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 |
|---|---|---|---|---|---|
d12b5094-3b3b-454d-a3b5-4b488eeee7c8 | sequences_series | false | null | Express the series $\sum_{n=1}^\infty \left(\frac{ 1 }{ (x-3)^{2 \cdot n-1} }-\frac{ 1 }{ (x-2)^{2 \cdot n-1} }\right)$ as a rational function. | $\sum_{n=1}^\infty \left(\frac{ 1 }{ (x-3)^{2 \cdot n-1} }-\frac{ 1 }{ (x-2)^{2 \cdot n-1} }\right)$ = $\frac{x-3}{(x-3)^2-1}-\frac{x-2}{(x-2)^2-1}$ |
d1429fff-1216-40ad-8946-3dd3a5f2e620 | multivariable_calculus | false | null | Find the equations of the common tangent lines to the following ellipses:
1. $ rac{ x^2 }{ 6 } + y^2 = 1$
2. $ rac{ x^2 }{ 4 } + rac{ y^2 }{ 9 } = 1$ | The equations are: $2\cdot x+y-5=0 \lor 2\cdot x+y+5=0 \lor 2\cdot x-y-5=0 \lor 2\cdot x-y+5=0$ |
d154f14e-4cc7-40dd-ae67-66d209b8bf88 | integral_calc | false | null | Compute the integral:
$$
\int \sin(5 \cdot x)^6 \cdot \cos(5 \cdot x)^2 \, dx
$$ | Answer is: $\frac{1}{32}\cdot x-\frac{1}{32}\cdot\frac{1}{20}\cdot\sin(20\cdot x)-\frac{1}{8}\cdot\frac{1}{10}\cdot\frac{1}{3}\cdot\sin(10\cdot x)^3+\frac{1}{128}\cdot x-\frac{1}{128}\cdot\frac{1}{40}\cdot\sin(40\cdot x)+C$ |
d16d84d7-3932-44f0-89ff-521caa493041 | precalculus_review | false | null | Recall the following double-angle formulas:
1. $\sin\left(2 \cdot \theta\right) = 2 \cdot \sin\left(\theta\right) \cdot \cos\left(\theta\right)$
2. $\cos\left(2 \cdot \theta\right) = 2 \cdot \cos\left(\theta\right)^2 - 1$
3. $\cos\left(2 \cdot \theta\right) = 1 - 2 \cdot \sin\left(\theta\right)^2$
From the formulae, ... | Part (a): $\sin\left(\theta\right)^2=\frac{1}{2}\cdot\left(1-\cos\left(2\cdot\theta\right)\right)$ Part (b): $\cos\left(\theta\right)^2=\frac{1}{2}\cdot\left(1+\cos\left(2\cdot\theta\right)\right)$ |
d1fe21df-ee7f-40c2-9655-6bd6a7a23ff1 | sequences_series | false | null | Compute the first 4 nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of $f(x) = e^x \cdot \cos(x)$. | $f(x)$ = $1+x-\frac{x^3}{3}-\frac{x^4}{6}+\cdots$ |
d23ad7e7-ae9b-468a-b854-8452c64e0896 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAVUAAAEcCAIAAAD4ObDdAAB/yElEQVR4nO29d3wTx/M/fKdebEvuvRtj0wwG0zuYGiB0QggtgYQSSgJplIRQQ0lCyYeWQCAk1ITeMb0YjME2uGDADXe5SLa6Trrnj/mxz31PspG7EHr/4Zd8t7c7t7ezOzs7BSdJErPBBhveSTCamgAbbLChyWDjfxtseHdh438bbHh3YeN/G2x4d2HjfxtseHdh438bbHh3YeN/G2x4d2HjfxtseHdh438bbHh3YeN/G2x4d2HjfxtseH... | The ellipse $x^2 + 4 \cdot y^2 = 4$ and the parabola $4 \cdot y = 4 - 5 \cdot x^2$ intersect at the points $A$, $B$, and $C$ (see the figure below). Compute the tangent of the angle between the curves at all three intersection points. | 1. The tangent of the angle at $A$ is $-27$
2. The tangent of the angle at $B$ is $0$
3. The tangent of the angle at $C$ is $27$ |
d24d7863-cfff-4e13-9077-843af80b36a4 | differential_calc | false | null | Find the local minimum and local maximum values of the function $f(x) = \frac{ 3 }{ 4 } \cdot x^4 - 10 \cdot x^3 + 24 \cdot x^2 - 4$. | The point(s) where the function has a local minimum: $P(0,-4)$, $P(8,-516)$
The point(s) where the function has a local maximum: $P(2,24)$ |
d2a729e3-d012-4283-b8b4-1874c307c5b2 | precalculus_review | false | null | Find zeros of $f(x) = \sqrt{x-3} + \sqrt{x+2} - \sqrt{2 \cdot x-5} - 2$ | The final answer: $x=7$ |
d33534d4-e0dd-4da6-9c8e-02e5c4837211 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ x^3-2 \cdot x^2+x }{ 3+2 \cdot x-x^2 } \, dx
$$ | $\int \frac{ x^3-2 \cdot x^2+x }{ 3+2 \cdot x-x^2 } \, dx$ = $-\frac{1}{2}\cdot x^2-2\cdot\ln\left(\left|x^2-2\cdot x-3\right|\right)-\ln\left(\left|\frac{x-3}{x+1}\right|\right)+C$ |
d34c2468-5dd2-4918-ac13-a2f46633fb66 | 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 around the neighborhood of $x=1$. | $\lim_{{x \to 1}} f(x) = $\lim_{x\to1}f(x)=0$$ |
d3cdc4b9-7a58-4198-ba50-e9f278bf2aa3 | multivariable_calculus | false | null | Evaluate $L=\lim_{(x,y) \to (-1,1)}\left(\frac{ x \cdot y \cdot (x+y)-5 \cdot y \cdot (y+5)+5 \cdot x \cdot (x-5) }{ x^2+y^2+2 \cdot x \cdot y+5 \cdot x+5 \cdot y }\right)$. | The final answer: $L=-\frac{36}{5}$ |
d41983e6-eea1-4dfa-8060-ce53dcd25940 | precalculus_review | false | null | A rental car company rents cars for a flat fee of $\$20$ and an hourly charge of $\$10.25$. Therefore, the total cost $C$ to rent a car is a function of the hours $t$ the car is rented plus the flat fee.
1. Write the formula for the function that models this situation.
2. Find the total cost to rent a car for 2 days a... | 1. The formula for the function is $C(t)$ = $10.25\cdot t+20$
2. The total cost to rent a car is $ $583.75$
3. The time of car rental in hours is $40.26634146$ |
d430a0dd-708d-4daa-8072-d39638d29242 | differential_calc | false | null | A box is to be made with the following properties:
1. The length of the base, $l$, is twice the length of a width $w$.
2. The cost of material to be used for the lateral faces and the top of the box is three times the cost of the material to be used for the lower base.
Find the dimensions of the box in terms of its f... | The final answer: $w=\frac{1}{2}\cdot\sqrt[3]{\frac{9\cdot V}{2}}$, $l=\sqrt[3]{\frac{9\cdot V}{2}}$, $y=\frac{2}{3}\cdot\sqrt[3]{\frac{4\cdot V}{3}}$ |
d4ce3600-3ef0-47d0-8550-0158a474ed33 | 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)$ = $1$ |
d4f54a6b-e604-4bf3-8782-acbc409ea7d2 | 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 + z = 6 \cdot z^2$ | $z$ = $\frac{2}{3}$, $-\frac{1}{2}$ |
d503c1fb-2754-4803-b9ac-6462fc93e8db | differential_calc | false | null | Find the derivative of the 27th order $y^{(27)}$ for a function $y = 2 \cdot x^2 \cdot \sin(x)$. | $y^{(27)}$ = $1404\cdot\cos(x)-2\cdot x^2\cdot\cos(x)-108\cdot x\cdot\sin(x)$ |
d541eb2d-198c-47dd-8811-79633c9e701e | integral_calc | false | null | Solve the integral:
$$
\int \frac{ 1 }{ \sin(x)^5 } \, dx
$$ | The final answer: $C+\frac{1}{16}\cdot\left(2\cdot\left(\tan\left(\frac{x}{2}\right)\right)^2+6\cdot\ln\left(\left|\tan\left(\frac{x}{2}\right)\right|\right)+\frac{1}{4}\cdot\left(\tan\left(\frac{x}{2}\right)\right)^4-\frac{2}{\left(\tan\left(\frac{x}{2}\right)\right)^2}-\frac{1}{4\cdot\left(\tan\left(\frac{x}{2}\right... |
d57c2f7d-971f-43c9-bb9e-25e15c923b21 | multivariable_calculus | false | null | The solid $Q$ has the mass given by the triple integral:
$$
\int_{0}^1 \left( \int_{0}^{\frac{ \pi }{ 2 }} \left( \int_{0}^{r^2} \left( r^4 + r \right) dz \right) d \theta \right) dr
$$
1. Find the density of the solid in rectangular coordinates.
2. Find the moment $M_{xy}$ about the xy-plane. | 1. $\rho(x,y,z)$ = $1+\left(x^2+y^2\right)^{\frac{3}{2}}$
2. $M_{xy}$ = $\frac{5\cdot\pi}{72}$ |
d5eda09b-3de0-441e-a722-ccb5244adbb9 | integral_calc | false | null | Solve the integral:
$$
\int \frac{ \sqrt{9 \cdot x+4} }{ -3 \cdot x^2 } \, dx
$$ | $\int \frac{ \sqrt{9 \cdot x+4} }{ -3 \cdot x^2 } \, dx$ = $C+\frac{\sqrt{9\cdot x+4}}{3\cdot x}-\frac{3}{4}\cdot\ln\left(\frac{\left|\sqrt{9\cdot x+4}-2\right|}{2+\sqrt{9\cdot x+4}}\right)$ |
d5fcba67-c072-4cb0-8f4d-9da61c993923 | algebra | false | null | Evaluate the expression $\frac{ (2+i) \cdot (4-2 \cdot i) }{ (1+i) }$ and write the result as a simplified complex number. | $\frac{ (2+i) \cdot (4-2 \cdot i) }{ (1+i) }$ = $5-5\cdoti$ |
d602c133-9e68-402e-9b69-01ca11fa01fd | integral_calc | false | null | Compute the integral:
$$
\int \frac{ 2 \cdot x+1 }{ (x-1) \cdot \sqrt{x^2-4 \cdot x+2} } \, dx
$$ | $\int \frac{ 2 \cdot x+1 }{ (x-1) \cdot \sqrt{x^2-4 \cdot x+2} } \, dx$ = $2\cdot\ln\left(\left|x-2+\sqrt{x^2-4\cdot x+2}\right|\right)-3\cdot\arcsin\left(\frac{x}{(x-1)\cdot\sqrt{2}}\right)+C$ |
d666f55d-1b0e-4b1c-821c-ddf1d1ea87df | integral_calc | false | null | Compute the integral:
$$
\int \frac{ -1 }{ 3 \cdot \sin\left(\frac{ x }{ 3 }\right)^6 } \, dx
$$ | $\int \frac{ -1 }{ 3 \cdot \sin\left(\frac{ x }{ 3 }\right)^6 } \, dx$ = $\frac{\cos\left(\frac{x}{3}\right)}{5\cdot\sin\left(\frac{x}{3}\right)^5}-\frac{4}{15}\cdot\left(-\frac{\cos\left(\frac{x}{3}\right)}{\sin\left(\frac{x}{3}\right)^3}-2\cdot\cot\left(\frac{x}{3}\right)\right)+C$ |
d6cfc281-9a27-43ac-b0d7-17e6d64a8bf9 | multivariable_calculus | false | null | Find the equation of a quadratic curve, which passes through the origin, and is tangent to $4 \cdot x + 3 \cdot y + 2 = 0$ at $(1,-2)$ and to $x - y - 1 = 0$ at $(0,-1)$. | The final answer: $6\cdot x^2-y^2+3\cdot x\cdot y+2\cdot x-y=0$ |
d6d00db5-5433-4e2e-be1f-07df6d0782c8 | multivariable_calculus | false | null | Find the first derivative $y_{x}'$ of the function:
$$
x = \arcsin\left(\frac{ t }{ \sqrt{2+2 \cdot t^2} }\right), \quad y = \arccos\left(\frac{ 1 }{ \sqrt{2+2 \cdot t^2} }\right), \quad t \ge 0
$$ | $y_{x}'$ = $\frac{t\cdot\sqrt{t^2+2}}{\sqrt{2\cdot t^2+1}}$ |
d6d0e931-8b06-4a31-80b1-2abb2700d680 | precalculus_review | false | null | 1. Find the inverse function of $f(x) = (x-1)^2$, for $x \le 1$.
2. Find the domain of the inverse function.
3. Find the range of the inverse function. | 1. The inverse function is: $1-\sqrt{x}$
2. The domain of the inverse function is: $x\ge0$
3. The range of the inverse function is: $y\le1$ |
d6e1f663-40a9-40f9-a0c7-53c225984de0 | multivariable_calculus | false | null | Find the area of the region $R$ bounded by circles $x^2+y^2=1$ and $x^2+y^2=4$ and the lines $y=x$ and $y=\sqrt{3} \cdot x$. | The final answer: $\frac{\pi}{8}$ |
d6ec8d67-1fdb-4d5b-ba6c-40d38b33cabb | algebra | false | null | Solve the following equations:
1. $\frac{ 3 }{ 4 } (x+9) = 15$
2. $\frac{ 2 }{ 3 } (m-8) = 4$
3. $\frac{ 1 }{ 8 } (b+2) = \frac{ 3 }{ 8 }$
4. $\frac{ 1 }{ 7 } (x-4) = -\frac{ 1 }{ 2 }$
5. $\frac{ 2 }{ 5 } (-3-2 x) = \frac{ 18 }{ 5 }$
6. $\frac{ 4 }{ 5 } (j+5) = \frac{ 2 }{ 5 }$ | The solutions to the given equations are:
1. $x=11$
2. $m=14$
3. $b=1$
4. $x=\frac{ 1 }{ 2 }$
5. $x=-6$
6. $j=\frac{ -9 }{ 2 }$ |
d6fcf747-13d5-42bb-99f4-1c6e55d5cdb5 | multivariable_calculus | false | null | Consider a rectangular box of width $w$, length $l$, and height $h$ without a lid. If the box is to be made from $12$ sq. cm of cardboard, find the dimensions that maximize the volume of the box and the maximum possible volume itself. | Optimal dimensions:
$w$: $2$
$l$: $2$
$h$: $1$
Maximum volume: $4$ |
d7685a03-c915-479a-bc22-d9a0106fca54 | precalculus_review | false | null | State the domain and range of the function: $f(x) = \ln(x-5)$
1. Domain
2. Range | 1. Domain: $(5,\infty)$
2. Range: $(-\infty,\infty)$ |
d78475ef-cb8a-41af-9269-a2327c4a2fff | differential_calc | false | null | Make full curve sketching of $y = \sqrt[3]{9 \cdot x^2 - x^3}$. 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 functi... | 1. The domain (in interval notation) $(-1\cdot\infty,\infty)$
2. Vertical asymptotes None
3. Horizontal asymptotes None
4. Slant asymptotes $y=-x+3$
5. Intervals where the function is increasing $(0,6)$
6. Intervals where the function is decreasing $(-\infty,0)$, $(6,\infty)$
7. Intervals where the function is concave ... |
d8e23310-2d6a-4413-9633-55f92b63775c | algebra | false | null | Recall the formula for calculating the magnitude of an earthquake, $M=\frac{ 2 }{ 3 } \cdot \log_{10}\left(\frac{ S }{ S_{0} }\right)$. One earthquake has magnitude $3.9$ on the MMS scale. If a second earthquake has $750$ times as much energy as the first, find the magnitude of the second quake. Round to the nearest hu... | Magnitude of the second quake: $5.82$ |
d906874a-47e3-404c-9e09-885002ec79f3 | algebra | false | null | Find the solution to the following inequality and express it in interval notation:
$$
5 (x-7) (x+3) (x-2) < 0
$$ | The solution set to the inequality is: $\left(-\infty,\ -3\right)\cup\left(2,\ 7\right)$ |
d920bf7f-8420-4c43-9f6a-e8ac4d3f2423 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAACBZklEQVR4nO3deXhU9d3//1eAQAJhZBEQnBQlIpCwSRBIRMWFom21KpIggmyCS91ppVX0dmnvW2xppfrTr4BLZQ2rohaQWCFiEEzYAy6IYqI2IIpDgCzE/P6gmRLmM1ln5syZ83xcF9dFPmc4eecwk5nX+WxRFRUVFQqA3NxcSVJycnIgThdy1G+tWbNmSZImT55scSX1E87XPz8/XwMHDvRp/+abb7x/D+f6a4P6rTVjxgzNmDFDkjRlyh... | The functions $f$ and $g$ are differentiable functions. The graphs of $f'$ and $g'$, the derivatives of $f$ and $g$ respectively, are shown. Let $h(x) = \frac{ f(x) }{ 2 } - \frac{ 3 \cdot g(x) }{ 4 } + 2 \cdot x^2$. Find $h'(2)$.
| |
| --- | | $h'(2)$ = $10$ |
d930869f-c932-4aba-a4ea-e00bc94237a8 | integral_calc | false | null | Compute the area of the figure bounded by curves $y = 8 \cdot x^2$, $y = 4 + 4 \cdot x^2$, lines $x = 3$, $x = -2$, and the $x$-axis. | Area = $\frac{184}{3}$ |
d982ac94-be51-47d3-a37f-8d59be82c81e | precalculus_review | false | null | Find points on a coordinate plane that satisfy the following equation:
$$
10 \cdot x^2 + 29 \cdot y^2 + 34 \cdot x \cdot y + 8 \cdot x + 14 \cdot y + 2 = 0
$$ | The final answer: $(3,-2)$ |
d9a85aa2-237a-48ee-aa50-dadd42070cd8 | sequences_series | false | null | Find the radius of convergence and sum of the series:
$$
\frac{ 3 }{ 2 }+\frac{ 3 \cdot x }{ 1 \cdot 3 }+\frac{ 3 \cdot x^2 }{ 1 \cdot 2 \cdot 4 }+\cdots+\frac{ 3 \cdot x^n }{ \left(n!\right) \cdot (n+2) }+\cdots
$$ | 1. Radius of convergence: $R=\infty$
2. Sum: $f(x)=\begin{cases}\frac{3}{x^2}+\frac{3\cdot x\cdot e^x-3\cdot e^x}{x^2},&x\ne0\\\frac{3}{2},&x=0\end{cases}$ |
da100f17-a491-4014-8c00-0061a65cc2b4 | precalculus_review | false | null | 1. Find the inverse function of $f(x) = \frac{ 1 }{ x+2 }$
2. Find the domain of the inverse function
3. Find the range of the inverse function | 1. The inverse function is: $\frac{1}{x}-2$
2. The domain of the inverse function is: $(-\infty,0)\cup(0,\infty)$
3. The range of the inverse function is: $(-\infty,-2)\cup(-2,\infty)$ |
da29a7ac-6d10-4e13-9fc6-088951921221 | multivariable_calculus | false | null | Evaluate $I=\int_{0}^4 \int_{0}^x \int_{0}^{x+y} \left(3 \cdot e^x \cdot (y+2 \cdot z)\right) dz dy dx$. | The final answer: $I=19\cdot\left(17\cdot e^4+3\right)$ |
da7e571f-2201-495c-a7df-955283c258ef | algebra | false | null | Divide rational expressions and simplify it:
$$
\frac{ x^2-9 }{ 3 } \div \frac{ x-3 }{ 4 }
$$ | The final answer: $\frac{4}{3}\cdot x+4$ |
daa2fcb5-aa37-4baf-9417-863cc372bd78 | differential_calc | false | null | For the curve $x = a \cdot \left(t - \sin(t)\right)$, $y = a \cdot \left(1 - \cos(t)\right)$, determine the curvature. Use $a = 12$. | The curvature is: $\frac{1}{48\cdot\left|\sin\left(\frac{t}{2}\right)\right|}$ |
dac50753-b5d3-4c11-a1c7-1575dde6e606 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAywAAAEnCAYAAABL1sd3AAAYU2lDQ1BJQ0MgUHJvZmlsZQAAeJyVeQVUVF8X77mTzDAM3d0l3SAxdHeDwNAdQ4NBiYSKIKCUCioIIliEiIUgooigAgYiIUqpoIIiIO8S+v2//7fWe+udWefe3+yzz46zT+0ZADhTyJGRoQg6AMLCYyi2Rrp8zi6ufNgpAMEfIkACcbJPdCTJ2tocwOXP+7/L8hDMC5fnUpuy/rf9/1roff2ifQCArGHs7RvtEwbj6wCgMnwiKTEAYFRhum... | The following table lists the $2013$ schedule of federal income tax versus taxable income:
Federal Income Tax Versus Taxable Income
Suppose that Steve just received a $10,000 raise. How much of this raise is left after federal taxes if Steve's salary before receiving the raise was
1. $\$40,000$,
2. $\$90,000$, and
... | 1. If Steve's salary before receiving the raise is $\$40,000$, $$7500$$ is left.
2. If Steve's salary before receiving the raise is $\$90,000$, $$7200$$ is left.
3. If Steve's salary before receiving the raise is $\$385,000$, $$6700$$ is left. |
dacc8400-999f-45b5-9169-bbc3631b4aeb | multivariable_calculus | false | null | Let $z = x^2 + 3 \cdot x \cdot y - y^2$. Find the exact change $\Delta z = f\left(x+\Delta x,y+\Delta y\right) - f(x,y)$ in the function and find the approximate change $d z$ in the function as $x$ changes from $2$ to $2.05$ and $y$ changes from $3$ to $2.96$. | The final answer:
1. The exact change $\Delta z$ is $0.6449$
2. The approximate change $d z$ is $0.65$ |
db0e5db4-a7a3-45e3-abc6-9ae4b77fcf97 | algebra | false | null | Add and subtract the rational expressions, and then simplify:
$$
\frac{ x-4 }{ x+4 } - \frac{ 2 \cdot x+6 }{ 2 \cdot x+1 }
$$ | The final answer: $\frac{-21\cdot x-28}{2\cdot x^2+9\cdot x+4}$ |
db737faf-9359-40c5-9af9-bd828d9cb054 | precalculus_review | false | null | Simplify the expression $\frac{ \cos(t) }{ \sin(t) }+\frac{ \sin(t) }{ 1+\cos(t) }$ by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. | $\frac{ \cos(t) }{ \sin(t) }+\frac{ \sin(t) }{ 1+\cos(t) }$ = $\csc(t)$ |
db73d0f5-3e61-4495-9063-c8cb58f9e4b3 | multivariable_calculus | false | null | Evaluate $L=\lim_{(x,y) \to (3,3)}\left(\frac{ x^6-y^6 }{ x^2+x^2 \cdot y-x \cdot y^2-y^2 }\right)$ | The final answer: $L=\frac{486}{5}$ |
db7ee4b1-97eb-441e-8ab6-9f71a248e8be | integral_calc | false | null | Compute the integral:
$$
\int \frac{ 10 }{ \sin(4 \cdot x)^6 } \, dx
$$ | $\int \frac{ 10 }{ \sin(4 \cdot x)^6 } \, dx$ = $C-\frac{1}{2}\cdot\left(\cot(4\cdot x)\right)^5-\frac{5}{2}\cdot\cot(4\cdot x)-\frac{5}{3}\cdot\left(\cot(4\cdot x)\right)^3$ |
db996e6d-cc8a-4fb9-84e6-66007524a88c | precalculus_review | false | null | Find the zeros of the function $f(x) = \left(1 - \tan(x)\right) \cdot \left(1 + \sin(2 \cdot x)\right) - \tan(x) + \cos(2 \cdot x) - 1$. | The final answer: $x_1=-\frac{\pi}{4}+\pi\cdot n$, $x_2=\arctan\left(\frac{1}{2}\right)+\pi\cdot n$ |
dc06d10a-40f2-42cc-a038-53faf5f27d1c | sequences_series | false | null | Find the Fourier series of the periodic function $f(x) = x^2$ in the interval $-\pi \le x < \pi$ if $f(x) = f(x + 2 \cdot \pi)$. | The Fourier series is: $x^2=\frac{\pi^2}{3}-4\cdot\left(\frac{\cos(x)}{1^2}-\frac{\cos(2\cdot x)}{2^2}+\frac{\cos(3\cdot x)}{3^2}-\cdots\right)$ |
dc20dd10-80d5-4ed4-8e8b-1f706dc8ffb6 | sequences_series | false | null | Evaluate the telescoping series or state whether the series diverges:
$$
\sum_{n=1}^\infty \left(\sin(n)-\sin(n+1)\right)
$$ | $S_{k}$ -> None as $k$ -> $\infty$. |
dc221026-1425-4caa-9b02-8ddde198eebe | multivariable_calculus | false | null | Express the region $D$ in polar coordinates. $D$ = $\left\{(x,y)|x^2+y^2 \le 4 \cdot x\right\}$ ($a \le \theta \le b$, $a \ge 0$, $b \ge 0$). | 1. The interval for $r$ is $\left[0,4\cdot\cos\left(\theta\right)\right]$;
2. The interval for $\theta$ is $\left[\frac{3}{2}\cdot\pi,\frac{5\cdot\pi}{2}\right]$. |
dc3838be-dabf-496c-90ab-3fe766bda473 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAY8AAAGlCAYAAAACrXq3AADNDElEQVR4nOy9Z5gdx3Um/HbfHCfPYJADwQAmMIhiEoNESqJEURYlSlaW12HXu7JlW2vZn7N3P+tby17ba3tty0GUJVuSRSWaJsEkMYhBYoIIgASIDEy6cebm27fj92N4Cn3vdFX3YAYz94L9Pg8fDqbPVJ2uqq5T4bznSJZlWfDhw4cPHz4WAXm1FfDhw4cPH70H33j48OHDh49FwzcePnz48OFj0fCNhw8fPnz4WDR84+HDhw8fPhYN33... | The graph below is a function, $j$, whose domain is the set of all real numbers and is continuous everywhere. Determine the x-values for points of inflection. | There is a point of inflection at x-value(s): $x=-2.5$, $x=0$, $x=2.5$ |
dc71824f-02b4-4210-a32a-0d141c9fcf5c | precalculus_review | false | null | Simplify the expression $\frac{ 1+\tan\left(\alpha\right)^2 }{ 1+\cot\left(\alpha\right)^2 }$ by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. | $\frac{ 1+\tan\left(\alpha\right)^2 }{ 1+\cot\left(\alpha\right)^2 }$ = $\tan\left(\alpha\right)^2$ |
dcb60112-7be2-4d1a-b711-5f77dd0dc4bd | integral_calc | false | null | Find the integral:
$$
\int \frac{ 1 }{ \sqrt[3]{\left(\sin(x)\right)^{11} \cdot \cos(x)} } \, dx
$$ | $\int \frac{ 1 }{ \sqrt[3]{\left(\sin(x)\right)^{11} \cdot \cos(x)} } \, dx$ = $-\frac{3\cdot\left(1+4\cdot\left(\tan(x)\right)^2\right)}{8\cdot\left(\tan(x)\right)^2\cdot\sqrt[3]{\left(\tan(x)\right)^2}}+C$ |
dce6ec64-2f61-4db6-9f12-8e5de1da1c22 | multivariable_calculus | false | null | Find the tangential and normal components of acceleration for $\vec{r}(t) = \left\langle a \cdot \cos\left(\omega \cdot t\right), b \cdot \sin\left(\omega \cdot t\right) \right\rangle$ at $t=0$. | The final answer:
1. The tangential component of acceleration, $a_{t}$, is: $0$
2. The normal component of acceleration, $a_{N}$, is: $\omega^2\cdot|a|$ |
dd266316-efe9-4f84-ba9d-09ebb753bc3c | integral_calc | false | null | Consider the Karun-$3$ dam in Iran. Its shape can be approximated as an isosceles triangle with height $205$ m and width $388$ m. Assume the current depth of the water is $180$ m. The density of water is $1000$ kg/$m^3$. Find the total force on the wall of the dam. | The total force : $18028940487.8049$ |
dd3412c0-c4c3-4b9f-8035-8467e18b9f5f | precalculus_review | false | null | Calculate $\cos\left(\frac{ \pi }{ 12 }\right)$. | The final answer: $\cos\left(\frac{\pi}{12}\right)=\frac{\sqrt{2}+\sqrt{6}}{4}$ |
de2bd512-089e-4d1e-b577-dd185962e7cf | differential_calc | false | null | Sketch the curve:
$y = 3 \cdot x \cdot \sqrt{2-x^2}$.
Submit as your final answer:
1. The domain (in interval notation)
2. Vertical asymptotes
3. Horizontal asymptotes
4. Slant asymptotes
5. Intervals where the function is increasing
6. Intervals where the function is decreasing
7. Intervals where the function is c... | 1. The domain (in interval notation): $\left[-1\cdot2^{2^{-1}},2^{2^{-1}}\right]$
2. Vertical asymptotes: None
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: $(-1,1)$
6. Intervals where the function is decreasing: $\left(-2^{2^{-1}},-1\right)$, $\left(1,2^{2^{-1}... |
df12f49b-9f83-4e83-bf35-dcfb2c0375ce | sequences_series | false | null | Find the sum of the series $\sum_{n=1}^\infty \frac{ x^{4 \cdot n-3} }{ 4 \cdot n-3 }$. (Use differentiation of the 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)$ |
df82e2b4-2fde-4ea7-8256-2cefd78de08c | differential_calc | false | null | For the function $y = \sqrt[3]{2-x} + \frac{ 1 }{ 9 } \cdot x^2$ determine the intervals, where the function is concave up and concave down and points of inflection. Submit as your final answer:
1. Interval(s) where the function is concave up
2. Interval(s) where the function is concave down
3. Point(s) of inflection | 1. Interval(s) where the function is concave up: $(-\infty,1)$, $(2,\infty)$
2. Interval(s) where the function is concave down: $(1,2)$
3. Point(s) of inflection: $P\left(1,\frac{10}{9}\right)$, $P\left(2,\frac{4}{9}\right)$ |
df9d5d54-0fe2-4815-8a98-f454e166f165 | sequences_series | false | null | Determine the Taylor series for $f(x) = \frac{ 2 \cdot x - 1 }{ x^2 - 3 \cdot x + 2 }$, centered at $x_{0} = 4$. Write out the sum of the first four non-zero terms, followed by dots. | The final answer: $\frac{7}{6}+\left(\frac{1}{3^2}-\frac{3}{2^2}\right)\cdot(x-4)-\left(\frac{1}{3^3}-\frac{3}{2^3}\right)\cdot(x-4)^2+\left(\frac{1}{3^4}-\frac{3}{2^4}\right)\cdot(x-4)^3+\cdots$ |
dfce03fb-fa8b-46ea-94b9-343e9c0ada11 | integral_calc | false | null | Compute the length of the arc $y = 2 \cdot \ln(3 \cdot x)$ between the points $x = \sqrt{5}$ and $x = 2 \cdot \sqrt{3}$. | Arc Length: $1+\ln\left(\frac{5}{3}\right)$ |
e01089ef-d282-4c55-b0cc-0fb6bc6d2bdf | precalculus_review | false | null | Solve $2 \cdot \left(\arcsin(x)\right)^2 - \arcsin(x) - 6 = 0$. | The final answer: $x=-\sin(1.5)$ |
e09be213-4d6a-4283-860d-c2f57bdaf5d4 | algebra | false | null | Let $f(x) = \frac{ 1 }{ x }$. Find the number $b$ such that the average rate of change of $f$ on the interval $(2,b)$ is $-\frac{ 1 }{ 10 }$. | The final answer: $b=5$ |
e12228fb-4501-47ee-83fc-3c74da98cc45 | multivariable_calculus | false | null | Use two circular permutations of the variables $x$, $y$, and $z$ to write new integral whose value equal the value of the original integral. A circular permutation of $x$, $y$, and $z$ is the arrangement of the numbers in one of the following orders: $y$, $z$, and $x$ or $z$, $x$, and $y$:
$$
\int_{0}^1 \int_{1}^3 \int... | 1. $I$ with $d z \cdot d x \cdot d y$ is $\frac{148}{9}$
2. $I$ with $d y \cdot d z \cdot d x$ is $\frac{148}{9}$ |
e1380c07-189c-47d2-92c7-e5456c086a22 | algebra | false | null | Solve the following equations:
1. $-15c = -75$
2. $-8 + r = 27$
3. $19 + m = 3$
4. $-\frac{w}{8} = 1$
5. $p - 4.8 = 11.2$
6. $\frac{g}{3.4} = 2.2$
7. $3.6m = 25.2$
8. $\frac{2}{5}t = \frac{8}{25}$
9. $5\frac{5}{6} + m = 3\frac{5}{12}$ | The solutions to the given equations are:
1. $c=5$
2. $r=35$
3. $m=-16$
4. $w=-8$
5. $p=16$
6. $g=7.48$
7. $m=7$
8. $t=\frac{ 4 }{ 5 }$
9. $m=-\frac{ 29 }{ 12 }$ |
e13c1022-2adc-4e50-a481-b6d868cea5be | multivariable_calculus | false | null | The volume of a right circular cylinder is given by $V(x,y) = \pi \cdot x^2 \cdot y$ where $x$ is the radius of the cylinder and $y$ is the cylinder height. Suppose $x$ and $y$ are functions of $t$ given by $x = \frac{ 1 }{ 2 } \cdot t$ and $y = \frac{ 1 }{ 3 } \cdot t$ so that $x$ and $y$ are both increasing with time... | The volume is increasing at a rate of $\frac{34\cdot\pi}{3}$ |
e159cbdd-10fe-4676-a0b0-6425fc80a525 | differential_calc | false | null | For the function $y = \frac{ 2 \cdot x + 3 }{ 4 \cdot x + 5 }$ find the derivative $y^{(n)}$. | The General Form of the Derivative of $y = \frac{ 2 \cdot x + 3 }{ 4 \cdot x + 5 }$: $y^{(n)}=\frac{1}{2}\cdot(-1)^n\cdot\left(n!\right)\cdot4^n\cdot(4\cdot x+5)^{-(n+1)}$ |
e16f36eb-5794-4d0a-9b53-77b2e1730e59 | differential_calc | false | null | Find the $x$-coordinate for each critical point of $f(x) = e^{-x} \cdot \sqrt{x}$ | $x$ = $\frac{1}{2}$, $0$ |
e1c6ee2e-873b-4879-a3ef-f06229679527 | sequences_series | false | null | Find the 3rd order Taylor polynomial $P_{3}(x)$ for the function $f(x) = \arctan(x)$ in powers of $x-1$ and give the Lagrange form of the remainder. | $P_{3}(x)$ = $\frac{ \pi }{ 4 }+\frac{ 1 }{ 2 } \cdot (x-1)-\frac{ 1 }{ 4 } \cdot (x-1)^2+\frac{ 1 }{ 12 } \cdot (x-1)^3$
$R_{3}(x)$ = $\frac{ -\frac{ 48 \cdot c^3 }{ \left(1+c^2\right)^4 }+\frac{ 24 \cdot c }{ \left(1+c^2\right)^3 } }{ 4! } \cdot (x-1)^4$ |
e1d09bc3-2f9a-4138-9ee0-fd559f60f732 | differential_calc | false | null | Find $\frac{ d y }{d x}$ if $y = \frac{ 5 \cdot x^2 - 3 \cdot x }{ \left( 3 \cdot x^7 + 2 \cdot x^6 \right)^4 }$. | $\frac{ d y }{d x}$ = $\frac{-390\cdot x^2+23\cdot x+138}{x^{24}\cdot(3\cdot x+2)^5}$ |
e20ecd1f-1245-4ec5-959b-ea8ea17a5d1e | sequences_series | false | null | Find $\lim_{n \to \infty} \left(x_{n}\right)$, where $x_{n} = \left(\frac{ 2 \cdot n^3 }{ 2 \cdot n^2+3 }+\frac{ 1-5 \cdot n^2 }{ 5 \cdot n+1 }\right)$ is the general term of a sequence. | The final answer: $L=\frac{1}{5}$ |
e26ffc23-804d-4eaf-9729-1820407e0c76 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPwAAAEjCAIAAACZ1vt+AABQeklEQVR4nO2deXwURdrHn6d6ZnJf5CIhhCMBQoJAuE9RFLkEZL0QxBVddN11V1f3VVfd1WU91mtFxftEdr1FEBCUQzAcAuEICeEKCZD7ziRzz3Q97x/VMxlyEWByTNLfjx+Z9HRXV/f8uvqpp556CokI2gsiQsTW7NOaPVVULg3WnidrTsfuD56qdZW2pj1E3+BlUlVVtXz58ueee85qtYotzz333HvvvVdVVeXax2Kx7Nq1qx3qptINaQ... | Find the average value of the function $f(x,y) = 3 \cdot x \cdot y$ on the region $D$: | The average value is $\frac{27}{20}$ |
e2afa39f-a784-407d-be4f-00e90b4ec92e | precalculus_review | false | null | Is the statement True or False?
The following linear programming problem is infeasible.
Maximize $f = 4 \cdot x_{1} - x_{2}$ subject to
$x_{1} - x_{2} \le 3$
$-x_{1} + 3 \cdot x_{2} \le -5$
$x_{1} \ge 0$, $x_{2} \ge 0$ | The final answer: $T\cdot r\cdot u\cdot e$ |
e2ec6409-4e11-40cc-a4a2-6c6b8f85fc24 | precalculus_review | false | null | Solve the exponential equation exactly: $4^{x+1} - 32 = 0$ | $x$ = $\frac{3}{2}$ |
e33483e5-a385-4ce3-91af-54df50ca062b | differential_calc | false | null | Sketch the curve:
$$
y = 4 \cdot x^2 \cdot e^{\frac{ 1 }{ 2 \cdot x }}
$$
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 wher... | 1. The domain (in interval notation): $(-\infty,0)\cup(0,\infty)$
2. Vertical asymptotes: $x=0$
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: $\left(\frac{1}{4},\infty\right)$
6. Intervals where the function is decreasing: $\left(0,\frac{1}{4}\right)$, $(-\infty... |
e39d239e-1f98-4224-8c4c-dc39999c7faf | differential_calc | false | null | Make full curve sketching of $f(x) = \frac{ 3 \cdot x^3 }{ 2 \cdot x^2 - 3 }$.
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) $\left(-1\cdot\infty,-1\cdot3^{2^{-1}}\cdot2^{-1\cdot2^{-1}}\right)\cup\left(-1\cdot3^{2^{-1}}\cdot2^{-1\cdot2^{-1}},3^{2^{-1}}\cdot2^{-1\cdot2^{-1}}\right)\cup\left(3^{2^{-1}}\cdot2^{-1\cdot2^{-1}},\infty\right)$
2. Vertical asymptote(s) $x=\frac{\sqrt{3}}{\sqrt{2}}$, $x=-\frac{\sq... |
e3f282dc-5d88-47c9-a17b-5900c9e45805 | integral_calc | false | null | Find the area of the surface formed by rotating the arc of the circle $x^2 + y^2 = 1$ between the points $(1,0)$ and $(0,1)$ in the first quadrant, around the line $x + y = 1$. | The final answer: $\frac{4\cdot\pi-\pi^2}{\sqrt{2}}$ |
e47cac1c-64b3-44fb-a64c-e30da6bea8bd | precalculus_review | false | null | Use the double-angle formulas to evaluate the integral:
$$
\int \sin(x)^2 \, dx + \int \cos(x)^2 \, dx
$$ | $\int \sin(x)^2 \, dx + \int \cos(x)^2 \, dx$ = $x+C$ |
e4cbc0e5-b562-4a04-b2af-dab4fd6cba29 | differential_calc | false | null | Calculate the derivative of the function $r = 9 \cdot \ln\left(\sqrt[3]{\frac{ 1+\tan\left(\frac{ \varphi }{ 3 }\right) }{ 1-\tan\left(\frac{ \varphi }{ 3 }\right) }}\right)$. | $r'$ = $2\cdot\sec\left(\frac{2\cdot\varphi}{3}\right)$ |
e4e2cd9f-3550-4c70-ae5f-41f38df496e0 | sequences_series | false | null | Find the radius of convergence and sum of the series:
$$
\frac{ 1 }{ 3 }+\frac{ x }{ 1 \cdot 4 }+\frac{ x^2 }{ 1 \cdot 2 \cdot 5 }+\cdots+\frac{ x^n }{ \left(n!\right) \cdot (n+3) }+\cdots
$$ | 1. Radius of convergence: $R=\infty$
2. Sum: $f(x)=\begin{cases}\frac{x\cdot\left(2\cdot e^x-2\cdot x\cdot e^x\right)+e^x\cdot x^3}{x^4}-\frac{2}{x^3},&x\ne0\\\frac{1}{3},&x=0\end{cases}$ |
e50e9658-dc0f-4685-8298-ba3fdc11e9f0 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ 1 }{ \sin(x)^6 } \, dx
$$ | $\int \frac{ 1 }{ \sin(x)^6 } \, dx$ = $-\frac{\cos(x)}{5\cdot\sin(x)^5}+\frac{4}{5}\cdot\left(-\frac{\cos(x)}{3\cdot\sin(x)^3}-\frac{2}{3}\cdot\cot(x)\right)$ |
e5720838-c669-4dd3-a936-ff2b5c33436f | sequences_series | false | null | Consider the power series $\sum_{k=1}^\infty \frac{ (2 \cdot x+1)^k }{ 2^k \cdot \sqrt{k} }$.
1. Find the center of convergence of the series.
2. Find the radius of convergence of the series. | 1. The center of convergence is $a=-\frac{1}{2}$
2. The radius of convergence is $R=1$ |
e580db01-e32b-49d4-810e-5a37e87204b8 | sequences_series | false | null | Express the series $\sum_{k=0}^\infty\left(x^k-x^{2 \cdot k+1}\right)$ in terms of elementary functions and find the radius of convergence of the sum. | 1. $\sum_{k=0}^\infty\left(x^k-x^{2 \cdot k+1}\right)$ = $\frac{1}{1-x^2}$
2. radius of convergence $R$ = $1$ |
e589347b-40ae-472d-bd48-78cbed301827 | multivariable_calculus | false | null | Determine a definite integral that represents the area enclosed by the inner loop of $r = 3 - 4 \cdot \cos(\theta)$. | The final answer: $\int_0^{\arccos\left(\frac{3}{4}\right)}\left(3-4\cdot\cos\left(\theta\right)\right)^2d\theta$ |
e613669c-e828-4933-8ad1-68615e2e0b67 | multivariable_calculus | false | null | The position of a particle at time $t$ is given by $\vec{r}(t) = \left\langle \frac{ t^2 }{ 2 },\frac{ 2 \cdot \sqrt{2} }{ 3 } \cdot t^{\frac{ 3 }{ 2 }},t \right\rangle$.
1. Find the velocity vector of the particle.
2. Find the acceleration vector of the particle.
3. Find the distance traveled by the particle between ... | 1. $\left\langlet,\sqrt{2}\cdot\sqrt{t},1\right\rangle$
2. $\left\langle1,\frac{1}{\sqrt{2}\cdot\sqrt{t}},0\right\rangle$
3. $\frac{3}{2}$ |
e694511e-0873-4b17-82a8-6e13ecc589c1 | sequences_series | false | null | Expand the function $y = \ln\left(x + \sqrt{1 + x^2}\right)$ in a power series. | The final answer: $x-\frac{1}{2}\cdot\frac{x^3}{3}+\frac{1\cdot3}{4\cdot2}\cdot\frac{x^5}{5}-\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\cdot\frac{x^7}{7}+\cdots+\frac{(2\cdot n-1)!!}{(2\cdot n)!!}\cdot\frac{x^{2\cdot n+1}}{2\cdot n+1}+\cdots$ |
e6a45a85-fdbb-4381-9f51-63e7d1c32fcd | sequences_series | false | null | Compute $\sqrt[4]{80}$ with accuracy $0.0001$. | The final answer: $2.9907$ |
e78bdde6-d3e2-47e8-81e3-b17177ba3818 | precalculus_review | false | null | Use properties of the natural logarithm to write the expression $\ln\left(x^4 \cdot y\right)$ as an expression of $\ln(x)$ and $\ln(y)$. | $\ln\left(x^4 \cdot y\right)$ = $4\cdot\ln(x)+\ln(y)$ |
e7c924e6-d60f-43b4-b2ef-d83524eb6886 | integral_calc | false | null | Find the moment of inertia of the figure bounded by the arc of the semicircle $x^2 + y^2 = 9$, $y > 0$ relative to the x-axis. | The moment of inertia is: $\frac{243}{8}\cdot\pi$ |
e7e29834-4747-4aee-a690-1635add2dd35 | multivariable_calculus | false | null | Find the maximum volume of the parallelepiped inscribed in the half-sphere of radius $R$. | $V$ = $\frac{4}{3\cdot\sqrt{3}}\cdot R^3$ |
e8395a31-7b69-4c48-a257-c31ad220222f | algebra | false | null | Determine all rational values of $x$ at which $y = \sqrt{x^2 + x + 3}$ is a rational number. | The final answer: $x=\frac{q^2-3}{(1-2\cdot q)}$ |
e83a5d89-d36a-426f-96ca-3f62759e1d1a | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAv8AAACKCAIAAACy6bIuAAAa5ElEQVR4nO3d4Wsb9/0H8K9+7GmanvWssBF8yoNAh8N8SsCLH9jQSLVLn4xUMoURyIhz2vCDrbUWeWOULG5PlPbRfLZpwQ/GdFoYCaPSfDY4MGljjZVikUIG84lQsj3SRUv7B2gPPvS430mW5Ph0X0nf9+uRJZ9On/vodPfR9/u974WazSYDAAAAEMb/8Q4AAAAAIFCofgAAAEAsqH4AAABALKh+AAAAQCyofgAAAEAsqH4AAABALKh+AA... | A large fish tank owner notices that the filter on his tank loses effectiveness over time. He measures the rate the pollutants increase in the water every day until he changes the filter. Using the right rectangular approximation method (RRAM) with $n=3$, and the data below, what is the total amount of pollutants the w... | Total amount of pollutants: $200$ |
e88de257-9aa0-4277-b404-bce62070bf07 | differential_calc | false | null | Find the second derivative $\frac{d ^2y}{ d x^2}$ of the function $x = \left(4 \cdot \sin(t)\right)^3$, $y = 2 \cdot \sin(2 \cdot t)$. | $\frac{d ^2y}{ d x^2}$ = $\frac{\left(2304\cdot\left(\sin(t)\right)^3-1536\cdot\sin(t)\right)\cdot\cos(2\cdot t)-1536\cdot\left(\sin(t)\right)^2\cdot\cos(t)\cdot\sin(2\cdot t)}{7077888\cdot\left(\cos(t)\right)^3\cdot\left(\sin(t)\right)^6}$ |
e8e1d2cd-2424-4468-aaad-ab1de689cf41 | multivariable_calculus | false | null | Evaluate the area of the figure bounded by $\rho = a \cdot \cos(2 \cdot \varphi)$. | The area is $\frac{\pi}{2}\cdot a^2$ |
e96b0661-863a-46a1-9573-432fb705ee4b | algebra | false | null | 1. On a map, the scale is 1.5 inches : 20 miles. If two cities are 115 miles apart, how far apart are they on the map?
2. A model of a dinosaur built to a scale of 3 cm : 1 m is 24 centimeters tall. How tall was the actual dinosaur?
3. What are the dimensions of a building whose dimensions on a blueprint are 1.8 cm b... | 1. $8.625$ inches
2. $8$ m
3. $27$ feet by $36$ feet
4. $7.2$ inches |
e9c1a928-3748-44fc-8791-eaf192d07f88 | multivariable_calculus | false | null | Compute the area of the region bounded by the graph of $\left(x^2+y^2\right)^2=2 \cdot y^3$. (Hint: use the polar coordinates). | $S$ = $\frac{5}{4}\cdot\pi$ |
ea099411-f1fd-44e5-b9db-54271bc9afd6 | algebra | false | null | Find an exponential function that passes through the points:
| $x$ | $1$ | $2$ | $3$ | $4$ |
| --- | --- | --- | --- | --- |
| $f(x)$ | $10$ | $20$ | $40$ | $80$ | | $f(x)$ = $5\cdot2^x$ |
ea191942-29a9-43a1-ba80-9d4753b80368 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAT4AAAENCAYAAACW1gkzAADYeElEQVR4nOz9V6xlR5bnB/9iH3/O9TdvekeXdMli0RTJKrJZLMeqIqu6p9Xq6elRaaQHDSABMwNBwGgwDwPpTS+CIOlJkCAVoE+QIHWrBWi6Rz3TNd3VXdVlyGQWfRqmN9d7c/ze8T3sMCv2OTcNmUzeJM9K3Lz37BM7IraJf/yXiRVKa60ZyECEJEnC1tYW8/PzTE9P8+GHH3Lt2jWOHz/OV77yFfbv30+pVEIpBcDFixd55513eO6559... | Evaluate the triple integral of the function $f(x,y,z) = 1 - \sqrt{x^2+y^2+z^2}$ over the solid $B$ bounded by $x^2+y^2+z^2 \le 9$, $z \ge 0$, and $y \ge 0$. | $\int\int\int_{B}{f(x,y,z) dV}$ = $-\frac{45\cdot\pi}{4}$ |
ea1fe8c2-0c26-4940-8333-08a978425882 | differential_calc | false | null | Find the local extrema of the function $f(x) = 3 \cdot \left(x^2\right)^{\frac{ 1 }{ 3 }} - 4 \cdot x^2$ using the First Derivative Test. | Local maxima: $x=-\frac{1}{2\cdot\sqrt[4]{4}}$, $x=\frac{1}{2\cdot\sqrt[4]{4}}$
Local minima: $x=0$ |
ea4c115a-3f35-4c2e-bdb8-5b92cb2e7e31 | precalculus_review | false | null | Solve the exponential equation exactly: $3^{\frac{ x }{ 14 }} = \frac{ 1 }{ 10 }$ | $x$ = $-\frac{14}{\log_{10}(3)}$ |
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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.