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college_math.Calculus | exercise.9.1.8 | Find the area bounded by the curves: $y=\sqrt{x}$ and $y=\sqrt{x+1}, 0 \leq x \leq 4 $ | $10 \sqrt{5} / 3-6$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 10 \sqrt{5} / 3-6 |
college_math.Calculus | exercise.8.2.3 | Find the antiderivative: $\int \sin ^{4} x d x $ | $3 x / 8-(\sin 2 x) / 4+(\sin 4 x) / 32+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 3 x / 8-(\sin 2 x) / 4+(\sin 4 x) / 32+C |
college_math.Calculus | exercise.5.2.3 | Find all critical points of the function $y=x^{3}-9 x^{2}+24 x $. Identify them as local maximum points, local minimum points, or neither. | $\max$ at $x=2$, min at $x=4$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \max$ at $x=2$, min at $x=4 |
college_math.Calculus | exercise.4.8.4 | Compute the limit of $\lim _{x \rightarrow \infty} \frac{\ln x}{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.4.5.15 | Find an equation for the tangent line to $\sin ^{2}(x)$ at $x=\pi / 3$. | $\sqrt{3} x / 2+3 / 4-\sqrt{3} \pi / 6$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \sqrt{3} x / 2+3 / 4-\sqrt{3} \pi / 6 |
college_math.Calculus | exercise.7.2.20 | Find the derivative of the function: $G(x)=\int_{1}^{x^{2}} e^{t^{2}} d t $ | $2 x e^{x^{4}}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 2 x e^{x^{4}} |
college_math.Calculus | exercise.8.1.14 | Find the antiderivative of the function: $\int \frac{\sin (\tan x)}{\cos ^{2} x} d x $ | $-\cos (\tan x)+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -\cos (\tan x)+C |
college_math.Calculus | exercise.10.8.5 | Find the radius and interval of convergence for the series: $\sum_{n=1}^{\infty} \frac{(n !)^{2}}{n^{n}}(x-2)^{n} $ | $R=0$, converges only when $x=2$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | R=0$, converges only when $x=2 |
college_math.Calculus | exercise.3.2.5 | Find the derivative of the function: $(x+1)\left(x^{2}+2 x-3\right) $ | $3 x^{2}+6 x-1$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 3 x^{2}+6 x-1 |
college_math.Calculus | exercise.4.8.7 | The function $f(x)=\frac{x}{\sqrt{x^{2}+1}}$ has two horizontal asymptotes. Find them and give a rough sketch of $f$ with its horizontal asymptotes. | $y=1$ and $y=-1$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | y=1$ and $y=-1 |
college_math.Calculus | exercise.1.3.9 | Find the domain of the function: $y=f(x)=1 / \sqrt{1-(3 x)^{2}} $ | $\{x \mid-1 / 3<x<1 / 3\}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \{x \mid-1 / 3<x<1 / 3\} |
college_math.Calculus | exercise.1.3.4 | Find the domain of the function: $y=f(x)=\sqrt{-1 / x} $ | $\{x \mid x<0\}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \{x \mid x<0\} |
college_math.Calculus | exercise.4.7.5 | Find the derivative of the function: $e^{\sin x} $ | $\cos (x) e^{\sin x}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \cos (x) e^{\sin x} |
college_math.Calculus | exercise.9.5.1 | How much work is done in lifting a 100 kilogram weight from the surface of the earth to an orbit 35,786 kilometers above the surface of the earth? | $\approx 5,305,028,517 \mathrm{~N}-\mathrm{m}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \approx 5,305,028,517 \mathrm{~N}-\mathrm{m} |
college_math.Calculus | exercise.3.1.6 | Find the derivative of the function: $x^{-9 / 7} $ | $-(9 / 7) x^{-16 / 7}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -(9 / 7) x^{-16 / 7} |
college_math.Calculus | exercise.6.2.2 | A cylindrical tank standing upright (with one circular base on the ground) has radius 1 meter. Find the rate at which the water level in the tank drops when the water is being drained at 3 liters per second. | $3 /(1000 \pi)$ meters $/$ second | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 3 /(1000 \pi)$ meters $/$ second |
college_math.Calculus | exercise.3.4.5 | Find an equation for the tangent line to $f(x)=\left(x^{2}-4\right) /(5-x)$ at $x=3$. | $y=17 x / 4-41 / 4$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | y=17 x / 4-41 / 4 |
college_math.Calculus | exercise.5.4.8 | Describe the concavity of the function: $y=\sin x+\cos x $ | concave down on $((8 n-1) \pi / 4,(8 n+$ $3) \pi / 4)$, concave up on $((8 n+$ $3) \pi / 4,(8 n+7) \pi / 4)$, for integer $n$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | concave down on $((8 n-1) \pi / 4,(8 n+$ $3) \pi / 4)$, concave up on $((8 n+$ $3) \pi / 4,(8 n+7) \pi / 4)$, for integer $n |
college_math.Calculus | exercise.8.1.4 | Find the antiderivative of the function: $\int \frac{1}{\sqrt[3]{1-5 t}} d t $ | $-3(1-5 t)^{2 / 3} / 10+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -3(1-5 t)^{2 / 3} / 10+C |
college_math.Calculus | exercise.8.5.7 | Find the antiderivative: $\int \frac{x^{3}}{4+x^{2}} d x $ | $x^{2} / 2-2 \ln \left(4+x^{2}\right)+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | x^{2} / 2-2 \ln \left(4+x^{2}\right)+C |
college_math.Calculus | exercise.9.7.13 | Determine whether the volume of the solid obtained by rotating the curve $y=1 / x$ around the x-axis, from $x=1$ to infinity, is finite or infinite. If it is finite, compute the volume. | $\pi$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \pi |
college_math.Calculus | exercise.6.2.6 | A baseball diamond is a square $90 \mathrm{ft}$ on a side. A player runs from first base to second base at a speed of $15 \mathrm{ft} / \mathrm{sec}$. At what rate is the player's distance from third base decreasing when she is halfway from first to second base? | $3 \sqrt{5} \mathrm{ft} / \mathrm{s}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 3 \sqrt{5} \mathrm{ft} / \mathrm{s} |
college_math.Calculus | exercise.8.4.12 | Find the antiderivative: $\int \sin (\sqrt{x}) d x $ | $2 \sin (\sqrt{x})-2 \sqrt{x} \cos (\sqrt{x})+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 2 \sin (\sqrt{x})-2 \sqrt{x} \cos (\sqrt{x})+C |
college_math.Calculus | exercise.2.4.2 | Find the derivative of the function: $y=f(t)=80-4.9 t^{2}$. | $-9.8 t$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -9.8 t |
college_math.Calculus | exercise.7.2.13 | Compute the value of the integral: $\int_{1}^{10} \frac{1}{x} d x $ | $\ln (10)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \ln (10) |
college_math.Calculus | exercise.8.4.13 | Find the antiderivative: $\int \sec ^{2} x \csc ^{2} x d x $ | $\sec x \csc x-2 \cot x+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \sec x \csc x-2 \cot x+C |
college_math.Calculus | exercise.2.4.1 | Find the derivative of the function: $y=f(x)=\sqrt{169-x^{2}}$. | $-x / \sqrt{169-x^{2}}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -x / \sqrt{169-x^{2}} |
college_math.Calculus | exercise.4.7.20 | Find the value of $a$ so that the tangent line to $y=\ln (x)$ at $x=a$ is a line through the origin. Sketch the resulting situation. | $e$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | e |
college_math.Calculus | exercise.5.3.1 | Find all local maximum and minimum points of the function: $y=x^{2}-x $ | $\min$ at $x=1 / 2$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \min$ at $x=1 / 2 |
college_math.Calculus | exercise.7.1.7 | Let $f(x)=x^{2}+3 x+2$. Approximate the area under the curve between $x=0$ and $x=2$ using 4 rectangles and also using 8 rectangles. | 4 rectangles: $41 / 4=10.25$, 8 rectangles: $183 / 16=11.4375$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 4 rectangles: $41 / 4=10.25$, 8 rectangles: $183 / 16=11.4375 |
college_math.Calculus | exercise.3.5.24 | Find the derivative of the function: $\frac{1}{1+1 / x} $ | $1 /(x+1)^{2}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 1 /(x+1)^{2} |
college_math.Calculus | exercise.5.4.12 | Describe the concavity of the function: $y=6 x+\sin 3 x $ | concave down on $(2 n \pi / 3,(2 n+$ 1) $\pi / 3)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | concave down on $(2 n \pi / 3,(2 n+$ 1) $\pi / 3) |
college_math.Calculus | exercise.10.10.6 | Find the Maclaurin series or Taylor series centered at $a$ and the radius of convergence for the function: $1 / x^{2}, a=1 $ | $\sum_{n=0}^{\infty}(-1)^{n}(n+1)(x-1)^{n}, R=1$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \sum_{n=0}^{\infty}(-1)^{n}(n+1)(x-1)^{n}, R=1 |
college_math.Calculus | exercise.8.3.2 | Find the antiderivative: $\int \csc ^{3} x d x $ | $-\csc x \cot x / 2-(1 / 2) \ln \mid \csc x+$ $\cot x \mid+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -\csc x \cot x / 2-(1 / 2) \ln \mid \csc x+$ $\cot x \mid+C |
college_math.Calculus | exercise.3.4.9 | If $f^{\prime}(4)=5, g^{\prime}(4)=12,(f g)(4)=f(4) g(4)=2$, and $g(4)=6$, compute $f(4)$ and $\frac{d}{d x} \frac{f}{g}$ at 4 . | $13 / 18$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 13 / 18 |
college_math.Calculus | exercise.3.5.37 | Find an equation for the tangent line to $y=9 x^{-2}$ at $(3,1)$. | $y=3-2 x / 3$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | y=3-2 x / 3 |
college_math.Calculus | exercise.10.4.2 | Determine whether the series $\sum_{n=4}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n-3}} $ 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.5.4.15 | Describe the concavity of the function: $y=(x+5)^{1 / 4} $ | concave down everywhere | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | concave down everywhere |
college_math.Calculus | exercise.8.6.19 | Evaluate the integral: $\int \frac{t^{3}}{\left(2-t^{2}\right)^{5 / 2}} d t $ | $\frac{2}{3\left(2-t^{2}\right)^{3 / 2}}-\frac{1}{\left(2-t^{2}\right)^{1 / 2}}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{2}{3\left(2-t^{2}\right)^{3 / 2}}-\frac{1}{\left(2-t^{2}\right)^{1 / 2}}+C |
college_math.Calculus | exercise.10.3.10 | Find an $N$ such that $\sum_{n=0}^{\infty} \frac{1}{e^{n}}=\sum_{n=0}^{N} \frac{1}{e^{n}} \pm 10^{-4}$. | $N=10$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | N=10 |
college_math.Calculus | exercise.10.6.6 | Determine whether the series $\sum_{n=0}^{\infty}(-1)^{n} \frac{3^{n}}{2^{n}+5^{n}} $ converges absolutely, converges conditionally, or diverges. | converges absolutely | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | converges absolutely |
college_math.Calculus | exercise.10.2.5 | Compute the value of the series $\sum_{n=0}^{\infty} \frac{3}{2^{n}}+\frac{4}{5^{n}}$. | 11 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 11 |
college_math.Calculus | exercise.3.5.29 | Find the derivative of the function: $\frac{x^{2}-1}{x^{2}+1} $ | $4 x /\left(x^{2}+1\right)^{2}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 4 x /\left(x^{2}+1\right)^{2} |
college_math.Calculus | exercise.9.7.10 | Does the improper integral $\int_{-\infty}^{\infty} x d x$ converge or diverge? If it converges, find the value. Also, find the Cauchy Principal Value, if it exists. | diverges, 0 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | diverges, 0 |
college_math.Calculus | exercise.5.3.3 | Find all local maximum and minimum points of the function: $y=x^{3}-9 x^{2}+24 x $ | $\max$ at $x=2$, min at $x=4$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \max$ at $x=2$, min at $x=4 |
college_math.Calculus | exercise.3.5.12 | Find the derivative of the function: $\sqrt{\frac{169}{x}-x} $ | $\frac{1}{2}\left(\frac{-169}{x^{2}}-1\right) / \sqrt{\frac{169}{x}-x}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{1}{2}\left(\frac{-169}{x^{2}}-1\right) / \sqrt{\frac{169}{x}-x} |
college_math.Calculus | exercise.2.3.8 | Compute the limit: $\lim _{x \rightarrow 4} 3 x^{3}-5 x $. | 172 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 172 |
college_math.Calculus | exercise.4.5.16 | Find an equation for the tangent line to $\sec ^{2} x$ at $x=\pi / 3$. | $8 \sqrt{3} x+4-8 \sqrt{3} \pi / 3$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 8 \sqrt{3} x+4-8 \sqrt{3} \pi / 3 |
college_math.Calculus | exercise.9.2.11 | 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.3.5.1 | Find the derivative of the function: $x^{4}-3 x^{3}+(1 / 2) x^{2}+7 x-\pi $ | $4 x^{3}-9 x^{2}+x+7$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 4 x^{3}-9 x^{2}+x+7 |
college_math.Calculus | exercise.4.7.7 | Find the derivative of the function: $x^{3} e^{x} $ | $3 x^{2} e^{x}+x^{3} e^{x}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 3 x^{2} e^{x}+x^{3} e^{x} |
college_math.Calculus | exercise.3.5.17 | Find the derivative of the function: $(x+8)^{5} $ | $5(x+8)^{4}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 5(x+8)^{4} |
college_math.Calculus | exercise.3.5.3 | Find the derivative of the function: $\left(x^{2}+1\right)^{3} $ | $6\left(x^{2}+1\right)^{2} x$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 6\left(x^{2}+1\right)^{2} x |
college_math.Calculus | exercise.1.2.6 | Find the standard equation of the circle passing through $(-2,1)$ and tangent to the line $3 x-2 y=6$ at the point $(4,3)$. Sketch. (Hint: The line through the center of the circle and the point of tangency is perpendicular to the tangent line.) | $(x+2 / 7)^{2}+(y-41 / 7)^{2}=1300 / 49$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | (x+2 / 7)^{2}+(y-41 / 7)^{2}=1300 / 49 |
college_math.Calculus | exercise.10.1.3 | Determine whether the sequence $\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}$ converges or diverges. If it converges, compute the limit. | 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.3.5.13 | Find the derivative of the function: $\sqrt{x^{3}-x^{2}-(1 / x)} $ | $\frac{3 x^{2}-2 x+1 / x^{2}}{2 \sqrt{x^{3}-x^{2}-(1 / x)}}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{3 x^{2}-2 x+1 / x^{2}}{2 \sqrt{x^{3}-x^{2}-(1 / x)}} |
college_math.Calculus | exercise.5.4.6 | Describe the concavity of the function: $y=\left(x^{2}-1\right) / x $ | concave up when $x<0$, concave down when $x>0$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | concave up when $x<0$, concave down when $x>0 |
college_math.Calculus | exercise.10.4.3 | Determine whether the series $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{n}{3 n-2} $ 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.8.4.2 | Find the antiderivative: $\int x^{2} \cos x d x $ | $x^{2} \sin x-2 \sin x+2 x \cos x+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | x^{2} \sin x-2 \sin x+2 x \cos x+C |
college_math.Calculus | exercise.6.5.9 | Describe all functions with derivative $\sin (2 x)$. | $-\cos (2 x) / 2+k$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -\cos (2 x) / 2+k |
college_math.Calculus | exercise.6.1.30 | If you fit the cone with the largest possible surface area (lateral area plus area of base) into a sphere, what percent of the volume of the sphere is occupied by the cone? | The ratio of the volume of the sphere to the volume of the cone is $1033 / 4096+33 / 4096 \sqrt{17} \approx 0.2854$, so the cone occupies approximately $28.54 \%$ of the sphere. | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | The ratio of the volume of the sphere to the volume of the cone is $1033 / 4096+33 / 4096 \sqrt{17} \approx 0.2854$, so the cone occupies approximately $28.54 \%$ of the sphere. |
college_math.Calculus | exercise.9.6.10 | A thin plate lies in the region contained by $\sqrt{x}+\sqrt{y}=1$ and the axes in the first quadrant. Find the centroid. | $\bar{x}=\bar{y}=1 / 5$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \bar{x}=\bar{y}=1 / 5 |
college_math.Calculus | exercise.6.3.2 | Use Newton's Method to approximate the cube root of 10 to two decimal places. | 2.15 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 2.15 |
college_math.Calculus | exercise.9.4.5 | An object moves with velocity $v(t)=-t^{2}+1$ feet per second between $t=0$ and $t=2$. Find the average velocity and the average speed of the object between $t=0$ and $t=2 . $ | $-1 / 3,1$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -1 / 3,1 |
college_math.Calculus | exercise.10.3.6 | Determine whether the series converges or diverges: $\sum_{n=1}^{\infty} \frac{n}{e^{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.4.1.3 | Use an angle sum identity to compute $\cos (\pi / 12)$. | $(\sqrt{2}+\sqrt{6}) / 2$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | (\sqrt{2}+\sqrt{6}) / 2 |
college_math.Calculus | exercise.2.4.3 | Find the derivative of the function: $y=f(x)=x^{2}-(1 / x)$. | $2 x+1 / x^{2}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 2 x+1 / x^{2} |
college_math.Calculus | exercise.3.2.2 | Find the derivative of the function: $-4 x^{5}+3 x^{2}-5 / x^{2} $ | $-20 x^{4}+6 x+10 / x^{3}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -20 x^{4}+6 x+10 / x^{3} |
college_math.Calculus | exercise.9.2.4 | For the velocity function $v=\sin (\pi t / 3)-t$, find both the net distance and the total distance traveled during the time interval $0 \leq t \leq 1$. | $(3-\pi) /(2 \pi),(18-12 \sqrt{3}+\pi) /(4 \pi)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | (3-\pi) /(2 \pi),(18-12 \sqrt{3}+\pi) /(4 \pi) |
college_math.Calculus | exercise.3.5.18 | Find the derivative of the function: $(4-x)^{3} $ | $-3(4-x)^{2}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -3(4-x)^{2} |
college_math.Calculus | exercise.8.6.18 | Evaluate the integral: $\int\left(t^{3 / 2}+47\right)^{3} \sqrt{t} d t $ | $\frac{\left(t^{3 / 2}+47\right)^{4}}{6}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{\left(t^{3 / 2}+47\right)^{4}}{6}+C |
college_math.Calculus | exercise.8.6.7 | Evaluate the integral: $\int \frac{1}{t\left(t^{2}-4\right)} d t $ | $\frac{1}{8} \ln \left|1-4 / t^{2}\right|+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{1}{8} \ln \left|1-4 / t^{2}\right|+C |
college_math.Calculus | exercise.3.5.5 | Find the derivative of the function: $\left(x^{2}-4 x+5\right) \sqrt{25-x^{2}} $ | $(2 x-4) \sqrt{25-x^{2}}-$ $\left(x^{2}-4 x+5\right) x / \sqrt{25-x^{2}}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | (2 x-4) \sqrt{25-x^{2}}-$ $\left(x^{2}-4 x+5\right) x / \sqrt{25-x^{2}} |
college_math.Calculus | exercise.5.4.18 | Describe the concavity of the function: $y=\sin ^{3} x $ | inflection points at $n \pi$, $\pm \arcsin (\sqrt{2 / 3})+n \pi$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | inflection points at $n \pi$, $\pm \arcsin (\sqrt{2 / 3})+n \pi |
college_math.Calculus | exercise.7.2.16 | Compute the value of the integral: $\int_{1}^{2} x^{5} d x $ | $2^{6} / 6-1 / 6$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 2^{6} / 6-1 / 6 |
college_math.Calculus | exercise.4.10.5 | Find the derivative of $\arctan \left(e^{x}\right)$. | $\frac{e^{x}}{1+e^{2 x}}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{e^{x}}{1+e^{2 x}} |
college_math.Calculus | exercise.6.3.1 | Approximate the fifth root of 7 , using $x_{0}=1.5$ as a first guess. Use Newton's method to find $x_{3}$ as your approximation. | $x_{3}=1.475773162$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | x_{3}=1.475773162 |
college_math.Calculus | exercise.3.2.3 | Find the derivative of the function: $5\left(-3 x^{2}+5 x+1\right) $ | $-30 x+25$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -30 x+25 |
college_math.Calculus | exercise.8.5.10 | Find the antiderivative: $\int \frac{1}{x^{2}+3 x} d x $ | $(1 / 3) \ln |x|-(1 / 3) \ln |x+3|+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | (1 / 3) \ln |x|-(1 / 3) \ln |x+3|+C |
college_math.Calculus | exercise.10.8.1 | Find the radius and interval of convergence for the series: $\sum_{n=0}^{\infty} n x^{n} $ | $R=1, I=(-1,1)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | R=1, I=(-1,1) |
college_math.Calculus | exercise.7.2.5 | Find the antiderivative of the function: $7 s^{-1} $ | $7 \ln s+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 7 \ln s+C |
college_math.Calculus | exercise.9.4.6 | The observation deck on the 102nd floor of the Empire State Building is 1,224 feet above the ground. If a steel ball is dropped from the observation deck its velocity at time $t$ is approximately $v(t)=-32 t$ feet per second. Find the average speed between the time it is dropped and the time it hits the ground, and fin... | $-4 \sqrt{1224} \mathrm{ft} / \mathrm{s} ;-8 \sqrt{1224} \mathrm{ft} / \mathrm{s}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -4 \sqrt{1224} \mathrm{ft} / \mathrm{s} ;-8 \sqrt{1224} \mathrm{ft} / \mathrm{s} |
college_math.Calculus | exercise.3.5.31 | Find the derivative of the function: $\frac{2 x^{-1}-x^{-2}}{3 x^{-1}-4 x^{-2}} $ | $-5 /(3 x-4)^{2}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -5 /(3 x-4)^{2} |
college_math.Calculus | exercise.7.1.8 | Let $f(x)=x^{2}-2 x+3$. Approximate the area under the curve between $x=1$ and $x=3$ using 4 rectangles. | $23 / 4$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 23 / 4 |
college_math.Calculus | exercise.10.1.1 | Compute the limit: $\lim _{x \rightarrow \infty} x^{1 / x} . $ | 1 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 1 |
college_math.Calculus | exercise.9.4.4 | Find the average height of $\sqrt{1-x^{2}}$ over the interval $[-1,1] . $ | $\pi / 4$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \pi / 4 |
college_math.Calculus | exercise.10.1.5 | Determine whether the sequence $\left\{\frac{n+47}{\sqrt{n^{2}+3 n}}\right\}_{n=1}^{\infty}$ converges or diverges. If it converges, compute the limit. | 1 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 1 |
college_math.Calculus | exercise.2.3.15 | Compute the limit: $\lim _{x \rightarrow 1}\left\{\begin{array}{ll}x-5 & x \neq 1, \\ 7 & x=1 .\end{array} \right.$. If a limit does not exist, explain why. | -4 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -4 |
college_math.Calculus | exercise.8.1.1 | Find the antiderivative of the function: $\int(1-t)^{9} d t $ | $-(1-t)^{10 / 10+C}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -(1-t)^{10 / 10+C} |
college_math.Calculus | exercise.6.2.8 | A boat is pulled in to a dock by a rope with one end attached to the front of the boat and the other end passing through a ring attached to the dock at a point $5 \mathrm{ft}$ higher than the front of the boat. The rope is being pulled through the ring at a rate of $0.6 \mathrm{ft} / \mathrm{sec}$. Find the rate at whi... | $13 / 20 \mathrm{ft} / \mathrm{s}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 13 / 20 \mathrm{ft} / \mathrm{s} |
college_math.Calculus | exercise.6.1.23 | You are designing a poster to contain a fixed amount $A$ of printing (measured in square centimeters) and have margins of $a$ centimeters at the top and bottom and $b$ centimeters at the sides. Find the ratio of vertical dimension to horizontal dimension of the printed area on the poster if you want to minimize the amo... | $a / b$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | a / b |
college_math.Calculus | exercise.7.3.3 | An object moves so that its velocity at time $t$ is $v(t)=1+2 \sin t \mathrm{~m} / \mathrm{s}$. Find the net distance traveled by the object between $t=0$ and $t=2 \pi$, and find the total distance traveled during the same period. | net: $2 \pi$, total: $2 \pi / 3+4 \sqrt{3}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | net: $2 \pi$, total: $2 \pi / 3+4 \sqrt{3} |
college_math.Calculus | exercise.9.1.4 | Find the area bounded by the curves: $x=3 y-y^{2}$ and $x+y=3 $ | $4 / 3$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 4 / 3 |
college_math.Calculus | exercise.3.2.8 | Find an equation for the tangent line to $f(x)=3 x^{2}-\pi^{3}$ at $x=4$. | $y=24 x-48-\pi^{3}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | y=24 x-48-\pi^{3} |
college_math.Calculus | exercise.4.7.15 | Find the derivative of the function: $x^{\sin (x)} $ | $x^{\sin (x)}(\cos (x) \ln (x)+\sin (x) / x)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | x^{\sin (x)}(\cos (x) \ln (x)+\sin (x) / x) |
college_math.Calculus | exercise.8.3.3 | Find the antiderivative: $\int \sqrt{x^{2}-1} d x $ | $x \sqrt{x^{2}-1} / 2-\ln \left|x+\sqrt{x^{2}-1}\right| / 2+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | x \sqrt{x^{2}-1} / 2-\ln \left|x+\sqrt{x^{2}-1}\right| / 2+C |
college_math.Calculus | exercise.7.2.9 | Find the antiderivative of the function: $\frac{2}{x \sqrt{x}} $ | $-4 / \sqrt{x}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -4 / \sqrt{x}+C |
college_math.Calculus | exercise.7.3.6 | Evaluate the three integrals:
$$
A=\int_{0}^{3}-x^{2}+9 d x \quad B=\int_{0}^{4}-x^{2}+9 d x \quad C=\int_{4}^{3}-x^{2}+9 d x,
$$
and verify that $A=B+C$. $$ | $A=18, B=44 / 3, C=10 / 3$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | A=18, B=44 / 3, C=10 / 3 |
college_math.Calculus | exercise.6.1.10 | Find the area of the largest rectangle that fits inside a semicircle of radius 10 (one side of the rectangle is along the diameter of the semicircle). | 100 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 100 |
college_math.Calculus | exercise.9.7.4 | Does the improper integral $\int_{1}^{\infty} 1 / \sqrt{x} d x$ converge or diverge? 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 |
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