Calculus

Problem 24101

Calculate the limit: $\lim _{x \rightarrow 0} \frac{10 x^{2}+x}{\tan x}$ and fill in the table values for $f(x)$ .

See Solution

Problem 24102

Find the volume of the solid formed by revolving the region defined by $y = 27\sin(x)\cos(x)$ from $x = 0$ to $x = \frac{\pi}{2}$ around the x-axis. Use $\pi$ as needed.

See Solution

Problem 24103

Find points on the curve $y=x^{3}+3x+6$ where the tangent is parallel to $5x-y=9$ . Are there multiple points?

See Solution

Problem 24104

Find the horizontal asymptote of the function $f(x)=\frac{x^{2}-2}{4x-x^{4}}$ .

See Solution

Problem 24105

Find the volume of the solid formed by revolving the shaded area defined by $3x + 4y = 12$ around the x-axis.

See Solution

Problem 24106

Find the integral for the volume of a cylindrical tank (length 15 ft, radius 5 ft) filled to a depth of 9 ft.
$v=\int_{-5}^{h} A(y) \, dy$

See Solution

Problem 24107

Find the integral for the volume of a horizontal cylinder tank (length 13 ft, radius 4 ft) filled to 6 ft deep.
$V=\int_{-2}^{2} A(x) \, dx$

See Solution

Problem 24108

Berechne die Anzahl der Lkw-Fahrten für Baumaterial eines Deichs mit $f(x) = \frac{5}{64} x^2 + 1$ , $\rho = 1.8 \, \text{g/cm}^3$ , Länge 12 m, und 20 t Kapazität.

See Solution

Problem 24109

Find the value of $f^{\prime}\left(\frac{\sqrt{21}}{5}\right)$ if $f(x)=\arcsin (x)$ .

See Solution

Problem 24110

Find the value of $f^{\prime}\left(\frac{1}{5}\right)$ if $f(x)=\sin^{-1}(x)$ .

See Solution

Problem 24111

Find the derivative of the function $f(x)=\tan^{-1}(2x)$ in its simplest form.

See Solution

Problem 24112

A ball is thrown down at $8 \mathrm{~m/s}$ from $30 \mathrm{~m}$ . How long until it hits the ground?

See Solution

Problem 24113

Show that the reproduction rate $r(C)$ approaches zero as sugar level $C$ approaches zero: $\lim_{C \rightarrow 0} r(C)=0$ .

See Solution

Problem 24114

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ for the equation $-2-3 x^{2}+3 y^{2}=-x y$ .

See Solution

Problem 24115

Find the volume using the shell method for the region bounded by $y = \sqrt{6}$ , $x = 12 - 2y^2$ , and $x = 12$ . Answer in terms of $\pi$ .

See Solution

Problem 24116

Find the integral for the volume of a cylindrical gas tank (length 11 ft, radius 3 ft) filled to a depth of 4 ft.
$V=\int_{-3}^{1} A(x) \, dx$

See Solution

Problem 24117

Find when the speed of a particle, defined by $x(t)=t^{3}-27 t^{2}+51 t$ , is decreasing for $t \geq 0$ .

See Solution

Problem 24118

Show that the reproduction rate $r(C)$ approaches 0 as sugar level $C$ approaches 0: $\lim_{C \rightarrow 0} r(C) = 0$ .

See Solution

Problem 24119

Calculate the limit: $\lim _{x \rightarrow 0} \frac{4-\sqrt{16+x^{2}}}{2 x^{2}}$

See Solution

Problem 24120

Show that the function $f(x)$ is continuous at $x=-2$ : $f(x)=\frac{2x^2+3x-2}{x+2}$ for $x \neq -2$ , $f(-2)=-5$ .

See Solution

Problem 24121

Show that $\lim_{N \rightarrow 0} R(N) = 0$ using the formula $R(N) = N \ln \left(\frac{1}{N}\right)$ for $d=A=1$ . Complete the table for $N$ : 0.1, 0.01, 0.001.

See Solution

Problem 24122

Find when the acceleration of a particle, with position $x(t)=5 t^{4}-30 t^{2}$ , is positive for $t \geq 0$ .

See Solution

Problem 24123

Find the intervals where the speed of a particle, with velocity $v(t)=3 t^{2}-36 t-84$ , is decreasing for $t \geq 0$ .

See Solution

Problem 24124

Find the volume using the shell method by revolving the shaded area defined by $y = \sqrt{6}$ and $x = 3y^2$ around the $x$ -axis. Set up the integral for the volume.

See Solution

Problem 24125

Find the limit: $\lim _{x \rightarrow 0^{+}} \frac{1-\cos x}{x^{2} \sin x}$

See Solution

Problem 24126

Find the position of a particle at time $t=8$ given its velocity $v(t)=\frac{\pi}{2} \sin \left(\frac{\pi}{6} t\right)$ and $x=-6$ at $t=12$ .

See Solution

Problem 24127

Find the slope of the curve defined by $x=-6 t^{3}+4$ and $y=2 t^{2}$ at $t=5$ . Slope $=\square$ .

See Solution

Problem 24128

Find $\frac{d^{2} y}{d x^{2}}$ at the point $(2,-5)$ given $-5 x^{2}=5-y^{2}$ .

See Solution

Problem 24129

Find the position of a particle at time $t=0$ given $v(t)=2 \pi \sin (\pi t)$ and $x=4$ at $t=\frac{1}{2}$ .

See Solution

Problem 24130

Find $\frac{d^{2} y}{d x^{2}}$ at the point $(-1,2)$ given $-y=-y^{2}-2 x^{3}$ .

See Solution

Problem 24131

Find the area of the shaded region using two integrals with respect to $y$ for $y = 3 - x$ and $y = x + 1$ .

See Solution

Problem 24132

Find the limit: $\lim _{x \rightarrow \infty}(1+2 x)^{\frac{1}{2 \ln x}}$ .

See Solution

Problem 24133

Identify a convergence test for the series $\sum_{k=1}^{\infty}(-1)^{k}\left(5+\frac{1}{k^{2}}\right)^{k}$ .

See Solution

Problem 24134

Find when the particle at $x(t)=3 t^{3}-9 t^{2}-72 t$ moves right for $t \geq 0$ .

See Solution

Problem 24135

Find $\frac{d^{2} y}{d x^{2}}$ for $2 x^{3}+y+y^{2}=0$ at the point $(-1,-2)$ .

See Solution

Problem 24136

Find the volume when the region bounded by $y=15-\sqrt{x}, y=15, x=225$ is revolved around $y=15$ .

See Solution

Problem 24137

1. Bestimme die Fläche zwischen $f(x)=0.5 x^{2}$ und der $x$ -Achse für $x=0$ bis $x=2$ mit $n=4$ Säulen.
2. Finde Nullstellen und Extremstellen von $f(x)=x^{3}-x^{2}-x+2$ und berechne die Fläche zwischen dem Graphen und den Achsen.
3. Bestimme $a$ so, dass $\int_{1}^{a} \frac{1}{3} x^{2}+x \, dx=\frac{8}{9}$ gilt.
4. Skizziere einen möglichen Graphen von $g(x)$ , wenn $\int_{0}^{2} g(x) \, dx=0$ ist, und erläutere die Bedeutung.

See Solution

Problem 24138

Skizzieren Sie einen Graphen der stetigen Funktion $g$ auf $[0 ; 2]$ , wenn $\int_{0}^{2} g(x) d x=0$ . Erklären Sie das Integral.

See Solution

Problem 24139

Find the derivative $\frac{dy}{dx}$ of the function $-\frac{4}{2y+1}$ .

See Solution

Problem 24140

Find the average rate of change of $y = (-5x + 15)^{1}$ on the interval $[1,3]$ using the Mean Value Theorem.

See Solution

Problem 24141

Zeigen Sie, dass die Tangente von $f$ an $x_{0}$ parallel zur $x$ -Achse ist. a) $f(x)=x^{3}-4x^{2}+5x; x_{0}=1$ b) $f(x)=x^{3}-6x^{2}+12x; x_{0}=2$

See Solution

Problem 24142

Find $p$ such that $\int_{1}^{5} p f(x) dx = 83$ , where $f(x)=3x-2$ for $x \leq 2$ and $f(x)=x^2$ for $x > 2$ .

See Solution

Problem 24143

Find values of $t$ in $[0, 2]$ where instantaneous velocity $x'(t)$ equals average velocity $\frac{x(2)-x(0)}{2-0}$ .

See Solution

Problem 24144

Given $f^{\prime \prime}(x)>0$ , which could be $f^{\prime}(5)$ : (A) 0.5, (B) 0.7, (C) 0.9, (D) 1.1, (E) 1.3?

See Solution

Problem 24145

Berechne $a$ so, dass $\int_{1}^{a} \left( \frac{1}{3} x^{2}+x \right) dx=\frac{8}{9}$ . Skizziere einen Graphen für $g(x)$ mit $\int_{0}^{2} g(x) dx=0$ .

See Solution

Problem 24146

Gegeben ist die Funktion $f(x)=x^{3}-x^{2}-x+2$ .
(a) Bestimme Nullstellen und Extremstellen mit dem Taschenrechner.
(b) Berechne das Flächenstück zwischen $f$ , x- und y-Achse.
(c) Im Tiefpunkt von $f$ eine Parallele $g$ zur x-Achse zeichnen.
Berechne das Flächenstück zwischen $g$ und $f$ und zeichne es ein.

See Solution

Problem 24147

Bestimme den Grenzwert von $P(x)=\frac{x^{3}-2x+1}{x-1}$ mit der $\mathrm{H}$ -Methode.

See Solution

Problem 24148

Find the Riemann sum formula for $f(x)=4x^{3}$ on $[0,4]$ using right endpoints, then take the limit as $n \to \infty$ .

See Solution

Problem 24149

Bestimme den Grenzwert von $f(x)=\frac{x^{3}-2 x+1}{x-1}$ mit der H-Methode.

See Solution

Problem 24150

Find the limit of the sequence $a_{n}=\frac{\sqrt{n}-n}{2 n}$ as $n$ approaches infinity. Answer: $\lim _{n \rightarrow \infty} a_{n}=\square$ .

See Solution

Problem 24151

Wie viel Schwefel-37 bleibt nach 40 Minuten von 120 mg übrig, wenn die Halbwertszeit 5 Minuten beträgt?

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Problem 24152

Untersuchen Sie die Funktionen $a(x)=e^{x}+e^{-x}$ , $b(x)=e^{x} \cdot e^{-x}$ und $c(x)=\frac{e^{x}}{e^{-x}}$ auf Asymptoten und Symmetrien.

See Solution

Problem 24153

Find the limit of the sequence $a_{n}=\cos \left(\frac{13}{n+14}\right)$ as $n$ approaches infinity. What is the value?

See Solution

Problem 24154

Find the limit of the sequence $a_{n}=5\left(-\frac{1}{2}\right)^{n}$ . Answer: $\lim _{n \rightarrow \infty} 5\left(-\frac{1}{2}\right)^{n}=\square$

See Solution

Problem 24155

Find the limit of the sequence $a_{n}=\frac{2 \sin (n)}{n}$ as $n$ approaches infinity. What is $\lim _{n \rightarrow \infty} \frac{2 \sin (n)}{n}=\square$ ?

See Solution

Problem 24156

Does the sequence $a_{n}=4 e^{\left(\frac{1}{n-7}\right)}$ converge? If yes, find the limit; if no, enter $\varnothing$ .

See Solution

Problem 24157

Does the series $\sum_{n=1}^{\infty}(8 \sqrt{n+1}-8 \sqrt{n})$ converge or diverge?

See Solution

Problem 24158

Check if the sequence $a_{n}=\left(\frac{4}{7}\right)^{n}$ converges using the Monotone Convergence Theorem.

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Problem 24159

Një trup bie nga një lartësi dhe arrin tokën pas 4 sekondash. a) Cila është lëvizja e trupit? b) Sa është lartësia fillestare? c) Cila është shpejtësia në rënie?

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Problem 24160

Find the first five partial sums $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ of the series $\sum_{n=1}^{\infty} \frac{10}{3^{n}-18}$ . Round to the nearest thousandth.

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Problem 24161

Determine if the series $\sum_{n=1}^{\infty}(-8)^{4 n}(-9)^{1-n}$ converges or diverges. Choose: converges or diverges.

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Problem 24162

Find the sum of the series: $\sum_{n=1}^{\infty} e^{2 n}(-10)^{1-n}$ as a fraction in terms of $e$ .

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Problem 24163

Find the sum of the series: $\sum_{n=1}^{\infty}\left(\frac{5}{\sqrt{n+7}}-\frac{5}{\sqrt{n+8}}\right)$ . If it diverges, answer with $\varnothing$ .

See Solution

Problem 24164

Überprüfen Sie die Aussagen zur Extremstellen und geben Sie Gegenbeispiele an, falls sie falsch sind: a) $f^{\prime}(x_{0})=0 \Rightarrow$ Extremstelle, b) ganzrationale Funktion 2. Grades hat immer Extremstelle, c) globales Maximum ist lokal, d) $g(x)=5(x-2)^{3}+4$ .

See Solution

Problem 24165

Find the sum of the series: $\sum_{n=1}^{\infty} 2^{4 n} 20^{1-n}$ .
Answer: $\sum_{n=1}^{\infty} 2^{4 n} 20^{1-n}=\square$

See Solution

Problem 24166

Apply the divergence test to the series $\sum_{n=1}^{\infty}\left[\frac{(n+7) !}{(n+9) !}\right]$ . What is the result?

See Solution

Problem 24167

Check if the sequence $a_{n}=\frac{1-7 n^{2}}{3 n^{4}-2}$ converges and find its limit or state $\varnothing$ if it diverges.

See Solution

Problem 24168

Use the divergence test on the series $\sum_{n=1}^{\infty}(4 \sin^{2}(n-3))$ and choose the correct conclusion.

See Solution

Problem 24169

Calculate the limit: $\lim _{n \rightarrow \infty}\left(\frac{-3 n^{4}-6}{\sqrt{n^{4}+3 n^{3}-4}}\right)$ . If it doesn't exist, enter $\varnothing$ .

See Solution

Problem 24170

Find the sum of the series $\sum_{n=1}^{\infty}\left(\frac{17}{8^{n}}-\frac{15}{3^{n}}\right)$ using given sums.

See Solution

Problem 24171

Determine if the series $\sum_{n=1}^{\infty} \frac{5}{e^{n / 2}}$ converges or diverges using the integral test.

See Solution

Problem 24172

Find the value of $p$ for the series $\sum_{n=1}^{\infty} \frac{\sqrt[4]{n}}{3 \sqrt{n}}$ . Provide your answer: $p=$

See Solution

Problem 24173

Does the series $3+\frac{6}{(2)^{5 / 4}}+\frac{9}{(3)^{5 / 4}}+\cdots$ converge or diverge?

See Solution

Problem 24174

Use the integral test to check if the series $\sum_{n=1}^{\infty} \frac{n+5}{n+4}$ converges or diverges.

See Solution

Problem 24175

Can the integral test be applied to the series $\sum_{n=1}^{\infty}\left(-\frac{1}{7}\right)^{n}$ ?

See Solution

Problem 24176

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{4}{\sqrt[3]{n+1}}$ using the integral test: $\int \frac{4}{\sqrt[3]{x+1}} dx$

See Solution

Problem 24177

Find the sum of the series: $\sum_{n=1}^{\infty}\left(\frac{3}{\sqrt{n+5}}-\frac{3}{\sqrt{n+6}}\right)$ . If it diverges, answer with $\varnothing$ .

See Solution

Problem 24178

Find the value of $p$ for the series: $\frac{1}{2}+\frac{2}{2(2)^{3 / 2}}+\frac{3}{2(3)^{3 / 2}}+\cdots$
Answer: $p=$

See Solution

Problem 24179

Bestimme das charakteristische Polynom $p(X)$ der Gleichung $y^{(3)}-6 y^{(2)}+12 y^{(1)}-8 y^{(0)}=0$ , berechne $p(2)$ und zerlege $p(X)$ in Linearfaktoren. Finde ein Fundamentalsystem und löse das Anfangswertproblem mit $y(1)=0, y^{\prime}(1)=0, y^{\prime \prime}(1)=2$ .

See Solution

Problem 24180

Find the limit: $\lim _{n \rightarrow \infty}\left[\frac{(n+8) !}{(n+2) !}\right]$ . If none, enter $\varnothing$ .

See Solution

Problem 24181

Find the sum of the series: $\sum_{n=1}^{\infty}\left(8^{8 / n}-8^{8 /(n+1)}\right)$ . If it diverges, enter $\varnothing$ .

See Solution

Problem 24182

Determine if the series $\sum_{n=1}^{\infty} n\left(3^{-3 \ln (n)}\right)$ converges or diverges.

See Solution

Problem 24183

Which series helps determine the divergence of $\sum_{n=1}^{\infty} \frac{\sqrt{3 n^{2}+2 n+3}}{n}$ ? Choose one: $\sum_{n=1}^{\infty} \frac{1}{n}$ , $\sum_{n=1}^{\infty} \frac{1}{n^{5}}$ , $\sum_{n=1}^{\infty} \frac{1}{n^{4}}$ , $\sum_{n=1}^{\infty} \frac{1}{n^{3}}$ .

See Solution

Problem 24184

Determine if the series $\sum_{n=1}^{\infty} \frac{8 n^{7}+2 n^{6}+3 n^{2}}{5 n^{5}-2 n^{3}-n}$ converges or diverges using the Comparison Test.

See Solution

Problem 24185

Which series helps check the convergence of $\sum_{n=1}^{\infty} \frac{n}{3 n^{4}+5 n^{3}+5 n}$ ? Choose one:
1. $\sum_{n=1}^{\infty} \frac{1}{n}$
2. $\sum_{n=1}^{\infty} \frac{1}{n^{4}}$
3. $\sum_{n=1}^{\infty} \frac{1}{n^{3}}$
4. $\sum_{n=1}^{\infty} \frac{1}{n^{5}}$

See Solution

Problem 24186

Use the limit comparison test with $\sum_{n=1}^{\infty} \frac{8^{n}}{6^{n}}$ to check if $\sum_{n=1}^{\infty} \frac{8^{n}+8}{6^{n}}$ converges or diverges.

See Solution

Problem 24187

Check if the series $\sum_{n=2}^{\infty} \frac{\ln (n)}{3 n^{7}}$ converges or diverges using the Comparison Test with a $p$ -series.

See Solution

Problem 24188

Use the divergence test on the series $\sum_{n=1}^{\infty}\left[\frac{(n+2) !}{(n+9) !}\right]$ and choose the correct conclusion.

See Solution

Problem 24189

Determine if the series $\sum_{n=1}^{\infty} \frac{7}{\sqrt{25 n}-2}$ converges or diverges using the limit comparison test with $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ .

See Solution

Problem 24190

Use the limit comparison test with $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ to check if $\sum_{n=1}^{\infty} \frac{\ln (n)^{8}}{n^{2}}$ converges or diverges.

See Solution

Problem 24191

Determine if the series $\sum_{n=5}^{\infty} \frac{2+\cos ^{2}(n)}{n-4}$ converges or diverges using the Comparison Test.

See Solution

Problem 24192

Use the limit comparison test with $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ to check if $\sum_{n=1}^{\infty} \frac{\ln (n)^{6}}{n^{2}}$ converges or diverges.

See Solution

Problem 24193

Evaluate the integral $\int_{-8}^{8} \sqrt{64-x^{2}} dx$ . What is its value? Choices: $-12 / 3$ , $32 \pi$ , $64 \pi$ , $12 \pi$ .

See Solution

Problem 24194

Compute the limit:
$\lim _{n \rightarrow \infty} \frac{\frac{3 \sqrt{n}}{n^{8}+5}}{\frac{1}{n^{15 / 2}}}$
Provide your answer below: $\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=\square$

See Solution

Problem 24195

Determine if the series $\sum_{n=1}^{\infty} \frac{25 n^{4}}{25 n^{10}+1}$ converges or diverges using the integral test.

See Solution

Problem 24196

Determine if the series $\sum_{n=1}^{\infty} \frac{14 e^{1/n}}{n^{2}}$ converges or diverges using the limit comparison test with $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ .

See Solution

Problem 24197

Find the value of $c$ for $f(x)=x^{3}+1$ on $[-1,2]$ that satisfies the Intermediate Value Theorem. Options: $c=3$ , $c=2$ , $c=1$ , $c=-1$ .

See Solution

Problem 24198

Calculate the integral $\int_{0}^{4} \frac{x}{\sqrt{x^{2}+9}} d x$ . What is the result?

See Solution

Problem 24199

Find the value of $c$ that satisfies the Intermediate Value Theorem for $f(x)=x^{3}+1$ on $[-1,2]$ with $w=2$ .

See Solution

Problem 24200

Find the horizontal asymptote of the function $f(x)=\frac{12 x^{6}-9 x^{3}+5}{2 x^{6}-81 x-1}$ .

See Solution

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Calculus

Problem 24101

Calculate the limit: lim⁡x→010x2+xtan⁡x \lim _{x \rightarrow 0} \frac{10 x^{2}+x}{\tan x} limx→0​tanx10x2+x​ and fill in the table values for f(x)f(x)f(x).

Problem 24102

Find the volume of the solid formed by revolving the region defined by y=27sin⁡(x)cos⁡(x)y = 27\sin(x)\cos(x)y=27sin(x)cos(x) from x=0x = 0x=0 to x=π2x = \frac{\pi}{2}x=2π​ around the x-axis. Use π\piπ as needed.

Problem 24103

Find points on the curve y=x3+3x+6y=x^{3}+3x+6y=x3+3x+6 where the tangent is parallel to 5x−y=95x-y=95x−y=9. Are there multiple points?

Problem 24104

Find the horizontal asymptote of the function f(x)=x2−24x−x4f(x)=\frac{x^{2}-2}{4x-x^{4}}f(x)=4x−x4x2−2​.

Problem 24105

Find the volume of the solid formed by revolving the shaded area defined by 3x+4y=123x + 4y = 123x+4y=12 around the x-axis.

Problem 24106

Find the integral for the volume of a cylindrical tank (length 15 ft, radius 5 ft) filled to a depth of 9 ft. v=∫−5hA(y) dy v=\int_{-5}^{h} A(y) \, dy v=∫−5h​A(y)dy

Problem 24107

Find the integral for the volume of a horizontal cylinder tank (length 13 ft, radius 4 ft) filled to 6 ft deep. V=∫−22A(x) dx V=\int_{-2}^{2} A(x) \, dx V=∫−22​A(x)dx

Problem 24108

Berechne die Anzahl der Lkw-Fahrten für Baumaterial eines Deichs mit f(x)=564x2+1f(x) = \frac{5}{64} x^2 + 1f(x)=645​x2+1, ρ=1.8 g/cm3\rho = 1.8 \, \text{g/cm}^3ρ=1.8g/cm3, Länge 12 m, und 20 t Kapazität.

Problem 24109

Find the value of f′(215)f^{\prime}\left(\frac{\sqrt{21}}{5}\right)f′(521​​) if f(x)=arcsin⁡(x)f(x)=\arcsin (x)f(x)=arcsin(x).

Problem 24110

Find the value of f′(15)f^{\prime}\left(\frac{1}{5}\right)f′(51​) if f(x)=sin⁡−1(x)f(x)=\sin^{-1}(x)f(x)=sin−1(x).

Problem 24111

Find the derivative of the function f(x)=tan⁡−1(2x)f(x)=\tan^{-1}(2x)f(x)=tan−1(2x) in its simplest form.

Problem 24112

A ball is thrown down at 8 m/s8 \mathrm{~m/s}8 m/s from 30 m30 \mathrm{~m}30 m. How long until it hits the ground?

Problem 24113

Show that the reproduction rate r(C)r(C)r(C) approaches zero as sugar level CCC approaches zero: lim⁡C→0r(C)=0\lim_{C \rightarrow 0} r(C)=0limC→0​r(C)=0.

Problem 24114

Find dydx\frac{d y}{d x}dxdy​ in terms of xxx and yyy for the equation −2−3x2+3y2=−xy-2-3 x^{2}+3 y^{2}=-x y−2−3x2+3y2=−xy.

Problem 24115

Find the volume using the shell method for the region bounded by y=6y = \sqrt{6}y=6​, x=12−2y2x = 12 - 2y^2x=12−2y2, and x=12x = 12x=12. Answer in terms of π\piπ.

Problem 24116

Find the integral for the volume of a cylindrical gas tank (length 11 ft, radius 3 ft) filled to a depth of 4 ft. V=∫−31A(x) dx V=\int_{-3}^{1} A(x) \, dx V=∫−31​A(x)dx

Problem 24117

Find when the speed of a particle, defined by x(t)=t3−27t2+51tx(t)=t^{3}-27 t^{2}+51 tx(t)=t3−27t2+51t, is decreasing for t≥0t \geq 0t≥0.

Problem 24118

Show that the reproduction rate r(C)r(C)r(C) approaches 0 as sugar level CCC approaches 0: lim⁡C→0r(C)=0\lim_{C \rightarrow 0} r(C) = 0limC→0​r(C)=0.

Problem 24119

Calculate the limit: lim⁡x→04−16+x22x2\lim _{x \rightarrow 0} \frac{4-\sqrt{16+x^{2}}}{2 x^{2}}x→0lim​2x24−16+x2​​

Problem 24120

Show that the function f(x)f(x)f(x) is continuous at x=−2x=-2x=−2: f(x)=2x2+3x−2x+2f(x)=\frac{2x^2+3x-2}{x+2}f(x)=x+22x2+3x−2​ for x≠−2x \neq -2x=−2, f(−2)=−5f(-2)=-5f(−2)=−5.

Problem 24121

Show that lim⁡N→0R(N)=0\lim_{N \rightarrow 0} R(N) = 0limN→0​R(N)=0 using the formula R(N)=Nln⁡(1N)R(N) = N \ln \left(\frac{1}{N}\right)R(N)=Nln(N1​) for d=A=1d=A=1d=A=1. Complete the table for NNN: 0.1, 0.01, 0.001.

Problem 24122

Find when the acceleration of a particle, with position x(t)=5t4−30t2x(t)=5 t^{4}-30 t^{2}x(t)=5t4−30t2, is positive for t≥0t \geq 0t≥0.

Problem 24123

Find the intervals where the speed of a particle, with velocity v(t)=3t2−36t−84v(t)=3 t^{2}-36 t-84v(t)=3t2−36t−84, is decreasing for t≥0t \geq 0t≥0.

Problem 24124

Find the volume using the shell method by revolving the shaded area defined by y=6y = \sqrt{6}y=6​ and x=3y2x = 3y^2x=3y2 around the xxx-axis. Set up the integral for the volume.

Problem 24125

Find the limit: lim⁡x→0+1−cos⁡xx2sin⁡x\lim _{x \rightarrow 0^{+}} \frac{1-\cos x}{x^{2} \sin x}limx→0+​x2sinx1−cosx​

Problem 24126

Find the position of a particle at time t=8t=8t=8 given its velocity v(t)=π2sin⁡(π6t)v(t)=\frac{\pi}{2} \sin \left(\frac{\pi}{6} t\right)v(t)=2π​sin(6π​t) and x=−6x=-6x=−6 at t=12t=12t=12.

Problem 24127

Find the slope of the curve defined by x=−6t3+4x=-6 t^{3}+4x=−6t3+4 and y=2t2y=2 t^{2}y=2t2 at t=5t=5t=5. Slope =□=\square=□.

Problem 24128

Find d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ at the point (2,−5)(2,-5)(2,−5) given −5x2=5−y2-5 x^{2}=5-y^{2}−5x2=5−y2.

Problem 24129

Find the position of a particle at time t=0t=0t=0 given v(t)=2πsin⁡(πt)v(t)=2 \pi \sin (\pi t)v(t)=2πsin(πt) and x=4x=4x=4 at t=12t=\frac{1}{2}t=21​.

Problem 24130

Find d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ at the point (−1,2)(-1,2)(−1,2) given −y=−y2−2x3-y=-y^{2}-2 x^{3}−y=−y2−2x3.

Problem 24131

Find the area of the shaded region using two integrals with respect to yyy for y=3−xy = 3 - xy=3−x and y=x+1y = x + 1y=x+1.

Problem 24132

Find the limit: lim⁡x→∞(1+2x)12ln⁡x\lim _{x \rightarrow \infty}(1+2 x)^{\frac{1}{2 \ln x}}limx→∞​(1+2x)2lnx1​.

Problem 24133

Identify a convergence test for the series ∑k=1∞(−1)k(5+1k2)k\sum_{k=1}^{\infty}(-1)^{k}\left(5+\frac{1}{k^{2}}\right)^{k}∑k=1∞​(−1)k(5+k21​)k.

Problem 24134

Find when the particle at x(t)=3t3−9t2−72tx(t)=3 t^{3}-9 t^{2}-72 tx(t)=3t3−9t2−72t moves right for t≥0t \geq 0t≥0.

Problem 24135

Find d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ for 2x3+y+y2=02 x^{3}+y+y^{2}=02x3+y+y2=0 at the point (−1,−2)(-1,-2)(−1,−2).

Problem 24136

Find the volume when the region bounded by y=15−x,y=15,x=225y=15-\sqrt{x}, y=15, x=225y=15−x​,y=15,x=225 is revolved around y=15y=15y=15.

Problem 24137

Problem 24138

Skizzieren Sie einen Graphen der stetigen Funktion ggg auf [0;2][0 ; 2][0;2], wenn ∫02g(x)dx=0\int_{0}^{2} g(x) d x=0∫02​g(x)dx=0. Erklären Sie das Integral.

Problem 24139

Find the derivative dydx\frac{dy}{dx}dxdy​ of the function −42y+1-\frac{4}{2y+1}−2y+14​.

Problem 24140

Find the average rate of change of y=(−5x+15)1y = (-5x + 15)^{1}y=(−5x+15)1 on the interval [1,3][1,3][1,3] using the Mean Value Theorem.

Calculate the limit: $\lim _{x \rightarrow 0} \frac{10 x^{2}+x}{\tan x}$ and fill in the table values for $f(x)$ .

Find the volume of the solid formed by revolving the region defined by $y = 27\sin(x)\cos(x)$ from $x = 0$ to $x = \frac{\pi}{2}$ around the x-axis. Use $\pi$ as needed.

Find points on the curve $y=x^{3}+3x+6$ where the tangent is parallel to $5x-y=9$ . Are there multiple points?

Find the horizontal asymptote of the function $f(x)=\frac{x^{2}-2}{4x-x^{4}}$ .

Find the volume of the solid formed by revolving the shaded area defined by $3x + 4y = 12$ around the x-axis.

Find the integral for the volume of a cylindrical tank (length 15 ft, radius 5 ft) filled to a depth of 9 ft.
$v=\int_{-5}^{h} A(y) \, dy$

Find the integral for the volume of a horizontal cylinder tank (length 13 ft, radius 4 ft) filled to 6 ft deep.
$V=\int_{-2}^{2} A(x) \, dx$

Berechne die Anzahl der Lkw-Fahrten für Baumaterial eines Deichs mit $f(x) = \frac{5}{64} x^2 + 1$ , $\rho = 1.8 \, \text{g/cm}^3$ , Länge 12 m, und 20 t Kapazität.

Find the value of $f^{\prime}\left(\frac{\sqrt{21}}{5}\right)$ if $f(x)=\arcsin (x)$ .

Find the value of $f^{\prime}\left(\frac{1}{5}\right)$ if $f(x)=\sin^{-1}(x)$ .

Find the derivative of the function $f(x)=\tan^{-1}(2x)$ in its simplest form.

A ball is thrown down at $8 \mathrm{~m/s}$ from $30 \mathrm{~m}$ . How long until it hits the ground?

Show that the reproduction rate $r(C)$ approaches zero as sugar level $C$ approaches zero: $\lim_{C \rightarrow 0} r(C)=0$ .

Find $\frac{d y}{d x}$ in terms of $x$ and $y$ for the equation $-2-3 x^{2}+3 y^{2}=-x y$ .

Find the volume using the shell method for the region bounded by $y = \sqrt{6}$ , $x = 12 - 2y^2$ , and $x = 12$ . Answer in terms of $\pi$ .

Find the integral for the volume of a cylindrical gas tank (length 11 ft, radius 3 ft) filled to a depth of 4 ft.
$V=\int_{-3}^{1} A(x) \, dx$

Find when the speed of a particle, defined by $x(t)=t^{3}-27 t^{2}+51 t$ , is decreasing for $t \geq 0$ .

Show that the reproduction rate $r(C)$ approaches 0 as sugar level $C$ approaches 0: $\lim_{C \rightarrow 0} r(C) = 0$ .

Calculate the limit: $\lim _{x \rightarrow 0} \frac{4-\sqrt{16+x^{2}}}{2 x^{2}}$

Show that the function $f(x)$ is continuous at $x=-2$ : $f(x)=\frac{2x^2+3x-2}{x+2}$ for $x \neq -2$ , $f(-2)=-5$ .

Show that $\lim_{N \rightarrow 0} R(N) = 0$ using the formula $R(N) = N \ln \left(\frac{1}{N}\right)$ for $d=A=1$ . Complete the table for $N$ : 0.1, 0.01, 0.001.

Find when the acceleration of a particle, with position $x(t)=5 t^{4}-30 t^{2}$ , is positive for $t \geq 0$ .

Find the intervals where the speed of a particle, with velocity $v(t)=3 t^{2}-36 t-84$ , is decreasing for $t \geq 0$ .

Find the volume using the shell method by revolving the shaded area defined by $y = \sqrt{6}$ and $x = 3y^2$ around the $x$ -axis. Set up the integral for the volume.

Find the limit: $\lim _{x \rightarrow 0^{+}} \frac{1-\cos x}{x^{2} \sin x}$

Find the position of a particle at time $t=8$ given its velocity $v(t)=\frac{\pi}{2} \sin \left(\frac{\pi}{6} t\right)$ and $x=-6$ at $t=12$ .

Find the slope of the curve defined by $x=-6 t^{3}+4$ and $y=2 t^{2}$ at $t=5$ . Slope $=\square$ .

Find $\frac{d^{2} y}{d x^{2}}$ at the point $(2,-5)$ given $-5 x^{2}=5-y^{2}$ .

Find the position of a particle at time $t=0$ given $v(t)=2 \pi \sin (\pi t)$ and $x=4$ at $t=\frac{1}{2}$ .

Find $\frac{d^{2} y}{d x^{2}}$ at the point $(-1,2)$ given $-y=-y^{2}-2 x^{3}$ .

Find the area of the shaded region using two integrals with respect to $y$ for $y = 3 - x$ and $y = x + 1$ .

Find the limit: $\lim _{x \rightarrow \infty}(1+2 x)^{\frac{1}{2 \ln x}}$ .

Identify a convergence test for the series $\sum_{k=1}^{\infty}(-1)^{k}\left(5+\frac{1}{k^{2}}\right)^{k}$ .

Find when the particle at $x(t)=3 t^{3}-9 t^{2}-72 t$ moves right for $t \geq 0$ .

Find $\frac{d^{2} y}{d x^{2}}$ for $2 x^{3}+y+y^{2}=0$ at the point $(-1,-2)$ .

Find the volume when the region bounded by $y=15-\sqrt{x}, y=15, x=225$ is revolved around $y=15$ .

Skizzieren Sie einen Graphen der stetigen Funktion $g$ auf $[0 ; 2]$ , wenn $\int_{0}^{2} g(x) d x=0$ . Erklären Sie das Integral.

Find the derivative $\frac{dy}{dx}$ of the function $-\frac{4}{2y+1}$ .

Find the average rate of change of $y = (-5x + 15)^{1}$ on the interval $[1,3]$ using the Mean Value Theorem.

Zeigen Sie, dass die Tangente von $f$ an $x_{0}$ parallel zur $x$ -Achse ist. a) $f(x)=x^{3}-4x^{2}+5x; x_{0}=1$ b) $f(x)=x^{3}-6x^{2}+12x; x_{0}=2$

Find $p$ such that $\int_{1}^{5} p f(x) dx = 83$ , where $f(x)=3x-2$ for $x \leq 2$ and $f(x)=x^2$ for $x > 2$ .

Find values of $t$ in $[0, 2]$ where instantaneous velocity $x'(t)$ equals average velocity $\frac{x(2)-x(0)}{2-0}$ .

Given $f^{\prime \prime}(x)>0$ , which could be $f^{\prime}(5)$ : (A) 0.5, (B) 0.7, (C) 0.9, (D) 1.1, (E) 1.3?

Berechne $a$ so, dass $\int_{1}^{a} \left( \frac{1}{3} x^{2}+x \right) dx=\frac{8}{9}$ . Skizziere einen Graphen für $g(x)$ mit $\int_{0}^{2} g(x) dx=0$ .

Bestimme den Grenzwert von $P(x)=\frac{x^{3}-2x+1}{x-1}$ mit der $\mathrm{H}$ -Methode.

Find the Riemann sum formula for $f(x)=4x^{3}$ on $[0,4]$ using right endpoints, then take the limit as $n \to \infty$ .

Bestimme den Grenzwert von $f(x)=\frac{x^{3}-2 x+1}{x-1}$ mit der H-Methode.

Find the limit of the sequence $a_{n}=\frac{\sqrt{n}-n}{2 n}$ as $n$ approaches infinity. Answer: $\lim _{n \rightarrow \infty} a_{n}=\square$ .

Untersuchen Sie die Funktionen $a(x)=e^{x}+e^{-x}$ , $b(x)=e^{x} \cdot e^{-x}$ und $c(x)=\frac{e^{x}}{e^{-x}}$ auf Asymptoten und Symmetrien.

Find the limit of the sequence $a_{n}=\cos \left(\frac{13}{n+14}\right)$ as $n$ approaches infinity. What is the value?

Find the limit of the sequence $a_{n}=5\left(-\frac{1}{2}\right)^{n}$ . Answer: $\lim _{n \rightarrow \infty} 5\left(-\frac{1}{2}\right)^{n}=\square$

Find the limit of the sequence $a_{n}=\frac{2 \sin (n)}{n}$ as $n$ approaches infinity. What is $\lim _{n \rightarrow \infty} \frac{2 \sin (n)}{n}=\square$ ?

Does the sequence $a_{n}=4 e^{\left(\frac{1}{n-7}\right)}$ converge? If yes, find the limit; if no, enter $\varnothing$ .

Does the series $\sum_{n=1}^{\infty}(8 \sqrt{n+1}-8 \sqrt{n})$ converge or diverge?

Check if the sequence $a_{n}=\left(\frac{4}{7}\right)^{n}$ converges using the Monotone Convergence Theorem.

Find the first five partial sums $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ of the series $\sum_{n=1}^{\infty} \frac{10}{3^{n}-18}$ . Round to the nearest thousandth.

Determine if the series $\sum_{n=1}^{\infty}(-8)^{4 n}(-9)^{1-n}$ converges or diverges. Choose: converges or diverges.

Find the sum of the series: $\sum_{n=1}^{\infty} e^{2 n}(-10)^{1-n}$ as a fraction in terms of $e$ .

Find the sum of the series: $\sum_{n=1}^{\infty}\left(\frac{5}{\sqrt{n+7}}-\frac{5}{\sqrt{n+8}}\right)$ . If it diverges, answer with $\varnothing$ .

Find the sum of the series: $\sum_{n=1}^{\infty} 2^{4 n} 20^{1-n}$ .
Answer: $\sum_{n=1}^{\infty} 2^{4 n} 20^{1-n}=\square$

Apply the divergence test to the series $\sum_{n=1}^{\infty}\left[\frac{(n+7) !}{(n+9) !}\right]$ . What is the result?

Check if the sequence $a_{n}=\frac{1-7 n^{2}}{3 n^{4}-2}$ converges and find its limit or state $\varnothing$ if it diverges.

Use the divergence test on the series $\sum_{n=1}^{\infty}(4 \sin^{2}(n-3))$ and choose the correct conclusion.

Calculate the limit: $\lim _{n \rightarrow \infty}\left(\frac{-3 n^{4}-6}{\sqrt{n^{4}+3 n^{3}-4}}\right)$ . If it doesn't exist, enter $\varnothing$ .

Find the sum of the series $\sum_{n=1}^{\infty}\left(\frac{17}{8^{n}}-\frac{15}{3^{n}}\right)$ using given sums.

Determine if the series $\sum_{n=1}^{\infty} \frac{5}{e^{n / 2}}$ converges or diverges using the integral test.

Find the value of $p$ for the series $\sum_{n=1}^{\infty} \frac{\sqrt[4]{n}}{3 \sqrt{n}}$ . Provide your answer: $p=$

Does the series $3+\frac{6}{(2)^{5 / 4}}+\frac{9}{(3)^{5 / 4}}+\cdots$ converge or diverge?

Use the integral test to check if the series $\sum_{n=1}^{\infty} \frac{n+5}{n+4}$ converges or diverges.