Calculus

Problem 18501

Find the tangent line equation to the curve $x - y^3 = 0$ at the point $(1, 1)$ .

See Solution

Problem 18502

Bestimme den Definitionsbereich der Funktion $f(x)=\ln \left(\frac{x}{x^{2}+4}\right)$ und bilde die Ableitung $f^{\prime}(x)$ . Zeige, dass $f$ nur einen Extrempunkt bei $H(2 \mid f(2))$ hat und begründe, dass $f$ keine Nullstelle besitzt.

See Solution

Problem 18503

Beweisen Sie, dass wenn der Graph von $f(x)=x^{n}, n \in \mathbb{N}$ punktsymmetrisch ist, dann gilt: $\int_{-1}^{1} x^{n} d x=0$ .

See Solution

Problem 18504

Find the derivative of $f(t)=x^{3}$ . Choose the correct answer: A. $3 x^{2}$ B. $x^{4}$ C. $3 x$ D. 0

See Solution

Problem 18505

A 10 ft ladder leans against a wall, with the base 6 ft away. It slides out at 2 ft/sec. Find the top's velocity.

See Solution

Problem 18506

Find the resistance $r(x)=\frac{3}{8} x-5$ of an elastic band when the stretched length is 19 inches.

See Solution

Problem 18507

Find the derivative of $f(x)=\frac{-1}{\tan x}$ . Which option is correct for $f^{\prime}(x)$ ? A, B, C, or D?

See Solution

Problem 18508

Find the derivative of $f(x)=\frac{4}{x \sqrt{x}}$ . Which option is correct? A. B. C. D.

See Solution

Problem 18509

Find the derivative of $f(x)=x \sin x+\cos x$ . What is $f'(x)$ ? A. $x \sin x$ B. $x \sin x + 2 \cos x$ C. $x \cos x$ D. $\cos x - \sin x$

See Solution

Problem 18510

Find the limit: $\lim _{x \rightarrow \infty} \frac{15-26 x^{3}-21 x^{6}}{25 x^{2}-10 x^{4}}$ . State if it DNE.

See Solution

Problem 18511

Lösen Sie die Klammer in $g(x)=-\frac{1}{2}(x-2)^{2}+1,5$ auf und leiten Sie sie ab.

See Solution

Problem 18512

Given $f(1)=-3$ and $f(7)=3$ , use the Intermediate Value Theorem to analyze $f(x)$ on $(1,7)$ .

See Solution

Problem 18513

Find when a \$1000 deposit grows to \$1677.41 with continuous compounding, given it reaches \$1366.56 in 5 years. Round to the nearest tenth.

See Solution

Problem 18514

Gegeben ist die Funktion $f(t)=-\frac{1}{350} t^{3}+\frac{1}{25} t^{2}$ . Analysiere die Eigenschaften und berechne die Stickstoffbindung für 7 und 14 Stunden.

See Solution

Problem 18515

Find $\lim _{x \rightarrow 2} f(x)$ for the piecewise function $f(x)=\begin{cases}2 x^{2}-7 & x>2 \\ 2-4 x & x<2\end{cases}$ .

See Solution

Problem 18516

Untersuchen Sie die Punktarten (Hoch-, Tief-, Wendepunkt, Sattelpunkt) für $f(x)=2 x^{3}-6 x^{2}$ bei $x=1,2$ und $f(x)=\frac{1}{6} x^{6}-\frac{1}{3} x^{3}$ bei $x=0,1$ .

See Solution

Problem 18517

Use the Intermediate Value Theorem on $f(x)$ in $(-7,0)$ with $f(-7)=4$ and $f(0)=-3$ . What can you conclude?

See Solution

Problem 18518

Berechnen Sie das Integral $\int_{0}^{3} \frac{1}{9} x^{2} d x$ und interpretieren Sie es für Geschwindigkeit und Produktionsrate.

See Solution

Problem 18519

Calculate the integral of the function $(2 x^{3}-7)^{2}$ with respect to $x$ .

See Solution

Problem 18520

Find the instantaneous rate of change of height $h(t)=\frac{1.4 \sqrt{t}}{a}$ at $t=9$ seconds.

See Solution

Problem 18521

Evaluate the integral $\int\left(7 \sec x \tan x-6 \sec ^{2} x\right)dx$ .

See Solution

Problem 18522

Bestimme die durchschnittliche Stickstoffbindung von $f(t)=-\frac{1}{350} t^{3}+\frac{1}{25} t^{2}$ über 7, 7 und 14 Stunden.

See Solution

Problem 18523

Untersuchen Sie die Funktionen auf Wendestellen und analysieren Sie Nullstellen, Extrema und Wendepunkte.

See Solution

Problem 18524

Bestimme die Ableitung von $f(x)=\frac{1}{25}(x-5)^{3}-3x$ .

See Solution

Problem 18525

A substance decays at 0.073/min. How much of a 120g sample remains after 30 min? Round to the nearest tenth.

See Solution

Problem 18526

A sculptor forms a cylinder with height 8 in and radius 3 in. Radius decreases at $1/2$ in/min. Find:
(a) Rate of area decrease. (b) Rate of height increase. (c) Expression for radius change rate in terms of height and radius.

See Solution

Problem 18527

Find the tangent and normal equations to the curve $y=x^{4}-6 x+3$ at the point where $x=2$ .

See Solution

Problem 18528

Bestimmen Sie die zweite Ableitung von $f(x)$ , wenn $f^{\prime}(x)=3 x^{2}-3 x$ gegeben ist.

See Solution

Problem 18529

Find the value of $x$ that maximizes revenue given the demand function $p=D(x)=85 e^{-0.01 x}$ .

See Solution

Problem 18530

Find the max revenue value given $b$ . Also, evaluate $\lim _{x \rightarrow-\infty}\left(\sqrt{x^{2}+x-1}+x\right)$ .

See Solution

Problem 18531

Find the elasticity function for the demand function $p=D(x)=\sqrt{472.5-3.5 x}$ .

See Solution

Problem 18532

Find the elasticity function for the demand $p=D(x)=\sqrt{148.5-1.5 x}$ .

See Solution

Problem 18533

Find the value of $m$ to ensure the function $f(x)$ is continuous at $x=0$ , where $f(x)$ is defined as: $f(x)=\begin{cases} x\left[\cos ^{2}\left(\frac{1}{x}\right)-\sin ^{2}\left(\frac{1}{x}\right)\right] & \text{if } x \neq 0 \\ m & \text{if } x=0 \end{cases}$

See Solution

Problem 18534

Bestimme die Tangentengleichung $t_{1}$ von $f(x)=\frac{1}{3} x^{2}$ bei $x_{0}=-2$ und die orthogonale Tangente.

See Solution

Problem 18535

Finde die Stammfunktion von $f(x)=4 x^{3}+2 x^{2}-1$ .

See Solution

Problem 18536

Find the area represented by the integral $\int_{-8}^{0}\left(2+\sqrt{64-x^{2}}\right) d x$ .

See Solution

Problem 18537

Calculate the integral: $\int_{-8}^{0}\left(2+\sqrt{64-x^{2}}\right) d x$

See Solution

Problem 18538

Untersuchen Sie die Funktion f auf Wendestellen und Nullstellen für die folgenden Funktionen: a) $f(x)=x^{3}+6 x^{2}-1$ , b) $f(x)=0,5 x^{4}-12 x^{2}$ , c) $f(x)=x^{4}+4 x^{2}$ , d) $f(x)=x^{5}+5 x^{2}$ , e) $f(x)=4-4 x-x^{2}$ , f) $f(x)=\frac{1}{3} x^{6}-\frac{1}{8} x^{4}$ .

See Solution

Problem 18539

Gegeben ist $f_{a}(x)=\frac{1}{a} x^{2}-4 x \quad(a \neq 0)$ . Zeigen Sie, dass $P(-2,0)$ und $Q(2,0)$ auf den Graphen liegen. Bestimmen Sie die Wendetangente mit $m=8$ .

See Solution

Problem 18540

$\lim _{x \rightarrow \pi} \frac{\sin ^{2} x}{1+\cos ^{3} x}$ nedir? A) -1 B) $-\frac{1}{2}$ C) $\frac{2}{3}$ D) 1 E) $\frac{4}{3}$

See Solution

Problem 18541

Find the antiderivative of $f(x)=x^{5}-4 x^{3}$ .

See Solution

Problem 18542

Express the limit as a definite integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{x_{i}^{}}{(x_{i}^{})^{2}+8} \Delta x$ over $[1,9]$ .

See Solution

Problem 18543

Find the differential of the function $A=4 \pi x^{2}$ .

See Solution

Problem 18544

Calculate the antiderivative of $f(x)=\frac{3}{4} x^{2}+\frac{8}{7} x^{3}$ .

See Solution

Problem 18545

Approximate $\sqrt[3]{42}$ using differentials as $a \pm \frac{b}{c}$ , with $a$ being the closest integer.

See Solution

Problem 18546

Find the differential of the function $P=-0.7 x^{2}-12 x-11$ .

See Solution

Problem 18547

Find the differential of the function $y=9 x^{3}-x^{2}-4 x-2$ .

See Solution

Problem 18548

Approximate $\sqrt[3]{50}$ using differentials. Express as $a \pm \frac{b}{c}$ where $a$ is the nearest integer.

See Solution

Problem 18549

Find the differential of the function $P=-0.8 x^{2}-2 x-7$ .

See Solution

Problem 18550

Approximate $\sqrt[3]{152}$ using differentials. Express as $a \pm \frac{b}{c}$ with $a$ being the nearest integer.

See Solution

Problem 18551

Find the horizontal asymptote of the function $f(x)=\frac{2-x}{x^{2}-6 x+8}$ . Options: $y=-1$ , $y=0$ , $y=1$ , $y=2$ .

See Solution

Problem 18552

Find the differential of the function $V=14 \pi r^{3}$ .

See Solution

Problem 18553

Approximate $\sqrt[3]{155}$ using differentials as $a \pm \frac{b}{c}$ , with $a$ as the nearest integer.

See Solution

Problem 18554

Approximate $\sqrt[3]{12}$ using differentials as $a \pm \frac{b}{c}$ , where $a$ is the closest integer.

See Solution

Problem 18555

Find the differential of the function $u=(4 t+7)^{2}$ .

See Solution

Problem 18556

Find the differential of the function $u=(-9 t+2)^{2}$ .

See Solution

Problem 18557

Find the integral of the function: $\int\left(\sin y+\sec ^{2} y\right) d y$ .

See Solution

Problem 18558

Approximate $\sqrt{17}$ using differentials as $a \pm \frac{b}{c}$ with $a$ as the closest integer.

See Solution

Problem 18559

$\lim _{a \rightarrow 1} \frac{\sqrt[3]{a}-1}{\sqrt{a}-1}$ nedir? A) $\frac{1}{6}$ B) $\frac{1}{3}$ C) $\frac{2}{3}$ D) 1 E) $\frac{4}{3}$

See Solution

Problem 18560

Leiten Sie die Funktionen ab: a) $f(x)=x^{a}$ , b) $f(x)=x^{t+1}$ , c) $f(x)=\frac{1}{x^{r}}$ , d) $g(t)=t^{2 k} \cdot t$ .

See Solution

Problem 18561

In einem Gezeitenkraftwerk fließt Wasser in den Speicher. Beantworte:
a) Was bedeutet 1 FE unter dem Graphen von $d$ ?
b) Wann steigt die Wassermenge am schnellsten, wann ist sie maximal und minimal? Was passiert nach 12 Stunden?
c) Bei Springflut fließen $25 \%$ mehr Wasser. Wie verändert sich die Fläche zwischen $d$ und der X-Achse?

See Solution

Problem 18562

Calculate $\triangle y - d y$ for the function $y=(-3x-2)^{2}$ using $x=2$ and $\triangle x=-0.2$ . Round to three decimals.

See Solution

Problem 18563

Find the marginal revenue function $R^{\prime}(x)$ for the revenue $R(x)=86 x+\ln(7 x^{3}+17)$ .

See Solution

Problem 18564

Find the derivative of the function $g(x)=9 x^{5} e^{4-3 x^{4}}$ .

See Solution

Problem 18565

Find the integral of the function $e^{e^{t}+t}$ with respect to $t$ : $\int e^{e^{t}+t} d t$ .

See Solution

Problem 18566

Calculate $\triangle y - d y$ for the function $y=(-3 x-5)^{2}$ at $x=2$ and $\triangle x=-0.2$ . Round to three decimal places.

See Solution

Problem 18567

Approximate $\sqrt{5}$ using differentials as $a \pm \frac{b}{c}$ , where $a$ is the closest integer.

See Solution

Problem 18568

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

See Solution

Problem 18569

Find the differential of the function $A=9 \pi x^{2}$ .

See Solution

Problem 18570

Solve the integral: $\int v \sqrt{v+4} \, dv$

See Solution

Problem 18571

Find the integral of the function: $\int\left(\frac{7}{4} x^{5}-\sqrt{x-1}\right) d x$

See Solution

Problem 18572

Find the derivative of $f(x)=\ln \left(\left(\frac{23 x-25 x^{2}}{19 x^{2}+7 x}\right)^{2}\right)$ .

See Solution

Problem 18573

Find the integral of $\tan \gamma$ with respect to $\gamma$ .

See Solution

Problem 18574

Find the differential of the function $P=-0.2 x^{2}-20 x+16$ .

See Solution

Problem 18575

Maximize revenue $R(x) = x * \sqrt{351 - 3x}$ using the demand function $D(x) = \sqrt{351 - 3x}$ . Find $x$ .

See Solution

Problem 18576

Find the first derivative of the function $g(x)=-5 x^{4}-4 e^{x}$ .

See Solution

Problem 18577

Find the difference quotient, $\frac{f(x+h)-f(x)}{h}$ , for the function $f(x)=-7 x^{2}-6 x-8$ .

See Solution

Problem 18578

Find the elasticity function for the demand $p=D(x)=\sqrt{351-3x}$ .

See Solution

Problem 18579

Find the difference quotient for the function $f(x)=-x-7$ : $\frac{f(x+h)-f(x)}{h}$ .

See Solution

Problem 18580

Find the derivatives of these functions:
a) $f(x)=2 x^{3}+5 x^{2}$ , b) $f(x)=4 x^{5}-3 x$ , c) $f(x)=2 x^{7}-3 x^{4}$ , d) $f(x)=0.25 x^{4}-x^{3}$ , e) $f(t)=-5 t^{4}-4 t^{3}+t$ , f) $f(z)=-\frac{2}{3} z^{3}+2 z^{2}+3$ , g) $f(x)=\frac{1}{5} x^{5}-3 x^{3}-2$ , h) $f(t)=t^{4}-2 t^{3}+\frac{1}{4} t^{2}$ .

See Solution

Problem 18581

Find the difference quotient, $\frac{f(x+h)-f(x)}{h}$ , for the function $f(x)=-x-7$ .

See Solution

Problem 18582

Find the production level that maximizes revenue for the demand function $p=D(x)=49284-0.03 x^{2}$ .

See Solution

Problem 18583

Find the second partial derivatives $z_{xx}$ , $z_{yy}$ , and $z_{xy}$ for the function $z=x^{2}(1+2y)$ .

See Solution

Problem 18584

Calculate the integral from 0 to $\pi$ of $\sin x$ with respect to $x$ : $\int_{0}^{\pi} \sin x \, dx$ .

See Solution

Problem 18585

Finde die Stammfunktion von $f(x)=\frac{1}{4} x^{-2}$ .

See Solution

Problem 18586

Find the units sold for max revenue from demand $D(x)=160 e^{-0.04 x}$ , where $0 \leq x \leq 95$ . Round to nearest unit.

See Solution

Problem 18587

Find the Laplace transform of $f(t)=12 e^{-1.5 t} \cos (500 t-0.785) u(t)$ .

See Solution

Problem 18588

Find the price per unit for maximum revenue given the demand $D(x)=160 e^{-0.04 x}$ for $0 \leq x \leq 95$ . Round to the nearest cent.

See Solution

Problem 18589

Trouver la transformée de Laplace $\mathrm{F}(\mathrm{s})$ pour les fonctions suivantes : a) $f(t)=A(1-e^{-\alpha t}) u(t)$ , b) $f(t)=A$ pour $0 \leq t \leq T$ , c) $f(t)=A e^{-\alpha t}$ pour $0 \leq t \leq T$ , d) $f(t)=t^{2} e^{-5 t} u(t)$ , e) $f(t)=e^{-0.5 t} u(t-2)$ , f) $f(t)=12 e^{-1.5 t} \cos (500 t-0.785) u(t)$ .

See Solution

Problem 18590

Berechne die Steigung des Graphen von $f$ bei $A(2 \mid f(2))$ für die Funktionen a) bis d).

See Solution

Problem 18591

Find the production level where demand is elastic for $p=D(x)=\sqrt{169-13x}$ . Produce whole units only.

See Solution

Problem 18592

Estimate $y(0.6)$ using Euler's method with step size 0.2 for $y'=\cos(x+y)$ and $y(0)=0$ .

See Solution

Problem 18593

Evaluate the integral: $\int_{0}^{3} \frac{v^{2}+11}{\sqrt{v^{3}+33 v+7}} d v = \square$ (Exact answer with radicals).

See Solution

Problem 18594

Find the antiderivative of $f(x)=\frac{3}{x^{4}}$ .

See Solution

Problem 18595

Evaluate the integral using a change of variables: $\int_{0}^{3} \frac{v^{2}+11}{\sqrt{v^{3}+33 v+7}} d v$ .

See Solution

Problem 18596

Gegeben ist die Funktion $f_{a}(x)=\frac{1}{8} x^{4}-\frac{a}{12} x^{3}+2 x$ . Untersuchen Sie das Verhalten und die Ableitungen für $a=0$ .

See Solution

Problem 18597

Differentiate $y + \sin(y) = x$ to find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ in terms of $x$ and $y$ .

See Solution

Problem 18598

Differentiate $2 x^{2}-y^{2}=9$ to find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ in terms of $x$ and $y$ .

See Solution

Problem 18599

A stone is thrown upward from a 112 ft cliff with a speed of 96 ft/s. Height after $t$ seconds is $s(t)=-16t^2+96t+112$ . Find: a. Velocity $v(t)=\square$ b. Time to highest point: $\square$ c. Height at highest point: $\square$ d. Time when it hits ground: $\square$ e. Velocity when it hits ground: $\square$

See Solution

Problem 18600

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

See Solution

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Calculus

Problem 18501

Find the tangent line equation to the curve x−y3=0x - y^3 = 0x−y3=0 at the point (1,1)(1, 1)(1,1).

Problem 18502

Problem 18503

Beweisen Sie, dass wenn der Graph von f(x)=xn,n∈Nf(x)=x^{n}, n \in \mathbb{N}f(x)=xn,n∈N punktsymmetrisch ist, dann gilt: ∫−11xndx=0\int_{-1}^{1} x^{n} d x=0∫−11​xndx=0.

Problem 18504

Find the derivative of f(t)=x3f(t)=x^{3}f(t)=x3. Choose the correct answer: A. 3x23 x^{2}3x2 B. x4x^{4}x4 C. 3x3 x3x D. 0

Problem 18505

A 10 ft ladder leans against a wall, with the base 6 ft away. It slides out at 2 ft/sec. Find the top's velocity.

Problem 18506

Find the resistance r(x)=38x−5r(x)=\frac{3}{8} x-5r(x)=83​x−5 of an elastic band when the stretched length is 19 inches.

Problem 18507

Find the derivative of f(x)=−1tan⁡xf(x)=\frac{-1}{\tan x}f(x)=tanx−1​. Which option is correct for f′(x)f^{\prime}(x)f′(x)? A, B, C, or D?

Problem 18508

Find the derivative of f(x)=4xxf(x)=\frac{4}{x \sqrt{x}}f(x)=xx​4​. Which option is correct? A. B. C. D.

Problem 18509

Find the derivative of f(x)=xsin⁡x+cos⁡xf(x)=x \sin x+\cos xf(x)=xsinx+cosx. What is f′(x)f'(x)f′(x)? A. xsin⁡xx \sin xxsinx B. xsin⁡x+2cos⁡xx \sin x + 2 \cos xxsinx+2cosx C. xcos⁡xx \cos xxcosx D. cos⁡x−sin⁡x\cos x - \sin xcosx−sinx

Problem 18510

Find the limit: lim⁡x→∞15−26x3−21x625x2−10x4\lim _{x \rightarrow \infty} \frac{15-26 x^{3}-21 x^{6}}{25 x^{2}-10 x^{4}}x→∞lim​25x2−10x415−26x3−21x6​. State if it DNE.

Problem 18511

Lösen Sie die Klammer in g(x)=−12(x−2)2+1,5g(x)=-\frac{1}{2}(x-2)^{2}+1,5g(x)=−21​(x−2)2+1,5 auf und leiten Sie sie ab.

Problem 18512

Given f(1)=−3f(1)=-3f(1)=−3 and f(7)=3f(7)=3f(7)=3, use the Intermediate Value Theorem to analyze f(x)f(x)f(x) on (1,7)(1,7)(1,7).

Problem 18513

Find when a \$1000 deposit grows to \$1677.41 with continuous compounding, given it reaches \$1366.56 in 5 years. Round to the nearest tenth.

Problem 18514

Gegeben ist die Funktion f(t)=−1350t3+125t2f(t)=-\frac{1}{350} t^{3}+\frac{1}{25} t^{2}f(t)=−3501​t3+251​t2. Analysiere die Eigenschaften und berechne die Stickstoffbindung für 7 und 14 Stunden.

Problem 18515

Find lim⁡x→2f(x)\lim _{x \rightarrow 2} f(x)limx→2​f(x) for the piecewise function f(x)={2x2−7x>22−4xx<2f(x)=\begin{cases}2 x^{2}-7 & x>2 \\ 2-4 x & x<2\end{cases}f(x)={2x2−72−4x​x>2x<2​.

Problem 18516

Untersuchen Sie die Punktarten (Hoch-, Tief-, Wendepunkt, Sattelpunkt) für f(x)=2x3−6x2f(x)=2 x^{3}-6 x^{2}f(x)=2x3−6x2 bei x=1,2x=1,2x=1,2 und f(x)=16x6−13x3f(x)=\frac{1}{6} x^{6}-\frac{1}{3} x^{3}f(x)=61​x6−31​x3 bei x=0,1x=0,1x=0,1.

Problem 18517

Use the Intermediate Value Theorem on f(x)f(x)f(x) in (−7,0)(-7,0)(−7,0) with f(−7)=4f(-7)=4f(−7)=4 and f(0)=−3f(0)=-3f(0)=−3. What can you conclude?

Problem 18518

Berechnen Sie das Integral ∫0319x2dx\int_{0}^{3} \frac{1}{9} x^{2} d x∫03​91​x2dx und interpretieren Sie es für Geschwindigkeit und Produktionsrate.

Problem 18519

Calculate the integral of the function (2x3−7)2(2 x^{3}-7)^{2}(2x3−7)2 with respect to xxx.

Problem 18520

Find the instantaneous rate of change of height h(t)=1.4tah(t)=\frac{1.4 \sqrt{t}}{a}h(t)=a1.4t​​ at t=9t=9t=9 seconds.

Problem 18521

Evaluate the integral ∫(7sec⁡xtan⁡x−6sec⁡2x)dx\int\left(7 \sec x \tan x-6 \sec ^{2} x\right)dx∫(7secxtanx−6sec2x)dx.

Problem 18522

Bestimme die durchschnittliche Stickstoffbindung von f(t)=−1350t3+125t2f(t)=-\frac{1}{350} t^{3}+\frac{1}{25} t^{2}f(t)=−3501​t3+251​t2 über 7, 7 und 14 Stunden.

Problem 18523

Untersuchen Sie die Funktionen auf Wendestellen und analysieren Sie Nullstellen, Extrema und Wendepunkte.

Problem 18524

Bestimme die Ableitung von f(x)=125(x−5)3−3xf(x)=\frac{1}{25}(x-5)^{3}-3xf(x)=251​(x−5)3−3x.

Problem 18525

A substance decays at 0.073/min. How much of a 120g sample remains after 30 min? Round to the nearest tenth.

Problem 18526

A sculptor forms a cylinder with height 8 in and radius 3 in. Radius decreases at 1/21/21/2 in/min. Find:(a) Rate of area decrease. (b) Rate of height increase. (c) Expression for radius change rate in terms of height and radius.

Problem 18527

Find the tangent and normal equations to the curve y=x4−6x+3y=x^{4}-6 x+3y=x4−6x+3 at the point where x=2x=2x=2.

Problem 18528

Bestimmen Sie die zweite Ableitung von f(x)f(x)f(x), wenn f′(x)=3x2−3xf^{\prime}(x)=3 x^{2}-3 xf′(x)=3x2−3x gegeben ist.

Problem 18529

Find the value of xxx that maximizes revenue given the demand function p=D(x)=85e−0.01xp=D(x)=85 e^{-0.01 x}p=D(x)=85e−0.01x.

Problem 18530

Find the max revenue value given bbb. Also, evaluate lim⁡x→−∞(x2+x−1+x)\lim _{x \rightarrow-\infty}\left(\sqrt{x^{2}+x-1}+x\right)limx→−∞​(x2+x−1​+x).

Problem 18531

Find the elasticity function for the demand function p=D(x)=472.5−3.5xp=D(x)=\sqrt{472.5-3.5 x}p=D(x)=472.5−3.5x​.

Problem 18532

Find the elasticity function for the demand p=D(x)=148.5−1.5xp=D(x)=\sqrt{148.5-1.5 x}p=D(x)=148.5−1.5x​.

Problem 18533

Problem 18534

Bestimme die Tangentengleichung t1t_{1}t1​ von f(x)=13x2f(x)=\frac{1}{3} x^{2}f(x)=31​x2 bei x0=−2x_{0}=-2x0​=−2 und die orthogonale Tangente.

Problem 18535

Finde die Stammfunktion von f(x)=4x3+2x2−1f(x)=4 x^{3}+2 x^{2}-1f(x)=4x3+2x2−1.

Problem 18536

Find the area represented by the integral ∫−80(2+64−x2)dx\int_{-8}^{0}\left(2+\sqrt{64-x^{2}}\right) d x∫−80​(2+64−x2​)dx.

Problem 18537

Calculate the integral: ∫−80(2+64−x2)dx\int_{-8}^{0}\left(2+\sqrt{64-x^{2}}\right) d x∫−80​(2+64−x2​)dx

Problem 18538

Problem 18539

Gegeben ist fa(x)=1ax2−4x(a≠0)f_{a}(x)=\frac{1}{a} x^{2}-4 x \quad(a \neq 0)fa​(x)=a1​x2−4x(a=0). Zeigen Sie, dass P(−2,0)P(-2,0)P(−2,0) und Q(2,0)Q(2,0)Q(2,0) auf den Graphen liegen. Bestimmen Sie die Wendetangente mit m=8m=8m=8.

Problem 18540

lim⁡x→πsin⁡2x1+cos⁡3x\lim _{x \rightarrow \pi} \frac{\sin ^{2} x}{1+\cos ^{3} x}limx→π​1+cos3xsin2x​ nedir? A) -1 B) −12-\frac{1}{2}−21​ C) 23\frac{2}{3}32​ D) 1 E) 43\frac{4}{3}34​

Problem 18541

Find the antiderivative of f(x)=x5−4x3f(x)=x^{5}-4 x^{3}f(x)=x5−4x3.

Find the tangent line equation to the curve $x - y^3 = 0$ at the point $(1, 1)$ .

Beweisen Sie, dass wenn der Graph von $f(x)=x^{n}, n \in \mathbb{N}$ punktsymmetrisch ist, dann gilt: $\int_{-1}^{1} x^{n} d x=0$ .

Find the derivative of $f(t)=x^{3}$ . Choose the correct answer: A. $3 x^{2}$ B. $x^{4}$ C. $3 x$ D. 0

Find the resistance $r(x)=\frac{3}{8} x-5$ of an elastic band when the stretched length is 19 inches.

Find the derivative of $f(x)=\frac{-1}{\tan x}$ . Which option is correct for $f^{\prime}(x)$ ? A, B, C, or D?

Find the derivative of $f(x)=\frac{4}{x \sqrt{x}}$ . Which option is correct? A. B. C. D.

Find the derivative of $f(x)=x \sin x+\cos x$ . What is $f'(x)$ ? A. $x \sin x$ B. $x \sin x + 2 \cos x$ C. $x \cos x$ D. $\cos x - \sin x$

Find the limit: $\lim _{x \rightarrow \infty} \frac{15-26 x^{3}-21 x^{6}}{25 x^{2}-10 x^{4}}$ . State if it DNE.

Lösen Sie die Klammer in $g(x)=-\frac{1}{2}(x-2)^{2}+1,5$ auf und leiten Sie sie ab.

Given $f(1)=-3$ and $f(7)=3$ , use the Intermediate Value Theorem to analyze $f(x)$ on $(1,7)$ .

Gegeben ist die Funktion $f(t)=-\frac{1}{350} t^{3}+\frac{1}{25} t^{2}$ . Analysiere die Eigenschaften und berechne die Stickstoffbindung für 7 und 14 Stunden.

Find $\lim _{x \rightarrow 2} f(x)$ for the piecewise function $f(x)=\begin{cases}2 x^{2}-7 & x>2 \\ 2-4 x & x<2\end{cases}$ .

Untersuchen Sie die Punktarten (Hoch-, Tief-, Wendepunkt, Sattelpunkt) für $f(x)=2 x^{3}-6 x^{2}$ bei $x=1,2$ und $f(x)=\frac{1}{6} x^{6}-\frac{1}{3} x^{3}$ bei $x=0,1$ .

Use the Intermediate Value Theorem on $f(x)$ in $(-7,0)$ with $f(-7)=4$ and $f(0)=-3$ . What can you conclude?

Berechnen Sie das Integral $\int_{0}^{3} \frac{1}{9} x^{2} d x$ und interpretieren Sie es für Geschwindigkeit und Produktionsrate.

Calculate the integral of the function $(2 x^{3}-7)^{2}$ with respect to $x$ .

Find the instantaneous rate of change of height $h(t)=\frac{1.4 \sqrt{t}}{a}$ at $t=9$ seconds.

Evaluate the integral $\int\left(7 \sec x \tan x-6 \sec ^{2} x\right)dx$ .

Bestimme die durchschnittliche Stickstoffbindung von $f(t)=-\frac{1}{350} t^{3}+\frac{1}{25} t^{2}$ über 7, 7 und 14 Stunden.

Bestimme die Ableitung von $f(x)=\frac{1}{25}(x-5)^{3}-3x$ .

A sculptor forms a cylinder with height 8 in and radius 3 in. Radius decreases at $1/2$ in/min. Find:
(a) Rate of area decrease. (b) Rate of height increase. (c) Expression for radius change rate in terms of height and radius.

Find the tangent and normal equations to the curve $y=x^{4}-6 x+3$ at the point where $x=2$ .

Bestimmen Sie die zweite Ableitung von $f(x)$ , wenn $f^{\prime}(x)=3 x^{2}-3 x$ gegeben ist.

Find the value of $x$ that maximizes revenue given the demand function $p=D(x)=85 e^{-0.01 x}$ .

Find the max revenue value given $b$ . Also, evaluate $\lim _{x \rightarrow-\infty}\left(\sqrt{x^{2}+x-1}+x\right)$ .

Find the elasticity function for the demand function $p=D(x)=\sqrt{472.5-3.5 x}$ .

Find the elasticity function for the demand $p=D(x)=\sqrt{148.5-1.5 x}$ .

Bestimme die Tangentengleichung $t_{1}$ von $f(x)=\frac{1}{3} x^{2}$ bei $x_{0}=-2$ und die orthogonale Tangente.

Finde die Stammfunktion von $f(x)=4 x^{3}+2 x^{2}-1$ .

Find the area represented by the integral $\int_{-8}^{0}\left(2+\sqrt{64-x^{2}}\right) d x$ .

Calculate the integral: $\int_{-8}^{0}\left(2+\sqrt{64-x^{2}}\right) d x$

Gegeben ist $f_{a}(x)=\frac{1}{a} x^{2}-4 x \quad(a \neq 0)$ . Zeigen Sie, dass $P(-2,0)$ und $Q(2,0)$ auf den Graphen liegen. Bestimmen Sie die Wendetangente mit $m=8$ .

$\lim _{x \rightarrow \pi} \frac{\sin ^{2} x}{1+\cos ^{3} x}$ nedir? A) -1 B) $-\frac{1}{2}$ C) $\frac{2}{3}$ D) 1 E) $\frac{4}{3}$

Find the antiderivative of $f(x)=x^{5}-4 x^{3}$ .

Express the limit as a definite integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{x_{i}^{}}{(x_{i}^{})^{2}+8} \Delta x$ over $[1,9]$ .

Find the differential of the function $A=4 \pi x^{2}$ .

Calculate the antiderivative of $f(x)=\frac{3}{4} x^{2}+\frac{8}{7} x^{3}$ .

Approximate $\sqrt[3]{42}$ using differentials as $a \pm \frac{b}{c}$ , with $a$ being the closest integer.

Find the differential of the function $P=-0.7 x^{2}-12 x-11$ .

Find the differential of the function $y=9 x^{3}-x^{2}-4 x-2$ .

Approximate $\sqrt[3]{50}$ using differentials. Express as $a \pm \frac{b}{c}$ where $a$ is the nearest integer.

Find the differential of the function $P=-0.8 x^{2}-2 x-7$ .

Approximate $\sqrt[3]{152}$ using differentials. Express as $a \pm \frac{b}{c}$ with $a$ being the nearest integer.

Find the horizontal asymptote of the function $f(x)=\frac{2-x}{x^{2}-6 x+8}$ . Options: $y=-1$ , $y=0$ , $y=1$ , $y=2$ .

Find the differential of the function $V=14 \pi r^{3}$ .

Approximate $\sqrt[3]{155}$ using differentials as $a \pm \frac{b}{c}$ , with $a$ as the nearest integer.

Approximate $\sqrt[3]{12}$ using differentials as $a \pm \frac{b}{c}$ , where $a$ is the closest integer.

Find the differential of the function $u=(4 t+7)^{2}$ .

Find the differential of the function $u=(-9 t+2)^{2}$ .

Find the integral of the function: $\int\left(\sin y+\sec ^{2} y\right) d y$ .

Approximate $\sqrt{17}$ using differentials as $a \pm \frac{b}{c}$ with $a$ as the closest integer.

$\lim _{a \rightarrow 1} \frac{\sqrt[3]{a}-1}{\sqrt{a}-1}$ nedir? A) $\frac{1}{6}$ B) $\frac{1}{3}$ C) $\frac{2}{3}$ D) 1 E) $\frac{4}{3}$

Leiten Sie die Funktionen ab: a) $f(x)=x^{a}$ , b) $f(x)=x^{t+1}$ , c) $f(x)=\frac{1}{x^{r}}$ , d) $g(t)=t^{2 k} \cdot t$ .

Calculate $\triangle y - d y$ for the function $y=(-3x-2)^{2}$ using $x=2$ and $\triangle x=-0.2$ . Round to three decimals.

Find the marginal revenue function $R^{\prime}(x)$ for the revenue $R(x)=86 x+\ln(7 x^{3}+17)$ .

Find the derivative of the function $g(x)=9 x^{5} e^{4-3 x^{4}}$ .

Find the integral of the function $e^{e^{t}+t}$ with respect to $t$ : $\int e^{e^{t}+t} d t$ .

Calculate $\triangle y - d y$ for the function $y=(-3 x-5)^{2}$ at $x=2$ and $\triangle x=-0.2$ . Round to three decimal places.

Approximate $\sqrt{5}$ using differentials as $a \pm \frac{b}{c}$ , where $a$ is the closest integer.

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

Find the differential of the function $A=9 \pi x^{2}$ .

Solve the integral: $\int v \sqrt{v+4} \, dv$

Find the integral of the function: $\int\left(\frac{7}{4} x^{5}-\sqrt{x-1}\right) d x$

Find the derivative of $f(x)=\ln \left(\left(\frac{23 x-25 x^{2}}{19 x^{2}+7 x}\right)^{2}\right)$ .

Find the integral of $\tan \gamma$ with respect to $\gamma$ .

Find the differential of the function $P=-0.2 x^{2}-20 x+16$ .