Calculus

Problem 16401

Beweisen Sie für $x \neq 0$ , dass $F(x) = \ln (|x|)$ eine Stammfunktion von $f(x) = \frac{1}{x}$ ist.

See Solution

Problem 16402

Berechne das Integral $\int_{1}^{5} \frac{1}{x} \, dx$ mit einer Stammfunktion.

See Solution

Problem 16403

Find the domain of the function $f(x)=\frac{6 x}{\sqrt{x^{2}+1}}$ and its transition points (max, min, inflection).

See Solution

Problem 16404

Beweisen Sie allgemein: Für $x \neq 0$ ist $F(x)=\ln (|x|)$ die Stammfunktion von $f(x)=\frac{1}{x}$ .

See Solution

Problem 16405

Find the transition points and asymptotes of the function $f(x)=\frac{x}{x^{2}-16}$ .

See Solution

Problem 16406

Find the derivatives of these functions:
1. $f(x)=e x-\pi$
2. $f(x)=-\frac{3}{4} x^{4}+6 x^{2}+3 x-5$
3. $g(x)=2(2 x-3)^{2}-\frac{2}{x^{2}}$
4. $f(x)=5 x (4 x^{2}-1)$

See Solution

Problem 16407

Evaluate the integral $\oint_{C} \frac{z e^{z}}{z^{2}-1} d z$ where $C$ is the contour $|z|=3$ .

See Solution

Problem 16408

Find the limit as $n$ approaches infinity: $\lim _{n \rightarrow \infty} \frac{12 n^{3}+m-12 m^{3}+5 m}{\sqrt{12 n^{3}+n}+\sqrt{12 m^{3}-5 n}}$

See Solution

Problem 16409

Find the limit: $\lim _{x \rightarrow \infty} \frac{12 n^{3}+x-12 n^{3}+5 n}{\sqrt{12 n^{3}+x}+\sqrt{12 n^{3}-5 x}}$

See Solution

Problem 16410

Find the limit: $\lim _{x \rightarrow \infty} \frac{12 x^{3}+x-12 x^{3}+5 x}{\sqrt{12 x^{3}+x}+\sqrt{12 x^{3}-5 x}}$ .

See Solution

Problem 16411

Berechnen Sie die Geschwindigkeit $v(t)$ eines Fallschirmspringers mit $v_{e}=0,048$ , $t=0,2$ , $g=9,81$ und $m_{k}=0,0261$ .

See Solution

Problem 16412

Evaluate the integral using integration by parts: $\int(5 x-1) e^{-x} d x$ (Include constant $C$ )

See Solution

Problem 16413

Differentiate $y=\cos (x+5 y)$ implicitly to find $y^{\prime}=\frac{d y}{d x}$ .

See Solution

Problem 16414

A pizza pan cools from $450^{\circ} \mathrm{F}$ to $74^{\circ} \mathrm{F}$ . After 5 min it's $300^{\circ} \mathrm{F}$ . Find when it's $130^{\circ} \mathrm{F}$ and $240^{\circ} \mathrm{F}$ . What happens to the temperature over time?

See Solution

Problem 16415

Evaluate the integral using integration by parts: $\int(1-x) e^{x} d x$ (use constant $C$ ).

See Solution

Problem 16416

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

See Solution

Problem 16417

Calculate the indefinite integral and include the constant of integration $C$ : $\int(u+3)(8u+1) du$

See Solution

Problem 16418

Find the indefinite integral: $\int \frac{9+\sqrt{x}+x}{x} d x$ (Use $C$ for the constant of integration.)

See Solution

Problem 16419

Find the integral: $\int \sec (5 x) \tan (5 x) \, dx$

See Solution

Problem 16420

Evaluate the integral from 0 to 1: $\int_{0}^{1}\left(x^{14}+14^{x}\right) d x$

See Solution

Problem 16421

Berechne die Geschwindigkeit $v(t)$ eines Körpers in der Luft mit $g=9,81$ , $t=0,2$ , $v_{e}=9,81$ und der Gleichung $v(t)=v_{e} \cdot\left(\frac{e^{\frac{2 \cdot g \cdot t}{v_{e}}}-1}{1+e^{\frac{2 \cdot g \cdot t}{v_{e}}}}\right)$ .

See Solution

Problem 16422

Find the indefinite integral: $\int \sec(t)(9 \sec(t)+2 \tan(t)) dt$ . Use $C$ for the constant of integration.

See Solution

Problem 16423

Evaluate the integral from 0 to $\frac{1}{\sqrt{3}}$ : $\int 4 \frac{t^{2}-1}{t^{4}-1} dt$

See Solution

Problem 16424

Given the velocity function $v(t)=5 t-9$ for $0 \leq t \leq 3$ , find: (a) displacement (m), (b) total distance (m).

See Solution

Problem 16425

Evaluate the integral: $\int \sin (t) \sqrt{1+\cos (t)} \, dt + C$ .

See Solution

Problem 16426

Evaluate the integral using the substitution $u=x^{3}+7$ . Find $\int x^{2} \sqrt{u} \, dx$ and include constant $C$ .

See Solution

Problem 16427

Berechne die Geschwindigkeit $v(0.2)$ mit $v_{e}=9,81$ und $g=9,81$ in der Formel $v(t)=v_{e} \cdot\left(\frac{e^{\frac{2 \cdot g \cdot t}{v_{e}}}-1}{1+e^{\frac{2 \cdot g \cdot t}{v_{e}}}}\right)$ .

See Solution

Problem 16428

Evaluate the integral using the substitution $u=\sin(\theta)$ : $\int \sin^{5}(\theta) \cos(\theta) d\theta, \, C = \text{constant}.$

See Solution

Problem 16429

Find the area above the $x$ -axis for the curve $y=1-x^{2}$ between $x=-1$ and $x=2$ .

See Solution

Problem 16430

Evaluate the integral with substitution: $\int \frac{x+1}{\sqrt{x^{2}+2 x}} d x=\square+C$

See Solution

Problem 16431

Find the velocity of a falling body at $t = 3$ and $t = 6$ with $g = 9.81 \, m/s^2$ and $v_e = 0.61 \, m/s$ using $v(t)=v_{e} \cdot\left(\frac{e^{\frac{2 \cdot g \cdot t}{v_{e}}}-1}{1+e^{\frac{2 \cdot g \cdot t}{v_{e}}}}\right)$

See Solution

Problem 16432

Find the increase in total cost when production changes from $x=30$ to $x=90$ boxes, given marginal cost $4+\frac{x^{2}}{1,000}$ .

See Solution

Problem 16433

Determine if the integral $\int_{6}^{\infty} \frac{6}{(x+7)^{3 / 2}} d x$ is convergent or divergent and evaluate if convergent.

See Solution

Problem 16434

Evaluate the integral using the substitution $u=7+x^{4}$ : $\int x^{3}(u^{4}) d x, \quad C \text{ is the constant of integration.}$

See Solution

Problem 16435

How much to invest now at 8\% interest to have \ $9000 in 13 years? Use the formula for continuous compounding:$ A = Pe^{rt}$.

See Solution

Problem 16436

Calculate the integral: $\int_{0}^{1} \frac{-3}{\sqrt{1-x^{2}}} d x$

See Solution

Problem 16437

How long does it take for a bacteria population to double with a growth rate of $2.7\%$ per hour? Round to the nearest hundredth.

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Problem 16438

Evaluate the integral from $e$ to $e^{36}$ of $\frac{1}{x \sqrt{\ln(x)}} \, dx$ .

See Solution

Problem 16439

Find the half-life of a substance with a decay rate of $7.2\%$ per day. Round to the nearest hundredth.

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Problem 16440

Is the integral $\int_{-\infty}^{\infty} x^{8} e^{-x^{8}} d x$ convergent or divergent? Evaluate if convergent.

See Solution

Problem 16441

Find the limit: $\lim _{x \rightarrow 1^{+}} \frac{5+\ln x}{5-e^{9 x}}$ .

See Solution

Problem 16442

Berechne die Geschwindigkeit eines Fallschirmspringers mit $t=7$ , $v_e=9,61$ und $g=9,81$ in der Formel $v(t)=v_{e} \cdot\left(\frac{e^{\frac{2 \cdot g \cdot t}{v_{e}}}-1}{1+e^{\frac{2 \cdot g \cdot t}{v_{e}}}}\right)$ .

See Solution

Problem 16443

Berechne die Geschwindigkeit $v(3,8)$ mit $g=9,81$ m/s² und $v_{e}=9,61$ m/s: $v(t)=v_{e} \cdot\left(\frac{e^{\frac{2 \cdot g \cdot t}{v_{e}}}-1}{1+e^{\frac{2 \cdot g \cdot t}{v_{e}}}}\right)$ .

See Solution

Problem 16444

Find the limit as $x$ approaches $\frac{\pi}{4}$ of $(\tan x - 1) \sec x$ .

See Solution

Problem 16445

Evaluate the limit using L'Hôpital's Rule if needed: $\lim _{x \rightarrow \infty} \frac{3+x e^{2 / x}}{x \arctan (x)}$

See Solution

Problem 16446

Find the limit as $x$ approaches infinity for $\frac{5x - 4}{13x + 7}$ .

See Solution

Problem 16447

Find the velocity $x'(t)$ , speed, and acceleration of $x(t)=t^{3}-12t^{2}+36t-27$ . When is the particle moving forward?

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Problem 16448

Find the derivatives of the functions: $g(w)=\int_{0}^{w} \sin(2+t^{5}) dt$ and $h(x)=\int^{\sqrt{x}} \frac{z^{2}}{z^{4}+4} dz$ .

See Solution

Problem 16449

Evaluate the integral: $\int_{-1}^{2}(3 u-5)(u+2) du$

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Problem 16450

Find $y$ from $x^{2/3} + y^{2/3} = 1$ , then find its derivatives for $x \neq 0, \pm 1$ .

See Solution

Problem 16451

Find the time $t$ for maximum concentration of the drug $C(t)=13.2 t \cdot e^{-0.15 t}$ and its value at that time.

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Problem 16452

Find the interval where the function $f(x)=\int_{0}^{x}(4-t^{2}) e^{t^{2}} dt$ is increasing.

See Solution

Problem 16453

Find the quantity $q$ that maximizes profit, given $R(q)=550q$ and $C(q)=10,000+5q^{2}$ . What is the profit? $q=$ Profit $=\$ \mathbf{i}$

See Solution

Problem 16454

Find the quantity $q$ that maximizes profit, given revenue $R(q)=450q$ and cost $C(q)=10,000+5q^2$ . What is the profit? $q=$ Profit $=\$$

See Solution

Problem 16455

Find the limit $\lim _{n \rightarrow \infty} \frac{F_{n+1}}{F_{n}}$ using the Fibonacci formula to 3 decimal places.

See Solution

Problem 16456

Find the maximum concentration of the drug modeled by $C(t)=8 t e^{-0.2 t}$ . Choices: 10.6, 5, 14.7, 32.2, 20.2.

See Solution

Problem 16457

Calculez $f^{\prime}(3)$ en utilisant la dérivée implicite pour les équations suivantes : a) $x^{2} y+x y^{2}=3 x$ , b) $\sqrt{x y}=1+x^{2} y$ , c) $x y^{4}+x^{2} y=x+3 y$ , d) $y e^{y}=x^{2} e^{x}$ .

See Solution

Problem 16458

Find the profit from the 51st and 91st items using $C'(50)$ , $R'(50)$ , $C'(90)$ , $R'(90)$ ; compare $C'(78)$ and $R'(78)$ .

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Problem 16459

Find the local maximum of $y = \frac{1}{3}x^3 - \frac{5}{2}x^2 + 6x - 5$ in the grid $[-10,10]$ . Round to the nearest hundredth.

See Solution

Problem 16460

Find the marginal revenue $M R$ when $q=50$ for $R=\ln(5+1000q^{2})$ . Round to two decimal places.

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Problem 16461

Bestimmen Sie die Grenzwerte für die folgenden Ausdrücke: a) $\lim _{x \rightarrow 3} \frac{x^{2}-25}{x-5}$ b) $\lim _{x \rightarrow 3} \frac{3 x^{2}-27}{x-3}$ c) $\lim _{x \rightarrow 1} \frac{x^{2}-x}{x-1}$ d) $\lim _{x \rightarrow-2} \frac{x^{4}-16}{x+2}$

See Solution

Problem 16462

Nach einem Brand steigt die PFT-Konzentration im See. Berechne max. Konzentration, Zeitpunkt unter $50 \frac{\mathrm{ng}}{\mathrm{l}}$ , stärkster Abfall und langfristige Konzentration. Funktion: $k(x)=250 x \cdot e^{-0,5 x}+20$ .

See Solution

Problem 16463

Find the price $p$ that maximizes revenue for the demand curve $q=1000-7p^{2}$ . Round to two decimal places.

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Problem 16464

Berechne 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 16465

Find the limit: $\lim _{n \rightarrow \infty} \frac{F_{n+1}}{F_{n}}$ using the Fibonacci formula to 3 decimal places.

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Problem 16466

A ball is thrown horizontally at $5 \mathrm{~m/s}$ from a $30 \mathrm{~m}$ cliff. How long until it hits the ground?

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Problem 16467

Find the limit: $\lim _{x \rightarrow 1^{+}} \frac{4+\ln x}{4-e^{5 x}}$ .

See Solution

Problem 16468

Identify which statements about series are implied by the $n$ -th term test. Select all that apply.

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Problem 16469

Identify which statements about series are implied by the $n$ -th term test. Select all that apply.

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Problem 16470

Show that the polynomial $f(x)=3 x^{3}-2 x+6$ has a real zero between -2 and -1 using the intermediate value theorem.

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Problem 16471

Find the sum $S=\sum_{n=1}^{\infty} a_{n}$ given $S_{T}=\frac{2 T-3}{T+5}$ .

See Solution

Problem 16472

Find the sum $S=\sum_{n=1}^{\infty} a_{n}$ if $S_{T}=\frac{2 T-3}{T+5}$ . Choices: 0, 1, 2, 3, 4, 5.

See Solution

Problem 16473

Find the infinite sum $S=\sum_{n=1}^{\infty} a_{n}$ given $S_{T}=\frac{2 T-3}{T+5}$ . Options: 0, 1, 2, 3, 4, 5.

See Solution

Problem 16474

Evaluate the integral $\oint_{C} \frac{2 z-1}{z^{2}(z^{3}+1)} d z$ using Cauchy's residue theorem over the rectangle $C$ .

See Solution

Problem 16475

Berechnen Sie das Volumen des Rotationskörpers von $f(x)=\frac{3}{2} \sqrt{x}$ über $[0 ; 9]$ um die x-Achse.

See Solution

Problem 16476

Find the limit: $\lim _{x \rightarrow-\infty} \frac{-2 e^{-x}}{x^{2}+2 x-5}$ .

See Solution

Problem 16477

Find the derivative of $g(x)=x^{2}+\cos(x)\sin(x)$ .

See Solution

Problem 16478

Find the derivative of $h(x)=\sqrt{x^{2}-5x}$ .

See Solution

Problem 16479

Find the derivative of $p(x)=x e^{\cos (x)}$ .

See Solution

Problem 16480

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{6^{x}-7^{x}}{x}=$ using L'Hôpital's Rule.

See Solution

Problem 16481

Find the derivative of the function $q(x)=\ln \left(\sqrt[3]{x^{2}+1}\right)$ .

See Solution

Problem 16482

Find the limit using l'Hôpital's rule: $\lim _{x \rightarrow 3} \frac{(4-x)^{1/(x-3)}}{x}$ . State DNE if it doesn't exist.

See Solution

Problem 16483

Find the indeterminate form of $r(x)=(4-x)^{\frac{1}{x-3}}$ as $x \to 3$ and evaluate the limit using l'Hôpital's rule.

See Solution

Problem 16484

Find the inflection point and asymptotes of $f(x) = \frac{x}{x^2-25}$ . Use exact numbers and equations for asymptotes.

See Solution

Problem 16485

Find the domain and transition points of $f(x) = \frac{x}{x^2-25}$ . Identify domain as an interval and transition points as a list.

See Solution

Problem 16486

Find the volume of the solid formed by rotating the area between $y=9x^{2}$ , $x=2$ , $x=3$ , and $y=0$ around the $x$ -axis. $V=$

See Solution

Problem 16487

Find the average rate of change of $f(x)=2 x^{2}+4 x-6$ from $x=-3$ to $x=3$ .

See Solution

Problem 16488

Find the average rate of change of $f(x)$ from $x = -5$ to $x = 5$ given values of $f(x)$ .

See Solution

Problem 16489

Find the average rate of change of $f(x)$ from $x = -4$ to $x = 4$ given $f(-4) = -27$ and $f(4) = -27$ .

See Solution

Problem 16490

Find the derivative at $x=0$ : $\frac{d}{d x}(u v + u + v)$ given $u(0)=1, v(0)=2, u'(0)=5, v'(0)=-3$ .

See Solution

Problem 16491

Given functions $u$ and $v$ with $u(0)=1$ , $v(0)=2$ , $u'(0)=5$ , $v'(0)=-3$ , find derivatives at $x=0$ for: (a) $u v + u + v$ , (b) $\frac{u}{v^2}$ , (c) $e^{u v}$ , (d) $\cos(2 \pi u)$ .

See Solution

Problem 16492

Given $u(0)=1$ , $v(0)=2$ , $u'(0)=5$ , $v'(0)=-3$ , find the derivatives at $x=0$ : (a) $\frac{d}{dx}(uv+u+v)$ , (b) $\frac{d}{dx}(\frac{u}{v^2})$ , (c) $\frac{d}{dx}(e^{uv})$ , (d) $\frac{d}{dx}(\cos(2\pi u))$ .

See Solution

Problem 16493

Set up a definite integral for the area under the curve from (0,2) to (4,0) without evaluating it.

See Solution

Problem 16494

Calculate the integral $\int_{-6}^{6} \sqrt{36-x^{2}} \, dx$ .

See Solution

Problem 16495

Find the integral of $f(x)=4-|x|$ from -2 to 2, where the graph forms an upside down "v".

See Solution

Problem 16496

In a series circuit, $R$ increases at 3 ohms/s and $X$ decreases at 2 ohms/s. Find $\frac{dZ}{dt}$ at $t=10$ , given $R=10$ , $X=20$ .

See Solution

Problem 16497

Calculate the expression: $\frac{\pi}{4} \int_{0}^{2} \frac{dx}{2-x} + 5 \int_{0}^{2} \frac{dx}{1+x}$ .

See Solution

Problem 16498

Calculate the amount from an investment of \$10 at a continuous compound rate of 10% over 2 years.

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Problem 16499

Calculate the expression: $\frac{11}{4} \int_{0}^{2} \frac{dx}{2-x} - \frac{29}{4} \int_{0}^{2} \frac{dx}{2+x} + 5 \int_{0}^{2} \frac{dx}{1+x}$ .

See Solution

Problem 16500

Evaluate the integral: $\int_{0}^{\pi / 4} \frac{6-6(\sin (\theta))^{2}}{8(\cos (\theta))^{2}} d \theta$

See Solution

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Calculus

Problem 16401

Beweisen Sie für x≠0x \neq 0x=0, dass F(x)=ln⁡(∣x∣)F(x) = \ln (|x|)F(x)=ln(∣x∣) eine Stammfunktion von f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ ist.

Problem 16402

Berechne das Integral ∫151x dx\int_{1}^{5} \frac{1}{x} \, dx∫15​x1​dx mit einer Stammfunktion.

Problem 16403

Find the domain of the function f(x)=6xx2+1f(x)=\frac{6 x}{\sqrt{x^{2}+1}}f(x)=x2+1​6x​ and its transition points (max, min, inflection).

Problem 16404

Beweisen Sie allgemein: Für x≠0x \neq 0x=0 ist F(x)=ln⁡(∣x∣)F(x)=\ln (|x|)F(x)=ln(∣x∣) die Stammfunktion von f(x)=1xf(x)=\frac{1}{x}f(x)=x1​.

Problem 16405

Find the transition points and asymptotes of the function f(x)=xx2−16f(x)=\frac{x}{x^{2}-16}f(x)=x2−16x​.

Problem 16406

Problem 16407

Evaluate the integral ∮Czezz2−1dz\oint_{C} \frac{z e^{z}}{z^{2}-1} d z∮C​z2−1zez​dz where CCC is the contour ∣z∣=3|z|=3∣z∣=3.

Problem 16408

Find the limit as nnn approaches infinity: lim⁡n→∞12n3+m−12m3+5m12n3+n+12m3−5n\lim _{n \rightarrow \infty} \frac{12 n^{3}+m-12 m^{3}+5 m}{\sqrt{12 n^{3}+n}+\sqrt{12 m^{3}-5 n}}limn→∞​12n3+n​+12m3−5n​12n3+m−12m3+5m​

Problem 16409

Find the limit: lim⁡x→∞12n3+x−12n3+5n12n3+x+12n3−5x\lim _{x \rightarrow \infty} \frac{12 n^{3}+x-12 n^{3}+5 n}{\sqrt{12 n^{3}+x}+\sqrt{12 n^{3}-5 x}}limx→∞​12n3+x​+12n3−5x​12n3+x−12n3+5n​

Problem 16410

Find the limit: lim⁡x→∞12x3+x−12x3+5x12x3+x+12x3−5x\lim _{x \rightarrow \infty} \frac{12 x^{3}+x-12 x^{3}+5 x}{\sqrt{12 x^{3}+x}+\sqrt{12 x^{3}-5 x}}limx→∞​12x3+x​+12x3−5x​12x3+x−12x3+5x​.

Problem 16411

Berechnen Sie die Geschwindigkeit v(t)v(t)v(t) eines Fallschirmspringers mit ve=0,048v_{e}=0,048ve​=0,048, t=0,2t=0,2t=0,2, g=9,81g=9,81g=9,81 und mk=0,0261m_{k}=0,0261mk​=0,0261.

Problem 16412

Evaluate the integral using integration by parts: ∫(5x−1)e−xdx\int(5 x-1) e^{-x} d x∫(5x−1)e−xdx (Include constant CCC)

Problem 16413

Differentiate y=cos⁡(x+5y)y=\cos (x+5 y)y=cos(x+5y) implicitly to find y′=dydxy^{\prime}=\frac{d y}{d x}y′=dxdy​.

Problem 16414

A pizza pan cools from 450∘F450^{\circ} \mathrm{F}450∘F to 74∘F74^{\circ} \mathrm{F}74∘F. After 5 min it's 300∘F300^{\circ} \mathrm{F}300∘F. Find when it's 130∘F130^{\circ} \mathrm{F}130∘F and 240∘F240^{\circ} \mathrm{F}240∘F. What happens to the temperature over time?

Problem 16415

Evaluate the integral using integration by parts: ∫(1−x)exdx\int(1-x) e^{x} d x∫(1−x)exdx (use constant CCC).

Problem 16416

Find the derivative of f(x)=(2x−x2)3f(x)=(2x-x^{2})^{3}f(x)=(2x−x2)3.

Problem 16417

Calculate the indefinite integral and include the constant of integration CCC: ∫(u+3)(8u+1)du\int(u+3)(8u+1) du∫(u+3)(8u+1)du

Problem 16418

Find the indefinite integral: ∫9+x+xxdx\int \frac{9+\sqrt{x}+x}{x} d x∫x9+x​+x​dx (Use CCC for the constant of integration.)

Problem 16419

Find the integral: ∫sec⁡(5x)tan⁡(5x) dx\int \sec (5 x) \tan (5 x) \, dx∫sec(5x)tan(5x)dx

Problem 16420

Evaluate the integral from 0 to 1: ∫01(x14+14x)dx\int_{0}^{1}\left(x^{14}+14^{x}\right) d x∫01​(x14+14x)dx

Problem 16421

Problem 16422

Find the indefinite integral: ∫sec⁡(t)(9sec⁡(t)+2tan⁡(t))dt\int \sec(t)(9 \sec(t)+2 \tan(t)) dt∫sec(t)(9sec(t)+2tan(t))dt. Use CCC for the constant of integration.

Problem 16423

Evaluate the integral from 0 to 13\frac{1}{\sqrt{3}}3​1​: ∫4t2−1t4−1dt\int 4 \frac{t^{2}-1}{t^{4}-1} dt∫4t4−1t2−1​dt

Problem 16424

Given the velocity function v(t)=5t−9v(t)=5 t-9v(t)=5t−9 for 0≤t≤30 \leq t \leq 30≤t≤3, find: (a) displacement (m), (b) total distance (m).

Problem 16425

Evaluate the integral: ∫sin⁡(t)1+cos⁡(t) dt+C\int \sin (t) \sqrt{1+\cos (t)} \, dt + C∫sin(t)1+cos(t)​dt+C.

Problem 16426

Evaluate the integral using the substitution u=x3+7u=x^{3}+7u=x3+7. Find ∫x2u dx\int x^{2} \sqrt{u} \, dx∫x2u​dx and include constant CCC.

Problem 16427

Problem 16428

Evaluate the integral using the substitution u=sin⁡(θ)u=\sin(\theta)u=sin(θ): ∫sin⁡5(θ)cos⁡(θ)dθ, C=constant.\int \sin^{5}(\theta) \cos(\theta) d\theta, \, C = \text{constant}.∫sin5(θ)cos(θ)dθ,C=constant.

Problem 16429

Find the area above the xxx-axis for the curve y=1−x2y=1-x^{2}y=1−x2 between x=−1x=-1x=−1 and x=2x=2x=2.

Problem 16430

Evaluate the integral with substitution: ∫x+1x2+2xdx=□+C\int \frac{x+1}{\sqrt{x^{2}+2 x}} d x=\square+C∫x2+2x​x+1​dx=□+C

Problem 16431

Problem 16432

Find the increase in total cost when production changes from x=30x=30x=30 to x=90x=90x=90 boxes, given marginal cost 4+x21,0004+\frac{x^{2}}{1,000}4+1,000x2​.

Problem 16433

Determine if the integral ∫6∞6(x+7)3/2dx\int_{6}^{\infty} \frac{6}{(x+7)^{3 / 2}} d x∫6∞​(x+7)3/26​dx is convergent or divergent and evaluate if convergent.

Problem 16434

Evaluate the integral using the substitution u=7+x4u=7+x^{4}u=7+x4: ∫x3(u4)dx,C is the constant of integration.\int x^{3}(u^{4}) d x, \quad C \text{ is the constant of integration.}∫x3(u4)dx,C is the constant of integration.

Problem 16435

How much to invest now at 8\% interest to have \9000in13years?Usetheformulaforcontinuouscompounding:9000 in 13 years? Use the formula for continuous compounding: 9000in13years?Usetheformulaforcontinuouscompounding:A = Pe^{rt}$.

Problem 16436

Calculate the integral: ∫01−31−x2dx\int_{0}^{1} \frac{-3}{\sqrt{1-x^{2}}} d x∫01​1−x2​−3​dx

Problem 16437

How long does it take for a bacteria population to double with a growth rate of 2.7%2.7\%2.7% per hour? Round to the nearest hundredth.

Problem 16438

Evaluate the integral from eee to e36e^{36}e36 of 1xln⁡(x) dx\frac{1}{x \sqrt{\ln(x)}} \, dxxln(x)​1​dx.

Problem 16439

Find the half-life of a substance with a decay rate of 7.2%7.2\%7.2% per day. Round to the nearest hundredth.

Problem 16440

Is the integral ∫−∞∞x8e−x8dx\int_{-\infty}^{\infty} x^{8} e^{-x^{8}} d x∫−∞∞​x8e−x8dx convergent or divergent? Evaluate if convergent.

Problem 16441

Find the limit: lim⁡x→1+5+ln⁡x5−e9x\lim _{x \rightarrow 1^{+}} \frac{5+\ln x}{5-e^{9 x}}limx→1+​5−e9x5+lnx​.

Problem 16442

Beweisen Sie für $x \neq 0$ , dass $F(x) = \ln (|x|)$ eine Stammfunktion von $f(x) = \frac{1}{x}$ ist.

Berechne das Integral $\int_{1}^{5} \frac{1}{x} \, dx$ mit einer Stammfunktion.

Find the domain of the function $f(x)=\frac{6 x}{\sqrt{x^{2}+1}}$ and its transition points (max, min, inflection).

Beweisen Sie allgemein: Für $x \neq 0$ ist $F(x)=\ln (|x|)$ die Stammfunktion von $f(x)=\frac{1}{x}$ .

Find the transition points and asymptotes of the function $f(x)=\frac{x}{x^{2}-16}$ .

Evaluate the integral $\oint_{C} \frac{z e^{z}}{z^{2}-1} d z$ where $C$ is the contour $|z|=3$ .

Find the limit as $n$ approaches infinity: $\lim _{n \rightarrow \infty} \frac{12 n^{3}+m-12 m^{3}+5 m}{\sqrt{12 n^{3}+n}+\sqrt{12 m^{3}-5 n}}$

Find the limit: $\lim _{x \rightarrow \infty} \frac{12 n^{3}+x-12 n^{3}+5 n}{\sqrt{12 n^{3}+x}+\sqrt{12 n^{3}-5 x}}$

Find the limit: $\lim _{x \rightarrow \infty} \frac{12 x^{3}+x-12 x^{3}+5 x}{\sqrt{12 x^{3}+x}+\sqrt{12 x^{3}-5 x}}$ .

Berechnen Sie die Geschwindigkeit $v(t)$ eines Fallschirmspringers mit $v_{e}=0,048$ , $t=0,2$ , $g=9,81$ und $m_{k}=0,0261$ .

Evaluate the integral using integration by parts: $\int(5 x-1) e^{-x} d x$ (Include constant $C$ )

Differentiate $y=\cos (x+5 y)$ implicitly to find $y^{\prime}=\frac{d y}{d x}$ .

A pizza pan cools from $450^{\circ} \mathrm{F}$ to $74^{\circ} \mathrm{F}$ . After 5 min it's $300^{\circ} \mathrm{F}$ . Find when it's $130^{\circ} \mathrm{F}$ and $240^{\circ} \mathrm{F}$ . What happens to the temperature over time?

Evaluate the integral using integration by parts: $\int(1-x) e^{x} d x$ (use constant $C$ ).

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

Calculate the indefinite integral and include the constant of integration $C$ : $\int(u+3)(8u+1) du$

Find the indefinite integral: $\int \frac{9+\sqrt{x}+x}{x} d x$ (Use $C$ for the constant of integration.)

Find the integral: $\int \sec (5 x) \tan (5 x) \, dx$

Evaluate the integral from 0 to 1: $\int_{0}^{1}\left(x^{14}+14^{x}\right) d x$

Find the indefinite integral: $\int \sec(t)(9 \sec(t)+2 \tan(t)) dt$ . Use $C$ for the constant of integration.

Evaluate the integral from 0 to $\frac{1}{\sqrt{3}}$ : $\int 4 \frac{t^{2}-1}{t^{4}-1} dt$

Given the velocity function $v(t)=5 t-9$ for $0 \leq t \leq 3$ , find: (a) displacement (m), (b) total distance (m).

Evaluate the integral: $\int \sin (t) \sqrt{1+\cos (t)} \, dt + C$ .

Evaluate the integral using the substitution $u=x^{3}+7$ . Find $\int x^{2} \sqrt{u} \, dx$ and include constant $C$ .

Evaluate the integral using the substitution $u=\sin(\theta)$ : $\int \sin^{5}(\theta) \cos(\theta) d\theta, \, C = \text{constant}.$

Find the area above the $x$ -axis for the curve $y=1-x^{2}$ between $x=-1$ and $x=2$ .

Evaluate the integral with substitution: $\int \frac{x+1}{\sqrt{x^{2}+2 x}} d x=\square+C$

Find the increase in total cost when production changes from $x=30$ to $x=90$ boxes, given marginal cost $4+\frac{x^{2}}{1,000}$ .

Determine if the integral $\int_{6}^{\infty} \frac{6}{(x+7)^{3 / 2}} d x$ is convergent or divergent and evaluate if convergent.

Evaluate the integral using the substitution $u=7+x^{4}$ : $\int x^{3}(u^{4}) d x, \quad C \text{ is the constant of integration.}$

How much to invest now at 8\% interest to have \ $9000 in 13 years? Use the formula for continuous compounding:$ A = Pe^{rt}$.

Calculate the integral: $\int_{0}^{1} \frac{-3}{\sqrt{1-x^{2}}} d x$

How long does it take for a bacteria population to double with a growth rate of $2.7\%$ per hour? Round to the nearest hundredth.

Evaluate the integral from $e$ to $e^{36}$ of $\frac{1}{x \sqrt{\ln(x)}} \, dx$ .

Find the half-life of a substance with a decay rate of $7.2\%$ per day. Round to the nearest hundredth.

Is the integral $\int_{-\infty}^{\infty} x^{8} e^{-x^{8}} d x$ convergent or divergent? Evaluate if convergent.

Find the limit: $\lim _{x \rightarrow 1^{+}} \frac{5+\ln x}{5-e^{9 x}}$ .

Find the limit as $x$ approaches $\frac{\pi}{4}$ of $(\tan x - 1) \sec x$ .

Evaluate the limit using L'Hôpital's Rule if needed: $\lim _{x \rightarrow \infty} \frac{3+x e^{2 / x}}{x \arctan (x)}$

Find the limit as $x$ approaches infinity for $\frac{5x - 4}{13x + 7}$ .

Find the velocity $x'(t)$ , speed, and acceleration of $x(t)=t^{3}-12t^{2}+36t-27$ . When is the particle moving forward?

Find the derivatives of the functions: $g(w)=\int_{0}^{w} \sin(2+t^{5}) dt$ and $h(x)=\int^{\sqrt{x}} \frac{z^{2}}{z^{4}+4} dz$ .

Evaluate the integral: $\int_{-1}^{2}(3 u-5)(u+2) du$

Find $y$ from $x^{2/3} + y^{2/3} = 1$ , then find its derivatives for $x \neq 0, \pm 1$ .

Find the time $t$ for maximum concentration of the drug $C(t)=13.2 t \cdot e^{-0.15 t}$ and its value at that time.

Find the interval where the function $f(x)=\int_{0}^{x}(4-t^{2}) e^{t^{2}} dt$ is increasing.

Find the quantity $q$ that maximizes profit, given $R(q)=550q$ and $C(q)=10,000+5q^{2}$ . What is the profit? $q=$ Profit $=\$ \mathbf{i}$

Find the quantity $q$ that maximizes profit, given revenue $R(q)=450q$ and cost $C(q)=10,000+5q^2$ . What is the profit? $q=$ Profit $=\$$

Find the limit $\lim _{n \rightarrow \infty} \frac{F_{n+1}}{F_{n}}$ using the Fibonacci formula to 3 decimal places.

Find the maximum concentration of the drug modeled by $C(t)=8 t e^{-0.2 t}$ . Choices: 10.6, 5, 14.7, 32.2, 20.2.

Calculez $f^{\prime}(3)$ en utilisant la dérivée implicite pour les équations suivantes : a) $x^{2} y+x y^{2}=3 x$ , b) $\sqrt{x y}=1+x^{2} y$ , c) $x y^{4}+x^{2} y=x+3 y$ , d) $y e^{y}=x^{2} e^{x}$ .

Find the profit from the 51st and 91st items using $C'(50)$ , $R'(50)$ , $C'(90)$ , $R'(90)$ ; compare $C'(78)$ and $R'(78)$ .

Find the local maximum of $y = \frac{1}{3}x^3 - \frac{5}{2}x^2 + 6x - 5$ in the grid $[-10,10]$ . Round to the nearest hundredth.

Find the marginal revenue $M R$ when $q=50$ for $R=\ln(5+1000q^{2})$ . Round to two decimal places.

Nach einem Brand steigt die PFT-Konzentration im See. Berechne max. Konzentration, Zeitpunkt unter $50 \frac{\mathrm{ng}}{\mathrm{l}}$ , stärkster Abfall und langfristige Konzentration. Funktion: $k(x)=250 x \cdot e^{-0,5 x}+20$ .

Find the price $p$ that maximizes revenue for the demand curve $q=1000-7p^{2}$ . Round to two decimal places.

Find the limit: $\lim _{n \rightarrow \infty} \frac{F_{n+1}}{F_{n}}$ using the Fibonacci formula to 3 decimal places.

A ball is thrown horizontally at $5 \mathrm{~m/s}$ from a $30 \mathrm{~m}$ cliff. How long until it hits the ground?

Find the limit: $\lim _{x \rightarrow 1^{+}} \frac{4+\ln x}{4-e^{5 x}}$ .

Identify which statements about series are implied by the $n$ -th term test. Select all that apply.

Identify which statements about series are implied by the $n$ -th term test. Select all that apply.

Show that the polynomial $f(x)=3 x^{3}-2 x+6$ has a real zero between -2 and -1 using the intermediate value theorem.

Find the sum $S=\sum_{n=1}^{\infty} a_{n}$ given $S_{T}=\frac{2 T-3}{T+5}$ .

Find the sum $S=\sum_{n=1}^{\infty} a_{n}$ if $S_{T}=\frac{2 T-3}{T+5}$ . Choices: 0, 1, 2, 3, 4, 5.

Find the infinite sum $S=\sum_{n=1}^{\infty} a_{n}$ given $S_{T}=\frac{2 T-3}{T+5}$ . Options: 0, 1, 2, 3, 4, 5.

Evaluate the integral $\oint_{C} \frac{2 z-1}{z^{2}(z^{3}+1)} d z$ using Cauchy's residue theorem over the rectangle $C$ .