Calculus

Problem 25101

Calculate the satellite's velocity for a $150 \mathrm{~km}$ orbit and the time to return to the cannon in seconds and minutes.

See Solution

Problem 25102

Find the critical numbers of the function $f(x)=(x^{2}-4)^{\frac{2}{3}}$ .

See Solution

Problem 25103

Woosterium-21 has a half-life of 3 hours. If you start with 100 mg, how much remains after 10 hours? Also, find the time to decay to 1/3 of its original amount. Round both answers to three decimal places.

See Solution

Problem 25104

Find the average velocity of a train with location $s(t)=\frac{50}{t}$ from $t=3$ to $t=8$ (in km/h).

See Solution

Problem 25105

Find the derivative of the Phillips curve equation $y=9.638 x^{-1.394}-0.900$ at $x=2\%$ and $x=7\%$ . Round to two decimal places. Interpret the results.

See Solution

Problem 25106

Find the critical numbers of the function $h(x) = \frac{x-3}{x+7}$ .

See Solution

Problem 25107

Linearize the function $f(x) = \sqrt{x}$ at the point $a = 9$ .

See Solution

Problem 25108

Find the absolute max and min of $f(x)=x^{3}+3x^{2}-9x$ on the interval $[-4,4]$ .

See Solution

Problem 25109

Define $f(x)$ piecewise and find $f(3)$ so that $f(3)=\lim_{x \rightarrow 3} f(x)$ . Compute limits as $x \to 3^{+}$ and $x \to 3^{-}$ . Also, find $f(1)$ for $\lim_{x \rightarrow 1} f(x)$ .

See Solution

Problem 25110

Given the function $f(x)=0.831 x^{2}-18.1 x+137.3$ for $10 \leq x \leq 16$ , find $f(12)$ and $f(16)$ , and their derivatives. Interpret results.

See Solution

Problem 25111

A 941-g mass on a spring ( $k = 23.6 \mathrm{~N/m}$ ) is stretched $2.1 \mathrm{~cm}$ . Find its position at $t=13.8 \mathrm{~s}$ .

See Solution

Problem 25112

A train's position is $s(t)=\frac{50}{t}$ for $3 \leq t \leq 8$ . Find its velocity at $t=4$ .

See Solution

Problem 25113

A bank offers $9\%$ continuous compounding. Find time for a deposit to: (a) quadruple, (b) increase by $55\%$ .

See Solution

Problem 25114

Gegeben ist die Funktion $f(x)=(x-6) \cdot \sqrt{x}$ . Finde Nullstellen, Ableitung, Extremum, Steigungswinkel bei $x=6$ und skizziere den Graphen.

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Problem 25115

Approximate the area under $f(x)=\sqrt{x}$ from $a=5$ to $b=8$ using 6 rectangles. Then find the exact area.

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Problem 25116

Estimate the integral $\int_{3}^{6} x e^{-\frac{x}{2}} d x$ using three methods: trapezoidal, midpoint, and Simpson's rule. Find the exact value using the antiderivative.

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Problem 25117

Find the limit as $x$ approaches 0 for $\mathcal{R} x$ and analyze $f(x)$ for $v \rightarrow 1^{n}$ in the intervals $(-3,1]$ and $[1,\infty)$ with $mx + b$ .

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Problem 25118

A craftsman makes a cylindrical jewelry box with volume $V=50$ in³.
a. Objective function: $C=250\left(pi r^2\right)+10(2 pi h r)+10\left(pi r^2\right)$
b. Constraint: $50=pi r^2 h$ Rewrite $h$ in terms of $r$ : $h=50/(pi r^2)$
c. Rewrite $C$ in terms of $r$ . d. Differentiate $C$ with respect to $r$ : $\frac{dC}{dr}=$
e. Find $r$ for potential extreme point of $C$ : $r=$
f. Find height $h$ : $h=$ $$

See Solution

Problem 25119

Find the increase rates for mp3 player sales and profit based on the functions $f(t)=0.4 t^{2}+2 t+3$ and $g(x)=0.02 x^{2}+5 x+40$ .

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Problem 25120

Find the derivative of $y=(f(x)+15 g(x)) g(x)$ where $f(x)$ and $g(x)$ are differentiable at $x$ .

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Problem 25121

Find the midpoint Riemann sum for $\int_{2}^{3.5} 3^{x} dx$ using 6 equal subintervals. Round to the nearest thousandth.

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Problem 25122

Integrate using substitution: $\int (x^{3}+4)^{8} 3 x^{2} dx$ . Find $u=x^{3}+4$ and transform the integral.

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Problem 25123

Find the arc length of the curve $y=\frac{1}{8} \ln x-x^{2}$ from $x=1$ to $x=3$ .

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Problem 25124

Zbadać zbieżność szeregów: (a) $\sum_{n=1}^{\infty} \frac{1}{n \sqrt[n]{n}}$ , (b) $\sum_{n=1}^{\infty} n(\sqrt{n^{2}+1}-\sqrt{n^{2}-1})$ , (c) $\sum_{n=1}^{\infty} \frac{1}{n}(\sqrt{n+1}-\sqrt{n})$ , (d) $\sum_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^{n(n+1)}$ , (e) $\sum_{n=1}^{\infty} \frac{\left(\frac{n+1}{n}\right)^{n^{2}}}{3^{n}}$ .

See Solution

Problem 25125

Zbadać zbieżność szeregu: $\sum_{n=1}^{\infty} \frac{1}{n \sqrt[n]{n}}$ .

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Problem 25126

Find how fast the volume, surface area, and diagonal length $s=\sqrt{x^{2}+y^{2}+z^{2}}$ change when $x=4, y=3, z=2$ . Rates: $\frac{d x}{d t}=1$ , $\frac{d y}{d t}=-2$ , $\frac{d z}{d t}=1$ .

See Solution

Problem 25127

Find the derivative of the function $f(x) = 2x^{2} + 5x + 3$ .

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Problem 25128

Find the antiderivative $F(x)$ if $f(x) = 2$ .

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Problem 25129

Find the limit: $\lim _{x \rightarrow 1} \frac{\ln (5-4 x)+e^{2 x-2}-1}{\frac{4}{\pi} \arctan (3 x-2)-x}$ .

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Problem 25130

Find the constant $c$ such that $F(x) = x^{5} + \frac{2}{x^{2}} + c$ matches $F(x) = \int 5e^{-3x} \, dx$ .

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Problem 25131

Berechne die Fahrstrecke $x$ (in $1000 \mathrm{~km}$ ), um die jährlichen Kosten minimal zu halten: $C(x) = 1450 + 150x + 22x \ln(x) + 12x^2$ .

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Problem 25132

Find the limits: $\lim_{t \to 12e} \ln\left(\frac{12}{t}\right)$ and $\lim_{x \to 5^+} \frac{2}{x-5}$ .

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Problem 25133

Solve the limits: 1) $\lim _{t \rightarrow 12 e} \ln \left(\frac{12}{t}\right)$ and 2) $\lim _{x \rightarrow 5^{+}} \frac{2}{x-5}$ .

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Problem 25134

Find the concavity regions for the function $y=\frac{2 x+7}{2 x}$ , where $x \neq 0$ .

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Problem 25135

Find values of $a, m$ , and $b$ so that the function $f(x)$ meets the Mean Value Theorem on $[0,2]$ .

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Problem 25136

Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for these functions: 1- $f(x, y)=\mathrm{e}^{x y^{3}+x}$ 2- $f(x, y)=\operatorname{Ln}(x+y)$ 3- $f(x, y)=(3 y+x y)^{4}$ 4- $f(x, y)=y \ln (x y)$ 5- $f(x, y)=y \cos (x y)$ .

See Solution

Problem 25137

Find $h(2)$ and $h'(2)$ for $h(x)=f(g(x))$ given tangents at $(2,7)$ and $(7,1)$ . Also, find the tangent line at $x=2$ .

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Problem 25138

Find the volume of the solid formed by revolving region $R$ around the line $x=2$ using the shell method, where $R$ is defined by $y=\frac{1}{1+x^{2}}$ , $y=0$ , and $0 \leq x \leq 2$ .

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Problem 25139

Zeigen Sie, dass der Differenzenquotient von $f(x)=2^{x}$ im Intervall $[a; a+2]$ gleich $\frac{3}{2} \cdot f(a)$ ist.

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Problem 25140

Evaluate the integral: $\int \frac{\ln \left(x^{2}+1\right)}{x^{2}} d x^{2}$

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Problem 25141

Which is an antiderivative of $f(x)=6 x^{2}-\sin (x)-\frac{1}{x+1}$ ? (a) (b) (c) (d) (e) None.

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Problem 25142

Find the limits: $\lim _{x \rightarrow 7}\left(\frac{x^{2}}{7}+\frac{7}{x^{2}}\right)$ and $\lim _{x \rightarrow 19} \frac{361-38 x+x^{2}}{19-x}$ .

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Problem 25143

Evaluate the integral: $\int \frac{\ln \left(x^{2}+1\right)}{x^{2}} d x$

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Problem 25144

Approximate the volume change of a sphere as radius goes from $70 \mathrm{ft}$ to $70.05 \mathrm{ft}$ . Find $\Delta V \approx \square \mathrm{ft}^{3}$ .

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Problem 25145

Express the change in $y$ for $f(x)=2x+4$ as $d y=f^{\prime}(x) d x$ : $d y=(\square) d x$ .

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Problem 25146

Find the limit: $\lim _{x \rightarrow 19} \frac{361 - 38x + x^{2}}{19 - x}$ .

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Problem 25147

Determine the convergence or divergence of these integrals: (1) $\int_{1}^{\infty} \frac{2}{x^{3/2}} dx$ , (2) $\int_{1}^{\infty} \frac{1}{x^{1/2}} dx$ , (3) $\int_{1}^{\infty} \frac{-2}{x^{5}} dx$ , (4) $\int_{1}^{\infty} \frac{2}{x^{4/5}} dx$ .

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Problem 25148

Find the relationship between a small change in $x$ and the change in $y$ for $f(x)=\cot 6x$ : $d y=\square d x$ .

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Problem 25149

Find the absolute maximum and minimum of $f(x)=\frac{\sqrt{x}}{27+x^{2}}$ for $0 \leq x \leq 5$ .

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Problem 25150

Find the relationship between a small change in $x$ and the change in $y$ for the function $f(x)=\frac{1}{x^{8}}$ : $d y=\square d x$ .

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Problem 25151

Find the approximate and exact average speed using $s(x)=\frac{3600}{60+x}$ for $x=6$ seconds.

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Problem 25152

Find the value of $r$ that minimizes the Morse potential $V(r) = e^{-r} - A e^{-a r}$ for $r > 0$ .

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Problem 25153

Check if the Mean Value Theorem applies to $f(x)=4 \sqrt{x}$ on $[1,81]$ . If yes, find the point(s) guaranteed by it.

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Problem 25154

Find the derivatives of these functions: a) $y=4 x^{3}+4 \pi^{3}-2 x^{2}+x$ , b) $g(t)=\sqrt[3]{t^{2}}+\frac{1}{t^{2}}$ , c) $h(x)=x^{2}\left(\sqrt{x}+\frac{1}{x}\right)$ .

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Problem 25155

Finde die Punkte, an denen der Graph von $f$ mit $f(x)=x^{5}$ eine Steigung von 80 hat.

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Problem 25156

Find the limits: $\lim _{x \rightarrow 0} \frac{4-\sqrt{16+x^{2}}}{2 x^{2}}$ and $\lim _{x \rightarrow \infty} \frac{2 e^{-x}}{7+8 e^{-x}}$ .

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Problem 25157

Find the population size at $t=4.1$ given a growth rate of $3\%$ and $N(4)=100$ . Use linear approximation.

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Problem 25158

Find the volume of the solid with square cross sections, bounded by $y=\sqrt{x^{2}+9}$ , $y=0$ , $x=0$ , and $x=4$ .

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Problem 25159

Find $c$ where $f'(c)=0$ for $f(x)=e^{1-x^{2}}$ and check if there's a local extremum at $x=c$ .

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Problem 25160

Find the limit as $x$ approaches -2 for $\frac{2 \cos (-4-2 x)+x}{3 \ln (2 x+5)}$ . Simplify your answer.

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Problem 25161

Déterminez le domaine de $f$ , les asymptotes verticales et horizontales, et les points critiques de $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ .

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Problem 25162

Find $\lim _{x \rightarrow-1} \frac{\ln (x+2)}{4-4 x^{2}}$ and express the answer in simplest form.

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Problem 25163

Déterminez le domaine de $f$ , les asymptotes verticales et horizontales de $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ .

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Problem 25164

Find $\lim _{x \rightarrow 3} \frac{\ln (x-2)}{\sin (6-2 x)}$ and simplify your answer.

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Problem 25165

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{3 e^{-x}-3}{3 \ln (1-x)+3 x}$ . Simplify your answer.

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Problem 25166

Find $\lim _{x \rightarrow \infty} \frac{6 x^{2}+4}{9 x+1-5 x^{2}}$ and $\lim _{x \rightarrow-\infty} \frac{4 x^{2}+x}{x+1}$ .

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Problem 25167

Find values of $a$ for convexity of $f(x, y, z)=x^{2}+y^{2}+z^{2}-x z+a y z$ . Then, for $a=1$ , find global minimum.

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Problem 25168

Take-home exam for MAT25: Create or solve problems using real data.
1. Drug decay: (a) Find half-life (min). (b) Model: amount after t hours. (c) Amount after 24 hours (g).
2. Roast cooling: (a) Model using Newton's Law. (b) Time to reach 125°F.
3. Rumor spread: (a) Initial people. (b) Carrying capacity. (c) Days to half capacity.

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Problem 25169

a) Trouvez le domaine de $f$ . b) Identifiez les asymptotes verticales. c) Identifiez les asymptotes horizontales. d) Trouvez les points critiques. e) Déterminez où $f$ est croissante. f) Déterminez où $f$ est décroissante. g) Trouvez les maximums locaux. h) Trouvez les minimums locaux.

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Problem 25170

Überprüfen Sie, ob $F$ eine Stammfunktion von $f$ ist für die folgenden Funktionen: a) $f(x)=x^{2} ; F(x)=\frac{1}{3} x^{3}+0,3$ b) $f(x)=3 x^{2}+x ; F(x)=x^{2} \cdot(x+0,5)$ c) $f(x)=0,1 x^{3} ; F(x)=0,25 x^{4}$ d) $f(x)=2 x^{5} ; F(x)=\frac{1}{3} x^{6}+\sqrt{5}$ e) $f(x)=5 x^{2} ; F(x)=1,\overline{3} x^{3}$ f) $f(x)=3 x+0,3 ; F(x)=1,5 x \cdot(0,2+x)$

See Solution

Problem 25171

Trouvez $f'(x)$ pour $f(x)=e^{-\frac{x^{2}}{2}}$ , les nombres critiques de $f$ , $f''(x)$ et les nombres critiques de $f'$ .

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Problem 25172

1) Trouver le domaine de $f$ . 2) Identifier les asymptotes verticales. 3) Identifier les asymptotes horizontales. 4) Trouver les points critiques. 5) Déterminer les intervalles où $f$ est croissante. 6) Déterminer les intervalles où $f$ est décroissante. 7) Trouver les maximums locaux de $f$ . 8) Trouver les minimums locaux de $f$ . 9) Identifier les intervalles de convexité de $f$ . 10) Identifier les intervalles de concavité de $f$ . 11) Trouver les points d'inflexion de $f$ . 12) Tracer la courbe de $f$ .

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Problem 25173

Déterminez le domaine de $f$ , les asymptotes, points critiques, intervalles croissants/décroissants, max/min locaux, convexité/concavité, points d'inflexion et tracez la courbe de $f$ .

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Problem 25174

The function $f(x)=x^{\frac{2}{3}}$ on $[-8,8]$ fails the Mean Value Theorem due to which reason? A. $f(0)$ undefined B. not continuous C. $f^{\prime}(-1)$ nonexistent D. not defined for $x<0$ E. $f^{\prime}(0)$ nonexistent

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Problem 25175

Find the volume of the solid with a base bounded by $y = \sqrt{x^{2}+9}$ , $y = 0$ , $x = 0$ , $x = 4$ , with semicircular cross sections.

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Problem 25176

Calculate the area under the standard normal curve from $z=0$ to $z=2.16$ .

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Problem 25177

Déterminez le domaine de définition de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ .

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Problem 25178

Find the $x$ -coordinates of the inflection points of $f(x)=x^{4}-18x^{2}$ . Critical points are $0, 3, -3$ . Options: A. $0,3,-3$ B. $3,-3$ C. $\sqrt{3},-\sqrt{3}$ D. $\sqrt{3}$ E. $0, \sqrt{3},-$

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Problem 25179

Find the volume of the solid formed by revolving the area under $y=6x-x^{2}$ in the first quadrant around the x-axis.

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Problem 25180

Déterminez les équations des asymptotes horizontales de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ .

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Problem 25181

Trouver les coordonnées des points critiques de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ .

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Problem 25182

Exercice 5 : Soit $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ . a) Trouver le domaine de $f$ . b) Trouver les asymptotes verticales.

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Problem 25183

Find the inflection points of $f(x)=x^{4}-18x^{2}$ . Options: A. $0,3,-3$ B. $3,-3$ C. $\sqrt{3},-\sqrt{3}$ D. $\sqrt{3}$ E. $0, \sqrt{3},-\sqrt{3}$ .

See Solution

Problem 25184

Déterminez les équations des asymptotes horizontales de $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ .

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Problem 25185

Trouver la valeur de $f(-1)$ pour la fonction $f(x)=e^{-\frac{x^{2}}{2}}$ .

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Problem 25186

Find the volume of the solid formed by revolving the area below $y=6x-x^{2}$ in the first quadrant around the $y$ -axis.

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Problem 25187

Déterminez les intervalles ouverts où la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ est croissante.

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Problem 25188

Find the limit: $\lim _{x \rightarrow 1}\left(\frac{x-1}{2}-\frac{1}{\ln x}\right)$ .

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Problem 25189

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

See Solution

Problem 25190

Déterminez les intervalles ouverts où la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ est décroissante.

See Solution

Problem 25191

Set up an integral for the volume of the solid formed by revolving the area below $y=6x-x^{2}$ in the first quadrant around $y=-2$ .

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Problem 25192

Find the limit as $x$ approaches 1 from the left: $\lim _{x \rightarrow 1^{-}}\left(\frac{x-1}{2}-\frac{1}{\ln x}\right)$ .

See Solution

Problem 25193

Find the average rate of change of $f(x)=x^{2}+3x-1$ over the interval $-3 \leq x \leq 4$ .

See Solution

Problem 25194

Set up an integral for the arc length of the curve $y=6x-x^{2}$ from $(0,0)$ to $(6,0)$ .

See Solution

Problem 25195

Déterminez les minimums locaux de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ en utilisant $f'(x)$ et $f''(x)$ .

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Problem 25196

Given the function $f(x)=2 x \sqrt{4 x^{2}+2}$ , find critical values and intervals where $f$ is increasing or decreasing.

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Problem 25197

Approximate displacement using $v=5t+6$ for $0 \leq t \leq 8$ with $n=2$ . Find the answer in $\mathrm{m}$ .

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Problem 25198

Find the volume of the solid formed by revolving region $R$ (bounded by $x=y^{2/3}$ and $y=x^{1/3}$ ) around $y=0$ using the shell method.

See Solution

Problem 25199

Déterminez les maximums locaux de la fonction $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ en utilisant $f^{\prime}(x)$ et $f^{\prime \prime}(x)$ .

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Problem 25200

16. (Avec calculatrice) Trouvez :
a) $\left.\frac{d Q}{d K}\right|_{t=\frac{\pi}{2}}$ et son interprétation. b) Moments où $\frac{d K}{d t} = 0$ . c) Taux de variation de $Q$ à $t=\frac{3 \pi}{2}$ .

See Solution

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Calculus

Problem 25101

Calculate the satellite's velocity for a 150 km150 \mathrm{~km}150 km orbit and the time to return to the cannon in seconds and minutes.

Problem 25102

Find the critical numbers of the function f(x)=(x2−4)23f(x)=(x^{2}-4)^{\frac{2}{3}}f(x)=(x2−4)32​.

Problem 25103

Woosterium-21 has a half-life of 3 hours. If you start with 100 mg, how much remains after 10 hours? Also, find the time to decay to 1/3 of its original amount. Round both answers to three decimal places.

Problem 25104

Find the average velocity of a train with location s(t)=50ts(t)=\frac{50}{t}s(t)=t50​ from t=3t=3t=3 to t=8t=8t=8 (in km/h).

Problem 25105

Find the derivative of the Phillips curve equation y=9.638x−1.394−0.900y=9.638 x^{-1.394}-0.900y=9.638x−1.394−0.900 at x=2%x=2\%x=2% and x=7%x=7\%x=7%. Round to two decimal places. Interpret the results.

Problem 25106

Find the critical numbers of the function h(x)=x−3x+7h(x) = \frac{x-3}{x+7}h(x)=x+7x−3​.

Problem 25107

Linearize the function f(x)=xf(x) = \sqrt{x}f(x)=x​ at the point a=9a = 9a=9.

Problem 25108

Find the absolute max and min of f(x)=x3+3x2−9xf(x)=x^{3}+3x^{2}-9xf(x)=x3+3x2−9x on the interval [−4,4][-4,4][−4,4].

Problem 25109

Define f(x)f(x)f(x) piecewise and find f(3)f(3)f(3) so that f(3)=lim⁡x→3f(x)f(3)=\lim_{x \rightarrow 3} f(x)f(3)=limx→3​f(x). Compute limits as x→3+x \to 3^{+}x→3+ and x→3−x \to 3^{-}x→3−. Also, find f(1)f(1)f(1) for lim⁡x→1f(x)\lim_{x \rightarrow 1} f(x)limx→1​f(x).

Problem 25110

Given the function f(x)=0.831x2−18.1x+137.3f(x)=0.831 x^{2}-18.1 x+137.3f(x)=0.831x2−18.1x+137.3 for 10≤x≤1610 \leq x \leq 1610≤x≤16, find f(12)f(12)f(12) and f(16)f(16)f(16), and their derivatives. Interpret results.

Problem 25111

A 941-g mass on a spring (k=23.6 N/mk = 23.6 \mathrm{~N/m}k=23.6 N/m) is stretched 2.1 cm2.1 \mathrm{~cm}2.1 cm. Find its position at t=13.8 st=13.8 \mathrm{~s}t=13.8 s.

Problem 25112

A train's position is s(t)=50ts(t)=\frac{50}{t}s(t)=t50​ for 3≤t≤83 \leq t \leq 83≤t≤8. Find its velocity at t=4t=4t=4.

Problem 25113

A bank offers 9%9\%9% continuous compounding. Find time for a deposit to: (a) quadruple, (b) increase by 55%55\%55%.

Problem 25114

Gegeben ist die Funktion f(x)=(x−6)⋅xf(x)=(x-6) \cdot \sqrt{x}f(x)=(x−6)⋅x​. Finde Nullstellen, Ableitung, Extremum, Steigungswinkel bei x=6x=6x=6 und skizziere den Graphen.

Problem 25115

Approximate the area under f(x)=xf(x)=\sqrt{x}f(x)=x​ from a=5a=5a=5 to b=8b=8b=8 using 6 rectangles. Then find the exact area.

Problem 25116

Estimate the integral ∫36xe−x2dx\int_{3}^{6} x e^{-\frac{x}{2}} d x∫36​xe−2x​dx using three methods: trapezoidal, midpoint, and Simpson's rule. Find the exact value using the antiderivative.

Problem 25117

Find the limit as xxx approaches 0 for Rx\mathcal{R} xRx and analyze f(x)f(x)f(x) for v→1nv \rightarrow 1^{n}v→1n in the intervals (−3,1](-3,1](−3,1] and [1,∞)[1,\infty)[1,∞) with mx+bmx + bmx+b.

Problem 25118

Problem 25119

Find the increase rates for mp3 player sales and profit based on the functions f(t)=0.4t2+2t+3f(t)=0.4 t^{2}+2 t+3f(t)=0.4t2+2t+3 and g(x)=0.02x2+5x+40g(x)=0.02 x^{2}+5 x+40g(x)=0.02x2+5x+40.

Problem 25120

Find the derivative of y=(f(x)+15g(x))g(x)y=(f(x)+15 g(x)) g(x)y=(f(x)+15g(x))g(x) where f(x)f(x)f(x) and g(x)g(x)g(x) are differentiable at xxx.

Problem 25121

Find the midpoint Riemann sum for ∫23.53xdx\int_{2}^{3.5} 3^{x} dx∫23.5​3xdx using 6 equal subintervals. Round to the nearest thousandth.

Problem 25122

Integrate using substitution: ∫(x3+4)83x2dx\int (x^{3}+4)^{8} 3 x^{2} dx∫(x3+4)83x2dx. Find u=x3+4u=x^{3}+4u=x3+4 and transform the integral.

Problem 25123

Find the arc length of the curve y=18ln⁡x−x2y=\frac{1}{8} \ln x-x^{2}y=81​lnx−x2 from x=1x=1x=1 to x=3x=3x=3.

Problem 25124

Problem 25125

Zbadać zbieżność szeregu: ∑n=1∞1nnn\sum_{n=1}^{\infty} \frac{1}{n \sqrt[n]{n}}∑n=1∞​nnn​1​.

Problem 25126

Find how fast the volume, surface area, and diagonal length s=x2+y2+z2s=\sqrt{x^{2}+y^{2}+z^{2}}s=x2+y2+z2​ change when x=4,y=3,z=2x=4, y=3, z=2x=4,y=3,z=2. Rates: dxdt=1\frac{d x}{d t}=1dtdx​=1, dydt=−2\frac{d y}{d t}=-2dtdy​=−2, dzdt=1\frac{d z}{d t}=1dtdz​=1.

Problem 25127

Find the derivative of the function f(x)=2x2+5x+3f(x) = 2x^{2} + 5x + 3f(x)=2x2+5x+3.

Problem 25128

Find the antiderivative F(x) F(x) F(x) if f(x)=2 f(x) = 2 f(x)=2.

Problem 25129

Find the limit: lim⁡x→1ln⁡(5−4x)+e2x−2−14πarctan⁡(3x−2)−x\lim _{x \rightarrow 1} \frac{\ln (5-4 x)+e^{2 x-2}-1}{\frac{4}{\pi} \arctan (3 x-2)-x}limx→1​π4​arctan(3x−2)−xln(5−4x)+e2x−2−1​.

Problem 25130

Find the constant c c c such that F(x)=x5+2x2+c F(x) = x^{5} + \frac{2}{x^{2}} + c F(x)=x5+x22​+c matches F(x)=∫5e−3x dx F(x) = \int 5e^{-3x} \, dx F(x)=∫5e−3xdx.

Problem 25131

Berechne die Fahrstrecke xxx (in 1000 km1000 \mathrm{~km}1000 km), um die jährlichen Kosten minimal zu halten: C(x)=1450+150x+22xln⁡(x)+12x2C(x) = 1450 + 150x + 22x \ln(x) + 12x^2C(x)=1450+150x+22xln(x)+12x2.

Problem 25132

Find the limits: lim⁡t→12eln⁡(12t)\lim_{t \to 12e} \ln\left(\frac{12}{t}\right)limt→12e​ln(t12​) and lim⁡x→5+2x−5\lim_{x \to 5^+} \frac{2}{x-5}limx→5+​x−52​.

Problem 25133

Solve the limits: 1) lim⁡t→12eln⁡(12t)\lim _{t \rightarrow 12 e} \ln \left(\frac{12}{t}\right)limt→12e​ln(t12​) and 2) lim⁡x→5+2x−5\lim _{x \rightarrow 5^{+}} \frac{2}{x-5}limx→5+​x−52​.

Problem 25134

Find the concavity regions for the function y=2x+72xy=\frac{2 x+7}{2 x}y=2x2x+7​, where x≠0x \neq 0x=0.

Problem 25135

Find values of a,ma, ma,m, and bbb so that the function f(x)f(x)f(x) meets the Mean Value Theorem on [0,2][0,2][0,2].

Problem 25136

Problem 25137

Find h(2)h(2)h(2) and h′(2)h'(2)h′(2) for h(x)=f(g(x))h(x)=f(g(x))h(x)=f(g(x)) given tangents at (2,7)(2,7)(2,7) and (7,1)(7,1)(7,1). Also, find the tangent line at x=2x=2x=2.

Problem 25138

Find the volume of the solid formed by revolving region RRR around the line x=2x=2x=2 using the shell method, where RRR is defined by y=11+x2y=\frac{1}{1+x^{2}}y=1+x21​, y=0y=0y=0, and 0≤x≤20 \leq x \leq 20≤x≤2.

Problem 25139

Zeigen Sie, dass der Differenzenquotient von f(x)=2xf(x)=2^{x}f(x)=2x im Intervall [a;a+2][a; a+2][a;a+2] gleich 32⋅f(a)\frac{3}{2} \cdot f(a)23​⋅f(a) ist.

Problem 25140

Evaluate the integral: ∫ln⁡(x2+1)x2dx2\int \frac{\ln \left(x^{2}+1\right)}{x^{2}} d x^{2}∫x2ln(x2+1)​dx2

Problem 25141

Which is an antiderivative of f(x)=6x2−sin⁡(x)−1x+1f(x)=6 x^{2}-\sin (x)-\frac{1}{x+1}f(x)=6x2−sin(x)−x+11​? (a) (b) (c) (d) (e) None.

Calculate the satellite's velocity for a $150 \mathrm{~km}$ orbit and the time to return to the cannon in seconds and minutes.

Find the critical numbers of the function $f(x)=(x^{2}-4)^{\frac{2}{3}}$ .

Find the average velocity of a train with location $s(t)=\frac{50}{t}$ from $t=3$ to $t=8$ (in km/h).

Find the derivative of the Phillips curve equation $y=9.638 x^{-1.394}-0.900$ at $x=2\%$ and $x=7\%$ . Round to two decimal places. Interpret the results.

Find the critical numbers of the function $h(x) = \frac{x-3}{x+7}$ .

Linearize the function $f(x) = \sqrt{x}$ at the point $a = 9$ .

Find the absolute max and min of $f(x)=x^{3}+3x^{2}-9x$ on the interval $[-4,4]$ .

Define $f(x)$ piecewise and find $f(3)$ so that $f(3)=\lim_{x \rightarrow 3} f(x)$ . Compute limits as $x \to 3^{+}$ and $x \to 3^{-}$ . Also, find $f(1)$ for $\lim_{x \rightarrow 1} f(x)$ .

Given the function $f(x)=0.831 x^{2}-18.1 x+137.3$ for $10 \leq x \leq 16$ , find $f(12)$ and $f(16)$ , and their derivatives. Interpret results.

A 941-g mass on a spring ( $k = 23.6 \mathrm{~N/m}$ ) is stretched $2.1 \mathrm{~cm}$ . Find its position at $t=13.8 \mathrm{~s}$ .

A train's position is $s(t)=\frac{50}{t}$ for $3 \leq t \leq 8$ . Find its velocity at $t=4$ .

A bank offers $9\%$ continuous compounding. Find time for a deposit to: (a) quadruple, (b) increase by $55\%$ .

Gegeben ist die Funktion $f(x)=(x-6) \cdot \sqrt{x}$ . Finde Nullstellen, Ableitung, Extremum, Steigungswinkel bei $x=6$ und skizziere den Graphen.

Approximate the area under $f(x)=\sqrt{x}$ from $a=5$ to $b=8$ using 6 rectangles. Then find the exact area.

Estimate the integral $\int_{3}^{6} x e^{-\frac{x}{2}} d x$ using three methods: trapezoidal, midpoint, and Simpson's rule. Find the exact value using the antiderivative.

Find the limit as $x$ approaches 0 for $\mathcal{R} x$ and analyze $f(x)$ for $v \rightarrow 1^{n}$ in the intervals $(-3,1]$ and $[1,\infty)$ with $mx + b$ .

Find the increase rates for mp3 player sales and profit based on the functions $f(t)=0.4 t^{2}+2 t+3$ and $g(x)=0.02 x^{2}+5 x+40$ .

Find the derivative of $y=(f(x)+15 g(x)) g(x)$ where $f(x)$ and $g(x)$ are differentiable at $x$ .

Find the midpoint Riemann sum for $\int_{2}^{3.5} 3^{x} dx$ using 6 equal subintervals. Round to the nearest thousandth.

Integrate using substitution: $\int (x^{3}+4)^{8} 3 x^{2} dx$ . Find $u=x^{3}+4$ and transform the integral.

Find the arc length of the curve $y=\frac{1}{8} \ln x-x^{2}$ from $x=1$ to $x=3$ .

Zbadać zbieżność szeregu: $\sum_{n=1}^{\infty} \frac{1}{n \sqrt[n]{n}}$ .

Find how fast the volume, surface area, and diagonal length $s=\sqrt{x^{2}+y^{2}+z^{2}}$ change when $x=4, y=3, z=2$ . Rates: $\frac{d x}{d t}=1$ , $\frac{d y}{d t}=-2$ , $\frac{d z}{d t}=1$ .

Find the derivative of the function $f(x) = 2x^{2} + 5x + 3$ .

Find the antiderivative $F(x)$ if $f(x) = 2$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{\ln (5-4 x)+e^{2 x-2}-1}{\frac{4}{\pi} \arctan (3 x-2)-x}$ .

Find the constant $c$ such that $F(x) = x^{5} + \frac{2}{x^{2}} + c$ matches $F(x) = \int 5e^{-3x} \, dx$ .

Berechne die Fahrstrecke $x$ (in $1000 \mathrm{~km}$ ), um die jährlichen Kosten minimal zu halten: $C(x) = 1450 + 150x + 22x \ln(x) + 12x^2$ .

Find the limits: $\lim_{t \to 12e} \ln\left(\frac{12}{t}\right)$ and $\lim_{x \to 5^+} \frac{2}{x-5}$ .

Solve the limits: 1) $\lim _{t \rightarrow 12 e} \ln \left(\frac{12}{t}\right)$ and 2) $\lim _{x \rightarrow 5^{+}} \frac{2}{x-5}$ .

Find the concavity regions for the function $y=\frac{2 x+7}{2 x}$ , where $x \neq 0$ .

Find values of $a, m$ , and $b$ so that the function $f(x)$ meets the Mean Value Theorem on $[0,2]$ .

Find $h(2)$ and $h'(2)$ for $h(x)=f(g(x))$ given tangents at $(2,7)$ and $(7,1)$ . Also, find the tangent line at $x=2$ .

Find the volume of the solid formed by revolving region $R$ around the line $x=2$ using the shell method, where $R$ is defined by $y=\frac{1}{1+x^{2}}$ , $y=0$ , and $0 \leq x \leq 2$ .

Zeigen Sie, dass der Differenzenquotient von $f(x)=2^{x}$ im Intervall $[a; a+2]$ gleich $\frac{3}{2} \cdot f(a)$ ist.

Evaluate the integral: $\int \frac{\ln \left(x^{2}+1\right)}{x^{2}} d x^{2}$

Which is an antiderivative of $f(x)=6 x^{2}-\sin (x)-\frac{1}{x+1}$ ? (a) (b) (c) (d) (e) None.

Find the limits: $\lim _{x \rightarrow 7}\left(\frac{x^{2}}{7}+\frac{7}{x^{2}}\right)$ and $\lim _{x \rightarrow 19} \frac{361-38 x+x^{2}}{19-x}$ .

Evaluate the integral: $\int \frac{\ln \left(x^{2}+1\right)}{x^{2}} d x$

Approximate the volume change of a sphere as radius goes from $70 \mathrm{ft}$ to $70.05 \mathrm{ft}$ . Find $\Delta V \approx \square \mathrm{ft}^{3}$ .

Express the change in $y$ for $f(x)=2x+4$ as $d y=f^{\prime}(x) d x$ : $d y=(\square) d x$ .

Find the limit: $\lim _{x \rightarrow 19} \frac{361 - 38x + x^{2}}{19 - x}$ .

Find the relationship between a small change in $x$ and the change in $y$ for $f(x)=\cot 6x$ : $d y=\square d x$ .

Find the absolute maximum and minimum of $f(x)=\frac{\sqrt{x}}{27+x^{2}}$ for $0 \leq x \leq 5$ .

Find the relationship between a small change in $x$ and the change in $y$ for the function $f(x)=\frac{1}{x^{8}}$ : $d y=\square d x$ .

Find the approximate and exact average speed using $s(x)=\frac{3600}{60+x}$ for $x=6$ seconds.

Find the value of $r$ that minimizes the Morse potential $V(r) = e^{-r} - A e^{-a r}$ for $r > 0$ .

Check if the Mean Value Theorem applies to $f(x)=4 \sqrt{x}$ on $[1,81]$ . If yes, find the point(s) guaranteed by it.

Find the derivatives of these functions: a) $y=4 x^{3}+4 \pi^{3}-2 x^{2}+x$ , b) $g(t)=\sqrt[3]{t^{2}}+\frac{1}{t^{2}}$ , c) $h(x)=x^{2}\left(\sqrt{x}+\frac{1}{x}\right)$ .

Finde die Punkte, an denen der Graph von $f$ mit $f(x)=x^{5}$ eine Steigung von 80 hat.

Find the limits: $\lim _{x \rightarrow 0} \frac{4-\sqrt{16+x^{2}}}{2 x^{2}}$ and $\lim _{x \rightarrow \infty} \frac{2 e^{-x}}{7+8 e^{-x}}$ .

Find the population size at $t=4.1$ given a growth rate of $3\%$ and $N(4)=100$ . Use linear approximation.

Find the volume of the solid with square cross sections, bounded by $y=\sqrt{x^{2}+9}$ , $y=0$ , $x=0$ , and $x=4$ .

Find $c$ where $f'(c)=0$ for $f(x)=e^{1-x^{2}}$ and check if there's a local extremum at $x=c$ .

Find the limit as $x$ approaches -2 for $\frac{2 \cos (-4-2 x)+x}{3 \ln (2 x+5)}$ . Simplify your answer.

Déterminez le domaine de $f$ , les asymptotes verticales et horizontales, et les points critiques de $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ .

Find $\lim _{x \rightarrow-1} \frac{\ln (x+2)}{4-4 x^{2}}$ and express the answer in simplest form.

Déterminez le domaine de $f$ , les asymptotes verticales et horizontales de $f(x)=\frac{-3 x^{2}}{x^{2}-4}$ .

Find $\lim _{x \rightarrow 3} \frac{\ln (x-2)}{\sin (6-2 x)}$ and simplify your answer.

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{3 e^{-x}-3}{3 \ln (1-x)+3 x}$ . Simplify your answer.

Find $\lim _{x \rightarrow \infty} \frac{6 x^{2}+4}{9 x+1-5 x^{2}}$ and $\lim _{x \rightarrow-\infty} \frac{4 x^{2}+x}{x+1}$ .

Find values of $a$ for convexity of $f(x, y, z)=x^{2}+y^{2}+z^{2}-x z+a y z$ . Then, for $a=1$ , find global minimum.

a) Trouvez le domaine de $f$ . b) Identifiez les asymptotes verticales. c) Identifiez les asymptotes horizontales. d) Trouvez les points critiques. e) Déterminez où $f$ est croissante. f) Déterminez où $f$ est décroissante. g) Trouvez les maximums locaux. h) Trouvez les minimums locaux.

Trouvez $f'(x)$ pour $f(x)=e^{-\frac{x^{2}}{2}}$ , les nombres critiques de $f$ , $f''(x)$ et les nombres critiques de $f'$ .

Déterminez le domaine de $f$ , les asymptotes, points critiques, intervalles croissants/décroissants, max/min locaux, convexité/concavité, points d'inflexion et tracez la courbe de $f$ .

The function $f(x)=x^{\frac{2}{3}}$ on $[-8,8]$ fails the Mean Value Theorem due to which reason? A. $f(0)$ undefined B. not continuous C. $f^{\prime}(-1)$ nonexistent D. not defined for $x<0$ E. $f^{\prime}(0)$ nonexistent