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