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college_math.Calculus
|
exercise.8.2.4
|
Find the antiderivative: $\int \cos ^{2} x \sin ^{3} x d x $
|
$\left(\cos ^{5} x\right) / 5-\left(\cos ^{3} x\right) / 3+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\left(\cos ^{5} x\right) / 5-\left(\cos ^{3} x\right) / 3+C
|
college_math.Calculus
|
exercise.10.2.4
|
Compute the value of the series $\sum_{n=0}^{\infty} \frac{4}{(-3)^{n}}-\frac{3}{3^{n}}$.
|
$-3 / 2$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-3 / 2
|
college_math.Calculus
|
exercise.4.7.11
|
Find the derivative of the function: $\ln \left(x^{3}+3 x\right) $
|
$\left(3 x^{2}+3\right) /\left(x^{3}+3 x\right)$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\left(3 x^{2}+3\right) /\left(x^{3}+3 x\right)
|
college_math.Calculus
|
exercise.5.2.14
|
Find the maxima and minima of the function $f(x)=\sec x $.
|
$\min$ at $2 n \pi, \max$ at $(2 n+1) \pi$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\min$ at $2 n \pi, \max$ at $(2 n+1) \pi
|
college_math.Calculus
|
exercise.4.5.11
|
Compute $\frac{d}{d t} t^{5} \cos (6 t)$.
|
$5 t^{4} \cos (6 t)-6 t^{5} \sin (6 t)$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
5 t^{4} \cos (6 t)-6 t^{5} \sin (6 t)
|
college_math.Calculus
|
exercise.6.5.7
|
Describe all functions with derivative $\frac{1}{1+x^{2}}$.
|
$\arctan x+k$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\arctan x+k
|
college_math.Calculus
|
exercise.8.1.18
|
Evaluate the definite integral: $\int_{-1}^{1}\left(2 x^{3}-1\right)\left(x^{4}-2 x\right)^{6} d x $
|
$-\left(3^{7}+1\right) / 14$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-\left(3^{7}+1\right) / 14
|
college_math.Calculus
|
exercise.3.5.6
|
Find the derivative of the function: $\sqrt{r^{2}-x^{2}}, r$ is a constant
|
$-x / \sqrt{r^{2}-x^{2}}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-x / \sqrt{r^{2}-x^{2}}
|
college_math.Calculus
|
exercise.10.7.8
|
Determine whether the series $\sum_{n=1}^{\infty} \frac{(n !)^{2}}{n^{n}} $ converges.
|
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.8.6.14
|
Evaluate the integral: $\int \frac{1}{t^{2} \sqrt{1+t^{2}}} d t $
|
$\frac{-1}{\sin \arctan t}+C=-\sqrt{1+t^{2}} / t+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{-1}{\sin \arctan t}+C=-\sqrt{1+t^{2}} / t+C
|
college_math.Calculus
|
exercise.8.3.6
|
Find the antiderivative: $\int x^{2} \sqrt{1-x^{2}} d x $
|
$\arcsin (x) / 8-\sin (4 \arcsin x) / 32+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\arcsin (x) / 8-\sin (4 \arcsin x) / 32+C
|
college_math.Calculus
|
exercise.8.1.3
|
Find the antiderivative of the function: $\int x\left(x^{2}+1\right)^{100} d x $
|
$\left(x^{2}+1\right)^{101} / 202+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\left(x^{2}+1\right)^{101} / 202+C
|
college_math.Calculus
|
exercise.6.1.16
|
Given a right circular cone, you put an upside-down cone inside it so that its vertex is at the center of the base of the larger cone and its base is parallel to the base of the larger cone. If you choose the upside-down cone to have the largest possible volume, what fraction of the volume of the larger cone does it occupy? (Let $H$ and $R$ be the height and base radius of the larger cone, and let $h$ and $r$ be the height and base radius of the smaller cone. Hint: Use similar triangles to get an equation relating $h$ and $r$.)
|
$4 / 27$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
4 / 27
|
college_math.Calculus
|
exercise.7.3.4
|
Consider the function $f(x)=(x+2)(x+1)(x-1)(x-2)$ on $[-2,2]$. Find the total area between the curve and the $x$-axis (measuring all area as positive).
|
8
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
8
|
college_math.Calculus
|
exercise.8.2.5
|
Find the antiderivative: $\int \cos ^{3} x d x $
|
$\sin x-\left(\sin ^{3} x\right) / 3+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\sin x-\left(\sin ^{3} x\right) / 3+C
|
college_math.Calculus
|
exercise.10.5.7
|
Determine whether the series $\sum_{n=1}^{\infty} \frac{\ln n}{n^{3}} $ converges or diverges.
|
converges
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
converges
|
college_math.Calculus
|
exercise.9.9.2
|
Find the arc length of $f(x)=x^{2} / 8-\ln x$ on the interval $[1,2]$.
|
$\ln (2)+3 / 8$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\ln (2)+3 / 8
|
college_math.Calculus
|
exercise.4.7.9
|
Find the derivative of the function: $(1 / 3)^{x^{2}} $
|
$-2 x \ln (3)(1 / 3)^{x^{2}}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-2 x \ln (3)(1 / 3)^{x^{2}}
|
college_math.Calculus
|
exercise.10.3.2
|
Determine whether the series converges or diverges: $\sum_{n=1}^{\infty} \frac{n}{n^{2}+1} $
|
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.10.3.5
|
Determine whether the series converges or diverges: $\sum_{n=1}^{\infty} \frac{1}{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.5.3.5
|
Find all local maximum and minimum points of the function: $y=3 x^{4}-4 x^{3} $
|
$\min$ 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.9.1.3
|
Find the area bounded by the curves: $x=1-y^{2}$ and $y=-x-1 $
|
$9 / 2$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
9 / 2
|
college_math.Calculus
|
exercise.4.8.3
|
Compute the limit of $\lim _{x \rightarrow \infty} \sqrt{x^{2}+x}-\sqrt{x^{2}-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.10.12.11
|
Determine whether the series converges: $\frac{1}{2 \cdot 3 \cdot 4}+\frac{2}{3 \cdot 4 \cdot 5}+\frac{3}{4 \cdot 5 \cdot 6}+\frac{4}{5 \cdot 6 \cdot 7}+\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.8.6.12
|
Evaluate the integral: $\int \cos ^{4} t d t $
|
$\frac{3 t}{8}+\frac{\sin 2 t}{4}+\frac{\sin 4 t}{32}+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{3 t}{8}+\frac{\sin 2 t}{4}+\frac{\sin 4 t}{32}+C
|
college_math.Calculus
|
exercise.8.3.9
|
Find the antiderivative: $\int \frac{1}{x^{2}\left(1+x^{2}\right)} d x $
|
$-\arctan x-1 / x+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-\arctan x-1 / x+C
|
college_math.Calculus
|
exercise.2.3.12
|
Compute the limit: $\lim _{x \rightarrow 0^{+}} \frac{\sqrt{2-x^{2}}}{x+1} $. If a limit does not exist, explain why.
|
$\sqrt{2}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\sqrt{2}
|
college_math.Calculus
|
exercise.10.10.9
|
Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for $x \cos \left(x^{2}\right)$.
|
$\sum_{n=0}^{\infty}(-1)^{n} x^{4 n+1} /(2 n)$ !
|
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^{4 n+1} /(2 n)$ !
|
college_math.Calculus
|
exercise.2.3.4
|
Compute the limit: $\lim _{x \rightarrow 2} \frac{x^{2}+x-12}{x-2} $. If a limit does not exist, explain why.
|
undefined
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
undefined
|
college_math.Calculus
|
exercise.3.5.4
|
Find the derivative of the function: $x \sqrt{169-x^{2}} $
|
$\sqrt{169-x^{2}}-x^{2} / \sqrt{169-x^{2}}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\sqrt{169-x^{2}}-x^{2} / \sqrt{169-x^{2}}
|
college_math.Calculus
|
exercise.9.6.5
|
A thin plate lies in the region between $y=x^{2}$ and the $x$-axis between $x=1$ and $x=2$. Find the centroid.
|
$\bar{x}=45 / 28, \bar{y}=93 / 70$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\bar{x}=45 / 28, \bar{y}=93 / 70
|
college_math.Calculus
|
exercise.7.2.14
|
Compute the value of the integral: $\int_{0}^{5} e^{x} d x $
|
$e^{5}-1$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
e^{5}-1
|
college_math.Calculus
|
exercise.1.3.7
|
Find the domain of the function: $y=f(x)=\sqrt{r^{2}-(x-h)^{2}}$, where $r$ and $h$ are positive constants.
|
$\{x \mid h-r \leq x \leq h+r\}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\{x \mid h-r \leq x \leq h+r\}
|
college_math.Calculus
|
exercise.9.9.1
|
Find the arc length of $f(x)=x^{3 / 2}$ on the interval $[0,2]$.
|
$(22 \sqrt{22}-8) / 27$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
(22 \sqrt{22}-8) / 27
|
college_math.Calculus
|
exercise.3.5.8
|
Find the derivative of the function: $\frac{1}{\sqrt{5-\sqrt{x}}} \cdot $
|
$\frac{1}{4 \sqrt{x}(5-\sqrt{x})^{3 / 2}}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{1}{4 \sqrt{x}(5-\sqrt{x})^{3 / 2}}
|
college_math.Calculus
|
exercise.6.1.12
|
For a cylinder with surface area 50, 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.1.1.9
|
Determine whether the lines $3 x+6 y=7$ and $2 x+4 y=5$ are parallel.
|
yes
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
yes
|
college_math.Calculus
|
exercise.5.3.13
|
Find all local maximum and minimum points of the function: $y=x+1 / x $
|
$\max$ at $-1, \min$ at 1
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\max$ at $-1, \min$ at 1
|
college_math.Calculus
|
exercise.5.3.17
|
Find all local maximum and minimum points of the function: $y=\cos ^{2} x-\sin ^{2} x $
|
$\max$ at $n \pi$, min at $\pi / 2+n \pi$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\max$ at $n \pi$, min at $\pi / 2+n \pi
|
college_math.Calculus
|
exercise.10.5.6
|
Determine whether the series $\sum_{n=1}^{\infty} \frac{\ln n}{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.3.5.38
|
Find an equation for the tangent line to $\left(x^{2}-4 x+5\right) \sqrt{25-x^{2}}$ at $(3,8)$.
|
$y=13 x / 2-23 / 2$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
y=13 x / 2-23 / 2
|
college_math.Calculus
|
exercise.8.1.5
|
Find the antiderivative of the function: $\int \sin ^{3} x \cos x d x $
|
$\left(\sin ^{4} x\right) / 4+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\left(\sin ^{4} x\right) / 4+C
|
college_math.Calculus
|
exercise.10.7.7
|
Determine whether the series $\sum_{n=1}^{\infty} \frac{n^{5}}{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.8.6.25
|
Evaluate the integral: $\int \frac{1}{t(\ln t)^{2}} d t $
|
$\frac{-1}{\ln t}+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{-1}{\ln t}+C
|
college_math.Calculus
|
exercise.7.2.17
|
Find the derivative of the function: $G(x)=\int_{1}^{x} t^{2}-3 t d t $
|
$x^{2}-3 x$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
x^{2}-3 x
|
college_math.Calculus
|
exercise.8.6.20
|
Evaluate the integral: $\int \frac{1}{t\left(9+4 t^{2}\right)} d t $
|
$\frac{\ln |\sin (\arctan (2 t / 3))|}{9}+C=$ $\left(\ln \left(4 t^{2}\right)-\ln \left(9+4 t^{2}\right)\right) / 18+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{\ln |\sin (\arctan (2 t / 3))|}{9}+C=$ $\left(\ln \left(4 t^{2}\right)-\ln \left(9+4 t^{2}\right)\right) / 18+C
|
college_math.Calculus
|
exercise.3.4.6
|
Find an equation for the tangent line to $f(x)=(x-2) /\left(x^{3}+4 x-1\right)$ at $x=1$.
|
$y=11 x / 16-15 / 16$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
y=11 x / 16-15 / 16
|
college_math.Calculus
|
exercise.6.2.3
|
A ladder 13 meters long rests on horizontal ground and leans against a vertical wall. The foot of the ladder is pulled away from the wall at a rate of $0.6 \mathrm{~m} / \mathrm{sec}$. Find the rate at which the top of the ladder is sliding down the wall when the foot of the ladder is $5 \mathrm{~m}$ from the wall.
|
$1 / 4 \mathrm{~m} / \mathrm{s}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
1 / 4 \mathrm{~m} / \mathrm{s}
|
college_math.Calculus
|
exercise.10.12.25
|
Find a series representation for the function: $2^{x} $
|
$\sum_{n=0}^{\infty} \frac{(\ln (2))^{n}}{n !} x^{n}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\sum_{n=0}^{\infty} \frac{(\ln (2))^{n}}{n !} x^{n}
|
college_math.Calculus
|
exercise.5.2.10
|
Find all critical points of the function $f(x)=\left|x^{2}-121\right| $. Identify them as local maximum points, local minimum points, or neither.
|
$\max$ at $x=0$, min at $x= \pm 11$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\max$ at $x=0$, min at $x= \pm 11
|
college_math.Calculus
|
exercise.10.6.1
|
Determine whether the series $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{1}{2 n^{2}+3 n+5} $ 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.5.2.8
|
Find all critical points of the function $y=\cos (2 x)-x $. Identify them as local maximum points, local minimum points, or neither.
|
$\min$ at $x=7 \pi / 12+k \pi, \max$ at $x=-\pi / 12+k \pi$, for integer $k$.
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\min$ at $x=7 \pi / 12+k \pi, \max$ at $x=-\pi / 12+k \pi$, for integer $k$.
|
college_math.Calculus
|
exercise.8.2.2
|
Find the antiderivative: $\int \sin ^{3} x d x $
|
$-\cos x+\left(\cos ^{3} x\right) / 3+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-\cos x+\left(\cos ^{3} x\right) / 3+C
|
college_math.Calculus
|
exercise.1.1.13
|
A car rental firm has the following charges for a certain type of car: $\$ 25$ per day with 100 free miles included, $\$ 0.15$ per mile for more than 100 miles. Suppose you want to rent a car for one day, and you know you'll use it for more than 100 miles. What is the equation relating the cost $y$ to the number of miles $x$ that you drive the car?
|
$y=0.15 x+10$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
y=0.15 x+10
|
college_math.Calculus
|
exercise.8.6.9
|
Evaluate the integral: $\int \frac{\cos 3 t}{\sqrt{\sin 3 t}} d t $
|
$\frac{2}{3} \sqrt{\sin 3 t}+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{2}{3} \sqrt{\sin 3 t}+C
|
college_math.Calculus
|
exercise.10.4.6
|
Approximate the value of the series $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{1}{n^{4}}$ to two decimal places.
|
0.95
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
0.95
|
college_math.Calculus
|
exercise.4.5.5
|
Find the derivative of the function: $\cot x $
|
$-\csc ^{2} x$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-\csc ^{2} x
|
college_math.Calculus
|
exercise.9.2.1
|
For the velocity function $v=\cos (\pi t)$, find both the net distance and the total distance traveled during the time interval $0 \leq t \leq 2.5$.
|
$1 / \pi, 5 / \pi$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
1 / \pi, 5 / \pi
|
college_math.Calculus
|
exercise.10.12.29
|
Find a series representation for the function: $\frac{1}{1+x^{2}} $
|
$\sum_{n=0}^{\infty}(-1)^{n} x^{2 n}$
|
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}
|
college_math.Calculus
|
exercise.9.5.7
|
A force of 2 Newtons will compress a spring from 1 meter (its natural length) to 0.8 meters. How much work is required to stretch the spring from 1.1 meters to 1.5 meters?
|
$6 / 5 \mathrm{~N}-\mathrm{m}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
6 / 5 \mathrm{~N}-\mathrm{m}
|
college_math.Calculus
|
exercise.6.2.14
|
A woman $5 \mathrm{~ft}$ tall walks at a speed of $3.5 \mathrm{~ft} / \mathrm{sec}$ away from a streetlight that is $12 \mathrm{~ft}$ above the ground. Find the rate at which the tip of her shadow is moving. Find the rate at which her shadow is lengthening.
|
tip: $6 \mathrm{ft} / \mathrm{s}$, length: $5 / 2 \mathrm{ft} / \mathrm{s}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
tip: $6 \mathrm{ft} / \mathrm{s}$, length: $5 / 2 \mathrm{ft} / \mathrm{s}
|
college_math.Calculus
|
exercise.8.6.17
|
Evaluate the integral: $\int e^{t} \sin t d t $
|
$\frac{e^{t} \sin t-e^{t} \cos t}{2}+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{e^{t} \sin t-e^{t} \cos t}{2}+C
|
college_math.Calculus
|
exercise.9.2.7
|
An object is shot upwards from ground level with an initial velocity of 100 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.
|
25000/49 meters, 1000/49 seconds
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
25000/49 meters, 1000/49 seconds
|
college_math.Calculus
|
exercise.8.6.16
|
Evaluate the integral: $\int t^{3} \sqrt{t^{2}+1} d t $
|
$\frac{\left(t^{2}+1\right)^{5 / 2}}{5}-\frac{\left(t^{2}+1\right)^{3 / 2}}{3}+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\frac{\left(t^{2}+1\right)^{5 / 2}}{5}-\frac{\left(t^{2}+1\right)^{3 / 2}}{3}+C
|
college_math.Calculus
|
exercise.10.12.30
|
Find a series representation for the function: $\arctan (x) $
|
$\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2 n+1} x^{2 n+1}$
|
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}}{2 n+1} x^{2 n+1}
|
college_math.Calculus
|
exercise.8.1.13
|
Find the antiderivative of the function: $\int_{0}^{\sqrt{\pi} / 2} x \sec ^{2}\left(x^{2}\right) \tan \left(x^{2}\right) d x $
|
$1 / 4$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
1 / 4
|
college_math.Calculus
|
exercise.1.3.3
|
Find the domain of the function: $y=f(x)=1 /\left(x^{2}-1\right) $
|
$\{x \mid x \neq 1$ 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 \neq 1$ and $x \neq-1\}
|
college_math.Calculus
|
exercise.5.4.3
|
Describe the concavity of the function: $y=x^{3}-9 x^{2}+24 x $
|
concave down when $x<3$, concave up when $x>3$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
concave down when $x<3$, concave up when $x>3
|
college_math.Calculus
|
exercise.2.3.14
|
Compute the limit: $\lim _{x \rightarrow 2}\left(x^{2}+4\right)^{3} $.
|
512
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
512
|
college_math.Calculus
|
exercise.5.3.4
|
Find all local maximum and minimum points of the function: $y=x^{4}-2 x^{2}+3 $
|
$\min$ at $x= \pm 1, \max$ at $x=0$.
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\min$ at $x= \pm 1, \max$ at $x=0$.
|
college_math.Calculus
|
exercise.4.7.6
|
Find the derivative of the function: $x^{\sin x} $
|
$x^{\sin x}\left(\cos x \ln x+\frac{\sin x}{x}\right)$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
x^{\sin x}\left(\cos x \ln x+\frac{\sin x}{x}\right)
|
college_math.Calculus
|
exercise.8.1.12
|
Find the antiderivative of the function: $\int \sec ^{2} x \tan x d x $
|
$\tan ^{2}(x) / 2+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\tan ^{2}(x) / 2+C
|
college_math.Calculus
|
exercise.4.1.2
|
Find all values of $\theta$ such that $\cos (2 \theta)=1 / 2$; give your answer in radians.
|
$n \pi \pm \pi / 6$, any integer $n$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
n \pi \pm \pi / 6$, any integer $n
|
college_math.Calculus
|
exercise.4.1.4
|
Use an angle sum identity to compute $\tan (5 \pi / 12)$.
|
$-(1+\sqrt{3}) /(1-\sqrt{3})$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-(1+\sqrt{3}) /(1-\sqrt{3})
|
college_math.Calculus
|
exercise.9.7.8
|
Does the improper integral $\int_{0}^{\pi / 2} \sec ^{2} x d x$ converge or diverge? If it converges, find the value.
|
diverges
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
diverges
|
college_math.Calculus
|
exercise.6.1.11
|
Find the area of the largest rectangle that fits inside a semicircle of radius $r$ (one side of the rectangle is along the diameter of the semicircle).
|
$r^{2}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
r^{2}
|
college_math.Calculus
|
exercise.9.5.8
|
A 20 meter long steel cable has density 2 kilograms per meter, and is hanging straight down. How much work is required to lift the entire cable to the height of its top end?
|
$3920 \mathrm{~N}-\mathrm{m}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
3920 \mathrm{~N}-\mathrm{m}
|
college_math.Calculus
|
exercise.7.2.6
|
Find the antiderivative of the function: $(5 x+1)^{2} $
|
$(5 x+1)^{3} / 15+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
(5 x+1)^{3} / 15+C
|
college_math.Calculus
|
exercise.9.6.2
|
A beam 10 meters long has density $\sigma(x)=\sin (\pi x / 10)$ at distance $x$ from the left end of the beam. Find the center of mass $\bar{x}$.
|
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.9.2.5
|
An object is shot upwards from ground level with an initial velocity of 2 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.
|
$10 / 49$ meters, $20 / 49$ seconds
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
10 / 49$ meters, $20 / 49$ seconds
|
college_math.Calculus
|
exercise.10.4.4
|
Determine whether the series $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{\ln n}{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.10.12.27
|
Find a series representation for the function: $\ln \left(\frac{1+x}{1-x}\right) $
|
$\sum_{n=0}^{\infty} \frac{2}{2 n+1} x^{2 n+1}$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\sum_{n=0}^{\infty} \frac{2}{2 n+1} x^{2 n+1}
|
college_math.Calculus
|
exercise.10.10.3
|
Find the Maclaurin series or Taylor series centered at $a$ and the radius of convergence for the function: $1 / x, a=5 $
|
$\sum_{n=0}^{\infty}(-1)^{n} \frac{(x-5)^{n}}{5^{n+1}}, R=5$
|
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} \frac{(x-5)^{n}}{5^{n+1}}, R=5
|
college_math.Calculus
|
exercise.10.12.16
|
Determine whether the series converges: $1+\frac{5^{2}}{2^{2}}+\frac{5^{4}}{(2 \cdot 4)^{2}}+\frac{5^{6}}{(2 \cdot 4 \cdot 6)^{2}}+\frac{5^{8}}{(2 \cdot 4 \cdot 6 \cdot 8)^{2}}+\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.4.9.9
|
A hyperbola passing through $(8,6)$ consists of all points whose distance from the origin is a constant more than its distance from the point $(5,2)$. Find the slope of the tangent line to the hyperbola at $(8,6) . $
|
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.5.2.4
|
Find all critical points of the function $y=x^{4}-2 x^{2}+3 $. Identify them as local maximum points, local minimum points, or neither.
|
$\min$ at $x= \pm 1, \max$ at $x=0$.
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\min$ at $x= \pm 1, \max$ at $x=0$.
|
college_math.Calculus
|
exercise.10.3.8
|
Determine whether the series converges or diverges: $\sum_{n=2}^{\infty} \frac{1}{n(\ln 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.9.1.12
|
Find the area bounded by the curves: $y=x^{2}-2 x$ and $y=x-2 $
|
$1 / 6$
|
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.9.1.7
|
Find the area bounded by the curves: $y=\sqrt{x}$ and $y=x^{2} $
|
$1 / 3$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
1 / 3
|
college_math.Calculus
|
exercise.4.3.3
|
Compute the limit: $\lim _{x \rightarrow 0} \frac{\cot (4 x)}{\csc (3 x)} $
|
$3 / 4$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
3 / 4
|
college_math.Calculus
|
exercise.6.1.3
|
Find the dimensions of the rectangle of largest area having fixed perimeter $P$.
|
$P / 4 \times P / 4$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
P / 4 \times P / 4
|
college_math.Calculus
|
exercise.4.5.8
|
Find the derivative of the function: $\sin ^{2} x+\cos ^{2} 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.6.1.4
|
A box with square base and no top is to hold a volume 100. Find 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.
|
$w=l=2 \cdot 5^{2 / 3}, h=5^{2 / 3}, h / w=$ $1 / 2$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
w=l=2 \cdot 5^{2 / 3}, h=5^{2 / 3}, h / w=$ $1 / 2
|
college_math.Calculus
|
exercise.8.1.7
|
Find the antiderivative of the function: $\int \frac{x^{2}}{\sqrt{1-x^{3}}} d x $
|
$-2 \sqrt{1-x^{3}} / 3+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-2 \sqrt{1-x^{3}} / 3+C
|
college_math.Calculus
|
exercise.2.4.4
|
Find the derivative of the function: $y=f(x)=a x^{2}+b x+c$ (where $a, b$, and $c$ are constants).
|
$2 a x+b$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
2 a x+b
|
college_math.Calculus
|
exercise.5.3.8
|
Find all local maximum and minimum points of the function: $y=\cos (2 x)-x $
|
$\min$ at $x=7 \pi / 12+n \pi$, $\max$ at $x=-\pi / 12+n \pi$, for integer $n$.
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
\min$ at $x=7 \pi / 12+n \pi$, $\max$ at $x=-\pi / 12+n \pi$, for integer $n$.
|
college_math.Calculus
|
exercise.5.4.5
|
Describe the concavity of the function: $y=3 x^{4}-4 x^{3} $
|
concave up when $x<0$ or $x>2 / 3$, concave down when $0<x<2 / 3$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
concave up when $x<0$ or $x>2 / 3$, concave down when $0<x<2 / 3
|
college_math.Calculus
|
exercise.8.1.6
|
Find the antiderivative of the function: $\int x \sqrt{100-x^{2}} d x $
|
$-\left(100-x^{2}\right)^{3 / 2} / 3+C$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
-\left(100-x^{2}\right)^{3 / 2} / 3+C
|
college_math.Calculus
|
exercise.2.3.6
|
Compute the limit: $\lim _{x \rightarrow 0} \sqrt{\frac{1}{x}+2}-\sqrt{\frac{1}{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.10.3.9
|
Find an $N$ such that $\sum_{n=1}^{\infty} \frac{1}{n^{4}}=\sum_{n=1}^{N} \frac{1}{n^{4}} \pm 0.005$.
|
$N=5$
|
Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0)
|
college_math.calculus
|
N=5
|
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