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college_math.Calculus
|
exercise.9.3.5
|
Use integration to find the volume of the solid obtained by revolving the region bounded by $x+y=2$ and the $x$ and $y$ axes around the $x$-axis.
|
$8 \pi / 3$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
8 \pi / 3
|
college_math.Calculus
|
exercise.10.3.11
|
Find an $N$ such that $\sum_{n=1}^{\infty} \frac{\ln n}{n^{2}}=\sum_{n=1}^{N} \frac{\ln n}{n^{2}} \pm 0.005$.
|
$N=1687$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
N=1687
|
college_math.Calculus
|
exercise.4.7.3
|
Find the derivative of the function: $\left(e^{x}\right)^{2} $
|
$2 e^{2 x}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
2 e^{2 x}
|
college_math.Calculus
|
exercise.6.2.15
|
A man 1.8 meters tall walks at a speed of 1 meter per second toward a streetlight that is 4 meters above the ground. Find the rate at which the tip of his shadow is moving. Find the rate at which his shadow is shortening.
|
tip: $20 / 11 \mathrm{~m} / \mathrm{s}$, length: $9 / 11 \mathrm{~m} / \mathrm{s}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
tip: $20 / 11 \mathrm{~m} / \mathrm{s}$, length: $9 / 11 \mathrm{~m} / \mathrm{s}
|
college_math.Calculus
|
exercise.4.3.1
|
Compute the limit: $\lim _{x \rightarrow 0} \frac{\sin (5 x)}{x} $
|
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.5.2.1
|
Find all critical points of the function $y=x^{2}-x $. Identify them as local maximum points, local minimum points, or neither.
|
$\min$ 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.6.3.3
|
The function $f(x)=x^{3}-3 x^{2}-3 x+6$ has a root between 3 and 4 , because $f(3)=-3$ and $f(4)=10$. Approximate the root to two decimal places.
|
3.36
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
3.36
|
college_math.Calculus
|
exercise.8.3.10
|
Find the antiderivative: $\int \frac{x^{2}}{\sqrt{4-x^{2}}} d x $
|
$2 \arcsin (x / 2)-x \sqrt{4-x^{2}} / 2+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
2 \arcsin (x / 2)-x \sqrt{4-x^{2}} / 2+C
|
college_math.Calculus
|
exercise.8.4.8
|
Find the antiderivative: $\int x^{2} \sin x d x $
|
$-x^{2} \cos x+2 x \sin x+2 \cos x+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-x^{2} \cos x+2 x \sin x+2 \cos x+C
|
college_math.Calculus
|
exercise.7.3.2
|
An object moves so that its velocity at time $t$ is $v(t)=\sin t$. Set up and evaluate a single definite integral to compute the net distance traveled between $t=0$ and $t=2 \pi$.
|
$\int_{0}^{2 \pi} \sin t d t=0$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\int_{0}^{2 \pi} \sin t d t=0
|
college_math.Calculus
|
exercise.5.4.13
|
Describe the concavity of the function: $y=x+1 / 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.10.8.6
|
Find the radius and interval of convergence for the series: $\sum_{n=1}^{\infty} \frac{(x+5)^{n}}{n(n+1)} $
|
$R=1, I=(-6,-4)$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
R=1, I=(-6,-4)
|
college_math.Calculus
|
exercise.10.12.1
|
Determine whether the series converges: $\sum_{n=0}^{\infty} \frac{n}{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.3.1.4
|
Find the derivative of the function: $x^{\pi} $
|
$\pi x^{\pi-1}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\pi x^{\pi-1}
|
college_math.Calculus
|
exercise.4.3.4
|
Compute the limit: $\lim _{x \rightarrow 0} \frac{\tan x}{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.3.5.7
|
Find the derivative of the function: $\sqrt{1+x^{4}} $
|
$2 x^{3} / \sqrt{1+x^{4}}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
2 x^{3} / \sqrt{1+x^{4}}
|
college_math.Calculus
|
exercise.9.6.3
|
A beam 4 meters long has density $\sigma(x)=x^{3}$ at distance $x$ from the left end of the beam. Find the center of mass $\bar{x}$.
|
$16 / 5$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
16 / 5
|
college_math.Calculus
|
exercise.1.3.14
|
A farmer wants to build a fence along a river. He has 500 feet of fencing and wants to enclose a rectangular pen on three sides (with the river providing the fourth side). If $x$ is the length of the side perpendicular to the river, determine the area of the pen as a function of $x$. What is the domain of this function?
|
$A=x(500-2 x),\{x \mid 0 \leq x \leq 250\}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
A=x(500-2 x),\{x \mid 0 \leq x \leq 250\}
|
college_math.Calculus
|
exercise.2.3.9
|
Compute the limit: $\lim _{x \rightarrow 0} \frac{4 x-5 x^{2}}{x-1} $. 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.10.10.8
|
Find the first four terms of the Maclaurin series for $\tan x$ (up to and including the $x^{3}$ term).
|
$x+x^{3} / 3$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
x+x^{3} / 3
|
college_math.Calculus
|
exercise.8.2.8
|
Find the antiderivative: $\int \sin x(\cos x)^{3 / 2} d x $
|
$-2(\cos x)^{5 / 2} / 5+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-2(\cos x)^{5 / 2} / 5+C
|
college_math.Calculus
|
exercise.8.2.6
|
Find the antiderivative: $\int \sin ^{2} x \cos ^{2} x d x $
|
$x / 8-(\sin 4 x) / 32+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
x / 8-(\sin 4 x) / 32+C
|
college_math.Calculus
|
exercise.3.5.16
|
Find the derivative of the function: $\sqrt{\left(x^{2}+1\right)^{2}+\sqrt{1+\left(x^{2}+1\right)^{2}}} $
|
$\left(4 x\left(x^{2}+1\right)+\frac{4 x^{3}+4 x}{2 \sqrt{1+\left(x^{2}+1\right)^{2}}}\right) /$ $2 \sqrt{\left(x^{2}+1\right)^{2}+\sqrt{1+\left(x^{2}+1\right)^{2}}}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\left(4 x\left(x^{2}+1\right)+\frac{4 x^{3}+4 x}{2 \sqrt{1+\left(x^{2}+1\right)^{2}}}\right) /$ $2 \sqrt{\left(x^{2}+1\right)^{2}+\sqrt{1+\left(x^{2}+1\right)^{2}}}
|
college_math.Calculus
|
exercise.7.2.7
|
Find the antiderivative of the function: $(x-6)^{2} $
|
$(x-6)^{3} / 3+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
(x-6)^{3} / 3+C
|
college_math.Calculus
|
exercise.10.9.4
|
Find a power series representation for $1 /(1-x)^{3}$. What is the radius of convergence?
|
$\sum_{n=0}^{\infty} \frac{(n+1)(n+2)}{2} x^{n}, R=1$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\sum_{n=0}^{\infty} \frac{(n+1)(n+2)}{2} x^{n}, R=1
|
college_math.Calculus
|
exercise.5.4.4
|
Describe the concavity of the function: $y=x^{4}-2 x^{2}+3 $
|
concave up when $x<-1 / \sqrt{3}$ or $x>1 / \sqrt{3}$, concave down when $-1 / \sqrt{3}<x<1 / \sqrt{3}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
concave up when $x<-1 / \sqrt{3}$ or $x>1 / \sqrt{3}$, concave down when $-1 / \sqrt{3}<x<1 / \sqrt{3}
|
college_math.Calculus
|
exercise.7.2.8
|
Find the antiderivative of the function: $x^{3 / 2} $
|
$2 x^{5 / 2} / 5+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
2 x^{5 / 2} / 5+C
|
college_math.Calculus
|
exercise.6.2.12
|
The sun is setting 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 25 meter wall is lengthening at the moment when the shadow is 50 meters.
|
$25 \pi / 144 \mathrm{~m} / \mathrm{min}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
25 \pi / 144 \mathrm{~m} / \mathrm{min}
|
college_math.Calculus
|
exercise.3.1.2
|
Find the derivative of the function: $x^{-100} $
|
$-100 x^{-101}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-100 x^{-101}
|
college_math.Calculus
|
exercise.9.1.9
|
Find the area bounded by the curves: $x=0$ and $x=25-y^{2} $
|
$500 / 3$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
500 / 3
|
college_math.Calculus
|
exercise.4.7.4
|
Find the derivative of the function: $\sin \left(e^{x}\right) $
|
$e^{x} \cos \left(e^{x}\right)$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
e^{x} \cos \left(e^{x}\right)
|
college_math.Calculus
|
exercise.8.6.6
|
Evaluate the integral: $\int \frac{2 t+1}{t^{2}+t+3} d t $
|
$\ln \left|t^{2}+t+3\right|+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\ln \left|t^{2}+t+3\right|+C
|
college_math.Calculus
|
exercise.2.3.13
|
Compute the limit: $\lim _{x \rightarrow a} \frac{x^{3}-a^{3}}{x-a} $.
|
$3 a^{2}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
3 a^{2}
|
college_math.Calculus
|
exercise.9.2.9
|
An object moves along a straight line with acceleration given by $a(t)=\sin (\pi t)$. Assume that when $t=0, s(t)=v(t)=0$. Find $s(t), v(t)$, and the maximum speed of the object. Describe the motion of the object.
|
$s(t)=-\sin (\pi t) / \pi^{2}+t / \pi$, $v(t)=-\cos (\pi t) / \pi+1 / \pi$, maximum speed is $2 / \pi$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
s(t)=-\sin (\pi t) / \pi^{2}+t / \pi$, $v(t)=-\cos (\pi t) / \pi+1 / \pi$, maximum speed is $2 / \pi
|
college_math.Calculus
|
exercise.5.3.14
|
Find all local maximum and minimum points of the function: $y=x^{2}+1 / x $
|
$\min$ at $2^{-1 / 3}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\min$ at $2^{-1 / 3}
|
college_math.Calculus
|
exercise.10.5.4
|
Determine whether the series $\sum_{n=1}^{\infty} \frac{3 n+4}{2 n^{2}+3 n+5} $ 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.8.6.21
|
Evaluate the integral: $\int \frac{\arctan 2 t}{1+4 t^{2}} d t $
|
$\frac{(\arctan (2 t))^{2}}{4}+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{(\arctan (2 t))^{2}}{4}+C
|
college_math.Calculus
|
exercise.6.1.19
|
A piece of carboard is 1 meter by $1 / 2$ meter. A square is to be cut from each corner and the sides folded up to make an open-top box. What are the dimensions of the box with maximum possible volume?
|
$\frac{\sqrt{3}}{6} \times \frac{\sqrt{3}}{6}+\frac{1}{2} \times \frac{1}{4}-\frac{\sqrt{3}}{12}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{\sqrt{3}}{6} \times \frac{\sqrt{3}}{6}+\frac{1}{2} \times \frac{1}{4}-\frac{\sqrt{3}}{12}
|
college_math.Calculus
|
exercise.10.12.4
|
Determine whether the series converges: $\sum_{n=0}^{\infty} \frac{n !}{8^{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.6.2.17
|
A police helicopter is flying at a speed of 200 kilometers per hour at a constant altitude of $1 \mathrm{~km}$ above a straight road. The pilot uses radar to determine that an oncoming car is at a distance of exactly 2 kilometers from the helicopter, and that this distance is decreasing at a rate of $250 \mathrm{~kph}$. Find the speed of the car.
|
$81 \mathrm{~km} / \mathrm{hr}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
81 \mathrm{~km} / \mathrm{hr}
|
college_math.Calculus
|
exercise.7.2.2
|
Find the antiderivative of the function: $3 t^{2}+1 $
|
$t^{3}+t+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
t^{3}+t+C
|
college_math.Calculus
|
exercise.7.2.12
|
Compute the value of the integral: $\int_{0}^{\pi} \sin t d t $
|
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.3.1.1
|
Find the derivative of the function: $x^{100} $
|
$100 x^{99}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
100 x^{99}
|
college_math.Calculus
|
exercise.7.1.1
|
Suppose an object moves in a straight line so that its speed at time $t$ is given by $v(t)=2 t+2$, and that at $t=1$ the object is at position 5 . Find the position of the object at $t=2 . $
|
10
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
10
|
college_math.Calculus
|
exercise.10.8.3
|
Find the radius and interval of convergence for the series: $\sum_{n=1}^{\infty} \frac{n !}{n^{n}} x^{n} $
|
$R=e, I=(-e, e)$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
R=e, I=(-e, e)
|
college_math.Calculus
|
exercise.3.5.28
|
Find the derivative of the function: $\frac{x+1}{x-1} $
|
$-2 /(x-1)^{2}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-2 /(x-1)^{2}
|
college_math.Calculus
|
exercise.3.5.36
|
Find an equation for the tangent line to $f(x)=(x-2)^{1 / 3} /\left(x^{3}+4 x-1\right)^{2}$ at $x=1$.
|
$y=23 x / 96-29 / 96$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
y=23 x / 96-29 / 96
|
college_math.Calculus
|
exercise.10.5.2
|
Determine whether the series $\sum_{n=2}^{\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.3.5.33
|
Find the derivative of the function: $\frac{1}{(2 x+1)(x-3)} $
|
$(5-4 x) /\left((2 x+1)^{2}(x-3)^{2}\right)$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
(5-4 x) /\left((2 x+1)^{2}(x-3)^{2}\right)
|
college_math.Calculus
|
exercise.8.5.1
|
Find the antiderivative: $\int \frac{1}{4-x^{2}} d x $
|
$-\ln |x-2| / 4+\ln |x+2| / 4+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-\ln |x-2| / 4+\ln |x+2| / 4+C
|
college_math.Calculus
|
exercise.6.2.9
|
A balloon is at a height of 50 meters and is rising at a constant rate of $5 \mathrm{~m} / \mathrm{sec}$. A bicyclist passes beneath it, traveling in a straight line at a constant speed of $10 \mathrm{~m} / \mathrm{sec}$. Find the rate at which the distance between the bicyclist and the balloon is increasing 2 seconds later.
|
$5 \sqrt{10} / 2 \mathrm{~m} / \mathrm{s}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
5 \sqrt{10} / 2 \mathrm{~m} / \mathrm{s}
|
college_math.Calculus
|
exercise.4.7.2
|
Find the derivative of the function: $\frac{\sin x}{e^{x}} $
|
$\frac{\cos x-\sin x}{e^{x}}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{\cos x-\sin x}{e^{x}}
|
college_math.Calculus
|
exercise.9.5.6
|
A spring has constant $k=10 \mathrm{~kg} / \mathrm{s}^{2}$. How much work is done in compressing it $1 / 10$ meter from its natural length?
|
$0.05 \mathrm{~N}-\mathrm{m}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
0.05 \mathrm{~N}-\mathrm{m}
|
college_math.Calculus
|
exercise.9.4.3
|
Find the average height of $1 / x^{2}$ over the interval $[1, A]$.
|
$1 / A$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
1 / A
|
college_math.Calculus
|
exercise.10.12.21
|
Find a series representation for the function: $x+\frac{1}{2} \frac{x^{3}}{3}+\frac{1 \cdot 3}{2 \cdot 4} \frac{x^{5}}{5}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{x^{7}}{7}+\cdots $
|
$(-1,1)$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
(-1,1)
|
college_math.Calculus
|
exercise.8.6.4
|
Evaluate the integral: $\int \sin t \cos 2 t d t $
|
$\cos t-\frac{2}{3} \cos ^{3} t+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\cos t-\frac{2}{3} \cos ^{3} t+C
|
college_math.Calculus
|
exercise.2.3.10
|
Compute the limit: $\lim _{x \rightarrow 1} \frac{x^{2}-1}{x-1} $. If a limit does not exist, explain why.
|
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.8.6.1
|
Evaluate the integral: $\int(t+4)^{3} d t $
|
$\frac{(t+4)^{4}}{4}+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{(t+4)^{4}}{4}+C
|
college_math.Calculus
|
exercise.9.2.3
|
For the velocity function $v=3(t-3)(t-1)$, find both the net distance and the total distance traveled during the time interval $0 \leq t \leq 5$.
|
20,28
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
20,28
|
college_math.Calculus
|
exercise.3.2.7
|
Find an equation for the tangent line to $f(x)=x^{3} / 4-1 / x$ at $x=-2$.
|
$y=13 x / 4+5$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
y=13 x / 4+5
|
college_math.Calculus
|
exercise.8.2.7
|
Find the antiderivative: $\int \cos ^{3} x \sin ^{2} x d x $
|
$\left(\sin ^{3} x\right) / 3-\left(\sin ^{5} x\right) / 5+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\left(\sin ^{3} x\right) / 3-\left(\sin ^{5} x\right) / 5+C
|
college_math.Calculus
|
exercise.2.3.1
|
Compute the limit: $\lim _{x \rightarrow 3} \frac{x^{2}+x-12}{x-3} $. If a limit does not exist, explain why.
|
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.6.4.4
|
Use differentials to estimate the amount of paint needed to apply a coat of paint $0.02 \mathrm{~cm}$ thick to a sphere with diameter 40 meters. (Recall that the volume of a sphere of radius $r$ is $V=(4 / 3) \pi r^{3}$. Notice that you are given that $d r=0.02$.)
|
$d V=32 \pi / 25$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
d V=32 \pi / 25
|
college_math.Calculus
|
exercise.10.10.2
|
Find the Maclaurin series or Taylor series centered at $a$ and the radius of convergence for the function: $e^{x} $
|
$\sum_{n=0}^{\infty} x^{n} / n !, R=\infty$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\sum_{n=0}^{\infty} x^{n} / n !, R=\infty
|
college_math.Calculus
|
exercise.10.12.3
|
Determine whether the series converges: $\sum_{n=0}^{\infty} \frac{n}{\left(n^{2}+4\right)^{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.7.2.15
|
Compute the value of the integral: $\int_{0}^{3} x^{3} d x $
|
$3^{4} / 4$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
3^{4} / 4
|
college_math.Calculus
|
exercise.10.6.8
|
Determine whether the series $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{\arctan n}{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.5.4.17
|
Describe the concavity of the function: $y=\cos ^{2} x-\sin ^{2} x $
|
concave up on $(\pi / 4+n \pi, 3 \pi / 4+n \pi)$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
concave up on $(\pi / 4+n \pi, 3 \pi / 4+n \pi)
|
college_math.Calculus
|
exercise.3.5.32
|
Find the derivative of the function: $3\left(x^{2}+1\right)\left(2 x^{2}-1\right)(2 x+3) $
|
$60 x^{4}+72 x^{3}+18 x^{2}+18 x-6$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
60 x^{4}+72 x^{3}+18 x^{2}+18 x-6
|
college_math.Calculus
|
exercise.8.6.3
|
Evaluate the integral: $\int\left(e^{t^{2}}+16\right) t e^{t^{2}} d t $
|
$\frac{\left(e^{t^{2}}+16\right)^{2}}{4}+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{\left(e^{t^{2}}+16\right)^{2}}{4}+C
|
college_math.Calculus
|
exercise.8.6.22
|
Evaluate the integral: $\int \frac{t}{t^{2}+2 t-3} d t $
|
$\frac{3 \ln |t+3|}{4}+\frac{\ln |t-1|}{4}+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{3 \ln |t+3|}{4}+\frac{\ln |t-1|}{4}+C
|
college_math.Calculus
|
exercise.6.1.27
|
Find the dimensions of the lightest cylindrical can containing 0.25 liter $\left(=250 \mathrm{~cm}^{3}\right)$ if the top and bottom are made of a material that is twice as heavy (per unit area) as the material used for the side.
|
$r=5 /(2 \pi)^{1 / 3} \approx 2.7 \mathrm{~cm}$, $h=5 \cdot 2^{5 / 3} / \pi^{1 / 3}=4 r \approx 10.8 \mathrm{~cm}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
r=5 /(2 \pi)^{1 / 3} \approx 2.7 \mathrm{~cm}$, $h=5 \cdot 2^{5 / 3} / \pi^{1 / 3}=4 r \approx 10.8 \mathrm{~cm}
|
college_math.Calculus
|
exercise.6.2.4
|
A ladder 13 meters long rests on horizontal ground and leans against a vertical wall. The top of the ladder is being pulled up the wall at a rate of 0.1 meters per second. Find the rate at which the foot of the ladder is approaching the wall when the foot of the ladder is $5 \mathrm{~m}$ from the wall.
|
$-6 / 25 \mathrm{~m} / \mathrm{s}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-6 / 25 \mathrm{~m} / \mathrm{s}
|
college_math.Calculus
|
exercise.8.4.6
|
Find the antiderivative: $\int \ln x d x $
|
$x \ln x-x+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
x \ln x-x+C
|
college_math.Calculus
|
exercise.3.2.9
|
Suppose the position of an object at time $t$ is given by $f(t)=-49 t^{2} / 10+5 t+10$. Find a function giving the speed of the object at time $t$. The acceleration of an object is the rate at which its speed is changing, which means it is given by the derivative of the speed function. Find the acceleration of the object at time $t . $
|
$-49 t / 5+5,-49 / 5$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-49 t / 5+5,-49 / 5
|
college_math.Calculus
|
exercise.5.4.9
|
Describe the concavity of the function: $y=4 x+\sqrt{1-x} $
|
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.10.12.8
|
Determine whether the series converges: $\sum_{n=0}^{\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.3.5.40
|
Find an equation for the tangent line to $\sqrt{\left(x^{2}+1\right)^{2}+\sqrt{1+\left(x^{2}+1\right)^{2}}}$ at $(1, \sqrt{4+\sqrt{5}})$.
|
$y=\frac{20+2 \sqrt{5}}{5 \sqrt{4+\sqrt{5}}} x+\frac{3 \sqrt{5}}{5 \sqrt{4+\sqrt{5}}}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
y=\frac{20+2 \sqrt{5}}{5 \sqrt{4+\sqrt{5}}} x+\frac{3 \sqrt{5}}{5 \sqrt{4+\sqrt{5}}}
|
college_math.Calculus
|
exercise.9.1.11
|
Find the area bounded by the curves: $y=x^{3 / 2}$ and $y=x^{2 / 3} $
|
$1 / 5$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
1 / 5
|
college_math.Calculus
|
exercise.10.3.1
|
Determine whether the series converges or diverges: $\sum_{n=1}^{\infty} \frac{1}{n^{\pi / 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.5.3.16
|
Find all local maximum and minimum points of the function: $y=\tan ^{2} x $
|
$\min$ at $n \pi$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\min$ at $n \pi
|
college_math.Calculus
|
exercise.4.7.13
|
Find the derivative of the function: $\sqrt{\ln \left(x^{2}\right)} / x $
|
$\left(1-\ln \left(x^{2}\right)\right) /\left(x^{2} \sqrt{\ln \left(x^{2}\right)}\right)$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\left(1-\ln \left(x^{2}\right)\right) /\left(x^{2} \sqrt{\ln \left(x^{2}\right)}\right)
|
college_math.Calculus
|
exercise.8.1.11
|
Find the antiderivative of the function: $\int_{0}^{\pi} \sin ^{5}(3 x) \cos (3 x) d 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.10.10.1
|
Find the Maclaurin series or Taylor series centered at $a$ and the radius of convergence for the function: $\cos x $
|
$\sum_{n=0}^{\infty}(-1)^{n} x^{2 n} /(2 n) !, R=\infty$
|
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} x^{2 n} /(2 n) !, R=\infty
|
college_math.Calculus
|
exercise.8.3.12
|
Find the antiderivative: $\int \frac{x^{3}}{\sqrt{4 x^{2}-1}} d x $
|
$\left(2 x^{2}+1\right) \sqrt{4 x^{2}-1} / 24+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\left(2 x^{2}+1\right) \sqrt{4 x^{2}-1} / 24+C
|
college_math.Calculus
|
exercise.2.4.5
|
Find the derivative of the function: $y=f(x)=x^{3}$.
|
$3 x^{2}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
3 x^{2}
|
college_math.Calculus
|
exercise.6.1.15
|
You want to make cylindrical containers of a given volume $V$ 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 (rather than $2 \pi r^{2}$, which is the total area of the top and bottom). Find the optimal ratio of height to radius.
|
$8 / \pi$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
8 / \pi
|
college_math.Calculus
|
exercise.3.5.23
|
Find the derivative of the function: $4\left(2 x^{2}-x+3\right)^{-2} $
|
$-8(4 x-1)\left(2 x^{2}-x+3\right)^{-3}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-8(4 x-1)\left(2 x^{2}-x+3\right)^{-3}
|
college_math.Calculus
|
exercise.8.5.6
|
Find the antiderivative: $\int \frac{1}{x^{2}+10 x+29} d x $
|
$(1 / 2) \arctan (x / 2+5 / 2)+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
(1 / 2) \arctan (x / 2+5 / 2)+C
|
college_math.Calculus
|
exercise.3.5.30
|
Find the derivative of the function: $\frac{(x-1)(x-2)}{x-3} $
|
$\left(x^{2}-6 x+7\right) /(x-3)^{2}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\left(x^{2}-6 x+7\right) /(x-3)^{2}
|
college_math.Calculus
|
exercise.10.12.19
|
Find a series representation for the function: $\sum_{n=0}^{\infty} \frac{x^{n}}{1+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.4.1.1
|
Find all values of $\theta$ such that $\sin (\theta)=-1$; give your answer in radians.
|
$2 n \pi-\pi / 2$, any integer $n$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
2 n \pi-\pi / 2$, any integer $n
|
college_math.Calculus
|
exercise.10.12.17
|
Determine whether the series converges: $\sum_{n=1}^{\infty} \sin (1 / n) $. Find the interval and radius of convergence; you need not check the endpoints of the intervals.
|
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.10.12.23
|
Find a series representation for the function: $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2} 3^{n}} x^{2 n} $
|
$(-\sqrt{3}, \sqrt{3})$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
(-\sqrt{3}, \sqrt{3})
|
college_math.Calculus
|
exercise.3.5.10
|
Find the derivative of the function: $\frac{\left(x^{2}+x+1\right)}{(1-x)} $
|
$\frac{2 x+1}{1-x}+\frac{x^{2}+x+1}{(1-x)^{2}}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{2 x+1}{1-x}+\frac{x^{2}+x+1}{(1-x)^{2}}
|
college_math.Calculus
|
exercise.8.1.19
|
Evaluate the definite integral: $\int_{-1}^{1} \sin ^{7} x d 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.9.4.2
|
Find the average height of $x^{2}$ over the interval $[-2,2] . $
|
$4 / 3$
|
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.10.9.2
|
Find a power series representation for $1 /(1-x)^{2}$.
|
$\sum_{n=0}^{\infty}(n+1) x^{n}$
|
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) x^{n}
|
college_math.Calculus
|
exercise.10.4.5
|
Approximate the value of the series $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{1}{n^{3}}$ to two decimal places.
|
0.90
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
0.90
|
college_math.Calculus
|
exercise.9.6.9
|
A thin plate lies in the region contained by $y=x^{1 / 3}$ and the $x$-axis between $x=0$ and $x=1$. Find the centroid.
|
$\bar{x}=4 / 7, \bar{y}=2 / 5$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\bar{x}=4 / 7, \bar{y}=2 / 5
|
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