Calculus

Problem 32801

How long does it take for a penny to fall from a height of 93 meters? Options: $176.2 \mathrm{~s}$ , $18.97 \mathrm{~s}$ , $4.4 \mathrm{~s}$ , $9.5 \mathrm{~s}$ .

See Solution

Problem 32802

Find the derivative of the function $f(x)=\tan (x)(6 \sin (x)+7 \cos (x))$ : $f^{\prime}(x)=$

See Solution

Problem 32803

If $\lim _{x \rightarrow-4}[f(x)+g(x)]=2$ and $\lim _{x \rightarrow-4} g(x)=13$ , find $\lim _{x \rightarrow-4} \frac{6 g(x)}{f(x)-g(x)}$ .

See Solution

Problem 32804

Find the limits: $\lim_{x \rightarrow 1^{-}} h(x)$ , $\lim_{x \rightarrow 1^{+}} h(x)$ , $\lim_{x \rightarrow 1} h(x)$ , $\lim_{x \rightarrow 2^{-}} h(x)$ , $\lim_{x \rightarrow 2^{+}} h(x)$ , $\lim_{x \rightarrow 2} h(x)$ for $h(x)=\left\{\begin{array}{ll}1 & \text { for } x<1 \\ x & \text { for } 1 \leq x<2 \\ 3 & \text { for } x \geq 2\end{array}\right.$

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Problem 32805

Find the values of $x$ where the tangents to $y=(1+x^{2})^{3}$ and $y=3x^{2}$ have equal slopes.

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Problem 32806

Find the tangent line equation for $f(x)=\frac{\sqrt{x}}{5x-6}$ at the point $(3, f(3))$ . $y=$

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Problem 32807

Find the second derivative $f^{\prime \prime}(\pi / 4)$ for the function $f(x)=6 \sec x$ .

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Problem 32808

Bestimmen Sie $u(x)$ und $v(x)$ und leiten Sie $a b$ für die Funktionen: a) $f(x)=(2 x^{3}-x) e^{x}$ , b) $f(x)=(x^{2}-1)(2 x^{2}+5)$ , c) $f(x)=(5 x^{2}-4) e^{x}$ .

See Solution

Problem 32809

Find the average rate of change of $h(x)=-x^{2}+8x+20$ on the interval $0 \leq x \leq 5$ .

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Problem 32810

Find the derivative of $f(x)=12 x(\sin (x)+\cos (x))$ and evaluate $f^{\prime}\left(-\frac{\pi}{6}\right)$ .

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Problem 32811

Find the tangent line equation to the curve $y=3 x \cos x$ at the point $(\pi, -3 \pi)$ in the form $y=m x+b$ .

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Problem 32812

Prove that $f(x)=\sqrt{x}$ is uniformly continuous but not Lipschitz continuous. Also, show that Lipschitz functions compose to be Lipschitz.

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Problem 32813

Max untersucht die Ableitung von $f(x)=x^{6}$ als Produkt. Vervollständigen Sie die Tabelle und vermuten Sie die Ableitungsformel.

See Solution

Problem 32814

Find the tangent line to $y=3 x \cos x$ at $(\pi,-3 \pi)$ in the form $y=m x+b$ , with $m=$ and $b=$ .

See Solution

Problem 32815

Differentiate the function $f(x)=4 e^{-2 x}$ using the chain rule to find $f'(x)$ .

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Problem 32816

A bullet is shot straight up at 140 m/s. How long will it stay in the air? Use $t = \frac{2v}{g}$ , where $g \approx 9.8 \mathrm{~m/s^2}$ .

See Solution

Problem 32817

Lara hat die Ableitung von $f(x)=(2-x)^{3}$ als $f'(x)=3(2-x)^{2}$ gefunden. Erklären Sie, überprüfen Sie und vermuten Sie eine Ableitungsformel.

See Solution

Problem 32818

Determine the convergence of $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}$ using comparison tests.

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Problem 32819

Determine if the series $\sum_{n=1}^{\infty}\left(\frac{1}{n^{2}}+1\right)^{2}$ converges or diverges.

See Solution

Problem 32820

Übungsaufgaben:
1. Ergänzen Sie die Ableitungen: a) $f(x)=\frac{1}{3} e^{3 x^{2}+1}$ , $f^{\prime}(x)=$ ? b) $g(x)=2 e^{5 x-3}$ , $g^{\prime}(x)=$ ? c) $f(x)=(0,5 x^{2}-5 x)^{4}$ , $f^{\prime}(x)=$ ?
2. Leiten Sie ab und vereinfachen: a) $f(x)=8 e^{1-x}$ b) $f(x)=(3 x-2)^{3}$ c) $f(x)=2 e^{0,5 x+1}$

See Solution

Problem 32821

Calculate $\langle x \rangle = \int_{0}^{L} x \left|\psi_n(x)\right|^2 \,dx$ for $\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$ .

See Solution

Problem 32822

Find the limit as $x$ approaches 0 for $\frac{\sin(9x)}{x}$ when $x$ is in degrees. Answer to 0.01 accuracy.

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Problem 32823

Finde die Gleichung der Normalen an den Graphen von $f$ in den Punkten A für die Funktionen a) bis d).

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Problem 32824

Find critical values of $f(x)=(x-1)^{3}$ on $[0,4]$ and determine if they are max, min, or neither. Analyze increasing/decreasing intervals.

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Problem 32825

Find the integral of $\frac{\sin(\ln(x))}{x}$ with respect to $x$ .

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Problem 32826

Find $f^{\prime \prime}(2)$ for the function $f(x)=x+\frac{16}{x}$ . Options: a. 4, b. 9, c. -3, d. -4.

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Problem 32827

-4

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Problem 32828

Calculate the limit: $\lim _{x \rightarrow 0^{+}} \operatorname{tg}(x) \ln (x)$ .

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Problem 32829

Find the limit as x approaches 0: $\lim _{x \rightarrow 0} \sqrt{x+4}$ .

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Problem 32830

Find the tangent line equation for $f(x)=\frac{\sqrt{x}}{5 x-6}$ at the point $(3, f(3))$ . $y=$

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Problem 32831

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{x+6}-\sqrt{6}}{x}$ .

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Problem 32832

Find the max and min of $f(x)=x \arcsin (x)+\sqrt{1-x^{2}}$ on the interval $[-1,1]$ .

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Problem 32833

Evaluate the limit: $\lim _{t \rightarrow 0} \frac{\sin 9 t}{8 t}$ .

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Problem 32834

Finde die Ableitung von $f(x) = e^{\sqrt{x}}$ .

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Problem 32835

Solve for $z$ in the equation $\int_{z}^{10} 2 x d x=19$ .

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Problem 32836

A boat is 3 feet from the dock, and the rope is pulled at 2 ft/s. How fast is the distance to the dock decreasing?

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Problem 32837

Find the tangent line equation for $f(x)=\frac{\sqrt{x}}{5 x-6}$ at the point $(3, f(3))$ . $y=$

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Problem 32838

Calculate the integral $\int_{1}^{\infty} \frac{2}{x^{2}} d x$ for each value of $t$ .

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Problem 32839

Calculate the integral: $\int_{0}^{2 z} 0.4 \, dx = 8$ .

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Problem 32840

A particle's velocity is $v(t)=1+2 \sin \left(\frac{t^{2}}{2}\right)$ and position $x=2$ at $t=4$ .
(a) Is it speeding up or slowing down at $t=4$ ?
(b) Find times $0<t<3$ when the particle changes direction.

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Problem 32841

Evaluate the integral: $\int_{0}^{9} \frac{6 \log _{10}(x+1)}{x+1} d x$ .

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Problem 32842

Find the derivative of $y=\ln (\ln (\ln x))$ .

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Problem 32843

Find the linearization of $y=f(x)=\frac{1}{8x}$ at $x=8$ . Provide exact coefficients. $L(x)=$

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Problem 32844

How fast is Butch's distance from the origin changing at $(8,15)$ if $\frac{d x}{d t}=-2$ m/sec and $\frac{d y}{d t}=-3$ m/sec?

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Problem 32845

Find the derivative of the function $f(x)=\frac{x^{3}+2}{x^{2}}$ . What is $f^{\prime}(x)$ ?

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Problem 32846

Find the tangent line equation for $y=2 x^{3}+3 x$ at $x=1$ using the derivative definition.

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Problem 32847

Evaluate the integral $\int_{1}^{\infty} \frac{d x}{\sqrt{x^{2}+3}}$ and determine its convergence or divergence.

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Problem 32848

Find the average position $\langle x \rangle$ of an electron in a 1-D box using $\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$ .

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Problem 32849

Finde die Punkte, wo die Tangente der Funktion $f$ parallel zur Geraden $y=2x-3$ ist. a) $f(x)=4x^{3}-x$ b) $f(x)=\frac{1}{3}x^{3}+\frac{1}{2}x^{2}-10x$

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Problem 32850

How fast is Butch's distance from the origin changing at $(12, 35)$ if $\frac{d x}{d t}=-2 \mathrm{~m/s}$ and $\frac{d y}{d t}=-3 \mathrm{~m/s}$ ?

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Problem 32851

Find the linearization of $f(x)=\frac{1}{8x}$ at $x=8$ : $L(x)=\frac{1}{32}-\frac{x}{512}$ . What is the area of the triangle formed?

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Problem 32852

Find the derivative of the function $f(x)=\frac{x^{3}+2}{x^{2}}$ . What is $f^{\prime}(x)=$ ?

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Problem 32853

Approximate $\int_{0}^{\pi} 5 \sin x \, dx$ using 4 equal intervals.

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Problem 32854

Solve the initial value problem: $\frac{dy}{dt}=3 e^{3t} \sin(e^{3t}-27)$ , with $y(\ln 3)=0$ .

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Problem 32855

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{6 x}{\ln \left(250 x+e^{x}\right)}$ .

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Problem 32856

How fast is Butch's distance from the origin changing at $(7,24)$ if $\frac{d x}{d t}=-5$ m/s and $\frac{d y}{d t}=-6$ m/s?

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Problem 32857

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $\sqrt[3]{x^{2}}+\sqrt[3]{y^{2}}=1$ .

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Problem 32858

Snødybde i fjellområde:
1. Tegn grafen til $s(x)=-0.017 x^{3}+2.0 x$ for $0 \leq x \leq 11$ .
2. Finn $s(2)$ , maksimum dybde, når $s(x)=3$ , og gjennomsnittlig endring fra kl. 06:00 til 10:00.
3. Beregn momentan vekst for kl. 03:00 og 09:00.
Bruk GeoGebra og lever inn som Word-dokument innen 24.1 kl. 22:00.

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Problem 32859

Show that for all $x>0$ , the inequalities $1-\frac{x^{2}}{2} \leq \cos(x)$ and $x-\frac{x^{3}}{6} \leq \sin(x)$ hold. Also, find max and min of $f(x)=x \arcsin(x)+\sqrt{1-x^{2}}$ on $[-1,1]$ .

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Problem 32860

Finde die Extrempunkte der Funktion $f(x)=x \cdot e^{-x}$ .

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Problem 32861

Finde die Extrempunkte der Funktionen: a) $f(x)=2 x-e^{x}$ , b) $f(x)=x \cdot e^{x}$ , c) $f(x)=x \cdot e^{-x}$ , d) $f(x)=x^{2} \cdot e^{2 x}$ , e) $f(x)=e^{3 x}-6 x$ , f) $f(x)=(x^{2}-3) \cdot e^{x}$ .

See Solution

Problem 32862

Bestimmen Sie die Steigung von $f$ an $x_{0}$ für die Funktionen a) bis j) mit den angegebenen $x_{0}$ -Werten.

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Problem 32863

Berechne die Fläche zwischen $G_f = \frac{1}{27}x^3 - \frac{4}{3}x$ und $t = \frac{8}{3}x - 16$ im Intervall $[-12, 6]$ . Wie viele neue Graphen entstehen durch drei Transformationen?

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Problem 32864

Find the derivative of the function $F(t)=t(\ln t)-t$ . What is $F^{\prime}(t)$ ? $F^{\prime}(t)=$

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Problem 32865

Bestimme die Steigung von $f$ an $x_{0}$ für: a) $f(x)=3 \cdot \sin (x)$ bei $x_{0}=0$ ; b) $f(x)=2-\cos (x)$ bei $x_{0}=\frac{\pi}{4}$ .

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Problem 32866

Evaluate the integral $\int_{2}^{9} \ln (t) \, dt = \square$ using the Fundamental Theorem of Calculus.

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Problem 32867

Find the average value $f_{\text{ave}}$ of $f(x)=14-|2x|$ from -7 to 7 and points $c$ where $f(c)=f_{\text{ave}}$ .

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Problem 32868

Evaluate the integral $\int_{4}^{7}\left(\frac{d}{d t} \sqrt{4+2 t^{4}}\right) d t$ using the Fundamental Theorem of Calculus.

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Problem 32869

Evaluate the integral $\int_{-5}^{5} \frac{1}{x^{3}} d x$ using the Fundamental Theorem of Calculus. If it doesn't exist, answer "DNE".

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Problem 32870

Bestimmen Sie die Menge an Plastikmüll $p(t)=0,2 t e^{-0,0625 t}+5$ in Mio. Tonnen und prüfen Sie $p(0)$ , $p(10)-p(5)$ und Ableitungen.

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Problem 32871

Find $f^{\prime}(e)$ if $f(x)=x^{\ln x}$ . Options: e, 2, 0, 1.

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Problem 32872

Determine if the series $\sum_{n=1} a_{n}$ with $a_{1}=4$ and $a_{n+1}=\sqrt[n]{n} a_{n}$ converges or diverges.

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Problem 32873

Sketch the area between $y=\frac{x^{3}}{9}$ and the x-axis over $[-4,-3]$ . Find the area. $\text { Area }=$

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Problem 32874

Find the speed and acceleration of a bug on a disk with radius $0.1 \mathrm{~m}$ spinning at $10 \mathrm{rad/sec}$ .

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Problem 32875

Find $f^{\prime}(e)$ for the function $f(x)=x^{\ln x}$ . Options: 2, 0, $e$ , 1.

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Problem 32876

Find the average angular speed of Earth around the sun in $\mathrm{rad} / \mathrm{s}$ , given it orbits in 365.25 days.

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Problem 32877

Evaluate the integral from -7 to 2: $\int_{-7}^{2}(3 x+8) d x$ . Use area formulas for help.

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Problem 32878

Evaluate the integral $\int_{-\pi}^{\pi} f(x) dx$ where $f(x) = \begin{cases} -4x & \text{if } -\pi \leq x \leq 0 \\ \sin(x) & \text{if } 0 < x \leq \pi \end{cases}$ . If it doesn't exist, write "DNE".

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Problem 32879

Find $f(7)$ given $f(1)=12$ , $f^{\prime}$ is continuous, and $\int_{1}^{7} f^{\prime}(t) dt=30$ .

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Problem 32880

Find the function $f(x)$ where $f'(x)=\frac{\tan (x)}{x}$ and $f(1)=3$ .

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Problem 32881

Find the minimum power needed for an escalator moving 20 people ( $60 \mathrm{~kg}$ each) per minute, $5 \mathrm{~m}$ high.

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Problem 32882

Berechnen Sie die folgenden Integrale: a) $\int_{0}^{2}(3 x^{2}-e^{x}) d x$ b) $\int_{1}^{4}(\sqrt{x}-1) d x$ c) $\int_{1}^{e}(\frac{1}{x}+\frac{1}{x^{2}}) d x$ d) $\int_{0}^{\pi}(\sin (x)+\cos (x)) d x$ e) $\int_{0}^{\pi} \sin (3 x-\pi) d x$ f) $\int_{-1}^{1} \frac{1}{5} e^{\frac{1}{2} x} d x$ g) $\int_{3}^{4} \frac{1}{2(x+1)} d x$ h) $\int_{1}^{4} \frac{3}{2 x-1} d x$

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Problem 32883

Berechnen Sie die folgenden Integrale: a) $\int_{0}^{2}(3 x^{2}-e^{x}) d x$ b) $\int_{1}^{4}(\sqrt{x}-1) d x$ c) $\int_{1}^{e}(\frac{1}{x}+\frac{1}{x^{2}}) d x$ d) $\int_{0}^{\pi}(\sin (x)+\cos (x)) d x$

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Problem 32884

Bestimmen Sie die Stammfunktion $\mathrm{F}$ von $\mathrm{f}$ mit $\mathrm{F}(0)=1$ für die Funktionen a) $f(x)=(x+2)^{2}$ , b) $f(x)=\frac{1}{x+1}$ , c) $f(t)=2 e^{0,5 t}$ , d) $f(t)=\cos (5 t)$ .

See Solution

Problem 32885

Determine the inflection points of the function $f(x)=3x^{3}+3x^{2}-36x$ .

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Problem 32886

حدد الفترات التي تكون فيها الدالة $f(x) = x \ln x - x$ متزايدة لـ $x > 0$ .

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Problem 32887

Berechne die Ableitung von $g(x) = 104 \cdot e^{0,22x}$ am Punkt $P$ .

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Problem 32888

Find $F^{\prime}(x)$ for $F(x)=\int_{x}^{8}-2 \cos(t^{2}) dt$ using the Fundamental Theorem of Calculus. $F^{\prime}(x)=$

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Problem 32889

Find the absolute minimum of the function $f(x)=x^{3}-3 x^{2}+1$ on the interval $[-1,1]$ .

See Solution

Problem 32890

Find $f^{\prime}(e)$ if $f(x)=x^{\ln x}$ . Choices: 2, 0, e, 1.

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Problem 32891

e) $\int_{0}^{1} \sin(3x - \pi) \, dx$ f) $\int_{-1}^{1} \frac{1}{5} e^{\frac{1}{2} x} \, dx$ g) $\int_{3}^{4} \frac{1}{2(x+1)} \, dx$ h) $\int_{1}^{4} \frac{3}{2x - 1} \, dx$ 5 Find an integral-free term for the integral function $J_{-1}$ of $f$ . a) $f(x)=(5x+3)^{3}$ b) $f(x)=2 e^{2x+2}$ c) $f(x)=\frac{1}{3+2x}, x>-\frac{2}{3}$ d) $f(x)=3 \cdot \cos(\pi x)$ 6 Correct the errors in finding an antiderivative.

See Solution

Problem 32892

Find the derivative of the function $\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$ .

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Problem 32893

Calculate the integral from 3 to 9 of $\frac{1}{x(\ln x)^{2}}$ dx.

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Problem 32894

Evaluate the integral $\int_{\pi / 3}^{\pi / 6} \frac{\cos x - x \sin x}{x \cos x} \, dx$ .

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Problem 32895

Find the intervals where the function $f(x)=\frac{x^{2}}{4 x+4}$ is concave up and down.

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Problem 32896

Find the derivative of $y=3^{x}$ . Options: $y'=\frac{3^{x}}{\ln 3}$ , $y'=3^{x} \ln x$ , None, $y'=3^{x} \ln$ , $y'=3^{-x} \ln 3$ .

See Solution

Problem 32897

Find the slope function $S(x)$ for these functions: 44. $f(x)=3$ , 45. $f(x)=2x+1$ , 46. $f(x)=|x|$ .

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Problem 32898

Find the normal boiling point of chloroform where $\Delta \mathrm{G}=0$ , given $\Delta \mathrm{H}=31.4 \mathrm{~kJ/mol}$ and $S=93.7 \mathrm{~J/K \cdot mol}$ . Options: 335 K, 567 K, 95.57 K, 126 K.

See Solution

Problem 32899

Find the sum of the series $\sum_{n=1}^{\infty}\left(\frac{9}{n^{2}}-\frac{9}{(n+1)^{2}}\right)$ . Options: a. 9 b. 0 c. Diverges d. 3

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Problem 32900

Find the sum: $\sum_{n=1}^{\infty} \frac{-1-(-2)^{n}}{3^{n}}$ . Choose from: a. 0.01, b. 1.1, c. 0.1, d. -0.1

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Calculus

Problem 32801

How long does it take for a penny to fall from a height of 93 meters? Options: 176.2 s176.2 \mathrm{~s}176.2 s, 18.97 s18.97 \mathrm{~s}18.97 s, 4.4 s4.4 \mathrm{~s}4.4 s, 9.5 s9.5 \mathrm{~s}9.5 s.

Problem 32802

Find the derivative of the function f(x)=tan⁡(x)(6sin⁡(x)+7cos⁡(x))f(x)=\tan (x)(6 \sin (x)+7 \cos (x))f(x)=tan(x)(6sin(x)+7cos(x)): f′(x)=f^{\prime}(x)=f′(x)=

Problem 32803

If lim⁡x→−4[f(x)+g(x)]=2\lim _{x \rightarrow-4}[f(x)+g(x)]=2limx→−4​[f(x)+g(x)]=2 and lim⁡x→−4g(x)=13\lim _{x \rightarrow-4} g(x)=13limx→−4​g(x)=13, find lim⁡x→−46g(x)f(x)−g(x)\lim _{x \rightarrow-4} \frac{6 g(x)}{f(x)-g(x)}limx→−4​f(x)−g(x)6g(x)​.

Problem 32804

Problem 32805

Find the values of xxx where the tangents to y=(1+x2)3y=(1+x^{2})^{3}y=(1+x2)3 and y=3x2y=3x^{2}y=3x2 have equal slopes.

Problem 32806

Find the tangent line equation for f(x)=x5x−6f(x)=\frac{\sqrt{x}}{5x-6}f(x)=5x−6x​​ at the point (3,f(3))(3, f(3))(3,f(3)). y=y=y=

Problem 32807

Find the second derivative f′′(π/4)f^{\prime \prime}(\pi / 4)f′′(π/4) for the function f(x)=6sec⁡xf(x)=6 \sec xf(x)=6secx.

Problem 32808

Bestimmen Sie u(x)u(x)u(x) und v(x)v(x)v(x) und leiten Sie aba bab für die Funktionen: a) f(x)=(2x3−x)exf(x)=(2 x^{3}-x) e^{x}f(x)=(2x3−x)ex, b) f(x)=(x2−1)(2x2+5)f(x)=(x^{2}-1)(2 x^{2}+5)f(x)=(x2−1)(2x2+5), c) f(x)=(5x2−4)exf(x)=(5 x^{2}-4) e^{x}f(x)=(5x2−4)ex.

Problem 32809

Find the average rate of change of h(x)=−x2+8x+20h(x)=-x^{2}+8x+20h(x)=−x2+8x+20 on the interval 0≤x≤50 \leq x \leq 50≤x≤5.

Problem 32810

Find the derivative of f(x)=12x(sin⁡(x)+cos⁡(x))f(x)=12 x(\sin (x)+\cos (x))f(x)=12x(sin(x)+cos(x)) and evaluate f′(−π6)f^{\prime}\left(-\frac{\pi}{6}\right)f′(−6π​).

Problem 32811

Find the tangent line equation to the curve y=3xcos⁡xy=3 x \cos xy=3xcosx at the point (π,−3π)(\pi, -3 \pi)(π,−3π) in the form y=mx+by=m x+by=mx+b.

Problem 32812

Prove that f(x)=xf(x)=\sqrt{x}f(x)=x​ is uniformly continuous but not Lipschitz continuous. Also, show that Lipschitz functions compose to be Lipschitz.

Problem 32813

Max untersucht die Ableitung von f(x)=x6f(x)=x^{6}f(x)=x6 als Produkt. Vervollständigen Sie die Tabelle und vermuten Sie die Ableitungsformel.

Problem 32814

Find the tangent line to y=3xcos⁡xy=3 x \cos xy=3xcosx at (π,−3π)(\pi,-3 \pi)(π,−3π) in the form y=mx+by=m x+by=mx+b, with m=m=m= and b=b=b=.

Problem 32815

Differentiate the function f(x)=4e−2xf(x)=4 e^{-2 x}f(x)=4e−2x using the chain rule to find f′(x)f'(x)f′(x).

Problem 32816

A bullet is shot straight up at 140 m/s. How long will it stay in the air? Use t=2vgt = \frac{2v}{g}t=g2v​, where g≈9.8 m/s2g \approx 9.8 \mathrm{~m/s^2}g≈9.8 m/s2.

Problem 32817

Lara hat die Ableitung von f(x)=(2−x)3f(x)=(2-x)^{3}f(x)=(2−x)3 als f′(x)=3(2−x)2f'(x)=3(2-x)^{2}f′(x)=3(2−x)2 gefunden. Erklären Sie, überprüfen Sie und vermuten Sie eine Ableitungsformel.

Problem 32818

Determine the convergence of ∑n=1∞n+1n2+1\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}∑n=1∞​n2+1n+1​​ using comparison tests.

Problem 32819

Determine if the series ∑n=1∞(1n2+1)2\sum_{n=1}^{\infty}\left(\frac{1}{n^{2}}+1\right)^{2}n=1∑∞​(n21​+1)2 converges or diverges.

Problem 32820

Problem 32821

Calculate ⟨x⟩=∫0Lx∣ψn(x)∣2 dx\langle x \rangle = \int_{0}^{L} x \left|\psi_n(x)\right|^2 \,dx⟨x⟩=∫0L​x∣ψn​(x)∣2dx for ψn(x)=2Lsin⁡(nπxL)\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)ψn​(x)=L2​​sin(Lnπx​).

Problem 32822

Find the limit as xxx approaches 0 for sin⁡(9x)x\frac{\sin(9x)}{x}xsin(9x)​ when xxx is in degrees. Answer to 0.01 accuracy.

Problem 32823

Finde die Gleichung der Normalen an den Graphen von fff in den Punkten A für die Funktionen a) bis d).

Problem 32824

Find critical values of f(x)=(x−1)3f(x)=(x-1)^{3}f(x)=(x−1)3 on [0,4][0,4][0,4] and determine if they are max, min, or neither. Analyze increasing/decreasing intervals.

Problem 32825

Find the integral of sin⁡(ln⁡(x))x\frac{\sin(\ln(x))}{x}xsin(ln(x))​ with respect to xxx.

Problem 32826

Find f′′(2)f^{\prime \prime}(2)f′′(2) for the function f(x)=x+16xf(x)=x+\frac{16}{x}f(x)=x+x16​. Options: a. 4, b. 9, c. -3, d. -4.

Problem 32827

-4

Problem 32828

Calculate the limit: lim⁡x→0+tg⁡(x)ln⁡(x)\lim _{x \rightarrow 0^{+}} \operatorname{tg}(x) \ln (x)limx→0+​tg(x)ln(x).

Problem 32829

Find the limit as x approaches 0: lim⁡x→0x+4\lim _{x \rightarrow 0} \sqrt{x+4}limx→0​x+4​.

Problem 32830

Find the tangent line equation for f(x)=x5x−6f(x)=\frac{\sqrt{x}}{5 x-6}f(x)=5x−6x​​ at the point (3,f(3))(3, f(3))(3,f(3)). y=y=y=

Problem 32831

Find the limit: lim⁡x→0x+6−6x\lim _{x \rightarrow 0} \frac{\sqrt{x+6}-\sqrt{6}}{x}x→0lim​xx+6​−6​​.

Problem 32832

Find the max and min of f(x)=xarcsin⁡(x)+1−x2f(x)=x \arcsin (x)+\sqrt{1-x^{2}}f(x)=xarcsin(x)+1−x2​ on the interval [−1,1][-1,1][−1,1].

Problem 32833

Evaluate the limit: lim⁡t→0sin⁡9t8t\lim _{t \rightarrow 0} \frac{\sin 9 t}{8 t}limt→0​8tsin9t​.

Problem 32834

Finde die Ableitung von f(x)=exf(x) = e^{\sqrt{x}}f(x)=ex​.

Problem 32835

Solve for zzz in the equation ∫z102xdx=19\int_{z}^{10} 2 x d x=19∫z10​2xdx=19.

Problem 32836

A boat is 3 feet from the dock, and the rope is pulled at 2 ft/s. How fast is the distance to the dock decreasing?

Problem 32837

Find the tangent line equation for f(x)=x5x−6f(x)=\frac{\sqrt{x}}{5 x-6}f(x)=5x−6x​​ at the point (3,f(3))(3, f(3))(3,f(3)). y=y=y=

Problem 32838

Calculate the integral ∫1∞2x2dx\int_{1}^{\infty} \frac{2}{x^{2}} d x∫1∞​x22​dx for each value of ttt.

Problem 32839

Calculate the integral: ∫02z0.4 dx=8\int_{0}^{2 z} 0.4 \, dx = 8∫02z​0.4dx=8.

Problem 32840

A particle's velocity is v(t)=1+2sin⁡(t22)v(t)=1+2 \sin \left(\frac{t^{2}}{2}\right)v(t)=1+2sin(2t2​) and position x=2x=2x=2 at t=4t=4t=4. (a) Is it speeding up or slowing down at t=4t=4t=4? (b) Find times 0<t<30<t<30<t<3 when the particle changes direction.

Problem 32841

How long does it take for a penny to fall from a height of 93 meters? Options: $176.2 \mathrm{~s}$ , $18.97 \mathrm{~s}$ , $4.4 \mathrm{~s}$ , $9.5 \mathrm{~s}$ .

Find the derivative of the function $f(x)=\tan (x)(6 \sin (x)+7 \cos (x))$ : $f^{\prime}(x)=$

If $\lim _{x \rightarrow-4}[f(x)+g(x)]=2$ and $\lim _{x \rightarrow-4} g(x)=13$ , find $\lim _{x \rightarrow-4} \frac{6 g(x)}{f(x)-g(x)}$ .

Find the values of $x$ where the tangents to $y=(1+x^{2})^{3}$ and $y=3x^{2}$ have equal slopes.

Find the tangent line equation for $f(x)=\frac{\sqrt{x}}{5x-6}$ at the point $(3, f(3))$ . $y=$

Find the second derivative $f^{\prime \prime}(\pi / 4)$ for the function $f(x)=6 \sec x$ .

Bestimmen Sie $u(x)$ und $v(x)$ und leiten Sie $a b$ für die Funktionen: a) $f(x)=(2 x^{3}-x) e^{x}$ , b) $f(x)=(x^{2}-1)(2 x^{2}+5)$ , c) $f(x)=(5 x^{2}-4) e^{x}$ .

Find the average rate of change of $h(x)=-x^{2}+8x+20$ on the interval $0 \leq x \leq 5$ .

Find the derivative of $f(x)=12 x(\sin (x)+\cos (x))$ and evaluate $f^{\prime}\left(-\frac{\pi}{6}\right)$ .

Find the tangent line equation to the curve $y=3 x \cos x$ at the point $(\pi, -3 \pi)$ in the form $y=m x+b$ .

Prove that $f(x)=\sqrt{x}$ is uniformly continuous but not Lipschitz continuous. Also, show that Lipschitz functions compose to be Lipschitz.

Max untersucht die Ableitung von $f(x)=x^{6}$ als Produkt. Vervollständigen Sie die Tabelle und vermuten Sie die Ableitungsformel.

Find the tangent line to $y=3 x \cos x$ at $(\pi,-3 \pi)$ in the form $y=m x+b$ , with $m=$ and $b=$ .

Differentiate the function $f(x)=4 e^{-2 x}$ using the chain rule to find $f'(x)$ .

A bullet is shot straight up at 140 m/s. How long will it stay in the air? Use $t = \frac{2v}{g}$ , where $g \approx 9.8 \mathrm{~m/s^2}$ .

Lara hat die Ableitung von $f(x)=(2-x)^{3}$ als $f'(x)=3(2-x)^{2}$ gefunden. Erklären Sie, überprüfen Sie und vermuten Sie eine Ableitungsformel.

Determine the convergence of $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}$ using comparison tests.

Determine if the series $\sum_{n=1}^{\infty}\left(\frac{1}{n^{2}}+1\right)^{2}$ converges or diverges.

Calculate $\langle x \rangle = \int_{0}^{L} x \left|\psi_n(x)\right|^2 \,dx$ for $\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$ .

Find the limit as $x$ approaches 0 for $\frac{\sin(9x)}{x}$ when $x$ is in degrees. Answer to 0.01 accuracy.

Finde die Gleichung der Normalen an den Graphen von $f$ in den Punkten A für die Funktionen a) bis d).

Find critical values of $f(x)=(x-1)^{3}$ on $[0,4]$ and determine if they are max, min, or neither. Analyze increasing/decreasing intervals.

Find the integral of $\frac{\sin(\ln(x))}{x}$ with respect to $x$ .

Find $f^{\prime \prime}(2)$ for the function $f(x)=x+\frac{16}{x}$ . Options: a. 4, b. 9, c. -3, d. -4.

Calculate the limit: $\lim _{x \rightarrow 0^{+}} \operatorname{tg}(x) \ln (x)$ .

Find the limit as x approaches 0: $\lim _{x \rightarrow 0} \sqrt{x+4}$ .

Find the tangent line equation for $f(x)=\frac{\sqrt{x}}{5 x-6}$ at the point $(3, f(3))$ . $y=$

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{x+6}-\sqrt{6}}{x}$ .

Find the max and min of $f(x)=x \arcsin (x)+\sqrt{1-x^{2}}$ on the interval $[-1,1]$ .

Evaluate the limit: $\lim _{t \rightarrow 0} \frac{\sin 9 t}{8 t}$ .

Finde die Ableitung von $f(x) = e^{\sqrt{x}}$ .

Solve for $z$ in the equation $\int_{z}^{10} 2 x d x=19$ .

Find the tangent line equation for $f(x)=\frac{\sqrt{x}}{5 x-6}$ at the point $(3, f(3))$ . $y=$

Calculate the integral $\int_{1}^{\infty} \frac{2}{x^{2}} d x$ for each value of $t$ .

Calculate the integral: $\int_{0}^{2 z} 0.4 \, dx = 8$ .

A particle's velocity is $v(t)=1+2 \sin \left(\frac{t^{2}}{2}\right)$ and position $x=2$ at $t=4$ .
(a) Is it speeding up or slowing down at $t=4$ ?
(b) Find times $0<t<3$ when the particle changes direction.

Evaluate the integral: $\int_{0}^{9} \frac{6 \log _{10}(x+1)}{x+1} d x$ .

Find the derivative of $y=\ln (\ln (\ln x))$ .

Find the linearization of $y=f(x)=\frac{1}{8x}$ at $x=8$ . Provide exact coefficients. $L(x)=$

How fast is Butch's distance from the origin changing at $(8,15)$ if $\frac{d x}{d t}=-2$ m/sec and $\frac{d y}{d t}=-3$ m/sec?

Find the derivative of the function $f(x)=\frac{x^{3}+2}{x^{2}}$ . What is $f^{\prime}(x)$ ?

Find the tangent line equation for $y=2 x^{3}+3 x$ at $x=1$ using the derivative definition.

Evaluate the integral $\int_{1}^{\infty} \frac{d x}{\sqrt{x^{2}+3}}$ and determine its convergence or divergence.

Find the average position $\langle x \rangle$ of an electron in a 1-D box using $\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$ .

Finde die Punkte, wo die Tangente der Funktion $f$ parallel zur Geraden $y=2x-3$ ist. a) $f(x)=4x^{3}-x$ b) $f(x)=\frac{1}{3}x^{3}+\frac{1}{2}x^{2}-10x$

How fast is Butch's distance from the origin changing at $(12, 35)$ if $\frac{d x}{d t}=-2 \mathrm{~m/s}$ and $\frac{d y}{d t}=-3 \mathrm{~m/s}$ ?

Find the linearization of $f(x)=\frac{1}{8x}$ at $x=8$ : $L(x)=\frac{1}{32}-\frac{x}{512}$ . What is the area of the triangle formed?

Find the derivative of the function $f(x)=\frac{x^{3}+2}{x^{2}}$ . What is $f^{\prime}(x)=$ ?

Approximate $\int_{0}^{\pi} 5 \sin x \, dx$ using 4 equal intervals.

Solve the initial value problem: $\frac{dy}{dt}=3 e^{3t} \sin(e^{3t}-27)$ , with $y(\ln 3)=0$ .

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{6 x}{\ln \left(250 x+e^{x}\right)}$ .

How fast is Butch's distance from the origin changing at $(7,24)$ if $\frac{d x}{d t}=-5$ m/s and $\frac{d y}{d t}=-6$ m/s?

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $\sqrt[3]{x^{2}}+\sqrt[3]{y^{2}}=1$ .

Finde die Extrempunkte der Funktion $f(x)=x \cdot e^{-x}$ .

Bestimmen Sie die Steigung von $f$ an $x_{0}$ für die Funktionen a) bis j) mit den angegebenen $x_{0}$ -Werten.

Berechne die Fläche zwischen $G_f = \frac{1}{27}x^3 - \frac{4}{3}x$ und $t = \frac{8}{3}x - 16$ im Intervall $[-12, 6]$ . Wie viele neue Graphen entstehen durch drei Transformationen?

Find the derivative of the function $F(t)=t(\ln t)-t$ . What is $F^{\prime}(t)$ ? $F^{\prime}(t)=$

Bestimme die Steigung von $f$ an $x_{0}$ für: a) $f(x)=3 \cdot \sin (x)$ bei $x_{0}=0$ ; b) $f(x)=2-\cos (x)$ bei $x_{0}=\frac{\pi}{4}$ .

Evaluate the integral $\int_{2}^{9} \ln (t) \, dt = \square$ using the Fundamental Theorem of Calculus.

Find the average value $f_{\text{ave}}$ of $f(x)=14-|2x|$ from -7 to 7 and points $c$ where $f(c)=f_{\text{ave}}$ .

Evaluate the integral $\int_{4}^{7}\left(\frac{d}{d t} \sqrt{4+2 t^{4}}\right) d t$ using the Fundamental Theorem of Calculus.

Evaluate the integral $\int_{-5}^{5} \frac{1}{x^{3}} d x$ using the Fundamental Theorem of Calculus. If it doesn't exist, answer "DNE".

Bestimmen Sie die Menge an Plastikmüll $p(t)=0,2 t e^{-0,0625 t}+5$ in Mio. Tonnen und prüfen Sie $p(0)$ , $p(10)-p(5)$ und Ableitungen.

Find $f^{\prime}(e)$ if $f(x)=x^{\ln x}$ . Options: e, 2, 0, 1.

Determine if the series $\sum_{n=1} a_{n}$ with $a_{1}=4$ and $a_{n+1}=\sqrt[n]{n} a_{n}$ converges or diverges.

Sketch the area between $y=\frac{x^{3}}{9}$ and the x-axis over $[-4,-3]$ . Find the area. $\text { Area }=$

Find the speed and acceleration of a bug on a disk with radius $0.1 \mathrm{~m}$ spinning at $10 \mathrm{rad/sec}$ .