3jnf+5zznOc8DxIRcDicMEYIdQM4p+GnHqP+1/lzVvZwlYYjgcJDxB89wP/6jx7AkRPKUDeA8yMw4bndbqvVqlAoYmJiAEAUxe7ubqVSGRUV5VemKI... | List all inflection points and intervals where the function $f$ is concave up and concave down: | Inflection points at: $x=0$, $x=1$
Intervals where function concaves up: $(-\infty,0)\cup(1,\infty)$
Intervals where function concaves down: $(0,1)$ |
177da421-b0f6-4ac2-9fe5-ca0ad9019fcc | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWgAAAD3CAIAAAC/wFmgAABmbklEQVR4nO39d3Qc55nniz9vhc6NDmjknEEEggQIkmAmRYqkoiWKljzSWOOxPbJlz97x7G/n7jk/756z9869e+7Z9ZzRXHm0lizbcpBlRZISJVKMIAFGMIEAkRORiAw0Ood67h9vVaOZuwmAhMj3Mx6K7K6ueru66lvP+6SXICLcEUQkhNA/77wlg8F4RODuugXVixtU465yw2AwHmLuLhwhmFgwGAxKFMJBJyy3excRmbIwGI8IkQrH/v... | Evaluate the integral of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the $x$-axis: | The final answer: $5\cdot\pi-4$ |
17da1e7d-97b0-4d23-b930-90c988aec14d | multivariable_calculus | false | null | Find the curvature for the vector function: $\vec{r}(t) = \left\langle 2 \cdot \sin(t), -4 \cdot t, 2 \cdot \cos(t) \right\rangle$. | The final answer: $\frac{1}{10}$ |
17e5455e-6eec-45ee-8ff2-6ed4855fe878 | algebra | false | null | Solve the inequality $|-6 x-6| \le 1$.
Express your answer using interval notation. | Solution using interval notation: $\left[-\frac{ 7 }{ 6 },\ -\frac{ 5 }{ 6 }\right]$
*Note: enter an interval or union of intervals. If there is no solution, leave empty or enter "none"* |
181d4ee1-d2bd-43d2-b77e-2d5c94992059 | sequences_series | false | null | Integrate the approximation $\sin(t) \approx t - \frac{ t^3 }{ 6 } + \frac{ t^5 }{ 120 } - \frac{ t^7 }{ 5040 } + \cdots$ evaluated at $\pi \cdot t$ to approximate $\int_{0}^1{\frac{ \sin(\pi \cdot t) }{ \pi \cdot t } d t}$ | $\int_{0}^1{\frac{ \sin(\pi \cdot t) }{ \pi \cdot t } d t}$ ≈ $0.58678687$ |
182fed94-4634-460f-a70e-7958a2033664 | sequences_series | false | null | Find the Fourier integral of the function $q(x) = \begin{cases} 0, & x < 0 \\ 2 \cdot \pi \cdot x, & 0 \le x \le 1 \\ 0, & x > 1 \end{cases}$ | $q(x) = $\int_0^\infty\left(\frac{2\cdot\left(\alpha\cdot\sin\left(\alpha\right)+\cos\left(\alpha\right)-1\right)\cdot\cos\left(\alpha\cdot x\right)+2\cdot\left(\sin\left(\alpha\right)-\alpha\cdot\cos\left(\alpha\right)\right)\cdot\sin\left(\alpha\cdot x\right)}{\alpha^2}\right)d\alpha$ |
18b1ee59-f5ad-4c3b-b9b7-fd2b634b16fd | differential_calc | false | null | Find the second derivative $\frac{d ^2y}{ d x^2}$ of the function $x = \left(2 \cdot \sin(3 \cdot t)\right)^2$, $y = \sin(2 \cdot t)$. | $\frac{d ^2y}{ d x^2}$ = $\frac{\left(288\cdot\left(\sin(3\cdot t)\right)^2-144\right)\cdot\cos(2\cdot t)-96\cdot\cos(3\cdot t)\cdot\sin(2\cdot t)\cdot\sin(3\cdot t)}{13824\cdot\left(\cos(3\cdot t)\right)^3\cdot\left(\sin(3\cdot t)\right)^3}$ |
18fc468f-42f0-4606-b4f3-92f1f552c344 | integral_calc | false | null | Calculate the integral:
$$
I = \int 4 \cdot \cos\left(3 \cdot \ln(2 \cdot x)\right) \, dx
$$ | The final answer: $\frac{1}{10}\cdot\left(C+4\cdot x\cdot\cos\left(3\cdot\ln(2\cdot x)\right)+12\cdot x\cdot\sin\left(3\cdot\ln(2\cdot x)\right)\right)$ |
1924526f-31e1-40a9-8f23-7a27b272596c | sequences_series | false | null | Find the sum of the series $\sum_{n=1}^\infty \frac{ 7 }{ 2 \cdot n \cdot 3^n }$. (Use the formula $\int_{3}^\infty \frac{ 7 }{ 2 \cdot x^{n+1} } \, dx = \frac{ 7 }{ 2 \cdot n \cdot 3^n }$) | The sum of the series is $-\frac{7\cdot\ln\left(\frac{2}{3}\right)}{2}$ |
193fcb07-99e8-4fd6-8954-283c505ec24e | precalculus_review | false | null | A vehicle with a 20-gallon tank gets 15 mpg. The number of miles $N$ that can be driven depends on the amount of gas $x$ in the tank.
1. Write a formula that models this situation.
2. Determine the number of miles the vehicle can travel on
* a full tank of gas and
* $\frac{ 3 }{ 4 }$ of a tank of gas.
3. Deter... | 1. The formula for the function is $N(x)$ = $15\cdot x$
2. The number of miles the vehicle can travel on
* full tank of gas is $300$
* $\frac{ 3 }{ 4 }$ of a tank of gas is $225$
3. Domain and range (in interval notation)
* domain is $[0,20]$
* range is $[0,300]$
4. Number of times that the driver had t... |
19877db1-15f2-43a7-962d-f116bd6032a8 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYoAAAGYCAIAAADTAjLXAABmPklEQVR4nO3deVwbZ54n/uepkqpK98F93xh8xNxgjI0dG2ywEzvO5SSOO91JOt3T0zOz/dt+7c4er9fO9O5rXrvbu6+e7Z7MdLo7nXR3EscnxveFARvfYC4DvrDNfQiBBAiQVPX8/ihJVgDjAwECvu8/ElsWUkmUPnrqqW99H0wIQQAA4H2oud4AAACYHMQTAMBLQTwBALwUxBMAwEtBPAEAvBTEEwDAS0E8AQC8FMQTAMBLQTwBALwUxB... | Use the following graph and find: $\lim_{x \to 2^{+}}\left(f(x)\right)$ | $\lim_{x \to 2^{+}}\left(f(x)\right)$: $2$ |
19e6fecd-10a0-4497-b9dd-72c176820215 | sequences_series | false | null | Find the radius of convergence of the series:
$$
\sum_{n=1}^\infty \left( \frac{ \prod_{i=1}^n(2 \cdot n) }{ (2 \cdot n)! } \cdot x^n \right)
$$ | $R$ = $\infty$ |
1a9033ba-a210-4168-b474-3d3b70ee389d | multivariable_calculus | false | null | Consider points $A(1,1)$, $B(2,-7)$, and $C(6,3)$.
1. Determine vectors $\vec{BA}$ and $\vec{BC}$. Express the answer in component form.
2. Determine the measure of angle $B$ in triangle $ABC$. Express the answer in radians, rounded to two decimal places. | 1. $\vec{BA}$ = $\left\langle-1,8\right\rangle$ , $\vec{BC}$ = $\left\langle4,10\right\rangle$
2. $B$ = $0.50$ |
1af0e1dd-7848-4c91-a8bd-ef00777094a3 | algebra | false | null | An epidemiological study of the spread of a certain influenza strain that hit a small school population found that the total number of students, $P$, who contracted the flu $t$ days after it broke out is given by the model $P=-t^2+13 \cdot t+130$, where $1 \le t \le 6$. Find the day that $160$ students had the flu. Rec... | The final answer: $3$ |
1af5b10f-762e-4cf3-a07d-e97b59ef24ed | multivariable_calculus | false | null | Evaluate $\int\int\int_{E}{(2 \cdot x+5 \cdot y+7 \cdot z) d V}$, where $E$ is the region defined by:
$$
E = \left\{(x,y,z) | 0 \le x \le 1, 0 \le y \le -x+1, 1 \le z \le 2\right\}
$$ | $I$ = $\frac{77}{12}$ |
1b031f17-e90a-462e-aee1-ff2751ce3e06 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUYAAADzCAIAAAB1xSjVAADcQUlEQVR4nOy9Z3hVVfY/fntCQAa7QnJbbm4KIXQEpQYQEHXUGRVHRKQLiBBCSEIqCSQkgYQiUlT4YkNUxIaKKIioiNITSM/NTQLYqCn3nrb/Lz7PXv8NMzrqzGj0x37BE849ZZ999uprfZaeMaa7Mq6MK+PPMgy/9wSujCvjyvhvjiskfWVcGX+qcYWkr4wr4081rpD0lXFl/KnGFZK+Mq6MP9W4QtJXxpXxpxpXSPrKuDL+VOMKSV8ZV8... | On a cylinder with a diameter of 6 cm, a channel is cut out along the surface, having an equilateral triangle with a side of 1 cm in cross section. Compute the volume of the cut out material. | $V$ = $\frac{6\cdot\sqrt{3}-1}{4}\cdot\pi$ |
1b15a0e5-a050-4df5-b065-654906cb02e0 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAArUAAADmCAIAAABER+pmAACEPklEQVR4nO29ebykRXXwf2p51t7uOhsMi4BARKNEVFxAI4oYxeVNopj4JqIiBmP8EYNLYqJGUeOSSDSaaDSaKGCMS5LXiEmIiooYIiDBBWQbmPVuvT17VZ3fH6f7mZ47MzAMc29331vfD5+Hvt1979RT56mqU6fOwtAUAACMAwAARwADe0EAAGAD73CwHEmotw/Wq/QpDrwjVrY5lgfBPPhX9sGOlyML229Kojf2HykGgAMwK4LhsUwoJL... | Given $f(t) = g(t) \cdot h(t)$, find $f'(1)$ using the table below: | $f'(1)$ = $-18$ |
1b5e0373-2602-49cd-8ef8-a49f3828292d | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAaoAAAEbCAIAAAAEe1dxAAAKMWlDQ1BJQ0MgUHJvZmlsZQAAeJydlndUU9kWh8+9N71QkhCKlNBraFICSA29SJEuKjEJEErAkAAiNkRUcERRkaYIMijggKNDkbEiioUBUbHrBBlE1HFwFBuWSWStGd+8ee/Nm98f935rn73P3Wfvfda6AJD8gwXCTFgJgAyhWBTh58WIjYtnYAcBDPAAA2wA4HCzs0IW+EYCmQJ82IxsmRP4F726DiD5+yrTP4zBAP+flLlZIjEAUJiM5/... | Use the graph in the figure, which shows the profit, $y$, in thousands of dollars, of a company in a given year, $t$, where $t$ represents the number of years since 1980:
Find the $t$-intercept (also known as the $x$-intercept). | $t$-intercept: $P(10,0)$ |
1bd5a8d0-b561-403f-b059-e34ad635cec0 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUAAAADLCAIAAACQ8t1SAABC10lEQVR4nO29eXgc1ZX/fc6t7ta+7/suWbtkecG72TcTh4Q1CWExLw8QCCHbZJLMEDJDwiSQ/GYI6zNMSNgChAQCtsHBxsYrtmVbtrXv+9rau9WSuu55/7hVrbZkyZbckizrfp4H0+qurrpdVd+65557zrlIRCCRSBYmbL4bIJFIZo4UsESygJEClkgWMFLAEskCRgpYIlnAGFyyl//93/8lok2bNkVERACA3W7fs2dPbW1tSEjI5s2bHZ... | The graph of $y=\int_{0}^x{f(t) d t}$, where $f$ is a piecewise constant function, is shown here:
1. Determine:
1. Over which intervals is $f$ positive?
2. Over which intervals is $f$ negative?
3. Over which intervals, if any, is $f$ equal to zero?
2. What are the maximum values of $f$?
3. What are the ... | 1. Intervals when function:
1. is positive: $(1,2)$, $(5,6)$
2. is negative: $(0,1)$, $(3,4)$
3. equal to zero: $(2,3)$, $(4,5)$
2. The maximum value of $f$: $2$
3. The minimum value of $f$: $-3$
4. The average value of $f$: $0$ |
1bfa31b8-3fee-48fc-94d6-22cb63f39329 | multivariable_calculus | false | null | Let $Q$ be the solid situated outside the sphere $x^2+y^2+z^2=z$ and inside the upper hemisphere $x^2+y^2+z^2=R^2$, where $R>1$. If the density of the solid is $\rho(x,y,z) = \frac{ 1 }{ \sqrt{x^2+y^2+z^2} }$, find $R$ such that the mass of the solid is $\frac{ 7 \cdot \pi }{ 2 }$. | $R$ = $\frac{\sqrt{138}}{6}$ |
1c5ea0f7-a6c7-4a84-a172-9fca7a4b4b37 | differential_calc | false | null | Find the minimum value of $y = \frac{ \left(\cos(x)\right)^2 - 4 \cdot \cos(x) + 5 }{ 3 - 2 \cdot \cos(x) }$. | The final answer: $\frac{1+\sqrt{5}}{2}$ |
1ccc052c-9604-4459-a752-98ebdf3e0764 | sequences_series | false | null | Compute the first 5 nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of $f(x) = e^{\sin(x)}$. | $f(x)$ = $1+x+\frac{x^2}{2}-\frac{x^4}{8}-\frac{x^5}{15}+\cdots$ |
1ce0ac9d-1bd3-4e51-8bd1-90c41444fb8e | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAa4AAAGpCAIAAABakkV1AABhf0lEQVR4nO39d1RbaZ4n/j/PvVcR5UCONrbJGZtgjMEB52yXq6a7enp6enpnZue7e3bPmT/3390z58zZ3+5Od3Wscjl2OSfAAedIBoNxwAGDySCBQFm6z++PK2QZHABLgODz+sOFVVi6kq7eesLnPg8mhCAAAFjYqNk+AAAAmH0QhQAAAFEIAAAQhQAAgCAKAQAAQRQCAACCKAQAAARRCAAACKIQAAAQRCEAACCIQgAAQBCFAACAIAoBAA... | Use the following graph and find: $\lim_{x \to 2}\left(f(x)\right)$ | $\lim_{x \to 2}\left(f(x)\right)$: $1$ |
1d6724cb-e2de-4789-bf24-cab32e9438d1 | differential_calc | false | null | Compute the limits:
1. $\lim_{x \to \infty}\left(x \cdot \left(\sqrt{x^2+1}-x\right)\right)$
2. $\lim_{x \to -\infty}\left(x \cdot \left(\sqrt{x^2+1}-x\right)\right)$ | 1. $\frac{1}{2}$
2. $-\infty$ |
1db212f0-2fac-410d-969d-fe3b5b55d076 | integral_calc | false | null | Solve the integral:
$$
\int \frac{ 3 }{ \sin(2 \cdot x)^7 \cdot \cos(-2 \cdot x) } \, dx
$$ | $\int \frac{ 3 }{ \sin(2 \cdot x)^7 \cdot \cos(-2 \cdot x) } \, dx$ = $C+\frac{3}{2}\cdot\left(\ln\left(\left|\tan(2\cdot x)\right|\right)-\frac{3}{2\cdot\left(\tan(2\cdot x)\right)^2}-\frac{3}{4\cdot\left(\tan(2\cdot x)\right)^4}-\frac{1}{6\cdot\left(\tan(2\cdot x)\right)^6}\right)$ |
1db350b9-5d62-4e83-9751-fccda80f11cc | differential_calc | false | null | Find the extrema of a function $y = \frac{ 3 \cdot x^4 }{ 4 } - \frac{ 4 \cdot x^3 }{ 3 } - \frac{ 3 \cdot x^2 }{ 2 } + 2$. Then determine the largest and smallest value of the function when $-2 \le x \le 4$. | 1. Extrema points: $P\left(\frac{4-2\cdot\sqrt{13}}{6},1.8363\right)$, $P(0,2)$, $P\left(\frac{4+2\cdot\sqrt{13}}{6},-2.7931\right)$
2. The largest value: $\frac{254}{3}$
3. The smallest value: $-2.7931$ |
1e317fe8-3370-43d7-a9fe-efa7691e872b | sequences_series | false | null | Compute $\lim_{x \to 0}\left(\frac{ 2 \cdot \cos(x)+4 }{ 3 \cdot x^3 \cdot \sin(x) }-\frac{ 6 }{ 3 \cdot x^4 }\right)$. Use the expansion of the function in the Taylor series. | The final answer: $\frac{1}{90}$ |
1e67594e-d4d6-420e-8e63-caa6f3d285fa | algebra | false | null | Rewrite the quadratic expression $x^2 - 9 \cdot x - 22$ by completing the square. | $x^2 - 9 \cdot x - 22$ = $(x-4.5)^2-42.25$ |
1e6b9832-43f4-4214-bfa2-5884ac2d279f | precalculus_review | false | null | Find the degree, $y$-intercept, and zeros for the polynomial function $f(x) = x^3 + 2 \cdot x^2 - 2 \cdot x$.
1. Degree
2. $y$-intercept
3. Zeros | 1. Degree: $3$
2. $y$-intercept: $0$, $\sqrt{3}-1$, $-1-\sqrt{3}$
3. Zeros: $0$, $\sqrt{3}-1$, $-1-\sqrt{3}$ |
1e99cf2f-e67e-45fd-9560-c7d532daf02a | sequences_series | false | null | Find the Fourier series of the periodic function $f(x) = \frac{ x^2 }{ 3 }$ in the interval $-4 \cdot \pi \le x < 4 \cdot \pi$ if $f(x) = f(x + 8 \cdot \pi)$. | The Fourier series is: $\frac{16\cdot\pi^2}{9}+\sum_{n=1}^\infty\left(\frac{64\cdot(-1)^n}{3\cdot n^2}\cdot\cos\left(\frac{n\cdot x}{4}\right)\right)$ |
1ea61b4b-45e2-493c-8d9a-14357d27796d | algebra | false | null | Rewrite the quadratic function $h(x) = 2 \cdot x^2 + 12 \cdot x - 4$ in standard form and give the vertex.
### Forms of quadratic functions
A quadratic function is a polynomial function of degree two. The graph of a quadratic function is a parabola.
The general form of a quadratic function is $f(x) = a \cdot x^2 + b... | The final answer: The standard form of the quadratic function: $2\cdot(x+3)^2-22$ Vertex $(h,k)$ = $P(-3,-22)$ |
1ecd4b7b-49ad-4fad-848b-95a689622f1a | multivariable_calculus | false | null | Evaluate $L=\lim_{P(x,y) \to P(a,2 \cdot b)}\left(\frac{ a^2 \cdot x \cdot y-2 \cdot a \cdot b \cdot x^2+a \cdot b \cdot y^2-2 \cdot b^2 \cdot x \cdot y }{ a^2 \cdot x \cdot y-2 \cdot a \cdot b \cdot x^2-a \cdot b \cdot y^2+2 \cdot b^2 \cdot x \cdot y }\right)$ | The final answer: $L=\frac{a^2+2\cdot b^2}{a^2-2\cdot b^2}$ |
1f898401-6ced-46c3-963d-b2312f6050a7 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAhgAAAIVCAIAAAAceBGZAACs8UlEQVR4nOydZ0ATWdeALyV0EKSqFAsqYMdewAaKgAWUVVHsrr27rmLBhu7a17bWFQHLqoCIitgRG3ZsgCigoEiXDgGS78d93/vdNwkxQJJJwnl+3bkzmTkMyZw59zQlLpeLAAAAAKCuKDMtAAAAACDfgCIBAAAA6gUoEgAAAKBegCIBAAAA6gUoEgAAAKBegCIBAAAA6gUoEgAAAKBegCIBAAAA6gUoEgAAAKBegCIBAAAA6gUoEgAAAK... | Find the surface area of the cylinder $x^2 + y^2 = x$, which is contained within the sphere $x^2 + y^2 + z^2 = 1$. | The final answer: $4$ |
1fb06482-1ecc-444f-b93d-73a9a2e17ebe | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAtsAAADdCAIAAADl1CnLAACf5UlEQVR4nO2ddXwT2ff3b5K6u7dIcSlWCou77uKyy7KLuy6wsLg7i9vi0AJFi0sFaQt16t7UvUnaxm0yzx+zvzz5liZNcwKhy7z/4FWa6aefzmTu3Jx77jkUHMcRCQkJCQkJCYlOoeraAAkJCQkJCQkJOSMhISEhISEh+QYgZyQkJCQkJCQkuoeckZCQkJCQkJDoHnJGQkJCQkJCQqJ7yBkJCQkJCQkJie4hZyQkJCQkJCQkuoeckZCQkJ... | Using the given graphs of $f(x)$ and $g(x)$, find $g\left(f(3)\right)$. | The final answer:
$g\left(f(3)\right)$ = $0$ |
1fd375be-b4ea-4035-b84a-c91779dafe26 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ x-\sqrt[3]{4 \cdot x^2}-\sqrt[6]{2 \cdot x} }{ x \cdot \left(4+\sqrt[3]{2 \cdot x}\right) } \, dx
$$ | $\int \frac{ x-\sqrt[3]{4 \cdot x^2}-\sqrt[6]{2 \cdot x} }{ x \cdot \left(4+\sqrt[3]{2 \cdot x}\right) } \, dx$ = $C+36\cdot\ln\left(\left|4+\sqrt[3]{2}\cdot\sqrt[3]{x}\right|\right)+\frac{\left(3\cdot2^{\frac{2}{3}}\right)}{4}\cdot x^{\frac{2}{3}}-3\cdot\arctan\left(\frac{1}{2^{\frac{5}{6}}}\cdot\sqrt[6]{x}\right)-9\c... |
2086d032-660b-4d65-bdc3-fe27a82e8ee2 | multivariable_calculus | false | null | A person is standing 8 feet from the nearest wall in a whispering gallery. If that person is at one focus and the other focus is 80 feet away, what is the length and the height at the center of the gallery? | Length is $96$ feet and height is approximately $26.53$ feet. |
20e1df0a-f00d-4545-8ad0-f44adad1d83e | sequences_series | false | null | Find the radius of convergence $R$ and interval of convergence for the power series $\sum_{n=0}^\infty a_{n} \cdot x^n$:
$$
\sum_{n=1}^\infty (-1)^n \cdot \frac{ x^n }{ \sqrt{n} }
$$ | * $R$ = $1$
* $I$ = $(-1,1]$ |
21207f35-779e-4510-91cd-89f0afe29a82 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAACr2UlEQVR4nOzdd3hU1d7F8TXpJJDQQ4DQIQQw9F5FEcXey71W7PVeee1iFxVFLKCAIAKCAipFEARFihSRDtKlt9AJIaTn/cM7Q4ZzAgGS2VO+n+e5j5NfQrL06kzWnLP3duTl5eUJAIBC6tevn/r16ydJ6tWrl3r16mU4EVB8unfvrtWrV7vNxowZoy5dupgJBPiBINMBAAAAvNXGjRsts8TERANJAP9BAQEAALCxfv16ZWRkuM1iYmIUGx... | The graphs of $y=4-x^2$ and $y=3^x$ are shown in the figure above. Find the combined area of the shaded regions. | $8.013$ |
213c8058-a2a8-43e2-b8f6-91371fe234b2 | algebra | false | null | Using the scoring system in the game of Jeopardy (you earn points for correct responses and lose points for incorrect responses), what would be the final score after the following 5 responses:
1. Player 1 answers a 200-point question correct, a 50-point question wrong, a 250-point question correct, a 50-point question... | 1. The final score for Player 1 is $500$
2. The final score for Player 2 is $-250$
3. The final score for Player 3 is $350$ |
217dbc0c-03b5-4b38-b4df-60f8ce6951d3 | multivariable_calculus | false | null | Determine a definite integral that represents the region common to $r = 3 \cdot \cos\left(\theta\right)$ and $r = 3 \cdot \sin\left(\theta\right)$. | The final answer: $9\cdot\int_0^{\frac{\pi}{4}}\sin\left(\theta\right)^2d\theta$ |
21b8ca26-51d9-4663-b145-8e74540fe9c2 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAT0AAAEXCAIAAABQzgSaAABecUlEQVR4nO29Z3Qc15nn/dzKnTNyDgQIghAoZlGkKEqmRYXxyJZkz0qeWR+PJ7zvbDhn1/Pufnm/7AfPmbN7dtbHZ+zj1zMjj+yRPB4HSSQlkVSimINIghkEkXOj0bm7urqq7vvhVhWayAAD0I362QKB7gq3wv+G5z73eRDGGExMTPIKaqULYGJismRM3ZqY5B+mbk1M8g9TtyYm+Yep27wh14K4eGsi2dK0PhYYpm7zAKI6hJDxCUJoQR... | Determine where the local and absolute maxima and minima occur on the graph given. Assume domains are closed intervals unless otherwise specified. | Absolute minimum at: $-2$, $2$
Absolute maximum at: $-2.5$, $2.5$
Local minimum at: $0$
Local maximum at: $-1$, $1$ |
2210a30a-0583-4c62-8efb-026cf9df28be | algebra | false | null | Solve the following equations:
1. $4 a - 15 = -23$
2. $44 = 5 x - 6$
3. $-3.7 x + 5.6 = -5.87$
4. $\frac{ x }{ 5 } + 3.5 = 4.16$
5. $-\frac{ x }{ 4 } + (-9.8) = -5.6$
6. $3.7 x - 2.2 = 16.3$
7. $\frac{ x }{ 6 } - 3.7 = 11$ | The solutions to the given equations are:
1. $a=-2$
2. $x=10$
3. $x=\frac{ 31 }{ 10 }$
4. $x=\frac{ 33 }{ 10 }$
5. $x=-\frac{ 84 }{ 5 }$
6. $x=5$
7. $x=\frac{ 441 }{ 5 }$ |
221b4d54-e3dd-4490-b838-5b045216ed65 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAAC3lklEQVR4nOzdd3wcd50//tfMbN/Vrnq1LLnITe5xie00J6SHJISSQODoRznuOLjjx30pB8dxx/EF7uA4ctxBaIFQAkkI6Q3iFNtx7022ZUu2JKuvtu/OzO8Pf2fY0ciJ7WhntLOv5+Ohh5W3HO3745n5zLxnPvP5CKqqqiAiIiIiIrKAaHcCRERERERUOliAEBERERGRZViAEBERERGRZViAEBERERGRZViAEBERERGRZViAEBERERGRZV... | The graph of $f'(x)$, the derivative of $f$, is shown. For what values of $x$ on the closed interval $[-3,3]$ does $f(x)$ have a horizontal tangent line? | The function $f$ has horizontal tangent lines at $x$ = $-3$, $-1$, $2$ |
222eb5cb-a7b1-4579-9f0a-2955d79998d4 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAW8AAAEJCAYAAABbkaZTAAAYWGlDQ1BJQ0MgUHJvZmlsZQAAeJyVeQVUVF8X77nTMzB0N9LdLd3dKSpDNzg0KiIgEioiIRKigAiCYFIiICGKSEkoKqKAgKJiACIg7xL6/b//t9Z76521zr2/2WefHaf22XcAYI8nhYYGImgACAoOJ9sY6vA4ObvwYKcABNAAB/iAHMkjLFTbysoMwOXP+7/L8ijMDZdnEluy/rf9/1poPb3CPACArGDs7hnmEQTjOwCgkjxCyeEAYJRhOl... | Find the coordinates of point $P$ and determine its distance to the origin. | $P$ : $P(2,3,1)$
$d$ = $\sqrt{14}$ |
22637bf3-6a19-4afe-bf21-7635c9389518 | differential_calc | false | null | Evaluate $\lim_{x \to \frac{ \pi }{ 2 }}\left(\frac{ \tan(3 \cdot x) }{ \tan(5 \cdot x) }\right)$ using L'Hopital's Rule. | $\lim_{x \to \frac{ \pi }{ 2 }}\left(\frac{ \tan(3 \cdot x) }{ \tan(5 \cdot x) }\right)$ = $\frac{5}{3}$ |
2282fbd1-3761-499d-91d7-97a38c7d2371 | sequences_series | false | null | Let $f(x) = \ln\left(x^2+1\right)$.
1. Find the 4th order Taylor polynomial $P_{4}(x)$ of $f(x)$ about $a=0$.
2. Compute $\left|f(1)-P_{4}(1)\right|$.
3. Compute $\left|f(0.1)-P_{4}(0.1)\right|$. | 1. $x^2-\frac{x^4}{2}$
2. $0.1931$
3. $0.0003\cdot10^{-3}$ |
22af6a45-4daa-492f-af67-2ddea807e859 | differential_calc | false | null | Differentiate $y = b \cdot \tan\left(\sqrt{1-x}\right)$. | The final answer: $\frac{dy}{dx}=\frac{b\left(\sec\left(\sqrt{1-x}\right)\right)^2}{2\cdot\sqrt{1-x}}$ |
22cc1731-e35c-4492-b6dc-7cbe9bc1a9a7 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAADghklEQVR4nOzdd1hT99sG8DtMWQoOFBW3KA7csw7q3ntvW7fiVtCq4F61Wq2j7t1aW22t/twD91bce+JCRWTJCOT9wzeniScoSMg34/5c13u95zwJyW1/Ss6T8x0KlUqlQgoGDRqEv//+W6vm5uaGixcvwt7ePqUfIz2YN28e5s2bBwAYNWoURo0aJTgREREREVH6WX3uwbFjx8LKSvsp7969w6pVqzI0FBERERERmafPNiAFChRA69atZf... | Let $g$ and $h$ be the functions given by $g(x) = \frac{ 1 }{ 5 } + \sin(\pi \cdot x)$ and $h(x) = 5^{-x}$. Let $T$ be the shaded region in the first quadrant bounded by the $y$-axis and the graphs of $g$ and $h$, and let $M$ be the shaded region in the first quadrant enclosed by the graphs of $g$ and $h$, as shown in ... | The total area is $0.481$ units². |
23010800-a63a-4bf5-85f6-314fa321b6f8 | precalculus_review | false | null | Use the double-angle formulas to evaluate the integral:
$$
\int \sin(x)^2 \cdot \cos(x)^2 \, dx
$$ | $\int \sin(x)^2 \cdot \cos(x)^2 \, dx$ = $\frac{1}{8}\cdot x-\frac{1}{32}\cdot\sin(4\cdot x)+C$ |
23883ab4-3177-4e59-9298-2e6f1717d1fb | differential_calc | false | null | Compute the derivative of the implicit function $e^{\frac{ y }{ 2 }} = x^{\frac{ x }{ 3 } - \frac{ y }{ 4 }}$. | $y'$: $y'=\frac{4\cdot x\cdot\left(\ln(x)+1\right)-3\cdot y}{3\cdot x\cdot\left(\ln(x)+2\right)}$ |
23ae23ae-f790-409a-bbdd-48cfeb156dc4 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUQAAADDCAIAAAB1Sv9FAAAODklEQVR4nO3dPXKrPBsGYPHNWQpMxoUXIO/AHhepXKaE1hQp3yplCmjtMqWrFB7YQViAC09G7EVfgfkJAfNjgeDJfTXnWMYJT6TbgIgjQ0rJAGD+/qd7BwBAjX/JP4Zh6N0PAOgtOb/+V3o8d4Zh0CikB9Suey/0yI7EOM0GIAJhBiACYQYgAmEGIAJhBiACYQYgAmEGIAJhBiACYQYgAmFWKfb9MG6/edhtc5iqifQ7wqxM6Djn7X5ttn... | Complete the following table for the function $f(x)= rac{ x^2-1 }{ |x-1| }$. Round your solutions to four decimal places.
The table is given. | 1. $-1.9000$
2. $-1.9900$
3. $-1.9990$
4. $-1.9999$
5. $2.1000$
6. $2.0100$
7. $2.0010$
8. $2.0001$ |
23f5fc3d-68f0-42c8-82a4-e91a293a9ecc | sequences_series | false | null | Solve the following series via differentiation of the geometric series:
1. $\sum_{n=1}^\infty\left(\frac{ n }{ 6^{n-1} }\right)$
2. $\sum_{n=2}^\infty\left(\frac{ n }{ 3^{n-1} }\right)$
3. $\sum_{n=3}^\infty\left(\frac{ n }{ 2^{n+1} }\right)$
4. $\sum_{n=2}^\infty\left(\frac{ n+1 }{ 5^n }\right)$
5. $\sum_{n=2}^\infty... | 1. $\frac{36}{25}$
2. $\frac{5}{4}$
3. $\frac{1}{2}$
4. $\frac{13}{80}$
5. $\frac{8}{27}$
6. $\frac{53}{108}$ |
2416152a-a967-4e43-9ad3-c412f0ee1d0a | multivariable_calculus | false | null | Determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbit of the comet or planet. Distance is given in astronomical units (AU):
1. Halley’s Comet: length of major axis = $35.88$, eccentricity = $0.967$. | 1. $r$ = $\frac{1.16450334}{1+0.967\cdot\cos(t)}$ |
24830b9a-f570-4094-9ca3-77216d3081a1 | precalculus_review | false | null | Find the zeros of the function $f(x) = x^3 - (3 + \sqrt{3}) \cdot x + 3$. | The final answer: $x_1=\sqrt{3}$, $x_2=\frac{-\sqrt{3}-\sqrt{3+4\cdot\sqrt{3}}}{2}$, $x_3=\frac{-\sqrt{3}+\sqrt{3+4\cdot\sqrt{3}}}{2}$ |
24ca0d32-0edc-4da0-8284-8a9ff5367ce8 | multivariable_calculus | false | null | Evaluate the triple integral $\int_{0}^2 \int_{4}^6 \int_{3 \cdot z}^{3 \cdot z+2} (5-4 \cdot y) \, dx \, dz \, dy$ by using the transformation $u=x-3 \cdot z$, $v=4 \cdot y$, and $w=z$. | $I$ = $8$ |
24f973aa-2278-4395-a6d8-04f362b5b698 | precalculus_review | false | null | Solve the trigonometric equation $6 \cdot \cos(x)^2 - 3 = 0$ on the interval $[-2 \cdot \pi, 2 \cdot \pi]$ exactly. | $x$ = $\frac{\pi}{4}$, $-\frac{\pi}{4}$, $\frac{3}{4}\cdot\pi$, $-\frac{3}{4}\cdot\pi$, $\frac{5}{4}\cdot\pi$, $-\frac{5}{4}\cdot\pi$, $\frac{7}{4}\cdot\pi$, $-\frac{7}{4}\cdot\pi$ |
2516247b-6387-402e-ac1c-88774804cefc | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAoAAAADQCAIAAAAYvkeqAACwLElEQVR4nOx9eZwsV1X/95x7q6q7Z3l79n0HI7KFhCVIWATZ90CIgoiAQFRQQNSfIIoBUQQVxAVQXBBZlS0KsilLhACyBcgjC0le3kte3jJLd1fVvef8/rhV1dU9My9vmZmuecw379OZ6enl1K17z75Q7vtExDCqCgCAEgMgr2yMwqsqERGRVxARFa9ax/JACapKnohI4Jk53AglAZiUATAAqJIA5W+NgRIANGdXlPRI9TMrFFBVJRg14U... | Given $s(t) = \left(h(t)\right)^2$, find $s'(0)$ using the table below: | $s'(0)$ = $18$ |
2522fa02-210e-4fa6-a658-73f26d3df1a2 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAACWc0lEQVR4nOzdeVxU9f7H8ffAsC+K+1ZohUruGqFpamVlaZqWmktW5k3LNBO7/rTsVqZWhnZTS7Mys+zmklmalZWZe+4r7ooiijuyr/P7g8tcxgEFhDkz+Ho+Hjwew5kzZz4zDHDe57uZLBaLRQAkSZGRkYqMjJQkRUREKCIiwuCKAAAAyhY3owsAAAAAcOMggAAAAABwGAIIAAAAAIchgAAAAABwGAIIAAAAAIchgAAAAABwGAIIAAAAAI... | Let $R$ be the region in the first quadrant bounded by the graph of $y = 3 \cdot \arctan(x)$ and the lines $x = \pi$ and $y = 1$, as shown in the figure above.
Find the volume of the solid generated when $R$ is revolved about the line $x = \pi$. | The volume of the solid is $36.736$ units³. |
2590af47-b424-44fb-9ea3-cd88319511c1 | integral_calc | false | null | Calculate $I=\int_{\pi}^{\frac{ 5 }{ 4 } \cdot \pi}{\frac{ \sin(2 \cdot x) }{ \left(\cos(x)\right)^4+\left(\sin(x)\right)^4 } \, dx}$ | The final answer: $I=\frac{\pi}{4}$ |
268ca3a8-6509-4de9-8ad8-2c035609fae6 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAckAAAEzCAYAAABaNvgOAAA9yUlEQVR4nO3de1xUdf4/8NfAwAAOaCCQiICSgJqaiHkplbykZiZa4dptM3Vds8uiaVv7LbF+W5mKbWVt2n7dLl8p87pp611TK7yheUUNBFTkoqPCCMzAML8/iMOMMDADc86Zy+v5ePR4nDNzZt7v04fPvD2fc87nIC8vzyi25cuXix6jNftRVllmHLD4QeG/gpuFosSxFtvE8eKwTRwvDtvEPnEKbhaa/f6VVZbZPYY9Sd0myoKCAohNo9... | Find a formula for $q(x)$, the quadratic polynomial whose graph is shown below: | The final answer: $q(x)=\frac{1}{2}\cdot(x-6)\cdot(x-10)$ |
2690fe65-791c-427f-909a-f4de8a01874c | multivariable_calculus | false | null | The period $T$ of a simple pendulum with small oscillations is calculated from the formula $T = 2 \cdot \pi \cdot \sqrt{\frac{ L }{ g }}$, where $L$ is the length of the pendulum and $g$ is the acceleration resulting from gravity. Suppose that $L$ and $g$ have errors of, at most, $0.5\%$ and $0.1\%$ respectively. Use d... | The maximum percentage error is: $0.003$ |
26f9dc9c-b9b4-4a40-9d75-4433f718752a | algebra | false | null | Solve the quadratic equation by completing the square. Show each step. (Give your answer either exactly or rounded to two decimal places).
$2 \cdot x^2 - 8 \cdot x - 5 = 0$ | $x$ = $\sqrt{\frac{13}{2}}+2$, $-\sqrt{\frac{13}{2}}+2$ |
272ed29b-1de2-48d4-a63d-c8c7173b76ef | differential_calc | false | null | Compute the derivative of the function $f(x) = \sqrt[x]{3} + \frac{ 1 }{ 6^{5 \cdot x} } + 4^{\sqrt{x}}$ at $x = 1$. | $f'(1)$ = $-0.5244$ |
273e613f-b3e1-4221-8dfb-a1364757f4ae | differential_calc | false | null | For the curve $x = a \cdot \left(t - \sin(t)\right)$, $y = a \cdot \left(1 - \cos(t)\right)$, determine the curvature. Use $a = 4$. | The curvature is: $\frac{1}{16\cdot\left|\sin\left(\frac{t}{2}\right)\right|}$ |
2745c073-ef92-4748-9387-783f4937bf15 | sequences_series | false | null | Determine the Taylor series for $y = \left(\cos(2 \cdot x)\right)^2$, centered at $x_{0} = 0$. Write out the sum of the first three non-zero terms, followed by dots. | The final answer: $1-\frac{2^3}{2!}\cdot x^2+\frac{2^7}{4!}\cdot x^4+\cdots$ |
2756b56e-7460-417d-8887-1069f48cab9e | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYoAAAGYCAIAAADTAjLXAABmPklEQVR4nO3deVwbZ54n/uepkqpK98F93xh8xNxgjI0dG2ywEzvO5SSOO91JOt3T0zOz/dt+7c4er9fO9O5rXrvbu6+e7Z7MdLo7nXR3EscnxveFARvfYC4DvrDNfQiBBAiQVPX8/ihJVgDjAwECvu8/ElsWUkmUPnrqqW99H0wIQQAA4H2oud4AAACYHMQTAMBLQTwBALwUxBMAwEtBPAEAvBTEEwDAS0E8AQC8FMQTAMBLQTwBALwUxB... | Use the following graph and find: $\lim_{x \to -2^{+}}\left(f(x)\right)$ | $\lim_{x \to -2^{+}}\left(f(x)\right)$: $2$ |
275f7ceb-f331-4a3f-96ec-346e6d81b32a | integral_calc | false | null | Solve the integral:
$$
\int \frac{ 1 }{ \sin(8 \cdot x)^5 } \, dx
$$ | The final answer: $C+\frac{1}{128}\cdot\left(2\cdot\left(\tan(4\cdot x)\right)^2+6\cdot\ln\left(\left|\tan(4\cdot x)\right|\right)+\frac{1}{4}\cdot\left(\tan(4\cdot x)\right)^4-\frac{2}{\left(\tan(4\cdot x)\right)^2}-\frac{1}{4\cdot\left(\tan(4\cdot x)\right)^4}\right)$ |
27bf3a57-31a0-4228-99e0-097019b8eb40 | precalculus_review | false | null | Find the domain of the function $g(x) = \ln(2 \cdot x - 3) + \sqrt{1 - x}$. Give your answer using interval notation. | The final answer: $\varnothing$ |
28349b87-6fef-4032-a4fa-3293f8f8f962 | algebra | false | null | Rewrite the quadratic expression $2 \cdot x^2 - 10 \cdot x - 15$ by completing the square. | $2 \cdot x^2 - 10 \cdot x - 15$ = $2\cdot(x-2.5)^2-27.5$ |
28aee58e-6261-4432-b15d-ef0d23395d05 | integral_calc | false | null | Leaves of deciduous trees fall in an exponential rate during autumn. A large maple has 10,000 leaves and the wind starts blowing. If there are 8,000 leaves left after 3 hours, how many will be left after 4 hours? | There will be $7426$ leaves after 4 hours. |
29315747-d98d-4c5c-8399-22390eec8c48 | algebra | false | null | The fox population in a certain region has an annual growth rate of 9 percent. In the year 2012, there were 23 900 fox counted in the area. Create a function representing the population.
What is the fox population to be in the year 2020? Round your answer to the nearest whole number. | The final answer: $47622$ |
296b39cb-0c02-4c19-80e9-77e35ea51b02 | precalculus_review | false | null | Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{0}^\pi \left(\sin(3 \cdot x) \cdot \sin(5 \cdot x)\right) \, dx
$$ | $\int_{0}^\pi \left(\sin(3 \cdot x) \cdot \sin(5 \cdot x)\right) \, dx$ = $0$ |
29933b1c-f96c-4df4-9f53-d9b8900e1597 | sequences_series | false | null | The $k$ th term of the given series has a factor $x^k$. Find the range of $x$ for which the ratio test implies that the series converges:
$$\sum_{k=1}^\infty \left(\frac{ x^k }{ k^2 }\right)$$ | $|x|$ < $1$ |
2a797d22-d902-4bf9-afea-9ff2d5690b7c | differential_calc | false | null | 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. ... | 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 asy... |
2aafde85-809f-4cc5-a0e7-ef098bfc2e4c | algebra | false | null | Find the degree and leading coefficient for the given polynomial $f(x) = x^3 \cdot (4 \cdot x - 3)^2$. | The degree of the polynomial: $5$
The leading coefficient of the polynomial: $16$ |
2ac1f00d-e481-43d0-8af8-3a40003255ab | differential_calc | false | null | Compute the limit using L'Hopital's Rule:
$$
\lim_{x \to 0} \left( \frac{ 1 }{ x^2 } - \cot(x)^2 \right)
$$ | The final answer: $\frac{2}{3}$ |
2ac945ef-fd61-419c-8564-32e46fef157c | differential_calc | false | null | Evaluate $\lim_{x \to 0}\left(\left(\frac{ \sin(2 \cdot x) }{ 2 \cdot x }\right)^{\frac{ 1 }{ 4 \cdot x^2 }}\right)$. | $\lim_{x \to 0}\left(\left(\frac{ \sin(2 \cdot x) }{ 2 \cdot x }\right)^{\frac{ 1 }{ 4 \cdot x^2 }}\right)$ = $e^{-\frac{1}{6}}$ |
2aed35f2-99fd-4a0a-ba04-4f6f9c2b3096 | sequences_series | false | null | Using the fact that:
$$
e^x = \sum_{n=0}^\infty \left(\frac{ x^n }{ n! }\right)
$$
Evaluate:
$$
\sum_{n=0}^\infty \left(\frac{ 3 \cdot n }{ n! } \cdot 7^{1-3 \cdot n}\right)
$$ | The final answer: $\frac{3}{49}\cdot e^{\frac{1}{343}}$ |
2aee6b67-afa7-427f-90a3-daffc1044c8f | integral_calc | false | null | Compute the integral:
$$
\int \sin(2 \cdot x)^6 \cdot \cos(2 \cdot x)^2 \, dx
$$ | Answer is: $\frac{1}{32}\cdot x-\frac{1}{32}\cdot\frac{1}{8}\cdot\sin(8\cdot x)-\frac{1}{8}\cdot\frac{1}{4}\cdot\frac{1}{3}\cdot\sin(4\cdot x)^3+\frac{1}{128}\cdot x-\frac{1}{128}\cdot\frac{1}{16}\cdot\sin(16\cdot x)+C$ |
2b05093d-b9a0-447b-87af-c2ffe13e010a | precalculus_review | false | null | Simplify the following expression:
$$
E = 2 \cdot \left( \left( \sin(x) \right)^6 + \left( \cos(x) \right)^6 \right) - 3 \cdot \left( \left( \sin(x) \right)^4 + \left( \cos(x) \right)^4 \right)
$$ | The final answer: $E=-1$ |
2b0b5339-ba8d-460a-991d-c0a4b8452626 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWYAAAEnCAYAAACaIunfAAAYU2lDQ1BJQ0MgUHJvZmlsZQAAeJyVeQVUVF8X77mTzDAM3d0l3SAxdHeDwNAdQ4NKioSKIKCUCioIIliEiIUgooigAgYiYVAqqKAIyLuEfv/v/6313npn1rn3N/vss+PsU3sGAM5UcmRkKIIOgLDwGIqtkS6fs4srH/YdgAAO/ggBEbJPdCTJ2tocwOXP+7/L8jDMDZdnUpuy/rf9/1roff2ifQCArGHs7RvtEwbjawCgMn0iKTEAYFRhum... | Using the graph of $f(x)$ given below, find $f^{-1}(3)$. | $f^{-1}(3)$ = $0$ |
2b575acf-e40a-4b0a-9ad7-099c38d6206c | precalculus_review | false | null | 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. | Coordinates of $a^x$: $x$ = $-\frac{\ln\left(\ln(a)\right)}{\ln(a)}$ , $y$ = $y=\frac{1}{\ln(a)}$
Coordinates of $\log_{a}(x)$: $x$ = $x=\frac{1}{\ln(a)}$ , $y$ = $-\frac{\ln\left(\ln(a)\right)}{\ln(a)}$ |
2b811576-774b-43ed-bed4-15f5b2cd794b | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAa4AAAGpCAIAAABakkV1AABhf0lEQVR4nO39d1RbaZ4n/j/PvVcR5UCONrbJGZtgjMEB52yXq6a7enp6enpnZue7e3bPmT/3390z58zZ3+5Od3Wscjl2OSfAAedIBoNxwAGDySCBQFm6z++PK2QZHABLgODz+sOFVVi6kq7eesLnPg8mhCAAAFjYqNk+AAAAmH0QhQAAAFEIAAAQhQAAgCAKAQAAQRQCAACCKAQAAARRCAAACKIQAAAQRCEAACCIQgAAQBCFAACAIAoBAA... | Use the following graph and find: $\lim_{x \to 1^{-}}\left(f(x)\right)$ | $\lim_{x \to 1^{-}}\left(f(x)\right)$: $1$ |
2b8e2958-1ef8-4d3f-adfb-a05baa324f8a | differential_calc | false | null | Find the first derivative of the function: $y = (x + 11)^5 \cdot (3 \cdot x - 7)^4 \cdot (x - 12) \cdot (x + 4)$. | $y'$ = $\left(\frac{5}{x+11}+\frac{12}{3\cdot x-7}+\frac{1}{x-12}+\frac{1}{x+4}\right)\cdot(x+11)^5\cdot(3\cdot x-7)^4\cdot(x-12)\cdot(x+4)$ |
2c34f270-20d2-4a0f-ba88-2883a71b7e30 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAqwAAADdCAIAAADxQbcqAACJMUlEQVR4nO29d5wlR3Xof05Vdbp50kbtSkJZIkhCPGSMCCYHE2TAD2wD5tnEB+bZ/ATYZGPEA9s88sNgJEwwGYPB8CSiEIgoRA6SFsUNE2/sWFXn98e5t/furCTSztww/R19Wnd6emaru7qqTp2IZDPQGpQErUEpAIAsAccBAgAAsJBDAKhIOAYKjhkSwAIQAD9VAhD8H0GaxL7vakuZlUKBBgAAB0CMsr1bnby/MgAB0OpljbKDg17DQV... | Given $h(x) = f(x) \cdot g(x)$, find $h'(2)$ using the table below: | $h'(2)$ = $-29$ |
2ce8f8fe-faae-4c62-9c73-692fbc4aeeb9 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABEHElEQVR4nO3dfZTV9X0n8M/oKIPAbYyS+MBEj7E8zJjUrjkhQ6TRKIJLxrMeEZKjjSmVqMd1I8HaNKmyYFy1eyaQxGNrYXVNNBTUPA2NULApVUYdE7tRByWuTSZjQzU+zgiMyMP+wd6pPNw7A8x8772/+3qdw5E733uHz5ffcP297/epZteuXbsiQ1pbWyMiorm5ucSVDK0rrrgiIiLuvPPOElcytKrlema9n1nvX17W+3nCCSfs8fi3v/... | Let $f$ be a differentiable function defined onthe closed interval $-2 \le x \le 3$ with $f(2) = -4$.
The graph of $f'(x)$, the derivative of $f$, is shown.
Write an equation for the line tangentto the graph of $f$ at $x=2$. | The equation of the tangent line is: $y+4=-(x-2)$ |
2cec29f7-4a99-4553-9211-8f39de5520b5 | multivariable_calculus | false | null | 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
$\vec{r}(t) = t^2 \cdot \vec{i} + 3 \cdot t \cdot \vec{j} + \left(1-t^3\right) \cdot \vec{k}$. | $f_{u}(P)$ = $\frac{11}{\sqrt{22}}$ |
2d167fb0-1096-474b-96fe-75a2a95ddc78 | algebra | false | null | Find the solution to the following inequality and express it in interval notation:
$$-4 (x-5) (x+4) (x-1) \le 0$$ | The solution set to the inequality is: $\left[-4,\ 1\right]\cup\left[5,\ \infty\right)$ |
2d16ea33-5993-4872-a76a-a21c70e522b5 | integral_calc | false | null | Compute the integral:
$$
\int \cos\left(\frac{ x }{ 2 }\right)^4 \, dx
$$ | $\int \cos\left(\frac{ x }{ 2 }\right)^4 \, dx$ = $\frac{\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)^3}{2}+\frac{3}{4}\cdot\left(\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)+\frac{x}{2}\right)+C$ |
2d1a9f91-c6a7-458f-b4b7-f764d6826606 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYoAAAGYCAIAAADTAjLXAABmPklEQVR4nO3deVwbZ54n/uepkqpK98F93xh8xNxgjI0dG2ywEzvO5SSOO91JOt3T0zOz/dt+7c4er9fO9O5rXrvbu6+e7Z7MdLo7nXR3EscnxveFARvfYC4DvrDNfQiBBAiQVPX8/ihJVgDjAwECvu8/ElsWUkmUPnrqqW99H0wIQQAA4H2oud4AAACYHMQTAMBLQTwBALwUxBMAwEtBPAEAvBTEEwDAS0E8AQC8FMQTAMBLQTwBALwUxB... | Use the following graph and find: $\lim_{x \to -2^{-}}\left(f(x)\right)$ | $\lim_{x \to -2^{-}}\left(f(x)\right)$: $0$ |
2d2c80a0-257a-411f-ade4-12e1906ab93b | algebra | false | null | Find the inverse function of $f(x) = \frac{ 2 \cdot x + 3 }{ 5 \cdot x + 4 }$. | $f^{-1}(x)$ = $\frac{3-4\cdot x}{5\cdot x-2}$ |
2d3c3b27-060d-45bc-8f0f-3ff0be69b15b | precalculus_review | false | null | Find zeros of $f(x) = \sqrt{(x+3)^2} + \sqrt{(x-2)^2} + \sqrt{(2 \cdot x-8)^2} - 9$ | The final answer: $x\in[2,4]$ |
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Differential Calc Problems
Retrieves specific math problems related to differential calculus, providing basic filtering but limited analytical value beyond finding relevant entries.