Calculus

Problem 5401

Find the derivative of $f(x)=\frac{6}{1+e^{x}}$ using the Quotient Rule. Calculate $f^{\prime}(x)=$ .

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Problem 5402

Find the derivative of $f(x)=\frac{e^{x}}{x^{5}+1}$ using the Quotient Rule. $f^{\prime}(x)=$

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Problem 5403

Find the limit as $t$ approaches 0 of the vector $\langle e^{-t}, e^{-2t}, \ln(4-t) \rangle$ .

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Problem 5404

Evaluate the limit using the Squeeze Theorem: $\lim _{x \rightarrow 0} x \cos (4 / x)=$

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Problem 5405

Find the derivative of $h(t)=\frac{t}{(t+8)(t^{2}+7)}$ . What is $h^{\prime}(t)=?$

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Problem 5406

Find the limit as $t$ approaches 0 for the vector $\langle 2 e^{t}, \frac{\sin (t)}{-2 t},(t+3)^{3}\rangle$ .

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Problem 5407

Find the derivative of $f(x)=\frac{e^{x}}{x^{8}+1}$ using the Quotient Rule. $f^{\prime}(x)=$

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Problem 5408

Find the derivative of $h(t)=\frac{t}{(t+8)(r^{2}+2)}$ .

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Problem 5409

Calculate $F^{\prime}(3)$ for $F(x)=x^{2} f(x)$ using $f(3)=4$ and $f^{\prime}(3)=-5$ . $F^{\prime}(3)=$

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Problem 5410

Calculate $F^{\prime}(2)$ for $F(x)=x^{2} f(x)$ given $f(2)=2$ and $f^{\prime}(2)=-4$ . $F^{\prime}(2)=$

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Problem 5411

Find the derivative of $g(z)=\left(\frac{z^{2}-16}{z-3}\right)\left(\frac{z^{2}-9}{z-4}\right)$ for $z \neq 4, 3$ . Simplify first.

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Problem 5412

Find $H^{\prime}(4)$ for $H(x)=\frac{x}{g(x) f(x)}$ given $f(4)=7$ , $f^{\prime}(4)=-2$ , $g(4)=4$ , $g^{\prime}(4)=-5$ .

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Problem 5413

Find the marginal revenue function for $R(x)=7 x-0.02 x^{2}$ .

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Problem 5414

Find the positive value $a$ where the tangent line to $f(x)=x^{2} e^{-8 x}$ at $x=a$ passes through the origin. $a=$

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Problem 5415

Find the derivative of (a x + b)(a b x³ + 2) with respect to x, where a and b are constants.

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Problem 5416

Determine the marginal cost function from $C(x)=172+0.2 x$ .

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Problem 5417

Find the marginal cost function for $C(x)=250+3.1x-0.01x^{2}$ .

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Problem 5418

Find the instantaneous rate of change of $f(x)=1+2t-0.5t^{2}$ at $t=1$ . Why is it 0.9995?

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Problem 5419

Using Ohm's Law $I=\frac{V}{R}$ , find $\left.\frac{d I}{d R}\right|_{R=10}$ for $V=5$ and $\left.\frac{d V}{d R}\right|_{R=10}$ for $I=6$ .

See Solution

Problem 5420

Find $G^{\prime}(4)$ for $G(x)=x \cdot g(x) \cdot f(x)$ using $f(4)=8$ , $f^{\prime}(4)=-10$ , $g(4)=15$ , $g^{\prime}(4)=-2$ .

See Solution

Problem 5421

Find the derivative of $(a x+b)(a b x^{4}+5)$ with respect to $x$ , where $a$ and $b$ are constants.

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Problem 5422

Find the tangent line equations for the curve $y=\frac{1}{x^{2}+1}$ at $x=9$ and $x=-9$ .

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Problem 5423

Calculate $H^{\prime}(3)$ for $H(x)=\frac{x}{g(x) f(x)}$ given $f(3)=6$ , $f^{\prime}(3)=-9$ , $g(3)=6$ , $g^{\prime}(3)=-2$ . $H^{\prime}(3)=$

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Problem 5424

Find the rate of change of $P(x)=40+35 \ln (x+2)$ at $x=20$ . Round to the nearest hundredth.

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Problem 5425

Find $\frac{d P}{d r}$ for power $P=\frac{V^{2} R}{(R+r)^{2}}$ with $V$ , $R$ constant and $r$ variable.

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Problem 5426

Find the revenue increase for Couture clothing chain with $R(t)=N(t) S(t)$ , $N(0)=50$ , $S(0)=\$150,000$ . Compute $\left.\frac{d R}{d t}\right|_{t=0}$ .

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Problem 5427

See Solution

Problem 5428

Find the derivative of the function $\frac{a x - 7}{x^8 - a}$ with respect to $x$ .

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Problem 5429

Find $\frac{d P}{d r}$ for power $P=\frac{V^{2} R}{(R+r)^{2}}$ with variable internal resistance $r$ and constant load $R$ .

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Problem 5430

Logistic growth function $f(t)=\frac{109,000}{1+4800 e^{-t}}$ models flu cases.
a. Initial cases? b. Cases after 4 weeks? c. Maximum ill population?

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Problem 5431

Find $f(0.1)$ , $f'(0.1)$ , $f(0.8)$ , and $f'(0.8)$ for $f(x)=\frac{100 x^{2}}{x^{2}+0.13}$ . Round to the nearest tenth.

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Problem 5432

Find the instantaneous rate of change of $f(x)=1+2t-0.5t^{2}$ at $t=1$ . Result: $0.9995$ .

See Solution

Problem 5433

Couture's revenue is $R(t)=N(t) S(t)$ . With $N(0)=50$ and $S(0)=\$150,000$ , find $\left.\frac{d R}{d t}\right|_{t=0}$ .

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Problem 5434

Find $a>0$ so the tangent line of $f(x)=x^{2} e^{-2 x}$ at $x=a$ goes through the origin.

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Problem 5435

Find $\left.\frac{d I}{d R}\right|_{R=2}$ for $I=\frac{V}{R}$ with $V=9$ and $\left.\frac{d V}{d R}\right|_{R=2}$ for $I=3$ .

See Solution

Problem 5436

Find the derivative of the function $\frac{a x-2}{x^{3}-a}$ with respect to $x$ .

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Problem 5437

Calculate the rate of change at $t=1$ for $f(t)=1+2t-0.5t^2$ using the formula: $\frac{f(1.001)-f(1)}{0.001}$ .

See Solution

Problem 5438

Find $f(0.2)$ and $f(0.9)$ for $f(x)=\frac{100 x^{2}}{x^{2}+0.42}$ , and calculate $f^{\prime}(0.2)$ and $f^{\prime}(0.9)$ . Which dose gives more relief? Where is the 0.1g increase most beneficial?

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Problem 5439

Find $G^{\prime}(2)$ for $G(x)=x \cdot g(x) \cdot f(x)$ given $f(2)=7$ , $f^{\prime}(2)=-12$ , $g(2)=15$ , $g^{\prime}(2)=-3$ .

See Solution

Problem 5440

Calculate $f(0.1)$ , $f'(0.1)$ , $f(0.8)$ , and $f'(0.8)$ for $f(x)=\frac{100 x^{2}}{x^{2}+0.13}$ . Round to the nearest tenth.

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Problem 5441

Find the derivative of $g(t)=\frac{t}{3-t}+5 t^{3}$ .

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Problem 5442

Find values of $a$ and $b$ for which the function $f(x)$ is differentiable at $x=1$ . $f(x)=\left\{\begin{array}{lll} 2 a x-7 & \text { for } & x<1 \\ 2 b x^{2}+x+7 & \text { for } & x \geq 1 \end{array}\right.$

See Solution

Problem 5443

Untersuchen Sie die Bestandsfunktionen für ein Elektroauto und ein Wasserbecken. Zeigen Sie Änderungen und neue Funktionen bei Anfangsbeständen.

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Problem 5444

Find values of $a$ and $b$ for the function $f(x)$ to be differentiable at $x=4$ :
$f(x)=\begin{cases} 2ax+5 & x \leq 4 \\ 2bx^2-x-6 & x > 4 \end{cases}$

See Solution

Problem 5445

Ein Radfahrer hat die Geschwindigkeit $f(t)=(-t+6) \cdot e^{t-3}$ . Zeigen Sie, dass $F(t)=(-t+7) \cdot e^{t-3}$ eine Stammfunktion ist und bestimmen Sie die Strecke.

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Problem 5446

Finde die Funktion $f(t)=30 \cdot\left(-0,3 \cdot e^{-0,3 t}+0,7 \cdot e^{-0,7 t}\right)$ und beantworte folgende Fragen: a) Konzentration nach t Stunden, b) Werte nach 3 und 7 Stunden, c) maximale Konzentration, d) Zeitpunkt der schnellsten Abnahme, e) Änderung zwischen 4 und 8 Stunden, f) Zeitpunkte für $7 \mathrm{mg} / \mathrm{l}$ .

See Solution

Problem 5447

Finde die Ableitungen der Funktionen $f(x)=2x^3-x$ und $f(x)=\frac{2}{3}x^4-x^3+2$ .

See Solution

Problem 5448

Bestimme die 1. Ableitung von $f(x)=2x^{3}-x$ und $f(x)=\frac{2}{2}x^{4}-x^{3}+2$ .

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Problem 5449

Analyze the continuity and differentiability of the piecewise function $f$ at $x=-1$ given by:
$f(x)=\left\{\begin{array}{lll} 5 \ln (-x)-2 & \text { for } & x<-1 \\ 4 x^{2}+6 x & \text { for } & x \geq-1 \end{array}\right.$
Find the limits and function values at $x=-1$ .

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Problem 5450

Find values of $a$ and $b$ for which the function $f(x)$ is differentiable at $x=4$ :
$f(x)=\left\{\begin{array}{lll} 2 a x+10 & \text { for } & x \leq 4 \\ 2 b x^{2}-x-4 & \text { for } & x>4 \end{array}\right.$

See Solution

Problem 5451

1. Ein Radfahrer hat die Geschwindigkeit $f(t)=(-t+6) \cdot e^{t-3}$ in den ersten 6 Sekunden.
a) Zeigen Sie, dass $F(t)=(-t+7) \cdot e^{t-3}$ eine Stammfunktion von $f$ ist und finden Sie die Strecke.
b) Bestimmen Sie die Strecke bis zur maximalen Geschwindigkeit.
c) Bestimmen Sie die Strecke, wenn der Radfahrer stoppt und seine Durchschnittsgeschwindigkeit in den ersten 6 Sekunden.

See Solution

Problem 5452

Berechne die Untersumme $U_{4}$ und Obersumme $\mathrm{O}_{4}$ für $f(x)=2 x^{2}+2$ im Intervall $\mathrm{I}=[0,2]$ .

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Problem 5453

Berechne die Integrale und interpretiere sie. Zeichne die Graphen von f im Integrationsintervall. a) $\int_{1}^{2}\left(x^{2}+1\right) d x$ b) $\int_{-1}^{2}(x-2) d x$ c) $\int_{0}^{3}\left(2-\frac{1}{2} x^{2}\right) d x$

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Problem 5454

Find the turning points of $f(x)=-x^{3}+x^{2}+x+2$ given that $f'(x)=m x^{2}+n x+k$ passes through points P, Q, R.

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Problem 5455

Bestimme die unbestimmten Integrale: a) $\int(2 x^{3}-6 x+3) dx$ , b) $\int(-2 x-\frac{4}{x^{3}}) dx$ , c) $\int 3 e^{-4 x+2} dx$ .

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Problem 5456

Zeigen Sie ohne Rechnung, dass $\int_{-4}^{4}\left(-x^{2}+16\right) d x>0$ gilt.

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Problem 5457

Find the turning points of the function $f(x)=-x^{3}+x^{2}+x+2$ given its derivative $y=f^{\prime}(x)=m x^{2}+n x+k$ .

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Problem 5458

Find the turning points of $f(x)=-x^{3}+x^{2}+x+2$ given $f^{\prime}(x)=m x^{2}+n x+k$ passes points P, Q, R.

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Problem 5459

Finde die Funktion $f$ , wenn $f'(x) = -6$ , $x=5$ Extremstelle und $T(2, -10)$ Wendepunkt ist.

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Problem 5460

Bestimmen Sie die Ableitungen von $f(x)$ für a) bis h) und zeichnen Sie die Graphen für Übung 6 a) bis d).

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Problem 5461

Leite die Funktion $f$ ab für: a) $f(x)=2 \sin (x)+x^{-3}$ , b) $f(x)=2 \sqrt{x}-2 x$ , c) $f(x)=x^{\frac{2}{3}}$ , d) $f(x)=-x^{-2}+3 x$ , e) $f(x)=10 x^{0,1}-5 \cos (x)$ , f) $f(x)=x^{2}-x^{-\frac{1}{7}}$ .

See Solution

Problem 5462

Bestimmen Sie die Ableitungsfunktion für die folgenden Funktionen: a) $f(x)=2 x+x^{3}$ , b) $f(x)=5 x$ , c) $f(x)=a x^{2}$ , d) $f(x)=2 x^{2}+4 x$ , e) $f(x)=\frac{1}{2} x^{2}+5$ , f) $f(x)=2 x^{3}-3 x^{2}+2$ , g) $f(x)=a x^{n}$ , h) $f(x)=a x^{3}+b x+c$ . Dann bestimmen Sie $f^{\prime}$ und zeichnen Sie die Graphen für: a) $f(x)=\frac{1}{2} x^{2}-2 x+2$ , b) $f(x)=4-x^{2}$ , c) $f(x)=\frac{1}{2} x^{3}-2 x$ , d) $f(x)=3 x-\frac{1}{3} x^{3}$ .

See Solution

Problem 5463

Berechne die erste und zweite Ableitung für die Funktionen: a) $f(x)=4 x^{2}+2 x+1$ b) $f(t)=t^{100}-t^{99}$ c) $f(t)=\frac{1}{2} t^{5}-\frac{2}{3} t^{4}+2 t$ d) $f(x)=-\sqrt{2} x^{3}+\frac{8}{7} x^{7}-x$ e) $f(x)=5$ f) $f(x)=x+2 x^{2}-\sqrt{3} x^{3}$ g) $f(t)=\sqrt{2}$ h) $f(t)=-2 t^{12}-\frac{1}{12} t^{6}+12$ i) $f(x)=-x^{8}+\sqrt{7} x^{5}-2$

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Problem 5464

Ein Wagen wird durch Luft aus einem Ballon angetrieben. Ersetze im $t-v$ -Diagramm Kurven durch Linien und bestimme die Beschleunigung. Beschreibe die zeitliche Veränderung.

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Problem 5465

A particle moves with $s=t^{3}-9 t^{2}+18 t$ . Find when acceleration is 0 and the displacement and velocity then.

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Problem 5466

Produktionsfunktion $P(x)=-0,5 x^{3}+1,5 x^{2}+0,075 x ; D(P)=[0 ; 3,05]$ : a) Wo wächst $P$ ? b) Maximales Wachstum und Betrag? c) Wo wächst $P$ progressiv? d) Wo wächst $P$ ?

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Problem 5467

Find the velocity of the particle at $t=1$ s and $t=3$ s for $s=t^{2}-4t+4$ . When is it at rest and moving positively?

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Problem 5468

Find when the car's velocity, given by $v = \frac{ds}{dt} = 320t + 20$ , equals $100 \mathrm{~km/h}$ .

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Problem 5469

Erlöse eines Monopolisten: $E(x)=-x^{2}+4x$ , $D_{\text{ök}}(E)=[0;4]$ .
a) Finde $E^{\prime}$ . b) Zeichne $E(x)$ und $E^{\prime}(x)$ . c) Beschreibe Erlösentwicklung. d) Was bedeutet die Änderungsrate? e) Interpretiere $E^{\prime}$ -Verlauf. f) Berechne und interpretiere $E^{\prime}(1)$ und $E^{\prime}(4)$ .

See Solution

Problem 5470

Untersuchen Sie die Funktion $h(x)=\frac{1}{x^{2}}-2$ für $x \rightarrow \pm \infty$ und $x \rightarrow 0$ . Erklären Sie die Grenzwerte.

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Problem 5471

A bungee jumper falls at $25 \mathrm{~m/s}$ , slows to $11 \mathrm{~m/s}$ in $2.5 \mathrm{~s}$ . Find acceleration and time to stop.

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Problem 5472

Bestimme die Ableitungen von $f(x)=-5 x^{3}+75 x^{2}$ für $D(f)=[0 ; 15]$ und analysiere die Epidemieausbreitung.

See Solution

Problem 5473

Berechnen Sie das Integral $\int_{-5}^{10}\left(x^{4}-2 x\right) d x$ mit den Stammfunktionen $F_{L}(x)=0,2 x^{5}-x^{2}+10$ und $F_{F}(x)=0,2 x^{5}-x^{2}-10$ . Führen beide zu demselben Ergebnis?

See Solution

Problem 5474

Calculate the integral $\int_{0}^{3}\left(x^{2}-2\right) dx$ .

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Problem 5475

Gegeben ist die Funktion $f(x)=\frac{4 x}{x^{2}+1}$ im Intervall $I=[1 ; 3]$ . a) Zeichnen Sie $f(x)$ und die Untersumme $U_{4}$ . b) Berechnen Sie $U_{4}$ .

See Solution

Problem 5476

Temperaturverlauf: Modell $f(x)=-0,009 x^{3}+0,21 x^{2}+8$ . a) Beschreiben Sie den Verlauf und die Änderungsrate. b) Wo steigt/fällt die Temperatur? Wann ist die Änderungsrate am größten? c) Wie könnte die Kurve außerhalb von 5.00-21.00 Uhr aussehen?

See Solution

Problem 5477

Gegeben ist $f(x)=4-x^{2}$ . Bestimmen Sie die Flächeninhalte $A_{1}$ und $A_{2}$ mit Integralen und finden Sie $b>0$ , sodass $\int_{-2}^{b} f(x) dx=0$ .

See Solution

Problem 5478

Find the third derivative $f^{\prime \prime \prime}(x)$ given $f^{\prime}(x)=-\frac{1}{2} x^{2}+2$ and $f^{\prime \prime}(x)=-x$ .

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Problem 5479

Berechnen Sie die Ableitungen der Funktionen: a) $f(t)=\sqrt{7 t-3}$ , b) $f(x)=(x+1)^{-4}$ , d) $f(t)=-2 \cdot \sqrt{4 t+6}$ , e) $f(x)=(-x+2)^{-6}$ , g) $f(x)=-\sqrt{3-8 x}$ , h) $f(t)=3 \cdot(5 t-4)^{-3}$ .

See Solution

Problem 5480

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

See Solution

Problem 5481

Zeigen Sie, dass die Folge $a_n = \frac{1-2n}{3n}$ konvergent ist und bestimmen Sie $G$ für Abweichungen < $\frac{1}{100}$ und $10^{-6}$ .

See Solution

Problem 5482

Solve for $y$ given that $2 \frac{d y}{d x}=6 x$ .

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Problem 5483

Gegeben ist die Funktion $f(x)=-5 x^{3}+75 x^{2}$ für $D(f)=[0 ; 15]$ .
a) Finde die 1. bis 3. Ableitung und skizziere die Graphen. b) Beschreibe die Epidemie mit $f$ , 1. und 2. Ableitung. c) Bestimme den Zeitpunkt, an dem die Ausbreitungsgeschwindigkeit maximal war.

See Solution

Problem 5484

Gegeben ist die Funktion $f(x)=-5 x^{3}+75 x^{2}$ für $D(f)=[0 ; 15]$ .
a) Bestimme die 1. bis 3. Ableitungen und skizziere deren Graphen. b) Analysiere die Epidemie mit $f$ , $f'$ und $f''$ . c) Finde den Zeitpunkt, an dem die Ausbreitungsgeschwindigkeit maximal war.

See Solution

Problem 5485

Calculate the area under these curves:
a) $f(x)=x^{2}-x+1$ on $[0, 2]$ b) $f(x)=\frac{1}{x^{2}}$ on $[0, 2]$ c) $f(x)=x^{3}-x$ on $[1, 3]$ d) $f(x)=x^{3}-x$ in the 4th quadrant e) Area between $f(x)=x^{2}$ and x-axis on $[0, 2]$ f) Area for an unspecified function.

See Solution

Problem 5486

A radioactive substance with an initial amount of 80 grams decays every hour. How much remains after 2 hours? What happens as $t \rightarrow \infty$ ?

See Solution

Problem 5487

Finde $a$ , sodass die Fläche unter $f(x)=x^{2}$ von $[0; 4]$ durch $x=a$ im Verhältnis $1:7$ geteilt wird.

See Solution

Problem 5488

Berechnen Sie den Flächeninhalt zwischen dem Graphen von $f$ und der X-Achse für die Intervalle I in a) bis d).

See Solution

Problem 5489

Calculate $P(20)$ , $P(50)$ , $P(80)$ , and their derivatives $P'(20)$ , $P'(50)$ , $P'(80)$ using $P(t)=0.00238 e^{0.0956 t}$ . Discuss limitations.

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Problem 5490

Which statements about the polynomial $h$ could be true based on its values? Consider AROC and concavity.

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Problem 5491

Find the tangent vector of the curve $\mathbf{q}(t)=\langle t^{2}, \sin(2t), \cos(3t)\rangle$ at $t=\frac{\pi}{3}$ .

See Solution

Problem 5492

Differentiate the function j(x) = x^{2.6} + e^{2.6}. Find j'(x).

See Solution

Problem 5493

Differentiate the function $y=\frac{4 x}{e^{x}}$ and find $y^{\prime}=\square$ .

See Solution

Problem 5494

Evaluate the integral: $\int\left(\frac{1}{\sqrt{1-t^{2}}} \mathbf{i}+\frac{1}{t} \mathbf{j}-4 \sec ^{2} t \mathbf{k}\right) dt$ .

See Solution

Problem 5495

Differentiate the function $f(x)=(9x^{2}-19x)e^{x}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 5496

Find the tangent line equation to the curve $y=3 e^{x}+x$ at the point $(0,3)$ .

See Solution

Problem 5497

Find the unit tangent vector $\mathbf{T}(t)=\frac{\mathbf{m}^{\prime}(t)}{\left\|\mathbf{m}^{\prime}(t)\right\|}$ for the curve $\mathbf{m}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+2 t \mathbf{k}, 0 \leq t \leq 3 \pi$ .

See Solution

Problem 5498

Find the critical points of the function where $f'(x)=0$ and solve $x^{3}-4x=0$ .

See Solution

Problem 5499

Find the extrema of $f'(x) = x^3 - 4x$ .

See Solution

Problem 5500

Evaluate the integral $\int_{4}^{9}\left\langle\sqrt{t}, 2 t-1, e^{t}\right\rangle d t$ .

See Solution

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Calculus

Problem 5401

Find the derivative of f(x)=61+exf(x)=\frac{6}{1+e^{x}}f(x)=1+ex6​ using the Quotient Rule. Calculate f′(x)=f^{\prime}(x)=f′(x)=.

Problem 5402

Find the derivative of f(x)=exx5+1f(x)=\frac{e^{x}}{x^{5}+1}f(x)=x5+1ex​ using the Quotient Rule. f′(x)=f^{\prime}(x)=f′(x)=

Problem 5403

Find the limit as ttt approaches 0 of the vector ⟨e−t,e−2t,ln⁡(4−t)⟩\langle e^{-t}, e^{-2t}, \ln(4-t) \rangle⟨e−t,e−2t,ln(4−t)⟩.

Problem 5404

Evaluate the limit using the Squeeze Theorem: lim⁡x→0xcos⁡(4/x)=\lim _{x \rightarrow 0} x \cos (4 / x)=limx→0​xcos(4/x)=

Problem 5405

Find the derivative of h(t)=t(t+8)(t2+7)h(t)=\frac{t}{(t+8)(t^{2}+7)}h(t)=(t+8)(t2+7)t​. What is h′(t)=?h^{\prime}(t)=?h′(t)=?

Problem 5406

Find the limit as ttt approaches 0 for the vector ⟨2et,sin⁡(t)−2t,(t+3)3⟩\langle 2 e^{t}, \frac{\sin (t)}{-2 t},(t+3)^{3}\rangle⟨2et,−2tsin(t)​,(t+3)3⟩.

Problem 5407

Find the derivative of f(x)=exx8+1f(x)=\frac{e^{x}}{x^{8}+1}f(x)=x8+1ex​ using the Quotient Rule. f′(x)=f^{\prime}(x)=f′(x)=

Problem 5408

Find the derivative of h(t)=t(t+8)(r2+2)h(t)=\frac{t}{(t+8)(r^{2}+2)}h(t)=(t+8)(r2+2)t​.

Problem 5409

Calculate F′(3)F^{\prime}(3)F′(3) for F(x)=x2f(x)F(x)=x^{2} f(x)F(x)=x2f(x) using f(3)=4f(3)=4f(3)=4 and f′(3)=−5f^{\prime}(3)=-5f′(3)=−5. F′(3)=F^{\prime}(3)= F′(3)=

Problem 5410

Calculate F′(2)F^{\prime}(2)F′(2) for F(x)=x2f(x)F(x)=x^{2} f(x)F(x)=x2f(x) given f(2)=2f(2)=2f(2)=2 and f′(2)=−4f^{\prime}(2)=-4f′(2)=−4. F′(2)=F^{\prime}(2)= F′(2)=

Problem 5411

Find the derivative of g(z)=(z2−16z−3)(z2−9z−4)g(z)=\left(\frac{z^{2}-16}{z-3}\right)\left(\frac{z^{2}-9}{z-4}\right)g(z)=(z−3z2−16​)(z−4z2−9​) for z≠4,3z \neq 4, 3z=4,3. Simplify first.

Problem 5412

Find H′(4)H^{\prime}(4)H′(4) for H(x)=xg(x)f(x)H(x)=\frac{x}{g(x) f(x)}H(x)=g(x)f(x)x​ given f(4)=7f(4)=7f(4)=7, f′(4)=−2f^{\prime}(4)=-2f′(4)=−2, g(4)=4g(4)=4g(4)=4, g′(4)=−5g^{\prime}(4)=-5g′(4)=−5.

Problem 5413

Find the marginal revenue function for R(x)=7x−0.02x2R(x)=7 x-0.02 x^{2}R(x)=7x−0.02x2.

Problem 5414

Find the positive value aaa where the tangent line to f(x)=x2e−8xf(x)=x^{2} e^{-8 x}f(x)=x2e−8x at x=ax=ax=a passes through the origin. a= a= a=

Problem 5415

Find the derivative of (a x + b)(a b x³ + 2) with respect to x, where a and b are constants.

Problem 5416

Determine the marginal cost function from C(x)=172+0.2xC(x)=172+0.2 xC(x)=172+0.2x.

Problem 5417

Find the marginal cost function for C(x)=250+3.1x−0.01x2C(x)=250+3.1x-0.01x^{2}C(x)=250+3.1x−0.01x2.

Problem 5418

Find the instantaneous rate of change of f(x)=1+2t−0.5t2f(x)=1+2t-0.5t^{2}f(x)=1+2t−0.5t2 at t=1t=1t=1. Why is it 0.9995?

Problem 5419

Using Ohm's Law I=VRI=\frac{V}{R}I=RV​, find dIdR∣R=10\left.\frac{d I}{d R}\right|_{R=10}dRdI​∣∣​R=10​ for V=5V=5V=5 and dVdR∣R=10\left.\frac{d V}{d R}\right|_{R=10}dRdV​∣∣​R=10​ for I=6I=6I=6.

Problem 5420

Find G′(4)G^{\prime}(4)G′(4) for G(x)=x⋅g(x)⋅f(x)G(x)=x \cdot g(x) \cdot f(x)G(x)=x⋅g(x)⋅f(x) using f(4)=8f(4)=8f(4)=8, f′(4)=−10f^{\prime}(4)=-10f′(4)=−10, g(4)=15g(4)=15g(4)=15, g′(4)=−2g^{\prime}(4)=-2g′(4)=−2.

Problem 5421

Find the derivative of (ax+b)(abx4+5)(a x+b)(a b x^{4}+5)(ax+b)(abx4+5) with respect to xxx, where aaa and bbb are constants.

Problem 5422

Find the tangent line equations for the curve y=1x2+1y=\frac{1}{x^{2}+1}y=x2+11​ at x=9x=9x=9 and x=−9x=-9x=−9.

Problem 5423

Calculate H′(3)H^{\prime}(3)H′(3) for H(x)=xg(x)f(x)H(x)=\frac{x}{g(x) f(x)}H(x)=g(x)f(x)x​ given f(3)=6f(3)=6f(3)=6, f′(3)=−9f^{\prime}(3)=-9f′(3)=−9, g(3)=6g(3)=6g(3)=6, g′(3)=−2g^{\prime}(3)=-2g′(3)=−2. H′(3)=H^{\prime}(3)=H′(3)=

Problem 5424

Find the rate of change of P(x)=40+35ln⁡(x+2)P(x)=40+35 \ln (x+2)P(x)=40+35ln(x+2) at x=20x=20x=20. Round to the nearest hundredth.

Problem 5425

Find dPdr\frac{d P}{d r}drdP​ for power P=V2R(R+r)2P=\frac{V^{2} R}{(R+r)^{2}}P=(R+r)2V2R​ with VVV, RRR constant and rrr variable.

Problem 5426

Find the revenue increase for Couture clothing chain with R(t)=N(t)S(t)R(t)=N(t) S(t)R(t)=N(t)S(t), N(0)=50N(0)=50N(0)=50, S(0)=$150,000S(0)=\$150,000S(0)=$150,000. Compute dRdt∣t=0\left.\frac{d R}{d t}\right|_{t=0}dtdR​∣∣​t=0​.

Problem 5427

Problem 5428

Find the derivative of the function ax−7x8−a\frac{a x - 7}{x^8 - a}x8−aax−7​ with respect to xxx.

Problem 5429

Find dPdr\frac{d P}{d r}drdP​ for power P=V2R(R+r)2P=\frac{V^{2} R}{(R+r)^{2}}P=(R+r)2V2R​ with variable internal resistance rrr and constant load RRR.

Problem 5430

Logistic growth function f(t)=109,0001+4800e−tf(t)=\frac{109,000}{1+4800 e^{-t}}f(t)=1+4800e−t109,000​ models flu cases. a. Initial cases? b. Cases after 4 weeks? c. Maximum ill population?

Problem 5431

Find f(0.1)f(0.1)f(0.1), f′(0.1)f'(0.1)f′(0.1), f(0.8)f(0.8)f(0.8), and f′(0.8)f'(0.8)f′(0.8) for f(x)=100x2x2+0.13f(x)=\frac{100 x^{2}}{x^{2}+0.13}f(x)=x2+0.13100x2​. Round to the nearest tenth.

Problem 5432

Find the instantaneous rate of change of f(x)=1+2t−0.5t2f(x)=1+2t-0.5t^{2}f(x)=1+2t−0.5t2 at t=1t=1t=1. Result: 0.99950.99950.9995.

Problem 5433

Couture's revenue is R(t)=N(t)S(t)R(t)=N(t) S(t)R(t)=N(t)S(t). With N(0)=50N(0)=50N(0)=50 and S(0)=$150,000S(0)=\$150,000S(0)=$150,000, find dRdt∣t=0\left.\frac{d R}{d t}\right|_{t=0}dtdR​∣∣​t=0​.

Problem 5434

Find a>0a>0a>0 so the tangent line of f(x)=x2e−2xf(x)=x^{2} e^{-2 x}f(x)=x2e−2x at x=ax=ax=a goes through the origin.

Problem 5435

Find dIdR∣R=2\left.\frac{d I}{d R}\right|_{R=2}dRdI​∣∣​R=2​ for I=VRI=\frac{V}{R}I=RV​ with V=9V=9V=9 and dVdR∣R=2\left.\frac{d V}{d R}\right|_{R=2}dRdV​∣∣​R=2​ for I=3I=3I=3.

Problem 5436

Find the derivative of the function ax−2x3−a\frac{a x-2}{x^{3}-a}x3−aax−2​ with respect to xxx.

Problem 5437

Calculate the rate of change at t=1t=1t=1 for f(t)=1+2t−0.5t2f(t)=1+2t-0.5t^2f(t)=1+2t−0.5t2 using the formula: f(1.001)−f(1)0.001\frac{f(1.001)-f(1)}{0.001}0.001f(1.001)−f(1)​.

Problem 5438

Find f(0.2)f(0.2)f(0.2) and f(0.9)f(0.9)f(0.9) for f(x)=100x2x2+0.42f(x)=\frac{100 x^{2}}{x^{2}+0.42}f(x)=x2+0.42100x2​, and calculate f′(0.2)f^{\prime}(0.2)f′(0.2) and f′(0.9)f^{\prime}(0.9)f′(0.9). Which dose gives more relief? Where is the 0.1g increase most beneficial?

Problem 5439

Find G′(2)G^{\prime}(2)G′(2) for G(x)=x⋅g(x)⋅f(x)G(x)=x \cdot g(x) \cdot f(x)G(x)=x⋅g(x)⋅f(x) given f(2)=7f(2)=7f(2)=7, f′(2)=−12f^{\prime}(2)=-12f′(2)=−12, g(2)=15g(2)=15g(2)=15, g′(2)=−3g^{\prime}(2)=-3g′(2)=−3.

Problem 5440

Calculate f(0.1)f(0.1)f(0.1), f′(0.1)f'(0.1)f′(0.1), f(0.8)f(0.8)f(0.8), and f′(0.8)f'(0.8)f′(0.8) for f(x)=100x2x2+0.13f(x)=\frac{100 x^{2}}{x^{2}+0.13}f(x)=x2+0.13100x2​. Round to the nearest tenth.

Find the derivative of $f(x)=\frac{6}{1+e^{x}}$ using the Quotient Rule. Calculate $f^{\prime}(x)=$ .

Find the derivative of $f(x)=\frac{e^{x}}{x^{5}+1}$ using the Quotient Rule. $f^{\prime}(x)=$

Find the limit as $t$ approaches 0 of the vector $\langle e^{-t}, e^{-2t}, \ln(4-t) \rangle$ .

Evaluate the limit using the Squeeze Theorem: $\lim _{x \rightarrow 0} x \cos (4 / x)=$

Find the derivative of $h(t)=\frac{t}{(t+8)(t^{2}+7)}$ . What is $h^{\prime}(t)=?$

Find the limit as $t$ approaches 0 for the vector $\langle 2 e^{t}, \frac{\sin (t)}{-2 t},(t+3)^{3}\rangle$ .

Find the derivative of $f(x)=\frac{e^{x}}{x^{8}+1}$ using the Quotient Rule. $f^{\prime}(x)=$

Find the derivative of $h(t)=\frac{t}{(t+8)(r^{2}+2)}$ .

Calculate $F^{\prime}(3)$ for $F(x)=x^{2} f(x)$ using $f(3)=4$ and $f^{\prime}(3)=-5$ . $F^{\prime}(3)=$

Calculate $F^{\prime}(2)$ for $F(x)=x^{2} f(x)$ given $f(2)=2$ and $f^{\prime}(2)=-4$ . $F^{\prime}(2)=$

Find the derivative of $g(z)=\left(\frac{z^{2}-16}{z-3}\right)\left(\frac{z^{2}-9}{z-4}\right)$ for $z \neq 4, 3$ . Simplify first.

Find $H^{\prime}(4)$ for $H(x)=\frac{x}{g(x) f(x)}$ given $f(4)=7$ , $f^{\prime}(4)=-2$ , $g(4)=4$ , $g^{\prime}(4)=-5$ .

Find the marginal revenue function for $R(x)=7 x-0.02 x^{2}$ .

Find the positive value $a$ where the tangent line to $f(x)=x^{2} e^{-8 x}$ at $x=a$ passes through the origin. $a=$

Determine the marginal cost function from $C(x)=172+0.2 x$ .

Find the marginal cost function for $C(x)=250+3.1x-0.01x^{2}$ .

Find the instantaneous rate of change of $f(x)=1+2t-0.5t^{2}$ at $t=1$ . Why is it 0.9995?

Using Ohm's Law $I=\frac{V}{R}$ , find $\left.\frac{d I}{d R}\right|_{R=10}$ for $V=5$ and $\left.\frac{d V}{d R}\right|_{R=10}$ for $I=6$ .

Find $G^{\prime}(4)$ for $G(x)=x \cdot g(x) \cdot f(x)$ using $f(4)=8$ , $f^{\prime}(4)=-10$ , $g(4)=15$ , $g^{\prime}(4)=-2$ .

Find the derivative of $(a x+b)(a b x^{4}+5)$ with respect to $x$ , where $a$ and $b$ are constants.

Find the tangent line equations for the curve $y=\frac{1}{x^{2}+1}$ at $x=9$ and $x=-9$ .

Calculate $H^{\prime}(3)$ for $H(x)=\frac{x}{g(x) f(x)}$ given $f(3)=6$ , $f^{\prime}(3)=-9$ , $g(3)=6$ , $g^{\prime}(3)=-2$ . $H^{\prime}(3)=$

Find the rate of change of $P(x)=40+35 \ln (x+2)$ at $x=20$ . Round to the nearest hundredth.

Find $\frac{d P}{d r}$ for power $P=\frac{V^{2} R}{(R+r)^{2}}$ with $V$ , $R$ constant and $r$ variable.

Find the revenue increase for Couture clothing chain with $R(t)=N(t) S(t)$ , $N(0)=50$ , $S(0)=\$150,000$ . Compute $\left.\frac{d R}{d t}\right|_{t=0}$ .

Find the derivative of the function $\frac{a x - 7}{x^8 - a}$ with respect to $x$ .

Find $\frac{d P}{d r}$ for power $P=\frac{V^{2} R}{(R+r)^{2}}$ with variable internal resistance $r$ and constant load $R$ .

Logistic growth function $f(t)=\frac{109,000}{1+4800 e^{-t}}$ models flu cases.
a. Initial cases? b. Cases after 4 weeks? c. Maximum ill population?

Find $f(0.1)$ , $f'(0.1)$ , $f(0.8)$ , and $f'(0.8)$ for $f(x)=\frac{100 x^{2}}{x^{2}+0.13}$ . Round to the nearest tenth.

Find the instantaneous rate of change of $f(x)=1+2t-0.5t^{2}$ at $t=1$ . Result: $0.9995$ .

Couture's revenue is $R(t)=N(t) S(t)$ . With $N(0)=50$ and $S(0)=\$150,000$ , find $\left.\frac{d R}{d t}\right|_{t=0}$ .

Find $a>0$ so the tangent line of $f(x)=x^{2} e^{-2 x}$ at $x=a$ goes through the origin.

Find $\left.\frac{d I}{d R}\right|_{R=2}$ for $I=\frac{V}{R}$ with $V=9$ and $\left.\frac{d V}{d R}\right|_{R=2}$ for $I=3$ .

Find the derivative of the function $\frac{a x-2}{x^{3}-a}$ with respect to $x$ .

Calculate the rate of change at $t=1$ for $f(t)=1+2t-0.5t^2$ using the formula: $\frac{f(1.001)-f(1)}{0.001}$ .

Find $f(0.2)$ and $f(0.9)$ for $f(x)=\frac{100 x^{2}}{x^{2}+0.42}$ , and calculate $f^{\prime}(0.2)$ and $f^{\prime}(0.9)$ . Which dose gives more relief? Where is the 0.1g increase most beneficial?

Find $G^{\prime}(2)$ for $G(x)=x \cdot g(x) \cdot f(x)$ given $f(2)=7$ , $f^{\prime}(2)=-12$ , $g(2)=15$ , $g^{\prime}(2)=-3$ .

Calculate $f(0.1)$ , $f'(0.1)$ , $f(0.8)$ , and $f'(0.8)$ for $f(x)=\frac{100 x^{2}}{x^{2}+0.13}$ . Round to the nearest tenth.

Find the derivative of $g(t)=\frac{t}{3-t}+5 t^{3}$ .

Find values of $a$ and $b$ for the function $f(x)$ to be differentiable at $x=4$ :
$f(x)=\begin{cases} 2ax+5 & x \leq 4 \\ 2bx^2-x-6 & x > 4 \end{cases}$

Ein Radfahrer hat die Geschwindigkeit $f(t)=(-t+6) \cdot e^{t-3}$ . Zeigen Sie, dass $F(t)=(-t+7) \cdot e^{t-3}$ eine Stammfunktion ist und bestimmen Sie die Strecke.

Finde die Ableitungen der Funktionen $f(x)=2x^3-x$ und $f(x)=\frac{2}{3}x^4-x^3+2$ .

Bestimme die 1. Ableitung von $f(x)=2x^{3}-x$ und $f(x)=\frac{2}{2}x^{4}-x^{3}+2$ .

Berechne die Untersumme $U_{4}$ und Obersumme $\mathrm{O}_{4}$ für $f(x)=2 x^{2}+2$ im Intervall $\mathrm{I}=[0,2]$ .

Find the turning points of $f(x)=-x^{3}+x^{2}+x+2$ given that $f'(x)=m x^{2}+n x+k$ passes through points P, Q, R.

Bestimme die unbestimmten Integrale: a) $\int(2 x^{3}-6 x+3) dx$ , b) $\int(-2 x-\frac{4}{x^{3}}) dx$ , c) $\int 3 e^{-4 x+2} dx$ .

Zeigen Sie ohne Rechnung, dass $\int_{-4}^{4}\left(-x^{2}+16\right) d x>0$ gilt.

Find the turning points of the function $f(x)=-x^{3}+x^{2}+x+2$ given its derivative $y=f^{\prime}(x)=m x^{2}+n x+k$ .

Find the turning points of $f(x)=-x^{3}+x^{2}+x+2$ given $f^{\prime}(x)=m x^{2}+n x+k$ passes points P, Q, R.

Finde die Funktion $f$ , wenn $f'(x) = -6$ , $x=5$ Extremstelle und $T(2, -10)$ Wendepunkt ist.

Bestimmen Sie die Ableitungen von $f(x)$ für a) bis h) und zeichnen Sie die Graphen für Übung 6 a) bis d).

Ein Wagen wird durch Luft aus einem Ballon angetrieben. Ersetze im $t-v$ -Diagramm Kurven durch Linien und bestimme die Beschleunigung. Beschreibe die zeitliche Veränderung.

A particle moves with $s=t^{3}-9 t^{2}+18 t$ . Find when acceleration is 0 and the displacement and velocity then.

Produktionsfunktion $P(x)=-0,5 x^{3}+1,5 x^{2}+0,075 x ; D(P)=[0 ; 3,05]$ : a) Wo wächst $P$ ? b) Maximales Wachstum und Betrag? c) Wo wächst $P$ progressiv? d) Wo wächst $P$ ?

Find the velocity of the particle at $t=1$ s and $t=3$ s for $s=t^{2}-4t+4$ . When is it at rest and moving positively?

Find when the car's velocity, given by $v = \frac{ds}{dt} = 320t + 20$ , equals $100 \mathrm{~km/h}$ .

Untersuchen Sie die Funktion $h(x)=\frac{1}{x^{2}}-2$ für $x \rightarrow \pm \infty$ und $x \rightarrow 0$ . Erklären Sie die Grenzwerte.

A bungee jumper falls at $25 \mathrm{~m/s}$ , slows to $11 \mathrm{~m/s}$ in $2.5 \mathrm{~s}$ . Find acceleration and time to stop.

Bestimme die Ableitungen von $f(x)=-5 x^{3}+75 x^{2}$ für $D(f)=[0 ; 15]$ und analysiere die Epidemieausbreitung.

Calculate the integral $\int_{0}^{3}\left(x^{2}-2\right) dx$ .