Calculus

Problem 8501

Find the derivative of the function $f(x)=12 \sinh^{-1}(x)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 8502

Determine the absolute max and min of $f(x)=x \sqrt{4-x^{2}}$ on $[-1,2]$ .

See Solution

Problem 8503

Differentiate $h(x)=2 \operatorname{sech}^{-1}(x^{2})$ and provide the exact form of $h^{\prime}(x)=$ .

See Solution

Problem 8504

Differentiate $g(x) = \tanh^{-1}(5x)$ and provide the exact form of $g'(x)$ .

See Solution

Problem 8505

Find the derivative of $f(x)=9 \sinh^{-1}(x^{2})-9 \ln(\sqrt{x^{4}+1})$ . What is $f'(x)$ ?

See Solution

Problem 8506

Differentiate the function $f(x)=-8 \cosh^{-1}(x)$ and provide the exact form of $f'(x)$ .

See Solution

Problem 8507

Differentiate the function $g(x)=\operatorname{coth}^{-1}(2 x+1)$ . Provide the exact form of $g^{\prime}(x)$ .

See Solution

Problem 8508

Differentiate the function $h(x)=14 \operatorname{csch}^{-1}\left(x^{3}\right)$ and provide $h^{\prime}(x)=$ .

See Solution

Problem 8509

Find the derivative of $y=\cosh^{-1}(21x)$ . What is $\frac{dy}{dx}=?$

See Solution

Problem 8510

Find the derivative of $y=\cosh^{-1}(4x)$ for $x > \frac{1}{4}$ . What is $y'$ ?

See Solution

Problem 8511

Analyze the function $f(x)=2 x^{3}-3 x^{2}-12 x+5$ for increasing/decreasing, local max/min, concavity, and inflection points, then sketch the curve.

See Solution

Problem 8512

Can Rolle's Theorem apply to $f(x)=\frac{x^{4}-4 x^{2}+3 x}{x-1}$ on $[0,3]$ ?

See Solution

Problem 8513

Check if $f(x)=x^{2}-2 x-2$ meets the Mean Value Theorem on $[-3,-1]$ and find values of $c$ if it does.

See Solution

Problem 8514

Find the derivatives of these inverse trig functions: 1. $y=\arcsin \frac{x}{2}$ , 2. $y=\cos^{-1} \frac{5}{x}$ , 3. $y=\arctan \frac{2x}{5}$ .

See Solution

Problem 8515

Identify which functions satisfy the MVT: A. $f(x)=x^{1/3}$ on $[0,1]$ , B. $f(x)=|x|$ on $[-1,1]$ , C. $f(x)=\frac{1}{x-1}$ on $[0,2]$ .

See Solution

Problem 8516

Find the derivatives of these functions:
4. $y = \cot^{-1} \left(\frac{5}{2x}\right)$
5. $y = \sin^{-1} \left(\frac{x}{2}\right) + \cos^{-1} \left(\frac{x}{2}\right)$
Provide the answers for each derivative.

See Solution

Problem 8517

Find the limit: $\lim _{x \rightarrow \infty} \frac{20 x^{2}-13 x+5}{5-4 x^{3}}$ . What is the value?

See Solution

Problem 8518

Find the value of $c$ so that the limit of the piecewise function $f(x)$ exists as $x \rightarrow 8$ . Round to two decimal places.

See Solution

Problem 8519

Find the average rate of change of $C(t)=60 e^{-0.95 t}$ from $t=0$ to $t=9$ . Round your answer to three decimal places.

See Solution

Problem 8520

Find the average rate of change of caffeine $C(t)=14 e^{-0.04 t}$ from $t=0$ to $t=7$ . Round to three decimal places.

See Solution

Problem 8521

Find the $x$ values for relative maxima of the function with derivative $f'(x)=x^2(x+1)^3(x-4)^2$ . Options: (A) $\{0,-1,4\}$ (B) $\{-1\}$ (C) $\{0,4\}$ (D) $\{1\}$ (E) none.

See Solution

Problem 8522

Find the value of $c$ so that the limit of $f(x)$ exists as $x \rightarrow 5$ for the piecewise function defined by $f(x)=8 x^{2}+c x$ if $x<5$ and $f(x)=1+5 c x$ if $x>5$ . Round to two decimal places.

See Solution

Problem 8523

Find $y$ at $x=0.1$ to five decimal places using Taylor's series, given $\frac{dy}{dx} = x^2 y - 1$ and $y(0) = 1$ .

See Solution

Problem 8524

Evaluate the integral $\int \frac{(x^{1/3}-4)^{5}}{6x^{2/3}} \, dx$ and find the correct answer choice.

See Solution

Problem 8525

For a strictly decreasing function, is a right hand Riemann Sum an overestimate, underestimate, exact, or undetermined?

See Solution

Problem 8526

Find $\frac{d^{2}y}{dx^{2}}$ for $x(t)=t^{2}+4$ and $y(t)=t^{4}+3$ , where $t>0$ .

See Solution

Problem 8527

Find the order and degree of the differential equation: $\frac{d^3 y}{dx^3} + \cos\left(\frac{d^2 y}{dx^2}\right) = 0$ .

See Solution

Problem 8528

Kostenfunktion $K(x)$ 3. Grades für Druckfix GmbH: Zeigen Sie, dass sie positive y-Achse, Wendepunkt hat, und stellen Sie $K(x)$ auf. Berechnen Sie Wendepunkt und zeigen Sie strenge Monotonie. Analysieren Sie $K_a(x)$ für betriebsminimale Ausbringungsmenge. Erläutern Sie fallende Preisabsatzfunktion.

See Solution

Problem 8529

Water flows from a tank at a rate $\mathrm{d}y/\mathrm{dt} = k y$ . Initially $10,000 \mathrm{ft}^3$ , $8000 \mathrm{ft}^3$ after 4 hours. Find $k$ . A. -0.050 B. -0.056 C. -0.169 D. -0.200

See Solution

Problem 8530

Find the third term in the Taylor series expansion of $\frac{z-1}{z+1}$ at $z=1$ . Options are: 1. $\frac{(z-1)^{2}}{2}$ , 2. $\frac{(z-1)^{2}}{4}$ , 3. $\frac{(z-1)^{3}}{8}$ , 4. $\frac{(z-1)^{3}}{4}$ .

See Solution

Problem 8531

Find the limit as $x$ approaches infinity for $x \sin(1/x)$ .

See Solution

Problem 8532

Bestimme die Ableitung der Funktion $f(x)=4-x^{2}$ mit der h-Methode.

See Solution

Problem 8533

Find the volume of the solid formed by rotating the area between $y=\sqrt{x}$ , $y=2$ , and the y-axis around the y-axis. Options: (A) $\frac{32}{5}\pi$ (B) $\frac{16}{3}\pi$ (C) $\frac{10}{3}\pi$ (D) $\frac{8}{3}\pi$ .

See Solution

Problem 8534

Bestimmen Sie die Dimensionen eines rechteckigen Fußballfeldes, um die Fläche $A$ innerhalb einer 400 m langen Laufbahn zu maximieren.

See Solution

Problem 8535

Betrachte die Funktion $f(x)=\frac{1}{(1-x)^{2}}$ . a) Bestimme die Steigung des Graphen an der $y$ -Achse. b) Finde die Stelle, an der die Steigung von $f$ gleich -4 ist.

See Solution

Problem 8536

Find the slope of the curve $y^{\wedge} 3 - x y^{\wedge} 2 = 4$ at the point where $y = 2$ . Choices: (A) -2, (B) -2, (C) 1, (D) 2, (E) $1/2$ .

See Solution

Problem 8537

Bestimme die Distanz vom Westufer zum östlichen Fußpunkt der Funktion $f(x)=\frac{1}{5}(3x-x^3)$ und finde die tiefste Stelle des Kanals sowie den Gipfel des Erdwalls.

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Problem 8538

Find the derivative $\mathrm{dy} / \mathrm{dx}$ for the equation $x^{\wedge} 3 - y^{\wedge} 3 = 1$ . Options: (A) $x$ (B) $3 x^{\wedge} 2$ (C) $3 \sqrt{ } 3 x^{\wedge} 2$ (D) $x^{\wedge} 2 / y^{\wedge} 2$ (E) $(3 x^{\wedge} 2 - 1) / y^{\wedge} 2$ .

See Solution

Problem 8539

Maximieren Sie die Flächeninhalte und Umfänge von Rechtecken und einem Dreieck, eingeschrieben in die Funktionen $f(x)=-2x^{2}+54$ , $f(x)=-x^{4}+80$ , und $f(x)=-6x^{2}+112,5$ .

See Solution

Problem 8540

Find the derivative of $C(x)=\ln (3x+2)$ .

See Solution

Problem 8541

Find $\frac{dy}{dx}$ for $3x - \tan y = 4$ in terms of $y$ . Options: (A) $3 \sin^2 y$ , (B) $3 \cos^2 y$ , (C) $3 \cos y \cot y$ , (D) $\frac{3}{1 + 9y^2}$ .

See Solution

Problem 8542

Berechne die Seitenlängen einer Schachtel mit maximalem Volumen, wenn aus $16 \mathrm{~cm} \times 10 \mathrm{~cm}$ Pappe Ecken ausgeschnitten werden.

See Solution

Problem 8543

Find the derivative $\frac{d y}{d x}$ of $y=x^{3} \sqrt{x+1}$ and simplify.

See Solution

Problem 8544

Calculate the integral from 0 to 2π of $\cos ^{2} x$ with respect to $x$ .

See Solution

Problem 8545

Bestimme die Stammfunktion von $f^{\prime}(x)=(2-3 x)^{2}$ .

See Solution

Problem 8546

Find the sum of the infinite geometric series: $\sum_{n=0}^{\infty}\left(\frac{1}{6}\right)^{n}$ .

See Solution

Problem 8547

Find the derivative of $2 x^{2}+3 \sqrt{y}=-4 x+y$ and the slope at the point (-2,9).

See Solution

Problem 8548

Calculate the integral $\int_{2}^{2+\pi / 2}|\sin 2 x| d x$ and verify $\int_{1}^{1+\pi}|\cos x| d x$ .

See Solution

Problem 8549

Find the function $G(x)$ for the integral $G(x)=\int_{x}^{x^{3}} \frac{1}{t^{2}+1} dt$ .

See Solution

Problem 8550

Finde den Winkel in Grad, bei dem die Ableitung von $f(x)=\sin (x)$ gleich $-\frac{1}{2}$ ist.

See Solution

Problem 8551

Die Bakterienzahl wird durch $f(x)=800 \cdot 1,20^x$ beschrieben. Bestimme: a) Bakterienzahl nach 3h und 3h vorher. b) Verdopplungszeit. c) Ableitung von $f$ . d) $f^{\prime}(0)$ und $f^{\prime}(5)$ mit Bedeutung. e) Zeitpunkt, ab dem Zunahme > 1000 Bakterien/h.

See Solution

Problem 8552

Find the first and second derivatives of the function $f(x)=\frac{1}{4} x^{4}+x+9$ .

See Solution

Problem 8553

Find the first and second derivatives of the function $f(x)=x^{4}-18 x^{2}+6$ .

See Solution

Problem 8554

Find the first and second derivatives of the function $f(x)=-\frac{2}{3} x^{3}-9 x^{2}-36 x+13$ .

See Solution

Problem 8555

Untersuchen Sie die Monotonie der Funktion $f(x)=(x-4) \cdot \sqrt{x}$ für $x \geq 0$ und identifizieren Sie die Extremstelle.

See Solution

Problem 8556

Bestimme die Tangentenfunktion an den Graphen von $f(x)=\frac{x^{3}}{6}-2 x^{2}+2$ im Punkt $P(1 / f(1))$ .

See Solution

Problem 8557

Find the tangent line to the hyperbola $\frac{x^{2}}{6}-\frac{y^{2}}{8}=1$ at the point (3,-2) using implicit differentiation.

See Solution

Problem 8558

Find the derivative $\frac{d y}{d x}$ for the equation $x y^{2}+2 x y=12$ .

See Solution

Problem 8559

Bestimme die mittlere Änderungsrate von $f(t)=-4 t^{2}+12 t+31$ in der ersten Stunde, die Zeit über $10 \frac{\mu g}{m^{3}}$ und das Maximum.

See Solution

Problem 8560

Find the rate of change of a cube's surface area when edges are 2 cm and 10 cm, given edges expand at 6 cm/s.

See Solution

Problem 8561

Find the limit $\lim _{x \rightarrow 4} \frac{2 x^{2}-5 x-12}{x-4}$ and solve for $k$ in $k(4)-5$ .

See Solution

Problem 8562

Find the tangent line slope at $x_{0}$ for these functions: (a) $f(x)=7-x^{2}$ at $x_{0}=2$ , (b) $g(x)=x^{3}+2$ at $x_{0}=-3$ .

See Solution

Problem 8563

Leiten Sie die Funktion $f$ ab: a) $f(x)=(5 x+2)^{3}$ b) $f(t)=2 \cdot(3 t-1)^{2}$ c) $f(x)=-3 \cdot(4-x)^{7}$ d) $f(x)=\left(\frac{1}{4} x-2\right)^{4}$ e) $f(x)=\frac{1}{2} \cdot\left(\frac{1}{3} x+5\right)^{6}$ f) $f(t)=(6-2 t)^{9}$ g) $f(x)=(6 x+7)^{5}$ h) $f(t)=4 \cdot(t+2)^{8}$ i) $f(x)=-5(-3 x+4)^{11}$

See Solution

Problem 8564

Estimate $\frac{1}{\sqrt{1+x}}$ using the binomial series for $x=2$ . Provide a linear approximation.

See Solution

Problem 8565

A 10m ladder leans against a wall. Base moves away at $\frac{2}{3}$ m/s. Find rates for (a) top height, (b) area change, (c) angle change when base is 2m, 6m, 8m from the wall.

See Solution

Problem 8566

Find the equations of horizontal tangent lines for the curve $y - 3y^2 = 2x - x^2$ . If none, explain why.

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Problem 8567

Find $G^{\prime}(x)$ if $G(x) = \int_{1}^{x} 2t \, dt$ . What is $G^{\prime}(x)$ ?

See Solution

Problem 8568

Find $x$ where the derivative of $f(x)=\sqrt{x} \cdot (x-9)$ is zero, for $x>0$ .

See Solution

Problem 8569

Find the second derivative of the function $f(x)=\sqrt{6x-2}$ .

See Solution

Problem 8570

Determine the absolute max and min of $f(x)=4 x^{3}+24 x^{2}-252 x$ on $[-9,5]$ .

See Solution

Problem 8571

Find the second derivative of the function $f(x)=2 x^{2}-6 \sqrt{x}-3$ .

See Solution

Problem 8572

Find $f^{\prime \prime}(3), f^{\prime \prime}(2)$ , and $f^{\prime \prime}(6)$ for $f(x)=\sqrt{9x+7}$ .

See Solution

Problem 8573

Evaluate the second derivative $f^{\prime \prime}(x)$ of $f(x)=\sqrt{8x+8}$ at $x=5, 6, 7$ . State "Does Not Exist" if not applicable.

See Solution

Problem 8574

Find the derivative of these functions without simplifying: (a) $f(x)=\frac{x e^{-x}}{2 x+1}$ (b) $f(x)=\ln \left(\left(3 x^{2}+x\right)^{6}(1-x)^{3}\right)$ .

See Solution

Problem 8575

19. Find $G^{\prime}(x)$ if $G(x)=\int_{0}^{x}(2 t^{2}+\sqrt{t}) dt$ .
20. Find $G^{\prime}(x)$ if $G(x)=\int_{1}^{x} \cos^{3}(2t) \tan(t) dt$ .

See Solution

Problem 8576

Find the limit: $\lim _{x \rightarrow 2}(x-2)^{4} \sin \left(\frac{3}{(x-2)^{2}}\right)$ .

See Solution

Problem 8577

Find the limit: $\lim _{t \rightarrow 6} 8(\sin (2 \pi t)-5)(t-7)$ .

See Solution

Problem 8578

Find the derivatives without simplification: (a) $f(x)=\frac{x e^{-x}}{2 x+1}$ , (b) $f(x)=\ln \left(\left(3 x^{2}+x\right)^{6}(1-x)^{3}\right)$ .

See Solution

Problem 8579

Find $G'(x)$ given $G'(x) = - (x-2) \cot(2x)$ for $x \in [-\pi/4, x]$ .

See Solution

Problem 8580

Find $G^{\prime}(x)$ if $G^{\prime}(x)=D_{x}\left[\int_{1}^{x^{2}} \sin t d t\right]=2 x \sin \left(x^{2}\right)$ .

See Solution

Problem 8581

Find the derivatives of these functions:
d) $f(x)=\frac{x^{5}-x}{x^{3}}$
e) $f(x)=\frac{x}{x+1}$

See Solution

Problem 8582

Find where the function $f(x)=\frac{x^{4}}{4}-\frac{10 x^{3}}{3}+\frac{21}{2} x^{2}+\frac{6}{5}$ has a horizontal tangent.

See Solution

Problem 8583

Find the derivative $G'(x)$ of the function defined by $G(x) = \int_{1}^{x} x t \, dt$ .

See Solution

Problem 8584

Find the limit as $x$ approaches 2: $(x-2)^{4} \sin \left(\frac{3}{(x-2)^{2}}\right)$ .

See Solution

Problem 8585

Bestimmen Sie die erste Ableitung für die folgenden Funktionen: a) $f(x)=(5-4 x^{2})^{3}$ , b) $f(t)=t \cdot \sqrt{2 t+1}$ , c) $f_{a}(x)=(a x^{2}-1)^{3}$ , d) $f(x)=\frac{x^{5}-x}{x^{3}}$ , e) $f(x)=\frac{x}{x+1}$ , f) $f_{k}(t)=(k t)^{2} \cdot \cos (k t)$ .

See Solution

Problem 8586

Bestimmen Sie die Werte für a, sodass der Graph $G_{f_{a}}$ rechtsgekrümmt ist für: a) $f_{a}(x)=a \cdot(2 x-3)^{4}$ , b) $f_{a}(x)=\sqrt{a^{2} x}$ , c) $f_{a}(x)=\frac{a}{x^{2}}$ .

See Solution

Problem 8587

Gravel is dumped at 40 ft³/min into a cone with equal base diameter and height. Find height increase rate when 12 ft high.

See Solution

Problem 8588

Find the Laplace transform $Y(s)=\mathcal{L}\{y\}$ for the initial value problem with $y'' + 16y = t$ (0≤t<1), 1 (1≤t<∞), $y(0)=6$ , $y'(0)=2$ .

See Solution

Problem 8589

Untersuchen Sie die Extrem- und Sattelstellen der Funktionen: a) $f(x)=x^{2}-4$ , b) $f(x)=\frac{1}{3} x^{3}-2 x^{2}+4 x$ , c) $f(x)=x^{3}+3 x^{2}+1$ , d) $f(x)=\frac{1}{9} x^{3}-x^{2}+3 x$ , e) $f(t)=t^{4}-2 t^{2}$ , f) $f(a)=-a^{3}+2,5 a^{2}+3,5$ , g) $f(x)=x^{5}-15 x^{3}-30$ , h) $f(x)=-3 x^{4}+4 x^{3}+8$ .

See Solution

Problem 8590

Solve the nonhomogeneous system using variation of parameters:
$\frac{d x}{d t}=5 x-5 y+5, \quad \frac{d y}{d t}=4 x-4 y-1$
with solution form $\langle x(t), y(t)\rangle=c_{1}+\frac{5}{4} c_{2} e^{t}-25 t-6$ .

See Solution

Problem 8591

What is the speed of a ball dropped from a height of $15 \mathrm{~m}$ when it hits the ground?

See Solution

Problem 8592

A cylinder has a radius of 9 ft and height of 22 ft. Find the rate of change of the radius given $dA/dt = 320 \mathrm{ft}^{2}/\mathrm{sec}$ and $dh/dt = -4 \mathrm{ft}/\mathrm{sec}$ .

See Solution

Problem 8593

Bestimme den Wert von $x$ , der das Volumen einer offenen Schachtel mit Länge $16 \mathrm{~cm}$ und Breite $10 \mathrm{~cm}$ maximiert.

See Solution

Problem 8594

Consider motion with constant acceleration $\vec{a}>0$ and initial conditions $\vec{r}(0)=0$ , $\vec{v}(0)>0$ .
(a) Draw a graph of acceleration vs. time with labeled axes.
(b) Find $\vec{v}(t)$ using $\vec{a}=\frac{d \vec{v}}{d t}$ . Show your work.

See Solution

Problem 8595

Berechne die Integrale: e.) $\int_{-3}^{3} x^{2} d x$ , f.) $\int_{0}^{2} e^{x} d x$ , g.) $\int_{4}^{9} \frac{1}{2 \sqrt{x}} d x$ , h.) $\int_{0}^{\frac{\pi}{2}} \pi \sin x d x$ .

See Solution

Problem 8596

Calculate the average rate of change of $f(x)=\sin \left(\frac{x}{2}\right)$ from 0 to $\frac{\pi}{2}$ .

See Solution

Problem 8597

A cannonball is fired at 20 m/s, 22 degrees. Find $v_x$ and $v_y$ at $t=0, 1, 2, 3, 4, 5, 6$ and direction at $t=35s$ .

See Solution

Problem 8598

Berechne die Integrale: 0) $\int_{-3}^{2} 2 e^{x} d x$ p) $\int_{-3}^{2}(e^{x}+1) d x$ q) $\int_{0}^{2} 0,5 x^{3} d x$ r) $\int_{1}^{e} \frac{2}{x} d x$

See Solution

Problem 8599

Find the first derivative of $f(x)=\frac{5}{(2-x)^{3}}$ .

See Solution

Problem 8600

Find the derivative of $f(x)=\frac{1}{x^{3}}$ using the limit: $\lim_{h\to0} \frac{f(x+h) - f(x)}{h}$ .

See Solution

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Calculus

Problem 8501

Find the derivative of the function f(x)=12sinh⁡−1(x)f(x)=12 \sinh^{-1}(x)f(x)=12sinh−1(x). What is f′(x)f^{\prime}(x)f′(x)?

Problem 8502

Determine the absolute max and min of f(x)=x4−x2f(x)=x \sqrt{4-x^{2}}f(x)=x4−x2​ on [−1,2][-1,2][−1,2].

Problem 8503

Differentiate h(x)=2sech⁡−1(x2)h(x)=2 \operatorname{sech}^{-1}(x^{2})h(x)=2sech−1(x2) and provide the exact form of h′(x)=h^{\prime}(x)=h′(x)=.

Problem 8504

Differentiate g(x)=tanh⁡−1(5x)g(x) = \tanh^{-1}(5x)g(x)=tanh−1(5x) and provide the exact form of g′(x)g'(x)g′(x).

Problem 8505

Find the derivative of f(x)=9sinh⁡−1(x2)−9ln⁡(x4+1)f(x)=9 \sinh^{-1}(x^{2})-9 \ln(\sqrt{x^{4}+1})f(x)=9sinh−1(x2)−9ln(x4+1​). What is f′(x)f'(x)f′(x)?

Problem 8506

Differentiate the function f(x)=−8cosh⁡−1(x)f(x)=-8 \cosh^{-1}(x)f(x)=−8cosh−1(x) and provide the exact form of f′(x)f'(x)f′(x).

Problem 8507

Differentiate the function g(x)=coth⁡−1(2x+1)g(x)=\operatorname{coth}^{-1}(2 x+1)g(x)=coth−1(2x+1). Provide the exact form of g′(x)g^{\prime}(x)g′(x).

Problem 8508

Differentiate the function h(x)=14csch⁡−1(x3)h(x)=14 \operatorname{csch}^{-1}\left(x^{3}\right)h(x)=14csch−1(x3) and provide h′(x)=h^{\prime}(x)=h′(x)=.

Problem 8509

Find the derivative of y=cosh⁡−1(21x)y=\cosh^{-1}(21x)y=cosh−1(21x). What is dydx=?\frac{dy}{dx}=?dxdy​=?

Problem 8510

Find the derivative of y=cosh⁡−1(4x)y=\cosh^{-1}(4x)y=cosh−1(4x) for x>14x > \frac{1}{4}x>41​. What is y′y'y′?

Problem 8511

Analyze the function f(x)=2x3−3x2−12x+5f(x)=2 x^{3}-3 x^{2}-12 x+5f(x)=2x3−3x2−12x+5 for increasing/decreasing, local max/min, concavity, and inflection points, then sketch the curve.

Problem 8512

Can Rolle's Theorem apply to f(x)=x4−4x2+3xx−1f(x)=\frac{x^{4}-4 x^{2}+3 x}{x-1}f(x)=x−1x4−4x2+3x​ on [0,3][0,3][0,3]?

Problem 8513

Check if f(x)=x2−2x−2f(x)=x^{2}-2 x-2f(x)=x2−2x−2 meets the Mean Value Theorem on [−3,−1][-3,-1][−3,−1] and find values of ccc if it does.

Problem 8514

Find the derivatives of these inverse trig functions: 1. y=arcsin⁡x2y=\arcsin \frac{x}{2}y=arcsin2x​, 2. y=cos⁡−15xy=\cos^{-1} \frac{5}{x}y=cos−1x5​, 3. y=arctan⁡2x5y=\arctan \frac{2x}{5}y=arctan52x​.

Problem 8515

Identify which functions satisfy the MVT: A. f(x)=x1/3f(x)=x^{1/3}f(x)=x1/3 on [0,1][0,1][0,1], B. f(x)=∣x∣f(x)=|x|f(x)=∣x∣ on [−1,1][-1,1][−1,1], C. f(x)=1x−1f(x)=\frac{1}{x-1}f(x)=x−11​ on [0,2][0,2][0,2].

Problem 8516

Find the derivatives of these functions: 4. y=cot⁡−1(52x)y = \cot^{-1} \left(\frac{5}{2x}\right)y=cot−1(2x5​) 5. y=sin⁡−1(x2)+cos⁡−1(x2)y = \sin^{-1} \left(\frac{x}{2}\right) + \cos^{-1} \left(\frac{x}{2}\right)y=sin−1(2x​)+cos−1(2x​) Provide the answers for each derivative.

Problem 8517

Find the limit: lim⁡x→∞20x2−13x+55−4x3\lim _{x \rightarrow \infty} \frac{20 x^{2}-13 x+5}{5-4 x^{3}}limx→∞​5−4x320x2−13x+5​. What is the value?

Problem 8518

Find the value of ccc so that the limit of the piecewise function f(x)f(x)f(x) exists as x→8x \rightarrow 8x→8. Round to two decimal places.

Problem 8519

Find the average rate of change of C(t)=60e−0.95tC(t)=60 e^{-0.95 t}C(t)=60e−0.95t from t=0t=0t=0 to t=9t=9t=9. Round your answer to three decimal places.

Problem 8520

Find the average rate of change of caffeine C(t)=14e−0.04tC(t)=14 e^{-0.04 t}C(t)=14e−0.04t from t=0t=0t=0 to t=7t=7t=7. Round to three decimal places.

Problem 8521

Find the xxx values for relative maxima of the function with derivative f′(x)=x2(x+1)3(x−4)2f'(x)=x^2(x+1)^3(x-4)^2f′(x)=x2(x+1)3(x−4)2. Options: (A) {0,−1,4}\{0,-1,4\}{0,−1,4} (B) {−1}\{-1\}{−1} (C) {0,4}\{0,4\}{0,4} (D) {1}\{1\}{1} (E) none.

Problem 8522

Find the value of ccc so that the limit of f(x)f(x)f(x) exists as x→5x \rightarrow 5x→5 for the piecewise function defined by f(x)=8x2+cxf(x)=8 x^{2}+c xf(x)=8x2+cx if x<5x<5x<5 and f(x)=1+5cxf(x)=1+5 c xf(x)=1+5cx if x>5x>5x>5. Round to two decimal places.

Problem 8523

Find yyy at x=0.1x=0.1x=0.1 to five decimal places using Taylor's series, given dydx=x2y−1\frac{dy}{dx} = x^2 y - 1dxdy​=x2y−1 and y(0)=1y(0) = 1y(0)=1.

Problem 8524

Evaluate the integral ∫(x1/3−4)56x2/3 dx\int \frac{(x^{1/3}-4)^{5}}{6x^{2/3}} \, dx∫6x2/3(x1/3−4)5​dx and find the correct answer choice.

Problem 8525

For a strictly decreasing function, is a right hand Riemann Sum an overestimate, underestimate, exact, or undetermined?

Problem 8526

Find d2ydx2\frac{d^{2}y}{dx^{2}}dx2d2y​ for x(t)=t2+4x(t)=t^{2}+4x(t)=t2+4 and y(t)=t4+3y(t)=t^{4}+3y(t)=t4+3, where t>0t>0t>0.

Problem 8527

Find the order and degree of the differential equation: d3ydx3+cos⁡(d2ydx2)=0 \frac{d^3 y}{dx^3} + \cos\left(\frac{d^2 y}{dx^2}\right) = 0dx3d3y​+cos(dx2d2y​)=0.

Problem 8528

Problem 8529

Water flows from a tank at a rate dy/dt=ky\mathrm{d}y/\mathrm{dt} = k ydy/dt=ky. Initially 10,000ft310,000 \mathrm{ft}^310,000ft3, 8000ft38000 \mathrm{ft}^38000ft3 after 4 hours. Find kkk. A. -0.050 B. -0.056 C. -0.169 D. -0.200

Problem 8530

Find the third term in the Taylor series expansion of z−1z+1\frac{z-1}{z+1}z+1z−1​ at z=1z=1z=1. Options are: 1. (z−1)22\frac{(z-1)^{2}}{2}2(z−1)2​, 2. (z−1)24\frac{(z-1)^{2}}{4}4(z−1)2​, 3. (z−1)38\frac{(z-1)^{3}}{8}8(z−1)3​, 4. (z−1)34\frac{(z-1)^{3}}{4}4(z−1)3​.

Problem 8531

Find the limit as xxx approaches infinity for xsin⁡(1/x)x \sin(1/x)xsin(1/x).

Problem 8532

Bestimme die Ableitung der Funktion f(x)=4−x2f(x)=4-x^{2}f(x)=4−x2 mit der h-Methode.

Problem 8533

Find the volume of the solid formed by rotating the area between y=xy=\sqrt{x}y=x​, y=2y=2y=2, and the y-axis around the y-axis. Options: (A) 325π\frac{32}{5}\pi532​π (B) 163π\frac{16}{3}\pi316​π (C) 103π\frac{10}{3}\pi310​π (D) 83π\frac{8}{3}\pi38​π.

Problem 8534

Bestimmen Sie die Dimensionen eines rechteckigen Fußballfeldes, um die Fläche AAA innerhalb einer 400 m langen Laufbahn zu maximieren.

Problem 8535

Betrachte die Funktion f(x)=1(1−x)2f(x)=\frac{1}{(1-x)^{2}}f(x)=(1−x)21​. a) Bestimme die Steigung des Graphen an der yyy-Achse. b) Finde die Stelle, an der die Steigung von fff gleich -4 ist.

Problem 8536

Find the slope of the curve y∧3−xy∧2=4y^{\wedge} 3 - x y^{\wedge} 2 = 4y∧3−xy∧2=4 at the point where y=2y = 2y=2. Choices: (A) -2, (B) -2, (C) 1, (D) 2, (E) 1/21/21/2.

Problem 8537

Bestimme die Distanz vom Westufer zum östlichen Fußpunkt der Funktion f(x)=15(3x−x3)f(x)=\frac{1}{5}(3x-x^3)f(x)=51​(3x−x3) und finde die tiefste Stelle des Kanals sowie den Gipfel des Erdwalls.

Problem 8538

Problem 8539

Maximieren Sie die Flächeninhalte und Umfänge von Rechtecken und einem Dreieck, eingeschrieben in die Funktionen f(x)=−2x2+54f(x)=-2x^{2}+54f(x)=−2x2+54, f(x)=−x4+80f(x)=-x^{4}+80f(x)=−x4+80, und f(x)=−6x2+112,5f(x)=-6x^{2}+112,5f(x)=−6x2+112,5.

Problem 8540

Find the derivative of C(x)=ln⁡(3x+2)C(x)=\ln (3x+2)C(x)=ln(3x+2).

Problem 8541

Find the derivative of the function $f(x)=12 \sinh^{-1}(x)$ . What is $f^{\prime}(x)$ ?

Determine the absolute max and min of $f(x)=x \sqrt{4-x^{2}}$ on $[-1,2]$ .

Differentiate $h(x)=2 \operatorname{sech}^{-1}(x^{2})$ and provide the exact form of $h^{\prime}(x)=$ .

Differentiate $g(x) = \tanh^{-1}(5x)$ and provide the exact form of $g'(x)$ .

Find the derivative of $f(x)=9 \sinh^{-1}(x^{2})-9 \ln(\sqrt{x^{4}+1})$ . What is $f'(x)$ ?

Differentiate the function $f(x)=-8 \cosh^{-1}(x)$ and provide the exact form of $f'(x)$ .

Differentiate the function $g(x)=\operatorname{coth}^{-1}(2 x+1)$ . Provide the exact form of $g^{\prime}(x)$ .

Differentiate the function $h(x)=14 \operatorname{csch}^{-1}\left(x^{3}\right)$ and provide $h^{\prime}(x)=$ .

Find the derivative of $y=\cosh^{-1}(21x)$ . What is $\frac{dy}{dx}=?$

Find the derivative of $y=\cosh^{-1}(4x)$ for $x > \frac{1}{4}$ . What is $y'$ ?

Analyze the function $f(x)=2 x^{3}-3 x^{2}-12 x+5$ for increasing/decreasing, local max/min, concavity, and inflection points, then sketch the curve.

Can Rolle's Theorem apply to $f(x)=\frac{x^{4}-4 x^{2}+3 x}{x-1}$ on $[0,3]$ ?

Check if $f(x)=x^{2}-2 x-2$ meets the Mean Value Theorem on $[-3,-1]$ and find values of $c$ if it does.

Find the derivatives of these inverse trig functions: 1. $y=\arcsin \frac{x}{2}$ , 2. $y=\cos^{-1} \frac{5}{x}$ , 3. $y=\arctan \frac{2x}{5}$ .

Identify which functions satisfy the MVT: A. $f(x)=x^{1/3}$ on $[0,1]$ , B. $f(x)=|x|$ on $[-1,1]$ , C. $f(x)=\frac{1}{x-1}$ on $[0,2]$ .

Find the derivatives of these functions:
4. $y = \cot^{-1} \left(\frac{5}{2x}\right)$
5. $y = \sin^{-1} \left(\frac{x}{2}\right) + \cos^{-1} \left(\frac{x}{2}\right)$
Provide the answers for each derivative.

Find the limit: $\lim _{x \rightarrow \infty} \frac{20 x^{2}-13 x+5}{5-4 x^{3}}$ . What is the value?

Find the value of $c$ so that the limit of the piecewise function $f(x)$ exists as $x \rightarrow 8$ . Round to two decimal places.

Find the average rate of change of $C(t)=60 e^{-0.95 t}$ from $t=0$ to $t=9$ . Round your answer to three decimal places.

Find the average rate of change of caffeine $C(t)=14 e^{-0.04 t}$ from $t=0$ to $t=7$ . Round to three decimal places.

Find the $x$ values for relative maxima of the function with derivative $f'(x)=x^2(x+1)^3(x-4)^2$ . Options: (A) $\{0,-1,4\}$ (B) $\{-1\}$ (C) $\{0,4\}$ (D) $\{1\}$ (E) none.

Find the value of $c$ so that the limit of $f(x)$ exists as $x \rightarrow 5$ for the piecewise function defined by $f(x)=8 x^{2}+c x$ if $x<5$ and $f(x)=1+5 c x$ if $x>5$ . Round to two decimal places.

Find $y$ at $x=0.1$ to five decimal places using Taylor's series, given $\frac{dy}{dx} = x^2 y - 1$ and $y(0) = 1$ .

Evaluate the integral $\int \frac{(x^{1/3}-4)^{5}}{6x^{2/3}} \, dx$ and find the correct answer choice.

Find $\frac{d^{2}y}{dx^{2}}$ for $x(t)=t^{2}+4$ and $y(t)=t^{4}+3$ , where $t>0$ .

Find the order and degree of the differential equation: $\frac{d^3 y}{dx^3} + \cos\left(\frac{d^2 y}{dx^2}\right) = 0$ .

Water flows from a tank at a rate $\mathrm{d}y/\mathrm{dt} = k y$ . Initially $10,000 \mathrm{ft}^3$ , $8000 \mathrm{ft}^3$ after 4 hours. Find $k$ . A. -0.050 B. -0.056 C. -0.169 D. -0.200

Find the third term in the Taylor series expansion of $\frac{z-1}{z+1}$ at $z=1$ . Options are: 1. $\frac{(z-1)^{2}}{2}$ , 2. $\frac{(z-1)^{2}}{4}$ , 3. $\frac{(z-1)^{3}}{8}$ , 4. $\frac{(z-1)^{3}}{4}$ .

Find the limit as $x$ approaches infinity for $x \sin(1/x)$ .

Bestimme die Ableitung der Funktion $f(x)=4-x^{2}$ mit der h-Methode.

Find the volume of the solid formed by rotating the area between $y=\sqrt{x}$ , $y=2$ , and the y-axis around the y-axis. Options: (A) $\frac{32}{5}\pi$ (B) $\frac{16}{3}\pi$ (C) $\frac{10}{3}\pi$ (D) $\frac{8}{3}\pi$ .

Bestimmen Sie die Dimensionen eines rechteckigen Fußballfeldes, um die Fläche $A$ innerhalb einer 400 m langen Laufbahn zu maximieren.

Betrachte die Funktion $f(x)=\frac{1}{(1-x)^{2}}$ . a) Bestimme die Steigung des Graphen an der $y$ -Achse. b) Finde die Stelle, an der die Steigung von $f$ gleich -4 ist.

Find the slope of the curve $y^{\wedge} 3 - x y^{\wedge} 2 = 4$ at the point where $y = 2$ . Choices: (A) -2, (B) -2, (C) 1, (D) 2, (E) $1/2$ .

Bestimme die Distanz vom Westufer zum östlichen Fußpunkt der Funktion $f(x)=\frac{1}{5}(3x-x^3)$ und finde die tiefste Stelle des Kanals sowie den Gipfel des Erdwalls.

Maximieren Sie die Flächeninhalte und Umfänge von Rechtecken und einem Dreieck, eingeschrieben in die Funktionen $f(x)=-2x^{2}+54$ , $f(x)=-x^{4}+80$ , und $f(x)=-6x^{2}+112,5$ .

Find the derivative of $C(x)=\ln (3x+2)$ .

Find $\frac{dy}{dx}$ for $3x - \tan y = 4$ in terms of $y$ . Options: (A) $3 \sin^2 y$ , (B) $3 \cos^2 y$ , (C) $3 \cos y \cot y$ , (D) $\frac{3}{1 + 9y^2}$ .

Berechne die Seitenlängen einer Schachtel mit maximalem Volumen, wenn aus $16 \mathrm{~cm} \times 10 \mathrm{~cm}$ Pappe Ecken ausgeschnitten werden.

Find the derivative $\frac{d y}{d x}$ of $y=x^{3} \sqrt{x+1}$ and simplify.

Calculate the integral from 0 to 2π of $\cos ^{2} x$ with respect to $x$ .

Bestimme die Stammfunktion von $f^{\prime}(x)=(2-3 x)^{2}$ .

Find the sum of the infinite geometric series: $\sum_{n=0}^{\infty}\left(\frac{1}{6}\right)^{n}$ .

Find the derivative of $2 x^{2}+3 \sqrt{y}=-4 x+y$ and the slope at the point (-2,9).

Calculate the integral $\int_{2}^{2+\pi / 2}|\sin 2 x| d x$ and verify $\int_{1}^{1+\pi}|\cos x| d x$ .

Find the function $G(x)$ for the integral $G(x)=\int_{x}^{x^{3}} \frac{1}{t^{2}+1} dt$ .

Finde den Winkel in Grad, bei dem die Ableitung von $f(x)=\sin (x)$ gleich $-\frac{1}{2}$ ist.

Find the first and second derivatives of the function $f(x)=\frac{1}{4} x^{4}+x+9$ .

Find the first and second derivatives of the function $f(x)=x^{4}-18 x^{2}+6$ .

Find the first and second derivatives of the function $f(x)=-\frac{2}{3} x^{3}-9 x^{2}-36 x+13$ .

Untersuchen Sie die Monotonie der Funktion $f(x)=(x-4) \cdot \sqrt{x}$ für $x \geq 0$ und identifizieren Sie die Extremstelle.

Bestimme die Tangentenfunktion an den Graphen von $f(x)=\frac{x^{3}}{6}-2 x^{2}+2$ im Punkt $P(1 / f(1))$ .

Find the tangent line to the hyperbola $\frac{x^{2}}{6}-\frac{y^{2}}{8}=1$ at the point (3,-2) using implicit differentiation.

Find the derivative $\frac{d y}{d x}$ for the equation $x y^{2}+2 x y=12$ .

Bestimme die mittlere Änderungsrate von $f(t)=-4 t^{2}+12 t+31$ in der ersten Stunde, die Zeit über $10 \frac{\mu g}{m^{3}}$ und das Maximum.

Find the limit $\lim _{x \rightarrow 4} \frac{2 x^{2}-5 x-12}{x-4}$ and solve for $k$ in $k(4)-5$ .

Find the tangent line slope at $x_{0}$ for these functions: (a) $f(x)=7-x^{2}$ at $x_{0}=2$ , (b) $g(x)=x^{3}+2$ at $x_{0}=-3$ .

Estimate $\frac{1}{\sqrt{1+x}}$ using the binomial series for $x=2$ . Provide a linear approximation.

A 10m ladder leans against a wall. Base moves away at $\frac{2}{3}$ m/s. Find rates for (a) top height, (b) area change, (c) angle change when base is 2m, 6m, 8m from the wall.

Find the equations of horizontal tangent lines for the curve $y - 3y^2 = 2x - x^2$ . If none, explain why.

Find $G^{\prime}(x)$ if $G(x) = \int_{1}^{x} 2t \, dt$ . What is $G^{\prime}(x)$ ?

Find $x$ where the derivative of $f(x)=\sqrt{x} \cdot (x-9)$ is zero, for $x>0$ .

Find the second derivative of the function $f(x)=\sqrt{6x-2}$ .

Determine the absolute max and min of $f(x)=4 x^{3}+24 x^{2}-252 x$ on $[-9,5]$ .

Find the second derivative of the function $f(x)=2 x^{2}-6 \sqrt{x}-3$ .

Find $f^{\prime \prime}(3), f^{\prime \prime}(2)$ , and $f^{\prime \prime}(6)$ for $f(x)=\sqrt{9x+7}$ .

Evaluate the second derivative $f^{\prime \prime}(x)$ of $f(x)=\sqrt{8x+8}$ at $x=5, 6, 7$ . State "Does Not Exist" if not applicable.

Find the derivative of these functions without simplifying: (a) $f(x)=\frac{x e^{-x}}{2 x+1}$ (b) $f(x)=\ln \left(\left(3 x^{2}+x\right)^{6}(1-x)^{3}\right)$ .