Calculus

Problem 20201

Find the dimensions of a cylindrical can with volume 950 cm³ to minimize metal cost ( $0.01/cm² sides,$ 0.02/cm² top/bottom). What is the height/radius ratio?

See Solution

Problem 20202

Ein Snowboarder fährt mit der Bewegung $s(t)=1,5 t^2$ . Berechne: a) Weg nach 1s und 5s, b) mittlere Geschwindigkeit in 5s, c) Momentangeschwindigkeit nach 5s.

See Solution

Problem 20203

Calculate the integral $A=\int_{9}^{16} 15 \sqrt{4-\sqrt{x}} d x$ .

See Solution

Problem 20204

Berechnen Sie die Fläche zwischen den Kurven $F(x) = -\frac{1}{x^2}$ und $g(x) = 2.5x - 5.25$ .

See Solution

Problem 20205

Find $\frac{dy}{dx}$ for $2x^2 + 3x^3 y - 5y^3 = 16$ and the tangent line at $(2,2)$ .

See Solution

Problem 20206

A particle moves in the $xy$ -plane with position $\langle x(t), y(t)\rangle=\langle\sin(2t), t^2-t\rangle$ for $0 \leq t \leq 8$ .
a. Find the speed at $t=3$ seconds.
b. Is the speed increasing or decreasing at $t=5$ seconds?
c. Calculate the total distance traveled from $t=0$ to $t=6$ seconds.
d. Find the position at $t=11$ seconds if the particle moves straight after $t=8$ .

See Solution

Problem 20207

Find the derivative $\frac{d y}{d x}$ for $y=\int_{0}^{x} \sqrt{12+13 t^{2}} dt$ . What is $\frac{d y}{d x}=\square$ ?

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Problem 20208

Calculate the integral from 1 to 4 of $x^{2}+\sqrt{x}$ . Find $\int_{1}^{4}\left(x^{2}+\sqrt{x}\right) d x=\square($ exact answer).

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Problem 20209

Evaluate the integral using the fundamental theorem of calculus: $\int_{1}^{4}(4 x^{2}+5) dx=\square$

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Problem 20210

A car's velocity is $v(t)=20t$ for $0 \leq t \leq 6$ . Estimate distance with $\Delta t=2$ and find the exact distance.

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Problem 20211

Estimate the total water volume used by a factory from 6 am to 9 am, given flow rates from $70 \mathrm{~m}^{3}/\mathrm{hr}$ to $250 \mathrm{~m}^{3}/\mathrm{hr}$ . $\text { Best estimate }=\mathbf{i} \quad \mathrm{m}^{3}$

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Problem 20212

Evaluate the integral using substitution: $\int\left(x^{2}-2 x\right)\left(x^{3}-3 x^{2}+6\right)^{\frac{2}{3}} d x$

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Problem 20213

Find $f^{\prime}(2)$ given that $40 + 4f(x) + x^{2}(f(x))^{3} = 0$ and $f(2) = -2$ .

See Solution

Problem 20214

Find $g^{\prime}\left(-\frac{8}{9}\right)$ given $8 g(x)-9 x \sin (g(x))=72 x^{2}-17 x-72$ and $g\left(-\frac{8}{9}\right)=0$ .

See Solution

Problem 20215

Evaluate the integral using substitution: $\int \frac{(1+4 \ln x)^{10}}{x} d x$ .

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Problem 20216

Evaluate the integral: $\int \frac{8}{\sqrt{x}(7+8 \sqrt{x})^{5}} dx$

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Problem 20217

Use L'Hôpital's rule to show: (a) $\lim _{x \rightarrow \infty} \frac{\ln \left(x^{n}\right)}{x^{k}}=0$ ; (b) $\lim _{x \rightarrow \infty} \frac{\ln \left(\ln \left(x^{n}\right)\right)}{\ln \left(x^{k}\right)}=0$ .

See Solution

Problem 20218

Ein Behälter hat zu Beginn $2 m^{3}$ Öl. Zuflussrate ist $f(t)=0,1 e^{-0,1 t}$ . Bestimmen Sie die Ölmenge $g(t)$ für $t>0$ und die Zeit bis $2,5 m^{3}$ . Warum nie mehr als $3 m^{3}$ ?

See Solution

Problem 20219

Find the shortest ladder length $L(\theta)$ that reaches over a 4 ft fence, 2 ft from a building.
[A] Formula for $L(\theta)$ :
$L(\theta)=$
[B] Derivative $L^{\prime}(\theta)$ :
$L^{\prime}(\theta)=$
[C] Find $\theta$ where $L^{\prime}(\theta)=0$ and compute $L\left(\theta_{\min }\right) \approx$ (to 5 decimal places) feet.

See Solution

Problem 20220

Evaluate the integral $\int \frac{8}{t^{9}} \cos \left(\frac{1}{t^{8}}-1\right) dt$ .

See Solution

Problem 20221

Bestimmen Sie das Verhalten von $f$ für $x \rightarrow \infty$ und $x \rightarrow -\infty$ für die folgenden Funktionen: a) $f(x)=\frac{e^{x}}{x^{4}+1}$ b) $f(x)=e^{x} \cdot x^{2}$ c) $f(x)=\frac{-x^{2}}{e^{x}}$ d) $f(x)=2 e^{x}-10 x^{3}$ e) $f(x)=100 x^{4}+0,1 \cdot e^{x}$ f) $f(x)=\frac{-1-x^{2}}{e^{x}}$

See Solution

Problem 20222

Bestimmen Sie das Verhalten von $f$ für $x \rightarrow \infty$ und $x \rightarrow -\infty$ . Geben Sie die waagerechte Asymptote an. a) $f(x)=\frac{e^{x}}{x^{4}+1}$ b) $f(x)=e^{x} \cdot x^{2}$ c) $f(x)=\frac{-x^{2}}{e^{x}}$

See Solution

Problem 20223

Show why $\lim _{x \rightarrow \infty} x^{n} e^{-x}=0$ using L'Hôpital's rule for positive integer $n$ .

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Problem 20224

Projectile fired vertically with initial velocity $550 \mathrm{ft/sec}$ and $g=-32 \mathrm{ft}^2/sec$ . Find: A) velocity eqn B) displacement eqn C) velocity at 3s D) time to max height E) max height.

See Solution

Problem 20225

Find the volume of the solid formed by revolving the region between $y=x^{2}+2$ and $y=x^{1/2}$ from $x=2$ to $x=4$ around the $y$ -axis. Round to the tenths place.

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Problem 20226

Find the volume of the cylinder formed by revolving $f(x)=e^{5 x^{2}}$ from $x=0.8$ to $x=1$ . Round to tenths.

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Problem 20227

Bestimme die Woche mit den meisten Brötchenverkäufen, den langfristigen Verkauf und die Verkäufe in den ersten 8 Wochen. Funktion: $f(x)=2000 \cdot x \cdot e^{-0,5 x}+2500$ .

See Solution

Problem 20228

Find the indefinite integral of $x^{2} \cos (3 x) \, dx$ .

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Problem 20229

Find the position function $s(t)$ given $a(t)=16 t-6$ , $v(1)=4$ , and $s(3)=6(10 p t)$ .

See Solution

Problem 20230

Evaluate the integral using integration by parts: $\int \frac{\ln (x)}{x^{5}} \mathrm{~d} x=$

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Problem 20231

An object at $150^{\circ} \mathrm{F}$ cools in water at $70^{\circ} \mathrm{F}$ . Find $F(t)$ with cooling constant $k=1.4$ . $F(t)=$

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Problem 20232

Calculate the integral: $\int x^{2} \sin (6 x) d x$ .

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Problem 20233

What is the best u-substitution for $\int 4 x^{4} \sin (7 x^{5}+4) dx$ ? Choose one: $7 x^{5}+4$ , $u=\sin (x)$ , $u=x+4$ , $u=x$ .

See Solution

Problem 20234

What is the new integral in terms of $u$ after performing $u$ -sub for $\int x^{3} \sin(x^{4}) \mathrm{dx}$ ?

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Problem 20235

Find the average value of $x^{9}+7$ over the interval $[4,8]$ . Provide an exact or decimal answer with 0.1% accuracy.

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Problem 20236

Find the best u-substitution for $\int \sec ^{2} x \tan ^{4} x d x$ . Choose one: $u=\tan (x)$ , $u=\sec (x)$ , $u=\tan \hat{\uparrow} x$ , $u=x$ , $u=\sec ^{2} x$ .

See Solution

Problem 20237

Find the limit as $x$ approaches 0 from the right: $\lim _{x \rightarrow 0^{+}} \sin (x)^{\sin (x)^{\sin (x)}}$ .

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Problem 20238

3. (a) Why is $\int \ln (|x|) d x=x \ln (|x|)+C$ incorrect? (b) What is the correct value of $\int \ln (|x|) d x$ ?

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Problem 20239

Gegeben sind die Funktionen $f_{t}(x)=e^{t x}-1(t>0)$ . Bestimmen Sie $t$ für folgende Aufgaben: a) Geht durch den Ursprung? b) $f_{t}(2)=5$ ? c) $f_{t}^{\prime}$ schneidet $y$ -Achse bei $S(0 \mid 3)$ ? d) $f_{t}=g(x)=8^{x}-1$ ? e) Normale von $f_{1}$ schneidet $x$ -Achse? f) Normale von $f_{t}$ schneidet $x$ -Achse bei $Q(2 \mid 0)$ ? g) Ist $f_{t}$ monoton wachsend? h) Fläche für $t>0$ ist endlich. Wie ist $t$ zu wählen, damit der Flächeninhalt 2 ist?

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Problem 20240

Identify the TRUE statements regarding these integrals without evaluating them:
1. $\int_{-2}^{2} x^{3} d x=2 \int_{0}^{2} x^{3} d x$
2. $\int_{-3}^{2} x^{2} d x=2 \int_{0}^{2} x^{2} d x$
3. $\int_{-4}^{4} x^{2} d x=2 \int_{0}^{4} x^{2} d x$

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Problem 20241

Une entreprise fabrique des objets avec un coût $C(q)=0,0000025 q^{3}-0,03 q^{2}+120 q+40000$ .
1) Quel est le prix maximal en \ $? 2) Coût marginal à 100 unités, arrondi à deux décimales. 3) Trouvez$ R(q) $pour le revenu. 4) Revenu marginal à 100 unités, arrondi à deux décimales. 5) Établissez$ \epsilon(p) $sans$ q$.

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Problem 20242

Une entreprise fabrique $q$ objets avec un coût $C(q)=0,0000025 q^{3}-0,03 q^{2}+120 q+40000$ . Trouvez le prix maximal en \$ et le coût marginal à 100 unités.

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Problem 20243

Find the marginal cost in \ $for producing 100 units using the cost function$ C(q) = 0.0000025q^3 - 0.03q^2 + 120q + 4000$. Round to two decimal places.

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Problem 20244

Une entreprise fabrique $q$ objets avec un coût $C(q)=0,0000025 q^{3}-0,03 q^{2}+120 q+40000$ . La demande est $q=\frac{180000-1000 p}{6}$ .
1) Quel est le prix maximal (en \$) pour le produit? 2) Calculez le coût marginal (en \$) à 100 unités, arrondi à deux décimales.

See Solution

Problem 20245

Fonction de coût : $C(q)=0,0000025 q^{3}-0,03 q^{2}+120 q+40000$ .
1) Prix maximal en \ $? 2) Coût marginal à 100 unités, arrondi à deux décimales ? 3) Fonction de revenu$ R(q) $? 4) Revenu marginal à 100 unités, arrondi à deux décimales ? 5) Élasticité de la demande$ \epsilon(p) $sans$ q$ ? 6) Élasticité à \$100 ? 7) Prix de vente pour une demande unitaire ? 8) Quantité vendue quand la demande est unitaire ?

See Solution

Problem 20246

Evaluate the integral of $\frac{\tan^{-1}(5x)}{1+25x^2}$ with respect to $x$ .

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Problem 20247

Find the derivative of $y=\frac{2 x^{2}+9}{x^{2}+7}$ . What is $y^{\prime}=\square$ ?

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Problem 20248

Find the derivative of $y=\frac{9 x+2}{9 x-5}$ . Which option correctly shows the derivative process?

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Problem 20249

Find the limit: $\lim _{x \rightarrow 8} \frac{\frac{1}{x}-\frac{1}{8}}{x-8}$ . Choose A or B.

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Problem 20250

Find the derivative of $f(y)=\frac{4 y-7}{7 \sqrt{y}-4}$ . What is the result?

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Problem 20251

Find the limit: $\lim _{z \rightarrow 4} \frac{\frac{1}{z}-\frac{1}{4}}{z-4}$ . Choose A or B.

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Problem 20252

Find the limits of $f(x)=\frac{8 x^{5}+8 x^{4}+4}{8 x^{6}}$ as $x \rightarrow \infty$ and $x \rightarrow -\infty$ .

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Problem 20253

Le nombre de personnes $q$ utilisant le service d'autobus est donné par $q(p)=3000 \sqrt{9-p}$ .
1) Calculez l'élasticité de la demande $\epsilon(p)$ à $p=4$ et arrondissez à deux décimales. 2) Trouvez le prix pour lequel la demande est unitaire, arrondissez à deux décimales.

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Problem 20254

Find the derivative of $v=\frac{9+x-6 \sqrt{x}}{x}$ . Which option correctly shows the derivative process?

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Problem 20255

A soccer ball is kicked from 3 ft with an initial speed of 48 ft/s. What is its maximum height?

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Problem 20256

Find the derivative of the function $y=2 e^{-x}+e^{3 x}$ . What is $\frac{d y}{d x}$ ?

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Problem 20257

Find all derivatives of the function $y=(x-5)(x+3)(x+4)$ . What is $y'=\square$ ?

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Problem 20258

Find the first and second derivatives of $y=\frac{2 x^{4}+7}{x^{2}}$ . What is $\frac{d y}{d x}$ ?

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Problem 20259

Evaluate the integral: $\int \frac{\sin (\ln 29 x)}{x} \, dx$

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Problem 20260

Deux compagnies produisent un bien. Coûts: $7$ pour C1 et $9$ pour C2. Prix: $p=515-0.4Q$ . Trouvez les profits et dérivées.

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Problem 20261

Find the first and second derivatives of the function $w=\left(\frac{1+15 z}{5 z}\right)(15-z)$ . Calculate $\frac{d w}{d z}=\square$ .

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Problem 20262

Find the derivative of $y=3 x^{8} e^{x}$ . Which option shows the correct derivative calculation?

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Problem 20263

Ein Ortsteil hat 500 Einwohner und will das Wachstum modellieren. Bestimmen Sie die $e$ -Funktion und beantworten Sie folgende Fragen:
e) Gleichung der $e$ -Funktion? f) Einwohnerzahl vor 10 Jahren? g) Wann verdoppelt sich die Einwohnerzahl? h) Anfangswachstumsgeschwindigkeit? i) Wann halbiert sich die Wachstumsgeschwindigkeit?

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Problem 20264

Evaluate the integral from 1 to 2: $\int_{1}^{2} \frac{x^{3}+4 x}{x^{4}+8 x^{2}+6} d x$

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Problem 20265

Find the position of a particle at time $t=8$ given $a(t)=18t+14$ , $s(0)=10$ , and $v(0)=11$ .

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Problem 20266

Find the derivative $\frac{d y}{d x}$ for the function $y=\sqrt[8]{x} \csc x+5$ .

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Problem 20267

Find the derivative $\frac{d y}{d x}$ for the function $y=-19 x+2 \cos x$ . What is $\frac{d}{d x}(-19 x+2 \cos x)$ ?

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Problem 20268

Deux compagnies produisent un bien. Compagnie 1: coût \ $7, Compagnie 2: coût \$9. Prix:$ p=515-0,4 Q $. Trouvez les profits$ \Pi_{1} $,$ \Pi_{2}$ et leurs dérivées.

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Problem 20269

Find derivatives at $x=0$ : a. $\frac{d}{d x}(u v)$ , b. $\frac{d}{d x}\left(\frac{u}{v}\right)$ , c. $\frac{d}{d x}\left(\frac{v}{u}\right)$ , d. $\frac{d}{d x}(7 v-3 u)$ , given $u(0)=9$ , $u'(0)=4$ , $v(0)=-8$ , $v'(0)=6$ .

See Solution

Problem 20270

Is the integral $\int_{0}^{\infty} \frac{1}{\sqrt[6]{1+x}} d x$ convergent or divergent? If convergent, find its value.

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Problem 20271

Find the derivative of $f(x)=\frac{5 x}{x-9}$ using $f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$ .

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Problem 20272

Find the derivative $\frac{d y}{d x}$ for $y=\frac{\tan x}{1+\tan x}$ . What is $\frac{\mathrm{dy}}{\mathrm{dx}}$ ?

See Solution

Problem 20273

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{4 \cos x}{1-\sin x}$ .

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Problem 20274

Evaluate the integral: $\int \frac{x^{2}-x+12}{x^{3}+3 x} d x$ (include absolute values and use $C$ for the constant).

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Problem 20275

Is the integral $\int_{0}^{\infty} e^{-8 x} d x$ convergent or divergent? If convergent, find its value.

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Problem 20276

Find the value of $F(2)$ where $F(x)=\int_{2}^{x} t^{4} \sin ^{2}(t) d t$ .

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Problem 20277

Is the integral $\int_{3}^{\infty} \frac{1}{x^{2}+x} d x$ convergent or divergent? If convergent, evaluate it.

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Problem 20278

Determine if the integral $\int_{-\infty}^{\infty} 17 x e^{-x^{2}} d x$ converges or diverges; evaluate if convergent.

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Problem 20279

Evaluate the integral from 0 to $\sqrt[10]{\pi}$ of $7 x^{9} \cos(x^{10}) \, dx$ .

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Problem 20280

Find the area between $f(x)=x^{3}$ and the $x$ -axis for $x \in[-1,0]$ . What is it?

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Problem 20281

Find the value of $\int_{0}^{6} f(x) dx$ where $f(x)=\{2x \text{ for } x<1, 3 \text{ for } x \geq 1\}$ .

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Problem 20282

Use Newton's method to find a root of $e^{-x}=3+x$ accurate to eight decimal places.

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Problem 20283

Is the integral $\int_{-2}^{3} \frac{21}{x^{4}} d x$ convergent or divergent? If convergent, evaluate it.

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Problem 20284

Find $u$ if $\int \sqrt{5 t+6} d t=\frac{1}{5} \int u^{1 / 2} d u$ . Choices: a. $t-6$ , b. $5 t+6$ , c. $t+6$ , d. $5 t-6$ , e. $x-6$ , f. $x+6$ .

See Solution

Problem 20285

Find the inner product of $f(x) = 15x$ and $g(x) = 3x^{3}$ over $[-3,1]$ , given $\langle f, g\rangle = 2 \times 5$ .

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Problem 20286

Approximate the root of $e^{-x}=3+x$ using Newton's method to eight decimal places. The root is 1.

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Problem 20287

If $\int(\sin (\theta))^{5} \cos (\theta) d \theta=\int u^{5} d u$ , what is the value of $u$ ?

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Problem 20288

Set up an integral for the curve length of $y=x-5 \ln (x)$ from $x=1$ to $x=4$ : $\int_{1}^{4}(\sqrt{1 + (y')^2}) \, dx$

See Solution

Problem 20289

Find the inner product of $f(x)=15x$ and $g(x)=3x^{3}$ on $C[-2,1]$ .

See Solution

Problem 20290

Find $\lim _{x \rightarrow \infty} \frac{-2 \ln (x)}{3 x}$ . What is the answer?

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Problem 20291

Find the distance travelled by an elephant with velocity $v(t)=2t-2$ from $t=0$ to $t=5$ . Options: a. 15m b. 12s c. -17s d. -15m e. -15s f. 17m

See Solution

Problem 20292

Find the derivative $\frac{d r}{d \theta}$ for the equation $r=2-\theta^{2} \sin \theta$ .

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Problem 20293

Find the area of the surface formed by rotating $y=\cos \left(\frac{1}{6} x\right)$ , for $0 \leq x \leq 3 \pi$ , about the $x$ -axis.

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Problem 20294

Calculate the integration norm of the polynomial $f(x)=-3x+6$ on the interval $C[0,1]$ .

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Problem 20295

Calculate the $L1$ norm of $f(x) = -9x - 4$ on the interval $[0,1]$ using the integral of its absolute value.

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Problem 20296

Ein Snowboarder fährt den Hang hinunter. Gegeben ist $s(t)=1,5 t^2$ . Berechne: a) Weg nach 1s und 5s, b) mittlere Geschwindigkeit in 5s, c) Momentangeschwindigkeit nach 5s.

See Solution

Problem 20297

Find the derivative $\frac{d y}{d x}$ for $y=\frac{\tan x}{1+\tan x}$ . $\frac{\mathrm{dy}}{\mathrm{dx}}=\square$

See Solution

Problem 20298

Calculate the weighted integration norm $\langle f, f\rangle=\int_{0}^{1} (x-3)(x-3)(8x) dx$ for $f(x)=x-3$ and $w(x)=8x$ .

See Solution

Problem 20299

Find the derivative $\frac{d p}{d q}$ for $p=\frac{\sin q-\cos q}{\sin q}$ . What is $\frac{d p}{d q}=\square$ ?

See Solution

Problem 20300

Find the derivative $\frac{d r}{d \theta}$ for the function $r=(3+\sec \theta) \sin \theta$ . What is $\frac{\mathrm{dr}}{\mathrm{d} \theta}$ ?

See Solution

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Calculus

Problem 20201

Find the dimensions of a cylindrical can with volume 950 cm³ to minimize metal cost (0.01/cm²sides,0.01/cm² sides, 0.01/cm²sides,0.02/cm² top/bottom). What is the height/radius ratio?

Problem 20202

Ein Snowboarder fährt mit der Bewegung s(t)=1,5t2s(t)=1,5 t^2s(t)=1,5t2. Berechne: a) Weg nach 1s und 5s, b) mittlere Geschwindigkeit in 5s, c) Momentangeschwindigkeit nach 5s.

Problem 20203

Calculate the integral A=∫916154−xdxA=\int_{9}^{16} 15 \sqrt{4-\sqrt{x}} d xA=∫916​154−x​​dx.

Problem 20204

Berechnen Sie die Fläche zwischen den Kurven F(x)=−1x2F(x) = -\frac{1}{x^2}F(x)=−x21​ und g(x)=2.5x−5.25g(x) = 2.5x - 5.25g(x)=2.5x−5.25.

Problem 20205

Find dydx\frac{dy}{dx}dxdy​ for 2x2+3x3y−5y3=162x^2 + 3x^3 y - 5y^3 = 162x2+3x3y−5y3=16 and the tangent line at (2,2)(2,2)(2,2).

Problem 20206

Problem 20207

Find the derivative dydx\frac{d y}{d x}dxdy​ for y=∫0x12+13t2dty=\int_{0}^{x} \sqrt{12+13 t^{2}} dty=∫0x​12+13t2​dt. What is dydx=□\frac{d y}{d x}=\squaredxdy​=□?

Problem 20208

Calculate the integral from 1 to 4 of x2+xx^{2}+\sqrt{x}x2+x​. Find ∫14(x2+x)dx=□(\int_{1}^{4}\left(x^{2}+\sqrt{x}\right) d x=\square(∫14​(x2+x​)dx=□( exact answer).

Problem 20209

Evaluate the integral using the fundamental theorem of calculus: ∫14(4x2+5)dx=□\int_{1}^{4}(4 x^{2}+5) dx=\square∫14​(4x2+5)dx=□

Problem 20210

A car's velocity is v(t)=20tv(t)=20tv(t)=20t for 0≤t≤60 \leq t \leq 60≤t≤6. Estimate distance with Δt=2\Delta t=2Δt=2 and find the exact distance.

Problem 20211

Estimate the total water volume used by a factory from 6 am to 9 am, given flow rates from 70 m3/hr70 \mathrm{~m}^{3}/\mathrm{hr}70 m3/hr to 250 m3/hr250 \mathrm{~m}^{3}/\mathrm{hr}250 m3/hr. Best estimate =im3 \text { Best estimate }=\mathbf{i} \quad \mathrm{m}^{3} Best estimate =im3

Problem 20212

Evaluate the integral using substitution: ∫(x2−2x)(x3−3x2+6)23dx\int\left(x^{2}-2 x\right)\left(x^{3}-3 x^{2}+6\right)^{\frac{2}{3}} d x∫(x2−2x)(x3−3x2+6)32​dx

Problem 20213

Find f′(2)f^{\prime}(2)f′(2) given that 40+4f(x)+x2(f(x))3=040 + 4f(x) + x^{2}(f(x))^{3} = 040+4f(x)+x2(f(x))3=0 and f(2)=−2f(2) = -2f(2)=−2.

Problem 20214

Find g′(−89)g^{\prime}\left(-\frac{8}{9}\right)g′(−98​) given 8g(x)−9xsin⁡(g(x))=72x2−17x−728 g(x)-9 x \sin (g(x))=72 x^{2}-17 x-728g(x)−9xsin(g(x))=72x2−17x−72 and g(−89)=0g\left(-\frac{8}{9}\right)=0g(−98​)=0.

Problem 20215

Evaluate the integral using substitution: ∫(1+4ln⁡x)10xdx\int \frac{(1+4 \ln x)^{10}}{x} d x∫x(1+4lnx)10​dx.

Problem 20216

Evaluate the integral: ∫8x(7+8x)5dx\int \frac{8}{\sqrt{x}(7+8 \sqrt{x})^{5}} dx∫x​(7+8x​)58​dx

Problem 20217

Problem 20218

Ein Behälter hat zu Beginn 2m32 m^{3}2m3 Öl. Zuflussrate ist f(t)=0,1e−0,1tf(t)=0,1 e^{-0,1 t}f(t)=0,1e−0,1t. Bestimmen Sie die Ölmenge g(t)g(t)g(t) für t>0t>0t>0 und die Zeit bis 2,5m32,5 m^{3}2,5m3. Warum nie mehr als 3m33 m^{3}3m3?

Problem 20219

Problem 20220

Evaluate the integral ∫8t9cos⁡(1t8−1)dt\int \frac{8}{t^{9}} \cos \left(\frac{1}{t^{8}}-1\right) dt∫t98​cos(t81​−1)dt.

Problem 20221

Problem 20222

Problem 20223

Show why lim⁡x→∞xne−x=0\lim _{x \rightarrow \infty} x^{n} e^{-x}=0limx→∞​xne−x=0 using L'Hôpital's rule for positive integer nnn.

Problem 20224

Projectile fired vertically with initial velocity 550ft/sec550 \mathrm{ft/sec}550ft/sec and g=−32ft2/secg=-32 \mathrm{ft}^2/secg=−32ft2/sec. Find: A) velocity eqn B) displacement eqn C) velocity at 3s D) time to max height E) max height.

Problem 20225

Find the volume of the solid formed by revolving the region between y=x2+2y=x^{2}+2y=x2+2 and y=x1/2y=x^{1/2}y=x1/2 from x=2x=2x=2 to x=4x=4x=4 around the yyy-axis. Round to the tenths place.

Problem 20226

Find the volume of the cylinder formed by revolving f(x)=e5x2f(x)=e^{5 x^{2}}f(x)=e5x2 from x=0.8x=0.8x=0.8 to x=1x=1x=1. Round to tenths.

Problem 20227

Bestimme die Woche mit den meisten Brötchenverkäufen, den langfristigen Verkauf und die Verkäufe in den ersten 8 Wochen. Funktion: f(x)=2000⋅x⋅e−0,5x+2500f(x)=2000 \cdot x \cdot e^{-0,5 x}+2500f(x)=2000⋅x⋅e−0,5x+2500.

Problem 20228

Find the indefinite integral of x2cos⁡(3x) dxx^{2} \cos (3 x) \, dxx2cos(3x)dx.

Problem 20229

Find the position function s(t)s(t)s(t) given a(t)=16t−6a(t)=16 t-6a(t)=16t−6, v(1)=4v(1)=4v(1)=4, and s(3)=6(10pt)s(3)=6(10 p t)s(3)=6(10pt).

Problem 20230

Evaluate the integral using integration by parts: ∫ln⁡(x)x5 dx=\int \frac{\ln (x)}{x^{5}} \mathrm{~d} x=∫x5ln(x)​ dx=

Problem 20231

An object at 150∘F150^{\circ} \mathrm{F}150∘F cools in water at 70∘F70^{\circ} \mathrm{F}70∘F. Find F(t)F(t)F(t) with cooling constant k=1.4k=1.4k=1.4. F(t)= F(t)= F(t)=

Problem 20232

Calculate the integral: ∫x2sin⁡(6x)dx\int x^{2} \sin (6 x) d x∫x2sin(6x)dx.

Problem 20233

What is the best u-substitution for ∫4x4sin⁡(7x5+4)dx\int 4 x^{4} \sin (7 x^{5}+4) dx∫4x4sin(7x5+4)dx? Choose one: 7x5+47 x^{5}+47x5+4, u=sin⁡(x)u=\sin (x)u=sin(x), u=x+4u=x+4u=x+4, u=xu=xu=x.

Problem 20234

What is the new integral in terms of uuu after performing uuu-sub for ∫x3sin⁡(x4)dx\int x^{3} \sin(x^{4}) \mathrm{dx}∫x3sin(x4)dx?

Problem 20235

Find the average value of x9+7x^{9}+7x9+7 over the interval [4,8][4,8][4,8]. Provide an exact or decimal answer with 0.1% accuracy.

Problem 20236

Find the best u-substitution for ∫sec⁡2xtan⁡4xdx\int \sec ^{2} x \tan ^{4} x d x∫sec2xtan4xdx. Choose one: u=tan⁡(x)u=\tan (x)u=tan(x), u=sec⁡(x)u=\sec (x)u=sec(x), u=tan⁡↑^xu=\tan \hat{\uparrow} xu=tan↑^​x, u=xu=xu=x, u=sec⁡2xu=\sec ^{2} xu=sec2x.

Problem 20237

Find the limit as xxx approaches 0 from the right: lim⁡x→0+sin⁡(x)sin⁡(x)sin⁡(x)\lim _{x \rightarrow 0^{+}} \sin (x)^{\sin (x)^{\sin (x)}}limx→0+​sin(x)sin(x)sin(x).

Problem 20238

3. (a) Why is ∫ln⁡(∣x∣)dx=xln⁡(∣x∣)+C\int \ln (|x|) d x=x \ln (|x|)+C∫ln(∣x∣)dx=xln(∣x∣)+C incorrect? (b) What is the correct value of ∫ln⁡(∣x∣)dx\int \ln (|x|) d x∫ln(∣x∣)dx?

Problem 20239

Problem 20240

Problem 20241

Problem 20242

Une entreprise fabrique qqq objets avec un coût C(q)=0,0000025q3−0,03q2+120q+40000C(q)=0,0000025 q^{3}-0,03 q^{2}+120 q+40000C(q)=0,0000025q3−0,03q2+120q+40000. Trouvez le prix maximal en \$ et le coût marginal à 100 unités.

Problem 20243

Find the marginal cost in \forproducing100unitsusingthecostfunction for producing 100 units using the cost function forproducing100unitsusingthecostfunctionC(q) = 0.0000025q^3 - 0.03q^2 + 120q + 4000$. Round to two decimal places.

Problem 20244

Find the dimensions of a cylindrical can with volume 950 cm³ to minimize metal cost ( $0.01/cm² sides,$ 0.02/cm² top/bottom). What is the height/radius ratio?

Ein Snowboarder fährt mit der Bewegung $s(t)=1,5 t^2$ . Berechne: a) Weg nach 1s und 5s, b) mittlere Geschwindigkeit in 5s, c) Momentangeschwindigkeit nach 5s.

Calculate the integral $A=\int_{9}^{16} 15 \sqrt{4-\sqrt{x}} d x$ .

Berechnen Sie die Fläche zwischen den Kurven $F(x) = -\frac{1}{x^2}$ und $g(x) = 2.5x - 5.25$ .

Find $\frac{dy}{dx}$ for $2x^2 + 3x^3 y - 5y^3 = 16$ and the tangent line at $(2,2)$ .

Find the derivative $\frac{d y}{d x}$ for $y=\int_{0}^{x} \sqrt{12+13 t^{2}} dt$ . What is $\frac{d y}{d x}=\square$ ?

Calculate the integral from 1 to 4 of $x^{2}+\sqrt{x}$ . Find $\int_{1}^{4}\left(x^{2}+\sqrt{x}\right) d x=\square($ exact answer).

Evaluate the integral using the fundamental theorem of calculus: $\int_{1}^{4}(4 x^{2}+5) dx=\square$

A car's velocity is $v(t)=20t$ for $0 \leq t \leq 6$ . Estimate distance with $\Delta t=2$ and find the exact distance.

Estimate the total water volume used by a factory from 6 am to 9 am, given flow rates from $70 \mathrm{~m}^{3}/\mathrm{hr}$ to $250 \mathrm{~m}^{3}/\mathrm{hr}$ . $\text { Best estimate }=\mathbf{i} \quad \mathrm{m}^{3}$

Evaluate the integral using substitution: $\int\left(x^{2}-2 x\right)\left(x^{3}-3 x^{2}+6\right)^{\frac{2}{3}} d x$

Find $f^{\prime}(2)$ given that $40 + 4f(x) + x^{2}(f(x))^{3} = 0$ and $f(2) = -2$ .

Find $g^{\prime}\left(-\frac{8}{9}\right)$ given $8 g(x)-9 x \sin (g(x))=72 x^{2}-17 x-72$ and $g\left(-\frac{8}{9}\right)=0$ .

Evaluate the integral using substitution: $\int \frac{(1+4 \ln x)^{10}}{x} d x$ .

Evaluate the integral: $\int \frac{8}{\sqrt{x}(7+8 \sqrt{x})^{5}} dx$

Ein Behälter hat zu Beginn $2 m^{3}$ Öl. Zuflussrate ist $f(t)=0,1 e^{-0,1 t}$ . Bestimmen Sie die Ölmenge $g(t)$ für $t>0$ und die Zeit bis $2,5 m^{3}$ . Warum nie mehr als $3 m^{3}$ ?

Evaluate the integral $\int \frac{8}{t^{9}} \cos \left(\frac{1}{t^{8}}-1\right) dt$ .

Show why $\lim _{x \rightarrow \infty} x^{n} e^{-x}=0$ using L'Hôpital's rule for positive integer $n$ .

Projectile fired vertically with initial velocity $550 \mathrm{ft/sec}$ and $g=-32 \mathrm{ft}^2/sec$ . Find: A) velocity eqn B) displacement eqn C) velocity at 3s D) time to max height E) max height.

Find the volume of the solid formed by revolving the region between $y=x^{2}+2$ and $y=x^{1/2}$ from $x=2$ to $x=4$ around the $y$ -axis. Round to the tenths place.

Find the volume of the cylinder formed by revolving $f(x)=e^{5 x^{2}}$ from $x=0.8$ to $x=1$ . Round to tenths.

Bestimme die Woche mit den meisten Brötchenverkäufen, den langfristigen Verkauf und die Verkäufe in den ersten 8 Wochen. Funktion: $f(x)=2000 \cdot x \cdot e^{-0,5 x}+2500$ .

Find the indefinite integral of $x^{2} \cos (3 x) \, dx$ .

Find the position function $s(t)$ given $a(t)=16 t-6$ , $v(1)=4$ , and $s(3)=6(10 p t)$ .

Evaluate the integral using integration by parts: $\int \frac{\ln (x)}{x^{5}} \mathrm{~d} x=$

An object at $150^{\circ} \mathrm{F}$ cools in water at $70^{\circ} \mathrm{F}$ . Find $F(t)$ with cooling constant $k=1.4$ . $F(t)=$

Calculate the integral: $\int x^{2} \sin (6 x) d x$ .

What is the best u-substitution for $\int 4 x^{4} \sin (7 x^{5}+4) dx$ ? Choose one: $7 x^{5}+4$ , $u=\sin (x)$ , $u=x+4$ , $u=x$ .

What is the new integral in terms of $u$ after performing $u$ -sub for $\int x^{3} \sin(x^{4}) \mathrm{dx}$ ?

Find the average value of $x^{9}+7$ over the interval $[4,8]$ . Provide an exact or decimal answer with 0.1% accuracy.

Find the best u-substitution for $\int \sec ^{2} x \tan ^{4} x d x$ . Choose one: $u=\tan (x)$ , $u=\sec (x)$ , $u=\tan \hat{\uparrow} x$ , $u=x$ , $u=\sec ^{2} x$ .

Find the limit as $x$ approaches 0 from the right: $\lim _{x \rightarrow 0^{+}} \sin (x)^{\sin (x)^{\sin (x)}}$ .

3. (a) Why is $\int \ln (|x|) d x=x \ln (|x|)+C$ incorrect? (b) What is the correct value of $\int \ln (|x|) d x$ ?

Une entreprise fabrique $q$ objets avec un coût $C(q)=0,0000025 q^{3}-0,03 q^{2}+120 q+40000$ . Trouvez le prix maximal en \$ et le coût marginal à 100 unités.

Find the marginal cost in \ $for producing 100 units using the cost function$ C(q) = 0.0000025q^3 - 0.03q^2 + 120q + 4000$. Round to two decimal places.

Evaluate the integral of $\frac{\tan^{-1}(5x)}{1+25x^2}$ with respect to $x$ .

Find the derivative of $y=\frac{2 x^{2}+9}{x^{2}+7}$ . What is $y^{\prime}=\square$ ?

Find the derivative of $y=\frac{9 x+2}{9 x-5}$ . Which option correctly shows the derivative process?

Find the limit: $\lim _{x \rightarrow 8} \frac{\frac{1}{x}-\frac{1}{8}}{x-8}$ . Choose A or B.

Find the derivative of $f(y)=\frac{4 y-7}{7 \sqrt{y}-4}$ . What is the result?

Find the limit: $\lim _{z \rightarrow 4} \frac{\frac{1}{z}-\frac{1}{4}}{z-4}$ . Choose A or B.

Find the limits of $f(x)=\frac{8 x^{5}+8 x^{4}+4}{8 x^{6}}$ as $x \rightarrow \infty$ and $x \rightarrow -\infty$ .

Find the derivative of $v=\frac{9+x-6 \sqrt{x}}{x}$ . Which option correctly shows the derivative process?

Find the derivative of the function $y=2 e^{-x}+e^{3 x}$ . What is $\frac{d y}{d x}$ ?

Find all derivatives of the function $y=(x-5)(x+3)(x+4)$ . What is $y'=\square$ ?

Find the first and second derivatives of $y=\frac{2 x^{4}+7}{x^{2}}$ . What is $\frac{d y}{d x}$ ?

Evaluate the integral: $\int \frac{\sin (\ln 29 x)}{x} \, dx$

Deux compagnies produisent un bien. Coûts: $7$ pour C1 et $9$ pour C2. Prix: $p=515-0.4Q$ . Trouvez les profits et dérivées.

Find the first and second derivatives of the function $w=\left(\frac{1+15 z}{5 z}\right)(15-z)$ . Calculate $\frac{d w}{d z}=\square$ .

Find the derivative of $y=3 x^{8} e^{x}$ . Which option shows the correct derivative calculation?

Evaluate the integral from 1 to 2: $\int_{1}^{2} \frac{x^{3}+4 x}{x^{4}+8 x^{2}+6} d x$

Find the position of a particle at time $t=8$ given $a(t)=18t+14$ , $s(0)=10$ , and $v(0)=11$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\sqrt[8]{x} \csc x+5$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=-19 x+2 \cos x$ . What is $\frac{d}{d x}(-19 x+2 \cos x)$ ?

Deux compagnies produisent un bien. Compagnie 1: coût \ $7, Compagnie 2: coût \$9. Prix:$ p=515-0,4 Q $. Trouvez les profits$ \Pi_{1} $,$ \Pi_{2}$ et leurs dérivées.

Is the integral $\int_{0}^{\infty} \frac{1}{\sqrt[6]{1+x}} d x$ convergent or divergent? If convergent, find its value.

Find the derivative of $f(x)=\frac{5 x}{x-9}$ using $f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$ .

Find the derivative $\frac{d y}{d x}$ for $y=\frac{\tan x}{1+\tan x}$ . What is $\frac{\mathrm{dy}}{\mathrm{dx}}$ ?

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{4 \cos x}{1-\sin x}$ .

Evaluate the integral: $\int \frac{x^{2}-x+12}{x^{3}+3 x} d x$ (include absolute values and use $C$ for the constant).