Calculus

Problem 18001

Bestimmen Sie eine Stammfunktion für:
a) $f(x)=\frac{1}{2 x^{3}}+\cos (0,5 x)$
b) $f(x)=2(5 x+1)^{7}+\sqrt{x}$

See Solution

Problem 18002

Bestimme die Steigung von $f(x)=10 x^{3}-9 x^{2}+2 x$ bei $x=5$ .

See Solution

Problem 18003

Untersuchen Sie die Funktion $f(x)=\frac{2 x+1}{x}$ :
1. Skizzieren Sie den Graphen für $0 \leq x \leq 12$ .
2. Vermuten Sie das Verhalten für große $x$ .
3. Berechnen Sie $f(1), f(10), f(100), f(1000)$ .

See Solution

Problem 18004

Gegeben ist die Funktion $f_{a}(x)=-x^{3}+a^{2} \cdot x, a>0$ . Finde Nullstellen, Extremstellen und Wendestellen.

See Solution

Problem 18005

Find the time intervals where the particle's speed is decreasing based on the velocity graph $v(t)$ for $0 \leq t \leq m$ .

See Solution

Problem 18006

Berechnen Sie die Integrale: $\int_{2}^{4} 4 e^{2 x} d x$ und $\int_{2}^{4} 5 x^{4} d x - \int_{2}^{4} 3 x^{2} d x + \int_{4}^{2} 5 x^{4} d x$ .

See Solution

Problem 18007

Find $t$ in $1 \leq t \leq 16$ where instantaneous velocity equals average velocity for $p(t)=\sqrt{t}-2$ . Choices: (A) 1, (B) $\frac{121}{25}$ , (C) $\frac{25}{4}$ , (D) 25.

See Solution

Problem 18008

Eine Familie nutzt eine Solaranlage. Vergleiche die Funktionen $f(t)$ und $g(t)$ und berechne den solaren Deckungsgrad.

See Solution

Problem 18009

Find when the particle's speed decreases given its velocity graph intersects at $t=0, k, m$ and has tangents at $t=j, l$ .

See Solution

Problem 18010

Gegeben ist die Funktion $f(x)=\frac{1}{8} x^{3}-\frac{1}{2} x$ .
1. Berechne die Fläche zwischen $f$ und der $x$ -Achse im Intervall $[-1, 2]$ .
2. Finde die Fläche zwischen $f$ und $g(x)=-\frac{1}{2} x^{2}+x$ .
3. Skizziere die Tangente $t$ an $f$ bei $P(2,5/0,7$ und erkläre die Flächenberechnung zwischen $f$ , $t$ und der $x$ -Achse.

See Solution

Problem 18011

Aufgabe 1: Gegeben ist die Funktionsschar $f_{a}(x)=-x^{3}+a^{2} \cdot x$ . Bestimme Nullstellen, Extremstellen und Wendestellen.
Aufgabe 2: Für $f_{a}(x)=-a x^{3}+4 a x$ zeige Symmetrie, Punkte $P(-2|0)$ und $Q(2|0)$ , Hoch- und Trefpunkt und Wendetangente.
Aufgabe 3: Beurteile Aussagen zu $f_{a}(x)=a x^{2}+a x+4$ bezüglich Wendestellen, Hochpunkten, y-Achsen-Schnitt, x-Achsen-Schnitten, Parabelöffnung, Steigungen und Punkt $P(10|0)$ .

See Solution

Problem 18012

Find the acceleration of the particle at the first point where the velocity $v(t)=\frac{dx}{dt}=0$ for $x(t)=\sin t-\cos t$ .

See Solution

Problem 18013

Beweisen Sie, dass $\int_{a}^{a} f(x) d x=0$ . Untersuchen Sie das Integral von $f(x)=e^{-\frac{1}{4} x}$ im Intervall $[1; \infty]$ .
Für $f(x)=\frac{1}{8} x^{3}-\frac{1}{2} x$ :
1. Berechnen Sie die Fläche im Intervall $[-1; 2]$ .
2. Bestimmen Sie die Fläche zwischen $f$ und $g(x)=-\frac{1}{2} x^{2}+x$ .
3. Skizzieren Sie die Tangente $t$ an $f$ bei $P(2, 5/0,7$ ) und erklären Sie, wie man die Fläche zwischen der $x$ -Achse, $f$ und $t$ berechnet.

See Solution

Problem 18014

Bestimmen Sie $p$ für $f(x)=3 x^{2}+p^{2}$ , sodass die Fläche im Intervall $[-1; 2]$ gleich 21 FE ist.

See Solution

Problem 18015

Find when the speed of a particle, with velocity $v(t)=-t^{3}+2 t^{2}+2^{-t}$ for $t \geq 0$ , is increasing.

See Solution

Problem 18016

A snail's velocity is $v(t)=1.4 \ln(1+t^{2})$ . Find its acceleration at $t=5$ minutes.

See Solution

Problem 18017

Find how many times the velocity, given by $y(t)=\frac{1}{6} \cos (5 t)-\frac{1}{4} \sin (5 t)$ , is 0 in 4 seconds.

See Solution

Problem 18018

A particle moves along the $x$ -axis with velocity $v(t)=\cos \left(\frac{\pi}{6} t\right)$ and starts at $x=-2$ .
(a) When is the particle moving left for $0 \leq t \leq 12$ ? (c) Find acceleration at time $t$ and determine if speed is increasing or decreasing at $t=4$ .

See Solution

Problem 18019

Bestimme die Stammfunktion $F$ für die Funktionen: a) $f(x)=5 x^{3}+7 x^{2}-4 x+9$ , b) $f(x)=\frac{1}{3} x^{6}+x$ , c) $f(x)=8 x^{2}-x+2$ , e) $f(x)=4 x^{3}+3 x+1$ .

See Solution

Problem 18020

Particle $X$ moves as $x(t)=5 t^{3}-9 t^{2}+7$ . Determine its direction at $t=1$ , when it's farthest left, and area change at $t=1$ for triangle with $Y$ .

See Solution

Problem 18021

Find the limit of the sequence $a_{n}=\frac{n^{2}+(-1)^{n} \cdot n}{(n+\sqrt{3})^{2}}$ as $n \to \infty$ .

See Solution

Problem 18022

Berechnen Sie den Grenzwert: $\lim _{n \rightarrow \infty}\left(\sqrt{4 n^{2}+11 n}-2 n\right)$ .

See Solution

Problem 18023

Skizzieren Sie die Flächen und berechnen Sie die Fläche A für die Funktionen: 44.1 $f(x)=x^2+1$ zwischen $x=1$ und $x=2$ ; 44.2 $f(x)=\frac{1}{2}x^2-8$ .

See Solution

Problem 18024

Find the limit: $\lim _{n \rightarrow \infty}\left(1+\frac{\pi}{2^{n+1}}\right)^{2^{n}}$ .

See Solution

Problem 18025

Evaluate the integral $\int_{-\infty}^{-5} \frac{-1}{(x-4)^{3}} dx$ or say if it diverges.

See Solution

Problem 18026

Evaluate the integral $I = \int_{0}^{\infty} 2 e^{-\frac{1}{3} x} dx$ or state if it diverges.

See Solution

Problem 18027

Ordnen Sie die Ableitungen A, B oder C den Funktionstermen 1-4 zu. $f(x)=x^{n} \cdot x^{n+1}, \, f(x)=\frac{x^{2 n}}{x}, \, f(x)=\left(x^{n}\right)^{2} \cdot x, \, f(x)=\left(\frac{\sqrt{x}}{x^{n}}\right)^{2}$

See Solution

Problem 18028

Evaluate the integral $\int_{e}^{\infty} \frac{-3}{2 x(\ln x)^{\frac{3}{2}}} dx$ or determine if it diverges.

See Solution

Problem 18029

Evaluate the integral $\int_{-\infty}^{-2} \frac{-4}{4x-9} \, dx$ or indicate if it diverges.

See Solution

Problem 18030

Evaluate the integral $\int_{e^{10}}^{\infty} \frac{-1}{2 x \ln x} d x$ or indicate if it diverges.

See Solution

Problem 18031

Berechne das Integral $\int_{-3}^{-1}\left(x^{3}+4 x^{2}+3 x\right) d x$ .

See Solution

Problem 18032

Hausaufgabe 13: Betrachten Sie die Differentialgleichung $y^{\prime}=-\frac{y^{3}}{x^{2}+1}$ .
(a) Finde konstante Lösungen. (b) Finde alle Lösungen. (c) Löse das Anfangswertproblem $y^{\prime}=-\frac{y^{3}}{x^{2}+1}, \quad y(1)=1$ .

See Solution

Problem 18033

Evaluate the integral $\int_{0}^{\infty} 2 e^{-\frac{1}{3} x} dx$ or state if it diverges.

See Solution

Problem 18034

Ein Getränk wird aus dem Kühlschrank (5°C) geholt. Raumtemperatur ist 22°C. Nutze $T(t)=22-17 \cdot e^{-0,02t}$ .
a) Zeichne $T(t)$ und finde, wann $T=12°C$ ist. b) Berechne, wann $T=12°C$ erreicht wird. c) Zeige, dass $T(t)$ die Form $S(1-c \cdot a^t)$ hat.

See Solution

Problem 18035

Finde die kleinste natürliche Zahl $n_{\varepsilon}$ , sodass $\left|a_{n}-\frac{9}{7}\right|<\frac{1}{1000}$ für alle $n>n_{\varepsilon}$ .

See Solution

Problem 18036

Find the time $t$ when the drug concentration $C(t)=\frac{0.6 t}{t^{2}+40}$ is maximized. Round your answer to 2 decimal places.

See Solution

Problem 18037

Find the first and second derivatives of $f(x)=(x^{2}-1) \cdot e^{-0.5 x}$ .

See Solution

Problem 18038

Find the average change in revenue for $R(q)=200 q-2 q^{2}$ when sales increase from 30 to 35 units.

See Solution

Problem 18039

Find the limit: $\lim _{n \rightarrow \infty}\left((\sin (\pi / 2))^{n}+(2 \sin (\pi / 4))^{n}+(1 / 10)^{n}\right)^{1 / n}$

See Solution

Problem 18040

Find the marginal revenue for the revenue function $R(q)=200 q-2 q^{2}$ when 30 units are sold.

See Solution

Problem 18041

Calculate the integral of $3 \sqrt{x}$ with respect to $x$ .

See Solution

Problem 18042

1. a) Find points where the slope of $y=-\sin x$ is i) zero ii) a local max iii) a local min b) Determine intervals for i) concave up ii) concave down

See Solution

Problem 18043

Temperatures in Michigan are given by $T(x)=-15 e^{-0.3857 x}+25$ .
a. Find the $y$ -intercept and its meaning. b. What was the temperature at noon? Round to the nearest tenth. c. Find $\lim_{x \rightarrow \infty} T(x)$ and its meaning. Round to the nearest tenth.

See Solution

Problem 18044

Berechne das Volumen des Körpers, der entsteht, wenn die Fläche zwischen $f(x)=3 \sqrt{x}$ und $g(x)=\sqrt{4-x}$ um die $x$ -Achse rotiert.

See Solution

Problem 18045

Find the area between the graph of $y=-2x+3$ over $[0,1]$ and the $x$ -axis using limits. Sketch the region.

See Solution

Problem 18046

Find the population in 13 years using the model $P(t)=197,000 e^{-0.0141t}$ . Round to the nearest person.

See Solution

Problem 18047

Find the derivative of $f(x) = 9 - \frac{1}{2}x$ using the limit definition: $f^{\prime}(x)=\lim _{\Delta x \rightarrow 0} \frac{9-\frac{1}{2}(x+\Delta x)-\left[9-\frac{1}{2}x\right]}{\Delta x}$ .

See Solution

Problem 18048

Find the derivative $f'(x)$ of $f(x) = \frac{4}{\sqrt{x}}$ using the limit definition.

See Solution

Problem 18049

Find the derivative of $y=\ln(x^{2}-4)$ .

See Solution

Problem 18050

Find the value of $\int_{2}^{5} 2|x-3| \, dx$ .

See Solution

Problem 18051

Use substitution to simplify the integral $\int(\sec x \tan x) \ln (\sec x+15) d x$ to $\int \ln \mathrm{u} \, du$ and evaluate.

See Solution

Problem 18052

Find the derivative of $y=e^{\sin x}$ with respect to $x$ .

See Solution

Problem 18053

Find the value of $\lim _{x \rightarrow-1^{-}} |x-1|$ .

See Solution

Problem 18054

Find the partial derivative $f_{x}(x, y)$ of $f(x, y)=4 x^{2} y^{2}-16 x^{2}+4 y$ .

See Solution

Problem 18055

What is the best choice for $\mathrm{dv}$ in $\int x^{n} e^{a x} \mathrm{dx}$ using integration by parts? A. $e^{a x} \mathrm{dx}$ B. $e^{a x}$ C. $x^{n} d x$ D. $x^{n}$

See Solution

Problem 18056

Evaluate the integral $\int \frac{\ln x}{x^{9}} d x = \square$ using integration by parts.

See Solution

Problem 18057

Evaluate the integral $\int_{0}^{\pi} 3 x \cos x \, dx$ using integration by parts. Which expression is simpler?

See Solution

Problem 18058

Find the increase in cost when producing 300 to 720 bikes using the marginal cost function $C^{\prime}(x)=200-\frac{x}{3}$ .

See Solution

Problem 18059

Evaluate the integral $\int_{0}^{\pi} 3 x \cos x \, dx$ using integration by parts. Choose the correct expression for the new integral.

See Solution

Problem 18060

a. Show that $\int r^{-1}(x) dx = \int y f^{\prime}(y) dy$ . What steps are needed to prove this?
b. Integrate $\int y^{\prime}(y) dy$ using $\int x f^{\prime}(x) dx = x f(x) - \int f(x) dx$ . Choose the correct answer.
c. Find $\int \ln x \, dx$ in terms of $x$ .
d. Find $\int \sin^{-1} x \, dx$ in terms of $x$ .
e. Find $\int \tan^{-1} x \, dx$ in terms of $x$ .

See Solution

Problem 18061

Maximize profit $P(x, y)=24 x-x^{2}-x y-2 y^{2}+33 y-43$ for goods $x$ and $y$ . What are the optimal values?

See Solution

Problem 18062

Find the partial derivative $f_{x}(x, y)$ for $f(x, y)=4 x^{2}+5 x^{2} y+12 x y^{2}$ and evaluate at $x=12$ , $y=4$ .

See Solution

Problem 18063

Is it true that to evaluate $\int \frac{7 x^{6}}{x^{4}+2 x^{2}} d x$ , you first need the partial fraction decomposition? A. No, numerator degree > denominator. B. No, denominator not factored. C. Yes, numerator degree > denominator. D. Yes, denominator can be factored.

See Solution

Problem 18064

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

See Solution

Problem 18065

Is it easier to evaluate $\int \frac{10 x+3}{5 x^{2}+3 x} d x$ using partial fractions? Choose A, B, C, or D.

See Solution

Problem 18066

Eva invests \$8000 at 4.7% annual interest, compounded continuously. Find the investment value after 8 years, rounded to the nearest cent.

See Solution

Problem 18067

Calculate the integral $\int_{0}^{6}\left(5 e^{0.5 x}+1\right) d x$ .

See Solution

Problem 18068

Calculate the integral from 0 to 2 of the function $6 e^{0.5 x} - 6 x$ .

See Solution

Problem 18069

Calculate the integral from 0 to 6 of the function $3 e^{-0.5 x} + 4$ .

See Solution

Problem 18070

Evaluate the integral $I=\int \frac{x^{2}}{x+8} d x$ using substitution $u=x+8$ and long division. Compare results.

See Solution

Problem 18071

Find the derivative of $y=x^{2} \ln x$ with respect to $x$ .

See Solution

Problem 18072

Find the average value of $f(x)=\frac{1}{2 x-11}$ from $x=3$ to $x=5$ , expressed as a constant times $\ln 5$ .

See Solution

Problem 18073

Evaluate the integral $\int \frac{2 x+5}{x^{2}+36} d x$ . Can partial fraction decomposition be used? A. Yes B. No. Find the integral's value.

See Solution

Problem 18074

Find the derivative of $y=\ln \sqrt[3]{x}$ with respect to $x$ .

See Solution

Problem 18075

Find the derivative of $y=\ln(x^{2}-4)$ with respect to $x$ .

See Solution

Problem 18076

Evaluate the integral: $\int \sin ^{2}\left(\theta-\frac{\pi}{5}\right) d \theta =$ (exact answer).

See Solution

Problem 18077

Calculate the integral: $\int_{6}^{10} \frac{2 d x}{2-x}$ .

See Solution

Problem 18078

Untersuche die Funktion $f(x)=\frac{1}{10000}(40 x^{3}-0,15 x^{4})$ für $x \in[0; 267]$ . Bestimme Nullstellen und Extrema.

See Solution

Problem 18079

Calculate $\int_{2}^{6} \frac{4}{x-10} dx$ and express the result as a constant multiplied by $\ln 2$ .

See Solution

Problem 18080

Find the first four nonzero terms of the Taylor series for $f(x)=\sqrt{1-2 x}$ around 0. $\sqrt{1-2 x}=\square+\square+\square+\square+\cdots$

See Solution

Problem 18081

Calculate the integral from 1 to 4 of $\frac{5 x+2}{\sqrt{x}}$ dx.

See Solution

Problem 18082

Find the integral of the function: $\int \frac{16}{x^{2}} d x$

See Solution

Problem 18083

Bestimme den mittleren Anstieg und den mittleren Anstiegswinkel der Skischanze mit $f(x)=\frac{1}{120} x^{2}-x+60$ für $0 \leq x \leq 30$ .

See Solution

Problem 18084

Évaluez les intégrales: a) $\int x \sqrt{1+4 x} d x$ b) $\int \frac{1+2 x}{1+16 x^{2}} d x$

See Solution

Problem 18085

Evaluate the integral $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{\cos x}{\sin ^{2} x} d x$ using a change of variables.

See Solution

Problem 18086

Evaluate the integral: $\int \sin^{2}\left(\theta-\frac{\pi}{5}\right) d\theta = \square$ (exact answer).

See Solution

Problem 18087

Bestimmen Sie das Krümmungsverhalten von $f(x)=\frac{1}{3} x^{3}-x^{2}-x+\frac{3}{2}$ durch die zweite Ableitung.

See Solution

Problem 18088

Berechne das Volumen des Rotationskörpers, der durch die Kurven $y=\frac{4}{x}$ und $y=5 x-x^{3}$ im ersten Quadranten um die $x$ -Achse entsteht.

See Solution

Problem 18089

Evaluate the integral $\int x^{2} e^{3 x} d x$ using integration by parts. Choose the correct answer from options A, B, C, D.

See Solution

Problem 18090

Find the limit: $\lim _{n \rightarrow \infty}\left(1+\frac{\pi}{\ln (n)}\right)^{\ln (3 n)}$ .

See Solution

Problem 18091

Evaluate the integral $\int \frac{16 x^{2}}{(x-24)(x+8)^{2}} d x$ and find its partial fraction decomposition.

See Solution

Problem 18092

Evaluate $\int_{3}^{5} \frac{4x^{2}-15x+11}{x-2} \, dx$ and simplify your answer, condensing logs if needed.

See Solution

Problem 18093

Find the projected world population in 2036 given a 2017 population of 7.5 billion and a growth rate of $1.11\%$ per year.

See Solution

Problem 18094

Calculate the integral: $\int\left(3^{\frac{t}{3}}+t^{2}\right) d t$

See Solution

Problem 18095

Find the projected world population in 2046, given a 2017 population of 7.5 billion and a growth rate of $1.11\%$ . Round to nearest ten millionth.

See Solution

Problem 18096

Calculate the sum: $\sum_{k=2}^{\infty} \frac{2^{k+3}}{3^{k}}$ .

See Solution

Problem 18097

Évaluez la limite suivante avec la règle de l'Hôpital si applicable: $\lim _{y \rightarrow \infty}\left(1+\frac{2}{y}\right)^{2 y}=\square$

See Solution

Problem 18098

Bestimmen Sie die Ableitung von $f(x)=2 x^{-2}-5$ .

See Solution

Problem 18099

Bestimmen Sie die Extremstellen von $f(x)=3 x \cdot e^{x}$ . Berechnen Sie $f'(x)$ , $f''(x)$ und finden Sie $x$ für $f'(x)=0$ .

See Solution

Problem 18100

Bestimmen Sie die Ableitung von $f(x) = x^{-1}$ oder $f(x) = 1/x$ .

See Solution

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Calculus

Problem 18001

Bestimmen Sie eine Stammfunktion für: a) f(x)=12x3+cos⁡(0,5x)f(x)=\frac{1}{2 x^{3}}+\cos (0,5 x)f(x)=2x31​+cos(0,5x) b) f(x)=2(5x+1)7+xf(x)=2(5 x+1)^{7}+\sqrt{x}f(x)=2(5x+1)7+x​

Problem 18002

Bestimme die Steigung von f(x)=10x3−9x2+2xf(x)=10 x^{3}-9 x^{2}+2 xf(x)=10x3−9x2+2x bei x=5x=5x=5.

Problem 18003

Untersuchen Sie die Funktion f(x)=2x+1xf(x)=\frac{2 x+1}{x}f(x)=x2x+1​: 1. Skizzieren Sie den Graphen für 0≤x≤120 \leq x \leq 120≤x≤12.2. Vermuten Sie das Verhalten für große xxx.3. Berechnen Sie f(1),f(10),f(100),f(1000)f(1), f(10), f(100), f(1000)f(1),f(10),f(100),f(1000).

Problem 18004

Gegeben ist die Funktion fa(x)=−x3+a2⋅x,a>0f_{a}(x)=-x^{3}+a^{2} \cdot x, a>0fa​(x)=−x3+a2⋅x,a>0. Finde Nullstellen, Extremstellen und Wendestellen.

Problem 18005

Find the time intervals where the particle's speed is decreasing based on the velocity graph v(t)v(t)v(t) for 0≤t≤m0 \leq t \leq m0≤t≤m.

Problem 18006

Berechnen Sie die Integrale: ∫244e2xdx\int_{2}^{4} 4 e^{2 x} d x∫24​4e2xdx und ∫245x4dx−∫243x2dx+∫425x4dx\int_{2}^{4} 5 x^{4} d x - \int_{2}^{4} 3 x^{2} d x + \int_{4}^{2} 5 x^{4} d x∫24​5x4dx−∫24​3x2dx+∫42​5x4dx.

Problem 18007

Find ttt in 1≤t≤161 \leq t \leq 161≤t≤16 where instantaneous velocity equals average velocity for p(t)=t−2p(t)=\sqrt{t}-2p(t)=t​−2. Choices: (A) 1, (B) 12125\frac{121}{25}25121​, (C) 254\frac{25}{4}425​, (D) 25.

Problem 18008

Eine Familie nutzt eine Solaranlage. Vergleiche die Funktionen f(t)f(t)f(t) und g(t)g(t)g(t) und berechne den solaren Deckungsgrad.

Problem 18009

Find when the particle's speed decreases given its velocity graph intersects at t=0,k,mt=0, k, mt=0,k,m and has tangents at t=j,lt=j, lt=j,l.

Problem 18010

Problem 18011

Problem 18012

Find the acceleration of the particle at the first point where the velocity v(t)=dxdt=0v(t)=\frac{dx}{dt}=0v(t)=dtdx​=0 for x(t)=sin⁡t−cos⁡tx(t)=\sin t-\cos tx(t)=sint−cost.

Problem 18013

Problem 18014

Bestimmen Sie ppp für f(x)=3x2+p2f(x)=3 x^{2}+p^{2}f(x)=3x2+p2, sodass die Fläche im Intervall [−1;2][-1; 2][−1;2] gleich 21 FE ist.

Problem 18015

Find when the speed of a particle, with velocity v(t)=−t3+2t2+2−tv(t)=-t^{3}+2 t^{2}+2^{-t}v(t)=−t3+2t2+2−t for t≥0t \geq 0t≥0, is increasing.

Problem 18016

A snail's velocity is v(t)=1.4ln⁡(1+t2)v(t)=1.4 \ln(1+t^{2})v(t)=1.4ln(1+t2). Find its acceleration at t=5t=5t=5 minutes.

Problem 18017

Find how many times the velocity, given by y(t)=16cos⁡(5t)−14sin⁡(5t)y(t)=\frac{1}{6} \cos (5 t)-\frac{1}{4} \sin (5 t)y(t)=61​cos(5t)−41​sin(5t), is 0 in 4 seconds.

Problem 18018

Problem 18019

Bestimme die Stammfunktion FFF für die Funktionen: a) f(x)=5x3+7x2−4x+9f(x)=5 x^{3}+7 x^{2}-4 x+9f(x)=5x3+7x2−4x+9, b) f(x)=13x6+xf(x)=\frac{1}{3} x^{6}+xf(x)=31​x6+x, c) f(x)=8x2−x+2f(x)=8 x^{2}-x+2f(x)=8x2−x+2, e) f(x)=4x3+3x+1f(x)=4 x^{3}+3 x+1f(x)=4x3+3x+1.

Problem 18020

Particle XXX moves as x(t)=5t3−9t2+7x(t)=5 t^{3}-9 t^{2}+7x(t)=5t3−9t2+7. Determine its direction at t=1t=1t=1, when it's farthest left, and area change at t=1t=1t=1 for triangle with YYY.

Problem 18021

Find the limit of the sequence an=n2+(−1)n⋅n(n+3)2a_{n}=\frac{n^{2}+(-1)^{n} \cdot n}{(n+\sqrt{3})^{2}}an​=(n+3​)2n2+(−1)n⋅n​ as n→∞n \to \inftyn→∞.

Problem 18022

Berechnen Sie den Grenzwert: lim⁡n→∞(4n2+11n−2n)\lim _{n \rightarrow \infty}\left(\sqrt{4 n^{2}+11 n}-2 n\right)n→∞lim​(4n2+11n​−2n).

Problem 18023

Skizzieren Sie die Flächen und berechnen Sie die Fläche A für die Funktionen: 44.1 f(x)=x2+1f(x)=x^2+1f(x)=x2+1 zwischen x=1x=1x=1 und x=2x=2x=2; 44.2 f(x)=12x2−8f(x)=\frac{1}{2}x^2-8f(x)=21​x2−8.

Problem 18024

Find the limit: lim⁡n→∞(1+π2n+1)2n\lim _{n \rightarrow \infty}\left(1+\frac{\pi}{2^{n+1}}\right)^{2^{n}}n→∞lim​(1+2n+1π​)2n.

Problem 18025

Evaluate the integral ∫−∞−5−1(x−4)3dx\int_{-\infty}^{-5} \frac{-1}{(x-4)^{3}} dx∫−∞−5​(x−4)3−1​dx or say if it diverges.

Problem 18026

Evaluate the integral I=∫0∞2e−13xdxI = \int_{0}^{\infty} 2 e^{-\frac{1}{3} x} dxI=∫0∞​2e−31​xdx or state if it diverges.

Problem 18027

Problem 18028

Evaluate the integral ∫e∞−32x(ln⁡x)32dx\int_{e}^{\infty} \frac{-3}{2 x(\ln x)^{\frac{3}{2}}} dx∫e∞​2x(lnx)23​−3​dx or determine if it diverges.

Problem 18029

Evaluate the integral ∫−∞−2−44x−9 dx\int_{-\infty}^{-2} \frac{-4}{4x-9} \, dx∫−∞−2​4x−9−4​dx or indicate if it diverges.

Problem 18030

Evaluate the integral ∫e10∞−12xln⁡xdx\int_{e^{10}}^{\infty} \frac{-1}{2 x \ln x} d x∫e10∞​2xlnx−1​dx or indicate if it diverges.

Problem 18031

Berechne das Integral ∫−3−1(x3+4x2+3x)dx\int_{-3}^{-1}\left(x^{3}+4 x^{2}+3 x\right) d x∫−3−1​(x3+4x2+3x)dx.

Problem 18032

Problem 18033

Evaluate the integral ∫0∞2e−13xdx\int_{0}^{\infty} 2 e^{-\frac{1}{3} x} dx∫0∞​2e−31​xdx or state if it diverges.

Problem 18034

Problem 18035

Finde die kleinste natürliche Zahl nεn_{\varepsilon}nε​, sodass ∣an−97∣<11000\left|a_{n}-\frac{9}{7}\right|<\frac{1}{1000}∣∣​an​−79​∣∣​<10001​ für alle n>nεn>n_{\varepsilon}n>nε​.

Problem 18036

Find the time ttt when the drug concentration C(t)=0.6tt2+40C(t)=\frac{0.6 t}{t^{2}+40}C(t)=t2+400.6t​ is maximized. Round your answer to 2 decimal places.

Problem 18037

Find the first and second derivatives of f(x)=(x2−1)⋅e−0.5xf(x)=(x^{2}-1) \cdot e^{-0.5 x}f(x)=(x2−1)⋅e−0.5x.

Problem 18038

Find the average change in revenue for R(q)=200q−2q2R(q)=200 q-2 q^{2}R(q)=200q−2q2 when sales increase from 30 to 35 units.

Problem 18039

Find the limit: lim⁡n→∞((sin⁡(π/2))n+(2sin⁡(π/4))n+(1/10)n)1/n\lim _{n \rightarrow \infty}\left((\sin (\pi / 2))^{n}+(2 \sin (\pi / 4))^{n}+(1 / 10)^{n}\right)^{1 / n}n→∞lim​((sin(π/2))n+(2sin(π/4))n+(1/10)n)1/n

Problem 18040

Find the marginal revenue for the revenue function R(q)=200q−2q2R(q)=200 q-2 q^{2}R(q)=200q−2q2 when 30 units are sold.

Problem 18041

Calculate the integral of 3x3 \sqrt{x}3x​ with respect to xxx.

Problem 18042

1. a) Find points where the slope of y=−sin⁡xy=-\sin xy=−sinx is i) zero ii) a local max iii) a local min b) Determine intervals for i) concave up ii) concave down

Problem 18043

Problem 18044

Bestimmen Sie eine Stammfunktion für:
a) $f(x)=\frac{1}{2 x^{3}}+\cos (0,5 x)$
b) $f(x)=2(5 x+1)^{7}+\sqrt{x}$

Bestimme die Steigung von $f(x)=10 x^{3}-9 x^{2}+2 x$ bei $x=5$ .

Untersuchen Sie die Funktion $f(x)=\frac{2 x+1}{x}$ :
1. Skizzieren Sie den Graphen für $0 \leq x \leq 12$ .
2. Vermuten Sie das Verhalten für große $x$ .
3. Berechnen Sie $f(1), f(10), f(100), f(1000)$ .

Gegeben ist die Funktion $f_{a}(x)=-x^{3}+a^{2} \cdot x, a>0$ . Finde Nullstellen, Extremstellen und Wendestellen.

Find the time intervals where the particle's speed is decreasing based on the velocity graph $v(t)$ for $0 \leq t \leq m$ .

Berechnen Sie die Integrale: $\int_{2}^{4} 4 e^{2 x} d x$ und $\int_{2}^{4} 5 x^{4} d x - \int_{2}^{4} 3 x^{2} d x + \int_{4}^{2} 5 x^{4} d x$ .

Find $t$ in $1 \leq t \leq 16$ where instantaneous velocity equals average velocity for $p(t)=\sqrt{t}-2$ . Choices: (A) 1, (B) $\frac{121}{25}$ , (C) $\frac{25}{4}$ , (D) 25.

Eine Familie nutzt eine Solaranlage. Vergleiche die Funktionen $f(t)$ und $g(t)$ und berechne den solaren Deckungsgrad.

Find when the particle's speed decreases given its velocity graph intersects at $t=0, k, m$ and has tangents at $t=j, l$ .

Find the acceleration of the particle at the first point where the velocity $v(t)=\frac{dx}{dt}=0$ for $x(t)=\sin t-\cos t$ .

Bestimmen Sie $p$ für $f(x)=3 x^{2}+p^{2}$ , sodass die Fläche im Intervall $[-1; 2]$ gleich 21 FE ist.

Find when the speed of a particle, with velocity $v(t)=-t^{3}+2 t^{2}+2^{-t}$ for $t \geq 0$ , is increasing.

A snail's velocity is $v(t)=1.4 \ln(1+t^{2})$ . Find its acceleration at $t=5$ minutes.

Find how many times the velocity, given by $y(t)=\frac{1}{6} \cos (5 t)-\frac{1}{4} \sin (5 t)$ , is 0 in 4 seconds.

Bestimme die Stammfunktion $F$ für die Funktionen: a) $f(x)=5 x^{3}+7 x^{2}-4 x+9$ , b) $f(x)=\frac{1}{3} x^{6}+x$ , c) $f(x)=8 x^{2}-x+2$ , e) $f(x)=4 x^{3}+3 x+1$ .

Particle $X$ moves as $x(t)=5 t^{3}-9 t^{2}+7$ . Determine its direction at $t=1$ , when it's farthest left, and area change at $t=1$ for triangle with $Y$ .

Find the limit of the sequence $a_{n}=\frac{n^{2}+(-1)^{n} \cdot n}{(n+\sqrt{3})^{2}}$ as $n \to \infty$ .

Berechnen Sie den Grenzwert: $\lim _{n \rightarrow \infty}\left(\sqrt{4 n^{2}+11 n}-2 n\right)$ .

Skizzieren Sie die Flächen und berechnen Sie die Fläche A für die Funktionen: 44.1 $f(x)=x^2+1$ zwischen $x=1$ und $x=2$ ; 44.2 $f(x)=\frac{1}{2}x^2-8$ .

Find the limit: $\lim _{n \rightarrow \infty}\left(1+\frac{\pi}{2^{n+1}}\right)^{2^{n}}$ .

Evaluate the integral $\int_{-\infty}^{-5} \frac{-1}{(x-4)^{3}} dx$ or say if it diverges.

Evaluate the integral $I = \int_{0}^{\infty} 2 e^{-\frac{1}{3} x} dx$ or state if it diverges.

Evaluate the integral $\int_{e}^{\infty} \frac{-3}{2 x(\ln x)^{\frac{3}{2}}} dx$ or determine if it diverges.

Evaluate the integral $\int_{-\infty}^{-2} \frac{-4}{4x-9} \, dx$ or indicate if it diverges.

Evaluate the integral $\int_{e^{10}}^{\infty} \frac{-1}{2 x \ln x} d x$ or indicate if it diverges.

Berechne das Integral $\int_{-3}^{-1}\left(x^{3}+4 x^{2}+3 x\right) d x$ .

Evaluate the integral $\int_{0}^{\infty} 2 e^{-\frac{1}{3} x} dx$ or state if it diverges.

Finde die kleinste natürliche Zahl $n_{\varepsilon}$ , sodass $\left|a_{n}-\frac{9}{7}\right|<\frac{1}{1000}$ für alle $n>n_{\varepsilon}$ .

Find the time $t$ when the drug concentration $C(t)=\frac{0.6 t}{t^{2}+40}$ is maximized. Round your answer to 2 decimal places.

Find the first and second derivatives of $f(x)=(x^{2}-1) \cdot e^{-0.5 x}$ .

Find the average change in revenue for $R(q)=200 q-2 q^{2}$ when sales increase from 30 to 35 units.

Find the limit: $\lim _{n \rightarrow \infty}\left((\sin (\pi / 2))^{n}+(2 \sin (\pi / 4))^{n}+(1 / 10)^{n}\right)^{1 / n}$

Find the marginal revenue for the revenue function $R(q)=200 q-2 q^{2}$ when 30 units are sold.

Calculate the integral of $3 \sqrt{x}$ with respect to $x$ .

1. a) Find points where the slope of $y=-\sin x$ is i) zero ii) a local max iii) a local min b) Determine intervals for i) concave up ii) concave down

Berechne das Volumen des Körpers, der entsteht, wenn die Fläche zwischen $f(x)=3 \sqrt{x}$ und $g(x)=\sqrt{4-x}$ um die $x$ -Achse rotiert.

Find the area between the graph of $y=-2x+3$ over $[0,1]$ and the $x$ -axis using limits. Sketch the region.

Find the population in 13 years using the model $P(t)=197,000 e^{-0.0141t}$ . Round to the nearest person.

Find the derivative $f'(x)$ of $f(x) = \frac{4}{\sqrt{x}}$ using the limit definition.

Find the derivative of $y=\ln(x^{2}-4)$ .

Find the value of $\int_{2}^{5} 2|x-3| \, dx$ .

Use substitution to simplify the integral $\int(\sec x \tan x) \ln (\sec x+15) d x$ to $\int \ln \mathrm{u} \, du$ and evaluate.

Find the derivative of $y=e^{\sin x}$ with respect to $x$ .

Find the value of $\lim _{x \rightarrow-1^{-}} |x-1|$ .

Find the partial derivative $f_{x}(x, y)$ of $f(x, y)=4 x^{2} y^{2}-16 x^{2}+4 y$ .

What is the best choice for $\mathrm{dv}$ in $\int x^{n} e^{a x} \mathrm{dx}$ using integration by parts? A. $e^{a x} \mathrm{dx}$ B. $e^{a x}$ C. $x^{n} d x$ D. $x^{n}$

Evaluate the integral $\int \frac{\ln x}{x^{9}} d x = \square$ using integration by parts.

Evaluate the integral $\int_{0}^{\pi} 3 x \cos x \, dx$ using integration by parts. Which expression is simpler?

Find the increase in cost when producing 300 to 720 bikes using the marginal cost function $C^{\prime}(x)=200-\frac{x}{3}$ .

Evaluate the integral $\int_{0}^{\pi} 3 x \cos x \, dx$ using integration by parts. Choose the correct expression for the new integral.

Maximize profit $P(x, y)=24 x-x^{2}-x y-2 y^{2}+33 y-43$ for goods $x$ and $y$ . What are the optimal values?

Find the partial derivative $f_{x}(x, y)$ for $f(x, y)=4 x^{2}+5 x^{2} y+12 x y^{2}$ and evaluate at $x=12$ , $y=4$ .

Is it true that to evaluate $\int \frac{7 x^{6}}{x^{4}+2 x^{2}} d x$ , you first need the partial fraction decomposition? A. No, numerator degree > denominator. B. No, denominator not factored. C. Yes, numerator degree > denominator. D. Yes, denominator can be factored.

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

Is it easier to evaluate $\int \frac{10 x+3}{5 x^{2}+3 x} d x$ using partial fractions? Choose A, B, C, or D.

Calculate the integral $\int_{0}^{6}\left(5 e^{0.5 x}+1\right) d x$ .

Calculate the integral from 0 to 2 of the function $6 e^{0.5 x} - 6 x$ .

Calculate the integral from 0 to 6 of the function $3 e^{-0.5 x} + 4$ .

Evaluate the integral $I=\int \frac{x^{2}}{x+8} d x$ using substitution $u=x+8$ and long division. Compare results.

Find the derivative of $y=x^{2} \ln x$ with respect to $x$ .

Find the average value of $f(x)=\frac{1}{2 x-11}$ from $x=3$ to $x=5$ , expressed as a constant times $\ln 5$ .

Evaluate the integral $\int \frac{2 x+5}{x^{2}+36} d x$ . Can partial fraction decomposition be used? A. Yes B. No. Find the integral's value.

Find the derivative of $y=\ln \sqrt[3]{x}$ with respect to $x$ .