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