Calculus

Problem 19201

Given the curve $(2 y+1)^{3}-24 x=-3$ , prove that $\frac{d y}{d x}=\frac{4}{(2 y+1)^{2}}$ .

See Solution

Problem 19202

Finde die Stammfunktionen für die folgenden Funktionen: a) $f(x)=x^{3}-2 x+1$ b) $f(x)=x+\sin (x)$ c) $f(x)=3 \sqrt{x}$ d) $f(x)=4 x^{6}$ e) $f(x)=3 a \cdot \sin (x)$ f) $f(x)=\frac{1}{\sqrt{x}}$

See Solution

Problem 19203

Find the tangent line equation to $(2y+1)^3 - 24x = -3$ at the point $(-1,-2)$ .

See Solution

Problem 19204

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ of $(2y+1)^3 = 24x-3$ at the point $(-1,-2)$ .

See Solution

Problem 19205

Find $g^{\prime}(-1)$ for the function $g(x)=\frac{f\left(x^{2}\right)}{e^{x}}$ using the given values of $f(x)$ and $f^{\prime}(x)$ .

See Solution

Problem 19206

Given a function $f$ with values in the table, find $g^{\prime}(-1)$ for $g(x)=\frac{f(x^{2})}{e^{x}}$ and $h^{\prime}(1)$ for $h(x)=f(f(-2x))$ .

See Solution

Problem 19207

Calculate the integral: $\int_{-1}^{1} x e^{-x^{2}} d x$

See Solution

Problem 19208

Gegeben ist $f_{a}(x)=a \cdot \ln \left(x^{2}+a\right)$ mit $a \neq 0$ . Bestimme den Definitionsbereich, Nullstellen und Extrempunkte für $a$ .

See Solution

Problem 19209

Find $k^{\prime}(-1)$ for the function $k(x)=f(x) \cdot \arcsin \left(\frac{x}{2}\right)$ using $f(-1)$ and $f'(-1)$ .

See Solution

Problem 19210

Bestimme die Ableitung von $f(x)=x^{3}(x^{2}+5x)$ mit der Produktregel.

See Solution

Problem 19211

Find $h^{\prime}(1)$ for the function $h(x)=f(f(-2 x))$ given $f(x)$ and $f^{\prime}(x)$ values in the table.

See Solution

Problem 19212

Erkläre die Produktregel anhand der Funktion $f(x)=x^{3}(x^{2}+5x)$ .

See Solution

Problem 19213

Find $\left(y^{-1}\right)^{\prime}(0)$ for the curve $(2y+1)^3 = 24x-3$ at the point $\left(\frac{1}{6}, 0\right)$ .

See Solution

Problem 19214

Bestimmen Sie $f^{\prime}(x)$ für $f(x)=5 x^{2} \cdot \ln x$ .

See Solution

Problem 19215

Untersuchen Sie die Funktion $v(t)=5 t-t^{2}$ für $t=0$ bis $t=5$ : a) Änderung der Zuflussgeschwindigkeit, b) Wassermenge bis $t$ , c) Menge in 4 Stunden.

See Solution

Problem 19216

Berechnen Sie $f^{\prime}(x)$ für $f(x)=5 x^{2} \cdot \ln x$ .

See Solution

Problem 19217

Estimate the change in drug concentration $C(t)=2+\frac{3t}{1+t^3}-e^{-0.05t}$ from $t=30$ to $t=40$ minutes.

See Solution

Problem 19218

A particle $P$ moves on the $x$ -axis with displacement $s=5t^{2}-t^{3}$ . Find: a) displacement change from $t=2$ to $t=4$ b) displacement change in the third second (between $t=2$ and $t=3$ ).

See Solution

Problem 19219

Find the position at time $t$ given $a(t)=12, v(0)=-7, s(0)=-6$ , and the displacement from $t=0$ to $t=2$ .

See Solution

Problem 19220

Find the derivative of $G(x)=\int_{4}^{x^{3}} \sin t \, dt$ . What is $G^{\prime}(x)$ ?

See Solution

Problem 19221

Bestimme die Ableitung von $f$ an den Stellen $x_{0}$ : a) $f(x)=x^{2}$ , $x_{0}=3$ ; b) $f(x)=2x^{2}$ , $x_{0}=1$ ; c) $f(x)=-x^{2}$ , $x_{0}=2$ ; d) $f(x)=-\frac{3}{x}$ , $x_{0}=4$ ; e) $f(x)=-x+2$ , $x_{0}=3$ ; f) $f(x)=4$ , $x_{0}=7$ .

See Solution

Problem 19222

Find the volume $v=\pi \cdot \int_{0}^{4}\left(2 x-\frac{1}{4} x^{2}\right)^{2} \, dx$ .

See Solution

Problem 19223

1. Bestimme den Grenzwert von $f(x)=\frac{1-2 x}{x+2}$ für $x \to \infty$ und $x \to -\infty$ .
2. Finde $\lim_{x \to 4} \frac{2x^2 - 32}{x - 4}$ durch Umformung oder h-Methode.
3. Ordne jedem Term seinen Grenzwert für $x \to \infty$ zu.
4. a) Skizziere $h(t)$ der Tulpe. b) Bestimme die mittlere Zuwachsrate. c) Wann wuchs die Tulpe am schnellsten?
5. a) Skizziere $h(t)$ des Segelflugzeugs. b) Bestimme die mittlere Steig-/Sinkgeschwindigkeit. c) Wann steigt das Flugzeug mit $400 \frac{m}{min}$ ?
6. Gegeben $f(x)=\frac{1}{2}x^2$ . a) Bestimme die mittlere Änderungsrate auf $[0;2]$ . b) Lokale Änderungsrate bei $x_0=-1$ zeichnerisch. c) Lokale Änderungsrate bei $x_0=2$ rechnerisch.
7. Ein Auto bremst: $s(t)=40t-4t^2$ . a) Skizziere $s$ für $0 \leq t \leq 5$ . b) Wann steht das Auto? c) Bestimme die mittlere Geschwindigkeit. d) Momentangeschwindigkeit zu Beginn $(t=0)$ ?

See Solution

Problem 19224

Find the new limits of integration for $u=\cos(x)$ in the integral $\int_{\pi}^{\frac{\pi}{4}}-8 \sin(x) \cos^{2}(x) dx$ .
The new lower bound is $\square$ and the new upper bound is $\square$ .

See Solution

Problem 19225

Find the new limits of integration for $u=3 x^{3}+3$ in the integral $\int_{-3}^{0} 3 x^{2}(3 x^{3}+3)^{6} dx$ .
The new lower bound is $\square$ and the new upper bound is $\square$ .

See Solution

Problem 19226

Evaluate the integral: $\int_{-5}^{2}\left(2(2 x+4)^{4}\right) d x$ . Provide the exact answer or round to two decimal places.

See Solution

Problem 19227

Find the average value of $f(x)=8 x^{2}+7$ on the interval $[-1,3]$ . Provide your answer exactly or rounded to two decimal places.

See Solution

Problem 19228

Evaluate the integral: $\int_{3}^{4}\left(3 \sqrt[3]{(3 x+6)^{2}}\right) d x$ . Provide exact or rounded answer.

See Solution

Problem 19229

Find the total distance traveled by the particle with velocity $v(t)=6+2 \ln (t+8)$ from $t=1$ to $t=6$ .

See Solution

Problem 19230

Find the number of fruit flies added from day 4 to day 9 given $g(t)=4 e^{0.04 t}$ and initial population 380. Round to nearest whole number.

See Solution

Problem 19231

Evaluate the integral from 2 to 9 of $\frac{1}{8x+3}$ dx. Provide the answer exactly or rounded to two decimal places.

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Problem 19232

Find the total distance traveled by the particle with velocity $v(t)=\frac{24 t^{2}+12}{4 t^{3}+6 t+9}$ from $t=1$ to $t=2$ .

See Solution

Problem 19233

How many fruit flies are in a population after 80 days if it starts at 360 and grows at $g(t)=5 e^{0.01 t}$ ?

See Solution

Problem 19234

Find the acceleration of particle $P$ when its velocity $v = 12 - t - t^2$ is zero.

See Solution

Problem 19235

Find the distance between the two points where the particle's displacement $x=4 t^{3}-39 t^{2}+120 t$ is at rest.

See Solution

Problem 19236

Two flies on a balloon inflate at 5 cm³/s. Find their distance over time and how fast they separate at various intervals.

See Solution

Problem 19237

Find the second derivative $f''(t)$ if $f'(t) = 1.5 \cos\left(\frac{2\pi}{12.6}(t-6.3)\right) \cdot \frac{2\pi}{12.6}$ .

See Solution

Problem 19238

Find the derivative of $f(x) = \frac{-3 x^{2}}{2\sqrt{-x^{3}+1}}$ .

See Solution

Problem 19239

Bestimme die Punkte, wo die Tangentensteigung von $f(x)=\frac{x^{2}+2}{x}$ gleich -1 oder 0 ist. Berechne $f^{\prime}(x)$ für $f(x)=\frac{k}{x}$ .

See Solution

Problem 19240

Find and simplify $\frac{p(q+h)-p(q)}{h}$ for $p(q)=q^{2}+2q-5$ . What happens as $h \to 0$ ?

See Solution

Problem 19241

Find and simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=1-x^{2}$ . What happens as $h \to 0$ ?

See Solution

Problem 19242

Maximize the volume given by the function $\frac{1}{3} \pi(x+8)^{2}(22-4 x)$ .

See Solution

Problem 19243

Find the equation of $F(x)$ if its slope is $6 e^{3 x}+5 e^{-x}$ and it has a $y$ -intercept of 5.

See Solution

Problem 19244

Find the average rate of change for $f(x)=x^{3}-2 x^{2}+x+1$ on $[0,2]$ and $[2,4]$ . Which interval has a greater rate?

See Solution

Problem 19245

Find the derivative of the function $\frac{-3 t^{2}+32 t}{8 \sqrt{-t^{3}+16 t^{2}}}$ .

See Solution

Problem 19246

Find $F(-2)$ , $F^{\prime}(-2)$ , and $F^{\prime}(2)$ for $F(x)=\int_{-2}^{x} \frac{d u}{u^{2}+1}$ .

See Solution

Problem 19247

Schreiben Sie die Ableitungen der Funktionen $f(x)=10^{x}$ und $f(x)=2 \cdot 0,1^{x}$ mit der Basis e.

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Problem 19248

Find and simplify $\frac{C(x+h)-C(x)}{h}$ for $C(x)=2x^{2}-4x+3$ . What happens as $h$ approaches 0?

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Problem 19249

Find the indefinite integral $\int \frac{\sqrt[7]{x}-5}{\sqrt[7]{x^{6}}} d x$ using the substitution $u=\sqrt[7]{x}-5$ . What is $d u$ ?

See Solution

Problem 19250

Evaluate the integral $\int x^{4}(x^{5}-11)^{26} dx$ using substitution $u = x^{5} - 11$ . Find the result in terms of $x$ .

See Solution

Problem 19251

Evaluate the integral: $\int x^{5} \sqrt{8+x^{6}} \, dx = \square + C$

See Solution

Problem 19252

Evaluate the integral $\int \frac{1}{(5 x+15)^{4}} d x$ using substitution: $u=$ , then find $\int \frac{1}{(5 x+15)^{4}} d x=\square+C$ .

See Solution

Problem 19253

Find where $f(x)=x^{3}-3x+1$ is increasing/decreasing on $[-3,3]$ and identify all extrema with labels.

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Problem 19254

Berechnen Sie die Ableitungen: a) $f(x)=\frac{x}{x+1}$ b) $g(x)=\frac{2 x}{1+3 x}$ c) $f(z)=\frac{1-z^{2}}{z+2}$ d) $f(t)=\frac{t^{2}+t+1}{t^{2}-1}$

See Solution

Problem 19255

Find the centripetal acceleration of a particle with total acceleration $5 \mathrm{~m} / \mathrm{s}^{2}$ and tangential acceleration $3 \mathrm{~m} / \mathrm{s}^{2}$ .

See Solution

Problem 19256

Find where $f(x)=x^{3}-12x+2$ is increasing/decreasing on $[-3,3]$ and identify all extrema with their types.

See Solution

Problem 19257

Berechne die Ableitung von $f(x)$ in e), f) und g) mit der Produktregel und durch Ausmultiplizieren.

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Problem 19258

Berechnen Sie die Ableitung von $f(x)=\left(0,25 x^{2}-2 x+4\right)\left(2 x^{3}+4 x-3\right)$ und $f(x)=\left(x^{2}-x\right)\left(\frac{1}{2} x^{2}+\frac{1}{2} x\right)$ mit Produktregel und Ausmultiplizieren.

See Solution

Problem 19259

Evaluate the integral: $6 \sin^{5}(x) \cos(x) \, dx =$

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Problem 19260

Evaluate the integral: $\int 6 \sin^{5}(x) \cos(x) \, dx = \square + C$

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Problem 19261

Express the limit $\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}(c_{k}^{6}-4 c_{k}) \Delta x_{k}$ as a definite integral.

See Solution

Problem 19262

Determine the domain, range, and asymptote of the function $f(x)=-e^{x}$ .

See Solution

Problem 19263

Calculate the integral: $\int_{1/2}^{1} 4 e^{2x-1} \, dx$

See Solution

Problem 19264

Integrate: $\int 4 t \sqrt{t-2} dt =$ (Hint: Use substitution $u=t-2$ ).

See Solution

Problem 19265

Evaluate these integrals: (a) $\int_{\frac{1}{2}}^{1} 4 e^{2 x-1} d x$ and (b) $\int_{-1}^{0} 6 x(1-3 x^{2})^{5} d x$ (use substitution).

See Solution

Problem 19266

Evaluate the integral from 1 to 4: $\int_{1}^{4}\left(5 x^{2}+\sqrt{x}\right) d x=\square($ Simplify your answer. $)$

See Solution

Problem 19267

Evaluate the surface integral $\iint_{S} x^{2} y z d S$ over the plane $z=1+4x+2y$ above the rectangle $0 \leq x \leq 4$ , $0 \leq y \leq 2$ .

See Solution

Problem 19268

Find the volume of the solid formed by rotating the triangle with vertices $(0,0),(3,3)$ , and $(0,3)$ about the $y$ axis.

See Solution

Problem 19269

Evaluate the integral: $\int x^{5} \sqrt{1+x^{2}} \, dx$

See Solution

Problem 19270

Evaluate the integral $C(x)=\int_{2}^{4}(x-2)^{2}\left[8-(x-2)^{3}\right]^{3 / 2} dx$ .

See Solution

Problem 19271

Evaluate the integral: $\int(2-x) \cdot e^{e^{x}} \, dx$

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Problem 19272

Evaluate the integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sec ^{2}(\theta) \tan (\theta) d \theta.$

See Solution

Problem 19273

Find the volume of the solid formed by rotating the region $Q$ (bounded by $g(y)=2\sqrt{y+4}$ , $y=-1$ , $y=2$ ) around the $y$ -axis using the disk method.

See Solution

Problem 19274

Calculate the area between the parabola $y=x^{2}-11$ and the line $y=10x$ using integration.

See Solution

Problem 19275

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

See Solution

Problem 19276

Find $\lim _{x \rightarrow \infty} \frac{e^{2 x}-x-1}{x^{2}}$ .

See Solution

Problem 19277

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan x}{x}$ .

See Solution

Problem 19278

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

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Problem 19279

Ein Stein fällt von einem Turm. Beantworte folgende Fragen: a) Höhe des Turms? b) Aufprallgeschwindigkeit? c) Zeit für die Hälfte des Fallweges? d) Zeit für die letzten $20 \mathrm{~m}$ ? e) Wann hört man den Aufprall (Schallgeschwindigkeit $320 \mathrm{~m/s}$ )?

See Solution

Problem 19280

Evaluate the integral: $\int_{0}^{2} \frac{1}{1-x^{2}} d x$ .

See Solution

Problem 19281

Find the limit: $\lim _{x \rightarrow 1} \frac{\ln x^{2}}{x^{2}-1}$ .

See Solution

Problem 19282

Calculez l'utilisation totale de l'ordinateur pour les 14 premières semaines avec $w(t)=620+900 \frac{e^{\frac{t}{4}}}{3+e^{\frac{t}{4}}}$ .

See Solution

Problem 19283

Find the volume of the solid formed by rotating the region $R$ (bounded by $f(x)=-2 e^{\frac{x}{2}}$ , $x=0$ , $x=2$ ) around the $x$ -axis.

See Solution

Problem 19284

Find the cylinder dimensions to hold 500 cm³ of soup while minimizing costs: sides at 0.03 cents/cm², top at 0.08 cents/cm².

See Solution

Problem 19285

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

See Solution

Problem 19286

Evaluate the integral $\int_{0}^{1}\left(2 x^{3}-6 x+\frac{3}{x^{2}+1}\right) d x$ .

See Solution

Problem 19287

Find the $x$ -coordinates of relative minima for the function $f$ on $[-3, 3]$ given its critical points and derivatives. Options: (A) $x=-1$ , (B) $x=0$ , (C) $x=1$ , (D) none.

See Solution

Problem 19288

Find the positive value of $x$ where the second derivative $f^{\prime \prime}(x)=0$ for $f(x)=x \exp(-x^{2}/4)$ . Round to three decimals.

See Solution

Problem 19289

Find the area between the curve $y=x^{2}-2x$ and the $x$ -axis for $x$ in $[0,3]$ .

See Solution

Problem 19290

Berechnen Sie die Integrale: a) $\int_{0}^{1} x^{3} d x$ , b) $\int_{0}^{0} x^{3} d x$ , c) $\int_{2}^{4} x^{3} d x$ , d) $\int_{4}^{4} x^{3} d x$ .

See Solution

Problem 19291

Find $\frac{d y}{d x}$ using implicit differentiation of $V=-4 x-2 \ln (x)+1.4 y-2.2 \ln (y)$ , then evaluate at $x=6$ , $y=2$ .

See Solution

Problem 19292

Estimez le coût total d'expédition en utilisant l'intégrale de $q(t) \cdot p(t)$ sur $[0, 10]$ .

See Solution

Problem 19293

Estimate $f(2.4)$ using linear approximation, given $f(2)=5$ and $f^{\prime}(2)=2.8$ . Round to three decimal places.

See Solution

Problem 19294

Find the derivative of $g(x)=\int_{5}^{x} \frac{1}{1+t^{5}} dt$ . What is $g'(x)$ ?

See Solution

Problem 19295

Evaluate the integral from 1 to $\sqrt{7}$ of $\frac{19 s^{7}+3 \sqrt{s}}{s^{7}}$ ds.

See Solution

Problem 19296

Find the integral for the area between $y=x^{2}$ , $x=2$ , and the $x$ -axis: options include $\int_{0}^{2} x^{2} d x$ .

See Solution

Problem 19297

Estimez le coût total des expéditions sur 10 mois avec $q(t)=3200+25t$ et $p(t)=4e^{0.04t}$ .

See Solution

Problem 19298

Näherungswerte für $f$ an den Stellen a berechnen und auf vier Dezimalen runden: a) $f(x)=2,3 \cdot e^{0,9 x}$ bei $a=2$ b) $f(x)=20 \cdot e^{-0,01 x}$ bei $a=2,2$ c) $f(x)=\ln (x)$ bei $a=0,5$

See Solution

Problem 19299

Évaluez les intégrales suivantes : a) $\int x \sqrt{1+4 x} d x$ , b) $\int \frac{1+2 x}{1+16 x^{2}} d x$ , c) $\int(4 t+2) e^{t^{2}+t} d x$ , d) $\int \frac{e^{x}}{e^{x}+e^{-x}} d x$ , e) $\int x^{2} 2^{x} d x$ , f) $\int x^{2} e^{4 x} d x$ , g) $\int 3 t^{3} \sin \left(t^{2}\right) d t$ , h) $\int x^{2}(\ln x)^{2} d x$ . Trouvez le profit maximal d'un appareil acheté à 2500 \ $avec$ R $et$ C $donnés par$ \frac{d R}{d t}=100(18-3 \sqrt{t}) $et$ \frac{d C}{d t}=100(2+\sqrt{t})$.

See Solution

Problem 19300

Bestimme die Ableitung von $f(x)=(x+1) e^{-x}$ .

See Solution

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Calculus

Problem 19201

Given the curve (2y+1)3−24x=−3(2 y+1)^{3}-24 x=-3(2y+1)3−24x=−3, prove that dydx=4(2y+1)2\frac{d y}{d x}=\frac{4}{(2 y+1)^{2}}dxdy​=(2y+1)24​.

Problem 19202

Problem 19203

Find the tangent line equation to (2y+1)3−24x=−3(2y+1)^3 - 24x = -3(2y+1)3−24x=−3 at the point (−1,−2)(-1,-2)(−1,−2).

Problem 19204

Find the second derivative d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ of (2y+1)3=24x−3(2y+1)^3 = 24x-3(2y+1)3=24x−3 at the point (−1,−2)(-1,-2)(−1,−2).

Problem 19205

Find g′(−1)g^{\prime}(-1)g′(−1) for the function g(x)=f(x2)exg(x)=\frac{f\left(x^{2}\right)}{e^{x}}g(x)=exf(x2)​ using the given values of f(x)f(x)f(x) and f′(x)f^{\prime}(x)f′(x).

Problem 19206

Given a function fff with values in the table, find g′(−1)g^{\prime}(-1)g′(−1) for g(x)=f(x2)exg(x)=\frac{f(x^{2})}{e^{x}}g(x)=exf(x2)​ and h′(1)h^{\prime}(1)h′(1) for h(x)=f(f(−2x))h(x)=f(f(-2x))h(x)=f(f(−2x)).

Problem 19207

Calculate the integral: ∫−11xe−x2dx\int_{-1}^{1} x e^{-x^{2}} d x∫−11​xe−x2dx

Problem 19208

Gegeben ist fa(x)=a⋅ln⁡(x2+a)f_{a}(x)=a \cdot \ln \left(x^{2}+a\right)fa​(x)=a⋅ln(x2+a) mit a≠0a \neq 0a=0. Bestimme den Definitionsbereich, Nullstellen und Extrempunkte für aaa.

Problem 19209

Find k′(−1)k^{\prime}(-1)k′(−1) for the function k(x)=f(x)⋅arcsin⁡(x2)k(x)=f(x) \cdot \arcsin \left(\frac{x}{2}\right)k(x)=f(x)⋅arcsin(2x​) using f(−1)f(-1)f(−1) and f′(−1)f'(-1)f′(−1).

Problem 19210

Bestimme die Ableitung von f(x)=x3(x2+5x)f(x)=x^{3}(x^{2}+5x)f(x)=x3(x2+5x) mit der Produktregel.

Problem 19211

Find h′(1)h^{\prime}(1)h′(1) for the function h(x)=f(f(−2x))h(x)=f(f(-2 x))h(x)=f(f(−2x)) given f(x)f(x)f(x) and f′(x)f^{\prime}(x)f′(x) values in the table.

Problem 19212

Erkläre die Produktregel anhand der Funktion f(x)=x3(x2+5x)f(x)=x^{3}(x^{2}+5x)f(x)=x3(x2+5x).

Problem 19213

Find (y−1)′(0)\left(y^{-1}\right)^{\prime}(0)(y−1)′(0) for the curve (2y+1)3=24x−3(2y+1)^3 = 24x-3(2y+1)3=24x−3 at the point (16,0)\left(\frac{1}{6}, 0\right)(61​,0).

Problem 19214

Bestimmen Sie f′(x)f^{\prime}(x)f′(x) für f(x)=5x2⋅ln⁡xf(x)=5 x^{2} \cdot \ln xf(x)=5x2⋅lnx.

Problem 19215

Untersuchen Sie die Funktion v(t)=5t−t2v(t)=5 t-t^{2}v(t)=5t−t2 für t=0t=0t=0 bis t=5t=5t=5: a) Änderung der Zuflussgeschwindigkeit, b) Wassermenge bis ttt, c) Menge in 4 Stunden.

Problem 19216

Berechnen Sie f′(x)f^{\prime}(x)f′(x) für f(x)=5x2⋅ln⁡xf(x)=5 x^{2} \cdot \ln xf(x)=5x2⋅lnx.

Problem 19217

Estimate the change in drug concentration C(t)=2+3t1+t3−e−0.05tC(t)=2+\frac{3t}{1+t^3}-e^{-0.05t}C(t)=2+1+t33t​−e−0.05t from t=30t=30t=30 to t=40t=40t=40 minutes.

Problem 19218

A particle PPP moves on the xxx-axis with displacement s=5t2−t3s=5t^{2}-t^{3}s=5t2−t3. Find: a) displacement change from t=2t=2t=2 to t=4t=4t=4 b) displacement change in the third second (between t=2t=2t=2 and t=3t=3t=3).

Problem 19219

Find the position at time ttt given a(t)=12,v(0)=−7,s(0)=−6a(t)=12, v(0)=-7, s(0)=-6a(t)=12,v(0)=−7,s(0)=−6, and the displacement from t=0t=0t=0 to t=2t=2t=2.

Problem 19220

Find the derivative of G(x)=∫4x3sin⁡t dtG(x)=\int_{4}^{x^{3}} \sin t \, dtG(x)=∫4x3​sintdt. What is G′(x)G^{\prime}(x)G′(x)?

Problem 19221

Problem 19222

Find the volume v=π⋅∫04(2x−14x2)2 dxv=\pi \cdot \int_{0}^{4}\left(2 x-\frac{1}{4} x^{2}\right)^{2} \, dxv=π⋅∫04​(2x−41​x2)2dx.

Problem 19223

Problem 19224

Find the new limits of integration for u=cos⁡(x)u=\cos(x)u=cos(x) in the integral ∫ππ4−8sin⁡(x)cos⁡2(x)dx\int_{\pi}^{\frac{\pi}{4}}-8 \sin(x) \cos^{2}(x) dx∫π4π​​−8sin(x)cos2(x)dx. The new lower bound is □\square□ and the new upper bound is □\square□.

Problem 19225

Find the new limits of integration for u=3x3+3u=3 x^{3}+3u=3x3+3 in the integral ∫−303x2(3x3+3)6dx\int_{-3}^{0} 3 x^{2}(3 x^{3}+3)^{6} dx∫−30​3x2(3x3+3)6dx. The new lower bound is □\square□ and the new upper bound is □\square□.

Problem 19226

Evaluate the integral: ∫−52(2(2x+4)4)dx\int_{-5}^{2}\left(2(2 x+4)^{4}\right) d x∫−52​(2(2x+4)4)dx. Provide the exact answer or round to two decimal places.

Problem 19227

Find the average value of f(x)=8x2+7f(x)=8 x^{2}+7f(x)=8x2+7 on the interval [−1,3][-1,3][−1,3]. Provide your answer exactly or rounded to two decimal places.

Problem 19228

Evaluate the integral: ∫34(3(3x+6)23)dx\int_{3}^{4}\left(3 \sqrt[3]{(3 x+6)^{2}}\right) d x∫34​(33(3x+6)2​)dx. Provide exact or rounded answer.

Problem 19229

Find the total distance traveled by the particle with velocity v(t)=6+2ln⁡(t+8)v(t)=6+2 \ln (t+8)v(t)=6+2ln(t+8) from t=1t=1t=1 to t=6t=6t=6.

Problem 19230

Find the number of fruit flies added from day 4 to day 9 given g(t)=4e0.04tg(t)=4 e^{0.04 t}g(t)=4e0.04t and initial population 380. Round to nearest whole number.

Problem 19231

Evaluate the integral from 2 to 9 of 18x+3\frac{1}{8x+3}8x+31​ dx. Provide the answer exactly or rounded to two decimal places.

Problem 19232

Find the total distance traveled by the particle with velocity v(t)=24t2+124t3+6t+9v(t)=\frac{24 t^{2}+12}{4 t^{3}+6 t+9}v(t)=4t3+6t+924t2+12​ from t=1t=1t=1 to t=2t=2t=2.

Problem 19233

How many fruit flies are in a population after 80 days if it starts at 360 and grows at g(t)=5e0.01tg(t)=5 e^{0.01 t}g(t)=5e0.01t?

Problem 19234

Find the acceleration of particle PPP when its velocity v=12−t−t2v = 12 - t - t^2v=12−t−t2 is zero.

Problem 19235

Find the distance between the two points where the particle's displacement x=4t3−39t2+120tx=4 t^{3}-39 t^{2}+120 tx=4t3−39t2+120t is at rest.

Problem 19236

Two flies on a balloon inflate at 5 cm³/s. Find their distance over time and how fast they separate at various intervals.

Problem 19237

Find the second derivative f′′(t)f''(t)f′′(t) if f′(t)=1.5cos⁡(2π12.6(t−6.3))⋅2π12.6f'(t) = 1.5 \cos\left(\frac{2\pi}{12.6}(t-6.3)\right) \cdot \frac{2\pi}{12.6}f′(t)=1.5cos(12.62π​(t−6.3))⋅12.62π​.

Problem 19238

Find the derivative of f(x)=−3x22−x3+1f(x) = \frac{-3 x^{2}}{2\sqrt{-x^{3}+1}}f(x)=2−x3+1​−3x2​.

Problem 19239

Bestimme die Punkte, wo die Tangentensteigung von f(x)=x2+2xf(x)=\frac{x^{2}+2}{x}f(x)=xx2+2​ gleich -1 oder 0 ist. Berechne f′(x)f^{\prime}(x)f′(x) für f(x)=kxf(x)=\frac{k}{x}f(x)=xk​.

Problem 19240

Find and simplify p(q+h)−p(q)h\frac{p(q+h)-p(q)}{h}hp(q+h)−p(q)​ for p(q)=q2+2q−5p(q)=q^{2}+2q-5p(q)=q2+2q−5. What happens as h→0h \to 0h→0?

Problem 19241

Find and simplify f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=1−x2f(x)=1-x^{2}f(x)=1−x2. What happens as h→0h \to 0h→0?

Given the curve $(2 y+1)^{3}-24 x=-3$ , prove that $\frac{d y}{d x}=\frac{4}{(2 y+1)^{2}}$ .

Find the tangent line equation to $(2y+1)^3 - 24x = -3$ at the point $(-1,-2)$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ of $(2y+1)^3 = 24x-3$ at the point $(-1,-2)$ .

Find $g^{\prime}(-1)$ for the function $g(x)=\frac{f\left(x^{2}\right)}{e^{x}}$ using the given values of $f(x)$ and $f^{\prime}(x)$ .

Given a function $f$ with values in the table, find $g^{\prime}(-1)$ for $g(x)=\frac{f(x^{2})}{e^{x}}$ and $h^{\prime}(1)$ for $h(x)=f(f(-2x))$ .

Calculate the integral: $\int_{-1}^{1} x e^{-x^{2}} d x$

Gegeben ist $f_{a}(x)=a \cdot \ln \left(x^{2}+a\right)$ mit $a \neq 0$ . Bestimme den Definitionsbereich, Nullstellen und Extrempunkte für $a$ .

Find $k^{\prime}(-1)$ for the function $k(x)=f(x) \cdot \arcsin \left(\frac{x}{2}\right)$ using $f(-1)$ and $f'(-1)$ .

Bestimme die Ableitung von $f(x)=x^{3}(x^{2}+5x)$ mit der Produktregel.

Find $h^{\prime}(1)$ for the function $h(x)=f(f(-2 x))$ given $f(x)$ and $f^{\prime}(x)$ values in the table.

Erkläre die Produktregel anhand der Funktion $f(x)=x^{3}(x^{2}+5x)$ .

Find $\left(y^{-1}\right)^{\prime}(0)$ for the curve $(2y+1)^3 = 24x-3$ at the point $\left(\frac{1}{6}, 0\right)$ .

Bestimmen Sie $f^{\prime}(x)$ für $f(x)=5 x^{2} \cdot \ln x$ .

Untersuchen Sie die Funktion $v(t)=5 t-t^{2}$ für $t=0$ bis $t=5$ : a) Änderung der Zuflussgeschwindigkeit, b) Wassermenge bis $t$ , c) Menge in 4 Stunden.

Berechnen Sie $f^{\prime}(x)$ für $f(x)=5 x^{2} \cdot \ln x$ .

Estimate the change in drug concentration $C(t)=2+\frac{3t}{1+t^3}-e^{-0.05t}$ from $t=30$ to $t=40$ minutes.

A particle $P$ moves on the $x$ -axis with displacement $s=5t^{2}-t^{3}$ . Find: a) displacement change from $t=2$ to $t=4$ b) displacement change in the third second (between $t=2$ and $t=3$ ).

Find the position at time $t$ given $a(t)=12, v(0)=-7, s(0)=-6$ , and the displacement from $t=0$ to $t=2$ .

Find the derivative of $G(x)=\int_{4}^{x^{3}} \sin t \, dt$ . What is $G^{\prime}(x)$ ?

Find the volume $v=\pi \cdot \int_{0}^{4}\left(2 x-\frac{1}{4} x^{2}\right)^{2} \, dx$ .

Find the new limits of integration for $u=\cos(x)$ in the integral $\int_{\pi}^{\frac{\pi}{4}}-8 \sin(x) \cos^{2}(x) dx$ .
The new lower bound is $\square$ and the new upper bound is $\square$ .

Find the new limits of integration for $u=3 x^{3}+3$ in the integral $\int_{-3}^{0} 3 x^{2}(3 x^{3}+3)^{6} dx$ .
The new lower bound is $\square$ and the new upper bound is $\square$ .

Evaluate the integral: $\int_{-5}^{2}\left(2(2 x+4)^{4}\right) d x$ . Provide the exact answer or round to two decimal places.

Find the average value of $f(x)=8 x^{2}+7$ on the interval $[-1,3]$ . Provide your answer exactly or rounded to two decimal places.

Evaluate the integral: $\int_{3}^{4}\left(3 \sqrt[3]{(3 x+6)^{2}}\right) d x$ . Provide exact or rounded answer.

Find the total distance traveled by the particle with velocity $v(t)=6+2 \ln (t+8)$ from $t=1$ to $t=6$ .

Find the number of fruit flies added from day 4 to day 9 given $g(t)=4 e^{0.04 t}$ and initial population 380. Round to nearest whole number.

Evaluate the integral from 2 to 9 of $\frac{1}{8x+3}$ dx. Provide the answer exactly or rounded to two decimal places.

Find the total distance traveled by the particle with velocity $v(t)=\frac{24 t^{2}+12}{4 t^{3}+6 t+9}$ from $t=1$ to $t=2$ .

How many fruit flies are in a population after 80 days if it starts at 360 and grows at $g(t)=5 e^{0.01 t}$ ?

Find the acceleration of particle $P$ when its velocity $v = 12 - t - t^2$ is zero.

Find the distance between the two points where the particle's displacement $x=4 t^{3}-39 t^{2}+120 t$ is at rest.

Find the second derivative $f''(t)$ if $f'(t) = 1.5 \cos\left(\frac{2\pi}{12.6}(t-6.3)\right) \cdot \frac{2\pi}{12.6}$ .

Find the derivative of $f(x) = \frac{-3 x^{2}}{2\sqrt{-x^{3}+1}}$ .

Bestimme die Punkte, wo die Tangentensteigung von $f(x)=\frac{x^{2}+2}{x}$ gleich -1 oder 0 ist. Berechne $f^{\prime}(x)$ für $f(x)=\frac{k}{x}$ .

Find and simplify $\frac{p(q+h)-p(q)}{h}$ for $p(q)=q^{2}+2q-5$ . What happens as $h \to 0$ ?

Find and simplify $\frac{f(x+h)-f(x)}{h}$ for $f(x)=1-x^{2}$ . What happens as $h \to 0$ ?

Maximize the volume given by the function $\frac{1}{3} \pi(x+8)^{2}(22-4 x)$ .

Find the equation of $F(x)$ if its slope is $6 e^{3 x}+5 e^{-x}$ and it has a $y$ -intercept of 5.

Find the average rate of change for $f(x)=x^{3}-2 x^{2}+x+1$ on $[0,2]$ and $[2,4]$ . Which interval has a greater rate?

Find the derivative of the function $\frac{-3 t^{2}+32 t}{8 \sqrt{-t^{3}+16 t^{2}}}$ .

Find $F(-2)$ , $F^{\prime}(-2)$ , and $F^{\prime}(2)$ for $F(x)=\int_{-2}^{x} \frac{d u}{u^{2}+1}$ .

Schreiben Sie die Ableitungen der Funktionen $f(x)=10^{x}$ und $f(x)=2 \cdot 0,1^{x}$ mit der Basis e.

Find and simplify $\frac{C(x+h)-C(x)}{h}$ for $C(x)=2x^{2}-4x+3$ . What happens as $h$ approaches 0?

Find the indefinite integral $\int \frac{\sqrt[7]{x}-5}{\sqrt[7]{x^{6}}} d x$ using the substitution $u=\sqrt[7]{x}-5$ . What is $d u$ ?

Evaluate the integral $\int x^{4}(x^{5}-11)^{26} dx$ using substitution $u = x^{5} - 11$ . Find the result in terms of $x$ .

Evaluate the integral: $\int x^{5} \sqrt{8+x^{6}} \, dx = \square + C$

Evaluate the integral $\int \frac{1}{(5 x+15)^{4}} d x$ using substitution: $u=$ , then find $\int \frac{1}{(5 x+15)^{4}} d x=\square+C$ .

Find where $f(x)=x^{3}-3x+1$ is increasing/decreasing on $[-3,3]$ and identify all extrema with labels.

Berechnen Sie die Ableitungen: a) $f(x)=\frac{x}{x+1}$ b) $g(x)=\frac{2 x}{1+3 x}$ c) $f(z)=\frac{1-z^{2}}{z+2}$ d) $f(t)=\frac{t^{2}+t+1}{t^{2}-1}$

Find the centripetal acceleration of a particle with total acceleration $5 \mathrm{~m} / \mathrm{s}^{2}$ and tangential acceleration $3 \mathrm{~m} / \mathrm{s}^{2}$ .

Find where $f(x)=x^{3}-12x+2$ is increasing/decreasing on $[-3,3]$ and identify all extrema with their types.

Berechne die Ableitung von $f(x)$ in e), f) und g) mit der Produktregel und durch Ausmultiplizieren.

Evaluate the integral: $6 \sin^{5}(x) \cos(x) \, dx =$

Evaluate the integral: $\int 6 \sin^{5}(x) \cos(x) \, dx = \square + C$

Express the limit $\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}(c_{k}^{6}-4 c_{k}) \Delta x_{k}$ as a definite integral.

Determine the domain, range, and asymptote of the function $f(x)=-e^{x}$ .

Calculate the integral: $\int_{1/2}^{1} 4 e^{2x-1} \, dx$

Integrate: $\int 4 t \sqrt{t-2} dt =$ (Hint: Use substitution $u=t-2$ ).

Evaluate these integrals: (a) $\int_{\frac{1}{2}}^{1} 4 e^{2 x-1} d x$ and (b) $\int_{-1}^{0} 6 x(1-3 x^{2})^{5} d x$ (use substitution).

Evaluate the integral from 1 to 4: $\int_{1}^{4}\left(5 x^{2}+\sqrt{x}\right) d x=\square($ Simplify your answer. $)$

Evaluate the surface integral $\iint_{S} x^{2} y z d S$ over the plane $z=1+4x+2y$ above the rectangle $0 \leq x \leq 4$ , $0 \leq y \leq 2$ .

Find the volume of the solid formed by rotating the triangle with vertices $(0,0),(3,3)$ , and $(0,3)$ about the $y$ axis.

Evaluate the integral: $\int x^{5} \sqrt{1+x^{2}} \, dx$

Evaluate the integral $C(x)=\int_{2}^{4}(x-2)^{2}\left[8-(x-2)^{3}\right]^{3 / 2} dx$ .

Evaluate the integral: $\int(2-x) \cdot e^{e^{x}} \, dx$

Evaluate the integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sec ^{2}(\theta) \tan (\theta) d \theta.$

Find the volume of the solid formed by rotating the region $Q$ (bounded by $g(y)=2\sqrt{y+4}$ , $y=-1$ , $y=2$ ) around the $y$ -axis using the disk method.

Calculate the area between the parabola $y=x^{2}-11$ and the line $y=10x$ using integration.