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college_math.Calculus | exercise.3.5.35 | Find the derivative of the function: $(2 x+1)^{3}\left(x^{2}+1\right)^{2} $ | $56 x^{6}+72 x^{5}+110 x^{4}+100 x^{3}+$ $60 x^{2}+28 x+6$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 56 x^{6}+72 x^{5}+110 x^{4}+100 x^{3}+$ $60 x^{2}+28 x+6 |
college_math.Calculus | exercise.9.6.11 | A thin plate lies in the region between the circle $x^{2}+y^{2}=4$ and the circle $x^{2}+y^{2}=1$, above the $x$-axis. Find the centroid. | $\bar{x}=0, \bar{y}=28 /(9 \pi)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \bar{x}=0, \bar{y}=28 /(9 \pi) |
college_math.Calculus | exercise.1.3.12 | Find the domain of the function: $h(x)=\left\{\begin{array}{ll}\left(x^{2}-9\right) /(x-3) & x \neq 3 \\ 6 & \text { if } x=3 .\end{array} \right.$ | $\mathbb{R}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \mathbb{R} |
college_math.Calculus | exercise.8.1.10 | Find the antiderivative of the function: $\int \tan x d x $ | $-\ln |\cos x|+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -\ln |\cos x|+C |
college_math.Calculus | exercise.10.4.1 | Determine whether the series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{2 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.7.2.1 | Find the antiderivative of the function: $8 \sqrt{x} $ | $(16 / 3) x^{3 / 2}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | (16 / 3) x^{3 / 2}+C |
college_math.Calculus | exercise.10.12.7 | Determine whether the series converges: $\sum_{n=0}^{\infty} \frac{\sin ^{3}(n)}{n^{2}} $ | 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.7.2.3 | Find the antiderivative of the function: $4 / \sqrt{x} $ | $8 \sqrt{x}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 8 \sqrt{x}+C |
college_math.Calculus | exercise.9.4.1 | Find the average height of $\cos x$ over the intervals $[0, \pi / 2],[-\pi / 2, \pi / 2]$, and $[0,2 \pi] . $ | $2 / \pi ; 2 / \pi ; 0$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 2 / \pi ; 2 / \pi ; 0 |
college_math.Calculus | exercise.5.4.11 | Describe the concavity of the function: $y=x^{5}-x $ | concave up on $(0, \infty)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | concave up on $(0, \infty) |
college_math.Calculus | exercise.9.2.2 | For the velocity function $v=-9.8 t+49$, find both the net distance and the total distance traveled during the time interval $0 \leq t \leq 10$. | 0,245 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 0,245 |
college_math.Calculus | exercise.4.7.12 | Find the derivative of the function: $\ln (\cos (x)) $ | $-\tan (x)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -\tan (x) |
college_math.Calculus | exercise.3.5.34 | Find the derivative of the function: $\left((2 x+1)^{-1}+3\right)^{-1} $ | $1 /\left(2(2+3 x)^{2}\right)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 1 /\left(2(2+3 x)^{2}\right) |
college_math.Calculus | exercise.3.5.25 | Find the derivative of the function: $\frac{-3}{4 x^{2}-2 x+1} $ | $3(8 x-2) /\left(4 x^{2}-2 x+1\right)^{2}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 3(8 x-2) /\left(4 x^{2}-2 x+1\right)^{2} |
college_math.Calculus | exercise.4.9.15 | Find an equation for the tangent line to $x^{2 / 3}+y^{2 / 3}=a^{2 / 3}$ at a point $\left(x_{1}, y_{1}\right)$ on the curve, with $x_{1} \neq 0$ and $y_{1} \neq 0$. (This curve is an astroid.) | $y=\left(-y_{1}^{1 / 3} x+y_{1}^{1 / 3} x_{1}+x_{1}^{1 / 3} y_{1}\right) / x_{1}^{1 / 3}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | y=\left(-y_{1}^{1 / 3} x+y_{1}^{1 / 3} x_{1}+x_{1}^{1 / 3} y_{1}\right) / x_{1}^{1 / 3} |
college_math.Calculus | exercise.8.2.10 | Find the antiderivative: $\int \tan ^{3} x \sec x d x $ | $\left(\sec ^{3} x\right) / 3-\sec x+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \left(\sec ^{3} x\right) / 3-\sec x+C |
college_math.Calculus | exercise.8.6.27 | Evaluate the integral: $\int t^{3} e^{t} d t $ | $\left(t^{3}-3 t^{2}+6 t-6\right) e^{t}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \left(t^{3}-3 t^{2}+6 t-6\right) e^{t}+C |
college_math.Calculus | exercise.4.5.3 | Find the derivative of the function: $\sqrt{x \tan x} $ | $\frac{\tan x+x \sec ^{2} x}{2 \sqrt{x \tan x}}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{\tan x+x \sec ^{2} x}{2 \sqrt{x \tan x}} |
college_math.Calculus | exercise.4.10.4 | Find the derivative of $\arcsin \left(x^{2}\right) . $ | $\frac{2 x}{\sqrt{1-x^{4}}}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{2 x}{\sqrt{1-x^{4}}} |
college_math.Calculus | exercise.5.3.18 | Find all local maximum and minimum points of the function: $y=\sin ^{3} x $ | $\max$ at $\pi / 2+2 n \pi$, min at $3 \pi / 2+2 n \pi$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \max$ at $\pi / 2+2 n \pi$, min at $3 \pi / 2+2 n \pi |
college_math.Calculus | exercise.10.7.5 | Determine whether the series $\sum_{n=0}^{\infty}(-1)^{n} \frac{3^{n}}{5^{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.6.4.1 | Let $f(x)=x^{4}$. If $a=1$ and $d x=\Delta x=1 / 2$, what are $\Delta y$ and $d y$ ? | $\Delta y=65 / 16, d y=2$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \Delta y=65 / 16, d y=2 |
college_math.Calculus | exercise.5.2.2 | Find all critical points of the function $y=2+3 x-x^{3} $. Identify them as local maximum points, local minimum points, or neither. | $\min$ 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.6.1.8 | You have $l$ feet of fence to make a rectangular play area alongside the wall of your house. The wall of the house bounds one side. What is the largest size possible (in square feet) for the play area? | $l^{2} / 8$ square feet | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | l^{2} / 8$ square feet |
college_math.Calculus | exercise.1.3.13 | Determine the domain of the composition $(g \circ f)(x)$ if $f(x)=3 x-9$ and $g(x)=\sqrt{x}$. What is the domain of $(f \circ g)(x) ? $ | $\{x \mid x \geq 3\},\{x \mid x \geq 0\}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \{x \mid x \geq 3\},\{x \mid x \geq 0\} |
college_math.Calculus | exercise.6.1.14 | You want to make cylindrical containers to hold 1 liter using the least amount of construction material. The side is made from a rectangular piece of material, and this can be done with no material wasted. However, the top and bottom are cut from squares of side $2 r$, so that $2(2 r)^{2}=8 r^{2}$ of material is needed... | $r=5, h=40 / \pi, h / r=8 / \pi$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | r=5, h=40 / \pi, h / r=8 / \pi |
college_math.Calculus | exercise.4.5.9 | Find the derivative of the function: $\sin (\cos (6 x)) $ | $-6 \cos (\cos (6 x)) \sin (6 x)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -6 \cos (\cos (6 x)) \sin (6 x) |
college_math.Calculus | exercise.7.2.19 | Find the derivative of the function: $G(x)=\int_{1}^{x} e^{t^{2}} d t $ | $e^{x^{2}}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | e^{x^{2}} |
college_math.Calculus | exercise.6.2.11 | The sun is rising at a rate of $1 / 4 \mathrm{deg} / \mathrm{min}$ and appears to be climbing into the sky perpendicular to the horizon. Find the rate at which the shadow of a 200 meter building is shrinking at the moment when the shadow is 500 meters long. | $145 \pi / 72 \mathrm{~m} / \mathrm{s}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 145 \pi / 72 \mathrm{~m} / \mathrm{s} |
college_math.Calculus | exercise.5.4.1 | Describe the concavity of the function: $y=x^{2}-x $ | concave up everywhere | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | concave up everywhere |
college_math.Calculus | exercise.9.7.11 | Does the improper integral $\int_{-\infty}^{\infty} \sin x d x$ 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.10.12.10 | Determine whether the series converges: $\sum_{n=1}^{\infty} \frac{1}{n \sqrt{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.10.2.2 | Explain why the series $\sum_{n=1}^{\infty} \frac{5}{2^{1 / n}+14}$ diverges. | $\lim _{n \rightarrow \infty} 5 /\left(2^{1 / n}+14\right)=1 / 3$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \lim _{n \rightarrow \infty} 5 /\left(2^{1 / n}+14\right)=1 / 3 |
college_math.Calculus | exercise.4.5.7 | Find the derivative of the function: $x^{3} \sin \left(23 x^{2}\right) $ | $3 x^{2} \sin \left(23 x^{2}\right)+46 x^{4} \cos \left(23 x^{2}\right)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 3 x^{2} \sin \left(23 x^{2}\right)+46 x^{4} \cos \left(23 x^{2}\right) |
college_math.Calculus | exercise.3.5.39 | Find an equation for the tangent line to $\frac{\left(x^{2}+x+1\right)}{(1-x)}$ at $(2,-7)$. | $y=2 x-11$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | y=2 x-11 |
college_math.Calculus | exercise.9.1.2 | Find the area bounded by the curves: $x=y^{3}$ and $x=y^{2} $ | $1 / 12$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 1 / 12 |
college_math.Calculus | exercise.5.4.14 | Describe the concavity of the function: $y=x^{2}+1 / x $ | concave up on $(-\infty,-1)$ and $(0, \infty)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | concave up on $(-\infty,-1)$ and $(0, \infty) |
college_math.Calculus | exercise.4.5.1 | Find the derivative of the function: $\sin x \cos x $ | $\cos ^{2} x-\sin ^{2} x$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \cos ^{2} x-\sin ^{2} x |
college_math.Calculus | exercise.8.5.3 | Find the antiderivative: $\int \frac{1}{x^{2}+10 x+25} d x $ | $-1 /(x+5)+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -1 /(x+5)+C |
college_math.Calculus | exercise.8.4.5 | Find the antiderivative: $\int \sin ^{2} x d x $ | $(x / 2)-\sin (2 x) / 4+C=$ $(x / 2)-(\sin x \cos x) / 2+C$ | 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=$ $(x / 2)-(\sin x \cos x) / 2+C |
college_math.Calculus | exercise.1.3.10 | Find the domain of the function: $y=f(x)=\sqrt{x}+1 /(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.9.9.3 | Find the arc length of $f(x)=(1 / 3)\left(x^{2}+2\right)^{3 / 2}$ on the interval $[0, a]$. | $a+a^{3} / 3$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | a+a^{3} / 3 |
college_math.Calculus | exercise.3.1.3 | Find the derivative of the function: $\frac{1}{x^{5}} $ | $-5 x^{-6}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -5 x^{-6} |
college_math.Calculus | exercise.8.1.20 | Find the antiderivative of the function: $\int f(x) f^{\prime}(x) d x $ | $f(x)^{2} / 2$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | f(x)^{2} / 2 |
college_math.Calculus | exercise.6.1.22 | A window consists of a rectangular piece of clear glass with a semicircular piece of colored glass on top. Suppose that the colored glass transmits only $k$ times as much light per unit area as the clear glass ( $k$ is between 0 and 1). If the distance from top to bottom (across both the rectangle and the semicircle) i... | If $k \leq 2 / \pi$ the ratio is $(2-k \pi) / 4$; if $k \geq 2 / \pi$, the ratio is zero: the window should be semicircular with no rectangular part. | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | If $k \leq 2 / \pi$ the ratio is $(2-k \pi) / 4$; if $k \geq 2 / \pi$, the ratio is zero: the window should be semicircular with no rectangular part. |
college_math.Calculus | exercise.5.2.5 | Find all critical points of the function $y=3 x^{4}-4 x^{3} $. Identify them as local maximum points, local minimum points, or neither. | $\min$ 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 |
college_math.Calculus | exercise.10.9.5 | Find a power series representation for $\int \ln (1-x) d x$. | $C+\sum_{n=0}^{\infty} \frac{-1}{(n+1)(n+2)} x^{n+2}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | C+\sum_{n=0}^{\infty} \frac{-1}{(n+1)(n+2)} x^{n+2} |
college_math.Calculus | exercise.10.1.4 | Determine whether the sequence $\left\{\frac{n^{2}+1}{(n+1)^{2}}\right\}_{n=0}^{\infty}$ 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.8.5.4 | Find the antiderivative: $\int \frac{x^{2}}{4-x^{2}} d x $ | $-x-\ln |x-2|+\ln |x+2|+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -x-\ln |x-2|+\ln |x+2|+C |
college_math.Calculus | exercise.9.7.5 | Does the improper integral $\int_{0}^{\infty} e^{-x} d x$ converge or diverge? If it converges, find the value. | 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.9.2.6 | An object is shot upwards from ground level with an initial velocity of 3 meters per second; it is subject only to the force of gravity (no air resistance). Find its maximum altitude and the time at which it hits the ground. | $45 / 98$ meters, $30 / 49$ seconds | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 45 / 98$ meters, $30 / 49$ seconds |
college_math.Calculus | exercise.2.3.2 | Compute the limit: $\lim _{x \rightarrow 1} \frac{x^{2}+x-12}{x-3} $. If a limit does not exist, explain why. | 5 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 5 |
college_math.Calculus | exercise.8.3.4 | Find the antiderivative: $\int \sqrt{9+4 x^{2}} d x $ | $x \sqrt{9+4 x^{2}} / 2+$ $(9 / 4) \ln \left|2 x+\sqrt{9+4 x^{2}}\right|+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | x \sqrt{9+4 x^{2}} / 2+$ $(9 / 4) \ln \left|2 x+\sqrt{9+4 x^{2}}\right|+C |
college_math.Calculus | exercise.8.1.8 | Find the antiderivative of the function: $\int \cos (\pi t) \cos (\sin (\pi t)) d t $ | $\sin (\sin \pi t) / \pi+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \sin (\sin \pi t) / \pi+C |
college_math.Calculus | exercise.6.1.21 | A window consists of a rectangular piece of clear glass with a semicircular piece of colored glass on top; the colored glass transmits only $1 / 2$ as much light per unit area as the the clear glass. If the distance from top to bottom (across both the rectangle and the semicircle) is 2 meters and the window may be no m... | 1.5 meters wide by 1.25 meters tall | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 1.5 meters wide by 1.25 meters tall |
college_math.Calculus | exercise.8.2.9 | Find the antiderivative: $\int \sec ^{2} x \csc ^{2} x d x $ | $\tan x-\cot x+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \tan x-\cot x+C |
college_math.Calculus | exercise.4.5.18 | Find the points on the curve $y=x+2 \cos x$ that have a horizontal tangent line. | $\pi / 6+2 n \pi, 5 \pi / 6+2 n \pi$, any integer $n$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \pi / 6+2 n \pi, 5 \pi / 6+2 n \pi$, any integer $n |
college_math.Calculus | exercise.8.5.9 | Find the antiderivative: $\int \frac{1}{2 x^{2}-x-3} d x $ | $(1 / 5) \ln |2 x-3|-(1 / 5) \ln |1+x|+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | (1 / 5) \ln |2 x-3|-(1 / 5) \ln |1+x|+C |
college_math.Calculus | exercise.9.1.5 | Find the area bounded by the curves: $y=\cos (\pi x / 2)$ and $y=1-x^{2}$ (in the first quadrant) | $2 / 3-2 / \pi$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 2 / 3-2 / \pi |
college_math.Calculus | exercise.2.3.7 | Compute the limit: $\lim _{x \rightarrow 2} 3 $. | 3 | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 3 |
college_math.Calculus | exercise.10.12.26 | Find a series representation for the function: $\ln (1+x) $ | $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n+1} x^{n+1}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n+1} x^{n+1} |
college_math.Calculus | exercise.6.1.2 | Find the dimensions of the rectangle of largest area having fixed perimeter $100 . $ | $25 \times 25$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 25 \times 25 |
college_math.Calculus | exercise.4.3.5 | Compute the limit: $\lim _{x \rightarrow \pi / 4} \frac{\sin x-\cos x}{\cos (2 x)} $ | $-\sqrt{2} / 2$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | -\sqrt{2} / 2 |
college_math.Calculus | exercise.10.11.1 | Find a polynomial approximation for $\cos x$ on $[0, \pi]$, accurate to $\pm 10^{-3} $ | $1-\frac{x^{2}}{2}+\frac{x^{4}}{24}-\frac{x^{6}}{720}+\cdots+\frac{x^{12}}{12 !}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 1-\frac{x^{2}}{2}+\frac{x^{4}}{24}-\frac{x^{6}}{720}+\cdots+\frac{x^{12}}{12 !} |
college_math.Calculus | exercise.10.12.20 | Find a series representation for the function: $\sum_{n=1}^{\infty} \frac{x^{n}}{n 3^{n}} $ | $(-3,3)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | (-3,3) |
college_math.Calculus | exercise.9.7.3 | Does the improper integral $\int_{0}^{\infty} x^{2}+2 x-1 d x$ 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 |
college_math.Calculus | exercise.7.3.5 | Consider the function $f(x)=x^{2}-3 x+2$ on $[0,4]$. Find the total area between the curve and the $x$-axis (measuring all area as positive). | $17 / 3$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 17 / 3 |
college_math.Calculus | exercise.4.4.2 | Find the derivative of the function: $\sqrt{x} \sin x $ | $\frac{\sin x}{2 \sqrt{x}}+\sqrt{x} \cos x$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{\sin x}{2 \sqrt{x}}+\sqrt{x} \cos x |
college_math.Calculus | exercise.8.6.11 | Evaluate the integral: $\int \frac{e^{t}}{\sqrt{e^{t}+1}} d t $ | $2 \sqrt{e^{t}+1}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 2 \sqrt{e^{t}+1}+C |
college_math.Calculus | exercise.9.6.7 | A thin plate lies in the region contained by $y=x$ and $y=x^{2}$. Find the centroid. | $\bar{x}=1 / 2, \bar{y}=2 / 5$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \bar{x}=1 / 2, \bar{y}=2 / 5 |
college_math.Calculus | exercise.4.5.4 | Find the derivative of the function: $\tan x /(1+\sin x) $ | $\frac{\sec ^{2} x(1+\sin x)-\tan x \cos x}{(1+\sin x)^{2}}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{\sec ^{2} x(1+\sin x)-\tan x \cos x}{(1+\sin x)^{2}} |
college_math.Calculus | exercise.9.5.2 | How much work is done in lifting a 100 kilogram weight from an orbit 1000 kilometers above the surface of the earth to an orbit 35,786 kilometers above the surface of the earth? | $\approx 4,457,854,041 \mathrm{~N}-\mathrm{m}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \approx 4,457,854,041 \mathrm{~N}-\mathrm{m} |
college_math.Calculus | exercise.10.12.18 | Find a series representation for the function: $\sum_{n=0}^{\infty} \frac{2^{n}}{n !} x^{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.10.6.4 | Determine whether the series $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{\ln n}{n^{3}} $ 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.4.3.7 | For all $x, 2 x \leq g(x) \leq x^{4}-x^{2}+2$. Find $\lim _{x \rightarrow 1} g(x)$. | 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.5.3.9 | Find all local maximum and minimum points of the function: $y=4 x+\sqrt{1-x} $ | $\max$ at $x=63 / 64$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \max$ at $x=63 / 64 |
college_math.Calculus | exercise.2.3.5 | Compute the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{x+8}-3}{x-1} $. If a limit does not exist, explain why. | $1 / 6$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 1 / 6 |
college_math.Calculus | exercise.4.7.8 | Find the derivative of the function: $x+2^{x} $ | $1+2^{x} \ln (2)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 1+2^{x} \ln (2) |
college_math.Calculus | exercise.10.3.3 | Determine whether the series converges or diverges: $\sum_{n=1}^{\infty} \frac{\ln n}{n^{2}} $ | 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.9.3.6 | Find the volume of the solid obtained by revolving the region bounded by $y=x-x^{2}$ and the $x$-axis around the $x$-axis. | $\pi / 30$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \pi / 30 |
college_math.Calculus | exercise.10.9.1 | Find a series representation for $\ln 2$. | the alternating harmonic series | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | the alternating harmonic series |
college_math.Calculus | exercise.6.2.5 | A rotating beacon is located 2 miles out in the water. Let $A$ be the point on the shore that is closest to the beacon. As the beacon rotates at $10 \mathrm{rev} / \mathrm{min}$, the beam of light sweeps down the shore once each time it revolves. Assume that the shore is straight. Find the speed at which the point wher... | $80 \pi \mathrm{mi} / \mathrm{min}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 80 \pi \mathrm{mi} / \mathrm{min} |
college_math.Calculus | exercise.8.6.10 | Evaluate the integral: $\int t \sec ^{2} t d t $ | $t \tan t+\ln |\cos t|+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | t \tan t+\ln |\cos t|+C |
college_math.Calculus | exercise.2.3.11 | Compute the limit: $\lim _{x \rightarrow 0^{+}} \frac{\sqrt{2-x^{2}}}{x} $. If a limit does not exist, explain why. | does not exist | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | does not exist |
college_math.Calculus | exercise.5.2.13 | Find all critical points of the function $f(x)=\sin ^{2} x $. Identify them as local maximum points, local minimum points, or neither. | $\min$ at $n \pi, \max$ at $\pi / 2+n \pi$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \min$ at $n \pi, \max$ at $\pi / 2+n \pi |
college_math.Calculus | exercise.5.4.7 | Describe the concavity of the function: $y=3 x^{2}-\left(1 / x^{2}\right) $ | concave up when $x<-1$ or $x>1$, concave down when $-1<x<0$ or $0<x<1$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | concave up when $x<-1$ or $x>1$, concave down when $-1<x<0$ or $0<x<1 |
college_math.Calculus | exercise.3.5.19 | Find the derivative of the function: $\left(x^{2}+5\right)^{3} $ | $6 x\left(x^{2}+5\right)^{2}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 6 x\left(x^{2}+5\right)^{2} |
college_math.Calculus | exercise.8.4.9 | Find the antiderivative: $\int x \sin ^{2} x d x $ | $x^{2} / 4-\left(\cos ^{2} x\right) / 4-(x \sin x \cos x) / 2+$ C | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | x^{2} / 4-\left(\cos ^{2} x\right) / 4-(x \sin x \cos x) / 2+$ C |
college_math.Calculus | exercise.10.12.2 | Determine whether the series converges: $\frac{1}{1 \cdot 2}+\frac{1}{3 \cdot 4}+\frac{1}{5 \cdot 6}+\frac{1}{7 \cdot 8}+\cdots $ | 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.5.4.16 | Describe the concavity of the function: $y=\tan ^{2} x $ | concave up everywhere | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | concave up everywhere |
college_math.Calculus | exercise.9.7.7 | Does the improper integral $\int_{0}^{1} 1 / \sqrt{x} d x$ converge or diverge? If it converges, find the value. | 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.8.6.15 | Evaluate the integral: $\int \frac{\sec ^{2} t}{(1+\tan t)^{3}} d t $ | $\frac{-1}{2(1+\tan t)^{2}}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \frac{-1}{2(1+\tan t)^{2}}+C |
college_math.Calculus | exercise.9.6.6 | A thin plate fills the upper half of the unit circle $x^{2}+y^{2}=1$. Find the centroid. | $\bar{x}=0, \bar{y}=4 /(3 \pi)$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \bar{x}=0, \bar{y}=4 /(3 \pi) |
college_math.Calculus | exercise.8.1.16 | Evaluate the definite integral: $\int_{0}^{\pi / 6}\left(\cos ^{2} x-\sin ^{2} x\right) d x $ | $\sqrt{3} / 4$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \sqrt{3} / 4 |
college_math.Calculus | exercise.8.4.3 | Find the antiderivative: $\int x e^{x} d x $ | $(x-1) e^{x}+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | (x-1) e^{x}+C |
college_math.Calculus | exercise.8.3.8 | Find the antiderivative: $\int \sqrt{x^{2}+2 x} d x $ | $(x+1) \sqrt{x^{2}+2 x} / 2-$ $\ln \left|x+1+\sqrt{x^{2}+2 x}\right| / 2+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | (x+1) \sqrt{x^{2}+2 x} / 2-$ $\ln \left|x+1+\sqrt{x^{2}+2 x}\right| / 2+C |
college_math.Calculus | exercise.1.3.8 | Find the domain of the function: $y=f(x)=\sqrt{1-(1 / x)} $ | $\{x \mid x \geq 1\}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | \{x \mid x \geq 1\} |
college_math.Calculus | exercise.3.5.21 | Find the derivative of the function: $\left(1-4 x^{3}\right)^{-2} $ | $24 x^{2}\left(1-4 x^{3}\right)^{-3}$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | 24 x^{2}\left(1-4 x^{3}\right)^{-3} |
college_math.Calculus | exercise.8.1.2 | Find the antiderivative of the function: $\int\left(x^{2}+1\right)^{2} d x $ | $x^{5} / 5+2 x^{3} / 3+x+C$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | x^{5} / 5+2 x^{3} / 3+x+C |
college_math.Calculus | exercise.2.1.6 | Find an algebraic expression for the difference quotient $(f(x+\Delta x)-f(x)) / \Delta x$ when $f(x)=$ $m x+b$. Simplify the expression as much as possible. Then determine what happens as $\Delta x$ approaches 0 . That value is $f^{\prime}(x)$. | $m$ | Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) | college_math.calculus | m |
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