Calculus

Problem 33301

Find the limit of $f(x)=\frac{-2+\frac{7}{x}}{8-\frac{3}{x^{2}}}$ as $x \to \infty$ and $x \to -\infty$ .

See Solution

Problem 33302

Differentiate $y+y^{2} \cos x=x^{2}$ implicitly to find $\frac{d y}{d x}$ . Then, find the gradient at $(0,2)$ .

See Solution

Problem 33303

Is it true that $\int \sec ^{2} 4 x \, dx = \frac{1}{4} \tan 4 x + c$ ? Select True or False.

See Solution

Problem 33304

Find the limit as $x$ approaches $-\infty$ :
$\lim _{x \rightarrow-\infty} \frac{\sqrt[3]{x}-4 x+1}{3 x+x^{\frac{2}{3}}-2}$
Write $\infty$ or $-\infty$ as needed.

See Solution

Problem 33305

Find the limits of $h(x)=\frac{6 x^{4}}{10 x^{4}+17 x^{3}+13 x^{2}}$ as $x \to \infty$ and $x \to -\infty$ .

See Solution

Problem 33306

Find critical points of $f(x)=x e^{x^{2}}$ . Choose: a. no points, b. $\frac{1}{3 \sqrt{e}}$ , c. $e^{-\frac{1}{2}}$ , d. $-\frac{1}{2}$ .

See Solution

Problem 33307

Find the area between $y=\ln x$ and $y=\ln 2 x$ from $x=1$ to $x=5$ : $A=\int^{5} \ln 2 x - \ln x \, dx$ .

See Solution

Problem 33308

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{(6-x)(3+9 x)}{(3-7 x)(10+6 x)}$ .

See Solution

Problem 33309

Find the slope of the tangent line of $y=3^{\sin x}$ at $x=0$ . Choices: $1 / \ln 2$ , $\ln 3$ , $1 / \ln 3$ , $\ln 2$ .

See Solution

Problem 33310

Finde drei Stammfunktionen für die folgenden Funktionen: a) $f(x)=3 x-1$ , b) $f(x)=5$ , c) $f(x)=x^{6}$ , d) $f(x)=a x^{n}$ , e) $f(x)=3 x^{2}$ , f) $f(x)=4 x^{3}-7 x+6$ , g) $f(x)=x(x-4)$ , h) $f(x)=-\frac{3}{x^{4}}$ , i) $f(x)=3 x^{-2}$ , j) $f(x)=\frac{1}{x^{2}}-x^{2}$ .

See Solution

Problem 33311

Find the horizontal asymptotes by calculating these limits:
1. $\lim _{x \rightarrow \infty} \frac{-9 x}{6+2 x}=\square$
2. $\lim _{x \rightarrow-\infty} \frac{11 x-3}{x^{3}+12 x-8}=\square$
3. $\lim _{x \rightarrow \infty} \frac{x^{2}-7 x-8}{14-15 x^{2}}=\square$
4. $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+13 x}}{12-14 x}=\square$
5. $\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}+13 x}}{12-14 x}=\square$

See Solution

Problem 33312

Städte:
1. Berechne die Einwohnerzahl nach 5 Jahren: $N_{1}(5)$ .
2. Bestimme, wann $N_{1}(t) = 20000$ .
3. Finde die Abnahmerate zu Beginn und nach 10 Jahren.
4. Finde, wann $N_{1}(t) = N_{2}(t)$ und deren Größe.
5. Bestimme den Zeitpunkt, an dem die Summe der Einwohnerzahlen minimal ist.

See Solution

Problem 33313

Evaluate the integral $\int_{1}^{2} \frac{2^{1 / x}}{x^{2}} d x$ and choose the correct answer.

See Solution

Problem 33314

Evaluate the integral $\int_{1}^{e} \frac{6 \ln x + 4}{x} \, dx$ .

See Solution

Problem 33315

Berechne die Ableitung der Funktion $f(x)=\frac{2}{x^{2}}$ .

See Solution

Problem 33316

Find the first derivative of $f(x) = \left(\ln(6 - x^2)\right)^3$ .

See Solution

Problem 33317

Berechne die Ableitung von $u(x)=(3x+4)^{4}$ und $v(x)=e^{3x+1}$ mit Produkt- und Kettenregel.

See Solution

Problem 33318

Use Laplace transform on the PDE $u_{x x}=u_{t t}$ with $u(x, 0)=0$ , $u_{t}(x, 0)=\sin x$ . Find the ODE.

See Solution

Problem 33319

Use Laplace transform on the PDE $t u_{tt} - t u_t + u = 2$ to find the transformed equation. Choose the correct option.

See Solution

Problem 33320

Bestimmen Sie die Extrem- und Wendepunkte der Funktion $f(x)=x \cdot e^{1-x}$ .

See Solution

Problem 33321

Find the integral of the function $x e^{x^{2}}$ with respect to $x$ : $\int x e^{x^{2}} d x=$

See Solution

Problem 33322

Solve $U_{x}=U_{t}+U$ with $U(x, 0)=e^{-x}$ using separation of variables. Choose the correct solution.

See Solution

Problem 33323

إذا كانت $u(x, y)=\sum_{n=1}^{\infty} a_{n} e^{n y} \sin (n x)$ هي الحل العام لمعادلة لابلاس بشرط الحدود $u(x, 0)=\sin x$ ، فهل يكون الحل $u(x, y)=e^{y} \sin x$ ؟ اختر: صحيح أم خطأ.

See Solution

Problem 33324

Find $\frac{d y}{d x}$ at the point $(1,1)$ for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ .

See Solution

Problem 33325

Find the derivative of $\ln y = \ln [t(t+1)(t-2)]$ .

See Solution

Problem 33326

Find $y$ given the equation $y = \frac{x^{\ln x}}{y'(e)}$ where $y' = \frac{dy}{dx}$ .

See Solution

Problem 33327

Find $f^{\prime}(\pi / 4)$ if $f(x) = x^2 \tan x$ . Choices: a, b, c, d.

See Solution

Problem 33328

Find the integral of $3^{x} \ln 3$ with respect to $x$ .

See Solution

Problem 33329

Find the derivative of $f(x)=\frac{x^{2}}{2x+1}$ and evaluate it at $x=2$ . What is $f^{\prime}(2)$ ?

See Solution

Problem 33330

Untersuchen Sie das Verhalten der Funktion $f(x)=2 \cdot e^{-0.5 x}+1$ für $x \rightarrow \infty$ .

See Solution

Problem 33331

Vereinfache die Funktionen $f(x)=\frac{2 x+1}{x}$ und $h(x)=\frac{0,5 x^{3}-13 x^{2}}{117 x}$ und finde die Grenzwerte. Bonus: Ändere sie für $x \rightarrow \infty$ . Finde Fehler in den Rechnungen und korrigiere sie.

See Solution

Problem 33332

Bestimme die Gleichung der Tangente an $f(x)=x \cdot e^{1-x}$ bei $P\left(2, \frac{2}{\mathrm{e}}\right)$ und wo sie den Fluss kreuzt.

See Solution

Problem 33333

Find the slope of the tangent line to the curve $y=x^{2} e^{2 x+1}$ at $x=1$ . Choose from: $4 e$ , $4 e^{3}$ , $2 e^{3}$ , $e^{3}$ , None.

See Solution

Problem 33334

Evaluate the integral $\int_{0}^{1} x^{2} 2^{x^{3}} d x$ and choose the correct answer from the options given.

See Solution

Problem 33335

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt[3]{7+x^{3}}-2}{x-1}$ .

See Solution

Problem 33336

Find the values of $x$ where the function $f(x)=3-x-2x^{2}$ is decreasing. Choose from: $x=-1$ , $x=0$ , $x=1$ , $x=2$ .

See Solution

Problem 33337

Find the values of $x$ where the function $\mathrm{f}(x)=3-x-2 x^{2}$ is decreasing. Options: $x=-1$ , $x=0$ , $x=1$ , $x=2$ .

See Solution

Problem 33338

Luftdruck:
a) Zeigen Sie, dass $p(h)=1013 e^{-0,00013 h}$ die Lösung ist. b) Bestimmen Sie den Luftdruck in $2000 \mathrm{~m}$ . c) Ab welcher Höhe ist bei $500 \mathrm{~mbar}$ Sauerstoffzufuhr nötig?

See Solution

Problem 33339

Bestimmen Sie die Ableitung von $f(x)=-\frac{1}{3 x^{6}}$ mit der Potenzregel für negative Exponenten.

See Solution

Problem 33340

Calculate the integral: $\int_{0}^{\frac{\pi}{6}} \tan 2 x \, dx$

See Solution

Problem 33341

The solution of $y' = e^y 2^x, y(0) = \ln(\ln(2))$ is $y = \ln(\ln(2)) - \ln(2 - 2^x)$ . True or False?

See Solution

Problem 33342

Bestimmen Sie die Ableitung der Funktion $f(x) = \frac{1}{x^{5}}$ mit der Potenzregel für negative Exponenten.

See Solution

Problem 33343

Find the derivative $y^{\prime}$ if $y=5^{3x+\ln x}$ .

See Solution

Problem 33344

Find the limit: $\lim _{x \rightarrow 1} \frac{\cos \frac{\pi}{2} x}{1-\sqrt{x}}$

See Solution

Problem 33345

Find $W\left(y_{1}, y_{2}\right)(4)$ given $W\left(y_{1}, y_{2}\right)(2)=3$ for the equation $t^{2} y^{\prime \prime}-2 y^{\prime}+(3+t) y=0$ .

See Solution

Problem 33346

Find the derivative of the function $y=\frac{t}{\ln t}$ .

See Solution

Problem 33347

Identify the differential equation for the function $y=c_{1} e^{x}+c_{2} x e^{x}$ .

See Solution

Problem 33348

Identify which function is NOT a solution to the equation $y^{\prime \prime}+y=0$ : a. $\cos t$ , b. $\cos (t+1)$ , c. $\tan t$ , d. $\sin t$ .

See Solution

Problem 33349

Given $W(y_{1}, y_{2})(2)=3$ , find $W(y_{1}, y_{2})(4)$ for the equation $t^{2} y^{\prime \prime}-2 y^{\prime}+(3+t) y=0$ .

See Solution

Problem 33350

Find the derivative of $y$ if $\ln y = \ln e^{-5x}$ . What is $y'$ ?

See Solution

Problem 33351

Bestimme die Ableitung der Funktion $f(x) = \frac{x^{4} - x + 1}{x^{3}}$ mit der Potenzregel für negative Exponenten.

See Solution

Problem 33352

Is the Wronskian of the solutions to the equation $t^{4} y^{\prime \prime} - 2 t^{3} y^{\prime} - t^{8} y = 0$ equal to $c t^{2}$ ? True or False?

See Solution

Problem 33353

Eine Funktion $f$ dritten Grades beschreibt die Sauerstoffproduktion einer Pflanze.
a) Finde die Funktionsgleichung $f$ . b) Berechne die Gesamtproduktion von 6 bis 20 Uhr. c) Bestimme die Zeiträume, in denen die Produktionsrate unter $10 \mathrm{l/h}$ liegt.

See Solution

Problem 33354

Finde und korrigiere die Fehler in den Ableitungen: a) $f(x)=4 x^{5}+2 x^{3}+2 x$ , $f^{\prime}(x)=20 x^{4}+6 x^{2}+2$ b) $f(x)=\frac{3}{x}+x^{2}-2$ , $f^{\prime}(x)=-\frac{3}{x^{2}}+2 x$

See Solution

Problem 33355

Finde und korrigiere die Fehler in den angegebenen Stammfunktionen: a) $\int(4 x+1)^{3} x$ b) $\int\left(4 e^{3 x+4}+5 x^{3}\right) d x$ c) $\int\left(3 x^{2}+2 a\right) d a$

See Solution

Problem 33356

If $y=5^{3 x+\ln x}$ , find $y^{\prime}$ . Choose from the options provided.

See Solution

Problem 33357

Bestimme die Tangentengleichung $t$ für $f(x)=0,04 x^{3}+0,6 x^{2}-0,3 x+3$ bei $P(-3 / 8,22)$ und berechne die Fläche A.

See Solution

Problem 33358

Which differential equation does the function $y=c_{1} e^{x}+c_{2} x e^{x}$ solve? Options: a. $y^{\prime \prime}-2 y^{\prime}+y=0$ b. $3 y^{\prime \prime}-2 y^{\prime}+y=0$ c. $y^{\prime \prime}-y^{\prime}+y=0$ d. $y^{\prime \prime}+2 y^{\prime}+y=0$

See Solution

Problem 33359

Find the equation satisfied by $v$ in $x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\frac{1}{4}\right) y=0$ . Select one: a, b, c, or d.

See Solution

Problem 33360

Berechnen Sie die Grenzwerte: a) $\lim _{x \rightarrow 4} \frac{x^{2}-16}{x-4}$ , b) $\lim _{x \rightarrow-1} \frac{x^{3}-x}{x+1}$ , c) $\lim _{x \rightarrow 3} \frac{3-x}{2 x^{2}-6 x}$ , d) $\lim _{x \rightarrow 2} \frac{x^{4}-16}{x-2}$ .

See Solution

Problem 33361

Bestimmen Sie die Tangentengleichung $t$ für $f(x)=0,04 x^{3}+0,6 x^{2}-0,3 x+3$ bei $P(-3 / 8,22)$ und den Flächeninhalt $A$ zwischen $f$ und $t$ .

See Solution

Problem 33362

Find the limit as $\alpha$ approaches $\pi$ for $\frac{1-\sin x}{1-\frac{\alpha^{2}}{\pi^{2}}}$ .

See Solution

Problem 33363

Find the derivative of $\ln y = \ln [t(t-1)(t-2)]$ . What is the correct expression?

See Solution

Problem 33364

Find $\frac{d y}{d x}$ for $\ln (2 x+5 y)=\sin y$ .

See Solution

Problem 33365

Find the Wronskian $W(y_{1}, y_{2})(t)$ for $y_{1}(t)=t \ln t$ and $y_{2}(t)=t^{2}$ , $t>0$ . Choices: a. $t \ln t+t^{2}$ b. $t^{2} \ln t$ c. $t^{2}(\ln t-1)$ d. $t^{2}(\ln t+1)$

See Solution

Problem 33366

حدد المعادلة التفاضلية التي تكون فيها الدالة $y=c_{1} e^{x}+c_{2} x e^{x}$ حلاً: a. $3 y^{\prime \prime}-2 y^{\prime}+y=0$ b. $y^{\prime \prime}-2 y^{\prime}+y=0$ c. $y^{\prime \prime}-y^{\prime}+y=0$ d. $y^{\prime \prime}+2 y^{\prime}+y=0$

See Solution

Problem 33367

Find the integrating factor for the linear differential equation $1-2xy=(1+y^{2})\frac{dx}{dy}$ . Options: a. $1+x$ , b. $1+y$ , c. $1+x^{2}$ , d. $1+y^{2}$ .

See Solution

Problem 33368

Find the slope of the tangent line of $f(x)=e^{\tan x}$ at $x=0$ .

See Solution

Problem 33369

Find the particular solution for $y^{\prime \prime}+3 y^{\prime}+2 y=t^{2} e^{-t}+e^{-2 t}$ . Choose the correct option.

See Solution

Problem 33370

Find the trapezoidal approximation of $\int_{2}^{8} f(x) dx$ using intervals $[2,5], [5,7], [7,8]$ from the table values.

See Solution

Problem 33371

Bestimme, wo die Funktionen monoton steigen oder fallen: a) $f(x)=x^{4}$ b) $f(x)=x^{3}$ c) $f(x)=x^{2}-8x$ d) $f(x)=e^{x}-x$

See Solution

Problem 33372

Find $f^{\prime}(x)$ for $f(x)=\ln(2e^{-x}\sin x)$ . Options: $\frac{-1+\cos x}{\sin x}$ or $\cot x-1$ .

See Solution

Problem 33373

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

See Solution

Problem 33374

Find the average rate of change of $P(\theta)=\sqrt{4 \theta+1}$ over $[0,2]$ . Options: a) 2 b) 0 c) -2 d) 1

See Solution

Problem 33375

Evaluate the integral $\int_{1}^{e} \frac{6 \ln x+4}{x} dx$ . What is the result? Choices: 8, 5, None, 10, 7.

See Solution

Problem 33376

Find $\lim _{x \rightarrow 2} f(x)$ if $\lim _{x \rightarrow 2} \frac{f(x)-5}{x^{2}-4}=3$ . Options: a) 0 b) 1 c) 5 d) 3

See Solution

Problem 33377

Find the limit: $\lim _{h \rightarrow 0} \frac{h}{\sin 3 h}$ . Choices: a) D N E b) $\frac{1}{3}$ c) 0 d) $\infty$

See Solution

Problem 33378

Rozstrzygnij o warunkowej i bezwzględnej zbieżności szeregu $\sum_{n=1}^{+\infty}\left(\sqrt[8]{19+\frac{1}{n^{\alpha}}}-\sqrt[8]{19}\right)^{(\alpha+1)} q^{n}$ dla $\alpha>0$ i $q \in \mathbb{R}$ .

See Solution

Problem 33379

Find the rate of change of $v=\frac{0.22 s}{0.25+s}$ with respect to $s$ at $s=0.05$ .

See Solution

Problem 33380

Find the derivative of the function: $-5 x^{7}+2 x^{3}-10$ .

See Solution

Problem 33381

Find the value of $\frac{d}{dx}\left(\frac{u}{v}\right)$ at $x=1$ given $u(1)=-3$ , $v(1)=2$ , $\dot{u}(1)=-3$ , $\dot{v}(1)=4$ . Options: a) $-\frac{2}{3}$ b) $-\frac{10}{2}$ c) $\frac{3}{2}$ d) $-\frac{3}{4}$ .

See Solution

Problem 33382

Find $f^{\prime}(e)$ for the function $f(x)=x^{\ln x}$ . Options: e, 0, 1, 2.

See Solution

Problem 33383

Find the derivative of the function $f(N) = \operatorname{Ln} 5 + N \operatorname{Ln} 7$ with respect to $N$ .

See Solution

Problem 33384

Find the tangent and normal lines to the curve $f(x)=2 x^{3}-3 x+1$ at the point $(-1,2)$ .

See Solution

Problem 33385

Find the rate of change of impedance $Z=\frac{4X}{4+X}$ as $X$ decreases at 2 ohms/sec in a parallel RL circuit.

See Solution

Problem 33386

Find the derivative of the function $f(x)=7 \sqrt{x}(x^{3}-8 \sqrt{x}+5)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 33387

Given $f^{\prime}(2.9) \approx 95$ (backward) and $f^{\prime}(2.9) \approx 40$ (central), find $A$ and $B$ .

See Solution

Problem 33388

Find the derivative $f'(x)$ of the function $f(x)=\ln(e^{-x}+1)$ .

See Solution

Problem 33389

Analyze the end behavior of $f(x)=-x^{3}-11x^{2}-40x-47$ . What happens as $x \rightarrow \pm \infty$ ?

See Solution

Problem 33390

Find the relative max and min points of the function $f(x)=x^{3}-3x^{2}+1$ .

See Solution

Problem 33391

Evaluate the integral from 2 to 4 of $\frac{1}{x(\ln x)^{2}} \, dx$ . What is the result?

See Solution

Problem 33392

Find $\frac{d y}{d x}$ for $\ln (3 x+4 y)=\sin y$ . Choose the correct derivative.

See Solution

Problem 33393

Evaluate the integral: $\int_{2}^{16} \frac{1}{2 x \sqrt{\ln x}} \, dx$ .

See Solution

Problem 33394

A snowball with radius $r$ cm melts at $800 \pi$ cm³/s. Find the radius change rate when $r = 20$ cm as $-\frac{a}{b}$ cm/s.

See Solution

Problem 33395

Find the maxima and minima of $f(x) = x^{3} - 8x^{2} + 20x - 16$ . Choose from the options provided.

See Solution

Problem 33396

Differentiate $\left(\sin(4x+3)\right)^{x^3-2}$ with respect to $x$ .

See Solution

Problem 33397

Find the limit: $\lim _{x \rightarrow 3} \frac{2 x^{2}-18 x}{x-3}$ .

See Solution

Problem 33398

Find $\frac{d y}{d x}$ for $\ln (3 x+4 y)=\sin y$ . What is the correct derivative?

See Solution

Problem 33399

Find the derivative of $f(x) = \ln\left((2x + 1)^{\cos(2x)}\right)$ .

See Solution

Problem 33400

Find the derivative $y^{\prime}$ if $y=\sqrt{\ln \sqrt{t}}$ . Options: a. $\frac{1}{4 t \sqrt{\ln (t)}}$ , b. 1

See Solution

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Calculus

Problem 33301

Find the limit of f(x)=−2+7x8−3x2f(x)=\frac{-2+\frac{7}{x}}{8-\frac{3}{x^{2}}}f(x)=8−x23​−2+x7​​ as x→∞x \to \inftyx→∞ and x→−∞x \to -\inftyx→−∞.

Problem 33302

Differentiate y+y2cos⁡x=x2y+y^{2} \cos x=x^{2}y+y2cosx=x2 implicitly to find dydx\frac{d y}{d x}dxdy​. Then, find the gradient at (0,2)(0,2)(0,2).

Problem 33303

Is it true that ∫sec⁡24x dx=14tan⁡4x+c\int \sec ^{2} 4 x \, dx = \frac{1}{4} \tan 4 x + c∫sec24xdx=41​tan4x+c? Select True or False.

Problem 33304

Find the limit as xxx approaches −∞-\infty−∞: lim⁡x→−∞x3−4x+13x+x23−2\lim _{x \rightarrow-\infty} \frac{\sqrt[3]{x}-4 x+1}{3 x+x^{\frac{2}{3}}-2}x→−∞lim​3x+x32​−23x​−4x+1​ Write ∞\infty∞ or −∞-\infty−∞ as needed.

Problem 33305

Find the limits of h(x)=6x410x4+17x3+13x2h(x)=\frac{6 x^{4}}{10 x^{4}+17 x^{3}+13 x^{2}}h(x)=10x4+17x3+13x26x4​ as x→∞x \to \inftyx→∞ and x→−∞x \to -\inftyx→−∞.

Problem 33306

Find critical points of f(x)=xex2f(x)=x e^{x^{2}}f(x)=xex2. Choose: a. no points, b. 13e\frac{1}{3 \sqrt{e}}3e​1​, c. e−12e^{-\frac{1}{2}}e−21​, d. −12-\frac{1}{2}−21​.

Problem 33307

Find the area between y=ln⁡xy=\ln xy=lnx and y=ln⁡2xy=\ln 2 xy=ln2x from x=1x=1x=1 to x=5x=5x=5: A=∫5ln⁡2x−ln⁡x dxA=\int^{5} \ln 2 x - \ln x \, dxA=∫5ln2x−lnxdx.

Problem 33308

Evaluate the limit as xxx approaches infinity: lim⁡x→∞(6−x)(3+9x)(3−7x)(10+6x)\lim _{x \rightarrow \infty} \frac{(6-x)(3+9 x)}{(3-7 x)(10+6 x)}limx→∞​(3−7x)(10+6x)(6−x)(3+9x)​.

Problem 33309

Find the slope of the tangent line of y=3sin⁡xy=3^{\sin x}y=3sinx at x=0x=0x=0. Choices: 1/ln⁡21 / \ln 21/ln2, ln⁡3\ln 3ln3, 1/ln⁡31 / \ln 31/ln3, ln⁡2\ln 2ln2.

Problem 33310

Problem 33311

Problem 33312

Problem 33313

Evaluate the integral ∫1221/xx2dx\int_{1}^{2} \frac{2^{1 / x}}{x^{2}} d x∫12​x221/x​dx and choose the correct answer.

Problem 33314

Evaluate the integral ∫1e6ln⁡x+4x dx\int_{1}^{e} \frac{6 \ln x + 4}{x} \, dx∫1e​x6lnx+4​dx.

Problem 33315

Berechne die Ableitung der Funktion f(x)=2x2f(x)=\frac{2}{x^{2}}f(x)=x22​.

Problem 33316

Find the first derivative of f(x)=(ln⁡(6−x2))3f(x) = \left(\ln(6 - x^2)\right)^3f(x)=(ln(6−x2))3.

Problem 33317

Berechne die Ableitung von u(x)=(3x+4)4u(x)=(3x+4)^{4}u(x)=(3x+4)4 und v(x)=e3x+1v(x)=e^{3x+1}v(x)=e3x+1 mit Produkt- und Kettenregel.

Problem 33318

Use Laplace transform on the PDE uxx=uttu_{x x}=u_{t t}uxx​=utt​ with u(x,0)=0u(x, 0)=0u(x,0)=0, ut(x,0)=sin⁡xu_{t}(x, 0)=\sin xut​(x,0)=sinx. Find the ODE.

Problem 33319

Use Laplace transform on the PDE tutt−tut+u=2t u_{tt} - t u_t + u = 2tutt​−tut​+u=2 to find the transformed equation. Choose the correct option.

Problem 33320

Bestimmen Sie die Extrem- und Wendepunkte der Funktion f(x)=x⋅e1−xf(x)=x \cdot e^{1-x}f(x)=x⋅e1−x.

Problem 33321

Find the integral of the function xex2x e^{x^{2}}xex2 with respect to xxx: ∫xex2dx=\int x e^{x^{2}} d x=∫xex2dx=

Problem 33322

Solve Ux=Ut+UU_{x}=U_{t}+UUx​=Ut​+U with U(x,0)=e−xU(x, 0)=e^{-x}U(x,0)=e−x using separation of variables. Choose the correct solution.

Problem 33323

Problem 33324

Find dydx\frac{d y}{d x}dxdy​ at the point (1,1)(1,1)(1,1) for the equation x23+y23=2yx^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}x32​+y32​=2y​.

Problem 33325

Find the derivative of ln⁡y=ln⁡[t(t+1)(t−2)]\ln y = \ln [t(t+1)(t-2)]lny=ln[t(t+1)(t−2)].

Problem 33326

Find yyy given the equation y=xln⁡xy′(e)y = \frac{x^{\ln x}}{y'(e)}y=y′(e)xlnx​ where y′=dydxy' = \frac{dy}{dx}y′=dxdy​.

Problem 33327

Find f′(π/4)f^{\prime}(\pi / 4)f′(π/4) if f(x)=x2tan⁡xf(x) = x^2 \tan xf(x)=x2tanx. Choices: a, b, c, d.

Problem 33328

Find the integral of 3xln⁡33^{x} \ln 33xln3 with respect to xxx.

Problem 33329

Find the derivative of f(x)=x22x+1f(x)=\frac{x^{2}}{2x+1}f(x)=2x+1x2​ and evaluate it at x=2x=2x=2. What is f′(2)f^{\prime}(2)f′(2)?

Problem 33330

Untersuchen Sie das Verhalten der Funktion f(x)=2⋅e−0.5x+1f(x)=2 \cdot e^{-0.5 x}+1f(x)=2⋅e−0.5x+1 für x→∞x \rightarrow \inftyx→∞.

Problem 33331

Vereinfache die Funktionen f(x)=2x+1xf(x)=\frac{2 x+1}{x}f(x)=x2x+1​ und h(x)=0,5x3−13x2117xh(x)=\frac{0,5 x^{3}-13 x^{2}}{117 x}h(x)=117x0,5x3−13x2​ und finde die Grenzwerte. Bonus: Ändere sie für x→∞x \rightarrow \inftyx→∞. Finde Fehler in den Rechnungen und korrigiere sie.

Problem 33332

Bestimme die Gleichung der Tangente an f(x)=x⋅e1−xf(x)=x \cdot e^{1-x}f(x)=x⋅e1−x bei P(2,2e)P\left(2, \frac{2}{\mathrm{e}}\right)P(2,e2​) und wo sie den Fluss kreuzt.

Problem 33333

Find the slope of the tangent line to the curve y=x2e2x+1y=x^{2} e^{2 x+1}y=x2e2x+1 at x=1x=1x=1. Choose from: 4e4 e4e, 4e34 e^{3}4e3, 2e32 e^{3}2e3, e3e^{3}e3, None.

Problem 33334

Evaluate the integral ∫01x22x3dx\int_{0}^{1} x^{2} 2^{x^{3}} d x∫01​x22x3dx and choose the correct answer from the options given.

Problem 33335

Find the limit: lim⁡x→17+x33−2x−1\lim _{x \rightarrow 1} \frac{\sqrt[3]{7+x^{3}}-2}{x-1}limx→1​x−137+x3​−2​.

Problem 33336

Find the values of xxx where the function f(x)=3−x−2x2f(x)=3-x-2x^{2}f(x)=3−x−2x2 is decreasing. Choose from: x=−1x=-1x=−1, x=0x=0x=0, x=1x=1x=1, x=2x=2x=2.

Problem 33337

Find the values of xxx where the function f(x)=3−x−2x2\mathrm{f}(x)=3-x-2 x^{2}f(x)=3−x−2x2 is decreasing. Options: x=−1x=-1x=−1, x=0x=0x=0, x=1x=1x=1, x=2x=2x=2.

Problem 33338

Luftdruck: a) Zeigen Sie, dass p(h)=1013e−0,00013hp(h)=1013 e^{-0,00013 h}p(h)=1013e−0,00013h die Lösung ist. b) Bestimmen Sie den Luftdruck in 2000 m2000 \mathrm{~m}2000 m. c) Ab welcher Höhe ist bei 500 mbar500 \mathrm{~mbar}500 mbar Sauerstoffzufuhr nötig?

Problem 33339

Bestimmen Sie die Ableitung von f(x)=−13x6f(x)=-\frac{1}{3 x^{6}}f(x)=−3x61​ mit der Potenzregel für negative Exponenten.

Problem 33340

Calculate the integral: ∫0π6tan⁡2x dx\int_{0}^{\frac{\pi}{6}} \tan 2 x \, dx∫06π​​tan2xdx

Problem 33341

The solution of y′=ey2x,y(0)=ln⁡(ln⁡(2))y' = e^y 2^x, y(0) = \ln(\ln(2))y′=ey2x,y(0)=ln(ln(2)) is y=ln⁡(ln⁡(2))−ln⁡(2−2x)y = \ln(\ln(2)) - \ln(2 - 2^x)y=ln(ln(2))−ln(2−2x). True or False?

Problem 33342

Find the limit of $f(x)=\frac{-2+\frac{7}{x}}{8-\frac{3}{x^{2}}}$ as $x \to \infty$ and $x \to -\infty$ .

Differentiate $y+y^{2} \cos x=x^{2}$ implicitly to find $\frac{d y}{d x}$ . Then, find the gradient at $(0,2)$ .

Is it true that $\int \sec ^{2} 4 x \, dx = \frac{1}{4} \tan 4 x + c$ ? Select True or False.

Find the limit as $x$ approaches $-\infty$ :
$\lim _{x \rightarrow-\infty} \frac{\sqrt[3]{x}-4 x+1}{3 x+x^{\frac{2}{3}}-2}$
Write $\infty$ or $-\infty$ as needed.

Find the limits of $h(x)=\frac{6 x^{4}}{10 x^{4}+17 x^{3}+13 x^{2}}$ as $x \to \infty$ and $x \to -\infty$ .

Find critical points of $f(x)=x e^{x^{2}}$ . Choose: a. no points, b. $\frac{1}{3 \sqrt{e}}$ , c. $e^{-\frac{1}{2}}$ , d. $-\frac{1}{2}$ .

Find the area between $y=\ln x$ and $y=\ln 2 x$ from $x=1$ to $x=5$ : $A=\int^{5} \ln 2 x - \ln x \, dx$ .

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{(6-x)(3+9 x)}{(3-7 x)(10+6 x)}$ .

Find the slope of the tangent line of $y=3^{\sin x}$ at $x=0$ . Choices: $1 / \ln 2$ , $\ln 3$ , $1 / \ln 3$ , $\ln 2$ .

Evaluate the integral $\int_{1}^{2} \frac{2^{1 / x}}{x^{2}} d x$ and choose the correct answer.

Evaluate the integral $\int_{1}^{e} \frac{6 \ln x + 4}{x} \, dx$ .

Berechne die Ableitung der Funktion $f(x)=\frac{2}{x^{2}}$ .

Find the first derivative of $f(x) = \left(\ln(6 - x^2)\right)^3$ .

Berechne die Ableitung von $u(x)=(3x+4)^{4}$ und $v(x)=e^{3x+1}$ mit Produkt- und Kettenregel.

Use Laplace transform on the PDE $u_{x x}=u_{t t}$ with $u(x, 0)=0$ , $u_{t}(x, 0)=\sin x$ . Find the ODE.

Use Laplace transform on the PDE $t u_{tt} - t u_t + u = 2$ to find the transformed equation. Choose the correct option.

Bestimmen Sie die Extrem- und Wendepunkte der Funktion $f(x)=x \cdot e^{1-x}$ .

Find the integral of the function $x e^{x^{2}}$ with respect to $x$ : $\int x e^{x^{2}} d x=$

Solve $U_{x}=U_{t}+U$ with $U(x, 0)=e^{-x}$ using separation of variables. Choose the correct solution.

Find $\frac{d y}{d x}$ at the point $(1,1)$ for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ .

Find the derivative of $\ln y = \ln [t(t+1)(t-2)]$ .

Find $y$ given the equation $y = \frac{x^{\ln x}}{y'(e)}$ where $y' = \frac{dy}{dx}$ .

Find $f^{\prime}(\pi / 4)$ if $f(x) = x^2 \tan x$ . Choices: a, b, c, d.

Find the integral of $3^{x} \ln 3$ with respect to $x$ .

Find the derivative of $f(x)=\frac{x^{2}}{2x+1}$ and evaluate it at $x=2$ . What is $f^{\prime}(2)$ ?

Untersuchen Sie das Verhalten der Funktion $f(x)=2 \cdot e^{-0.5 x}+1$ für $x \rightarrow \infty$ .

Vereinfache die Funktionen $f(x)=\frac{2 x+1}{x}$ und $h(x)=\frac{0,5 x^{3}-13 x^{2}}{117 x}$ und finde die Grenzwerte. Bonus: Ändere sie für $x \rightarrow \infty$ . Finde Fehler in den Rechnungen und korrigiere sie.

Bestimme die Gleichung der Tangente an $f(x)=x \cdot e^{1-x}$ bei $P\left(2, \frac{2}{\mathrm{e}}\right)$ und wo sie den Fluss kreuzt.

Find the slope of the tangent line to the curve $y=x^{2} e^{2 x+1}$ at $x=1$ . Choose from: $4 e$ , $4 e^{3}$ , $2 e^{3}$ , $e^{3}$ , None.

Evaluate the integral $\int_{0}^{1} x^{2} 2^{x^{3}} d x$ and choose the correct answer from the options given.

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt[3]{7+x^{3}}-2}{x-1}$ .

Find the values of $x$ where the function $f(x)=3-x-2x^{2}$ is decreasing. Choose from: $x=-1$ , $x=0$ , $x=1$ , $x=2$ .

Find the values of $x$ where the function $\mathrm{f}(x)=3-x-2 x^{2}$ is decreasing. Options: $x=-1$ , $x=0$ , $x=1$ , $x=2$ .

Luftdruck:
a) Zeigen Sie, dass $p(h)=1013 e^{-0,00013 h}$ die Lösung ist. b) Bestimmen Sie den Luftdruck in $2000 \mathrm{~m}$ . c) Ab welcher Höhe ist bei $500 \mathrm{~mbar}$ Sauerstoffzufuhr nötig?

Bestimmen Sie die Ableitung von $f(x)=-\frac{1}{3 x^{6}}$ mit der Potenzregel für negative Exponenten.

Calculate the integral: $\int_{0}^{\frac{\pi}{6}} \tan 2 x \, dx$

The solution of $y' = e^y 2^x, y(0) = \ln(\ln(2))$ is $y = \ln(\ln(2)) - \ln(2 - 2^x)$ . True or False?

Bestimmen Sie die Ableitung der Funktion $f(x) = \frac{1}{x^{5}}$ mit der Potenzregel für negative Exponenten.

Find the derivative $y^{\prime}$ if $y=5^{3x+\ln x}$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{\cos \frac{\pi}{2} x}{1-\sqrt{x}}$

Find $W\left(y_{1}, y_{2}\right)(4)$ given $W\left(y_{1}, y_{2}\right)(2)=3$ for the equation $t^{2} y^{\prime \prime}-2 y^{\prime}+(3+t) y=0$ .

Find the derivative of the function $y=\frac{t}{\ln t}$ .

Identify the differential equation for the function $y=c_{1} e^{x}+c_{2} x e^{x}$ .

Identify which function is NOT a solution to the equation $y^{\prime \prime}+y=0$ : a. $\cos t$ , b. $\cos (t+1)$ , c. $\tan t$ , d. $\sin t$ .

Given $W(y_{1}, y_{2})(2)=3$ , find $W(y_{1}, y_{2})(4)$ for the equation $t^{2} y^{\prime \prime}-2 y^{\prime}+(3+t) y=0$ .

Find the derivative of $y$ if $\ln y = \ln e^{-5x}$ . What is $y'$ ?

Bestimme die Ableitung der Funktion $f(x) = \frac{x^{4} - x + 1}{x^{3}}$ mit der Potenzregel für negative Exponenten.

Is the Wronskian of the solutions to the equation $t^{4} y^{\prime \prime} - 2 t^{3} y^{\prime} - t^{8} y = 0$ equal to $c t^{2}$ ? True or False?

Eine Funktion $f$ dritten Grades beschreibt die Sauerstoffproduktion einer Pflanze.
a) Finde die Funktionsgleichung $f$ . b) Berechne die Gesamtproduktion von 6 bis 20 Uhr. c) Bestimme die Zeiträume, in denen die Produktionsrate unter $10 \mathrm{l/h}$ liegt.

Finde und korrigiere die Fehler in den angegebenen Stammfunktionen: a) $\int(4 x+1)^{3} x$ b) $\int\left(4 e^{3 x+4}+5 x^{3}\right) d x$ c) $\int\left(3 x^{2}+2 a\right) d a$

If $y=5^{3 x+\ln x}$ , find $y^{\prime}$ . Choose from the options provided.

Bestimme die Tangentengleichung $t$ für $f(x)=0,04 x^{3}+0,6 x^{2}-0,3 x+3$ bei $P(-3 / 8,22)$ und berechne die Fläche A.

Find the equation satisfied by $v$ in $x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\frac{1}{4}\right) y=0$ . Select one: a, b, c, or d.

Bestimmen Sie die Tangentengleichung $t$ für $f(x)=0,04 x^{3}+0,6 x^{2}-0,3 x+3$ bei $P(-3 / 8,22)$ und den Flächeninhalt $A$ zwischen $f$ und $t$ .

Find the limit as $\alpha$ approaches $\pi$ for $\frac{1-\sin x}{1-\frac{\alpha^{2}}{\pi^{2}}}$ .

Find the derivative of $\ln y = \ln [t(t-1)(t-2)]$ . What is the correct expression?

Find $\frac{d y}{d x}$ for $\ln (2 x+5 y)=\sin y$ .

Find the integrating factor for the linear differential equation $1-2xy=(1+y^{2})\frac{dx}{dy}$ . Options: a. $1+x$ , b. $1+y$ , c. $1+x^{2}$ , d. $1+y^{2}$ .

Find the slope of the tangent line of $f(x)=e^{\tan x}$ at $x=0$ .

Find the particular solution for $y^{\prime \prime}+3 y^{\prime}+2 y=t^{2} e^{-t}+e^{-2 t}$ . Choose the correct option.

Find the trapezoidal approximation of $\int_{2}^{8} f(x) dx$ using intervals $[2,5], [5,7], [7,8]$ from the table values.

Bestimme, wo die Funktionen monoton steigen oder fallen: a) $f(x)=x^{4}$ b) $f(x)=x^{3}$ c) $f(x)=x^{2}-8x$ d) $f(x)=e^{x}-x$

Find $f^{\prime}(x)$ for $f(x)=\ln(2e^{-x}\sin x)$ . Options: $\frac{-1+\cos x}{\sin x}$ or $\cot x-1$ .

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

Find the average rate of change of $P(\theta)=\sqrt{4 \theta+1}$ over $[0,2]$ . Options: a) 2 b) 0 c) -2 d) 1