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 |
|---|---|---|---|---|---|
42ca2f4a-f9f4-4a24-9edf-76a32286a185 | algebra | false | null | Solve the following inequality: $|-12 x-6| \ge 2$.
Express your answer in the interval form. | Solution in the interval form: $\left(-\infty,\ -\frac{ 2 }{ 3 }\right] \cup \left[-\frac{ 1 }{ 3 },\ \infty\right)$
*Note: enter an interval or union of intervals. If there is no solution, leave empty or enter "none".* |
42e272d8-b9fa-4658-96c0-9c6d0121341d | differential_calc | false | null | Analyze the curve
$$
y = \frac{ 2 \cdot x^3 }{ x^2 - 4 }
$$
and submit as your final answer 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 wh... | 1. The domain (in interval notation): $(-\infty,-2)\cup(-2,2)\cup(2,\infty)$
2. Vertical asymptotes (leave blank if none): $x=-2$, $x=2$
3. Horizontal asymptotes (leave blank if none): None
4. Slant asymptotes (leave blank if none): $y=2\cdot x$
5. Intervals where the function is increasing (leave blank if none): $\lef... |
4301fe9b-6f1c-4157-a503-c9a4a63aac9b | differential_calc | false | null | Sketch the curve:
$$
y = \frac{ x^3 }{ 3 \cdot (x+2)^2 }
$$
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 co... | 1. The domain (in interval notation): $(-1\cdot\infty,-2)\cup(-2,\infty)$
2. Vertical asymptotes: $x=-2$
3. Horizontal asymptotes: None
4. Slant asymptotes: $y=\frac{x}{3}-\frac{4}{3}$
5. Intervals where the function is increasing: $(-\infty,-6)$, $(0,\infty)$, $(-2,0)$
6. Intervals where the function is decreasing: $(... |
4311c676-bbf4-4e98-8e11-adf7b06738e8 | precalculus_review | false | null | Solve the inequality:
$2 \le |x+5| \le 4$ | Write your final answer as intervals: $[-9,-7]\cup[-3,-1]$ |
432ae8b3-c454-430c-b444-f92da3c288d9 | sequences_series | false | null | Find the Taylor polynomial $P_{4}$ for the function $f(x) = \sec(x)$. | $P_{4}$ = $1+\frac{1}{2}\cdot x^2+\frac{5}{24}\cdot x^4$ |
436d359e-a6d2-4758-a930-95a641e3d787 | multivariable_calculus | false | null | Determine the equation of the ellipse using the information given: Endpoints of major axis at $(0,2)$, $(0,-2)$ and foci located at $(3,0)$, $(-3,0)$. | The equation is: $\frac{x^2}{13}+\frac{y^2}{4}=1$ |
43aab6e3-8669-4193-993c-885a518d1101 | differential_calc | false | null | Find the equation(s) of the tangent line(s) to the curve $y = 3 \cdot x^4 - 9 \cdot x^2 + 5$, that also passes through the point $\left(0,\frac{ 29 }{ 4 }\right)$. | Tangent Line(s): $4\cdot y-24\cdot\sqrt{2}\cdot x-29=0$, $4\cdot y+24\cdot\sqrt{2}\cdot x-29=0$ |
43c2c82e-a54b-40cb-8b95-07cf15365044 | precalculus_review | false | null | If a culture of bacteria doubles in $3$ hours, how many hours does it take to multiply by $10$? Use $y=y_{0} \cdot e^{k \cdot t}$, where $y_{0}$ is the beginning amount, $y$ is the ending amount, $k$ is the growth or decay rate, and $t$ is time. Leave your answer in exact form. | It takes $\frac{ 3 \cdot \ln(10) }{ \ln(2) }$ hours. |
43ce18b9-35be-43a7-b18f-bff62d27b1d5 | differential_calc | false | null | Sketch the curve:
$y = \sqrt{\frac{ 64-x^3 }{ 3 \cdot x }}$.
Submit as your final answer:
1. The domain (in interval notation)
2. Vertical asymptotes
3. Horizontal asymptotes
4. Slant asymptotes
5. Intervals where the function is increasing
6. Intervals where the function is decreasing
7. Intervals where the funct... | 1. The domain (in interval notation): $(0,4]$
2. Vertical asymptotes: $x=0$
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: None
6. Intervals where the function is decreasing: $(0,4]$
7. Intervals where the function is concave up: $\left(0,2\cdot\sqrt[3]{2}\right)... |
43d0031e-c751-4a2e-89b6-68e8cbc45eee | sequences_series | false | null | Find $\lim_{x \to 0}\frac{ 2 \cdot \left(\tan(x)-\sin(x)\right)-x^3 }{ x^5 }$, using the expansion of the function in the Maclaurin series. | $\lim_{x \to 0}\frac{ 2 \cdot \left(\tan(x)-\sin(x)\right)-x^3 }{ x^5 }$ = $\frac{1}{4}$ |
43f3644e-64db-4bac-bd39-fa50622f3e9c | sequences_series | false | null | Find the 5th order Taylor polynomial $P_{5}$ (in powers of $x$) for the function $f(x) = e^x \cdot \sin(x)$. | $P_{5}(x)$ = $x+x^2+\frac{1}{3}\cdot x^3-\frac{1}{30}\cdot x^5$ |
44151f67-d055-41b9-a33e-677471a36551 | differential_calc | false | null | For the function $f(x) = -\frac{ 3 }{ 2 } \cdot x^4 + 2 \cdot x^3 + 3 \cdot x^2 - 6 \cdot x + \frac{ 1 }{ 2 }$, specify the points where local maxima and minima of $f(x)$ 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: $P(-1,6)$
2. The point(s) where local minima occur: None |
4464c80e-9661-4b58-b9a1-744d565b7ea2 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXAAAAEKCAYAAAALoA6YAAAYWGlDQ1BJQ0MgUHJvZmlsZQAAeJyVeQVUVF8X77nTMzB0N9LdLd3dKSpDNzg0KiIgEioiIRKigAiCYFIiICGKSEkoKqKAgKJiACIg7xL6/b//t9Z76521zr2/2WefHaf22XcAYI8nhYYGImgACAoOJ9sY6vA4ObvwYKcABNAAB/iAHMkjLFTbysoMwOXP+7/L8ijMDZdnEluy/rf9/1poPb3CPACArGDs7hnmEQTjOwCgkjxCyeEAYJRhOl... | The values of the function $f$ on rectangle $R$ $[0,2] \times [7,9]$ are given in the table. Estimate the double integral $\int \int f(x,y) \, dx \, dy$ by using a Riemann sum with $m=2$, $n=2$. Select the sample points to be the upper right corners of the subsquares in $R$. | $\int \int f(x,y) \, dx \, dy$ is approximately: $35.42$ |
4495f45d-c9a9-468c-87c3-cb3701fe37a1 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABdqklEQVR4nO3deZzO9f7/8ec1MxczxowtpIhyhDZ1JloQRva1bHUi0ckhkqXO0XpKddKixbFEKYkiJI19mxGiMqcTZYqUOcYyiIzdNcvvj37X9e3yuRRm5vO+rs/ncb/dzu3w+lzj83z7TJfrOZ/rc308BQUFBbJBenq6JCkpKcmO3RkzadIkSVK/fv0MJ7GHW46rW9bpx3qdafTo0Ro9erQkafjw4Ro+fLjhRMXLLcfVLev0c8t63bJOP7... | Use the graph of $f(x)$ shown below to find $\lim_{x \to -2}f(x)$, if it exists. | $\lim_{x \to -2}f(x)$ is None |
453b6199-ab94-487a-b46d-44885715e8be | multivariable_calculus | false | null | Apply the gradient descent algorithm to the function $g(x,y) = \left(x^2-1\right) \cdot \left(x^2-3 \cdot x+1\right)+y^2$ with step size $\frac{ 1 }{ 5 }$ and initial guess $p_{0} = \left\langle -1,0 \right\rangle$ for three steps (so steps $p_{1}$, $p_{2}$, and $p_{3}$). | | $i$ | $1$ | $2$ | $3$ |
| --- | --- | --- | --- |
| $p_{i}$ | $\left\langle1,0\right\rangle$ | $\left\langle\frac{7}{5},0\right\rangle$ | $\left\langle\frac{1333}{625},0\right\rangle$ |
| $g\left(p_{i}\right)$ | $0$ | $-\frac{744}{625}$ | $-\frac{460\ 046\ 957\ 304}{152\ 587\ 890\ 625}$ | |
45844f47-7807-4d44-903a-819bbfd75778 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARsAAADPCAIAAABKl+L7AABFx0lEQVR4nO29d3ic1ZX4f859p6iM2ozaqIx6s+TeC9imGggtYanB8A37ywLLEgzsLpBAyLJPluoEAmTJw8ISIFm6HQKuyNiy5SJX2ZLVR713aTQjzbz3/P54Z8ZjSbaR/I5GI7+fB2Rpynvve9977j33nHPPBVKYOAcOHLjtttsEQQAPFi5ceODAgYu/eEFBgXTBp59+2uFwXPwFFaYSJCJQUFCQCebrCigozCgUiVJQkBNFohQU5ESRKA... | 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 value of $f$?
3. What are the mi... | 1. Intervals when function:
1. is positive: $(0,1)$, $(3,4)$
2. is negative: $(1,3)$, $(5,6)$
3. equal to zero: $(4,5)$
2. The maximum value of $f$: $3$
3. The minimum value of $f$: $-2$
4. The average value of $f$: $\frac{1}{3}$ |
45a1d7b5-f624-4a83-b813-aa80b29cd088 | algebra | false | null | Construct a polynomial $p(x)$ of least degree possible such that its zeros are $-2$, $\frac{ 1 }{ 2 }$ (each with multiplicity $2$), and $p(-3) = 5$. | $p(x)$ = $\frac{20}{49}\cdot(x+2)^2\cdot\left(x-\frac{1}{2}\right)^2$ |
45dc5791-4737-45d6-b1b2-81e807134f2f | sequences_series | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAW0AAAFTCAIAAAB9E8qoAACcXUlEQVR4nOydd5xdZbX+n7Xefc7MZGpmJo0EEtIIHdKIooKQ0JSWhKJguXp/1qu0BNR7r4jSJL3RFCkhHVFRUQnpAaSIqAhRQKQFSCghycxp+13P74/3zGEMiddIJpnJ7C+fTzizzz777FP2Outd7RGSaDNIighJkqoatgAIG0UkbNnmXgBmpqqtd0tISGi3aJseXUTMTETCjdLG0l3BTKgqybBD+Nc5B8B736anl5CQsEuQ3eCPlP40Mz... | Find the total length of the dashed path in the following figure: | $L$ = $\frac{3}{2}$ |
4608d6c8-00c7-42e7-ad2a-9b3ef61eece1 | differential_calc | false | null | Find $y'$ and $y''$ for $x^2 + 6 \cdot x \cdot y - 2 \cdot y^2 = 3$. | $y'$ = $\frac{x+3\cdot y}{2\cdot y-3\cdot x}$
$y''$ = $\frac{11\cdot\left(x^2+6\cdot x\cdot y-2\cdot y^2\right)}{(3\cdot x-2\cdot y)^3}$ |
46efe3d6-df69-43e8-a52b-d85c524aab17 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ -1 }{ \sqrt[3]{\tan\left(\frac{ x }{ 2 }\right)} } \, dx
$$ | $\int \frac{ -1 }{ \sqrt[3]{\tan\left(\frac{ x }{ 2 }\right)} } \, dx$ = $C+\frac{1}{2}\cdot\ln\left(\left|1+\sqrt[3]{\tan\left(\frac{1}{2}\cdot x\right)}\cdot\tan\left(\frac{1}{2}\cdot x\right)-\sqrt[3]{\tan\left(\frac{x}{2}\right)^2}\right|\right)-\sqrt{3}\cdot\arctan\left(\frac{1}{\sqrt{3}}\cdot\left(2\cdot\sqrt[3]{... |
475726a1-1caa-4fce-9a01-413a63f721d8 | sequences_series | false | null | Find the sum of the series $\sum_{n=1}^\infty \frac{ x^{4 \cdot n-1} }{ 4 \cdot n-1 }$ using the integration of series. | The sum of the series is $\frac{1}{4}\cdot\ln\left(\frac{|x+1|}{|x-1|}\right)-\frac{1}{2}\cdot\arctan(x)$ |
476fbe1a-47b2-4905-a5c9-c8f15fec9af0 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAA0YAAAE0CAIAAACHM7SsAABBpElEQVR4nO3deWBTZaI28DfJyZ50S7qmdC8tLbSUpeyUvawygMgyyqAfAg64IYq7jAN4r9cFvaCAIMo4M+wIwrBZULDsUFpooXvT0n1J2uzr+f6I9HbQgS6hp6c8v7+Sk5PjU0OSJ+9ZXg5N0wQAAAAA2IzLdAAAAAAA6ChUOgAAAADWQ6UDAAAAYD1UOgAAAADWQ6UDAAAAYD1UOgAAAADWQ6UDAAAAYD1UOgAAAADWQ6UDAAAAYD1UOg... | Given $k(x) = \frac{ f(x) }{ g(x) }$, find $k'(1)$ using the table below: | $k'(1)$ = $-2$ |
478c9874-1edb-498e-9111-048a658a240e | precalculus_review | false | null | Find the center of symmetry of the curve of $f(x) = \frac{ x^2 + 2 \cdot x - 1 }{ 1 - x }$. | The final answer: $(1,-4)$ |
47a11349-0386-4969-9263-d3cdfcc98cb9 | integral_calc | false | null | Evaluate the integral:
$$
I = \int \left(x^3 + 3\right) \cdot \cos(2 \cdot x) \, dx
$$ | The final answer: $\frac{1}{256}\cdot\left(384\cdot\sin(2\cdot x)+128\cdot x^3\cdot\sin(2\cdot x)+192\cdot x^2\cdot\cos(2\cdot x)-96\cdot\cos(2\cdot x)-256\cdot C-192\cdot x\cdot\sin(2\cdot x)\right)$ |
47c69ab5-9be0-4cef-8f3c-dd7fc78de350 | differential_calc | false | null | Given the function $y = \frac{ x^3 + x }{ x^4 - x^2 + 1 }$, identify the maxima and/or minima by using the Second Derivative Test. | Maxima: $P(1,2)$
Minima: $P(-1,-2)$ |
47dbe135-6e32-462c-a77d-8581fe5ed596 | algebra | false | null | The radius of the right circular cylinder is $\frac{ 1 }{ 3 }$ meter greater than the height. The volume is $\frac{ 98 \cdot \pi }{ 9 }$ cubic meters. Find the dimensions. | $r$: $\frac{7}{3}$ $h$: $2$ |
47fadf54-e0ee-45c5-8da1-4791534583a0 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABcgUlEQVR4nO3deXhU9d3+8TswCWsQ2cUoLiyiApHgE1TCEgKEVWwRihqtUFFsi9rQorb8qlVrQfMgVcGiaJVaRUEUwhoJA5QlSFjVkLAZSQgalCWQQLb5/eEzU4aZhCSTnHNm5v26Lq8rfuaceDM4HO58zxLicDgcsrD09HRJUlRUlMlJvCOfb6yeb968eZKkSZMmmZzEO6u/f+TzjVXzJSUlKSkpSZKUmJioxMREkxN5Z9X3z4l8vrF6Po... | Given the graph of $f(x)$ shown to the right, write the limit statement to describe the function's behavior as $x$ approaches $3$ from the left. | $\lim_{{x \to 3^-}} f(x) = $\lim_{x\to3^-}f(x)=-5$$ |
481d4a88-aaae-4cae-8e98-29ee8182edbc | precalculus_review | false | null | Solve the following inequality:
$$
\left| |x+3| + |x-2| \right| \le 5
$$
By using rules of the absolute value function. | The final answer: $[-3,2]$ |
4872f755-05b7-49f4-859b-c56ded668225 | precalculus_review | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAkYAAAFMCAYAAAAurnOXAABTHUlEQVR4nO3deXhTVfoH8O/NTdt0S0vZoS0ouzgqZRFUKBVEQRBFxEFBZ0B0HDcWl58MyqbjNiI4qDMgOrKOiCgqiIoURIZiEZBVdiilFAolhC5pm5v7+yPpbW73JclNbr6f5+EhJ03znvQ2p2/ee+45kGVZPnXqlOxtCxYs8HoMX7wOX8TwVRwed/+Lo5cYAOTp06d7PY5efl56iSHL+nnP6yWGr+Lo6bgbMzIykJWVBW/Lzc1FRkaGV2... | Find a formula for $f(x)$, the sinusoidal function whose graph is shown below: | The final answer: $f(x)=-4\cdot\cos\left(\frac{1}{2}\cdot\left(x-\frac{\pi}{4}\right)\right)+4$ |
4881a8a3-a6ab-40be-aa32-3b272f2dc704 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ -3 \cdot \tan(4 \cdot x) }{ \sqrt{\sin(4 \cdot x)^4+\cos(4 \cdot x)^4} } \, dx
$$ | $\int \frac{ -3 \cdot \tan(4 \cdot x) }{ \sqrt{\sin(4 \cdot x)^4+\cos(4 \cdot x)^4} } \, dx$ = $C-\frac{3}{8}\cdot\ln\left(\sqrt{1+\tan(4\cdot x)^4}+\tan(4\cdot x)^2\right)$ |
489ae381-47d0-4a7b-bce0-f7974248806b | precalculus_review | false | null | Transform $z = -3 + i$ to a trigonometric form. | The final answer: $z=\sqrt{10}\cdot\left(\cos\left(\arccos\left(-\frac{3}{\sqrt{10}}\right)\right)+i\cdot\sin\left(\arccos\left(-\frac{3}{\sqrt{10}}\right)\right)\right)$ |
490352f2-8fc0-4790-86a9-ff74e119f7b7 | algebra | false | null | Solve the following equations:
1. $\frac{ 3 }{ 4 } (8 x+12)=21$
2. $\frac{ 1 }{ 2 } (8 r+10)=17$
3. $\frac{ 5 x }{ 8 }+9=4$
4. $\frac{ 5 }{ 9 } x-\frac{ 4 }{ 3 }=\frac{ 7 }{ 9 }$
5. $2\frac{2}{3} p+3 \left(\frac{ 2 }{ 3 } p-8\right)=-10$
6. $-5\frac{1}{4} u+2 \left(\frac{ 3 }{ 8 } u-9\right)=\frac{ 5 }{ 2 }$ | The solutions to the given equations are:
1. $x=2$
2. $r=3$
3. $x=-8$
4. $x=\frac{ 19 }{ 5 }$
5. $p=3$
6. $u=-\frac{ 41 }{ 9 }$ |
49102ccc-ebb1-432a-b82a-0c7d6913d946 | sequences_series | false | null | Expand function $y = \sin\left(\frac{ \pi \cdot x }{ 4 }\right)$ in Taylor series at $x = 2$, given that $\cos(x) = \sum_{n=0}^\infty \left((-1)^n \cdot \frac{ x^{2 \cdot n} }{ (2 \cdot n)! }\right)$. | The final answer: $\sum_{n=0}^\infty\left((-1)^n\cdot\frac{\left(\frac{\pi}{4}\cdot(x-2)\right)^{2\cdot n}}{(2\cdot n)!}\right)$ |
4947b302-5fc0-437c-b39e-7235e8c4307d | precalculus_review | false | null | $P = \left(x, \frac{ \sqrt{7} }{ 3 }\right)$, $x<0$ is a point on the unit circle.
1. Find the (exact) missing coordinate value of the point.
2. Find the values of the six trigonometric functions for the angle $\theta$ with a terminal side that passes through point $P$.
Rationalize denominators. | 1. The (exact) missing coordinate value of the point is: $-\frac{\sqrt{2}}{3}$
2. The values of the six trigonometric functions is:
* $\sin\left(\theta\right)$ = $\frac{\sqrt{7}}{3}$
* $\cos\left(\theta\right)$ = $-\frac{\sqrt{2}}{3}$
* $\tan\left(\theta\right)$ = $-\frac{\sqrt{14}}{2}$
* $\csc\left(\th... |
49b9c6d5-e705-4fd6-ae01-f4731cbf0fa5 | integral_calc | false | null | Solve the integral:
$$
\int \frac{ \sin(x)^5 }{ \cos(x)^4 } \, dx
$$ | $\int \frac{ \sin(x)^5 }{ \cos(x)^4 } \, dx$ = $C+\frac{2}{\cos(x)}+\frac{1}{3\cdot\left(\cos(x)\right)^3}+\cos(x)$ |
49e7bb62-d218-413a-a026-fb3d83dd990a | algebra | false | null | Divide rational expressions and simplify it:
$$
\frac{ x^2-4 }{ 2 } \div \frac{ x+2 }{ 3 }
$$ | The final answer: $\frac{3}{2}\cdot x-3$ |
4a693b69-d42b-4359-a183-f84c09f12bb0 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQIAAAEBCAIAAAAcuTypAACpcklEQVR4nOz9d9wlR3EvjH+rumfmnPPEzTkqrHIWkkACSQSRcw4WFiLZJtlwMbaxcXjv5QZfv9f27/58ubaxMcFgwBhEEBlEUkB5Ja12tdqVNqcnnjDT3VXvHz1znvM8+6ykXVixElv7fJ49zzlzZnp6urrSt6pIVXFU6eDTkxzyU0KABIABAKZ8Lx7JAKAEou63VOEZRCBAoQYUj1cFASrdvwEBtDpjfDMoABiCxmsoiFSEmIP3xlogAA... | Bore a hole of radius $a$ down the axis of a right cone and through the base of radius $b$ as seen here. Find the volume of this solid. | Volume = $\frac{\pi\cdot b}{3}\cdot\left(b^2-3\cdot a^2\right)$ |
4ae78dde-37d1-48f7-a6a5-480cb979092b | precalculus_review | false | null | Simplify $E = \left(\sin(2 \cdot a)\right)^3 \cdot \left(\cos(6 \cdot a)\right) + \left(\cos(2 \cdot a)\right)^3 \cdot \left(\sin(6 \cdot a)\right)$. | The final answer: $E=\frac{3}{4}\cdot\sin(8\cdot a)$ |
4afe68d6-980e-4422-a4cb-58f6988fee7a | integral_calc | false | null | Compute the integral:
$$
8 \cdot \int \cot(-4 \cdot x)^5 \cdot \csc(4 \cdot x)^4 \, dx
$$ | $8 \cdot \int \cot(-4 \cdot x)^5 \cdot \csc(4 \cdot x)^4 \, dx$ = $C+\frac{1}{3}\cdot\left(\cot(4\cdot x)\right)^6+\frac{1}{4}\cdot\left(\cot(4\cdot x)\right)^8$ |
4b450811-e985-4a9a-981d-b5f7ab763312 | multivariable_calculus | false | null | Find vector $\vec{c} \times \left(\vec{a}+3 \cdot \vec{b}\right)$ where
$$
\vec{a}=\begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 5 & 0 & 9 \\ 0 & 1 & 0 \end{vmatrix}, \quad
\vec{b}=\begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 0 & -1 & 1 \\ 7 & 1 & -1 \end{vmatrix}, \quad \text{and} \quad
\vec{c}=\vec{i}-\vec{k}
$$ | $\vec{c} \times \left(\vec{a}+3 \cdot \vec{b}\right)$ = $21\cdot\vec{i}-17\cdot\vec{j}+21\cdot\vec{k}$ |
4b815f84-4919-4db1-9328-5412d4a3e0ea | precalculus_review | false | null | Solve $\left(\sin(x)\right)^2 + \left(\cos(3 \cdot x)\right)^2 = 1$. | The final answer: $x=\frac{n\cdot\pi}{4}$ |
4bd1631f-c515-4d4c-a71e-7d68591a7814 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAScAAAEYCAIAAACRFKduAAA6XUlEQVR4nO29eZxU1bX3vdY5NXVVVw/V88DQ0Ii0SCOzcUIURd+IU7wYvUHeR6MSjAPe6+s1PhKDr1MUh8Tca67xRb3XJ+YTzY1DJIIIKkIEZBJopmZoep6qa+gaz17vH/vU6aInqrurqut07++HD11VZ6hd55zfWfvstfZaSEQgEAiSiDTcDRAIRh1CdSlNdE+Ev9Y+EZ0U/YLi5KUgRISI/P/hbosg/ghbl4pwsfH/xW1x5CFUl4ocOX... | Examine the graph. Identify where the vertical asymptotes are located. | $x$ = $0$, $1$, $2$ |
4c1292e1-d4b3-4acf-afaf-eaac62f2662d | integral_calc | false | null | Compute the integral:
$$
\int \frac{ -1 }{ x^2 \cdot \left(3+x^3\right)^{\frac{ 5 }{ 3 }} } \, dx
$$ | $\int \frac{ -1 }{ x^2 \cdot \left(3+x^3\right)^{\frac{ 5 }{ 3 }} } \, dx$ = $C+\frac{1}{9}\cdot\sqrt[3]{1+\frac{3}{x^3}}+\frac{1}{18\cdot\left(1+\frac{3}{x^3}\right)^{\frac{2}{3}}}$ |
4c30d87c-c696-465b-a078-ec3b61d444a8 | multivariable_calculus | false | null | Change the order of integration by integrating first with respect to $z$, then $x$, then $y$:
$$
\int_{0}^1 \int_{1}^2 \int_{2}^3 \left( x^2 + \ln(y) + z \right) \, dx \, dy \, dz
$$ | $I$ = $2\cdot\ln(2)+\frac{35}{6}$ |
4c9566d0-2bc4-4a11-b0ca-36e6a42c2622 | algebra | false | null | Rewrite the quadratic expression $x^2 + \frac{ 2 }{ 5 } \cdot x - \frac{ 1 }{ 5 }$ by completing the square. | $x^2 + \frac{ 2 }{ 5 } \cdot x - \frac{ 1 }{ 5 }$ = $\left(x+\frac{1}{5}\right)^2-\frac{6}{25}$ |
4cd052e9-254c-4c7a-9c30-cbbb888769c0 | multivariable_calculus | false | null | Compute the second order derivative $\frac{d ^2y}{ d x^2}$ for the parametrically defined function
$$
x = \frac{ 1 }{ 2 } \cdot \left(\cos(t)\right)^3, \quad y = 6 \cdot \left(\sin(t)\right)^3
$$ | $\frac{d ^2y}{ d x^2}$ = $\frac{8}{\left(\cos(t)\right)^4\cdot\sin(t)}$ |
4d41436f-c675-45ef-87a1-009350e3953b | precalculus_review | false | null | Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{0}^\pi \left(\cos(99 \cdot x) \cdot \sin(101 \cdot x)\right) \, dx
$$ | $\int_{0}^\pi \left(\cos(99 \cdot x) \cdot \sin(101 \cdot x)\right) \, dx$ = $0$ |
4d4989ec-b4b8-4b1b-b20c-0861da90c835 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAtsAAADmCAYAAAAEC0nyAAAYU2lDQ1BJQ0MgUHJvZmlsZQAAeJyVeQVUVF8X77mTzDAM3d0l3SAxdHeDwNAdQ4NKioSKIKCUCioIIliEiIUgooigAgYiYVAqqKAIyLuEfv/v/6313npn1rn3N/vss+PsU3sGAM5UcmRkKIIOgLDwGIqtkS6fs4srH/YdgAAO/ggBEbJPdCTJ2tocwOXP+7/L8jDMDZdnUpuy/rf9/1roff2ifQCArGHs7RvtEwbjawCgMn0iKTEAYFRhum... | Using the given graphs of $f(x)$ and $g(x)$, find $f\left(g(3)\right)$. | $f\left(g(3)\right)$ = $0$ |
4d8dcf18-41d7-4085-97ab-f076a076978e | precalculus_review | false | null | Calculate the following indefinite integral:
$$
\int x \cdot \left(a^x\right)^x \, dx
$$
for $a > 0$ and $a \ne 1$. | $\int x \cdot \left(a^x\right)^x \, dx$ = $\frac{a^{x^2}}{2\cdot\ln(a)}+C$ |
4df2e972-39b8-4bc5-8726-98ce26fea8f8 | differential_calc | false | null | Find the local extrema of the function $f(x) = 2 \cdot \left(x^2\right)^{\frac{ 1 }{ 3 }} - x^2$ using the First Derivative Test. | Local maxima: $x=-\sqrt[4]{\frac{8}{27}}$, $x=\sqrt[4]{\frac{8}{27}}$
Local minima: $x=0$ |
4e00e130-7fca-4c2a-8d84-cdd7d0405967 | algebra | false | null | Solve the equation for $x$:
$12 - 5 \cdot (x + 3) = 2 \cdot x - 5$ | The final answer: $x=\frac{2}{7}$ |
4e3604b8-d25e-467e-bb05-b739b155a1e8 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ x-\sqrt[3]{x^2}-\sqrt[6]{x} }{ x \cdot \left(4+\sqrt[3]{x}\right) } \, dx
$$ | $\int \frac{ x-\sqrt[3]{x^2}-\sqrt[6]{x} }{ x \cdot \left(4+\sqrt[3]{x}\right) } \, dx$ = $C+\frac{3}{2}\cdot\sqrt[3]{x^2}+60\cdot\ln\left(\left|4+\sqrt[3]{x}\right|\right)-3\cdot\arctan\left(\frac{1}{2}\cdot\sqrt[6]{x}\right)-15\cdot\sqrt[3]{x}$ |
4e4e997a-4e1e-4694-96ec-bffc19506e79 | algebra | false | null | Using the Rational Zero Theorem, list all possible rational zeros of the following polynomial:
$p(x) = 2 \cdot x^3 + x - 1$ | Possible rational zeros are $1$, $-1$, $\frac{1}{2}$, $-\frac{1}{2}$ |
4e68083d-84c3-4070-9ae7-06b59dcb7717 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAACEUUlEQVR4nO3dd3RUhbv18T3pCSFU6QiiVKUryk9QUMQCqNgAsYF0RKRzrSCIFSxIQEVQRBRQuoB0AaUICKIU6aFILyGkTpL3D96MGc4kZJKZOVO+n7XuWplnJjOPufcCO+fscyyZmZmZAuBRo0eP1ujRoyVJAwYM0IABA0zeCAACW5cuXbRw4UK72aBBg9SvXz+TNiq4pKQkPfzww/rrr7/s5rfccovmzJljzlKApCCzFwAAADBTZmamfv... | Let $R$ be the region enclosed by the graphs of $y=\frac{ 1 }{ 2 } \cdot x^2$ and $y=x$, as shown in the image above.
Region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $y$-axis is an isosceles right triangle with a leg in $R$. Find the volume of the solid. | The volume of the solid is $\frac{2}{15}$ units³. |
4ec04fb6-2b14-47de-8edd-ab602686736d | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABm6klEQVR4nO3deXhU5cH//0+SkSVkQBSjQFCJCIg2LpRF44IIolKLoCLiBtXG5an+UOqC1dbHtlZUoE+LVVMVFEUQBAsUQSwqy9cAgiVQlrC5JCxRtkwICSQ5vz9wTjNkJuvM2eb9ui4uT3JOJvfthGHeOVuCYRiGGiE7O1uSlJWV1ZiHcRzm5S7My13GjRuncePGSZJGjx6t0aNH2zyi6PDq88W83IV5uZNX58e83MWqeSXG9NEBAAAAoA... | The graph of $g'(x)$, the derivative of $g$, is shown above. Find the value of $g(6)$ if $g(0)=-2$. | $g(6)$ = $8$ |
4ec668c6-0576-46de-a1b8-2bd85c15805a | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAV0AAAGCCAIAAADMgocaAACQJUlEQVR4nOydd3wc1dWG33Nn+6p3WZYsyVW23Hu3cQEbAwZM7wGSQEgBQhIS0vMlpJNCTQIk9OYGxoAr7r33bkuyZFlWb1vmnu+PO7OSKy6ypZXm+SVGWq1Wd2dnzpx7ynuImWHRemFmIjrl68YPnu3JFm0Z0dwLsLiMLF++fMqUKU8//bT69siRI2PGjPnDH/5QV1cXes727dvXrFmjvraMgoXCsgutmbi4uPXr18+YMSMQCPj9/m9961... | Find the volume of the solid under the graph of the function $f(x,y) = x \cdot y + 1$ and above the region in the figure: | The volume is $\frac{7+2\cdot\pi}{4}$ |
4edfa960-3d41-4fa9-88f9-e850ed2af94b | sequences_series | false | null | Find the radius of convergence of the series:
$$
\sum_{n=1}^\infty \left(\left(\frac{ n-1 }{ 2 \cdot n+3 }\right)^n \cdot x^n\right)
$$ | $R$ = $2$ |
4f311636-5a26-454b-a708-5f7ab2aeccbc | precalculus_review | false | null | Solve the following system of equations:
1. $\sin(x) \cdot \sin(y) = \frac{ \sqrt{3} }{ 4 }$
2. $\cos(x) \cdot \cos(y) = \frac{ \sqrt{3} }{ 4 }$ | The final answer: $x=\frac{\pi}{3}+\frac{\pi}{2}\cdot(2\cdot n+k)$, $x=\frac{\pi}{6}+\frac{\pi}{2}\cdot(2\cdot n+k)$, $y=\frac{\pi}{6}+\frac{\pi}{2}\cdot(k-2\cdot n)$, $y=\frac{\pi}{3}+\frac{\pi}{2}\cdot(k-2\cdot n)$ |
4f622685-24ee-4e55-a33b-6270bb8204e5 | precalculus_review | false | null | Solve $x^2 - 4 \cdot x - 6 = \sqrt{2 \cdot x^2 - 8 \cdot x + 12}$. | The final answer: $x=-2 \lor x=6$ |
4fabf73c-c71a-486d-a161-716b6ec6fd31 | precalculus_review | false | null | Calculate the following integral:
$$
\int \log_{a^x}(b) \, dx
$$
for $a > 0$, $b > 0$, $a \ne 1$, $b \ne 1$, $x \ne 0$. | $\int \log_{a^x}(b) \, dx$ = $\frac{\ln(x)\cdot\ln(b)}{\ln(a)}+C$ |
4faec218-e81d-4743-8372-242b474f1a9b | sequences_series | false | null | Suppose that we have a series: $1-\frac{ 1 }{ 2 }+\frac{ 1 }{ 3 }-\cdots+(-1)^{n+1} \cdot \frac{ 1 }{ n }+\cdots$ and its sum is equal to $\ln(2)$. Find the sum of the series: $1-\frac{ 1 }{ 2 }-\frac{ 1 }{ 4 }+\frac{ 1 }{ 3 }-\frac{ 1 }{ 6 }-\frac{ 1 }{ 8 }+\frac{ 1 }{ 5 }-\frac{ 1 }{ 10 }-\frac{ 1 }{ 12 }+\cdots$. | The final answer: $S=\frac{1}{2}\cdot\ln(2)$ |
4fbf3770-e3dd-4d3d-87ac-825d1fa83719 | integral_calc | false | null | Compute the integral:
$$
3 \cdot \int \left(\cos\left(\frac{ x }{ 6 }\right)\right)^6 \, dx
$$ | $3 \cdot \int{\left(\cos\left(\frac{ x }{ 6 }\right)\right)^6 d x}$ = $C+\frac{9}{2}\cdot\sin\left(\frac{x}{3}\right)+\frac{15}{16}\cdot x+\frac{27}{32}\cdot\sin\left(\frac{2\cdot x}{3}\right)-\frac{3}{8}\cdot\left(\sin\left(\frac{x}{3}\right)\right)^3$ |
509c8613-037d-4553-b210-f89cd8434288 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABsXElEQVR4nO3deZzNdf//8eeYGVlGv1S40oRLiagsM4xtSDEMgyhLxtgbcXUp0UJarlTaKJLRZBkMEZGlxjKVfbsoS5NCixpEKjG2WX9/nOuc74xzZj/n8znL4367dbvxOcu8vBtjnvP8vD8fv5ycnBxB8fHxkqTY2FiTJzHX3r17JUkhISEmT+IeWA8L1sGCdbCYPHmyJk+eLEkaM2aMxowZY/JE5uLzwoJ1yIv1sGAdLPg+M68yZg8AAA... | The graphs of the continuous functions $f$ and $g$ are shown. Let $h(x) = -5 \cdot f(x) - g(x) + 3 \cdot x^2$. Find $h'(2)$. | $h'(2)$ = $15$ |
50a65ac5-fd17-452d-a81f-fc9ce7492fe9 | differential_calc | false | null | Evaluate $\lim_{x \to 0^{+}} \left( \left( \frac{ \tan(4 \cdot x) }{ 4 \cdot x } \right)^{\frac{ 1 }{ 2 \cdot x^2 }} \right)$ using L'Hopital's Rule. | $\lim_{x \to 0^{+}} \left( \left( \frac{ \tan(4 \cdot x) }{ 4 \cdot x } \right)^{\frac{ 1 }{ 2 \cdot x^2 }} \right)$ = $e^{\frac{8}{3}}$ |
50b3292a-5159-410f-811e-6cefeab6f78a | precalculus_review | false | null | Given $k$ as an integer and $a \ne \pi \cdot k$, evaluate $P=\cos(a) \cdot \cos(2 \cdot a) \cdot \cos(4 \cdot a) \ldots \cos\left(2^n \cdot a\right)$. | The final answer: $P=\frac{1}{2^{n+1}}\cdot\frac{\sin\left(2^{n+1}\cdot a\right)}{\sin(a)}$ |
50c99276-f6a4-4769-86d3-1c5338bb62b1 | algebra | false | null | Multiply the rational expressions and express the product in simplest form:
$$
\frac{ 2 \cdot d^2+9 \cdot d-35 }{ d^2+10 \cdot d+21 } \cdot \frac{ 3 \cdot d^2+2 \cdot d-21 }{ 3 \cdot d^2+14 \cdot d-49 }
$$ | The final answer: $\frac{2\cdot d-5}{d+7}$ |
50f9b4f4-fe8f-40e1-8b8c-56f2c06fd385 | integral_calc | false | null | Solve the integral:
$$
\int \left(\frac{ x+6 }{ x-6 } \right)^{\frac{ 3 }{ 2 }} \, dx
$$ | $\int \left(\frac{ x+6 }{ x-6 } \right)^{\frac{ 3 }{ 2 }} \, dx$ = $C+\sqrt{\frac{x+6}{x-6}}\cdot(x-30)-18\cdot\ln\left(\left|\frac{\sqrt{x-6}-\sqrt{x+6}}{\sqrt{x-6}+\sqrt{x+6}}\right|\right)$ |
510a0ac3-d3f3-40e2-9d7a-8f0f487490c8 | sequences_series | false | null | Find the radius of convergence of the series:
$$
\sum_{n=1}^\infty \left( \left(n^{\frac{ 1 }{ n }}-1\right)^n \cdot x^n \right)
$$ | $R$ = $\infty$ |
51106f64-d305-4db0-b14f-2d0ec24181d3 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAfMAAAD4CAIAAAB33ZhrAABKPklEQVR4nO3dd1xT1/s48JMECDuAICCKbBkOBMVtFRyAogKOOoqiqG1dta5vHfXTuqvWOtsqCmrdigsVKs4KiAMHI7JHRQGRvbN+f5zP9/7uN4EQIORekuf9R18352Y8Fnhy7rnnPIchEokQAAAAJcKkOgAAAAByBpkdAACUDWR2AABQNpDZAQBA2UBmBwAAZQOZHQAAlA1kdgAAUDaQ2QEAQNlAZgcAAGWjRnUA/x+fzxcKhQwGQ11dne... | Consider the cycloid given by the parametric equations $x = 2 \cdot \left(t - \sin(t)\right)$, $y = 2 \cdot \left(1 - \cos(t)\right)$.
Find the area of the surface obtained by rotating the arc of the cycloid OBA about the segment DE. | The final answer: $\frac{64\cdot\pi}{3}$ |
517c1de0-1f6b-4762-8073-04faf90d2df8 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASkAAABRCAIAAAAB/cfNAAAKMWlDQ1BJQ0MgUHJvZmlsZQAAeJydlndUU9kWh8+9N71QkhCKlNBraFICSA29SJEuKjEJEErAkAAiNkRUcERRkaYIMijggKNDkbEiioUBUbHrBBlE1HFwFBuWSWStGd+8ee/Nm98f935rn73P3Wfvfda6AJD8gwXCTFgJgAyhWBTh58WIjYtnYAcBDPAAA2wA4HCzs0IW+EYCmQJ82IxsmRP4F726DiD5+yrTP4zBAP+flLlZIjEAUJiM5/... | A car is speeding up and its velocity is recorded in the following table.
Use the table to estimate the total distance traveled, giving an
* upper estimate, and
* lower estimate. | The final answer: $44$, $66$ |
51860641-1b22-460b-9fd9-5e074f620c93 | precalculus_review | false | null | Solve the differential equation: $\frac{ d y }{d x} = \sin(x)^2$. The curve passes through the point $(0,0)$. | $f(x)$ = $\frac{x}{2}-\frac{\sin(2\cdot x)}{4}$ |
51dae41a-dc02-4794-8d1a-c0cc2838d16d | multivariable_calculus | false | null | A small appliances company makes toaster ovens and pizza cookers, and have noticed their customer base's purchasing habits give them the price-demand equations given below, where $p$ is the price and $x$ is the quantity of toaster ovens, $q$ is the price and $y$ is the quantity of pizza cookers. The Cost function is $C... | 1. $R(x,y)$ = $R(x,y)=50\cdot x+100\cdot y+2\cdot x\cdot y-4\cdot x^2-5\cdot y^2$
2. $R(8,10)$ = $804$
3. $P(x,y)$ = $P(x,y)=38\cdot x+76\cdot y+2\cdot x\cdot y-160-4\cdot x^2-5\cdot y^2$
4. $P(8,10)$ = $308$ |
51f91d21-b2e5-408b-b549-aaec8249f58b | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQcAAACnCAIAAACTog6qAABUpklEQVR4nO19d3wVZfb+Oe/cmt4TEkgIaST03ruKgIiKihXsfm27+nPX1XXXdV1dXXtbu+KqoCJFRAURkB56CTWBFEjv7ebWmff8/nhnbkK/6Te7efyYhHvnzrx35j3ve8pzzkEigm6oIAAkIABEIkDs7PF0o3Og6+wBeA9IlQdAgO6V4n8arLMH4DUgRPfuQEjd+8T/MLqlwg1q/InUvV38L6Nbg9KAAAD1NucH320urrJOHBJ/zaSBnT... | The region $D$ is shown in the following figure. Evaluate the double integral $\int\int_{D}{\left(x^2+y\right) d A}$ by using the easier order of integration. | $\int\int_{D}{\left(x^2+y\right) d A}$ = $\frac{256}{15}$ |
526a43b4-e38a-43d7-8930-47b626fbf7ef | precalculus_review | false | null | 1. Find the inverse function of $f(x) = \sqrt{x-1}$
2. Find the domain of the inverse function
3. Find the range of the inverse function | 1. The inverse function is: $x^2+1$
2. The domain of the inverse function is: $x\ge0$
3. The range of the inverse function is: $y\ge1$ |
52ba9ace-9cce-481d-bc05-13c2eb4d5460 | differential_calc | false | null | Find the derivative of the function $y = \frac{ 3 \cdot \csc(x) - 4 \cdot \sin(x) }{ 8 \cdot \left(\cos(x)\right)^5 } - \frac{ 76 }{ 5 } \cdot \cot(3 \cdot x)$. | $y'$ = $\frac{228}{5\cdot\left(\sin(3\cdot x)\right)^2}+\frac{16\cdot\left(\cos(x)\right)^6-5\cdot\left(\cos(x)\right)^4-3\cdot\left(\cos(x)\right)^6\cdot\left(\csc(x)\right)^2}{8\cdot\left(\cos(x)\right)^{10}}$ |
52dac142-fce4-4cac-8d4c-02d6f91936b9 | sequences_series | false | null | Find the Fourier series of the function $f(x) = \frac{ 1 }{ 4 } \cdot x$ in the interval $[-3,3]$. | The Fourier series is: $\sum_{n=1}^\infty\left(\frac{\frac{3}{2}\cdot(-1)^{n+1}}{\pi\cdot n}\cdot\sin\left(\frac{\pi\cdot n\cdot x}{3}\right)\right)$ |
531fbc44-4d03-459d-bc47-0dd40bf62702 | integral_calc | false | null | Find the area bounded by the curves $y = 3 \cdot x + 3$, $y = 3 \cdot \cos(x)$, and $y = 0$. | Area: $\frac{9}{2}$ |
5381ca65-50cc-430f-b450-b4e00e2e6040 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ 1 }{ \left(\cos(5 \cdot x)\right)^3 } \, dx
$$ | $\int \frac{ 1 }{ \left(\cos(5 \cdot x)\right)^3 } \, dx$ = $C+\frac{\sin(5\cdot x)}{10\cdot\left(\cos(5\cdot x)\right)^2}+\frac{1}{10}\cdot\ln\left(\left|\tan\left(\left(\frac{5}{2}\right)\cdot x+\frac{\pi}{4}\right)\right|\right)$ |
541e5d98-324c-43bf-b2f7-56b3fdec55f0 | sequences_series | false | null | Find the domain of convergence of the series $\sum_{n=1}^\infty \left(\frac{ x^n }{ n \cdot 10^{n-1} }\right)$. | The final answer: $-10\le x<10$ |
548e5105-3474-4300-9672-0196f2a27c1a | integral_calc | false | null | Compute the integral:
$$
2 \cdot \int \left(\cos\left(\frac{ x }{ 4 }\right)\right)^6 \, dx
$$ | $2 \cdot \int{\left(\cos\left(\frac{ x }{ 4 }\right)\right)^6 d x}$ = $C+2\cdot\sin\left(\frac{x}{2}\right)+\frac{3}{8}\cdot\sin(x)+\frac{5}{8}\cdot x-\frac{1}{6}\cdot\left(\sin\left(\frac{x}{2}\right)\right)^3$ |
5491d76d-661b-40b8-8192-b4c98c1dd5b6 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAACOE0lEQVR4nO3deXiU9b3//9ckmewLYSckgCCERdwiIAooYBW11lprrbUV7Gk99dSe/sDW47c99LRVe9qj1tq6YrXiglpL3SsuLLIIASIgSICwJCSEkJCNkH2Z3x9xBsJMZibJzD33PfN8XFevozNh5u2c25hnPvfnvm0Oh8OhIFqyZIkk6Y477gjm2wRMXl6eJCknJyfEk/SM1ea22rwPP/ywHn74YUnS3XffrbvvvjvEE3lntc/XavM6WW... | Given the graphs of $f(x)$ and $g(x)$ above, find $m'(4)$ where $m(x) = f\left(g(x)\right)$. | $m'(4)$ = $1$ |
54b9cf44-c38c-4ccf-8e90-71e67b980af1 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAewAAAEzCAIAAAB8IKfNAABmLklEQVR4nO29d1xTyff4PZTQexFEijQFBWxYwAICYkORolixL7qWta0FlbUhumvvvaOorIptKYqoICqgAioqTaQISG8hoeT3xzyfee43jRCSmwTm/Yev62TunZOQnHvumVOkGAwGwGAwGIxkIi1qATAYDAbDP1iJYzAYjASDlTgGg8FIMFiJYzAYjASDlTgGg8FIMFiJYzAYjASDlTgGg8FIMFiJYzAYjASDlTgGg8FIMFiJYzAYjASDlT... | Find the area of the surface formed by rotating the loop of the parametric curve $x = t^2$, $y = \frac{ t \cdot \left(t^2 - 3\right) }{ 3 }$ between the points $P(0,0)$ and $P(3,0)$ about the x-axis (see the picture below): | The final answer: $3\cdot\pi$ |
551ab8c6-6717-445e-9eb3-a9b72ff35da6 | algebra | false | null | Use synthetic division to find the quotient and remainder. Ensure the equation is in the form required by synthetic division:
$$
\frac{ 4 \cdot x^3-12 \cdot x^2-5 \cdot x-1 }{ 2 \cdot x+1 }
$$
Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor. Solve on a paper, if it is more c... | Quotient: $2\cdot x^2-7\cdot x+1$ Remainder: $-2$ |
5564d511-3879-48af-baed-3005bc3a2261 | precalculus_review | false | null | Find the inverse of the function $f(x) = \frac{ 4-2 \cdot \sqrt{x} }{ 4+2 \cdot \sqrt{x} }$. | The final answer: $\frac{4\cdot(1-x)^2}{(x+1)^2}$ |
557fa740-6516-45d6-a45a-1bfc252ab6e1 | sequences_series | false | null | Using the series expansion for the function $(1+x)^m$, calculate approximately $\sqrt[4]{15}$ with an accuracy of 0.0001. | The final answer: $1.968$ |
558c9ebf-be00-40bc-a0cd-e1a4432bac21 | precalculus_review | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAlkAAAE9CAYAAAA4dXeWAAABSmlDQ1BJQ0MgUHJvZmlsZQAAeJxjYGASSSwoyGFhYGDIzSspCnJ3UoiIjFJgf8LAzSDMwMkgyyCfmFxc4BgQ4ANUwgCjUcG3awyMIPqyLsgsH+uW3+9WzlmodPBt3FW+e9qY6lEAV0pqcTKQ/gPEGckFRSUMDIwpQLZyeUkBiN0BZIsUAR0FZM8BsdMh7A0gdhKEfQSsJiTIGci+AWQLJGckAs1gfAFk6yQhiacjsaH2ggCfa0VJUX5ecH... | The graph of a function $f$ is given in the figure below:
1. What is the value or limit of $f(x)$ as $x$->$\infty$?
2. What is the value or limit of $x$->$b^{+}$?
3. What are the vertical asymptotes? (Enter as x=_)
4. What are the horizontal asymptotes? (Enter as y=_)
5. Give the value (letter), if any, at which the g... | 1. As $x$->$\infty$, $f(x)$->$d$
2. As $x$->$b^{+}$, $f(x)$->$c$
3. Vertical asymptotes at $x=a$
4. Horizontal asymptotes with $y=d$
5. $f$ has a vertical tangent at $x=p$
6. $f$ has a vertical cusp at $x=q$ |
5597a385-33cd-41a2-b5ad-dbc9390e795c | differential_calc | false | null | What are the points of inflection of the graph of $f(x) = \frac{ x+1 }{ x^2+1 }$? | The final answer: $x_1=1 \land x_2=-2+\sqrt{3} \land x_3=-2-\sqrt{3}$ |
55c912d9-77af-430d-8bc8-fde2929769da | differential_calc | false | null | Find the local minimum and local maximum values of the function $f(x) = \frac{ x^4 }{ 2 } - \frac{ 20 }{ 3 } \cdot x^3 + 24 \cdot x^2 + 13$. | The point(s) where the function has a local minimum: $P(0,13)$, $P(6,85)$
The point(s) where the function has a local maximum: $P\left(4,\frac{295}{3}\right)$ |
55dc25d0-d0df-47f6-ac8a-f4f29087033a | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAVIAAAGXCAIAAABukk+/AABEcElEQVR4nO2dd0AUV/f3z9KbiIiCHTsqYO8FG/aKikZNzGM0WKJJRINP1CRqFGOU5FGjJppEY4kVETvYKSqKCjbEFlCkY6OIlH3/OD/vu9mdGQFhZ+/u+fx1mHNnObD73Zm599xzFEqlEgiCMCSM5A6AIAhtQ7InCIODZE8QBgfJniAMDpI9QRgcJHuCMDhI9gRhcJDsCcLgINkThMFBsicIg4NkTxAGB8meIAwOkj1BGBwke4IwOEj2BG... | Compute the area enclosed by the curve $y = x^2 - 3 \cdot x$, the line $x = -4$, and the $x$-axis as shown below: | Area = $\frac{299}{6}$ |
568e4b30-a796-49d7-aaf8-5ca4d6418607 | differential_calc | false | null | Evaluate $\lim_{x \to 0}\left(\left(\frac{ \sin(4 \cdot x) }{ 4 \cdot x }\right)^{\frac{ 1 }{ 3 \cdot x^2 }}\right)$. | $\lim_{x \to 0}\left(\left(\frac{ \sin(4 \cdot x) }{ 4 \cdot x }\right)^{\frac{ 1 }{ 3 \cdot x^2 }}\right)$ = $e^{-\frac{8}{9}}$ |
56f22d87-6f2d-41cd-a939-1a15194856dc | multivariable_calculus | false | null | Find partial derivatives of the function $r = \sqrt{a \cdot x^2 - b \cdot y^2}$. | $\frac{\partial}{\partial x}(r)$ = $\frac{a\cdot x}{\sqrt{a\cdot x^2-b\cdot y^2}}$
$\frac{\partial}{\partial a}(r)$ = $\frac{x^2}{2\cdot\sqrt{a\cdot x^2-b\cdot y^2}}$
$\frac{\partial}{\partial y}(r)$ = $-\frac{b\cdot y}{\sqrt{a\cdot x^2-b\cdot y^2}}$
$\frac{\partial}{\partial b}(r)$ = $-\frac{y^2}{2\cdot\sqrt{a\cdot x^... |
570d1fef-3e41-428e-af8f-0eb4802e9667 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAVsAAADaCAIAAACHNnZzAABT+klEQVR4nO2dd3wUZfrAn+ed7SW9kx5aIIHQQpHeBAQsIGK9Qz04251n19/ZDvUsnJ6KigXL2cWCItKr9C5JgCQQShJSSK+b3Z33+f3xzmwCBA1tE8h8P3eSnZ2dfWd23mee96lIRKDhLYgIEX9/i4ZGK4KaRNDQ0PDAWnsA7R0hkTW5rNFG0CSCt3nkkUf+8pe/ZGRkAIDD4XjttdemTZuWm5vbVCjs27evoqICNEmh4XU0ieBViMjtdv... | Split the region between the two curves $y = x^3$ and $y = x^2 + x$ into two smaller regions, then determine the area by integrating over the $x$-axis. Note that you will have two integrals to solve. | Area = $\frac{13}{12}$ |
575785fa-191f-4c8c-bb7c-9ca9c825b568 | sequences_series | false | null | Find the Fourier series of the function $f(x) = \frac{ -1 }{ 3 } \cdot x$ in the interval $[-3,3]$. | The Fourier series is: $\sum_{n=1}^\infty\left(\frac{2\cdot(-1)^n}{\pi\cdot n}\cdot\sin\left(\frac{\pi\cdot n\cdot x}{3}\right)\right)$ |
57a3be6d-92a6-4afd-9f1e-c15fc64195dc | algebra | false | null | Solve the following equations:
1. $2 m - 8 = -28$
2. $\frac{ x }{ 9 } - 3 = 8$
3. $12 m + 20 = -40$
4. $-\frac{ x }{ 3 } + 5 = 21$
5. $8 r - 27 = -19$
6. $6 + \frac{ k }{ 3 } = 33$
7. $15 = -4 y - 9$
8. $8 w + 4 = -36$ | The solutions to the given equations are:
1. $m=-10$
2. $x=99$
3. $m=-5$
4. $x=-48$
5. $r=1$
6. $k=81$
7. $y=-6$
8. $w=-5$ |
57b00837-b0bf-47f4-b0ce-76f29fbd9c0f | precalculus_review | false | null | Solve the following inequality by using rules of the absolute value function. Express your answer as an interval or union of intervals.
$\left| |x+3| + |x-2| \right| \le 9$ | The final answer: $[-5,4]$ |
57f16359-5178-4dc3-9c70-6dc3b425a540 | sequences_series | false | null | Find the Fourier series of the function $u = \left| \frac{ \sin\left( \frac{ x }{ 2 } \right) }{ 2 } \right|$ in the interval $[-\pi, \pi]$. | The Fourier series is: $\frac{1}{\pi}-\frac{2}{\pi}\cdot\sum_{n=1}^\infty\left(\frac{1}{\left(4\cdot n^2-1\right)}\cdot\cos(n\cdot x)\right)$ |
5876988d-f7b7-4106-aae1-88d2bd872ed3 | differential_calc | false | null | Find the derivative of the 25th order $y^{(25)}$ for a function $y = 2 \cdot x^2 \cdot \sin(x)$. | $y^{(25)}$ = $\left(2\cdot x^2-1200\right)\cdot\cos(x)+100\cdot x\cdot\sin(x)$ |
58a92fd4-6288-4849-8902-9d02c154e0c2 | differential_calc | false | null | Find the second derivative $\frac{d ^2y}{ d x^2}$ of the function $x = 5 \cdot \left(\cos(t)\right)^3$, $y = 6 \cdot \left(\sin(3 \cdot t)\right)^3$. | $\frac{d ^2y}{ d x^2}$ = $\frac{-15\cdot\left(\cos(t)\right)^2\cdot\sin(t)\cdot\left(324\cdot\sin(3\cdot t)-486\cdot\left(\sin(3\cdot t)\right)^3\right)-\left(30\cdot\cos(t)-45\cdot\left(\cos(t)\right)^3\right)\cdot54\cdot\left(\sin(3\cdot t)\right)^2\cdot\cos(3\cdot t)}{\left(-15\cdot\left(\cos(t)\right)^2\cdot\sin(t)... |
58e2d06c-dce7-4705-aa36-ca5e519312ac | algebra | false | null | Rewrite the quadratic expression $x^2 - 10 \cdot x + 7$ by completing the square. | $x^2 - 10 \cdot x + 7$ = $(x-5)^2-18$ |
5903bf68-557c-4045-b8f0-5601f1e040ac | precalculus_review | false | null | Solve $\sin(x) + 7 \cdot \cos(x) + 7 = 0$. | The final answer: $x=2\cdot\pi\cdot k-2\cdot\arctan(7) \lor x=\pi+2\cdot\pi\cdot k$ |
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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.