Calculus

Problem 13501

Bestimme die Stammfunktionen für: a) $f(x)=2 x-1$ , b) $f(x)=5$ , c) $f(x)=x^{6}$ , d) $f(x)=4 x^{3}-7 x+6$ , e) $f(x)=\sin x$ .

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Problem 13502

Graph $f(x)=\frac{e^{2 x}}{2 x-1}$ . Find inflection points using $f^{\prime \prime}(x)=\frac{4(4 x^{2}-8 x+5)e^{2 x}}{(2 x-1)^{3}}$ . A. No inflection points. B. Inflection point(s) at $x=$

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Problem 13503

Graph $f(x)=\frac{e^{2 x}}{2 x-1}$ using its derivatives: $f^{\prime}(x)=\frac{4(x-1)e^{2 x}}{(2 x-1)^{2}}$ , $f^{\prime \prime}(x)=\frac{4(4x^{2}-8x+5)e^{2 x}}{(2 x-1)^{3}}$ . Select the correct graph from options.

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Problem 13504

Evaluate the integral: $\int \frac{x d x}{\sqrt{1+x^{2}}}$ .

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Problem 13505

What is the graph of function for case D, focusing on its concavity given that $f$ is -, $f'$ is +, and $f''$ is -?

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Problem 13506

Find the critical points of the function $f(x)=-2 x^{2} \ln x+19 x^{2}$ .

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Problem 13507

Calculate the integral: $\int \frac{2}{\sqrt{x}} d x=$

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Problem 13508

Calculate the integral: $\int \frac{x \, dx}{1+x^{4}}$ .

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Problem 13509

Zeige, dass $F(x)=5 x^{4}+2$ eine Stammfunktion von $f(x)=20 x^{3}$ ist, und nenne drei weitere Stammfunktionen.

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Problem 13510

Ein ICE hat das Weg-Zeit-Gesetz $s(t)=0,7 t^{2}$ . Berechne den Weg und die mittlere Geschwindigkeit in den ersten 20 s.

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Problem 13511

John deposits \$12,000 at 3% APR. Find the value after 3 years for yearly, semiannual, quarterly, monthly, daily, and continuous compounding. Graph continuous growth.

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Problem 13512

Find the constant for the surface area element $d \sigma$ in the plane defined by $G(s, t) = \begin{pmatrix} 3s + 2t \\ 4s - t \\ -2s + t \end{pmatrix}$ .

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Problem 13513

Find the integral: $\int\left(t^{2}+1\right)^{2} dt$

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Problem 13514

How long for \$50,000 to grow to \$60,000 at a continuous 3.5% interest rate? Round to nearest 0.01 year.

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Problem 13515

Find the critical points of $f(x)=-2 x^{2} \ln x+19 x^{2}$ . Are there any local maxima?

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Problem 13516

Find where the function $f(x)=-e^{-x^{2} / 32}$ is concave up or down and identify inflection points.

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Problem 13517

Find the integral: $\int \frac{x^{3}+3}{x^{2}} d x$

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Problem 13518

Find the concavity intervals and inflection points of $f(x)=5 x^{4}+60 x^{3}+15$ .

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Problem 13519

Find the derivative $f^{\prime}(4)$ for the function $f(x)=-\frac{1}{x}+5 \sqrt{x}$ . Provide the answer as a simplified fraction.

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Problem 13520

Find future value, effective rate, and time to reach \$17,000 for \$5400 at 3.9% interest compounded continuously.

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Problem 13521

Find critical points of $f(x)=-x^{\frac{2}{3}}(x-5)$ on $[-5,5]$ and use the First Derivative Test for max/min values.

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Problem 13522

Bestimmen Sie die Intervalle, in denen die Funktionen $f$ streng monoton wachsend oder fallend sind und untersuchen Sie die Krümmung.
a) $f(x)=x^{2}-x-6$ b) $f(x)=x^{3}+x$ c) $f(x)=\frac{1}{9} x^{3}-x$

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Problem 13523

Two cars start from the same point, one going east at $75 \mathrm{~km/h}$ and the other north at $40 \mathrm{~km/h}$ . Find the distance increase rate after 3 hours. Options: $72 \mathrm{~km/h}$ , $85 \mathrm{~km/h}$ , $60 \mathrm{~km/h}$ , $146 \mathrm{~km/h}$ .

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Problem 13524

Pancake batter is poured at $52 \mathrm{~cm}^3/\mathrm{s}$ . Find the radius growth rate when the radius is $7 \mathrm{~cm}$ . Use $V=\pi r^{2} h$ .

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Problem 13525

Check if the Mean Value Theorem applies to $f(x)=5 x^{\frac{1}{9}}$ on $[-512,512]$ and find the point(s) if it does.

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Problem 13526

An ant's position is $(x, y) = (t^2 - 7, 15 - 4t)$ . Find the temperature change rate $A$ when $t=3$ using $\frac{\partial T}{\partial x}$ and $\frac{\partial T}{\partial y}$ .

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Problem 13527

Given $y=\left(x+\sqrt{1+x^{2}}\right)^{\frac{1}{2}}$ , prove that $(1+x^{2})\left(\frac{dy}{dx}\right)^{2}=\frac{1}{4} y^{2}$ and show $(1+x^{2})\left(\frac{d^{2}y}{dx^{2}}\right)+x \frac{dy}{dx}-\frac{1}{4} y=0$ .

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Problem 13528

Two cars start from the same point: one goes east at $72 \mathrm{~km/h}$ , the other north at $30 \mathrm{~km/h}$ . Find the distance increase rate after 1 hour.

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Problem 13529

A person $1.9 \mathrm{~m}$ tall walks away from a $4.75 \mathrm{~m}$ pole at $1 \mathrm{~m/s}$ . Find shadow tip speed at $8 \mathrm{~m}$ .

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Problem 13530

Evaluate $\lim _{x \rightarrow \infty} \frac{2 x^{2}-7 x}{9 x^{2}+7}$ using l'Hôpital's Rule and limit laws. Rewrite the limit: $\lim _{x \rightarrow \infty}(\square)$

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Problem 13531

Schreiben Sie die Funktionstermine um und leiten Sie dann $f(x)$ ab für: a) $f(x)=\frac{x^{4}}{2}$ , b) $f(x)=\frac{7}{x}$ , c) $f(x)=-\frac{3}{x^{2}}$ , d) $f(x)=\frac{5}{3 x^{3}}$ , e) $f(x)=(3 x)^{3}$ , f) $f(x)=\sqrt{9 x}$ .

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Problem 13532

Given Boyle's Law $P V=C$ , find the rate of volume decrease when $V=325 \mathrm{~cm}^{3}$ , $P=130 \mathrm{kPa}$ , and $\frac{dP}{dt}=12 \frac{\mathrm{kPa}}{\min}$ .

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Problem 13533

Skater-Parcours:
a) Bestimme den durchschnittlichen Anstieg zwischen A(-12, f(-12)) und B. b) Welchen Winkel hat der Skater im Ursprung? c) Zeige, dass $f$ und $g$ ohne Knick übergehen. d) Nachweis, dass $g$ in C(15,0) waagerecht ist. e) Finde die tiefste Stelle des Parcours. f) Wo ist die maximale Steigung von $g$ ?

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Problem 13534

A ball is hit up at $25.6 \mathrm{~m} / \mathrm{s}$ . Find its maximum height and total time in the air.

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Problem 13535

Pancake batter is poured at $61 \mathrm{~cm}^{3}/\mathrm{s}$ . Find the radius growth rate (in $\mathrm{cm}/\mathrm{s}$ ) at $r=9 \mathrm{~cm}$ with thickness $1 \mathrm{~cm}$ . Use $V=\pi r^{2} h$ .

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Problem 13536

Two cars start from the same point, one going east at $60 \mathrm{~km/h}$ and the other north at $45 \mathrm{~km/h}$ . Find the distance increase rate after 3 hours. Options: $75 \mathrm{~km/h}$ , $59 \mathrm{~km/h}$ , $105 \mathrm{~km/h}$ , $135 \mathrm{~km/h}$ .

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Problem 13537

Find the derivative of $y=\sqrt[4]{\frac{x^{2}+1}{x^{2}-1}}$ .

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Problem 13538

A 2 m tall person walks away from a 7 m pole at 1.25 m/s. Find the speed of their shadow's tip when they are 14 m from the pole.

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Problem 13539

Find the derivative of $y=\sqrt[4]{\frac{x^{2}+1}{x^{2}-1}}$ using logarithmic differentiation.

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Problem 13540

Bestimme die Ableitung der Funktionen an den angegebenen Stellen: a) $f(x)=-4 x^{2}, x_{0}=-3$ b) $f(x)=3 \sqrt{x}, x_{0}=9$ c) $f(x)=\frac{3}{x^{\prime}}, x_{0}=6$ d) $f(x)=\frac{1}{3 x^{\prime}}, x_{0}=2$ e) $f(x)=8, x_{0}=2$ f) $f(x)=5 x, x_{0}=-9$ .

See Solution

Problem 13541

Bestimmen Sie die Ableitungen der Funktionen: a) $f(x)=a x^{2}$ , b) $g(y)=-2 y^{3}$ , c) $h(t)=v t$ , d) $f(a)=5 a b$ , e) $f(x)=k^{2}$ , f) $s(t)=\frac{3}{a \sqrt{t}}$ .

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Problem 13542

Find the osculating circle at the local minimum of $f(x)=3 x^{3}-6 x^{2}+\frac{32}{9} x+6$ . Use $x=t, y=f(t)$ . Equation: $\left((x-(8 / 9))^{\wedge} 2\right)+\left((y-(1586 / 243))^{\wedge} 2\right)=(1 /(20))^{\wedge} 2$ .

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Problem 13543

Find the total cost of producing 560 units if fixed cost is \ $11,000 and marginal costs are given as$ C^{\prime}(q)$.

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Problem 13544

Untersuchen Sie das Verhalten von $f$ für die Grenzprozesse: a) $f(x)=\frac{2x+1}{x}, x<0$ , $x \rightarrow-\infty$ ; b) $f(x)=\frac{x+1}{x^{2}}, x>0$ , $x \rightarrow \infty$ . Skizzieren Sie die Graphen.

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Problem 13545

Führe die Kurvendiskussion für die Funktionen $f(x)=\frac{x^{4}}{5}-x^{2}$ und $f(x)=x^{5}-4 x^{3}$ durch. Zeichne auch Graphen und Ableitungen!

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Problem 13546

Berechnen Sie den Differenzenquotienten von $g(x)=3 x^{2}-1$ für die Intervalle I: a) $I=[1; 4]$ , b) $I=[-3; -1]$ .

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Problem 13547

A balloon's radius decreases at $15 \mathrm{~cm} / \mathrm{min}$ . Find the volume change rate when $V=972 \pi \mathrm{cm}^{3}$ .

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Problem 13548

Evaluate $\lim _{x \rightarrow 0} \frac{\sin 9 x}{x}$ using l'Hôpital's Rule and known limits. Rewrite the limit as $\lim _{x \rightarrow 0}(\square$ .

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Problem 13549

Find the value of $C$ for the joint density $\rho(x, y)=C\left(x^{2} y+y^{2}\right)$ over $D: -1 \leq x \leq 1$ , $0 \leq y \leq 1$ . Then, compute $P(X \leq 0, Y \geq 1/2)$ .

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Problem 13550

Evaluate the integral $\int y^{3} \cos y \, dy$ .

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Problem 13551

Bestimmen Sie den Differenzenquotienten für die Funktionen in den angegebenen Intervallen: a) $f(x)=(x-2)^{2}, \quad I=[1 ; 6]$ b) $f(x)=\frac{9}{x^{2}}-3, \quad I=[-3 ;-1]$ c) $f(x)=\sqrt{x+5}+x, \quad I=[-4 ;-1]$ d) $f(x)=x^{3}+x^{2}, \quad I=[-2 ; 4]$

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Problem 13552

Arjun walks south at 1.5 m/s and Maya east at 2 m/s. Find the rate of distance change after 2 minutes.

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Problem 13553

Find the second derivative $y^{\prime \prime}$ of the function $y=f(x)$ given $y^{\prime}=x(x-9)^{2}$ and sketch the graph. $y^{\prime \prime}=\square$

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Problem 13554

Bestimme das Volumen eines Körpers mit quadratischer Querschnittsfläche $A(z)=36-2 z^{2}+\frac{1}{36} z^{4}$ für $z \in[0 ; 6]$ .

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Problem 13555

Bestimme das Volumen eines Körpers mit Querschnittsfläche $A(z)=-2 \sqrt{z}+4$ und Höhe $h=4$ als Integral und berechne es.

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Problem 13556

Calculate the integral: $\int -\frac{1}{3} \sin(2 - 3x) \, dx$ .

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Problem 13557

Berechnen Sie die Ableitung $f^{\prime}(3)$ für die Funktion $f(x)=4 x^{2}-2$ .

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Problem 13558

Find $f'(3)$ given $\frac{f(x)-f(3)}{x-3} = 4(x+3)$ and evaluate the limit as $x \to 3$ .

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Problem 13559

Find the 4th term in the Taylor series for $f(x)=4 e^{3 x}$ around $x=4$ . $4 \text{th term} =$

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Problem 13560

Find the 4th term in the Taylor series for $f(x)=4 e^{3 x}$ around $x=4$ . 4th term =

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Problem 13561

Trouver les points de discontinuité de $f$ et déterminer la continuité à droite, à gauche ou ni l'un ni l'autre. $f(x)=\left\{\begin{array}{lrr} 1+x^{2} & \text { si } & x \leq 0 \\ 2-x & \text { si } & 0<x \leq 2 \end{array}\right.$

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Problem 13562

Trouvez les points de discontinuité de $f$ et indiquez si $f$ est continue à droite, à gauche ou nulle. $f(x)=\left\{\begin{array}{lrr} 1+x^{2} & \text { si } & x \leq 0 \\ 2-x & \text { si } & 0<x \leq 2 \\ (x-2)^{2} & \text { si } & x>2 \end{array}\right.$

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Problem 13563

Gegeben ist die Funktion $f(x)=x^{3}-3x^{2}$ .
a) Finde die Gleichung der Sekante durch $A(-1, f(-1))$ und $B(4, f(4))$ und den Schnittpunkt $C$ mit $f$ . b) Bestimme die Ableitungsfunktion $f'$ und zeichne ihren Graphen. c) Zeige, dass $f$ einen Hochpunkt $H$ , einen Tiefpunkt $T$ und einen Wendepunkt $W$ hat. Finde deren Koordinaten. d) Finde die Tangentengleichung durch $W$ und den Schnittpunkt mit der Sekante aus Teil a).

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Problem 13564

Trouver l'équation de la tangente à $y=\sqrt[3]{x}$ en $x=8$ .

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Problem 13565

Bestimme die Stammfunktionen von $f(x)$ für: a) $f(x)=3 \cdot x^{4}-5 x+4$ , b) $f(x)=3 a \cdot x^{4}$ .

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Problem 13566

Bestimme Hoch-, Tief- und Sattelpunkte für die Funktionen: $f(t)=t^{4}-2 t^{2}$ , $f(a)=-a^{3}+2,5 a^{2}+3,5$ , $f(x)=x^{5}-15 x^{3}-30$ , $f(x)=-3 x^{4}+4 x^{3}+8$ , $f(x)=\frac{1}{5} x^{5}-\frac{4}{2} x^{3}+2$ , $f(x)=2 x^{3}-6 x^{2}+6 x+4$ .

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Problem 13567

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for the curve defined by $x 2^{y}=\ln y$ in terms of $x$ and $y$ .

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Problem 13568

A person 1.7 m tall walks away from a 5.95 m pole at 1.5 m/s. Find the shadow's tip speed when 10 m from the pole.

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Problem 13569

Find the rate of change of distance $D$ at $P=13.5 \mathrm{KPa}$ and $dP/dt=220 \mathrm{KPa/s}$ when $D^2=\frac{2P}{900}+0.01$ .

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Problem 13570

Bestimme die Ableitungsfunktion $f^{\prime}$ für die folgenden Funktionen: a) $f(x)=4 x^{3}-8 x+17$ , b) $f(x)=-2 x^{2}-8 x+15$ , c) $f(x)=3 \sin (x)$ , d) $f(x)=2 x^{-4}$ , e) $f(x)=5 x^{\frac{1}{3}}$ , f) $f(x)=3 \cos (x)-\frac{1}{4} x^{-2}$ .

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Problem 13571

Compare the functions $f(x)=4^{x}$ and $g(x)=5x+6$ . Which statement is true about their rates of change?

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Problem 13572

Bestimme die Ableitungsfunktion $f^{\prime}$ für die Funktionen: a) $f(x)=(x-3)(x+5)$ , b) $f(x)=2(x-4)^{2}+5$ , c) $f(x)=(5 x)^{2}$ , d) $f(x)=2 x-\frac{1}{x}$ , e) $f(x)=\sqrt{x}$ , f) $f(x)=x^{\frac{1}{3}}$ , g) $f(u)=2 u+u^{\frac{1}{3}}$ , h) $f(x)=a \cdot x^{5}+\frac{a}{x^{-3}}$ , i) $f(a)=a \cdot x^{5}+\frac{a}{x^{-3}}$ .

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Problem 13573

Differentiate $3 \sqrt{x}-2 \sqrt{y}=4$ to find $\frac{d y}{d x}$ and the slope at (4,1).

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Problem 13574

Differentiate $x^{5}+y^{5}=244$ to find $\frac{d y}{d x}$ , then find the slope at the point (1,3).

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Problem 13575

Find the point $(x_{0}, y_{0})$ on $y=x^{3}+6x^{2}+3x-4$ where the tangent slope is minimized. Answer as $(, )$ .

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Problem 13576

Find the derivative of the function $f(x) = -7x^2 - 7x - 5$ using the limit definition: $\frac{f(x+h)-f(x)}{h}$ .

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Problem 13577

A person $1.9 \mathrm{~m}$ tall walks away from a $2.85 \mathrm{~m}$ pole at $2 \mathrm{~m/s}$ . Find shadow speed at $12 \mathrm{~m}$ .

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Problem 13578

Calculate the indefinite integral: $\int\left(x^{1 / 2}+\frac{4}{x}-7 e^{x}\right) d x$

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Problem 13579

Calculate the indefinite integral: $\int \frac{(\sqrt{x}-7)^{2}}{x^{2}} d x$

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Problem 13580

Differentiate $\sin(y) = 3x^3 - 3$ to find $\frac{dy}{dx}$ and the slope at (1, $\pi$ ).

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Problem 13581

Solve for $f(x)$ where $x>0$ : $f'(x)=3+2x+\frac{4}{x}$ and $f(1)=2$ . Find $f(x)=$ .

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Problem 13582

Find the time $t$ when the drug concentration $C(t)=533.3\left(e^{-0.5 t}-e^{-0.6 t}\right)$ is maximized. Round to the tenths.

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Problem 13583

Gravel is dumped at $18 \mathrm{~m}^{3} / \mathrm{h}$ forming a cone. Find the height increase rate when height is $5 \mathrm{~m}$ . Use $V=\frac{1}{3} \pi r^{2} h$ .

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Problem 13584

Maximize the volume of a cylinder with total surface area $110 \mathrm{~cm}^{2}$ . Find radius $r$ and height $h$ .

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Problem 13585

Evaluate the limit as $x$ approaches 2 for $\frac{3 x^{2}-7 x}{2 e^{2-x}+x^{2}}$ . Simplify your answer.

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Problem 13586

Étudier les limites suivantes : a) $\lim _{x \rightarrow 0^{+}}\left(\frac{1}{x(1-x)}-\frac{1}{x^{2}}\right)$ b) $\lim _{x \rightarrow+\infty}\left(x-\sqrt{x^{2}+2 x}\right)$

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Problem 13587

Find $\lim _{x \rightarrow 0} \frac{4 x-7 x^{2}}{\tan (-2 x)+x}$ and simplify your answer.

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Problem 13588

Find $\lim _{x \rightarrow-1} \frac{e^{-2-2 x}-1}{5 x^{2}-5}$ and simplify your answer.

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Problem 13589

Express the limit of $L_{n}=\frac{3}{n} \sum_{i=1}^{n}\left(2+3 \frac{i-1}{n}\right)$ as a definite integral.

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Problem 13590

Find the derivative of the function $f(x)=\sin^{-1}(3x)$ in its simplest form.

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Problem 13591

Find the derivative of the function $f(x)=\arctan(2x)$ , expressed in simplest form.

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Problem 13592

Find the derivative of $f(x)=\sin^{-1}(4x)$ , expressed in simplest form.

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Problem 13593

Find the derivative of the function $f(x)=\tan^{-1}(4x)$ in its simplest form.

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Problem 13594

Bestimme die Ableitungen $f^{\prime}(1)$ und $f^{\prime}(-2)$ sowie $f^{\prime}(-1), f^{\prime}(2), f^{\prime}(0)$ aus einer Grafik. Berechne $f^{\prime}(1)$ für $f(x)=2 x^{2}$ .

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Problem 13595

Déterminez la continuité de la fonction $f(x)$ au point $x_{0}=4$ avec $f(x)$ défini par : $f(x)=\left\{\begin{array}{lc} \frac{x^{2}-16}{2 x-8} & \text{si } x<4 \\ 4 & \text{si } x=4 \\ \frac{x-4}{\sqrt{x}-2} & \text{si } x>4 \end{array}\right.$

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Problem 13596

Express the limit of $R_{n}=\frac{1}{n} \sum_{i=1}^{n}\left(4+1 \frac{i}{n}\right)$ as a definite integral.
$\int_{0}^{1} (4+x) \, dx$

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Problem 13597

Which limit represents the integral $\int_{-10}^{-5} 10 x d x$ using a right-endpoint Riemann sum?

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Problem 13598

Differentiate $x^{2}+4 y^{2}=4$ two times with respect to $y$ .

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Problem 13599

Which expression shows the limit definition of the integral $\int_{-6}^{2} 6 x^{3} d x$ using a right-endpoint Riemann sum?

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Problem 13600

Find the net signed area between $f(x)=|x-1|-3$ and the $x$ -axis from $x=-5$ to $x=6$ . Answer as an exact value.

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Calculus

Problem 13501

Bestimme die Stammfunktionen für: a) f(x)=2x−1f(x)=2 x-1f(x)=2x−1, b) f(x)=5f(x)=5f(x)=5, c) f(x)=x6f(x)=x^{6}f(x)=x6, d) f(x)=4x3−7x+6f(x)=4 x^{3}-7 x+6f(x)=4x3−7x+6, e) f(x)=sin⁡xf(x)=\sin xf(x)=sinx.

Problem 13502

Graph f(x)=e2x2x−1f(x)=\frac{e^{2 x}}{2 x-1}f(x)=2x−1e2x​. Find inflection points using f′′(x)=4(4x2−8x+5)e2x(2x−1)3f^{\prime \prime}(x)=\frac{4(4 x^{2}-8 x+5)e^{2 x}}{(2 x-1)^{3}}f′′(x)=(2x−1)34(4x2−8x+5)e2x​. A. No inflection points. B. Inflection point(s) at x=x=x=

Problem 13503

Problem 13504

Evaluate the integral: ∫xdx1+x2\int \frac{x d x}{\sqrt{1+x^{2}}}∫1+x2​xdx​.

Problem 13505

What is the graph of function for case D, focusing on its concavity given that fff is -, f′f'f′ is +, and f′′f''f′′ is -?

Problem 13506

Find the critical points of the function f(x)=−2x2ln⁡x+19x2f(x)=-2 x^{2} \ln x+19 x^{2}f(x)=−2x2lnx+19x2.

Problem 13507

Calculate the integral: ∫2xdx=\int \frac{2}{\sqrt{x}} d x=∫x​2​dx=

Problem 13508

Calculate the integral: ∫x dx1+x4\int \frac{x \, dx}{1+x^{4}}∫1+x4xdx​.

Problem 13509

Zeige, dass F(x)=5x4+2F(x)=5 x^{4}+2F(x)=5x4+2 eine Stammfunktion von f(x)=20x3f(x)=20 x^{3}f(x)=20x3 ist, und nenne drei weitere Stammfunktionen.

Problem 13510

Ein ICE hat das Weg-Zeit-Gesetz s(t)=0,7t2s(t)=0,7 t^{2}s(t)=0,7t2. Berechne den Weg und die mittlere Geschwindigkeit in den ersten 20 s.

Problem 13511

John deposits \$12,000 at 3% APR. Find the value after 3 years for yearly, semiannual, quarterly, monthly, daily, and continuous compounding. Graph continuous growth.

Problem 13512

Find the constant for the surface area element dσd \sigmadσ in the plane defined by G(s,t)=(3s+2t4s−t−2s+t)G(s, t) = \begin{pmatrix} 3s + 2t \\ 4s - t \\ -2s + t \end{pmatrix}G(s,t)=⎝⎛​3s+2t4s−t−2s+t​⎠⎞​.

Problem 13513

Find the integral: ∫(t2+1)2dt\int\left(t^{2}+1\right)^{2} dt∫(t2+1)2dt

Problem 13514

How long for \$50,000 to grow to \$60,000 at a continuous 3.5% interest rate? Round to nearest 0.01 year.

Problem 13515

Find the critical points of f(x)=−2x2ln⁡x+19x2f(x)=-2 x^{2} \ln x+19 x^{2}f(x)=−2x2lnx+19x2. Are there any local maxima?

Problem 13516

Find where the function f(x)=−e−x2/32f(x)=-e^{-x^{2} / 32}f(x)=−e−x2/32 is concave up or down and identify inflection points.

Problem 13517

Find the integral: ∫x3+3x2dx\int \frac{x^{3}+3}{x^{2}} d x∫x2x3+3​dx

Problem 13518

Find the concavity intervals and inflection points of f(x)=5x4+60x3+15f(x)=5 x^{4}+60 x^{3}+15f(x)=5x4+60x3+15.

Problem 13519

Find the derivative f′(4)f^{\prime}(4)f′(4) for the function f(x)=−1x+5xf(x)=-\frac{1}{x}+5 \sqrt{x}f(x)=−x1​+5x​. Provide the answer as a simplified fraction.

Problem 13520

Find future value, effective rate, and time to reach \$17,000 for \$5400 at 3.9% interest compounded continuously.

Problem 13521

Find critical points of f(x)=−x23(x−5)f(x)=-x^{\frac{2}{3}}(x-5)f(x)=−x32​(x−5) on [−5,5][-5,5][−5,5] and use the First Derivative Test for max/min values.

Problem 13522

Bestimmen Sie die Intervalle, in denen die Funktionen fff streng monoton wachsend oder fallend sind und untersuchen Sie die Krümmung.a) f(x)=x2−x−6f(x)=x^{2}-x-6f(x)=x2−x−6 b) f(x)=x3+xf(x)=x^{3}+xf(x)=x3+x c) f(x)=19x3−xf(x)=\frac{1}{9} x^{3}-xf(x)=91​x3−x

Problem 13523

Problem 13524

Pancake batter is poured at 52 cm3/s52 \mathrm{~cm}^3/\mathrm{s}52 cm3/s. Find the radius growth rate when the radius is 7 cm7 \mathrm{~cm}7 cm. Use V=πr2hV=\pi r^{2} hV=πr2h.

Problem 13525

Check if the Mean Value Theorem applies to f(x)=5x19f(x)=5 x^{\frac{1}{9}}f(x)=5x91​ on [−512,512][-512,512][−512,512] and find the point(s) if it does.

Problem 13526

An ant's position is (x,y)=(t2−7,15−4t)(x, y) = (t^2 - 7, 15 - 4t)(x,y)=(t2−7,15−4t). Find the temperature change rate AAA when t=3t=3t=3 using ∂T∂x\frac{\partial T}{\partial x}∂x∂T​ and ∂T∂y\frac{\partial T}{\partial y}∂y∂T​.

Problem 13527

Problem 13528

Two cars start from the same point: one goes east at 72 km/h72 \mathrm{~km/h}72 km/h, the other north at 30 km/h30 \mathrm{~km/h}30 km/h. Find the distance increase rate after 1 hour.

Problem 13529

A person 1.9 m1.9 \mathrm{~m}1.9 m tall walks away from a 4.75 m4.75 \mathrm{~m}4.75 m pole at 1 m/s1 \mathrm{~m/s}1 m/s. Find shadow tip speed at 8 m8 \mathrm{~m}8 m.

Problem 13530

Evaluate lim⁡x→∞2x2−7x9x2+7\lim _{x \rightarrow \infty} \frac{2 x^{2}-7 x}{9 x^{2}+7}limx→∞​9x2+72x2−7x​ using l'Hôpital's Rule and limit laws. Rewrite the limit: lim⁡x→∞(□)\lim _{x \rightarrow \infty}(\square)x→∞lim​(□)

Problem 13531

Problem 13532

Given Boyle's Law PV=CP V=CPV=C, find the rate of volume decrease when V=325 cm3V=325 \mathrm{~cm}^{3}V=325 cm3, P=130kPaP=130 \mathrm{kPa}P=130kPa, and dPdt=12kPamin⁡\frac{dP}{dt}=12 \frac{\mathrm{kPa}}{\min}dtdP​=12minkPa​.

Problem 13533

Problem 13534

A ball is hit up at 25.6 m/s25.6 \mathrm{~m} / \mathrm{s}25.6 m/s. Find its maximum height and total time in the air.

Problem 13535

Pancake batter is poured at 61 cm3/s61 \mathrm{~cm}^{3}/\mathrm{s}61 cm3/s. Find the radius growth rate (in cm/s\mathrm{cm}/\mathrm{s}cm/s) at r=9 cmr=9 \mathrm{~cm}r=9 cm with thickness 1 cm1 \mathrm{~cm}1 cm. Use V=πr2hV=\pi r^{2} hV=πr2h.

Problem 13536

Problem 13537

Find the derivative of y=x2+1x2−14y=\sqrt[4]{\frac{x^{2}+1}{x^{2}-1}}y=4x2−1x2+1​​.

Problem 13538

A 2 m tall person walks away from a 7 m pole at 1.25 m/s. Find the speed of their shadow's tip when they are 14 m from the pole.

Problem 13539

Find the derivative of y=x2+1x2−14y=\sqrt[4]{\frac{x^{2}+1}{x^{2}-1}}y=4x2−1x2+1​​ using logarithmic differentiation.

Problem 13540

Problem 13541

Bestimmen Sie die Ableitungen der Funktionen: a) f(x)=ax2f(x)=a x^{2}f(x)=ax2, b) g(y)=−2y3g(y)=-2 y^{3}g(y)=−2y3, c) h(t)=vth(t)=v th(t)=vt, d) f(a)=5abf(a)=5 a bf(a)=5ab, e) f(x)=k2f(x)=k^{2}f(x)=k2, f) s(t)=3ats(t)=\frac{3}{a \sqrt{t}}s(t)=at​3​.

Problem 13542

Problem 13543

Find the total cost of producing 560 units if fixed cost is \11,000andmarginalcostsaregivenas11,000 and marginal costs are given as 11,000andmarginalcostsaregivenasC^{\prime}(q)$.

Problem 13544

Bestimme die Stammfunktionen für: a) $f(x)=2 x-1$ , b) $f(x)=5$ , c) $f(x)=x^{6}$ , d) $f(x)=4 x^{3}-7 x+6$ , e) $f(x)=\sin x$ .

Graph $f(x)=\frac{e^{2 x}}{2 x-1}$ . Find inflection points using $f^{\prime \prime}(x)=\frac{4(4 x^{2}-8 x+5)e^{2 x}}{(2 x-1)^{3}}$ . A. No inflection points. B. Inflection point(s) at $x=$

Evaluate the integral: $\int \frac{x d x}{\sqrt{1+x^{2}}}$ .

What is the graph of function for case D, focusing on its concavity given that $f$ is -, $f'$ is +, and $f''$ is -?

Find the critical points of the function $f(x)=-2 x^{2} \ln x+19 x^{2}$ .

Calculate the integral: $\int \frac{2}{\sqrt{x}} d x=$

Calculate the integral: $\int \frac{x \, dx}{1+x^{4}}$ .

Zeige, dass $F(x)=5 x^{4}+2$ eine Stammfunktion von $f(x)=20 x^{3}$ ist, und nenne drei weitere Stammfunktionen.

Ein ICE hat das Weg-Zeit-Gesetz $s(t)=0,7 t^{2}$ . Berechne den Weg und die mittlere Geschwindigkeit in den ersten 20 s.

Find the constant for the surface area element $d \sigma$ in the plane defined by $G(s, t) = \begin{pmatrix} 3s + 2t \\ 4s - t \\ -2s + t \end{pmatrix}$ .

Find the integral: $\int\left(t^{2}+1\right)^{2} dt$

Find the critical points of $f(x)=-2 x^{2} \ln x+19 x^{2}$ . Are there any local maxima?

Find where the function $f(x)=-e^{-x^{2} / 32}$ is concave up or down and identify inflection points.

Find the integral: $\int \frac{x^{3}+3}{x^{2}} d x$

Find the concavity intervals and inflection points of $f(x)=5 x^{4}+60 x^{3}+15$ .

Find the derivative $f^{\prime}(4)$ for the function $f(x)=-\frac{1}{x}+5 \sqrt{x}$ . Provide the answer as a simplified fraction.

Find critical points of $f(x)=-x^{\frac{2}{3}}(x-5)$ on $[-5,5]$ and use the First Derivative Test for max/min values.

Bestimmen Sie die Intervalle, in denen die Funktionen $f$ streng monoton wachsend oder fallend sind und untersuchen Sie die Krümmung.
a) $f(x)=x^{2}-x-6$ b) $f(x)=x^{3}+x$ c) $f(x)=\frac{1}{9} x^{3}-x$

Pancake batter is poured at $52 \mathrm{~cm}^3/\mathrm{s}$ . Find the radius growth rate when the radius is $7 \mathrm{~cm}$ . Use $V=\pi r^{2} h$ .

Check if the Mean Value Theorem applies to $f(x)=5 x^{\frac{1}{9}}$ on $[-512,512]$ and find the point(s) if it does.

An ant's position is $(x, y) = (t^2 - 7, 15 - 4t)$ . Find the temperature change rate $A$ when $t=3$ using $\frac{\partial T}{\partial x}$ and $\frac{\partial T}{\partial y}$ .

Two cars start from the same point: one goes east at $72 \mathrm{~km/h}$ , the other north at $30 \mathrm{~km/h}$ . Find the distance increase rate after 1 hour.

A person $1.9 \mathrm{~m}$ tall walks away from a $4.75 \mathrm{~m}$ pole at $1 \mathrm{~m/s}$ . Find shadow tip speed at $8 \mathrm{~m}$ .

Evaluate $\lim _{x \rightarrow \infty} \frac{2 x^{2}-7 x}{9 x^{2}+7}$ using l'Hôpital's Rule and limit laws. Rewrite the limit: $\lim _{x \rightarrow \infty}(\square)$

Given Boyle's Law $P V=C$ , find the rate of volume decrease when $V=325 \mathrm{~cm}^{3}$ , $P=130 \mathrm{kPa}$ , and $\frac{dP}{dt}=12 \frac{\mathrm{kPa}}{\min}$ .

A ball is hit up at $25.6 \mathrm{~m} / \mathrm{s}$ . Find its maximum height and total time in the air.

Pancake batter is poured at $61 \mathrm{~cm}^{3}/\mathrm{s}$ . Find the radius growth rate (in $\mathrm{cm}/\mathrm{s}$ ) at $r=9 \mathrm{~cm}$ with thickness $1 \mathrm{~cm}$ . Use $V=\pi r^{2} h$ .

Find the derivative of $y=\sqrt[4]{\frac{x^{2}+1}{x^{2}-1}}$ .

Find the derivative of $y=\sqrt[4]{\frac{x^{2}+1}{x^{2}-1}}$ using logarithmic differentiation.

Bestimmen Sie die Ableitungen der Funktionen: a) $f(x)=a x^{2}$ , b) $g(y)=-2 y^{3}$ , c) $h(t)=v t$ , d) $f(a)=5 a b$ , e) $f(x)=k^{2}$ , f) $s(t)=\frac{3}{a \sqrt{t}}$ .

Find the total cost of producing 560 units if fixed cost is \ $11,000 and marginal costs are given as$ C^{\prime}(q)$.

Untersuchen Sie das Verhalten von $f$ für die Grenzprozesse: a) $f(x)=\frac{2x+1}{x}, x<0$ , $x \rightarrow-\infty$ ; b) $f(x)=\frac{x+1}{x^{2}}, x>0$ , $x \rightarrow \infty$ . Skizzieren Sie die Graphen.

Führe die Kurvendiskussion für die Funktionen $f(x)=\frac{x^{4}}{5}-x^{2}$ und $f(x)=x^{5}-4 x^{3}$ durch. Zeichne auch Graphen und Ableitungen!

Berechnen Sie den Differenzenquotienten von $g(x)=3 x^{2}-1$ für die Intervalle I: a) $I=[1; 4]$ , b) $I=[-3; -1]$ .

A balloon's radius decreases at $15 \mathrm{~cm} / \mathrm{min}$ . Find the volume change rate when $V=972 \pi \mathrm{cm}^{3}$ .

Evaluate $\lim _{x \rightarrow 0} \frac{\sin 9 x}{x}$ using l'Hôpital's Rule and known limits. Rewrite the limit as $\lim _{x \rightarrow 0}(\square$ .

Find the value of $C$ for the joint density $\rho(x, y)=C\left(x^{2} y+y^{2}\right)$ over $D: -1 \leq x \leq 1$ , $0 \leq y \leq 1$ . Then, compute $P(X \leq 0, Y \geq 1/2)$ .

Evaluate the integral $\int y^{3} \cos y \, dy$ .

Find the second derivative $y^{\prime \prime}$ of the function $y=f(x)$ given $y^{\prime}=x(x-9)^{2}$ and sketch the graph. $y^{\prime \prime}=\square$

Bestimme das Volumen eines Körpers mit quadratischer Querschnittsfläche $A(z)=36-2 z^{2}+\frac{1}{36} z^{4}$ für $z \in[0 ; 6]$ .

Bestimme das Volumen eines Körpers mit Querschnittsfläche $A(z)=-2 \sqrt{z}+4$ und Höhe $h=4$ als Integral und berechne es.

Calculate the integral: $\int -\frac{1}{3} \sin(2 - 3x) \, dx$ .

Berechnen Sie die Ableitung $f^{\prime}(3)$ für die Funktion $f(x)=4 x^{2}-2$ .

Find $f'(3)$ given $\frac{f(x)-f(3)}{x-3} = 4(x+3)$ and evaluate the limit as $x \to 3$ .

Find the 4th term in the Taylor series for $f(x)=4 e^{3 x}$ around $x=4$ . $4 \text{th term} =$

Find the 4th term in the Taylor series for $f(x)=4 e^{3 x}$ around $x=4$ . 4th term =

Trouver l'équation de la tangente à $y=\sqrt[3]{x}$ en $x=8$ .

Bestimme die Stammfunktionen von $f(x)$ für: a) $f(x)=3 \cdot x^{4}-5 x+4$ , b) $f(x)=3 a \cdot x^{4}$ .

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for the curve defined by $x 2^{y}=\ln y$ in terms of $x$ and $y$ .

Find the rate of change of distance $D$ at $P=13.5 \mathrm{KPa}$ and $dP/dt=220 \mathrm{KPa/s}$ when $D^2=\frac{2P}{900}+0.01$ .

Compare the functions $f(x)=4^{x}$ and $g(x)=5x+6$ . Which statement is true about their rates of change?

Differentiate $3 \sqrt{x}-2 \sqrt{y}=4$ to find $\frac{d y}{d x}$ and the slope at (4,1).

Differentiate $x^{5}+y^{5}=244$ to find $\frac{d y}{d x}$ , then find the slope at the point (1,3).