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college_math.Calculus | exercise.10.6.2 | Determine whether the series $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{3 n^{2}+4}{2 n^{2}+3 n+5} $ converges absolutely, converges conditionally, or diverges. | diverges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | diverges |
college_math.Calculus | exercise.8.2.1 | Find the antiderivative: $\int \sin ^{2} x d x $ | $x / 2-\sin (2 x) / 4+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | x / 2-\sin (2 x) / 4+C |
college_math.Calculus | exercise.9.2.8 | An object moves along a straight line with acceleration given by $a(t)=-\cos (t)$, and $s(0)=1$ and $v(0)=0$. Find the maximum distance the object travels from zero, and find its maximum speed. Describe the motion of the object. | $s(t)=\cos t, v(t)=-\sin t$, maximum distance is 1 , maximum speed is 1 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | s(t)=\cos t, v(t)=-\sin t$, maximum distance is 1 , maximum speed is 1 |
college_math.Calculus | exercise.5.3.10 | Find all local maximum and minimum points of the function: $y=(x+1) / \sqrt{5 x^{2}+35} $ | $\max$ at $x=7$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \max$ at $x=7 |
college_math.Calculus | exercise.1.3.5 | Find the domain of the function: $y=f(x)=\sqrt[3]{x} $ | $\{x \mid x \in \mathbb{R}\}$, i.e., all $x$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \{x \mid x \in \mathbb{R}\}$, i.e., all $x |
college_math.Calculus | exercise.4.8.1 | Compute the limit of $\lim _{x \rightarrow 0} \frac{\cos x-1}{\sin x} $. | 0 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 0 |
college_math.Calculus | exercise.8.4.10 | Find the antiderivative: $\int x \sin x \cos x d x $ | $x / 4-\left(x \cos ^{2} x\right) / 2+(\cos x \sin x) / 4+$ C | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | x / 4-\left(x \cos ^{2} x\right) / 2+(\cos x \sin x) / 4+$ C |
college_math.Calculus | exercise.3.5.9 | Find the derivative of the function: $(1+3 x)^{2} $ | $6+18 x$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 6+18 x |
college_math.Calculus | exercise.4.10.3 | Find the derivative of $\operatorname{arccot} x$, the inverse cotangent. | $-1 /\left(1+x^{2}\right)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -1 /\left(1+x^{2}\right) |
college_math.Calculus | exercise.3.2.1 | Find the derivative of the function: $5 x^{3}+12 x^{2}-15 $ | $15 x^{2}+24 x$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 15 x^{2}+24 x |
college_math.Calculus | exercise.9.6.8 | A thin plate lies in the region contained by $y=4-x^{2}$ and the $x$-axis. Find the centroid. | $\bar{x}=0, \bar{y}=8 / 5$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \bar{x}=0, \bar{y}=8 / 5 |
college_math.Calculus | exercise.10.12.6 | Determine whether the series converges: $\sum_{n=0}^{\infty} \frac{1}{\sqrt{n^{2}+4}} $ | diverges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | diverges |
college_math.Calculus | exercise.9.3.11 | A hemispheric bowl of radius $r$ contains water to a depth $h$. Find the volume of water in the bowl. | $\pi h^{2}(3 r-h) / 3$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \pi h^{2}(3 r-h) / 3 |
college_math.Calculus | exercise.6.5.6 | Describe all functions with derivative $x^{2}+47 x-5$. | $x^{3} / 3+47 x^{2} / 2-5 x+k$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | x^{3} / 3+47 x^{2} / 2-5 x+k |
college_math.Calculus | exercise.3.5.2 | Find the derivative of the function: $x^{3}-2 x^{2}+4 \sqrt{x} $ | $3 x^{2}-4 x+2 / \sqrt{x}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 3 x^{2}-4 x+2 / \sqrt{x} |
college_math.Calculus | exercise.10.6.5 | Determine whether the series $\sum_{n=2}^{\infty}(-1)^{n} \frac{1}{\ln n} $ converges absolutely, converges conditionally, or diverges. | converges conditionally | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | converges conditionally |
college_math.Calculus | exercise.10.12.9 | Determine whether the series converges: $\sum_{n=0}^{\infty} \frac{n !}{1 \cdot 3 \cdot 5 \cdots(2 n-1)} $ | converges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | converges |
college_math.Calculus | exercise.10.12.5 | Determine whether the series converges: $1-\frac{3}{4}+\frac{5}{8}-\frac{7}{12}+\frac{9}{16}+\cdots $ | diverges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | diverges |
college_math.Calculus | exercise.8.6.2 | Evaluate the integral: $\int t\left(t^{2}-9\right)^{3 / 2} d t $ | $\frac{\left(t^{2}-9\right)^{5 / 2}}{5}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{\left(t^{2}-9\right)^{5 / 2}}{5}+C |
college_math.Calculus | exercise.10.7.6 | Determine whether the series $\sum_{n=1}^{\infty} \frac{n !}{n^{n}} $ converges. | converges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | converges |
college_math.Calculus | exercise.8.3.5 | Find the antiderivative: $\int x \sqrt{1-x^{2}} d x $ | $-\left(1-x^{2}\right)^{3 / 2} / 3+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -\left(1-x^{2}\right)^{3 / 2} / 3+C |
college_math.Calculus | exercise.1.3.6 | Find the domain of the function: $y=f(x)=\sqrt[4]{x} $ | $\{x \mid x \geq 0\}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \{x \mid x \geq 0\} |
college_math.Calculus | exercise.9.3.7 | Find the volume of the solid obtained by revolving the region bounded by $y=\sqrt{\sin x}$, the $y$-axis, and the lines $y=1$ and $x=\pi / 2$ around the $x$-axis. | $\pi(\pi / 2-1)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \pi(\pi / 2-1) |
college_math.Calculus | exercise.4.7.1 | Find the derivative of the function: $3^{x^{2}} $ | $2 \ln (3) x 3^{x^{2}}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 2 \ln (3) x 3^{x^{2}} |
college_math.Calculus | exercise.7.2.18 | Find the derivative of the function: $G(x)=\int_{1}^{x^{2}} t^{2}-3 t d t $ | $2 x\left(x^{4}-3 x^{2}\right)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 2 x\left(x^{4}-3 x^{2}\right) |
college_math.Calculus | exercise.3.1.5 | Find the derivative of the function: $x^{3 / 4} $ | $(3 / 4) x^{-1 / 4}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | (3 / 4) x^{-1 / 4} |
college_math.Calculus | exercise.6.1.6 | A box with square base and no top is to hold a volume $V$. Find (in terms of $V$ ) the dimensions of the box that requires the least material for the five sides. Also find the ratio of height to side of the base. (This ratio will not involve $V$.) | $w=l=2^{1 / 3} V^{1 / 3}, h=V^{1 / 3} / 2^{2 / 3}$, $h / w=1 / 2$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | w=l=2^{1 / 3} V^{1 / 3}, h=V^{1 / 3} / 2^{2 / 3}$, $h / w=1 / 2 |
college_math.Calculus | exercise.9.7.9 | Does the improper integral $\int_{-\infty}^{\infty} \frac{x^{2}}{4+x^{6}} d x$ converge or diverge? If it converges, find the value. | $\pi / 6$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \pi / 6 |
college_math.Calculus | exercise.9.5.5 | A water tank has the shape of the bottom half of a sphere with radius $r=1$ meter. If the tank is full, how much work is required to pump all the water out the top of the tank? | $2450 \pi \mathrm{N}-\mathrm{m}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 2450 \pi \mathrm{N}-\mathrm{m} |
college_math.Calculus | exercise.3.5.15 | Find the derivative of the function: $\sqrt[3]{x+x^{3}} $ | $\frac{1+3 x^{2}}{3\left(x+x^{3}\right)^{2 / 3}}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{1+3 x^{2}}{3\left(x+x^{3}\right)^{2 / 3}} |
college_math.Calculus | exercise.9.9.4 | Find the arc length of $f(x)=\ln (\sin x)$ on the interval $[\pi / 4, \pi / 3] . $ | $\ln ((\sqrt{2}+1) / \sqrt{3})$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \ln ((\sqrt{2}+1) / \sqrt{3}) |
college_math.Calculus | exercise.4.7.10 | Find the derivative of the function: $e^{4 x} / x $ | $e^{4 x}(4 x-1) / x^{2}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | e^{4 x}(4 x-1) / x^{2} |
college_math.Calculus | exercise.10.5.3 | Determine whether the series $\sum_{n=1}^{\infty} \frac{1}{2 n^{2}-3 n-5} $ converges or diverges. | converges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | converges |
college_math.Calculus | exercise.10.12.24 | Find a series representation for the function: $\sum_{n=0}^{\infty} \frac{(x-1)^{n}}{n !} $ | $(-\infty, \infty)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | (-\infty, \infty) |
college_math.Calculus | exercise.3.5.27 | Find the derivative of the function: $\left(3 x^{2}+1\right)(2 x-4)^{3} $ | $120 x^{4}-576 x^{3}+888 x^{2}-480 x+96$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 120 x^{4}-576 x^{3}+888 x^{2}-480 x+96 |
college_math.Calculus | exercise.9.6.1 | A beam 10 meters long has density $\sigma(x)=x^{2}$ at distance $x$ from the left end of the beam. Find the center of mass $\bar{x}$. | $15 / 2$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 15 / 2 |
college_math.Calculus | exercise.9.1.6 | Find the area bounded by the curves: $y=\sin (\pi x / 3)$ and $y=x$ (in the first quadrant) | $3 / \pi-3 \sqrt{3} /(2 \pi)-1 / 8$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 3 / \pi-3 \sqrt{3} /(2 \pi)-1 / 8 |
college_math.Calculus | exercise.8.3.1 | Find the antiderivative: $\int \csc x d x $ | $-\ln |\csc x+\cot x|+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -\ln |\csc x+\cot x|+C |
college_math.Calculus | exercise.4.3.6 | For all $x \geq 0,4 x-9 \leq f(x) \leq x^{2}-4 x+7$. Find $\lim _{x \rightarrow 4} f(x)$. | 7 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 7 |
college_math.Calculus | exercise.6.4.2 | Let $f(x)=\sqrt{x}$. If $a=1$ and $d x=\Delta x=1 / 10$, what are $\Delta y$ and $d y$ ? | $\Delta y=\sqrt{11 / 10}-1, d y=0.05$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \Delta y=\sqrt{11 / 10}-1, d y=0.05 |
college_math.Calculus | exercise.10.3.7 | Determine whether the series converges or diverges: $\sum_{n=2}^{\infty} \frac{1}{n \ln n} $ | diverges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | diverges |
college_math.Calculus | exercise.6.2.1 | A cylindrical tank standing upright (with one circular base on the ground) has radius 20 $\mathrm{cm}$. Find the rate at which the water level in the tank drops when the water is being drained at 25 $\mathrm{cm}^{3} / \mathrm{sec}$. | $1 /(16 \pi) \mathrm{cm} / \mathrm{s}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 1 /(16 \pi) \mathrm{cm} / \mathrm{s} |
college_math.Calculus | exercise.3.2.4 | Find the derivative of the function: $f(x)+g(x)$, where $f(x)=x^{2}-3 x+2$ and $g(x)=2 x^{3}-5 x $ | $6 x^{2}+2 x-8$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 6 x^{2}+2 x-8 |
college_math.Calculus | exercise.9.7.1 | Determine whether the area under the curve $y=1 / x$ from 1 to infinity is finite or infinite. If it is finite, compute the area. | $\infty$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \infty |
college_math.Calculus | exercise.7.2.10 | Find the antiderivative of the function: $|2 t-4| $ | $4 t-t^{2}+C, t<2 ; t^{2}-4 t+8+C$, $t \geq 2$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 4 t-t^{2}+C, t<2 ; t^{2}-4 t+8+C$, $t \geq 2 |
college_math.Calculus | exercise.4.4.3 | Find the derivative of the function: $\frac{1}{\sin x} $ | $-\frac{\cos x}{\sin ^{2} x}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -\frac{\cos x}{\sin ^{2} x} |
college_math.Calculus | exercise.8.4.7 | Find the antiderivative: $\int x \arctan x d x $ | $\left(x^{2} \arctan x+\arctan x-x\right) / 2+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \left(x^{2} \arctan x+\arctan x-x\right) / 2+C |
college_math.Calculus | exercise.8.4.11 | Find the antiderivative: $\int \arctan (\sqrt{x}) d x $ | $x \arctan (\sqrt{x})+\arctan (\sqrt{x})-\sqrt{x}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | x \arctan (\sqrt{x})+\arctan (\sqrt{x})-\sqrt{x}+C |
college_math.Calculus | exercise.10.9.3 | Find a power series representation for $2 /(1-x)^{3}$. | $\sum_{n=0}^{\infty}(n+1)(n+2) x^{n}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \sum_{n=0}^{\infty}(n+1)(n+2) x^{n} |
college_math.Calculus | exercise.10.12.12 | Determine whether the series converges: $\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{(2 n) !} $ | converges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | converges |
college_math.Calculus | exercise.5.2.11 | Find all critical points of the function $f(x)=x^{3} /(x+1) $. Identify them as local maximum points, local minimum points, or neither. | $\min$ at $x=-3 / 2$, neither at $x=0$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \min$ at $x=-3 / 2$, neither at $x=0 |
college_math.Calculus | exercise.5.3.2 | Find all local maximum and minimum points of the function: $y=2+3 x-x^{3} $ | $\min$ at $x=-1, \max$ at $x=1$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \min$ at $x=-1, \max$ at $x=1 |
college_math.Calculus | exercise.4.1.11 | Find all of the solutions of $2 \sin (t)-1-\sin ^{2}(t)=0$ in the interval $[0,2 \pi]$. | $t=\pi / 2$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | t=\pi / 2 |
college_math.Calculus | exercise.10.6.7 | Determine whether the series $\sum_{n=0}^{\infty}(-1)^{n} \frac{3^{n}}{2^{n}+3^{n}} $ converges absolutely, converges conditionally, or diverges. | diverges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | diverges |
college_math.Calculus | exercise.8.3.7 | Find the antiderivative: $\int \frac{1}{\sqrt{1+x^{2}}} d x $ | $\ln \left|x+\sqrt{1+x^{2}}\right|+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \ln \left|x+\sqrt{1+x^{2}}\right|+C |
college_math.Calculus | exercise.3.5.26 | Find the derivative of the function: $\left(x^{2}+1\right)(5-2 x) / 2 $ | $-3 x^{2}+5 x-1$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -3 x^{2}+5 x-1 |
college_math.Calculus | exercise.8.6.23 | Evaluate the integral: $\int \sin ^{3} t \cos ^{4} t d t $ | $\frac{\cos ^{7} t}{7}-\frac{\cos ^{5} t}{5}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{\cos ^{7} t}{7}-\frac{\cos ^{5} t}{5}+C |
college_math.Calculus | exercise.4.5.6 | Find the derivative of the function: $\csc x $ | $-\csc x \cot x$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -\csc x \cot x |
college_math.Calculus | exercise.6.1.9 | Marketing tells you that if you set the price of an item at $\$ 10$ then you will be unable to sell it, but that you can sell 500 items for each dollar below $\$ 10$ that you set the price. Suppose your fixed costs total $\$ 3000$, and your marginal cost is $\$ 2$ per item. What is the most profit you can make? | $\$ 5000$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \$ 5000 |
college_math.Calculus | exercise.10.1.6 | Determine whether the sequence $\left\{\frac{2^{n}}{n !}\right\}_{n=0}^{\infty}$ converges or diverges. | 0 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 0 |
college_math.Calculus | exercise.10.2.1 | Explain why the series $\sum_{n=1}^{\infty} \frac{n^{2}}{2 n^{2}+1}$ diverges. | $\lim _{n \rightarrow \infty} n^{2} /\left(2 n^{2}+1\right)=1 / 2$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \lim _{n \rightarrow \infty} n^{2} /\left(2 n^{2}+1\right)=1 / 2 |
college_math.Calculus | exercise.6.1.25 | What fraction of the volume of a sphere is taken up by the largest cylinder that can be fit inside the sphere? | $1 / \sqrt{3} \approx 58 \%$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 1 / \sqrt{3} \approx 58 \% |
college_math.Calculus | exercise.9.5.3 | A water tank has the shape of a cylinder with radius $r=1$ meter and height 10 meters. If the depth of the water is 5 meters, how much work is required to pump all the water out the top of the tank? | $367,500 \pi \mathrm{N}-\mathrm{m}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 367,500 \pi \mathrm{N}-\mathrm{m} |
college_math.Calculus | exercise.9.1.10 | Find the area bounded by the curves: $y=\sin x \cos x$ and $y=\sin x, 0 \leq x \leq \pi $ | 2 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 2 |
college_math.Calculus | exercise.10.5.8 | Determine whether the series $\sum_{n=2}^{\infty} \frac{1}{\ln n} $ converges or diverges. | diverges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | diverges |
college_math.Calculus | exercise.9.2.10 | An object moves along a straight line with acceleration given by $a(t)=1+\sin (\pi t)$. Assume that when $t=0, s(t)=v(t)=0$. Find $s(t)$ and $v(t)$. | $s(t)=t^{2} / 2-\sin (\pi t) / \pi^{2}+t / \pi$, $v(t)=t-\cos (\pi t) / \pi+1 / \pi$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | s(t)=t^{2} / 2-\sin (\pi t) / \pi^{2}+t / \pi$, $v(t)=t-\cos (\pi t) / \pi+1 / \pi |
college_math.Calculus | exercise.9.6.12 | A thin plate lies in the region between the circle $x^{2}+y^{2}=4$ and the circle $x^{2}+y^{2}=1$ in the first quadrant. Find the centroid. | $\bar{x}=\bar{y}=28 /(9 \pi)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \bar{x}=\bar{y}=28 /(9 \pi) |
college_math.Calculus | exercise.4.4.5 | Find the derivative of the function: $\sqrt{1-\sin ^{2} x} $ | $\frac{-\sin x \cos x}{\sqrt{1-\sin ^{2} x}}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{-\sin x \cos x}{\sqrt{1-\sin ^{2} x}} |
college_math.Calculus | exercise.8.6.24 | Evaluate the integral: $\int \frac{1}{t^{2}-6 t+9} d t $ | $\frac{-1}{t-3}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{-1}{t-3}+C |
college_math.Calculus | exercise.3.5.14 | Find the derivative of the function: $100 /\left(100-x^{2}\right)^{3 / 2} $ | $\frac{300 x}{\left(100-x^{2}\right)^{5 / 2}}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{300 x}{\left(100-x^{2}\right)^{5 / 2}} |
college_math.Calculus | exercise.8.5.5 | Find the antiderivative: $\int \frac{x^{4}}{4+x^{2}} d x $ | $-4 x+x^{3} / 3+8 \arctan (x / 2)+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -4 x+x^{3} / 3+8 \arctan (x / 2)+C |
college_math.Calculus | exercise.1.3.11 | Find the domain of the function: $y=f(x)=1 /(\sqrt{x}-1) $ | $\{x \mid x \geq 0$ and $x \neq 1\}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \{x \mid x \geq 0$ and $x \neq 1\} |
college_math.Calculus | exercise.5.4.10 | Describe the concavity of the function: $y=(x+1) / \sqrt{5 x^{2}+35} $ | concave up on $(-\infty,(21-\sqrt{497}) / 4)$ and $(21+\sqrt{497}) / 4, \infty)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | concave up on $(-\infty,(21-\sqrt{497}) / 4)$ and $(21+\sqrt{497}) / 4, \infty) |
college_math.Calculus | exercise.2.3.3 | Compute the limit: $\lim _{x \rightarrow-4} \frac{x^{2}+x-12}{x-3} $. If a limit does not exist, explain why. | 0 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 0 |
college_math.Calculus | exercise.10.12.13 | Determine whether the series converges: $\sum_{n=0}^{\infty} \frac{6^{n}}{n !} $ | converges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | converges |
college_math.Calculus | exercise.6.5.8 | Describe all functions with derivative $x^{3}-\frac{1}{x}$. | $x^{4} / 4-\ln x+k$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | x^{4} / 4-\ln x+k |
college_math.Calculus | exercise.4.9.14 | Find an equation for the tangent line to $x^{4}=y^{2}+x^{2}$ at $(2, \sqrt{12})$. (This curve is the kampyle of Eudoxus.) | $y=7 x / \sqrt{3}-8 / \sqrt{3}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | y=7 x / \sqrt{3}-8 / \sqrt{3} |
college_math.Calculus | exercise.7.1.2 | Suppose an object moves in a straight line so that its speed at time $t$ is given by $v(t)=t^{2}+2$, and that at $t=0$ the object is at position 5 . Find the position of the object at $t=2 . $ | $35 / 3$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 35 / 3 |
college_math.Calculus | exercise.10.8.2 | Find the radius and interval of convergence for the series: $\sum_{n=0}^{\infty} \frac{x^{n}}{n !} $ | $R=\infty, I=(-\infty, \infty)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | R=\infty, I=(-\infty, \infty) |
college_math.Calculus | exercise.6.1.13 | For a cylinder with given surface area $S$, including the top and the bottom, find the ratio of height to base radius that maximizes the volume. | $h / r=2$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | h / r=2 |
college_math.Calculus | exercise.9.7.2 | Determine whether the area under the curve $y=1 / x^{3}$ from 1 to infinity is finite or infinite. If it is finite, compute the area. | $1 / 2$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 1 / 2 |
college_math.Calculus | exercise.4.4.1 | Find the derivative of the function: $\sin ^{2}(\sqrt{x}) $ | $\sin (\sqrt{x}) \cos (\sqrt{x}) / \sqrt{x}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \sin (\sqrt{x}) \cos (\sqrt{x}) / \sqrt{x} |
college_math.Calculus | exercise.9.7.6 | Express the improper integral $\int_{0}^{1 / 2}(2 x-1)^{-3} d x$ as a limit and determine whether it converges or diverges. If it converges, find the value. | diverges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | diverges |
college_math.Calculus | exercise.10.3.12 | Find an $N$ such that $\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{2}}=\sum_{n=2}^{N} \frac{1}{n(\ln n)^{2}} \pm 0.005$. | any integer greater than $e^{200}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | any integer greater than $e^{200} |
college_math.Calculus | exercise.6.2.16 | A police helicopter is flying at a speed of $150 \mathrm{~mph}$ at a constant altitude of 0.5 mile above a straight road. The pilot uses radar to determine that an oncoming car is at a distance of exactly 1 mile from the helicopter, and that this distance is decreasing at a rate of $190 \mathrm{~mph}$. Find the speed o... | $380 / \sqrt{3}-150 \approx 69.4 \mathrm{mph}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 380 / \sqrt{3}-150 \approx 69.4 \mathrm{mph} |
college_math.Calculus | exercise.1.3.2 | Find the domain of the function: $y=f(x)=1 /(x+1) $ | $\{x \mid x \neq-1\}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \{x \mid x \neq-1\} |
college_math.Calculus | exercise.10.3.4 | Determine whether the series converges or diverges: $\sum_{n=1}^{\infty} \frac{1}{n^{2}+1} $ | converges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | converges |
college_math.Calculus | exercise.4.8.2 | Compute the limit of $\lim _{x \rightarrow \infty} \frac{e^{x}}{x^{3}} $. | $\infty$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \infty |
college_math.Calculus | exercise.10.5.9 | Determine whether the series $\sum_{n=1}^{\infty} \frac{3^{n}}{2^{n}+5^{n}} $ converges or diverges. | converges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | converges |
college_math.Calculus | exercise.6.2.24 | A light shines from the top of a pole $20 \mathrm{~m}$ high. An object is dropped from the same height from a point $10 \mathrm{~m}$ away, so that its height at time $t$ seconds is $h(t)=20-9.8 t^{2} / 2$. Find the rate at which the object's shadow is moving on the ground one second later. | $4000 / 49 \mathrm{~m} / \mathrm{s}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 4000 / 49 \mathrm{~m} / \mathrm{s} |
college_math.Calculus | exercise.4.4.4 | Find the derivative of the function: $\frac{x^{2}+x}{\sin x} $ | $\frac{(2 x+1) \sin x-\left(x^{2}+x\right) \cos x}{\sin ^{2} x}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{(2 x+1) \sin x-\left(x^{2}+x\right) \cos x}{\sin ^{2} x} |
college_math.Calculus | exercise.7.2.4 | Find the antiderivative of the function: $2 / z^{2} $ | $-2 / z+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -2 / z+C |
college_math.Calculus | exercise.8.1.9 | Find the antiderivative of the function: $\int \frac{\sin x}{\cos ^{3} x} d x $ | $1 /\left(2 \cos ^{2} x\right)=(1 / 2) \sec ^{2} x+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 1 /\left(2 \cos ^{2} x\right)=(1 / 2) \sec ^{2} x+C |
college_math.Calculus | exercise.3.5.22 | Find the derivative of the function: $5(x+1-1 / x) $ | $5+5 / x^{2}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 5+5 / x^{2} |
college_math.Calculus | exercise.10.5.10 | Determine whether the series $\sum_{n=1}^{\infty} \frac{3^{n}}{2^{n}+3^{n}} $ converges or diverges. | diverges | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | diverges |
college_math.Calculus | exercise.8.1.15 | Evaluate the definite integral: $\int_{3}^{4} \frac{1}{(3 x-7)^{2}} d x $ | $1 / 10$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 1 / 10 |
college_math.Calculus | exercise.8.4.4 | Find the antiderivative: $\int x e^{x^{2}} d x $ | $(1 / 2) e^{x^{2}}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | (1 / 2) e^{x^{2}}+C |
college_math.Calculus | exercise.8.6.13 | Evaluate the integral: $\int \frac{1}{t^{2}+3 t} d t $ | $\frac{\ln |t|}{3}-\frac{\ln |t+3|}{3}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{\ln |t|}{3}-\frac{\ln |t+3|}{3}+C |
college_math.Calculus | exercise.3.5.20 | Find the derivative of the function: $\left(6-2 x^{2}\right)^{3} $ | $-12 x\left(6-2 x^{2}\right)^{2}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -12 x\left(6-2 x^{2}\right)^{2} |
college_math.Calculus | exercise.5.3.11 | Find all local maximum and minimum points of the function: $y=x^{5}-x $ | $\max$ at $-5^{-1 / 4}$, min at $5^{-1 / 4}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \max$ at $-5^{-1 / 4}$, min at $5^{-1 / 4} |
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