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 |
|---|---|---|---|---|---|
2d64140b-ec41-43b2-aa47-a02a2b515f51 | integral_calc | false | null | Compute the integral:
$$
\int x^{-4} \cdot \left(3+x^2\right)^{\frac{ 1 }{ 2 }} \, dx
$$ | $\int x^{-4} \cdot \left(3+x^2\right)^{\frac{ 1 }{ 2 }} \, dx$ = $C-\frac{1}{9}\cdot\left(1+\frac{3}{x^2}\right)\cdot\sqrt{1+\frac{3}{x^2}}$ |
2d799998-115a-489b-a48b-57090954303e | differential_calc | false | null | Compute the limit:
$$
\lim_{x \to 5} \left( \frac{ 3 \cdot x }{ x-5 }-\frac{ 3 }{ \ln\left(\frac{ x }{ 5 }\right) } \right)
$$ | $\lim_{x \to 5} \left( \frac{ 3 \cdot x }{ x-5 }-\frac{ 3 }{ \ln\left(\frac{ x }{ 5 }\right) } \right)$ = $\frac{3}{2}$ |
2e1592e8-b882-4761-b04a-613d85f94fbd | precalculus_review | false | null | Find the domain of the function $f(x) = \frac{ 1 }{ \sqrt{ |x| - x } }$. | The final answer: $(-\infty,0)$ |
2e672f49-9aec-4635-895a-d3f19e391509 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARcAAAI5CAIAAAAE/+lYAAB4EklEQVR4nO2dd1wUx/vHn9m726sc5ei9igVRQUWxABaMNRpLjN0YY0lMTKIxxpjkm95Ms8aUn8bE3nsvsReQYgNEAekIUg64uvP7Y4/zRETggLuDeX9fX3Ps7c4+u7efnZln5nkGKRQKIFgsGGOEkKmtaO1wKYoytQ2EelBaWkrTtEAg0Gq1crlcIpGwvyDGWKvVcrlcUxvYGkEajcbUNhDqBMYYACZMmBAcHLx48eKtW7euWbNm+/btdn... | Use the following graphs and the limit laws to evaluate the limit $\lim_{x \to -9}\left(x \cdot f(x) + 2 \cdot g(x)\right)$. | $\lim_{x \to -9}\left(x \cdot f(x) + 2 \cdot g(x)\right)$ = $-46$ |
2e6afcbe-7883-4e94-9b96-cfc96e2841fc | integral_calc | false | null | Solve the integral:
$$
\int 3 \cdot \cot(-7 \cdot x)^6 \, dx
$$ | $\int 3 \cdot \cot(-7 \cdot x)^6 \, dx$ = $C-\frac{3}{7}\cdot\left(\frac{1}{5}\cdot\left(\cot(7\cdot x)\right)^5+\cot(7\cdot x)-\frac{1}{3}\cdot\left(\cot(7\cdot x)\right)^3-\arctan\left(\cot(7\cdot x)\right)\right)$ |
2e8c12b1-bd78-4c9e-a3f0-0e5888d2e71e | multivariable_calculus | false | null | Evaluate the triple integral of the function $f(x,y,z) = z$ over the solid $B$ bounded by the half-sphere $x^2+y^2+z^2=16$ with $z \ge 0$ and below by the cone $2 \cdot z^2 = x^2+y^2$. | $\int\int\int_{B}{f(x,y,z) d V}$ = $\frac{128\cdot\pi}{3}$ |
2ec21ca8-41a5-4d45-82cf-4ea1a390ce7b | multivariable_calculus | false | null | Find a normal vector and a tangent vector for $2 \cdot x^3 - x^2 \cdot y^2 = 3 \cdot x - y - 7$ at point $P : (1,-2)$ | Normal vector: $\vec{N}=\vec{i}-\vec{j}$
Tangent vector: $\vec{T}=\vec{i}+\vec{j}$ |
2ec678c4-d3d1-479f-9dc7-3117fb7bb232 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYwAAAGICAIAAADdykALAABhyUlEQVR4nO29aXAbV5bne29mAkjsO7ivWihxEaldohZqtWxtlteaKtfSNd3Vr6tffZiIFzEx0RMx0W/eRH3pnldTr6ba1S5XeZctyyWXrdW2ZEnWQlELSYkiJVLcd2Ih9h2Z931IAIRISpZEgASJ84sqmQTARCKR+c9zz/3fczAhBAEAAKQr1FzvAAAAwOMAkQIAIK0BkQIAIK0BkQIAIK0BkQIAIK0BkQIAIK0BkQIAIK0BkQIAIK0BkQ... | Use the following graph and find: $\lim_{x \to 0^{+}}\left(f(x)\right)$ | $\lim_{x \to 0^{+}}\left(f(x)\right)$: $-4$ |
2f336dde-88d0-403c-9bb7-60cb816dc6d7 | differential_calc | false | null | $f(x) = \frac{ 1 }{ 4 } \cdot \sqrt{x} + \frac{ 1 }{ x }$, $x > 0$. Determine:
1. Intervals where $f$ is increasing
2. Intervals where $f$ is decreasing
3. Local minima of $f$
4. Local maxima of $f$
5. Intervals where $f$ is concave up
6. Intervals where $f$ is concave down
7. The inflection points of $f$ | 1. Intervals where $f$ is increasing: $(4,\infty)$
2. Intervals where $f$ is decreasing: $(0,4)$
3. Local minima of $f$: $4$
4. Local maxima of $f$: None
5. Intervals where $f$ is concave up: $\left(0,8\cdot\sqrt[3]{2}\right)$
6. Intervals where $f$ is concave down: $\left(8\cdot\sqrt[3]{2},\infty\right)$
7. The inflec... |
2f7471d9-02df-48a7-8044-e5409141de5a | multivariable_calculus | false | null | Find the average value of the function $f(x,y) = \arctan(x \cdot y)$ over the region $R = [0,1] \times [0,1]$. | $f_{ave}$ = $\frac{\pi-\ln(4)}{4}-\frac{\pi^2}{48}$ |
2f7ca7bd-b50a-4cf6-9870-cec83f04faa0 | differential_calc | false | null | Find the extrema of a function $y = \frac{ 2 \cdot x^4 }{ 4 } - \frac{ x^3 }{ 3 } - \frac{ 3 \cdot x^2 }{ 2 } + 2$. Then determine the largest and smallest value of the function when $-2 \le x \le 4$. | 1. Extrema points: $P\left(\frac{3}{2},\frac{1}{32}\right)$, $P\left(-1,\frac{4}{3}\right)$, $P(0,2)$
2. The largest value: $\frac{254}{3}$
3. The smallest value: $\frac{1}{32}$ |
2fa3efa3-1a87-4757-879d-66fa65c0cf62 | algebra | false | null | Write an expression for a rational function with the given characteristics:
1. vertical asymptotes $x=-3$ and $x=2$,
2. x-intercepts at $P(1,0)$ and $P(-4,0)$,
3. y-intercept at $P(0,5)$. | The rational function satisfying the given conditions is $f(x)=\frac{15\cdot(x-1)\cdot(x+4)}{2\cdot(x+3)\cdot(x-2)}$ |
2fe7cf5b-86d4-417a-a942-46e2194625eb | integral_calc | false | null | Solve the integral:
$$
\int \cot(x)^4 \, dx
$$ | $\int \cot(x)^4 \, dx$ = $C+\cot(x)-\frac{1}{3}\cdot\left(\cot(x)\right)^3-\arctan\left(\cot(x)\right)$ |
2ff5a6e4-6a0a-48e5-adfc-bdaf179a7fc9 | multivariable_calculus | false | null | Use the method of Lagrange multipliers to maximize $U(x,y) = 8 \cdot x^{\frac{ 4 }{ 5 }} \cdot y^{\frac{ 1 }{ 5 }}$ subject to the constraint $4 \cdot x + 2 \cdot y = 12$. | Answer: maximum $16.715$ at $P(2.4,1.2)$ |
309063a4-cd05-4d2e-a3ff-4f5937d4df66 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAAB+8klEQVR4nOzdeZxN9ePH8fe9c2dfzGLJFq20CpVCi5IWpVKWiqSEhLIk3yLKTkiyZJmxS1laTLKkRIlESBqkKNtgLGP2u/z+8Js7rhn7zDl37n09H48eX3PumZl338nc+76fzeJyuVxCsTd8+HANHz5cktStWzd169bN5EQAAABAflazAwAAAADwHxQQAAAAAIahgAAAAAAwDAUEAAAAgGEoIAAAAAAMQwEBAAAAYBgKCAAAAADDUEAAAA... | Let $A$ be the region bounded by the graph $x=(y+2)^2$, $y=x-4$, $y=2$, and the $y$-axis as shown in the figure above.
1. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid generated when $A$ is revolved about the $y$-axis.
2. Find the value of the volume of th... | 1. $\int_{-2}^0\left(\pi\cdot\left((y+2)^2\right)^2\right)dy+\int_0^2\left(\pi\cdot(y+4)^2\right)dy$
2. $179.28$ units³ |
30b1fff3-c7e8-4443-8b21-ad150fba7da0 | algebra | false | null | Note that $1 \in (-3,7)$. Think of the solution of the inequality $|x-1|<\delta$, where $\delta$ (delta, a Greek letter) is a constant.
Find the greatest $\delta$ so that the solution is completely on the interval $(-3,7)$.
[Hint: How far you can go within $(-3,7)$, starting from $1$?] | The final answer: $4$ |
30b647a3-04b2-4224-8e1d-9a46f4fe6103 | integral_calc | false | null | Solve the integral:
$$
\int \left(\frac{ x+3 }{ x-3 }\right)^{\frac{ 3 }{ 2 }} \, dx
$$ | $\int \left(\frac{ x+3 }{ x-3 }\right)^{\frac{ 3 }{ 2 }} \, dx$ = $C+\sqrt{\frac{x+3}{x-3}}\cdot(x-15)-9\cdot\ln\left(\left|\frac{\sqrt{x-3}-\sqrt{x+3}}{\sqrt{x-3}+\sqrt{x+3}}\right|\right)$ |
30cec86e-4be2-4e33-b449-e66f79f364a2 | multivariable_calculus | false | null | $E$ is located above the xy-plane, below $z=1$, outside the one-sheeted hyperboloid $x^2+y^2-z^2=1$, and inside the cylinder $x^2+y^2=2$. Find the volume of $E$. | Volume = $\frac{2\cdot\pi}{3}$ |
30d63199-0a17-480a-9a4e-b3761098338b | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXYAAAGuCAYAAACEMy0nAAAYU2lDQ1BJQ0MgUHJvZmlsZQAAeJyVeQVUVF8X77mTzDAM3d0l3SAxdHeDwNAdQ4NKioSKIKCUCioIIliEiIUgooigAgYiYVAqqKAIyLuEfv/v/6313npn1rn3N/vss+PsU3sGAM5UcmRkKIIOgLDwGIqtkS6fs4srH/YdgAAO/ggBEbJPdCTJ2tocwOXP+7/L8jDMDZdnUpuy/rf9/1roff2ifQCArGHs7RvtEwbjawCgMn0iKTEAYFRhum... | Use the graph of the function to estimate the intervals on which the function is increasing or decreasing.
A function is increasing/decreasing on an interval when its values increase/decrease as $x$ values increase (moving to the right on the graph). | The final answer:
Interval(s) of increase: $(-1.5,2)$
Interval(s) of decrease: $(-\infty,-1.5)\cup(2,\infty)$ |
3105d97f-f772-43ee-ad96-a977b62b3322 | multivariable_calculus | false | null | Develop the Cartesian equation for the following parametric equations:
$x = 3 \cdot t + \frac{ 3 }{ 4 \cdot t }$
$y = 5 \cdot t - \frac{ 5 }{ 4 \cdot t }$ | The final answer: $\frac{x^2}{9}-\frac{y^2}{25}=1$ |
313f8edb-e1ee-4bc2-ac17-270dc7106f35 | multivariable_calculus | false | null | Evaluate the integral by choosing the order of integration:
$$
\int_{1}^e \int_{1}^e \left( \frac{ x \cdot \ln(y) }{ \sqrt{y} } + \frac{ y \cdot \ln(x) }{ \sqrt{x} } \right) \, dy \, dx
$$ | $\int_{1}^e \int_{1}^e \left( \frac{ x \cdot \ln(y) }{ \sqrt{y} } + \frac{ y \cdot \ln(x) }{ \sqrt{x} } \right) \, dy \, dx$ = $2\cdot\sqrt{e}+4\cdot e^2-4-2\cdot e^2\cdot\sqrt{e}$ |
31b62f10-54ed-4f45-a532-2220b453838d | differential_calc | false | null | Make full curve sketching of $y = \ln\left(\left|\frac{ 2 \cdot x-5 }{ 2 \cdot x+5 }\right|\right)$. Submit as your final answer:
1. The domain (in interval notation)
2. Vertical asymptotes
3. Horizontal asymptotes
4. Slant asymptotes
5. Intervals where the function is increasing
6. Intervals where the function is dec... | 1. The domain (in interval notation) $\left(-\infty,-\frac{5}{2}\right)\cup\left(-\frac{5}{2},\frac{5}{2}\right)\cup\left(\frac{5}{2},\infty\right)$
2. Vertical asymptotes $x=-\frac{5}{2}$, $x=\frac{5}{2}$
3. Horizontal asymptotes $y=0$
4. Slant asymptotes None
5. Intervals where the function is increasing $\left(-\inf... |
324f58d5-6dfb-461d-9750-6e713fedb6f6 | integral_calc | false | null | Compute the length of the arc $y = 3 \cdot \ln(2 \cdot x)$ between the points $x = \sqrt{7}$ and $x = 4$. | Arc Length: $1+\frac{3}{2}\cdot\ln\left(\frac{7}{4}\right)$ |
3265c2a2-779f-4ede-a31f-feb2dde5e7cf | sequences_series | false | null | Find $L=\lim_{n \to \infty}\left(x_{n}\right)$, where $x_{n} = \frac{ \sqrt{n^2+1} + \sqrt{n} }{ \sqrt[4]{n^3+n} - \sqrt{n} }$ is the general term of a sequence. | The final answer: $L=\infty$ |
32d27abf-2f33-48e4-84fb-756af840c821 | precalculus_review | false | null | Find points on a coordinate plane that satisfy the equation $x \cdot (x-2) + y \cdot (y+4) + 5 = 0$. | The final answer: $(1,-2)$ |
32f04626-b937-48c5-a7e7-708115124653 | precalculus_review | false | null | Find points on a coordinate plane that satisfy the equation $y = \sqrt{\ln\left(\cos(x)\right)}$. | The final answer: $(2\cdot k\cdot\pi,0)$ |
33478f29-8573-4922-a984-f7f20a3f68c9 | differential_calc | false | null | Compute the derivative $y^{(5)}$ of the function $y = e^{\frac{ x }{ 3 }} \cdot \sin\left(\frac{ x }{ 2 }\right)$. | $y^{(5)}$ = $\frac{1}{7776}\cdot e^{\frac{x}{3}}\cdot\left(122\cdot\sin\left(\frac{x}{2}\right)-597\cdot\cos\left(\frac{x}{2}\right)\right)$ |
3355c2b6-9095-4bb3-90ef-51b6fb9f3708 | algebra | false | null | Identify all points of removable discontinuity (singularity) of the function $f(x) = \frac{ x^2 - 4 }{ x - 2 }$. | $f(x)$ has removable discontinuities at $x=2$ |
337a55dc-e856-4e1e-b717-2c1ca37ca438 | differential_calc | false | null | The cost for printing a book can be given by the equation $C(x) = 1000 + 12 \cdot x + \frac{ 1 }{ 2 } \cdot x^{\frac{ 2 }{ 3 }}$. Use Newton’s method to find the break-even point if the printer sells each book for $\$20$. | $127$ books |
33d1cff7-b004-4c49-878f-f1056b5dd633 | multivariable_calculus | false | null | Use the method of Lagrange multipliers to find the maximum and minimum values of $f(x,y,z) = x^2 + y^2 + z^2$ subject to the constraint $x^4 + y^4 + z^4 = 1$. If there is no maximum or minimum, then write $\text{None}$ in the answer. | Maximum: $\sqrt{3}$
Minimum: $1$ |
34800a4c-8294-4b02-a23e-f4be6115f560 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABnmUlEQVR4nO3deXxV1b3///fJSAYCBJDBMA/BgUGCBZVBcEINaG+drdqq2Gqt91bt7a+2j9bW6fY6VGtFr7S9eqvWCa0mCKICghFEgoyGMAcigyhDQhIynJzfH36Tetg7kOScs9fe+7yej0cfD/sh6mdxTjy8sz5r7UAoFArJB4qLiyVJeXl5hjuJjF/W8eyzz0qSbrnlFsOdRObbr8ehQ4d00kknhf16YmKidu7caaK1NvHL+4p1uMOjjz... | Let $h$ be a function defined by $h(x) = \left( rac{ f(x) }{ g(x) }
ight)^2$.
Selected values of $f$ and $f'$ are shown in the table below.
The graph of $g$ is shown below.
If $f$ and $g$ are continuous functions, determine $h'(0)$ (else write None).
| $x$ | $f(x)$ | $f'(x)$ |
| --- | --- | --- |
| $0$ | $2$ | $5$ ... | $h'(0)$ = $\frac{4}{5}$ |
3497d594-978a-4c17-9210-5654e5e7023a | 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 $\$33$, they... | The final answer: $33$ |
34bdbf1b-9f1b-43b3-9195-dabc016eb1a7 | differential_calc | false | null | Find the derivative of $f(x) = \frac{ \left(\left(\tan(x)\right)^2-1\right) \cdot \left(\left(\tan(x)\right)^4+10 \cdot \left(\tan(x)\right)^2+1\right) }{ 3 \cdot \left(\tan(x)\right)^3 }$ | The final answer: $f'(x)=\left(\tan(x)\right)^4+4\cdot\left(\tan(x)\right)^2+\frac{4}{\left(\tan(x)\right)^2}+\frac{1}{\left(\tan(x)\right)^4}+6$ |
34ee9891-810d-43a5-9a82-9d2c37ec5f6d | sequences_series | false | null | Decompose the function $f(x) = \frac{ 1 }{ 2 } \cdot (\pi-x)$ into a trigonometric series on the interval $[0,2 \cdot \pi]$. | The final answer: $f(x)=\sin(x)+\frac{1}{2}\cdot\sin(2\cdot x)+\frac{1}{3}\cdot\sin(3\cdot x)+\cdots+\frac{1}{n}\cdot\sin(n\cdot x)+\cdots$ |
34f480a3-bc95-4865-8a8a-cf0132878a01 | integral_calc | false | null | Compute the integral:
$$
\int_{0}^1 \frac{ \sqrt{x}+1 }{ \sqrt[3]{x}+1 } \, dx
$$ | $\int_{0}^1 \frac{ \sqrt{x}+1 }{ \sqrt[3]{x}+1 } \, dx$ = $3\cdot\ln(2)+\frac{3\cdot\pi}{2}-\frac{409}{70}$ |
35180e65-8d1d-4399-86d3-ca456db0e24c | multivariable_calculus | false | null | Evaluate the integral by choosing the order of integration:
$$
\int_{0}^1 \int_{0}^{\frac{ 1 }{ 2 }} \left(\arcsin(x) + \arcsin(y)\right) \, dy \, dx
$$ | $\int_{0}^1 \int_{0}^{\frac{ 1 }{ 2 }} \left(\arcsin(x) + \arcsin(y)\right) \, dy \, dx$ = $\frac{\pi}{12}+\frac{\sqrt{3}}{2}+\frac{\arcsin(1)}{2}-\frac{3}{2}$ |
35505dd4-7798-48a2-9d3b-11edc71de275 | multivariable_calculus | false | null | Compute the partial derivatives of the implicit function $z(x,y)$, given by the equation $-x-6 \cdot y+z=3 \cdot \cos(-x-6 \cdot y+z)$.
Submit as your final answer:
1. $\frac{\partial z}{\partial x}$;
2. $\frac{\partial z}{\partial y}$. | 1. $1$
2. $6$ |
3592c6d2-0fb1-4557-b219-d88f5b8a7401 | sequences_series | false | null | Find the Fourier integral of the function $q(x) = \begin{cases} 0, & x < 0 \\ \pi \cdot x, & 0 \le x \le 1 \\ 0, & x > 1 \end{cases}$ | $q(x) = $\int_0^\infty\left(\frac{\left(\alpha\cdot\sin\left(\alpha\right)+\cos\left(\alpha\right)-1\right)\cdot\cos\left(\alpha\cdot x\right)+\left(\sin\left(\alpha\right)-\alpha\cdot\cos\left(\alpha\right)\right)\cdot\sin\left(\alpha\cdot x\right)}{\alpha^2}\right)d\alpha$ |
35cb0f7d-db0d-494e-980b-55bd98f0170b | sequences_series | false | null | Find the Maclaurin series for the function: $f(x) = \cos(x) - x \cdot \sin(x)$. | The series: $\sum_{n=0}^\infty\left(\frac{(-1)^n\cdot(2\cdot n+1)\cdot x^{2\cdot n}}{(2\cdot n)!}\right)$ |
35e4725a-56f7-46f8-8115-a4f3b75a09a2 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQYAAAHRCAIAAAAHbID8AABxAUlEQVR4nO29e5RtXVUf+Jtr73NOVd3vxUsUfEIAsRUQSfsgIQZ8AwJRUYhtDJ3WJG0bsB3RpPM0xnRGp9NJdERjHnZifIBRE4dD0diokYBCfAQJj483AkHlg+9x7606e++1Zv8xz/rV3PtUnVO37qmq81i/8Y37nTpn77XX3nvONd9zSUoJOwNRIAEBXUgRCKhqVUkJQFdVCggQEIMCADQAggAAbduORqOUUgjB/lXVq7yTggtDuOoJXD... | The region bounded by the parabola $y^2 = 2 \cdot p \cdot x$ and the line AB is revolved about the Y-axis. The line AB passes through the focus of the parabola and is perpendicular to the X-axis. Find the volume of this solid of revolution using integration with respect to $y$. Use $p = \frac{ 1 }{ 2 }$. | Volume: $\frac{\pi}{20}$ |
363dd580-f1fc-4867-a6ef-db2a03139745 | differential_calc | false | null | Evaluate $\lim_{x \to 0^{+}}\left(\left(\frac{ \tan\left(\frac{ x }{ 2 }\right) }{ \frac{ x }{ 2 } }\right)^{\frac{ 3 }{ x^2 }}\right)$ using L'Hopital's Rule. | $\lim_{x \to 0^{+}}\left(\left(\frac{ \tan\left(\frac{ x }{ 2 }\right) }{ \frac{ x }{ 2 } }\right)^{\frac{ 3 }{ x^2 }}\right)$ = $e^{\frac{1}{4}}$ |
3655d579-c27c-463e-8ece-662e1f8a0b02 | algebra | false | null | A phone company has a monthly cellular plan, where a customer pays a flat monthly fee and then a certain amount of money per minute used on the phone. If a customer uses 410 minutes, the monthly cost will be $71.5. If the customer uses 720 minutes, the monthly cost will be $118.1.
1. Find a linear equation for the mon... | 1. The monthly cost function is $C(x)=0.15\cdot x+10$
2. The flat monthly fee is $10$ and the fee for each additional minute used is $0.15$
3. The total monthly cost if 687 minutes are used is $113.05$ |
36585e1b-1b7a-4835-befc-4610d20674a1 | precalculus_review | false | null | A car is racing along a circular track with a diameter of 1 mile. A trainer standing in the center of the circle marks his progress every 5 seconds. After 5 seconds, the trainer has to turn $55^o$ to keep up with the car. How fast is the car traveling? | Velocity of car is $345.57519189$ mph. |
36e3fe5a-1d6e-4976-b51a-6eea157d5128 | precalculus_review | false | null | Find all values of $x$ that satisfy the following equation:
$$
\left|\left(x^2+4 \cdot x+9\right)+(2 \cdot x-3)\right|=\left|x^2+4 \cdot x+9\right|+|2 \cdot x-3|
$$ | The final answer: $x\ge\frac{3}{2}$ |
37440b24-a694-4066-acf5-e5dc13144552 | precalculus_review | false | null | Find the zeros of $\tan(x) + \tan\left(\frac{ \pi }{ 4 } + x\right) = -2$. | The final answer: $x_{1}=-\frac{ \pi }{ 3 }+\pi \cdot n$, $x_{2}=\frac{ \pi }{ 3 }+n \cdot \pi$ |
374da26c-e042-4624-9e44-edaf584b1909 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPMAAAEsCAIAAACZixImAABDoUlEQVR4nO2deZwU1dX3z6nqdXp6nX1fgGFYB2QTh019BIQAimISRTRxedQsblGTJ4u+GvMkJhoTYxI1PsrihhITgqAiOyowwz4wwwwDszD72j29L3XeP6q6aQYYZ+lhqnrq+0mwp7r61q2qX90699xzz0UiAhmZqIMZ6grIyAwKsrJlohNZ2TLRiaxsmehEVrZMdCIrWyY6kZUtE53IypaJTmRly0Qnw1HZ4cOu8hBstDK8lM3rGBFDW8... | Analyze the graph of $f'$, then list all inflection points and intervals on which $f$ is concave up and concave down. | Inflection points at: $x=0$, $x=1$
Intervals where function is concave up: $(-\infty,0)$, $(1,\infty)$
Intervals where function is concave down: $(0,1)$ |
3761ac12-4b27-4ca2-8fd6-345ca0a137fe | precalculus_review | false | null | Calculate $E = \left(\sin\left(\frac{ \pi }{ 8 }\right)\right)^4 + \left(\sin\left(\frac{ 3 \cdot \pi }{ 8 }\right)\right)^4 + \left(\sin\left(\frac{ 5 \cdot \pi }{ 8 }\right)\right)^4 + \left(\sin\left(\frac{ 7 \cdot \pi }{ 8 }\right)\right)^4$ | The final answer: $E=\frac{3}{2}$ |
37d1acdf-64e4-4c6d-a76d-c273e763ad18 | multivariable_calculus | false | null | Compute the partial derivatives of the implicit function $z(x,y)$, given by the equation
$$
-10 \cdot x-9 \cdot y+8 \cdot z=4 \cdot \cos(-10 \cdot x-9 \cdot y+8 \cdot z)
$$
Submit as your final answer:
1. $\frac{\partial z}{\partial x}$;
2. $\frac{\partial z}{\partial y}$. | 1. $\frac{5}{4}$
2. $\frac{9}{8}$ |
37de6838-f6c9-47b8-8257-12db6a7f5b6b | multivariable_calculus | false | null | For the following exercise, line $L$ is given.
1. Find point $P$ that belongs to the line and direction vector $\vec{v}$ of the line. Express $\vec{v}$ in component form.
2. Find the distance from the origin to line $L$.
Line $L$: $-x = y + 1$, $z = 2$ | 1. $P$: $P(0,-1,2)$ ; $\vec{v}$= $\left\langle-1,1,0\right\rangle$
2. $d$= $\frac{3}{\sqrt{2}}$ |
37e7e328-accc-4a28-98f7-7391204c2892 | integral_calc | false | null | Compute the integral:
$$
\int x^{-6} \cdot \left(1+x^2\right)^{\frac{ 1 }{ 2 }} \, dx
$$ | $\int x^{-6} \cdot \left(1+x^2\right)^{\frac{ 1 }{ 2 }} \, dx$ = $C+\frac{1}{3}\cdot\left(\frac{1}{x^2}+1\right)\cdot\sqrt{\frac{1}{x^2}+1}-\frac{1}{5}\cdot\left(\frac{1}{x^2}+1\right)^2\cdot\sqrt{\frac{1}{x^2}+1}$ |
37f31f02-4302-49b8-bd61-4bad89d04fe7 | precalculus_review | false | null | Use the Rational Zero Theorem to find all real zeros of the following polynomial:
$p(x) = x^3 - 3 \cdot x^2 - 10 \cdot x + 24$ | The real zeros are $2$, $-3$, $4$ |
3868940c-d8fc-4b5f-a82d-15c0853f15b8 | sequences_series | false | null | Find the radius of convergence and the interval of convergence for the series:
$$
\sum_{n=0}^\infty \left(\frac{ x^n }{ n^n }\right)
$$ | $R$ = $\infty$
$I$ = $(-\infty,\infty)$ |
3882cc71-2ff8-4a16-8ebb-3eeee2365bfa | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARcAAAI5CAIAAAAE/+lYAAB4EklEQVR4nO2dd1wUx/vHn9m726sc5ei9igVRQUWxABaMNRpLjN0YY0lMTKIxxpjkm95Ms8aUn8bE3nsvsReQYgNEAekIUg64uvP7Y4/zRETggLuDeX9fX3Ps7c4+u7efnZln5nkGKRQKIFgsGGOEkKmtaO1wKYoytQ2EelBaWkrTtEAg0Gq1crlcIpGwvyDGWKvVcrlcUxvYGkEajcbUNhDqBMYYACZMmBAcHLx48eKtW7euWbNm+/btdn... | Use the following graphs and the limit laws to evaluate the limit $\lim_{x \to -5}\left(\frac{ 2+g(x) }{ f(x) }\right)$: | $\lim_{x \to -5}\left(\frac{ 2+g(x) }{ f(x) }\right)$ = $1$ |
38defb82-0872-4e28-80bc-d425575cf1ab | precalculus_review | false | null | Find the period of $f(x) = \left| \cos(x) \right|$ | The final answer: $T=\pi$ |
3934b610-d7c8-4cec-bef7-ed06c0b05391 | algebra | false | null | Rewrite the quadratic expression $x^2 - 11 \cdot x - 20$ by completing the square. | $x^2 - 11 \cdot x - 20$ = $(x-5.5)^2-50.25$ |
394e56e3-055c-4209-b8b1-1f2283907c2a | algebra | false | null | Use the fact that the vertex of the graph of the quadratic function is $(2,-3)$ and the graph opens up to find the domain and range of the function. | The domain is $(-\infty,\infty)$ and the range is $[-3,\infty)$ |
39955d74-b26d-4d7d-ae27-b93ee45f74dc | integral_calc | false | null | Compute the area of the figure bounded by curves $y = 2 \cdot x^2$, $y = 1 + x^2$, lines $x = 3$, $x = -2$, and the $x$-axis. | Area = $\frac{46}{3}$ |
39c876c6-191f-4992-b50f-f15595d08e4e | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAc8AAAGxCAIAAACLMqrNAABc/UlEQVR4nO3dd3hc5Zk3/vucM71IGvVerd6rAeMiwBhICDEBDDHLkmz4mWRJYPMGEjbvRcJudrmyIW8SAxte8rIOCYQEiBNCTDVgjJus3txVrC6rTp857fn9caTxeFSsMlW6P1euYI1Gc86079zznPs8D0UIAYQQQj5GB3oHEEJoXcC0RUFhKd+x8HsYCmmYtigA5uYmRVGLX40QMu91EAoVFNYLCCHkB1jbIoSQP2DaotCGX85QqMC0RY... | The following graph of the function $f$ satisfies $\lim_{x \to 3}f(x) = 2$. For $\varepsilon = 3$, find a value of $\delta > 0$ such that the precise definition of limit holds true. | $\delta \leq $1$$ |
3a761457-5727-4faa-b9c8-a9e3d0e1881e | precalculus_review | false | null | An alternating current for outlets in a home has voltage given by the function $V(t) = 150 \cdot \cos(368 \cdot t)$, where $V$ is the voltage in volts at time $t$ in seconds.
1. Find the period of the function.
2. Determine the number of periods that occur when $1$ sec. has passed. | 1. The period of the function is $\frac{\pi}{184}$
2. The number of periods that occur when $1$ sec. has passed is $58.56901906$ |
3a9ca0e5-d5e8-41d3-a45c-7623bbbb2969 | precalculus_review | false | null | Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{-\pi}^\pi \left(\cos(3 \cdot x)\right)^2 \, dx
$$ | $\int_{-\pi}^\pi \left(\cos(3 \cdot x)\right)^2 \, dx$ = $\pi$ |
3ae4b0b6-dcdf-4733-a2ff-2f6cccad2598 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ -4 }{ 3+\sin(4 \cdot x)+\cos(4 \cdot x) } \, dx
$$ | $\int \frac{ -4 }{ 3+\sin(4 \cdot x)+\cos(4 \cdot x) } \, dx$ = $C-\frac{2}{\sqrt{7}}\cdot\arctan\left(\frac{2}{\sqrt{7}}\cdot\left(\frac{1}{2}+\tan(2\cdot x)\right)\right)$ |
3b116e5c-f065-4e95-8774-f422916f3e8a | differential_calc | false | null | Make full curve sketching of $y = \arcsin\left(\frac{ 1-x^2 }{ 1+x^2 }\right)$. Submit as your final answer:
1. The domain (in interval notation)
2. Vertical asymptotes
3. Horizontal asymptotes
4. Slant asymptotes
5. Intervals where the function is increasing
6. Intervals where the function is decreasing
7. Intervals ... | 1. The domain (in interval notation) $(-1\cdot\infty,\infty)$
2. Vertical asymptotes None
3. Horizontal asymptotes $y=-\frac{\pi}{2}$
4. Slant asymptotes None
5. Intervals where the function is increasing $(-\infty,0)$
6. Intervals where the function is decreasing $(0,\infty)$, $(-\infty,0)$
7. Intervals where the func... |
3b126d90-4cae-43c5-8f74-072d49ac068b | multivariable_calculus | false | null | Find the point on the surface $f(x,y) = x^2 + y^2 + 10$ nearest the plane $x + 2 \cdot y - z = 0$. Identify the point on the plane. | Answer: $P\left(\frac{47}{24},\frac{47}{12},\frac{235}{24}\right)$ |
3b3ba848-ee8f-4978-b0c2-ceb26a88779e | differential_calc | false | null | Sketch the curve:
$$
y = \frac{ x^3 }{ 4 \cdot (x+3)^2 }
$$
Provide the following:
1. The domain (in interval notation)
2. Vertical asymptotes
3. Horizontal asymptotes
4. Slant asymptotes
5. Intervals where the function is increasing
6. Intervals where the function is decreasing
7. Intervals where the function is co... | 1. The domain (in interval notation): $(-1\cdot\infty,-3)\cup(-3,\infty)$
2. Vertical asymptotes: $x=-3$
3. Horizontal asymptotes: None
4. Slant asymptotes: $y=\frac{x}{4}-\frac{3}{2}$
5. Intervals where the function is increasing: $(-\infty,-9)$, $(-3,0)$, $(0,\infty)$
6. Intervals where the function is decreasing: $(... |
3b944c43-38f7-4bfb-a805-43620f329cf8 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWUAAAEwCAIAAADgpPeoAABg4UlEQVR4nO2deUBM6//Hn2klRSWkKJWdVmUrFQklSbgkS3a6KGtybVlyo6xlLSGi5ZJUIjRFi1JSad9L+75Oy8z5/fHce37nOxUzmX3O6685n+eR93POmc886+cDEATJysqysrJCcP4XTU1NdktgKUlJSVu3bmW3CsbT2NhoaGjIbhXczbJly0pLSxEEEQA4ODg4tIH7CxwcHFoRoqXSp0+fenp6sBYxMbFJkyZ9/fqVqqaCgoKysjLD1H... | The following table lists the electrical power in gigawatts—the rate at which energy is consumed—used in a certain city for different hours of the day, in a typical $24$-hour period, with hour $1$ corresponding to midnight to $1$ a.m.
Find the total amount of power in gigawatt-hours (gW-h) consumed by the city in a ... | $911$ gW-h. |
3ba2efcf-1203-4eaa-a8c6-c69f28313fa3 | precalculus_review | false | null | $P = \left(\frac{ 7 }{ 25 },y\right)$, $y>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{24}{25}$
2. The values of the six trigonometric functions are:
* $\sin\left(\theta\right)$ = $\frac{24}{25}$
* $\cos\left(\theta\right)$ = $\frac{7}{25}$
* $\tan\left(\theta\right)$ = $\frac{24}{7}$
* $\csc\left(\theta\right)$ = $\frac{25}{... |
3bf5f05a-3a2f-4939-9cc2-52605b9db700 | integral_calc | false | null | Solve the integral:
$$
\int \sqrt{\frac{ -16 \cdot \sin(-10 \cdot x) }{ 25 \cdot \cos(-10 \cdot x)^9 }} \, dx
$$ | $\int \sqrt{\frac{ -16 \cdot \sin(-10 \cdot x) }{ 25 \cdot \cos(-10 \cdot x)^9 }} \, dx$ = $C+\frac{2}{25}\cdot\left(\frac{2}{3}\cdot\left(\tan(10\cdot x)\right)^{\frac{3}{2}}+\frac{2}{7}\cdot\left(\tan(10\cdot x)\right)^{\frac{7}{2}}\right)$ |
3c05e140-7754-4647-a453-9073ee403a29 | differential_calc | false | null | Find the derivative of $f(x) = \frac{ 1 }{ 15 } \cdot \left(\cos(x)\right)^3 \cdot \left(\left(\cos(x)\right)^2-5\right)$. | The final answer: $f'(x)=-\frac{\left(\cos(x)\right)^2}{15}\cdot\left(3\cdot\sin(x)\cdot\left(\left(\cos(x)\right)^2-5\right)+\cos(x)\cdot\sin(2\cdot x)\right)$ |
3c515761-f013-4a2c-b5f8-1a70be519118 | sequences_series | false | null | Compute $\sqrt[4]{90}$ with accuracy $0.0001$. | The final answer: $3.0801$ |
3c7f8f93-bdba-4b7d-8e4a-4908ee1082cd | algebra | false | null | Solve the following compound (double) inequality and write your final answer in interval notation:
$-4 < 3 \cdot x + 2 \le 18$ | The final answer: $\left(-2,\frac{16}{3}\right]$ |
3cc19800-9373-4d92-97b9-b6f37e135eba | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAT4AAABICAYAAABvNJlYAAAuNUlEQVR4nO2dd3gcxf3GP7vX1YtVrW5ZcpdcsI2NO9jYwaaZXpJQQgkBQgshkISEUAIJhIQfoQUSSEJJAGOMe++23JvcJVmWZKt3Xdv9/bG7dyonWeUkO7Dv88iW7nZnvjs78863zYwgy7KMDh06dHyHIJ5vAXTo0KGjr6ETnw4dOr5z0IlPhw4d3znoxKdDh47vHHTi06FDx3cOOvHp0KHjOwfj+Rbg2wxZlpFlGQQBURDOtzg6dOhQoR... | Find the values of $a$ and $b$ that make continuous and differentiable at $x=1$. | The final answer:
$a = $-4$$
$b = $2$$ |
3cf85296-5e1a-4d78-897d-eac3424d5992 | differential_calc | false | null | Find the local minimum and local maximum values of the function $f(x) = \frac{ x^4 }{ 4 } - \frac{ 11 }{ 3 } \cdot x^3 + 15 \cdot x^2 + 17$. | The point(s) where the function has a local minimum: $P(0,17)$, $P(6,89)$
The point(s) where the function has a local maximum: $P\left(5,\frac{1079}{12}\right)$ |
3cfb5ae6-689c-4020-9a5f-eb65dc86e266 | sequences_series | false | null | Compute the first 3 nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of $f(x) = \ln(\cos(x))$. | $f(x)$ = $-\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^6}{45}+\cdots$ |
3d4817ab-4cb4-447a-ac9f-43a2d7a16970 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ 8 }{ 7 \cdot x^2 \cdot \sqrt{5 \cdot x^2-2 \cdot x+1} } \, dx
$$ | Answer is: $\frac{8}{7}\cdot\left(C-\sqrt{5+\frac{1}{x}^2-\frac{1\cdot2}{x}}-\ln\left(\left|\frac{1}{x}+\sqrt{5+\frac{1}{x}^2-\frac{1\cdot2}{x}}-1\right|\right)\right)$ |
3d90964b-fc19-4336-ab11-3d7a1f653616 | algebra | false | null | Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeros of the following polynomial:
$p(x) = x^3 - 2 \cdot x^2 + x - 1$ | The number of positive zeros: $1$, $3$
The number of negative zeros: $0$ |
3dc0ac7f-8b58-4f46-a326-120c0d57d1dc | integral_calc | false | null | Compute the integral:
$$
\int \frac{ \sqrt{25+x^2} }{ 5 \cdot x } \, dx
$$ | $\int \frac{ \sqrt{25+x^2} }{ 5 \cdot x } \, dx$ = $C+\frac{1}{2}\cdot\ln\left(\left|\frac{\sqrt{25+x^2}-5}{5+\sqrt{25+x^2}}\right|\right)+\frac{1}{5}\cdot\sqrt{25+x^2}$ |
3e091f07-ec1a-4a3f-a368-62e4012f1399 | integral_calc | false | null | Compute the integral:
$$
\int \sin\left(\frac{ x }{ 2 }\right)^5 \, dx
$$ | $\int \sin\left(\frac{ x }{ 2 }\right)^5 \, dx$ = $-\frac{2\cdot\sin\left(\frac{x}{2}\right)^4\cdot\cos\left(\frac{x}{2}\right)}{5}+\frac{4}{5}\cdot\left(-\frac{2}{3}\cdot\sin\left(\frac{x}{2}\right)^2\cdot\cos\left(\frac{x}{2}\right)-\frac{4}{3}\cdot\cos\left(\frac{x}{2}\right)\right)+C$ |
3e0aa281-8701-4f9a-a1ce-c60021ecb125 | multivariable_calculus | false | null | Evaluate the integral $\int\int\int_{E}{(x+y) \, dV}$, where $E$ is the region defined by:
$$
E = \left\{ (x,y,z) \, | \, 0 \le x \le \sqrt{1-y^2}, \, 0 \le y \le 1, \, 0 \le z \le 1-x \right\}
$$ | $I$ = $\frac{1}{48}\cdot(26-3\cdot\pi)$ |
3e310e2e-ca05-4323-8356-e67852c3dc4a | multivariable_calculus | false | null | Use the second derivative test to identify any critical points of the function $f(x,y) = 7 \cdot x^2 \cdot y + 9 \cdot x \cdot y^2$, and determine whether each critical point is a maximum, minimum, saddle point, or none of these. | Maximum: None
Minimum: None
Saddle point: None
The second derivative test is inconclusive at: $P(0,0)$ |
3e35605b-4778-4579-a47e-2caf92caeb45 | differential_calc | false | null | Find any local extrema for $s = \frac{ \pi }{ 2 } - \left| \arctan\left( x^2 - 1 \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(-1,\frac{\pi}{2}\right)$, $P\left(1,\frac{\pi}{2}\right)$
2. Local Minimum(s) $P\left(0,\frac{\pi}{4}\right)$ |
3e60b21a-39a4-4bfb-a47c-3798fee97a94 | precalculus_review | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAV4AAAGuCAIAAABX2/lUAADHxElEQVR4nOz955Nd2XUeDq994j3nnnNz6tsRjQwMMMAMJoHkzHAYRFKURNGiPJZKskqhXPYHu0offrar/Bf4g/2WqiS/r1Vl2ZKoH0kxWeQwDTk5AzOYAQaxG53DzfHktN8PJ/Tt7tvdt9EBPdB9CtW4ffucnfez115r7bURxhj66KOPPlaDuN8F6KOPPg4i+tTQRx99dEGfGvroo48uoO53AfrYHViWNT8/ryhKJBJJJpM8zwOA4zitVq... | The given graph is of the form $y = A \cdot \sin(B \cdot x)$ or $y = A \cdot \cos(B \cdot x)$, where $B > 0$. Write the equation of the graph. | The equation of the graph: $y=-2\cdot\cos(\pi\cdot x)$ |
3eaa5161-484b-40de-982c-87a1aec7f5d5 | sequences_series | false | null | Evaluate the sum of the series $\sum_{n=1}^\infty\left(\frac{ 1 }{ n \cdot (n+1) \cdot (n+2) }\right)$.
(Hint: $\frac{ 1 }{ n \cdot (n+1) \cdot (n+2) }=\frac{ 1 }{ 2 \cdot n }-\frac{ 1 }{ n+1 }+\frac{ 1 }{ 2 \cdot (n+2) }$) | $\sum_{n=1}^\infty\left(\frac{ 1 }{ n \cdot (n+1) \cdot (n+2) }\right)$ = $\frac{1}{4}$ |
3ee1b980-1888-485a-ae03-fd08a495dbd8 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAtsAAADdCAIAAADl1CnLAACf5UlEQVR4nO2ddXwT2ff3b5K6u7dIcSlWCou77uKyy7KLuy6wsLg7i9vi0AJFi0sFaQt16t7UvUnaxm0yzx+zvzz5liZNcwKhy7z/4FWa6aefzmTu3Jx77jkUHMcRCQkJCQkJCYlOoeraAAkJCQkJCQkJOSMhISEhISEh+QYgZyQkJCQkJCQkuoeckZCQkJCQkJDoHnJGQkJCQkJCQqJ7yBkJCQkJCQkJie4hZyQkJCQkJCQkuoeckZCQkJ... | Using the given graphs of $f(x)$ and $g(x)$, find $g\left(f(1)\right)$. | The final answer:
$g\left(f(1)\right)$ = $4$ |
3ef90475-04a4-4e4f-9125-9db390b3a277 | multivariable_calculus | false | null | Determine the coordinates, if any, for which $f(x,y) = 24 \cdot x + 12 \cdot y - 8 \cdot x \cdot y - 4 \cdot x^2 - 6 \cdot y^2$ 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 $P(6,-3)$ with the value(s) $54$
3. The function $f(x,y)$ has a Saddle Point(s) at None |
3f0c9f6c-3adf-47e7-abca-d957dd1b58e5 | 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{2 \cdot x-x^2}$ with the center $a=1$. | $\sqrt{2 \cdot x-x^2}$ = $\sum_{n=0}^\infty\left((-1)^n\cdot C_{\frac{1}{2}}^n\cdot(x-1)^{2\cdot n}\right)$ |
3f25f089-28ec-4e69-9844-a0fec7031aee | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQEAAAEaCAIAAACen3XyAAB8pklEQVR4nO2dd1wUV/f/Z3thl87Se5VepEkRUESxd6PGlqhBY0msT0xM0+TRmESjMYmmGHvvDRFBiqB06Sy9l2WX7X3m98f9PvPa3wIrKijCvv+C2bt3Z2fnzL333HM+B4MgCKRFyygG+7ZPQIuWt4zWBrSMdrQ2oGW0o7UBLaMdrQ1oGe1obUDLaEdrA1pGO1ob0DLa0dqAltGO1ga0jHbwb/sERiZSqZTP5+PxeF1dXSxW+6AZ1mh/ns... | Find the surface area bounded by the curves $y=2^x$, $y=2^{-2 \cdot x}$, and $y=4$. | $S$ = $\frac{24-\frac{9}{\ln(2)}}{2}$ |
3f5c9b71-a742-4303-afc0-19085f5bde1e | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAboAAAGeCAIAAABtqFPyAABtj0lEQVR4nO2dd3wU5/H/Z/ZOvQsJFUASRRRRBKjQDdiY6p44phk7btiOnbgktvNN7LgkThyn2D877t3gDjY2HQymSKJ3JCEEAtEF6iddf+b3x7O3nE8SnMTd6RDzfiX4dLe3z7N7u5+d55l5ZpCIgGF8hbzeENH5T/kOEWn/Nt2SYdodfXt3gLmy2L9//9GjR+Pj4wcMGBASEoKIx48f37t3LwCMHj06PDxc08cDBw4kJydHRESwYjJ+gt... | Use the following graph to evaluate:
1. $f'(-0.5)$
2. $f'(0)$
3. $f'(1)$
4. $f'(2)$
5. $f'(3)$
If it doesn't exist, then write $\text{None}$. | 1. $f'(-0.5)$: $1$
2. $f'(0)$: None
3. $f'(1)$: $-1$
4. $f'(2)$: None
5. $f'(3)$: $2$ |
3f808070-9259-416b-a36a-b19055958dcb | integral_calc | false | null | Solve the integral:
$$
\int \tan(x)^4 \, dx
$$ | $\int \tan(x)^4 \, dx$ = $C+\frac{1}{3}\cdot\left(\tan(x)\right)^3+\arctan\left(\tan(x)\right)-\tan(x)$ |
3f809052-b6b6-47df-b4fc-9b20684a67e7 | sequences_series | false | null | Find the Fourier series of the function $u = \left| \sin(x) \right|$ in the interval $[-\pi, \pi]$. | The Fourier series is: $\frac{2}{\pi}-\frac{4}{\pi}\cdot\left(\frac{\cos(2\cdot x)}{1\cdot3}+\frac{\cos(4\cdot x)}{3\cdot5}+\frac{\cos(6\cdot x)}{5\cdot7}+\cdots\right)$ |
3fae81c2-34d5-40d0-a54b-acbf74cc2037 | precalculus_review | false | null | The cost to remove a toxin from a lake is modeled by the function $C(p) = \frac{ 75 \cdot p }{ 85-p }$, where $C$ is the cost (in thousands of dollars) and $p$ is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than $85$ ppb.
1. Find ... | 1. The cost to remove $25$ ppb is: $31.25$ thousand dollars
The cost to remove $40$ ppb is: $66.66666667$ thousand dollars
The cost to remove $50$ ppb is: $107.14285714$ thousand dollars
2. $p(C)=\frac{85\cdot C}{75+C}$
3. $34$ ppb |
3fc943b6-d10d-40d3-965c-2b98d04be288 | multivariable_calculus | false | null | Calculate the area of the surface formed by rotating the astroid $x^{\frac{ 2 }{ 3 }} + y^{\frac{ 2 }{ 3 }} = 2^{\frac{ 2 }{ 3 }}$ about the x-axis. | The final answer: $\frac{48\cdot\pi}{5}$ |
4050b38b-62c7-4952-b909-c0012fe5130d | algebra | false | null | Use the properties of logarithms to expand the logarithm $\ln\left(y \cdot \sqrt{\frac{ y }{ 1-y }}\right)$ as much as possible. Rewrite the expression as a sum, difference, or product of logs. | The final answer: $\frac{3}{2}\cdot\ln(y)-\frac{1}{2}\cdot\ln(1-y)$ |
405af248-578b-4765-bfd1-52a2223805d6 | algebra | false | null | Multiply the rational expressions and express the product in simplest form:
$$
\frac{ 2 \cdot d^2 + d - 45 }{ d^2 + 7 \cdot d + 10 } \cdot \frac{ 4 \cdot d^2 + 7 \cdot d - 2 }{ 4 \cdot d^2 + 31 \cdot d - 8 }
$$ | The final answer: $\frac{(2\cdot d-9)}{(d+8)}$ |
40a59d5b-7cc2-4527-8be9-d64c434be864 | multivariable_calculus | false | null | Evaluate $\int\int_{D}{\left(\int_{0}^{4 \cdot x^2+4 \cdot y^2}{y \, dz}\right) \, dA}$, where $D = \left\{(x,y) | x^2+y^2 \le 4, y \ge 1, x \ge 0\right\}$ is the projection of $E$ onto the $x \cdot y$-plane. | $I$ = $\frac{274}{15}$ |
412a48cb-7490-498f-ae2c-160d74d3e912 | sequences_series | false | null | Consider the function $y = \left| \cos\left( \frac{ x }{ 4 } \right) \right|$.
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... | 1. The Fourier series is $\frac{2}{\pi}+\sum_{n=1}^\infty\left(\frac{4\cdot(-1)^n}{\pi\cdot\left(1-4\cdot n^2\right)}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)$
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{ ... |
413e37c6-98fa-4a69-b221-df34e3edf581 | precalculus_review | false | null | Solve $\cos(2 \cdot t) - 5 \cdot \sin(t) - 3 = 0$. | The final answer: $t=(-1)^{n+1}\cdot\frac{\pi}{6}+n\cdot\pi$ |
4209a5fe-b497-4040-8ef0-966c0d49f267 | multivariable_calculus | false | null | Find the center of mass of the region $\rho(x,y,z) = z$ on the inverted cone with radius $2$ and height $2$. | Center of mass: $P\left(0,0,\frac{8}{5}\right)$ |
429cab49-5ba8-46af-84cc-70f96e481e6a | algebra | false | null | Solve the following equations:
1. $-10 c = -80$
2. $n - (-6) = 12$
3. $-82 + x = -20$
4. $- \frac{ r }{ 2 } = 5$
5. $r - 3.4 = 7.1$
6. $\frac{ g }{ 2.5 } = 1.8$
7. $4.8 m = 43.2$
8. $\frac{ 3 }{ 4 } t = \frac{ 9 }{ 20 }$
9. $3\frac{ 2 }{ 3 } + m = 5\frac{ 1 }{ 6 }$ | The solutions to the given equations are:
1. $c=8$
2. $n=6$
3. $x=62$
4. $r=-10$
5. $r=10.5$
6. $g=\frac{ 9 }{ 2 }$
7. $m=9$
8. $t=\frac{3}{5}$
9. $m=\frac{3}{2}$ |
42ad9bcd-2e08-48bf-b2c5-8cbbbf1603db | integral_calc | false | null | The region bounded by the arc of the curve $y = \sqrt{2} \cdot \sin(2 \cdot x)$, $0 \le x \le \frac{ \pi }{ 2 }$, is revolved around the x-axis. Compute the surface area of this solid of revolution. | Surface Area: $\frac{\pi}{4}\cdot\left(12\cdot\sqrt{2}+\ln\left(17+12\cdot\sqrt{2}\right)\right)$ |
42c46aac-2976-4b49-a356-6d6ad55f62b2 | sequences_series | false | null | Given that $\frac{ 1 }{ 1-x } = \sum_{n=0}^\infty x^n$, use term-by-term differentiation or integration to find a power series for the function $f(x) = \frac{ 2 \cdot x }{ \left(1-x^2\right)^2 }$ centered at $x=0$. | $\frac{ 2 \cdot x }{ \left(1-x^2\right)^2 }$ = $\sum_{n=0}^\infty\left(2\cdot(n+1)\cdot x^{2\cdot n+1}\right)$ |
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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.