Calculus

Problem 15201

Bestimme die erste Ableitung von $g(x)=\frac{\left(x^{2}+1\right)^{\frac{3}{2}}}{3}$ .

See Solution

Problem 15202

Find the derivative of $f(x)=2 \sqrt{x}-2 x$ .

See Solution

Problem 15203

Find the derivative of $f(x)=\frac{5}{x^{3}}-\frac{x^{3}}{5}$ .

See Solution

Problem 15204

Bestimme die erste Ableitung der Funktionen: $f(x)=-\frac{1}{3} x^{3}+x-4$ , $f(x)=\frac{1}{x^{3}}+\sqrt{x}$ , $f(x)=(3 x+1)^{2}$ , $f(x)=(3 x^{2}-1)^{5}$ .

See Solution

Problem 15205

Bestimme die Nullstellen der Funktion $f(x)=2x^{3}+6x^{2}$ und die Wendetangente am Punkt $W(-1 \mid f(-1))$ .

See Solution

Problem 15206

Gegeben ist $f(x)=2 x^{3}+6 x^{2}$ . Finde Nullstellen, Hoch- und Tiefpunkte, Wendetangente $t$ und Schnittwinkel mit der X-Achse. Zeichne $f$ im Intervall $I=[-3; 1]$ .

See Solution

Problem 15207

Find the derivative of $f(x)=(x+4)^{2}$ .

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Problem 15208

Find the instantaneous temperature change for a particle moving along $\bar{r}(t)=\langle t^{3}, t^{2}, t \rangle$ in a cube with $\Theta(x, y, z)=1000 x y z e^{-x} e^{-y} e^{-z}$ . What is it at $\bar{r}(0.7)$ ?

See Solution

Problem 15209

Find the limit: $L = \lim _{x \rightarrow 0} \frac{\sin ^{-1} x - \tan ^{-1} x}{x^{3}}$ .

See Solution

Problem 15210

Untersuche die Funktion $f(x)=-\frac{4}{9} x^{2}+\frac{8}{3} x$ im Intervall $[0 ; 6]$ : Nullstellen, Scheitelpunkt, mittlere und lokale Steigung.

See Solution

Problem 15211

Bestimmen Sie, wo die Tangente an $f(x)=\sqrt{x}$ bei $P(4|2)$ die x-Achse schneidet, um die Aufschüttung zu starten.

See Solution

Problem 15212

Find the derivative of $f(x) = \frac{e^{4 x} \sin x}{x \cos 2 x}$ .

See Solution

Problem 15213

Ein Heliumballon steigt mit der Geschwindigkeit $v(t)=\frac{10}{t^{2}+2 t+1}$ . Beantworte: a) Anfangsgeschwindigkeit, b) immer aufwärts? c) Maximalhöhe? d) Zeit und Geschwindigkeit bei Halbhöhe?

See Solution

Problem 15214

Approximate $\int_{0}^{3} \sqrt{1+x^{2}} dx$ using a midpoint Riemann sum with intervals $[0,1],[1,2],[2,3]$ .

See Solution

Problem 15215

Find the area in the first quadrant between $f(x)=4x-x^3$ and the $x$ -axis. Choices: (A) $\frac{11}{4}$ (B) $\frac{7}{2}$ (C) 4 (D) $\frac{11}{2}$ .

See Solution

Problem 15216

Find the average value of the function $h(x)=-x^{2}-9x+26$ over the interval $-10 \leq x \leq -2$ .

See Solution

Problem 15217

Find the limit as $n \to \infty$ for the series $a_{n}=\sum_{k=1}^{n} \frac{(2k-1)^{2}}{n^{3}}$ .

See Solution

Problem 15218

Determine the remainder term $R_{N}$ in the Taylor series for $f(x)=x^{2}$ centered at $a=-1$ over $[-1,5]$ .

See Solution

Problem 15219

Simplify the expression: $\frac{d}{d x} \int_{4}^{x^{3}} \frac{d p}{p^{2}}$

See Solution

Problem 15220

Differentiate $9 x^{2}-8 y^{3}=145$ to find $\frac{d y}{d x}$ and the slope at the point (3, -2).

See Solution

Problem 15221

Simplify the expression: $\frac{d}{d x} \int_{4}^{x^{3}} \frac{d p}{p^{2}}$ .

See Solution

Problem 15222

Simplify the expression: $\frac{d}{d x} \int_{x}^{1} \sqrt{t^{3}+4} d t = \square$

See Solution

Problem 15223

Differentiate to find $\frac{d y}{d x}$ for the equation $y^{7}=x^{9}$ .

See Solution

Problem 15224

Find the limit: $\lim _{x \rightarrow 4} \frac{4-\sqrt{12+x}}{x-4}$ .

See Solution

Problem 15225

Find $\Delta C$ and $C'(x)$ for the cost function $C(x)=0.01 x^{2}+0.4 x+30$ at $x=90$ and $\Delta x=1$ .

See Solution

Problem 15226

Evaluate the integral: $\int_{0}^{\pi / 4} 6 \sec ^{2} x \, dx = \square$ (Type an exact answer.)

See Solution

Problem 15227

Evaluate the integral from 1 to 27: $\int_{1}^{27} \frac{\sqrt[3]{x^{2}}-1}{\sqrt[3]{x}} d x=\square$ (Simplify your answer.)

See Solution

Problem 15228

Evaluate the integral from 1 to 27: $\int_{1}^{27} \frac{\sqrt[3]{x^{2}}-1}{\sqrt[3]{x}} d x=\square($ Simplify your answer.)

See Solution

Problem 15229

Alice the Amoeba travels with $s=t^{3}-6 t^{2}+9 t+1$ . Answer these: a) Velocity at 3s? b) Velocity when $a=0$ ? c) When at rest? d) When moving backwards? e) Diagram for 6s with distances.

See Solution

Problem 15230

Does the limit of $f(x)$ as $x$ approaches 4 exist given the values near 4?

See Solution

Problem 15231

Alice the Amoeba's motion is described by $s=t^{3}-6t^{2}+9t+1$ for $t \geq 0$ .
a) Find her velocity at $t=3$ . b) When is her acceleration $0 \mathrm{~m/s}^{2}$ ? c) When is she at rest? d) When is she moving backwards? e) Sketch her distance for the first 6 seconds and note the distance at each second.

See Solution

Problem 15232

Find if the limit of $f(x)$ as $x$ approaches 4 exists, given values near $x=4$ .

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Problem 15233

Find the rate of change of volume $V=\frac{4}{3} \pi r^{3}$ when $r=5.9 \mathrm{~cm}$ and $\frac{dr}{dt}=0.9 \mathrm{~cm/week}$ .

See Solution

Problem 15234

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{\pi / 6} 3 \sin x \, dx = \square$ (Exact answer with radicals.)

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Problem 15235

Berechne den Flächeninhalt zwischen der Funktion $f(x)=(x-1)(x+2)(x-3)$ und der $\mathrm{x}$ -Achse im Intervall $I=[-1 ; 3]$ .

See Solution

Problem 15236

Given the function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $|f(x, y)-7|<2 \sqrt{x^{2}+y^{2}}$ , is $\lim_{(x, y) \rightarrow(0,0)} f(x, y)=7$ true?

See Solution

Problem 15237

Check which functions are differentiable at $(0,0)$ :
1. $f(x, y)=\frac{x^{2} y^{2}+3 x y+1}{x^{3} y-y x}$
2. $f(x, y)=\frac{\cos (x+y)}{\sin (x-y)}$
3. $f(x, y)=\frac{\sqrt{x^{2}+y^{2}}}{2 x+3 y-1}$
4. $f(x, y)=\frac{\ln \left(x^{2}+y^{2}+1\right)}{e^{x+y}}$

See Solution

Problem 15238

Find the tangent line equation for the function $f(x)=x^{2}+7$ at the point where $x=9$ .

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Problem 15239

Find the tangent line equation for $f(x)=7 x^{2}+10 x-1$ at $x=-7$ .

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Problem 15240

Given $h(t)=f(x(t), y(t))$ , check if $\frac{dh}{dt} = 0$ at $t=0$ given certain conditions on $f$ , $x(t)$ , and $y(t)$ .

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Problem 15241

Find the tangent line equation for the function $f(x)=4 x^{2}+6 x-2$ at the point where $x=6$ .

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Problem 15242

Find the derivative of the function $f(x)=5 x^{2}$ using the limit definition, without simplifying your answer.

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Problem 15243

Find the tangent line equation for the function $f(x)=-10 x^{2}+4 x-7$ at the point where $x=-2$ .

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Problem 15244

Find the first four nonzero terms of the Taylor series for $\frac{1}{\sqrt{1-x^{2}}}$ at $x=0$ .

See Solution

Problem 15245

Analysiere den Testflug eines Modellhelikopters mit der Funktion $h(t)=-t^{2}-4t$ für $0 \leq t \leq 3$ .
a) Finde den Zeitpunkt, an dem der Helikopter am schnellsten steigt. b) Beschreibe den Zustand, in dem die Höhe konstant bleibt. c) Mache weitere mathematische Aussagen über den Flugverlauf.

See Solution

Problem 15246

Bestimme den Wert von a, sodass $\int_{1}^{2}(3 a x^{2}+6 x) d x=2$ und $\int_{2}^{a}(2 x+5) d x=0$ .

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Problem 15247

Evaluate the integral from 1 to 32 of $x^{-9/5}$ . What is $\int_{1}^{32} x^{-9/5} d x$ ? Simplify your answer.

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Problem 15248

Evaluate $q(x)=\sqrt{x+11}$ at $x=-3$ and $x=0$ , then find the average rate of change between these points.

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Problem 15249

Find the 1st and 2nd derivatives of $f(x)=\frac{3}{4} x^{4}-\frac{9}{2} x^{2}+3 x+4$ .

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Problem 15250

Bestimmen Sie die Häufungspunkte der Folge $a_{n}=\frac{(-1)^{n} \cdot n^{2}}{(3 \cdot n+3)^{2}}$ . Anzahl und größter Punkt?

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Problem 15251

Evaluate the integral from 1/2 to 2: $\int_{1 / 2}^{2}\left(3-\frac{3}{x^{2}}\right) d x$ .

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Problem 15252

Gegeben ist die Funktion $f(x)=e^{-x}(x^{2}-3)$ . Bestimmen Sie Definitionsbereich, Grenzverhalten, Ableitung, Nullstellen, Wendepunkte und Tangente.

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Problem 15253

Evaluate the integral from 1/2 to 2 of $3 - \frac{3}{x^2}$ using the Fundamental Theorem of Calculus.

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Problem 15254

Find the function $f$ where $f'(x) = x^2 + 7$ and $f(0) = 3.

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Problem 15255

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

See Solution

Problem 15256

Find the limit: $\lim _{x \rightarrow 0} \cot x\left(\frac{1}{\sin x}-\frac{1}{x}\right)$ .

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Problem 15257

Find the derivative, integral, or specific value of the function $f_{k}(x)=\left(3 x-2 x^{2}\right) \cdot e^{-k x}$ .

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Problem 15258

Find the derivative of $g_{a}(t)=(3t+2) \cdot e^{at^{2}}$ .

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Problem 15259

A bodybuilder lifts an $80 \mathrm{~kg}$ barbell to a height. It drops and reaches $4.5 \mathrm{~m/s}$ . Find the height.

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Problem 15260

Find the consumers' surplus when the price is set at 400 for the demand function $p(x)=1200-\left(\frac{1}{10000}\right) x^{3}$ .

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Problem 15261

Find the $\mathrm{x}$ -coordinate of the inflection point for the graph of $y=-x(x-2)(x+2)$ .

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Problem 15262

Calculate the Gini coefficient for the Lorenz curve $f(x)=\frac{1}{100}(40 x^{4}+24 x^{3}+24 x^{2}+12 x)$ .

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Problem 15263

Calculate the Gini coefficient for the Lorenz curve $f(x)=\frac{1}{100}(40 x^{4}+24 x^{3}+24 x^{2}+12 x)$ .

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Problem 15264

Find consumers' surplus when price is \ $400 for demand function$ p(x)=1200-\left(\frac{1}{10000}\right) x^{3} $. Answer:$ $199900$ $

See Solution

Problem 15265

Find the consumer surplus when the price is set at 400 for the demand function $p(x)=1200-\left(\frac{1}{10000}\right) x^{3}$ .

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Problem 15266

1. Trouvez les dérivées des fonctions suivantes : a) $f(x)=\sqrt{x} \arcsin x$ , b) $g(x)=\arcsin \left(x^{7}-3 x\right)$ , c) $y=\sqrt{\arcsin \left(x^{4}\right)}$ .

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Problem 15267

Evaluate the integral $\int_{1 / 3}^{3}\left(4-\frac{4}{x^{2}}\right) d x$ and sketch the area under the curve.

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Problem 15268

Find the linearization $L(x)$ of the function $f(x)=\sqrt{x+4}$ at the point $x=1$ .

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Problem 15269

Find the derivative of $f_{a}(x)=x(x-a)^{2}$ .

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Problem 15270

A satellite orbits Earth at $196 \mathrm{~km}$ . Find its orbital radius, speed, and period (in seconds and hours). Given: $\mathrm{G}=6.674 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} ; M_{\text {Earth }}=5.974 \times 10^{24} \mathrm{~kg} ; R_{\text {Earth }}=6.378 \times 10^{6} \mathrm{~m}$ .

See Solution

Problem 15271

Approximate $\sqrt{52}$ using the linearization of the square root function at $a=49$ .

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Problem 15272

Differentiate implicitly the equation $\frac{x}{y} - \frac{y}{x} = 1$ .

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Problem 15273

Simplify the limit as $x$ approaches $-\infty$ : $\lim _{x \rightarrow-\infty} \frac{x^{3}}{12 x^{6}}$

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Problem 15274

A motorcycle daredevil starts at $43 \mathrm{~m/s}$ and peaks at $36 \mathrm{~m/s}$ . Find the height using energy concepts.

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Problem 15275

Finde die Tangentengleichung am Wendepunkt für die Funktionen: a) $f(x)=\frac{1}{6} x^{3}-\frac{3}{4} x^{2}+2$ , b) $g(x)=9 x \cdot\left(\frac{1}{6} x-1\right)^{2}$ , c) $h(t)=\left(t^{2}-t\right) \cdot(2 t-1)$ .

See Solution

Problem 15276

Bestimme die Steigung $k$ der Funktion $f(x)=\frac{2}{3}-7 x$ mit dem Differenzialquotienten.

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Problem 15277

Find the derivative of the function $f(x)=5 e^{x \cos x}$ . What is $f^{\prime}(x)$ ?

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Problem 15278

Find the derivative $f^{\prime}(3)$ for the function $f(x)=\sqrt{3 x^{2}+4}$ .

See Solution

Problem 15279

What happens to demand $d(p)=30+460 e^{-3.77}$ as price $p$ increases? It approaches 25 items/month.

See Solution

Problem 15280

Find the limit: $\lim _{x \rightarrow \pi / 4} \frac{\cos (x)-\sin (x)}{\tan (x)-1}$ . Use l'Hospital's Rule if needed.

See Solution

Problem 15281

Find the derivative $\frac{d y}{d x}$ for $y=-9 \frac{1}{\sin ^{3}(\ln (x))}$ .

See Solution

Problem 15282

Find the critical point $c$ larger than zero for $y=x^{7/2}-4x^{2}$ , and determine intervals of increase/decrease.

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Problem 15283

Use the First Derivative Test to find if $f(x)=x^{2}$ has a local min or max at $x=0$ .

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Problem 15284

Check if $f(x)=x^{4}$ is concave up for $x>0$ and $x<0$ . Is $x=0$ an inflection point? Explain. [2pts]

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Problem 15285

As price $p$ increases, what happens to demand $d(p)=30+460 e^{-0.27 p}$ ? Choose the correct limit.

See Solution

Problem 15286

Berechnen Sie die Integrale: $\int_{0}^{3}(x^{2}-2) dx$ und $\int_{-2}^{0}(-2 x^{3}+3 x^{2}-4) dx$ .

See Solution

Problem 15287

1. Trouvez l'élasticité de la demande $E$ pour la fonction $p(q)=-0,25 q+60$ à $p=20\$,$ 30\ $, et$ 40\$.

See Solution

Problem 15288

Berechnen Sie die Integrale a), d), g), und h) mithilfe einer Stammfunktion.

See Solution

Problem 15289

Calculate the integral $\int_{-2}^{0}\left(\frac{1}{3} x^{3}-\frac{1}{2} x^{2}+1\right) d x$ .

See Solution

Problem 15290

Berechnen Sie die Flächeninhalte der von den Funktionen $f(x)=4-x^{2}$ , $f(x)=-x^{2}+6x+7$ , $f(x)=x(x-1)(3-x)$ und $f(x)=(x-1)(x-2)(x-3)$ mit der $x$ -Achse eingeschlossenen Flächen.

See Solution

Problem 15291

Find the general antiderivative of $f(x)=\sqrt[6]{2}+\sqrt[6]{x}$ and verify by differentiation. Use $C$ for the constant.

See Solution

Problem 15292

What is the definition of absolute extreme values for a function at point $c$ in interval $[a, b]$ ? Choose the correct option.

See Solution

Problem 15293

Find $y(4)$ given $\frac{d y}{d x}=3 x^{1/2}-6 x^{-1/2}$ and $y(1)=-12$ .

See Solution

Problem 15294

What conditions ensure a function has an absolute max and min on an interval? A. Continuous on $[a, b]$ . B. Continuous on a subinterval. C. Critical point in $[a, b]$ . D. Continuous on $(a, b)$ .

See Solution

Problem 15295

Find the general anti-derivative of $h(x)=e^{x}+x^{3}$ , using $C$ as the constant. What is $H(x)=?$

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Problem 15296

Find the price-supply equation if the marginal price for $x$ bottles is $p^{\prime}(x)=\frac{70}{(x+4)^{2}}$ and $p(3)=12$ .

See Solution

Problem 15297

Find $g_{x}$ for $g(x, y)=-\frac{5}{x^{5} y^{4}}-\frac{3}{x^{3}}+x 5^{y}-\frac{1}{y^{4}}+3$ . Options: a) b) c) d)

See Solution

Problem 15298

How to find absolute max/min of a continuous function on $[a, b]$ ?
A. Endpoints only. B. Critical points only. C. Endpoints & critical points. D. Open interval $(a, b)$ .

See Solution

Problem 15299

Find the second order partial derivative $\frac{\partial^{2} f}{\partial x \partial y}$ for $f=\sqrt{9 x^{4}+9 y^{3}}$ .

See Solution

Problem 15300

Find the derivative of $f(x) = e^{2x} \sin 3x$ .

See Solution

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Calculus

Problem 15201

Bestimme die erste Ableitung von g(x)=(x2+1)323g(x)=\frac{\left(x^{2}+1\right)^{\frac{3}{2}}}{3}g(x)=3(x2+1)23​​.

Problem 15202

Find the derivative of f(x)=2x−2xf(x)=2 \sqrt{x}-2 xf(x)=2x​−2x.

Problem 15203

Find the derivative of f(x)=5x3−x35f(x)=\frac{5}{x^{3}}-\frac{x^{3}}{5}f(x)=x35​−5x3​.

Problem 15204

Bestimme die erste Ableitung der Funktionen: f(x)=−13x3+x−4f(x)=-\frac{1}{3} x^{3}+x-4f(x)=−31​x3+x−4, f(x)=1x3+xf(x)=\frac{1}{x^{3}}+\sqrt{x}f(x)=x31​+x​, f(x)=(3x+1)2f(x)=(3 x+1)^{2}f(x)=(3x+1)2, f(x)=(3x2−1)5f(x)=(3 x^{2}-1)^{5}f(x)=(3x2−1)5.

Problem 15205

Bestimme die Nullstellen der Funktion f(x)=2x3+6x2f(x)=2x^{3}+6x^{2}f(x)=2x3+6x2 und die Wendetangente am Punkt W(−1∣f(−1))W(-1 \mid f(-1))W(−1∣f(−1)).

Problem 15206

Gegeben ist f(x)=2x3+6x2f(x)=2 x^{3}+6 x^{2}f(x)=2x3+6x2. Finde Nullstellen, Hoch- und Tiefpunkte, Wendetangente ttt und Schnittwinkel mit der X-Achse. Zeichne fff im Intervall I=[−3;1]I=[-3; 1]I=[−3;1].

Problem 15207

Find the derivative of f(x)=(x+4)2f(x)=(x+4)^{2}f(x)=(x+4)2.

Problem 15208

Problem 15209

Find the limit: L=lim⁡x→0sin⁡−1x−tan⁡−1xx3L = \lim _{x \rightarrow 0} \frac{\sin ^{-1} x - \tan ^{-1} x}{x^{3}}L=limx→0​x3sin−1x−tan−1x​.

Problem 15210

Untersuche die Funktion f(x)=−49x2+83xf(x)=-\frac{4}{9} x^{2}+\frac{8}{3} xf(x)=−94​x2+38​x im Intervall [0;6][0 ; 6][0;6]: Nullstellen, Scheitelpunkt, mittlere und lokale Steigung.

Problem 15211

Bestimmen Sie, wo die Tangente an f(x)=xf(x)=\sqrt{x}f(x)=x​ bei P(4∣2)P(4|2)P(4∣2) die x-Achse schneidet, um die Aufschüttung zu starten.

Problem 15212

Find the derivative of f(x)=e4xsin⁡xxcos⁡2xf(x) = \frac{e^{4 x} \sin x}{x \cos 2 x}f(x)=xcos2xe4xsinx​.

Problem 15213

Ein Heliumballon steigt mit der Geschwindigkeit v(t)=10t2+2t+1v(t)=\frac{10}{t^{2}+2 t+1}v(t)=t2+2t+110​. Beantworte: a) Anfangsgeschwindigkeit, b) immer aufwärts? c) Maximalhöhe? d) Zeit und Geschwindigkeit bei Halbhöhe?

Problem 15214

Approximate ∫031+x2dx\int_{0}^{3} \sqrt{1+x^{2}} dx∫03​1+x2​dx using a midpoint Riemann sum with intervals [0,1],[1,2],[2,3][0,1],[1,2],[2,3][0,1],[1,2],[2,3].

Problem 15215

Find the area in the first quadrant between f(x)=4x−x3f(x)=4x-x^3f(x)=4x−x3 and the xxx-axis. Choices: (A) 114\frac{11}{4}411​ (B) 72\frac{7}{2}27​ (C) 4 (D) 112\frac{11}{2}211​.

Problem 15216

Find the average value of the function h(x)=−x2−9x+26h(x)=-x^{2}-9x+26h(x)=−x2−9x+26 over the interval −10≤x≤−2-10 \leq x \leq -2−10≤x≤−2.

Problem 15217

Find the limit as n→∞n \to \inftyn→∞ for the series an=∑k=1n(2k−1)2n3a_{n}=\sum_{k=1}^{n} \frac{(2k-1)^{2}}{n^{3}}an​=∑k=1n​n3(2k−1)2​.

Problem 15218

Determine the remainder term RNR_{N}RN​ in the Taylor series for f(x)=x2f(x)=x^{2}f(x)=x2 centered at a=−1a=-1a=−1 over [−1,5][-1,5][−1,5].

Problem 15219

Simplify the expression: ddx∫4x3dpp2\frac{d}{d x} \int_{4}^{x^{3}} \frac{d p}{p^{2}}dxd​∫4x3​p2dp​

Problem 15220

Differentiate 9x2−8y3=1459 x^{2}-8 y^{3}=1459x2−8y3=145 to find dydx\frac{d y}{d x}dxdy​ and the slope at the point (3, -2).

Problem 15221

Simplify the expression: ddx∫4x3dpp2\frac{d}{d x} \int_{4}^{x^{3}} \frac{d p}{p^{2}}dxd​∫4x3​p2dp​.

Problem 15222

Simplify the expression: ddx∫x1t3+4dt=□\frac{d}{d x} \int_{x}^{1} \sqrt{t^{3}+4} d t = \squaredxd​∫x1​t3+4​dt=□

Problem 15223

Differentiate to find dydx\frac{d y}{d x}dxdy​ for the equation y7=x9y^{7}=x^{9}y7=x9.

Problem 15224

Find the limit: lim⁡x→44−12+xx−4\lim _{x \rightarrow 4} \frac{4-\sqrt{12+x}}{x-4}limx→4​x−44−12+x​​.

Problem 15225

Find ΔC\Delta CΔC and C′(x)C'(x)C′(x) for the cost function C(x)=0.01x2+0.4x+30C(x)=0.01 x^{2}+0.4 x+30C(x)=0.01x2+0.4x+30 at x=90x=90x=90 and Δx=1\Delta x=1Δx=1.

Problem 15226

Evaluate the integral: ∫0π/46sec⁡2x dx=□\int_{0}^{\pi / 4} 6 \sec ^{2} x \, dx = \square∫0π/4​6sec2xdx=□ (Type an exact answer.)

Problem 15227

Evaluate the integral from 1 to 27: ∫127x23−1x3dx=□\int_{1}^{27} \frac{\sqrt[3]{x^{2}}-1}{\sqrt[3]{x}} d x=\square∫127​3x​3x2​−1​dx=□ (Simplify your answer.)

Problem 15228

Evaluate the integral from 1 to 27: ∫127x23−1x3dx=□(\int_{1}^{27} \frac{\sqrt[3]{x^{2}}-1}{\sqrt[3]{x}} d x=\square(∫127​3x​3x2​−1​dx=□( Simplify your answer.)

Problem 15229

Alice the Amoeba travels with s=t3−6t2+9t+1s=t^{3}-6 t^{2}+9 t+1s=t3−6t2+9t+1. Answer these: a) Velocity at 3s? b) Velocity when a=0a=0a=0? c) When at rest? d) When moving backwards? e) Diagram for 6s with distances.

Problem 15230

Does the limit of f(x)f(x)f(x) as xxx approaches 4 exist given the values near 4?

Problem 15231

Problem 15232

Find if the limit of f(x)f(x)f(x) as xxx approaches 4 exists, given values near x=4x=4x=4.

Problem 15233

Find the rate of change of volume V=43πr3V=\frac{4}{3} \pi r^{3}V=34​πr3 when r=5.9 cmr=5.9 \mathrm{~cm}r=5.9 cm and drdt=0.9 cm/week\frac{dr}{dt}=0.9 \mathrm{~cm/week}dtdr​=0.9 cm/week.

Problem 15234

Evaluate the integral using the Fundamental Theorem of Calculus: ∫0π/63sin⁡x dx=□\int_{0}^{\pi / 6} 3 \sin x \, dx = \square∫0π/6​3sinxdx=□ (Exact answer with radicals.)

Problem 15235

Berechne den Flächeninhalt zwischen der Funktion f(x)=(x−1)(x+2)(x−3)f(x)=(x-1)(x+2)(x-3)f(x)=(x−1)(x+2)(x−3) und der x\mathrm{x}x-Achse im Intervall I=[−1;3]I=[-1 ; 3]I=[−1;3].

Problem 15236

Given the function f:R2→Rf: \mathbb{R}^{2} \rightarrow \mathbb{R}f:R2→R with ∣f(x,y)−7∣<2x2+y2|f(x, y)-7|<2 \sqrt{x^{2}+y^{2}}∣f(x,y)−7∣<2x2+y2​, is lim⁡(x,y)→(0,0)f(x,y)=7\lim_{(x, y) \rightarrow(0,0)} f(x, y)=7lim(x,y)→(0,0)​f(x,y)=7 true?

Problem 15237

Problem 15238

Find the tangent line equation for the function f(x)=x2+7f(x)=x^{2}+7f(x)=x2+7 at the point where x=9x=9x=9.

Problem 15239

Find the tangent line equation for f(x)=7x2+10x−1f(x)=7 x^{2}+10 x-1f(x)=7x2+10x−1 at x=−7x=-7x=−7.

Problem 15240

Given h(t)=f(x(t),y(t))h(t)=f(x(t), y(t))h(t)=f(x(t),y(t)), check if dhdt=0\frac{dh}{dt} = 0dtdh​=0 at t=0t=0t=0 given certain conditions on fff, x(t)x(t)x(t), and y(t)y(t)y(t).

Problem 15241

Find the tangent line equation for the function f(x)=4x2+6x−2f(x)=4 x^{2}+6 x-2f(x)=4x2+6x−2 at the point where x=6x=6x=6.

Bestimme die erste Ableitung von $g(x)=\frac{\left(x^{2}+1\right)^{\frac{3}{2}}}{3}$ .

Find the derivative of $f(x)=2 \sqrt{x}-2 x$ .

Find the derivative of $f(x)=\frac{5}{x^{3}}-\frac{x^{3}}{5}$ .

Bestimme die erste Ableitung der Funktionen: $f(x)=-\frac{1}{3} x^{3}+x-4$ , $f(x)=\frac{1}{x^{3}}+\sqrt{x}$ , $f(x)=(3 x+1)^{2}$ , $f(x)=(3 x^{2}-1)^{5}$ .

Bestimme die Nullstellen der Funktion $f(x)=2x^{3}+6x^{2}$ und die Wendetangente am Punkt $W(-1 \mid f(-1))$ .

Gegeben ist $f(x)=2 x^{3}+6 x^{2}$ . Finde Nullstellen, Hoch- und Tiefpunkte, Wendetangente $t$ und Schnittwinkel mit der X-Achse. Zeichne $f$ im Intervall $I=[-3; 1]$ .

Find the derivative of $f(x)=(x+4)^{2}$ .

Find the limit: $L = \lim _{x \rightarrow 0} \frac{\sin ^{-1} x - \tan ^{-1} x}{x^{3}}$ .

Untersuche die Funktion $f(x)=-\frac{4}{9} x^{2}+\frac{8}{3} x$ im Intervall $[0 ; 6]$ : Nullstellen, Scheitelpunkt, mittlere und lokale Steigung.

Bestimmen Sie, wo die Tangente an $f(x)=\sqrt{x}$ bei $P(4|2)$ die x-Achse schneidet, um die Aufschüttung zu starten.

Find the derivative of $f(x) = \frac{e^{4 x} \sin x}{x \cos 2 x}$ .

Ein Heliumballon steigt mit der Geschwindigkeit $v(t)=\frac{10}{t^{2}+2 t+1}$ . Beantworte: a) Anfangsgeschwindigkeit, b) immer aufwärts? c) Maximalhöhe? d) Zeit und Geschwindigkeit bei Halbhöhe?

Approximate $\int_{0}^{3} \sqrt{1+x^{2}} dx$ using a midpoint Riemann sum with intervals $[0,1],[1,2],[2,3]$ .

Find the area in the first quadrant between $f(x)=4x-x^3$ and the $x$ -axis. Choices: (A) $\frac{11}{4}$ (B) $\frac{7}{2}$ (C) 4 (D) $\frac{11}{2}$ .

Find the average value of the function $h(x)=-x^{2}-9x+26$ over the interval $-10 \leq x \leq -2$ .

Find the limit as $n \to \infty$ for the series $a_{n}=\sum_{k=1}^{n} \frac{(2k-1)^{2}}{n^{3}}$ .

Determine the remainder term $R_{N}$ in the Taylor series for $f(x)=x^{2}$ centered at $a=-1$ over $[-1,5]$ .

Simplify the expression: $\frac{d}{d x} \int_{4}^{x^{3}} \frac{d p}{p^{2}}$

Differentiate $9 x^{2}-8 y^{3}=145$ to find $\frac{d y}{d x}$ and the slope at the point (3, -2).

Simplify the expression: $\frac{d}{d x} \int_{4}^{x^{3}} \frac{d p}{p^{2}}$ .

Simplify the expression: $\frac{d}{d x} \int_{x}^{1} \sqrt{t^{3}+4} d t = \square$

Differentiate to find $\frac{d y}{d x}$ for the equation $y^{7}=x^{9}$ .

Find the limit: $\lim _{x \rightarrow 4} \frac{4-\sqrt{12+x}}{x-4}$ .

Find $\Delta C$ and $C'(x)$ for the cost function $C(x)=0.01 x^{2}+0.4 x+30$ at $x=90$ and $\Delta x=1$ .

Evaluate the integral: $\int_{0}^{\pi / 4} 6 \sec ^{2} x \, dx = \square$ (Type an exact answer.)

Evaluate the integral from 1 to 27: $\int_{1}^{27} \frac{\sqrt[3]{x^{2}}-1}{\sqrt[3]{x}} d x=\square$ (Simplify your answer.)

Evaluate the integral from 1 to 27: $\int_{1}^{27} \frac{\sqrt[3]{x^{2}}-1}{\sqrt[3]{x}} d x=\square($ Simplify your answer.)

Alice the Amoeba travels with $s=t^{3}-6 t^{2}+9 t+1$ . Answer these: a) Velocity at 3s? b) Velocity when $a=0$ ? c) When at rest? d) When moving backwards? e) Diagram for 6s with distances.

Does the limit of $f(x)$ as $x$ approaches 4 exist given the values near 4?

Find if the limit of $f(x)$ as $x$ approaches 4 exists, given values near $x=4$ .

Find the rate of change of volume $V=\frac{4}{3} \pi r^{3}$ when $r=5.9 \mathrm{~cm}$ and $\frac{dr}{dt}=0.9 \mathrm{~cm/week}$ .

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{\pi / 6} 3 \sin x \, dx = \square$ (Exact answer with radicals.)

Berechne den Flächeninhalt zwischen der Funktion $f(x)=(x-1)(x+2)(x-3)$ und der $\mathrm{x}$ -Achse im Intervall $I=[-1 ; 3]$ .

Given the function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $|f(x, y)-7|<2 \sqrt{x^{2}+y^{2}}$ , is $\lim_{(x, y) \rightarrow(0,0)} f(x, y)=7$ true?

Find the tangent line equation for the function $f(x)=x^{2}+7$ at the point where $x=9$ .

Find the tangent line equation for $f(x)=7 x^{2}+10 x-1$ at $x=-7$ .

Given $h(t)=f(x(t), y(t))$ , check if $\frac{dh}{dt} = 0$ at $t=0$ given certain conditions on $f$ , $x(t)$ , and $y(t)$ .

Find the tangent line equation for the function $f(x)=4 x^{2}+6 x-2$ at the point where $x=6$ .

Find the derivative of the function $f(x)=5 x^{2}$ using the limit definition, without simplifying your answer.

Find the tangent line equation for the function $f(x)=-10 x^{2}+4 x-7$ at the point where $x=-2$ .

Find the first four nonzero terms of the Taylor series for $\frac{1}{\sqrt{1-x^{2}}}$ at $x=0$ .

Bestimme den Wert von a, sodass $\int_{1}^{2}(3 a x^{2}+6 x) d x=2$ und $\int_{2}^{a}(2 x+5) d x=0$ .

Evaluate the integral from 1 to 32 of $x^{-9/5}$ . What is $\int_{1}^{32} x^{-9/5} d x$ ? Simplify your answer.

Evaluate $q(x)=\sqrt{x+11}$ at $x=-3$ and $x=0$ , then find the average rate of change between these points.

Find the 1st and 2nd derivatives of $f(x)=\frac{3}{4} x^{4}-\frac{9}{2} x^{2}+3 x+4$ .

Bestimmen Sie die Häufungspunkte der Folge $a_{n}=\frac{(-1)^{n} \cdot n^{2}}{(3 \cdot n+3)^{2}}$ . Anzahl und größter Punkt?

Evaluate the integral from 1/2 to 2: $\int_{1 / 2}^{2}\left(3-\frac{3}{x^{2}}\right) d x$ .

Gegeben ist die Funktion $f(x)=e^{-x}(x^{2}-3)$ . Bestimmen Sie Definitionsbereich, Grenzverhalten, Ableitung, Nullstellen, Wendepunkte und Tangente.

Evaluate the integral from 1/2 to 2 of $3 - \frac{3}{x^2}$ using the Fundamental Theorem of Calculus.

Find the function $f$ where $f'(x) = x^2 + 7$ and $f(0) = 3.

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

Find the limit: $\lim _{x \rightarrow 0} \cot x\left(\frac{1}{\sin x}-\frac{1}{x}\right)$ .

Find the derivative, integral, or specific value of the function $f_{k}(x)=\left(3 x-2 x^{2}\right) \cdot e^{-k x}$ .

Find the derivative of $g_{a}(t)=(3t+2) \cdot e^{at^{2}}$ .

A bodybuilder lifts an $80 \mathrm{~kg}$ barbell to a height. It drops and reaches $4.5 \mathrm{~m/s}$ . Find the height.

Find the consumers' surplus when the price is set at 400 for the demand function $p(x)=1200-\left(\frac{1}{10000}\right) x^{3}$ .

Find the $\mathrm{x}$ -coordinate of the inflection point for the graph of $y=-x(x-2)(x+2)$ .

Calculate the Gini coefficient for the Lorenz curve $f(x)=\frac{1}{100}(40 x^{4}+24 x^{3}+24 x^{2}+12 x)$ .

Calculate the Gini coefficient for the Lorenz curve $f(x)=\frac{1}{100}(40 x^{4}+24 x^{3}+24 x^{2}+12 x)$ .

Find consumers' surplus when price is \ $400 for demand function$ p(x)=1200-\left(\frac{1}{10000}\right) x^{3} $. Answer:$ $199900$ $

Find the consumer surplus when the price is set at 400 for the demand function $p(x)=1200-\left(\frac{1}{10000}\right) x^{3}$ .

1. Trouvez les dérivées des fonctions suivantes : a) $f(x)=\sqrt{x} \arcsin x$ , b) $g(x)=\arcsin \left(x^{7}-3 x\right)$ , c) $y=\sqrt{\arcsin \left(x^{4}\right)}$ .

Evaluate the integral $\int_{1 / 3}^{3}\left(4-\frac{4}{x^{2}}\right) d x$ and sketch the area under the curve.

Find the linearization $L(x)$ of the function $f(x)=\sqrt{x+4}$ at the point $x=1$ .

Find the derivative of $f_{a}(x)=x(x-a)^{2}$ .

Approximate $\sqrt{52}$ using the linearization of the square root function at $a=49$ .

Differentiate implicitly the equation $\frac{x}{y} - \frac{y}{x} = 1$ .

Simplify the limit as $x$ approaches $-\infty$ : $\lim _{x \rightarrow-\infty} \frac{x^{3}}{12 x^{6}}$

A motorcycle daredevil starts at $43 \mathrm{~m/s}$ and peaks at $36 \mathrm{~m/s}$ . Find the height using energy concepts.