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 |
|---|---|---|---|---|---|
6e202713-1c80-40df-9e36-7634560c9076 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABFRUlEQVR4nO3de/zX8/34//u7k6iUHDqhSKWDacn51KxJ6byi1VphmTIbtfZximHpY40xm5xSfVpRWd4VJWGGYjR8nBJSKCJSOh/fvz/203f7vF/vdHi/H6/34Xq9XHaZuj9fr/d9Rr1vPZ+v5zMnLy8vLwAAABIol+0FAACAskOAAAAAyQgQAAAgGQECAAAkI0AAAIBkBAgAAJCMAAEAAJIRIAAAQDICBAAASEaAAAAAyQgQAAAgGQECAA... | The shaded region $R$ is bounded by the graph of $y = \cos(x)$ and the line $y = \frac{ 1 }{ 2 }$, as shown in the figure above.
There exists a number $k$, $k > 1$, such that when $R$ is revolved about the horizontal line $y = k$, the resulting solid has the volume $C$. Write, but do not solve, an equation involving a... | An equation that can be used to find the value of $k$ is $C=\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}\pi\cdot\left(\left(k-\frac{1}{2}\right)^2-\left(k-\cos(x)\right)^2\right)dx$ |
6e2f7c2f-40c0-498b-8398-22df34f72ca0 | algebra | false | null | What are the intercepts and asymptotes of the function $P(t) = \frac{ 20 }{ 1+4 \cdot e^{-0.5 \cdot t} }$? If there are no asymptotes or intercepts, then write $\text{None}$. | x-intercept: None
y-intercept: $P(0,4)$
asymptotes: $y=20$, $y=0$ |
6e469c97-4911-48e7-af18-f91442686260 | multivariable_calculus | false | null | Evaluate the double integral:
$$
\int \int \rho^2 \, d \varphi \, d \rho
$$
where the integration area $Q$ is bounded by the curve $\rho = a \cdot \sin(2 \cdot \varphi)$. | $\int \int \rho^2 \, d \varphi \, d \rho$ = $\frac{8}{9}\cdot a^3$ |
6e565a1e-93c4-4477-9a63-92e2c6f2dcf8 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUYAAADzCAIAAAB1xSjVAADcQUlEQVR4nOy9Z3hVVfY/fntCQAa7QnJbbm4KIXQEpQYQEHXUGRVHRKQLiBBCSEIqCSQkgYQiUlT4YkNUxIaKKIioiNITSM/NTQLYqCn3nrb/Lz7PXv8NMzrqzGj0x37BE849ZZ999uprfZaeMaa7Mq6MK+PPMgy/9wSujCvjyvhvjiskfWVcGX+qcYWkr4wr4081rpD0lXFl/KnGFZK+Mq6MP9W4QtJXxpXxpxpXSPrKuDL+VOMKSV8ZV8... | On a cylinder with a diameter of 6 cm, a channel is cut out along the surface, having an equilateral triangle with a side of 0.25 cm in cross section. Compute the volume of the cut out material. | $V$ = $\frac{24\cdot\sqrt{3}-1}{256}\cdot\pi$ |
6eab7d06-a894-4b93-8479-b153c4ea7408 | precalculus_review | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAdAAAAGoCAYAAAD2LLSsAAAYU2lDQ1BJQ0MgUHJvZmlsZQAAeJyVeQVUVF8X77mTzDAM3d0l3SAxdHeDwNAdQ4NKioSKIKCUCioIIliEiIUgooigAgYiYVAqqKAIyLuEfv/v/6313npn1rn3N/vss+PsU3sGAM5UcmRkKIIOgLDwGIqtkS6fs4srH/YdgAAO/ggBEbJPdCTJ2tocwOXP+7/L8jDMDZdnUpuy/rf9/1roff2ifQCArGHs7RvtEwbjawCgMn0iKTEAYFRhum... | Estimate the average rate of change from $x=1$ to $x=4$ for the following function $f(x)$ whose graph is given below:
Note that the average rate of change of a function from $x=a$ to $x=b$ is $\frac{ f(b)-f(a) }{ b-a }$. | The final answer: $-0.5$ |
6eb2112a-d921-461f-bbd8-4e02c7e628ba | sequences_series | false | null | Let $f(x) = \ln\left(2 \cdot x^2 + 1\right)$.
1. Find the 4th order Taylor polynomial $P_{4}(x)$ of $f(x)$ about $a=0$.
2. Compute $\left|f(1) - P_{4}(1)\right|$.
3. Compute $\left|f(0.1) - P_{4}(0.1)\right|$. | 1. $2\cdot x^2-2\cdot x^4$
2. $1.09861229$
3. $2.62729\cdot10^{-6}$ |
6eb723e9-f3be-45fb-93d2-513d7ac5c99b | differential_calc | false | null | Sketch the curve:
$$
y = \frac{ x^3 }{ 6 \cdot (x+3)^2 }
$$
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: $y=\frac{x}{6}-1$
5. Intervals where the function is increasing: $(-\infty,-9)$, $(-3,0)$, $(0,\infty)$
6. Intervals where the function is decreasing: $(-9,-3)$
7.... |
6efac4ac-e374-438a-95a1-859cfaa849f7 | differential_calc | false | null | Given the function $y = \frac{ 1 }{ 4 - x^2 }$, find 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 concave up
8. Inter... | 1. Domain (in interval notation) $(-1\cdot\infty,-2)\cup(-2,2)\cup(2,\infty)$
2. Vertical asymptotes $x=2$, $x=-2$
3. Horizontal asymptotes $y=0$
4. Slant asymptotes None
5. Intervals where the function is increasing $(0,2)$, $(2,\infty)$
6. Intervals where the function is decreasing $(-\infty,-2)$, $(-2,0)$
7. Interva... |
6f098df0-77ae-4edd-bf05-80600c45367d | sequences_series | false | null | Find the Fourier series expansion of the function $f(x) = \frac{ x }{ 2 }$ with the period $4$ on the interval $[-2,2]$. | The Fourier series is: $f(x)=\sum_{n=1}^\infty\left(\frac{(-1)^{n+1}\cdot2}{\pi\cdot n}\cdot\sin\left(\frac{\pi\cdot n\cdot x}{2}\right)\right)$ |
6f3269e2-8a1e-4e41-932c-de8a74a48740 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAP8AAADmCAIAAACVs8TOAABe+UlEQVR4nO2deSBU3f/4z52FYexLlpBQShuVFi0UbZZWKlpIaU+otD6USmjfRJv2Twv1tCFJ5UnaRImKZN93xpj13t8f5/uZ33zGELN6yusv995zzzl3vO+557zPe0EwDAN/BhiGIQgi7V700I3ASbsDkmPDhg0lJSXS7kUP3Yg/RfrZbPajR4+SkpKk3ZEeuhF/ivQ3NDQUFxcnJiZKuyM9dCN+N+mn0Wh0Oh3+3dzczFnVNDQ0oCj64c... | Compute the area bounded by $r = a \cdot \cos(3 \cdot \varphi)$, which lies in the circle $r = \frac{ a }{ 2 }$ ($a > 0$). | $A$ = $\frac{864\cdot\pi-648\cdot\sqrt{3}}{5184}\cdot a^2$ |
6f89334a-66bb-4ebe-ac5b-66e0e30d45e4 | differential_calc | false | null | Find the derivative of the function: $y = -4 \cdot x^{\sqrt{5 \cdot x}}$ | $\frac{ d y }{d x}$ = $-\left(\frac{4\cdot\sqrt{5}}{\sqrt{x}}+\frac{2\cdot\sqrt{5}\cdot\ln(x)}{\sqrt{x}}\right)\cdot x^{\sqrt{5}\cdot\sqrt{x}}$ |
6f92ed01-c020-4669-aa75-6a1803ed5b3e | sequences_series | false | null | Write the Taylor series for the function $f(x) = x \cdot \sin(2 \cdot x)$ at the point $x = \pi$ up to the third term (zero or non-zero). | The final answer: $0+2\cdot\pi\cdot(x-\pi)+\frac{4}{2}\cdot(x-\pi)^2$ |
6fbded2f-1451-45fb-b45b-817b52db5081 | algebra | false | null | A substance has a half-life of $2.045$ minutes. If the initial amount of the substance was $132.8$ grams, find the exponential decay model formula. How many half-lives will have passed before the substance decays to $8.3$ grams? | Model formula: $A=132.8\cdot e^{\frac{\ln(0.5)}{2.045}\cdot t}$
Half-lives count: $4$ |
70213df7-9bd5-40b0-8ee8-80692374e39b | algebra | false | null | A town has an initial population of $53\ 000$. It grows at a constant rate of $2100$ per year for $5$ years. The linear function that models the town’s population $P$ as a function of the year is: $P(t) = 53\ 000 + 2100 \cdot t$, where $t$ is the number of years since the model began. Find the $x$- and $y$-intercepts. | $x$-intercept: $P\left(-\frac{530}{21},0\right)$
$y$-intercept: $P(0,53000)$ |
70d548f2-b9a7-4d27-ad68-b0e32d82f5f7 | differential_calc | false | null | Find the derivative of the function $y = \arcsin\left(\sqrt{1-x^2}\right)$. | $y'$ = $-\frac{x}{|x|\cdot\sqrt{1-x^2}}$ |
70d6b1ed-272e-406b-853f-826b497f9b81 | integral_calc | false | null | Solve the integral:
$$
\int 5 \cdot \cos(3 \cdot x)^6 \, dx
$$ | $\int 5 \cdot \cos(3 \cdot x)^6 \, dx$ = $\frac{5}{8}\cdot\left(\frac{5}{2}\cdot x+\frac{2\cdot\sin(6\cdot x)}{3}+\frac{\sin(12\cdot x)}{8}-\frac{\sin(6\cdot x)^3}{18}\right)+C$ |
70e0b33e-584d-4813-b358-4056044430ea | precalculus_review | false | null | Find the range of $f(x) = \frac{ x }{ 1+x^2 }$. | The final answer: $-\frac{1}{2}\le y\le\frac{1}{2}$ |
70ff52a6-d767-4315-9035-91c8cf38c9c5 | differential_calc | false | null | Consider a limousine that gets $m(v) = \frac{ 120-2 \cdot v }{ 5 }$ mi/gal at speed $v$, the chauffer costs $\$15$/h, and gas is $\$3.5$/gal. Find the cheapest driving speed. | approximately $34.02$ mph. |
7109a551-97df-40ff-9e0d-8973e82c6726 | precalculus_review | false | null | Find the degree, $y$-intercept, and zeros for the polynomial function $f(x) = 2 \cdot x^2 + 9 \cdot x - 5$.
1. Degree
2. $y$-intercept
3. Zeros | 1. Degree: $2$
2. $y$-intercept: $-5$, $\frac{1}{2}$
3. Zeros: $-5$, $\frac{1}{2}$ |
710e1a90-6ff1-4545-a62e-9c4e38632819 | integral_calc | false | null | Evaluate $\int_{0}^\pi \sin(5 \cdot x) \cdot \sqrt{\cos(5 \cdot x)} \, dx$ using substitution. | The final answer: $\frac{2+2\cdot\sqrt{-1}}{15}$ |
714057c2-e085-4f6f-b664-9c3fe27b6536 | precalculus_review | false | null | Given two functions $f(x) = \sqrt{x^2 - 1}$ and $g(x) = \sqrt{3 - x}$:
1. Compute $f\left(g(x)\right)$
2. Compute $\frac{ f(x) }{ g(x) }$ and find the domain of the new function. | 1. The new function $f\left(g(x)\right)$ is $f\left(g(x)\right)=\sqrt{2-x}$
2. The function $\frac{ f(x) }{ g(x) }$ is $\frac{\sqrt{x^2-1}}{\sqrt{3-x}}$, and the domain for the function is $1\le x<3 \lor x\le-1$ |
714ca4e7-c104-45fa-b50b-076018133858 | multivariable_calculus | false | null | Find the center of mass of the region $\rho(x,y) = x \cdot y$ on the circle with radius $1$ in the first quadrant only. | Center of mass: $P\left(\frac{8}{15},\frac{8}{15}\right)$ |
71669c2a-8586-4e3d-b006-4b18c6143243 | precalculus_review | false | null | Find the domain of the function $f(x) = \sqrt{\cos\left(\sin(x)\right)} + \arcsin\left(\frac{ 1 + x^2 }{ 2 \cdot x }\right)$. | The final answer: $x=-1 \lor x=1$ |
72196b45-ca12-4d25-8f07-dc1dba889917 | algebra | false | null | A town has an initial population of $80\ 000$. It grows at a constant rate of $2200$ per year for $5$ years. The linear function that models the town’s population $P$ as a function of the year is: $P(t) = 80\ 000 + 2200 \cdot t$, where $t$ is the number of years since the model began. When will the population reach $12... | $t$: $\frac{200}{11}$ |
7224831a-52ae-4ea0-9309-2dcc80603125 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAA0UAAAL1CAIAAACHWNgjAAEAAElEQVR4nOz9V2xk2bon+K1tw3tHMug9k2SSSW/SZ1ZWnXOqb/dMC4IwDw1hgMFAUD9IaAkQIA3QDQ3UAuZF0INm3iQBwgw00PTMveecOpWm0tF7k/TeBYPh7Y4dO7bTw6qKy0symaYyk0nm93u4uIcVZgfJZPxjrfV9H6GqKgIAAAAAAJcWedEXAAAAAAAAfhfIcwAAAAAAlxvkOQAAAACAyw3yHAAAAADA5QZ5DgAAAADgcoM8BwAAAA... | Find the areas enclosed by the curves $y^2 = 2 \cdot x$ and $y^2 = 4 \cdot x - x^2$ using integration with respect to $x$:
1. The area of the smaller, shaded region.
2. The area of the larger, yellow region. | 1. The area of the smaller, shaded region is $2\cdot\left(\pi-\frac{8}{3}\right)$
2. The area of the larger, yellow region is $2\cdot\left(\pi+\frac{8}{3}\right)$ |
722661f3-5831-4484-beba-4d3d1b7997eb | integral_calc | false | null | Find the arc length of the parametric curve given by $x(t) = \frac{ 1 }{ 6 } \cdot t^3$, $y(t) = \frac{ 1 }{ 9 } \cdot t^3$ on $[1,3]$. | The final answer: $\frac{1573}{90}$ |
72bf7325-f168-4eca-843c-f1a99eba2924 | algebra | false | null | Calculate the average rate of change of the function $f(x) = 20 \cdot x - 2 \cdot x^2$ near $a=3$ over the intervals $[a,a+h]$ for $h=0.1$, $0.01$, and $0.001$. Use these values to estimate the instantaneous rate of change. | 1. The average rate of change for $f$ near $a$ with $h=0.1$ is: $7.8$
2. The average rate of change for $f$ near $a$ with $h=0.01$ is: $7.98$
3. The average rate of change for $f$ near $a$ with $h=0.001$ is: $7.998$
4. An estimate of the instantaneous rate of change is: $8$ |
72f5d1f8-3991-417a-8065-2b5469b8c052 | algebra | false | null | Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeros of the following polynomial:
$p(x) = x^3 - 6 \cdot x^2 + 2 \cdot x - 3$ | The number of positive zeros: $1$, $3$
The number of negative zeros: $0$ |
73162ddc-c250-44af-bed6-5ca6cbd8440d | precalculus_review | false | null | A restaurant owner wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $\$440$ for $20$ shirts and $\$1000$ for $100$ shirts.
1. Find the equation $C=f(x)$ that describes ... | 1. $f(x)$ = $7\cdot x+300$
2. $100$ shirts. |
731b3efc-3845-4ff1-8226-56166a4e8859 | multivariable_calculus | false | null | Find the moment of inertia of one arch of the cycloid $x = a^2 \cdot \left(\frac{ t }{ 4 } - \sin\left(\frac{ t }{ 4 }\right)\right)$, $y = a^2 \cdot \left(1 - \cos\left(\frac{ t }{ 4 }\right)\right)$ relative to the x-axis. | Moment of Inertia: $\frac{2}{15}\cdot\left(64-43\cdot\sqrt{2}\right)\cdot a^6$ |
734307ce-146c-4611-8188-1e99a85f7008 | algebra | false | null | The radius and height of the right circular cylinder differ by two meters. The height is greater than the radius, and the volume is $15.125 \cdot \pi$ cubic meters. Find the dimensions of the cylinder. | $r$: $3.95546093-2$ $h$: $3.95546093$ |
73840a81-4100-4726-8539-d48e5f16acf9 | sequences_series | false | null | Find the sum of the series $\sum_{k=2}^\infty \left(\frac{ 3^{k-1} }{ 4^{3 \cdot k+1} }\right)$. | The final answer: $\frac{3}{15616}$ |
73d9da66-451f-4585-a1cb-d03ba5a29318 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ 6 \cdot x^3-7 \cdot x^2+3 \cdot x-1 }{ 2 \cdot x-3 \cdot x^2 } \, dx
$$ | Answer is: $-x^2+x-\frac{1}{3}\cdot\ln\left(\left|x-\frac{2}{3}\right|\right)+\frac{1}{2}\cdot\ln\left(\left|1-\frac{2}{3\cdot x}\right|\right)+C$ |
73de1a56-66c6-4eac-929a-6307d780e875 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ x+2 }{ \sqrt{6+10 \cdot x+25 \cdot x^2} } \, dx
$$ | $\int \frac{ x+2 }{ \sqrt{6+10 \cdot x+25 \cdot x^2} } \, dx$ = $\frac{1}{25}\cdot\sqrt{6+10\cdot x+25\cdot x^2}+\frac{9}{25}\cdot\ln\left(1+5\cdot x+\sqrt{1+(5\cdot x+1)^2}\right)+C$ |
742ba56b-3cf1-4e54-912d-1fe003842acb | sequences_series | false | null | Determine the Taylor series for the function $f(x) = rac{ 1-x }{ (1+2 \cdot x)^3 }$, centered at $x = 0$.
Use this decomposition to find the sum of the series $\sum_{n=0}^\infty rac{ (-1)^n \cdot (n+1) \cdot (3 \cdot n+4) }{ 2^{n+2} }$.
1. Submit the Taylor series in summation notation.
2. Submit the sum of the se... | 1. The Taylor series in summation notation is $\sum_{n=0}^\infty\left(\frac{(-1)^{n+2}\cdot\left((n+1)!\right)\cdot2^{n-2}\cdot(3\cdot n+4)}{n!}\cdot x^n\right)$
2. The sum of the series is $\frac{2}{9}$ |
745e79db-e211-4134-9a5f-2f6014fa502b | differential_calc | false | null | For the function $y = 2 \cdot x + \ln\left(\frac{ 1 }{ \frac{ x^2 }{ 2 } - 1 }\right)$, determine the intervals where the function is increasing and decreasing.
Submit as your final answer:
1. Interval(s) where the function is increasing
2. Interval(s) where the function is decreasing | 1. Interval(s) where the function is increasing: $\left(-\infty,-\sqrt{2}\right)$, $(2,\infty)$
2. Interval(s) where the function is decreasing: $\left(\sqrt{2},2\right)$ |
746dd4c5-1521-4d23-9036-77e4ded2dd2e | precalculus_review | false | null | Divide $C = \frac{ (3-i)^2 }{ (3+i)^2 }$. | The final answer: $C=\frac{7}{25}-\frac{24}{25}\cdot i$ |
7488ad33-1fbe-467c-afe4-e964b3e99a87 | algebra | false | null | If an investor invests $\$23\ 000$ into two bonds, the first one that pays $4\%$ in simple interest, and the second one paying $2\%$ simple interest, and the investor earns $\$710$ annual interest, how much was invested in each account? | Add both values to your final answer here: $x=12500$, $y=10500$ |
74f2e8e7-1487-4b76-8039-92dab68d4d30 | algebra | false | null | Solve the following equations:
1. $-t + (5t - 7) = -5$
2. $21 - 3(2 - w) = -12$
3. $9 = 8b - (2b - 3)$
4. $4.5r - 2r + 3(r - 1) = 10.75$
5. $1.2(x - 8) + 2.4(x + 1) = 7.2$
6. $4.9m + (-3.2m) - 13 = -2.63$
7. $4(2.25w + 3.1) - 2.75w = 44.9$ | The solutions to the given equations are:
1. $t=\frac{ 1 }{ 2 }$
2. $w=-9$
3. $b=1$
4. $r=\frac{ 5 }{ 2 }$
5. $x=4$
6. $m=\frac{ 61 }{ 10 }$
7. $w=\frac{ 26 }{ 5 }$ |
74fc23ba-f1e6-4bf3-8e2e-70d75414bf97 | sequences_series | false | null | Calculate $\sqrt[3]{30}$ with an estimate error of $0.001$, using series expansion. | The final answer: $\frac{755}{243}$ |
755ad080-c182-4f30-96a4-1a2cb0462824 | multivariable_calculus | false | null | Find $I=\int_{0}^\pi \int_{0}^{4 \cdot \sin\left(\theta\right)} \frac{ 4 }{ \sqrt{16-r^2} } \cdot r \, d r \, d \theta$. | The final answer: $I=32\cdot\left(\frac{\pi}{2}-1\right)$ |
756c8302-f766-4e1a-980a-9d56e3adf724 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ \sqrt{1+x^2} }{ x } \, dx
$$ | $\int \frac{ \sqrt{1+x^2} }{ x } \, dx$ = $\sqrt{x^2+1}+\frac{1}{2}\cdot\ln\left(\left|\frac{\sqrt{x^2+1}-1}{\sqrt{x^2+1}+1}\right|\right)+C$ |
75d9bd99-c5d9-4b47-8d12-616de1d6f158 | algebra | false | null | Newton’s Law of Cooling states that the temperature $T$ of an object at any time $t$ can be described by the equation $T = T_{s} + (T_{0} - T_{s}) \cdot e^{-k \cdot t}$, where $T_{s}$ is the temperature of the surrounding environment, $T_{0}$ is the initial temperature of the object, and $k$ is the cooling rate. Use th... | $t$ = $\ln\left(\left(\frac{T-T_s}{T_0-T_s}\right)^{-\frac{1}{k}}\right)$ |
772d7be7-01a3-4370-b536-4a7ceed86f31 | precalculus_review | false | null | Solve $x \cdot (x+1) \cdot (x+2) \cdot (x+3) = 0.5625$. | The final answer: $x=-\frac{3}{2} \lor x=\frac{-3+\sqrt{10}}{2} \lor x=\frac{-3-\sqrt{10}}{2}$ |
774c7757-4ed5-43ae-b623-dac59dd300e8 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ \cos(2 \cdot x)^4 }{ \sin(2 \cdot x)^3 } \, dx
$$ | $\int \frac{ \cos(2 \cdot x)^4 }{ \sin(2 \cdot x)^3 } \, dx$ = $C+\frac{3}{8}\cdot\ln\left(\frac{1+\cos(2\cdot x)}{1-\cos(2\cdot x)}\right)-\frac{\left(\cos(2\cdot x)\right)^3}{4-4\cdot\left(\cos(2\cdot x)\right)^2}-\frac{3}{4}\cdot\cos(2\cdot x)$ |
77d38bbc-e5c6-48da-9bc0-143258d0e51e | differential_calc | false | null | Make full curve sketching of $y = \ln\left(\left|\frac{ \frac{ x }{ 2 }-3 }{ \frac{ x }{ 2 }+3 }\right|\right)$ . 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 fu... | 1. The domain (in interval notation) $(-\infty,-6)\cup(-6,6)\cup(6,\infty)$
2. Vertical asymptotes $x=-6$, $x=6$
3. Horizontal asymptotes $y=0$
4. Slant asymptotes None
5. Intervals where the function is increasing $(-\infty,-6)$, $(6,\infty)$
6. Intervals where the function is decreasing $(-6,6)$
7. Intervals where th... |
7836bb59-a1e0-4aff-ba47-a97fb07b38df | integral_calc | false | null | Solve the integral:
$$
\int 3 \cdot \sin(2 \cdot x)^4 \cdot \cos(2 \cdot x)^2 \, dx
$$ | $\int 3 \cdot \sin(2 \cdot x)^4 \cdot \cos(2 \cdot x)^2 \, dx$ = $\frac{3}{16}\cdot\left(x-\frac{\sin(8\cdot x)}{8}-\frac{\sin(4\cdot x)}{8}+\frac{\sin(12\cdot x)}{24}\right)+C$ |
787ba86e-1894-49ad-9a6d-bb1607e13786 | differential_calc | false | null | Sketch the curve:
$y = 4 \cdot x \cdot \sqrt{5-x^2}$.
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): $\left[-1\cdot5^{2^{-1}},5^{2^{-1}}\right]$
2. Vertical asymptotes: None
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: $\left(-\frac{\sqrt{5}}{\sqrt{2}},\frac{\sqrt{5}}{\sqrt{2}}\right)$
6. Intervals where the function is de... |
7888c61c-8515-4146-b87d-2ca3f2030b2c | sequences_series | false | null | Compute $\sqrt[5]{240}$ with an accuracy of $0.0001$. | The final answer: $2.9926$ |
78a834ca-8f7b-4688-9315-52ce28145e29 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAoAAAADQCAIAAAAYvkeqAACwLElEQVR4nOx9eZwsV1X/95x7q6q7Z3l79n0HI7KFhCVIWATZ90CIgoiAQFRQQNSfIIoBUQQVxAVQXBBZlS0KsilLhACyBcgjC0le3kte3jJLd1fVvef8/rhV1dU9My9vmZmuecw379OZ6enl1K17z75Q7vtExDCqCgCAEgMgr2yMwqsqERGRVxARFa9ax/JACapKnohI4Jk53AglAZiUATAAqJIA5W+NgRIANGdXlPRI9TMrFFBVJRg14U... | Given $r(t) = \frac{ g(t) }{ h(t) }$, find $r'(3)$ using the table below: | $r'(3)$ = $-\frac{7}{4}$ |
78b5f141-fca2-40af-8f51-cf73f052b7db | differential_calc | false | null | Find the derivative of the function $y = \frac{ 3 \cdot \csc(x) - 2 \cdot \sin(x) }{ 5 \cdot \left(\cos(x)\right)^5 } - \frac{ 16 }{ 5 } \cdot \cot(2 \cdot x)$. | $y'$ = $\frac{1}{\left(\sin(x)\right)^2\cdot\left(\cos(x)\right)^6}$ |
790253e0-8f58-47bc-92b9-ac728a08533a | algebra | false | null | Subtract the rational expression, then simplify:
$$
\frac{ 12 }{ 2 \cdot q }-\frac{ 6 }{ 3 \cdot p }
$$ | The final answer: $\frac{6\cdot p-2\cdot q}{p\cdot q}$ |
795dc875-d474-4ccd-a036-0e9f12923e70 | precalculus_review | false | null | If $\tan(\theta) = \sqrt{x^2 - 1}$, find $\sec(\theta) + \tan(\theta)$. | $\sec(\theta) + \tan(\theta)$ = $x+\sqrt{x^2-1}$ (Enter your solution as an expression using only the variable $x$.) |
798ed40f-8175-4491-9920-ae86878a57a9 | 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 $\$34$, they... | The final answer: $\frac{101}{3}$ |
79cc6592-1b0c-4b62-ac57-f7501f01bfae | algebra | false | null | Solve the following equations:
1. $6 (m+4) - 2m = -8$
2. $44 = 4 (8+h)$
3. $0.75 (8t - 4) = -2$
4. $3 (5-t) - 4t = 18$
5. $2 (y-5) = 16$
6. $0.1 (h+20) = 3$
7. $7y + 3 (8-y) = -12$ | The solutions to the given equations are:
1. $m=-8$
2. $h=3$
3. $t=0.166666666666667$
4. $t=-\frac{ 3 }{ 7 }$
5. $y=13$
6. $h=10$
7. $y=-9$ |
79d8b155-285b-42e4-852c-e2080b752e13 | precalculus_review | false | null | Find the value $k$ that satisfies the following integral equation:
$$
\int_{1}^k \log_{2}(x) \, dx = 2 - \frac{ 1 }{ \ln(2) }
$$ | $k$ = $2$ |
7a051da0-75c2-4686-98b3-d008e37fda65 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUYAAAF5CAYAAADj+4OIAAAKxmlDQ1BJQ0MgUHJvZmlsZQAAeJyVlwdYU1kWgO976Y0WCEVK6B0pAgGkhB66dBCVkAQSSggJAcWODI7gWFARQWVAB0UUHJUiY8WChUGwYZ8gg4o6DhZARWUfsISZ3W93vz3vO9/933nnnnPu/e7NdwIABc8WiTJhJQCyhLniyABvenxCIh03ACCAByRgD3TZHImIGRERAhCZGf8uo3cRb0RuWU/G+vfv/1WUuTwJBwAoAuEUroSThfAJRN... | A guy-wire supports a pole that is 75 feet high. One end of the wire is attached to the top of the pole and the other end is anchored to the ground 50 feet from the base of the pole. Determine the horizontal and vertical components of the force of tension in the wire if its magnitude is 50 lb. (Round to the nearest int... | Horizontal component: $42$ Vertical component: $28$ |
7a75b4a6-486b-4925-b92c-8982e6bc632c | sequences_series | false | null | Find the radius of convergence of the series:
$$
\sum_{k=1}^\infty \left(\frac{ \left(k!\right)^3 \cdot x^k }{ (2 \cdot k)! }\right)
$$ | $R$ = $0$ |
7a7e986f-3a25-4d58-b29e-e84d61b1d013 | precalculus_review | false | null | Solve the following absolute value equation:
$x^2 - 2 \cdot |x| - 3 = 0$ | The final answer: $x=-3 \lor x=3$ |
7ab2c0ae-7112-4339-907c-e65bac3a2548 | multivariable_calculus | false | null | The surface of a large cup is formed by revolving the graph of the function $y = 0.25 \cdot x^{1.6}$ from $x = 0$ to $x = 5$ about the $y$-axis (measured in centimeters). Find the curvature $\kappa$ of the generating curve as a function of $x$. | $\kappa$ = $\frac{30}{x^{\frac{2}{5}}\cdot\left(25+4\cdot x^{\frac{6}{5}}\right)^{\frac{3}{2}}}$ |
7ad62465-33c4-44bc-83cf-3a52da5732c3 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAtsAAADdCAIAAADl1CnLAACf5UlEQVR4nO2ddXwT2ff3b5K6u7dIcSlWCou77uKyy7KLuy6wsLg7i9vi0AJFi0sFaQt16t7UvUnaxm0yzx+zvzz5liZNcwKhy7z/4FWa6aefzmTu3Jx77jkUHMcRCQkJCQkJCYlOoeraAAkJCQkJCQkJOSMhISEhISEh+QYgZyQkJCQkJCQkuoeckZCQkJCQkJDoHnJGQkJCQkJCQqJ7yBkJCQkJCQkJie4hZyQkJCQkJCQkuoeckZCQkJ... | Using the given graphs of $f(x)$ and $g(x)$, find $g\left(g(3)\right)$. | The final answer:
$g\left(g(3)\right)$ = $5$ |
7b259818-c9a0-4cac-9c03-146900fa791b | multivariable_calculus | false | null | Evaluate the integral:
$$
\int \int \int_{R} 3 \cdot y \, dV
$$
where $R$ is bounded by:
1. $0 \le x \le 1$
2. $0 \le y \le x$
3. $0 \le z \le \sqrt{9-y^2}$ | $\int \int \int_{R} 3 \cdot y \, dV$ = $\frac{216-86\cdot\sqrt{2}-243\cdot\arcsin\left(\frac{1}{3}\right)}{8}$ |
7b9cce25-460c-4712-bb49-a0d60ded955c | sequences_series | false | null | Use the substitution $(b+x)^r = (b+a)^r \cdot \left(1 + \frac{ x-a }{ b+a }\right)^r$ in the binomial expansion to find the Taylor series of the function $\sqrt{x+2}$ with the center $a=1$. | $\sqrt{x+2}$ = $\sum_{n=0}^\infty\left(3^{\frac{1}{2}-n}\cdot C_{\frac{1}{2}}^n\cdot(x-1)^n\right)$ |
7c945151-a3be-4cdf-ac65-effe67ac8fe4 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbUAAAG1CAIAAADxyP2KAAAKMWlDQ1BJQ0MgUHJvZmlsZQAAeJydlndUU9kWh8+9N71QkhCKlNBraFICSA29SJEuKjEJEErAkAAiNkRUcERRkaYIMijggKNDkbEiioUBUbHrBBlE1HFwFBuWSWStGd+8ee/Nm98f935rn73P3Wfvfda6AJD8gwXCTFgJgAyhWBTh58WIjYtnYAcBDPAAA2wA4HCzs0IW+EYCmQJ82IxsmRP4F726DiD5+yrTP4zBAP+flLlZIjEAUJiM5/... | Find the slope of the line pictured: | The final answer: $0$ |
7ce3adca-6f57-46b9-a522-ebaad3f0a35e | algebra | false | null | Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeros of the following polynomial:
$f(x) = 10 \cdot x^4 - 21 \cdot x^2 + 11$ | The possible number of positive zeros: $2$, $0$
The possible number of negative zeros: $2$, $0$ |
7d1cb715-71f2-4ea7-97ae-3598147c0dff | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJsAAAEECAIAAACwRNH5AABeFklEQVR4nO19Z1wVR/fw7O4twKVeuPRepEhHUVEBsWuwYa9YosYUe0FjjzVGfdQYO7aoqNHYW2yxd0VRUREbvfdbdnfeD+fPvBtQozcI1zzP+cBv2Z3dOzNnzpkzp1IYY/SRwHEcwzAIIYwxz/MMw7AsyzAMRVFV2mCMKYqCn4ALiqJ4nqcoStj4f1CDQGvzDk0jhDDGgDae50UiEWCIZVmEEKAZIUTQBhdk9fwPnZ8OtMEoRVEcx1EUxb... | Find the area between the curves $y^2 = x^3 - x^2$ and $x = 2$: | Area: $\frac{32}{15}$ |
7d29f145-f403-4580-92f3-f1edafc102c6 | sequences_series | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYkAAAC1CAIAAAAlcJSIAAA5wElEQVR4nO29d3wU1ff/f2d7y6Ys6QmppNENJPQuIOVN6AgCiqAoIEhRBLEg0hSQJiAqICoQJAhKCRBEegmBFAIhJIRkSd9kU7bvzHz/OD/3lw81hMzM7mSef/Ag2cmeM7szr7n33HPPwUiSRBwcT6KoqOjrr7/etGkT045wNEV4TDvAYb+IxeJx48Yx7QVHEwXjxk2OC0mSly5dioyMdHV1RQhVVFSYTCZvb294FcdxPp/PqIMcHA2HGz... | Find the Fourier series expansion for the periodic function $f(x) = x - 1$ (on $(-1, 1]$) with period $2$. | The Fourier series is: $f(x)\approx\sum_{n=1}^\infty-\frac{2\cdot(-1)^n\cdot\sin(n\cdot x)}{\pi\cdot n}-1$ |
7d2b09cc-83d7-43ce-b44f-8ae789286891 | sequences_series | false | null | Find the Fourier series of the function $\varphi(x) = \frac{ x }{ 4 }$ in the interval $(0,2 \cdot \pi)$. | The Fourier series is: $\frac{\pi}{4}-\sum_{n=1}^\infty\left(\frac{\sin(n\cdot x)}{2\cdot n}\right)$ |
7d6bd5ff-9338-4e3b-91bb-6e8b1553040e | precalculus_review | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABcI0lEQVR4nO3de3RcZ33v/89IM7rfr7Ylyxf5JsmxE4tYJo7l3BoIKfwoNOSUhJSSQkpKD4ueVaCHmOaQAF3QNouGSwKEFhKSQ+khQMhp8CFBkuPEciJfYkvyJbIteWzrbt2l0Whmfn8oEpb3yNZltPeerfdrLRbKo71nvo/GGu3PPPt5Htebb74Z0hw0NjZKkkpKSuZyuu384he/kCR96EMfsriSyHDa60N/ooPT+vXMM8/omWeekSTde+... | Determine all six trigonomeric functions for the angle $ heta$ formed by the x-axis and the line passing through the origin and the point $P$ =$(8,-15)$.
1. $\sin( heta)$
2. $\cos( heta)$
3. $ an( heta)$
4. $\csc( heta)$
5. $\sec( heta)$
6. $\cot( heta)$ | 1. $\sin( heta)$ = $\frac{15}{17}$
2. $\cos( heta)$ = $\frac{8}{17}$
3. $ an( heta)$ = $\frac{15}{8}$
4. $\csc( heta)$ = $\frac{17}{15}$
5. $\sec( heta)$ = $\frac{17}{8}$
6. $\cot( heta)$ = $\frac{8}{15}$ |
7db54c2b-7ca2-43b2-9fc5-f33d8b54d9b8 | multivariable_calculus | false | null | The function $y=f(x)$ in parametric form is:
$x = t^5 - 5 \cdot t^3 - 20 \cdot t + 7$
$y = 4 \cdot t^3 - 3 \cdot t^2 - 18 \cdot t + 3$ on the interval $\left(-2<t<2\right)$.
Find extreme values of this function. Submit as your final answer:
1. The point(s) where the function has a maximum or maxima
2. The point(... | 1. The point(s) where the function has a maximum: $P(31,14)$
2. The point(s) where the function has a minimum: $P\left(-\frac{1033}{32},-\frac{69}{4}\right)$ |
7dfe7434-cf7f-4e8b-bbf3-c870a7a0ef46 | differential_calc | false | null | The following polynomial functions are given:
$f(x) = x^5 + x^4 + x + 11$
$g(x) = x^3 + x^2 - 10$
What is the value of $\frac{ f'(2) }{ g'(-2) }$? | The final answer: $\frac{f'(2)}{g'(-2)}=\frac{123}{8}$ |
7e2fda44-9aea-49f6-88a4-cf14f2342966 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ x + \sqrt[3]{x^2} + \sqrt[6]{x} }{ x \cdot \left(1 + \sqrt[3]{x}\right) } \, dx
$$ | $\int \frac{ x + \sqrt[3]{x^2} + \sqrt[6]{x} }{ x \cdot \left(1 + \sqrt[3]{x}\right) } \, dx$ = $C+6\cdot\left(\frac{1}{4}\cdot\sqrt[6]{x}^4+\arctan\left(\sqrt[6]{x}\right)\right)$ |
7e6d17d0-89e3-4b9f-8c9c-48b2fae8878b | multivariable_calculus | false | null | For the function $z = \ln\left(e^x + e^t\right)$, find:
1. $\frac{ d z }{d t}$
2. $\frac{ d z }{d t}$ when $x = t^3$ | 1. $\frac{ d z }{d t}$: $\frac{e^t}{e^t+e^x}$
2. $\frac{ d z }{d t}$, when $x = t^3$: $\frac{3\cdot e^{t^3}\cdot t^2+e^t}{e^t+e^x}$ |
7edd29b3-3371-4552-8048-2d9b0fdc6006 | multivariable_calculus | false | null | The following vectors are given:
$u = 2 \cdot \mathbf{i} - 3 \cdot \mathbf{j} + 2 \cdot \mathbf{k}$
$v = 3 \cdot \mathbf{i} + 2 \cdot \mathbf{j} - 3 \cdot \mathbf{k}$
Find the following component:
Project of $u$ onto $v$ | The final answer: $w=\frac{9}{11}\cdot i+\frac{6}{11}\cdot j-\frac{9}{11}\cdot k$ |
7eedb03b-f388-4069-b2c8-b099c9c16fcf | multivariable_calculus | false | null | Find the equation of a circle which passes through $(-3,1)$ and its center is on the same line with the centers of the following circles:
1. $(x-5)^2 + y^2 = 5$
2. $x^2 + (y-10)^2 = 130$ | The final answer: $(x-2)^2+(y-6)^2=50$ |
7efd2ae1-d387-447f-ad12-4b0dc2ab7a22 | differential_calc | false | null | Sketch the curve:
$y = 3 \cdot x^2 - x^4 - 2$.
Submit as your final answer:
1. The domain (in interval notation)
2. Vertical asymptotes (Leave blank if there are no vertical asymptotes)
3. Horizontal asymptotes (Leave blank if there are no horizontal asymptotes)
4. Slant asymptotes (Leave blank if there are no sla... | 1. The domain (in interval notation): $(-1\cdot\infty,\infty)$
2. Vertical asymptotes: None
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: $\left(-\infty,-\frac{\sqrt{3}}{\sqrt{2}}\right)$, $\left(0,\frac{\sqrt{3}}{\sqrt{2}}\right)$
6. Intervals where the functio... |
7f8713f5-81ca-4ec4-9134-e58d8b8b2c1e | differential_calc | false | null | Make full curve sketching of $y = 2 \cdot \arcsin\left(\frac{ 1-7 \cdot x^2 }{ 1+7 \cdot x^2 }\right)$. Submit as your final answer:
1. The domain (in interval notation)
2. Vertical asymptotes
3. Horizontal asymptotes
4. Slant asymptotes
5. Intervals where the function is increasing
6. Intervals where the function is ... | 1. The domain (in interval notation) $(-1\cdot\infty,\infty)$
2. Vertical asymptotes None
3. Horizontal asymptotes $y=-\pi$
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 function is co... |
802213f1-06e1-4063-b572-53eecf9dc2fc | integral_calc | false | null | Evaluate the integral:
$$
I = \int 2 \cdot \ln\left(\sqrt{2-x}+\sqrt{2+x}\right) \, dx
$$ | The final answer: $2\cdot x\cdot\ln\left(\sqrt{2-x}+\sqrt{2+x}\right)-\left((C+x)-2\cdot\arcsin\left(\frac{x}{2}\right)\right)$ |
802382d0-0530-42f9-9a0a-48e12b08da84 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABSRElEQVR4nO3deZRX5Z0n/k9hCYKAggso7hsCKigqICiixVIUZVHdkzgZu5M/OnHGMcl0YmJ6nEl+6UxMd48xiZ3IJGRMd84k6ZPYnSq2ogpKERDZREEUBDdwYVEQtARFlvr9YaRd7mWtut/t9TrHc8zz+fLN23hT+uZ57r1lra2trQEZu/fee+Pee++NiIg77rgj7rjjjhwnohRNnjw5IiJuvfXWT80+eo1+VJcuXeL5559v92wUvwNdf9... | The graph of a function $f$ is shown above. If $c(x) = 3 \cdot x \cdot f(x) - \frac{ e^x }{ f(x) }$, determine the value of $c'(0)$. | $c'(0)$ = $5$ |
8061d2e1-8561-4f7f-9ae9-18f132ca2705 | precalculus_review | false | null | Solve $\frac{ 1 }{ x \cdot (x+2) }-\frac{ 1 }{ (x+1)^2 }=\frac{ 1 }{ 12 }$ | The final answer: $x=-3 \lor x=1$ |
80a14173-80b6-4e3b-ace4-a3ff3b2af70e | algebra | false | null | A formula for the normal systolic blood pressure for a man age $A$, measured in mmHg, is given as $P = 0.006 \cdot A^2 - 0.02 \cdot A + 120$. Find the age to the nearest year of a man whose normal blood pressure measures $125$ mmHg. | The final answer: $31$ |
80f9d7db-f542-4e35-b4cf-2748290a1ac3 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ \tan(x) }{ \sqrt{\sin(x)^4+\cos(x)^4} } \, dx
$$ | $\int \frac{ \tan(x) }{ \sqrt{\sin(x)^4+\cos(x)^4} } \, dx$ = $\frac{1}{2}\cdot\ln\left(\tan(x)^2+\sqrt{\tan(x)^4+1}\right)+C$ |
8221ca9b-f6a2-4cfb-bf1a-ae2a389b6481 | integral_calc | false | null | Use the table of integrals to evaluate the integral $\int{\left(\sin(y)\right)^2 \cdot \left(\cos(y)\right)^3 \, dy}$.
Use this link to access the table of integrals: [Table of Integrals](https://openstax.org/books/calculus-volume-2/pages/a-table-of-integrals) | 1. Submit the formula used: $\int{\left(\sin(u)\right)^n \cdot \left(\cos(u)\right)^m d u}=-\frac{ \left(\sin(u)\right)^{n-1} \cdot \left(\cos(u)\right)^{m+1} }{ n+m }+\frac{ n-1 }{ n+m } \cdot \int{\left(\sin(u)\right)^{n-2} \cdot \left(\cos(u)\right)^m d u}$, $\int{\left(\cos(u)\right)^3 d u}=\frac{ 1 }{ 3 } \cdot \l... |
82584e44-4d1a-47ba-99f9-7cf389b26994 | differential_calc | false | null | Find the first derivative of the function:
$$
y = \left(3 \cdot a^2 - 2 \cdot a \cdot b \cdot x + \frac{ 5 }{ 3 } \cdot b^2 \cdot x^2\right) \cdot \sqrt[3]{\left(\frac{ a }{ 3 } + \frac{ b }{ 3 } \cdot x\right)^2}
$$ | The first derivative is: $\frac{40\cdot b^3\cdot x^2}{9\cdot3^{\frac{2}{3}}\cdot\sqrt[3]{a+b\cdot x}}$ |
82e1e37e-a913-4df5-bfa6-6fc6880a6ae2 | sequences_series | false | null | Find the Taylor series of the given function $f(x) = \frac{ 1 }{ (x-1)^2 }$ centered at the indicated point: $a=0$ . (Hint: Differentiate $\frac{ 1 }{ 1-x }$.) | $\frac{ 1 }{ (x-1)^2 }$ = $\sum_{n=0}^\infty\left((n+1)\cdot x^n\right)$ |
83a37faf-1b91-4237-bf28-3ddc2db64ef4 | algebra | false | null | A biologist observes that a certain bacterial colony triples every 4 hours and after 12 hours occupies 1 square centimeter.
1. How much area was occupied by the colony when first observed?
2. What is the doubling time for the colony? | 1. $\frac{1}{27}$ cm². (Enter an exact solution as a fraction.)
2. $2.52$ hours. (Enter an approximate solution rounded to 2 decimal places.) |
840a022b-2ef4-4fad-868b-52d564d2f551 | sequences_series | false | null | Find a “reasonable” upper-bound on the error in approximating $f(x) = x \cdot \ln(x)$ by its 3rd order Taylor polynomial $P_{3}(x)$ at $a = 1$ valid for all values of $x$ such that $|x - 1| \leq 0.7$. | The final answer: $\frac{2}{(0.3)^3}\cdot\frac{(0.7)^4}{4!}$ |
84575e59-630e-42a7-8f74-0a98ced4f213 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYwAAAGICAIAAADdykALAABhyUlEQVR4nO29aXAbV5bne29mAkjsO7ivWihxEaldohZqtWxtlteaKtfSNd3Vr6tffZiIFzEx0RMx0W/eRH3pnldTr6ba1S5XeZctyyWXrdW2ZEnWQlELSYkiJVLcd2Ih9h2Z931IAIRISpZEgASJ84sqmQTARCKR+c9zz/3fczAhBAEAAKQr1FzvAAAAwOMAkQIAIK0BkQIAIK0BkQIAIK0BkQIAIK0BkQIAIK0BkQIAIK0BkQIAIK0BkQ... | Use the following graph and find: $\lim_{x \to 0^{-}}\left(f(x)\right)$ | $\lim_{x \to 0^{-}}\left(f(x)\right)$: $1$ |
849aee88-91bf-42c9-9241-e0e9af266311 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ \sqrt{4+x^2} }{ x } \, dx
$$ | $\int \frac{ \sqrt{4+x^2} }{ x } \, dx$ = $C+\sqrt{4+x^2}+\ln\left(\left|\frac{\sqrt{4+x^2}-2}{2+\sqrt{4+x^2}}\right|\right)$ |
84c6a419-c103-41d5-aad5-dd8e690c6e88 | differential_calc | false | null | Compute the limit:
$$
\lim_{x \to 0}\left(\frac{ -\sin(x) }{ x }\right)^{\frac{ 1 }{ 4 \cdot x^2 }}
$$ | $\lim_{x \to 0}\left(\frac{ -\sin(x) }{ x }\right)^{\frac{ 1 }{ 4 \cdot x^2 }}$ = $e^{-\frac{1}{24}}$ |
84ca0349-623e-4dd9-8420-f4ee6fb7c129 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAT4AAAEaCAYAAACIBTsQAAC1BElEQVR4nOy9d5Bc153f+zm3c+7JGTMDDDIRSJAIDAAJBlGURFHJu2vZXu9WuRzKoewql13W8x/v2WWX1+VXu/ustUur3fVKlnZFeSlRgQqMYAYpEiQIEiRIEBkYABO7ezr3Pe+Pm7tv9/QAg0EDMz9y0N33nnTPved7f/kIKaVkhW4q+vjjj/nLv/xLtm/fzoEDB/jKV77Cnj17CAQCAEgpKZVKFAoFCoUCxWKRTCZDJpMhnU4zPT3NzM... | Evaluate the triple integral $\int \int \int f(x,y,z) \, dx \, dy \, dz$ over the solid $f(x,y,z) = x \cdot y$, $x^2 + y^2 \le 1$, $x \ge 0$, $x \ge y$, $-1 \le z \le 1$: | $\int \int \int f(x,y,z) \, dx \, dy \, dz$ = $\frac{1}{8}$ |
84d0c8fe-04f1-412b-84a4-45f02aa9a4e2 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAACEQUlEQVR4nO3dd3iT9f7/8Vc6KC1lFMoURBBliKBMERCUpSwHCgq4EGQ4EFERRdybijJFUURAFBEQRATZQxwMGYcpsmVDoaW7ye8Pfr2/Se90kuZO0ufjus51kvd9J3nD4UBe/Sybw+FwCEBAiImJUUxMjCRp6NChGjp0qMUdAQAAuAqyugEAAAAAhQcBBAAAAIDXEEAAAAAAeA0BBAAAAIDXEEAAAAAAeA0BBAAAAIDXEEAAAAAAeA0BBA... | Let $R$ be the region in the first quadrant enclosed by the graph of $g(x) = \frac{ 12 }{ 1+x^2 }-2$ as shown in the figure above.
1. Write, but do not evaluate, an integral expression that gives the volume of the solid generated when $R$ is revolved about the $x$-axis.
2. Write, but do not evaluate, an integral expre... | 1. $\int_0^{2.236}\left(\pi\cdot\left(\frac{12}{1+x^2}-2\right)^2\right)dx$
2. $\int_0^{10}\left(\pi\cdot\left(\frac{12}{y+2}-1\right)\right)dy$ |
84faeb60-6ed1-4ef8-b2cc-23a767cfb04e | precalculus_review | false | null | Solve the following system of equations:
1. $x + y = \frac{ 2 \cdot \pi }{ 3 }$
2. $\frac{ \sin(x) }{ \sin(y) } = 2$ | The final answer: $x=\frac{\pi}{2}+\pi\cdot k$, $y=\frac{\pi}{6}-\pi\cdot k$ |
85a160ac-e54e-409a-ab21-06d3510b70c4 | differential_calc | false | null | Find the maximum and minimum values of the function $r = \frac{ 1 }{ 3 } \cdot \sin(x) + \sin\left(\frac{ x }{ 3 }\right)$ in the closed interval $\left[0, \frac{ 3 }{ 2 } \cdot \pi\right]$. | Maximum Value: $\frac{2\cdot\sqrt{2}}{3}$
Minimum Value: $0$ |
85de5956-0bf2-4223-96e5-7ef262790fe9 | algebra | false | null | For what values of $x$ and $y$ does the following expression attain the minimum possible value?
$E = 5 \cdot x^2 + 9 \cdot y^2 - 12 \cdot x \cdot y - 6 \cdot x + 14$ | The final answer: $x=3$, $y=2$ |
868594b4-04b6-4f77-abef-df1528a43633 | precalculus_review | false | null | Evaluate $\tan\left(\arcsin\left(\frac{ 3 }{ 5 }\right)+\arccos\left(\frac{ 5 }{ 13 }\right)\right)$. | The final answer: $\tan\left(\arcsin\left(\frac{3}{5}\right)+\arccos\left(\frac{5}{13}\right)\right)=-\frac{63}{16}$ |
868d5f3c-4350-4fe7-b65b-161b591491ae | precalculus_review | false | null | Vertices of a triangle are at the points $A(-1,3)$, $B(3,0)$, and $C(8,12)$. Calculate the area of the triangle. | The final answer: $A=\frac{63}{2}$ |
877845fa-0d38-4fcf-8281-44fe62452bb9 | precalculus_review | false | null | Find zeros of $f(x) = \sqrt[3]{2 \cdot x - 3} + \sqrt[3]{3 \cdot x + 2} - 3$ | The final answer: $x=2$ |
877e3f76-3b61-4e49-8196-680ebce22030 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAcAAAADrCAIAAABxW9FlAACEKElEQVR4nO1dd5wV1fX/3ntn5r1tLAs2pINioaOioHSxxKjE8lOjYG+JscQkdqUZTSzRRI3GjsYkdqPGgoIUxUQ6dnpREGnLlvdm5t57fn+c92Zndykp7LrgfD/vs/vevHnT5zvnnvM95wgiQkPCWiul5PdEZK1VSllrhRBCCCIiIimlMYaIHMfh+YMg8DwPgO/7qVSKiIQQDbqdCRIkSPCfQjQ0gTLCMHRdV2vtOA5PiTgxYlV+D6AOVz... | A projectile is shot in the air from ground level with an initial velocity of $500$ m/sec at an angle of $60^\circ$ with the horizontal. What is the maximum range? Round your answer to one decimal digit.
The graph is shown here: | Answer: $22092.5$ m |
87bef79a-5776-402a-9247-942662ea7739 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABlPklEQVR4nO3de3hNZ/7//9eOCEGUOIaaqJaEDlGG0FA19dF2OsPo8dPZoaOjaUeLGm2no8z40BptR1Wd4zBDqZpWVcvoFEVIk6hTVIUeFFVxVo0EsZP1+8N37590JwRrr5211/NxXbnKnZV13+/u2FmvvNfBZRiGoQps/PjxGj9+vCRp2LBhGjZsWJBXZI3U1FRJUkpKSpBXYi2n1e20er2o2xl1O61eL6fV7bR6vZxWt9Pq9aLuwNQdFp... | Graph of $a(t)$
A bug is traveling along a straight path such that its acceleration, $a(t)$, in centimeters per second per second, for $0 \le t \le 12$ is given by the graph above.
At $t=0$, the bug is traveling at a rate of $10$ centimeters per second. At what other time is the bug traveling at a rate of $10$ centim... | $t$ = $11$ |
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Differential Calc Problems
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