Calculus

Problem 12201

Ein Fiebermittel senkt eine Temperatur von $41^{\circ} \mathrm{C}$ um $60\%$ pro Tag. Bestimme die Funktion, skizziere den Graphen und finde den Zeitpunkt, wann die Temperatur unter $37,5^{\circ} \mathrm{C}$ sinkt. Berechne auch $f^{\prime}(0)$ und $f^{\prime}(1)$ .

See Solution

Problem 12202

Find the derivative of $f(x)=\sqrt{x} \cdot(x+1)^{2}$ using product and chain rules. Rearrange if needed.

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Problem 12203

Given the demand equation $p=300-\sqrt{q^{2}+50}$ , find:
(a) $\frac{d p}{d q}$ , the rate of change of $p$ with respect to $q$ .
(b) $\frac{d p / d q}{p}$ , the relative rate of change of $p$ .
(c) $\frac{d r}{d q}$ , the marginal-revenue function.

See Solution

Problem 12204

Wildschweine:
a) Bestimme $f(0.5)$ und interpretiere das Ergebnis. b) Zeige, dass $f'(t)=(200-100t)e^{-0.5t}$ gilt. c) Berechne die Wachstumsrate zu Beginn und die durchschnittliche Zuwachsrate in den ersten 2 Jahren. d) Finde die langfristige Population und den Zeitpunkt des maximalen Bestands. e) Bestimme, wann die Wildschweinpopulation am schnellsten abnimmt.

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Problem 12205

Find the limit of $f(x)=\frac{x^{2}+11}{x^{3}+9}$ as $x \rightarrow \infty$ . What is $\lim _{x \rightarrow \infty} f(x)=\square$ ?

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Problem 12206

Find the marginal-revenue function for the demand equation $p=\frac{8 q+5}{6 q+1}$ , where revenue $=p q$ .

See Solution

Problem 12207

Differentiate $f(x) = (x+2)^{3} \cdot x$ using the product and chain rules, or simplify first.

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Problem 12208

Find the rate of change of income $y=4 x^{7/2}+5200$ with respect to education $x$ at $x=10$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 12209

Find the difference quotient for $f(x)=-6x-1$ : a. $\frac{f(a+h)-f(a)}{h}$ ; b. $\lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ .

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Problem 12210

Find the derivative of the function $f(x) = \sqrt[3]{17x}$ .

See Solution

Problem 12211

Bestimme die Stammfunktion $F$ von $f(t)=0.2t^3 - 48t^2 + 2880t$ und das Wasservolumen im Becken nach 120 Tagen.

See Solution

Problem 12212

Aufgabe 2.1: Klettergarten
Gegeben sind die Funktionen $f_{a}(x)=e^{\frac{1}{2} a x-a}+e^{-\frac{1}{2} a x+a}$ , $a \neq 0$ .
a) Zeigen Sie, dass $f_{a}$ keine Nullstelle hat und beschreiben Sie das Verhalten für $x \rightarrow \pm \infty$ .
b) Bestimmen Sie die $y$ -Achsen-Schnittpunkte von $G_{a}$ , wenn $S_{y}\left(0, \frac{17}{4}\right)$ gilt.
c) Bestimmen Sie die Koordinaten des lokalen Extrempunkts von $G_{1}$ und zeigen Sie, dass dies für alle $G_{a}$ gilt.
d) Finden Sie die Gleichung der Tangente $t$ an $G_{1}$ im Punkt $P\left(0, e+\frac{1}{e}\right)$ und zeigen Sie, dass das Dreieck mit der $x$ -Achse nicht rechtwinklig ist.
e) Berechnen Sie den Höhenunterschied und den steilsten Winkel der Hängebrücke $G_{0,5}$ im Intervall [0;8].
f) Bestimmen Sie die Fläche unter der Hängebrücke, die von Aufstellern genutzt werden kann (60% der Fläche).
g) Finden Sie den Parameterwert $a>0$ für den Punkt $Q\left(-1, e^{1}+e^{-1}\right)$ des Seils.

See Solution

Problem 12213

Find the derivative of $f(x)=\frac{1}{x}(x^{4}+3x^{2})$ using product and chain rules or simplify first.

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Problem 12214

Find the derivative of $-\sqrt{2x^{2}}$ directly, avoiding the chain rule.

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Problem 12215

Find the limit as $x$ approaches -21 for $\frac{20-\sqrt{x^{2}-41}}{x+21}$ . What is the result?

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Problem 12216

Differentiate the function $y=\frac{9 x-8}{5 x+9}$ . Find $y^{\prime}$ .

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Problem 12217

Find the marginal cost function for the total-cost function $c=\frac{4 q^{2}}{\sqrt{q^{2}+2}}+7000$ .

See Solution

Problem 12218

Find the derivative of $f(w)=(7 w^{2}+5) e^{w^{2}}$ . What is $f^{\prime}(w)$ ?

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Problem 12219

Find the derivative $f'(x)$ of the function $f(x)=-2 \sqrt{x^{3}}+\frac{\sqrt{x}}{2}$ and calculate $f'(3)$ .

See Solution

Problem 12220

Find the tangent line equation for $f(x)=\frac{5 x-7}{x+1}$ at $x=0$ .

See Solution

Problem 12221

Find the derivative of $H(t)=\sin t \sec ^{2} t$ without applying the chain rule.

See Solution

Problem 12222

Find the slope of the tangent line for $f(x)=-3 x^{3}+2 x^{2}-x$ at $x=2$ .

See Solution

Problem 12223

Two cylindrical pools fill at the same rate. Smaller pool radius is $5 \mathrm{~m}$ , rising at $0.5 \mathrm{~m/min}$ . Find the rise rate for the larger pool with radius $8 \mathrm{~m}$ .

See Solution

Problem 12224

Find the rate of change of the cost function $c=0.8 q^{2}+2.6 q+6$ at $q=14$ and its percentage change.

See Solution

Problem 12225

Wildschweinpopulation:
a) Bestimme $f(0.5)$ und interpretiere das Ergebnis.
b) Zeige, dass $f^{\prime}(t)=(200-100 t) \cdot e^{-0,5 t}$ .
c) Wie schnell wächst die Population zu Beginn?
d) Berechne die durchschnittliche Zuwachsrate in den ersten 2 Jahren.

See Solution

Problem 12226

Find the marginal-revenue function from the demand equation $p=\frac{6 q+7}{4 q+1}$ , where revenue $=p q$ .

See Solution

Problem 12227

Find the derivative $f^{\prime}(5)$ for the function $f(x)=-\frac{5 \sqrt{x}}{3}+\sqrt{x^{3}}$ .

See Solution

Problem 12228

Find the derivative of the function $y=\sqrt[3]{64 x} \cos x$ , i.e., compute $\frac{d y}{d x}$ .

See Solution

Problem 12229

Given the cost function $c(q)=1300+90q-0.1q^{2}$ , find: (a) average cost for 140 appliances, (b) marginal cost at 140 appliances.

See Solution

Problem 12230

Find the derivative of the function $f(x)=x-2 x \cos x$ , denoted as $f^{\prime}(x)$ .

See Solution

Problem 12231

Find the derivative of $h(x)=9 x^{4}-\sqrt{3} x^{3}+5 x-7$ .

See Solution

Problem 12232

Find the volume $\mathrm{V}$ of the region defined by $z=26-\sqrt{x^{2}+y^{2}}$ where it's nonnegative, and set up the double integral in polar coordinates.

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Problem 12233

Find the derivative $f'(x)$ of the function $f(x) = x^{2} - x^{2} \sin x$ .

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Problem 12234

Find the derivative $f'(x)$ of the function $f(x)=\frac{5+x}{2-x^{4}}$ .

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Problem 12235

Bestimmen Sie die Ableitung der Funktion $f(x) = -\frac{1}{2} x^{2}$ .

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Problem 12236

Find the derivative of the function $y=\frac{x}{5-6 x^{2}}$ in simplified form.

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Problem 12237

Find the derivative of the function $y=-3 \tan (x)$ with respect to $x$ : $\frac{d y}{d x}$ .

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Problem 12238

Find the derivative of the function $f(x)=-3 \sin (x)$ . What is $f^{\prime}(x)$ ?

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Problem 12239

Bestimmen Sie die Ableitung von $f(x) = \frac{1}{x^{2}}$ .

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Problem 12240

Find the derivative of the function $y=\frac{x}{4-5 x^{4}}$ and simplify it.

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Problem 12241

Write an iterated integral of a continuous function $f$ over the region $R=\{(x, y): 0 \leq x \leq y(1-y)\}$ using $d y d x$ .

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Problem 12242

Bestimmen Sie die Mindestgeschwindigkeit $v_{0}$ , um Steine bis $h=2 \mathrm{~km}$ zu schleudern, bei $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ .

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Problem 12243

Find $f^{\prime}(2)$ for the function $f(x)=x^{3} \ln (x)$ .

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Problem 12244

Find $h^{\prime}(1)$ for $h(x)=f(x) \cdot g(x)$ using the values from the chart.

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Problem 12245

Find the tangent line equation for $g(x)=3x^{2}f(x)$ at $x=1$ , given $f(1)=2$ and $f'(1)=-5$ .

See Solution

Problem 12246

Find the tangent line equation for $g(x)=(x^{2}-6)f(x)$ at $x=4$ given $f(4)=2$ and $f^{\prime}(4)=-1$ .

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Problem 12247

Find the critical values of the function $f(x)=x \sqrt{4-x}$ for $x \leq 4$ .

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Problem 12248

Analyze the limits of $f(x)=-x^{8}$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ . What do they approach?

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Problem 12249

Two cylindrical pools fill at the same rate. Smaller pool radius is $5 \mathrm{~m}$ , rising at $0.5 \mathrm{~m/min}$ . Find the rise rate for the larger pool with radius $8 \mathrm{~m}$ .

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Problem 12250

Analyze the behavior of $f(x)=-x^{3}$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ . What are the limits?

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Problem 12251

Find the limits $\lim_{x \to +\infty} f(x)$ and $\lim_{x \to -\infty} f(x)$ , and determine where $f(x)$ is increasing or decreasing.

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Problem 12252

Analyze the limits of $f(x)=x^{5}$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ . What do they approach?

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Problem 12253

Analyze the limits of $f(x)=x^{2}$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ . What are the results?

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Problem 12254

Find $a$ and $b$ so that the function $f(x)$ is differentiable at $x=1$ :
$f(x)=\left\{\begin{array}{lll} a x-2 & \text { for } & x<1 \\ 3 b x^{2}-6 x+10 & \text { for } & x \geq 1 \end{array}\right.$

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Problem 12255

Evaluate the limit: $\lim _{x \rightarrow \frac{5 \pi}{4}} \frac{3 \sec (x)-3 \sec \left(\frac{5 \pi}{4}\right)}{x-\frac{5 \pi}{4}}$

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Problem 12256

Evaluate the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin (x)-1}{x-\frac{\pi}{2}}$ .

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Problem 12257

Find values of $a$ and $b$ for which the function $f(x)$ is differentiable at $x=4$ :
$f(x)=\left\{\begin{array}{lll} a x+9 & \text { for } & x \leq 4 \\ 2 b x^{2}+x-8 & \text { for } & x>4 \end{array}\right.$

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Problem 12258

Bestimmen Sie die 1. Ableitung der Funktionen: a) $f(x)=x^{t-3}$ , b) $f(x)=\frac{1}{3} x^{3}+x-1$ , c) $f(b)=4 a b$ , d) $f(z)=\frac{4}{z}+\sqrt{z}$ , e) $f(x)=-4 x^{2}+0,5 x+1$ .

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Problem 12259

Maximize the volume of a cylinder with a ribbon of length $168 \, \text{cm}$ (after bow) wrapping around it. Find radius and height.

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Problem 12260

Write the iterated integral of a continuous function $f$ over the region $R=\{(x, y): 0 \leq x \leq y(2-y)\}$ using $d y d x$ .

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Problem 12261

Set up the double integral for the volume $V$ under $z=7-\sqrt{1+x^{2}+y^{2}}$ and above region $R=\{(r, \theta): \sqrt{3} \leq r \leq 2 \sqrt{6}, 0 \leq \theta \leq 2 \pi\}$ .

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Problem 12262

Calculate the volume of the solid above region $R=\{(x, y): 0 \leq x \leq 3, 0 \leq y \leq 2-\frac{2}{3} x\}$ , bounded by $z=4 x^{2}+9 y^{2}$ and $z=72-4 x^{2}-9 y^{2}$ .

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Problem 12263

Estimate the area between the curves $y=1.5 x$ and $y=0.25 x^{2}$ using a graphing calculator.

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Problem 12264

Calculate the area under the curve $y=19 x^{2}+2$ from $x=0$ to $x=1$ using a calculator.

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Problem 12265

Find the time for a bacteria population to double with a growth rate of $1.6\%$ per hour using continuous exponential growth.

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Problem 12266

Graph the curves $y=3x$ and $y=8\sqrt{x}$ . Find the other intersection point's $\mathrm{x}$ -coordinate and the area between them.

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Problem 12267

Encuentra la vida media de una sustancia radiactiva con una tasa de descomposición del $7.9\%$ por día.

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Problem 12268

Find the center of mass coordinates $(\overline{\mathscr{B}}, \bar{y})$ for the region between $x=2$ and $y=x+2$ with density $\rho=3x$ .

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Problem 12269

Find the mass of the half annulus $R=\{(r, \theta): 1 \leq r \leq 2,0 \leq \theta \leq \pi\}$ with density $\rho(r, \theta)=6+r \sin \theta$ .
$M=\square$

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Problem 12270

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

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Problem 12271

Find the center of mass $(\bar{x}, \bar{y})$ for the region between the axes and lines $x=2$ , $y=x+2$ with density $\rho=3x$ .

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Problem 12272

Find the derivative of $D(x)=e^{-\frac{x^{2}}{2 \sigma^{2}}}$ .

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Problem 12273

Estimate $\sum_{n=1}^{\infty} \frac{8}{n^{3}}$ within 0.112 of its exact value.

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Problem 12274

Una sustancia crece a una tasa del $15\%$ diario. Si inicia con 198 g, ¿cuánto pesará después de 2 días? Round a la décima.

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Problem 12275

Encuentra la población de bacterias después de 6 horas, comenzando con 666 y creciendo al $19\%$ por hora.

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Problem 12276

A radioactive substance starts at $4814 \mathrm{~kg}$ and decays at $4\%$ per day. What's its mass after 5 days?

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Problem 12277

Find the distance traveled by a car with velocity $f(t)=9t$ m/s from $t=0$ to $t=10$ seconds. Distance $=$ (in meters)

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Problem 12278

Estimate how many terms of the series $\sum_{n=1}^{\infty} \frac{3}{n^{1.1}}$ are needed for an error of at most $0.0001$ .

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Problem 12279

Calculate the area between $y=x^{1/2}$ and $y=x^{1/4}$ for $0 \leq x \leq 1$ . Area =

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Problem 12280

Find if the function $f(x)=5 x^{2}+50 x+121$ has a max, min, or neither. If so, determine the value and location.

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Problem 12281

Find the volume change rate $V$ of a ball with radius $r=10.5 \mathrm{~m}$ , given $V=\frac{4}{3} \pi r^{3}$ .

See Solution

Problem 12282

Find the difference quotient, $\frac{f(x+h)-f(x)}{h}$ , for the function $f(x)=2x-6$ .

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Problem 12283

A school uses 120 meters of fencing for 3 sides of a playground.
(a) Find the area function $A(x)$ in terms of $x$ . $A(x)=\square$
(b) What length $x$ maximizes the area? Length $x: \square$ meters
(c) What is the maximum area? Maximum area: $\square$ square meters

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Problem 12284

Find a number $a$ so that the average rate of change of $f(x)=\frac{1}{x}$ on $[4, a]$ equals $-\frac{1}{19}$ .

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Problem 12285

Find the average cost per appliance for the first 150 appliances and the marginal cost at 150 appliances.

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Problem 12286

Find the long-run behavior of the following functions:
1. $\frac{x^{2}+1}{x^{2}+2}$
2. $\frac{x^{2}+1}{x^{3}+2}$
3. $\frac{x^{3}+1}{x^{2}+2}$

See Solution

Problem 12287

Skizziere den Funktionsgraphen, sodass $f'(x) < 0$ und $f''(x) < 0$ im gewählten Intervall erfüllt sind.

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Problem 12288

Find the average velocity of a particle with displacement $s(t)=-3t$ between $t=4$ and $t=4+h$ , where $h \neq 0$ .

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Problem 12289

Find the average velocity of a particle with displacement $s(t)=-t^{2}$ between $t=4$ and $t=4+h$ , where $h \neq 0$ .

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Problem 12290

Find the average velocity of a particle with displacement $s(t)=10 t^{2}$ from $t=1$ to $t=1+h$ , $h \neq 0$ .

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Problem 12291

Find the instantaneous velocity of the particle at $t=3$ given its displacement $s(t)=t^{3}$ in meters.

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Problem 12292

Find the average velocity of a particle with displacement $s(t)=t^{2}-11$ from $t=5$ to $t=5+h$ , where $h \neq 0$ .

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Problem 12293

Find the instantaneous velocity of a particle moving with displacement $s(t)=5 t^{2}$ at $t=1$ hour.

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Problem 12294

Find the instantaneous velocity of the particle at $t=0$ given its displacement $s(t)=t^{2}-9t-22$ .

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Problem 12295

Evaluate the integrals: 6. $\int \frac{1}{4+(x-3)^{2}} d x$ and 8. $\int \frac{1}{x \sqrt{x^{4}-4}} d x$ .

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Problem 12296

Bestimme die Stammfunktion $F(x)$ für $f(x)=x^{n}$ und vervollständige: $F(x)=\frac{1}{n+1} x^{n+1} + C$ .

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Problem 12297

Find the average velocity of a particle with displacement $s(t)=-t+8$ between $t=7$ and $t=7+h$ , where $h \neq 0$ .

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Problem 12298

Welche Funktionsterme sind Stammfunktionen von $f(x)=x^{3}$ ? Wähle die richtigen Antworten aus: $\frac{1}{4} x^{4}, 4 x^{4}+7, \frac{1}{4} x^{4}+7, \frac{1}{4} x^{4}+4, 3 x^{2}$ .

See Solution

Problem 12299

Find the slope of the secant line for $k(x)=-3 x^{2}-19 x+10$ from $x=0$ to $x=h$ where $h \neq 0$ .

See Solution

Problem 12300

Calculate the average rate of change of $g(x)=9 x^{2}$ from $x=1$ to $x=1+h$ , with $h \neq 0$ .

See Solution

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Calculus

Problem 12201

Problem 12202

Find the derivative of f(x)=x⋅(x+1)2f(x)=\sqrt{x} \cdot(x+1)^{2}f(x)=x​⋅(x+1)2 using product and chain rules. Rearrange if needed.

Problem 12203

Given the demand equation p=300−q2+50p=300-\sqrt{q^{2}+50}p=300−q2+50​, find:(a) dpdq\frac{d p}{d q}dqdp​, the rate of change of ppp with respect to qqq.(b) dp/dqp\frac{d p / d q}{p}pdp/dq​, the relative rate of change of ppp.(c) drdq\frac{d r}{d q}dqdr​, the marginal-revenue function.

Problem 12204

Problem 12205

Find the limit of f(x)=x2+11x3+9f(x)=\frac{x^{2}+11}{x^{3}+9}f(x)=x3+9x2+11​ as x→∞x \rightarrow \inftyx→∞. What is lim⁡x→∞f(x)=□\lim _{x \rightarrow \infty} f(x)=\squarelimx→∞​f(x)=□?

Problem 12206

Find the marginal-revenue function for the demand equation p=8q+56q+1p=\frac{8 q+5}{6 q+1}p=6q+18q+5​, where revenue =pq=p q=pq.

Problem 12207

Differentiate f(x)=(x+2)3⋅xf(x) = (x+2)^{3} \cdot xf(x)=(x+2)3⋅x using the product and chain rules, or simplify first.

Problem 12208

Find the rate of change of income y=4x7/2+5200y=4 x^{7/2}+5200y=4x7/2+5200 with respect to education xxx at x=10x=10x=10. What is dydx\frac{d y}{d x}dxdy​?

Problem 12209

Find the difference quotient for f(x)=−6x−1f(x)=-6x-1f(x)=−6x−1: a. f(a+h)−f(a)h\frac{f(a+h)-f(a)}{h}hf(a+h)−f(a)​; b. lim⁡h→0f(a+h)−f(a)h\lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}limh→0​hf(a+h)−f(a)​.

Problem 12210

Find the derivative of the function f(x)=17x3f(x) = \sqrt[3]{17x}f(x)=317x​.

Problem 12211

Bestimme die Stammfunktion FFF von f(t)=0.2t3−48t2+2880tf(t)=0.2t^3 - 48t^2 + 2880tf(t)=0.2t3−48t2+2880t und das Wasservolumen im Becken nach 120 Tagen.

Problem 12212

Problem 12213

Find the derivative of f(x)=1x(x4+3x2)f(x)=\frac{1}{x}(x^{4}+3x^{2})f(x)=x1​(x4+3x2) using product and chain rules or simplify first.

Problem 12214

Find the derivative of −2x2-\sqrt{2x^{2}}−2x2​ directly, avoiding the chain rule.

Problem 12215

Find the limit as xxx approaches -21 for 20−x2−41x+21\frac{20-\sqrt{x^{2}-41}}{x+21}x+2120−x2−41​​. What is the result?

Problem 12216

Differentiate the function y=9x−85x+9y=\frac{9 x-8}{5 x+9}y=5x+99x−8​. Find y′y^{\prime}y′.

Problem 12217

Find the marginal cost function for the total-cost function c=4q2q2+2+7000c=\frac{4 q^{2}}{\sqrt{q^{2}+2}}+7000c=q2+2​4q2​+7000.

Problem 12218

Find the derivative of f(w)=(7w2+5)ew2f(w)=(7 w^{2}+5) e^{w^{2}}f(w)=(7w2+5)ew2. What is f′(w)f^{\prime}(w)f′(w)?

Problem 12219

Find the derivative f′(x)f'(x)f′(x) of the function f(x)=−2x3+x2f(x)=-2 \sqrt{x^{3}}+\frac{\sqrt{x}}{2}f(x)=−2x3​+2x​​ and calculate f′(3)f'(3)f′(3).

Problem 12220

Find the tangent line equation for f(x)=5x−7x+1f(x)=\frac{5 x-7}{x+1}f(x)=x+15x−7​ at x=0x=0x=0.

Problem 12221

Find the derivative of H(t)=sin⁡tsec⁡2tH(t)=\sin t \sec ^{2} tH(t)=sintsec2t without applying the chain rule.

Problem 12222

Find the slope of the tangent line for f(x)=−3x3+2x2−xf(x)=-3 x^{3}+2 x^{2}-xf(x)=−3x3+2x2−x at x=2x=2x=2.

Problem 12223

Two cylindrical pools fill at the same rate. Smaller pool radius is 5 m5 \mathrm{~m}5 m, rising at 0.5 m/min0.5 \mathrm{~m/min}0.5 m/min. Find the rise rate for the larger pool with radius 8 m8 \mathrm{~m}8 m.

Problem 12224

Find the rate of change of the cost function c=0.8q2+2.6q+6c=0.8 q^{2}+2.6 q+6c=0.8q2+2.6q+6 at q=14q=14q=14 and its percentage change.

Problem 12225

Problem 12226

Find the marginal-revenue function from the demand equation p=6q+74q+1p=\frac{6 q+7}{4 q+1}p=4q+16q+7​, where revenue =pq=p q=pq.

Problem 12227

Find the derivative f′(5)f^{\prime}(5)f′(5) for the function f(x)=−5x3+x3f(x)=-\frac{5 \sqrt{x}}{3}+\sqrt{x^{3}}f(x)=−35x​​+x3​.

Problem 12228

Find the derivative of the function y=64x3cos⁡xy=\sqrt[3]{64 x} \cos xy=364x​cosx, i.e., compute dydx\frac{d y}{d x}dxdy​.

Problem 12229

Given the cost function c(q)=1300+90q−0.1q2c(q)=1300+90q-0.1q^{2}c(q)=1300+90q−0.1q2, find: (a) average cost for 140 appliances, (b) marginal cost at 140 appliances.

Problem 12230

Find the derivative of the function f(x)=x−2xcos⁡xf(x)=x-2 x \cos xf(x)=x−2xcosx, denoted as f′(x)f^{\prime}(x)f′(x).

Problem 12231

Find the derivative of h(x)=9x4−3x3+5x−7h(x)=9 x^{4}-\sqrt{3} x^{3}+5 x-7h(x)=9x4−3​x3+5x−7.

Problem 12232

Find the volume V\mathrm{V}V of the region defined by z=26−x2+y2z=26-\sqrt{x^{2}+y^{2}}z=26−x2+y2​ where it's nonnegative, and set up the double integral in polar coordinates.

Problem 12233

Find the derivative f′(x)f'(x)f′(x) of the function f(x)=x2−x2sin⁡xf(x) = x^{2} - x^{2} \sin xf(x)=x2−x2sinx.

Problem 12234

Find the derivative f′(x)f'(x)f′(x) of the function f(x)=5+x2−x4f(x)=\frac{5+x}{2-x^{4}}f(x)=2−x45+x​.

Problem 12235

Bestimmen Sie die Ableitung der Funktion f(x)=−12x2f(x) = -\frac{1}{2} x^{2}f(x)=−21​x2.

Problem 12236

Find the derivative of the function y=x5−6x2y=\frac{x}{5-6 x^{2}}y=5−6x2x​ in simplified form.

Problem 12237

Find the derivative of the function y=−3tan⁡(x)y=-3 \tan (x)y=−3tan(x) with respect to xxx: dydx\frac{d y}{d x}dxdy​.

Problem 12238

Find the derivative of the function f(x)=−3sin⁡(x)f(x)=-3 \sin (x)f(x)=−3sin(x). What is f′(x)f^{\prime}(x)f′(x)?

Problem 12239

Bestimmen Sie die Ableitung von f(x)=1x2f(x) = \frac{1}{x^{2}}f(x)=x21​.

Problem 12240

Find the derivative of the function y=x4−5x4y=\frac{x}{4-5 x^{4}}y=4−5x4x​ and simplify it.

Problem 12241

Write an iterated integral of a continuous function fff over the region R={(x,y):0≤x≤y(1−y)}R=\{(x, y): 0 \leq x \leq y(1-y)\}R={(x,y):0≤x≤y(1−y)} using dydxd y d xdydx.

Problem 12242

Find the derivative of $f(x)=\sqrt{x} \cdot(x+1)^{2}$ using product and chain rules. Rearrange if needed.

Given the demand equation $p=300-\sqrt{q^{2}+50}$ , find:
(a) $\frac{d p}{d q}$ , the rate of change of $p$ with respect to $q$ .
(b) $\frac{d p / d q}{p}$ , the relative rate of change of $p$ .
(c) $\frac{d r}{d q}$ , the marginal-revenue function.

Find the limit of $f(x)=\frac{x^{2}+11}{x^{3}+9}$ as $x \rightarrow \infty$ . What is $\lim _{x \rightarrow \infty} f(x)=\square$ ?

Find the marginal-revenue function for the demand equation $p=\frac{8 q+5}{6 q+1}$ , where revenue $=p q$ .

Differentiate $f(x) = (x+2)^{3} \cdot x$ using the product and chain rules, or simplify first.

Find the rate of change of income $y=4 x^{7/2}+5200$ with respect to education $x$ at $x=10$ . What is $\frac{d y}{d x}$ ?

Find the difference quotient for $f(x)=-6x-1$ : a. $\frac{f(a+h)-f(a)}{h}$ ; b. $\lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ .

Find the derivative of the function $f(x) = \sqrt[3]{17x}$ .

Bestimme die Stammfunktion $F$ von $f(t)=0.2t^3 - 48t^2 + 2880t$ und das Wasservolumen im Becken nach 120 Tagen.

Find the derivative of $f(x)=\frac{1}{x}(x^{4}+3x^{2})$ using product and chain rules or simplify first.

Find the derivative of $-\sqrt{2x^{2}}$ directly, avoiding the chain rule.

Find the limit as $x$ approaches -21 for $\frac{20-\sqrt{x^{2}-41}}{x+21}$ . What is the result?

Differentiate the function $y=\frac{9 x-8}{5 x+9}$ . Find $y^{\prime}$ .

Find the marginal cost function for the total-cost function $c=\frac{4 q^{2}}{\sqrt{q^{2}+2}}+7000$ .

Find the derivative of $f(w)=(7 w^{2}+5) e^{w^{2}}$ . What is $f^{\prime}(w)$ ?

Find the derivative $f'(x)$ of the function $f(x)=-2 \sqrt{x^{3}}+\frac{\sqrt{x}}{2}$ and calculate $f'(3)$ .

Find the tangent line equation for $f(x)=\frac{5 x-7}{x+1}$ at $x=0$ .

Find the derivative of $H(t)=\sin t \sec ^{2} t$ without applying the chain rule.

Find the slope of the tangent line for $f(x)=-3 x^{3}+2 x^{2}-x$ at $x=2$ .

Two cylindrical pools fill at the same rate. Smaller pool radius is $5 \mathrm{~m}$ , rising at $0.5 \mathrm{~m/min}$ . Find the rise rate for the larger pool with radius $8 \mathrm{~m}$ .

Find the rate of change of the cost function $c=0.8 q^{2}+2.6 q+6$ at $q=14$ and its percentage change.

Find the marginal-revenue function from the demand equation $p=\frac{6 q+7}{4 q+1}$ , where revenue $=p q$ .

Find the derivative $f^{\prime}(5)$ for the function $f(x)=-\frac{5 \sqrt{x}}{3}+\sqrt{x^{3}}$ .

Find the derivative of the function $y=\sqrt[3]{64 x} \cos x$ , i.e., compute $\frac{d y}{d x}$ .

Given the cost function $c(q)=1300+90q-0.1q^{2}$ , find: (a) average cost for 140 appliances, (b) marginal cost at 140 appliances.

Find the derivative of the function $f(x)=x-2 x \cos x$ , denoted as $f^{\prime}(x)$ .

Find the derivative of $h(x)=9 x^{4}-\sqrt{3} x^{3}+5 x-7$ .

Find the volume $\mathrm{V}$ of the region defined by $z=26-\sqrt{x^{2}+y^{2}}$ where it's nonnegative, and set up the double integral in polar coordinates.

Find the derivative $f'(x)$ of the function $f(x) = x^{2} - x^{2} \sin x$ .

Find the derivative $f'(x)$ of the function $f(x)=\frac{5+x}{2-x^{4}}$ .

Bestimmen Sie die Ableitung der Funktion $f(x) = -\frac{1}{2} x^{2}$ .

Find the derivative of the function $y=\frac{x}{5-6 x^{2}}$ in simplified form.

Find the derivative of the function $y=-3 \tan (x)$ with respect to $x$ : $\frac{d y}{d x}$ .

Find the derivative of the function $f(x)=-3 \sin (x)$ . What is $f^{\prime}(x)$ ?

Bestimmen Sie die Ableitung von $f(x) = \frac{1}{x^{2}}$ .

Find the derivative of the function $y=\frac{x}{4-5 x^{4}}$ and simplify it.

Write an iterated integral of a continuous function $f$ over the region $R=\{(x, y): 0 \leq x \leq y(1-y)\}$ using $d y d x$ .

Bestimmen Sie die Mindestgeschwindigkeit $v_{0}$ , um Steine bis $h=2 \mathrm{~km}$ zu schleudern, bei $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ .

Find $f^{\prime}(2)$ for the function $f(x)=x^{3} \ln (x)$ .

Find $h^{\prime}(1)$ for $h(x)=f(x) \cdot g(x)$ using the values from the chart.

Find the tangent line equation for $g(x)=3x^{2}f(x)$ at $x=1$ , given $f(1)=2$ and $f'(1)=-5$ .

Find the tangent line equation for $g(x)=(x^{2}-6)f(x)$ at $x=4$ given $f(4)=2$ and $f^{\prime}(4)=-1$ .

Find the critical values of the function $f(x)=x \sqrt{4-x}$ for $x \leq 4$ .

Analyze the limits of $f(x)=-x^{8}$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ . What do they approach?

Two cylindrical pools fill at the same rate. Smaller pool radius is $5 \mathrm{~m}$ , rising at $0.5 \mathrm{~m/min}$ . Find the rise rate for the larger pool with radius $8 \mathrm{~m}$ .

Analyze the behavior of $f(x)=-x^{3}$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ . What are the limits?

Find the limits $\lim_{x \to +\infty} f(x)$ and $\lim_{x \to -\infty} f(x)$ , and determine where $f(x)$ is increasing or decreasing.

Analyze the limits of $f(x)=x^{5}$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ . What do they approach?

Analyze the limits of $f(x)=x^{2}$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ . What are the results?

Evaluate the limit: $\lim _{x \rightarrow \frac{5 \pi}{4}} \frac{3 \sec (x)-3 \sec \left(\frac{5 \pi}{4}\right)}{x-\frac{5 \pi}{4}}$

Evaluate the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin (x)-1}{x-\frac{\pi}{2}}$ .

Bestimmen Sie die 1. Ableitung der Funktionen: a) $f(x)=x^{t-3}$ , b) $f(x)=\frac{1}{3} x^{3}+x-1$ , c) $f(b)=4 a b$ , d) $f(z)=\frac{4}{z}+\sqrt{z}$ , e) $f(x)=-4 x^{2}+0,5 x+1$ .

Maximize the volume of a cylinder with a ribbon of length $168 \, \text{cm}$ (after bow) wrapping around it. Find radius and height.

Write the iterated integral of a continuous function $f$ over the region $R=\{(x, y): 0 \leq x \leq y(2-y)\}$ using $d y d x$ .

Set up the double integral for the volume $V$ under $z=7-\sqrt{1+x^{2}+y^{2}}$ and above region $R=\{(r, \theta): \sqrt{3} \leq r \leq 2 \sqrt{6}, 0 \leq \theta \leq 2 \pi\}$ .

Calculate the volume of the solid above region $R=\{(x, y): 0 \leq x \leq 3, 0 \leq y \leq 2-\frac{2}{3} x\}$ , bounded by $z=4 x^{2}+9 y^{2}$ and $z=72-4 x^{2}-9 y^{2}$ .

Estimate the area between the curves $y=1.5 x$ and $y=0.25 x^{2}$ using a graphing calculator.

Calculate the area under the curve $y=19 x^{2}+2$ from $x=0$ to $x=1$ using a calculator.

Find the time for a bacteria population to double with a growth rate of $1.6\%$ per hour using continuous exponential growth.

Graph the curves $y=3x$ and $y=8\sqrt{x}$ . Find the other intersection point's $\mathrm{x}$ -coordinate and the area between them.

Encuentra la vida media de una sustancia radiactiva con una tasa de descomposición del $7.9\%$ por día.

Find the center of mass coordinates $(\overline{\mathscr{B}}, \bar{y})$ for the region between $x=2$ and $y=x+2$ with density $\rho=3x$ .

Find the mass of the half annulus $R=\{(r, \theta): 1 \leq r \leq 2,0 \leq \theta \leq \pi\}$ with density $\rho(r, \theta)=6+r \sin \theta$ .
$M=\square$

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

Find the center of mass $(\bar{x}, \bar{y})$ for the region between the axes and lines $x=2$ , $y=x+2$ with density $\rho=3x$ .

Find the derivative of $D(x)=e^{-\frac{x^{2}}{2 \sigma^{2}}}$ .

Estimate $\sum_{n=1}^{\infty} \frac{8}{n^{3}}$ within 0.112 of its exact value.

Una sustancia crece a una tasa del $15\%$ diario. Si inicia con 198 g, ¿cuánto pesará después de 2 días? Round a la décima.