Calculus

Problem 29101

Find the sum of the series: $\sum_{k=1}^{\infty} \frac{1}{(k+1)(k+2)}$ .

See Solution

Problem 29102

Find the average rate of change of the function $h(x)=-x^{2}+x+8$ over the interval $[-2,5]$ .

See Solution

Problem 29103

Bestimme die Asymptoten, Nullstellen und den Extrempunkt der Funktion $f(x)=\frac{25}{\left(x^{2}-4\right)^{2}}-1$ . Zeichne den Graphen für $-6<x<6$ .

See Solution

Problem 29104

Find $\sum_{r=1}^{\infty} \frac{1}{(k r+1)(k r-k+1)}$ and $\sum_{r=n}^{n^{2}} \frac{1}{(k r+1)(k r-k+1)}$ .

See Solution

Problem 29105

Find the derivative of $f(x)=\frac{4 \sqrt[3]{x}}{\sin (x)+\ln (x)}$ for $x > 0$ . Does $f$ have an inverse?

See Solution

Problem 29106

How long will it take for \$440 to grow to \$610 at a continuous interest rate of 2%? Round to the nearest tenth of a year.

See Solution

Problem 29107

Find $f^{\prime}(x)$ and calculate $f^{\prime}(-2) \cdot(15-20)$ for the functions:
15. $f(x)=3 x-10$
16. $f(x)=x^{2}-3 x+e$
17. $f(x)=-x^{3}-2 x^{2}+3 x-5$
18. $f(x)=-4 x\left(x^{3}-\frac{1}{x^{3}}\right)$
19. $f(x)=\frac{3 x^{3}-2 x^{2}+4 x}{2 x}$
20. $f(x)=x^{3}+8\left(\frac{1}{x}-\frac{1}{x^{2}}\right)$

See Solution

Problem 29108

Find the left and right endpoint Riemann sums for $f(x)=\frac{13}{x}$ on $[2, 4]$ using 4 rectangles.

See Solution

Problem 29109

Find the tangent line to $f(x)=\ln(x^{2}+e^{x})$ at the point where $x=1$ .

See Solution

Problem 29110

Differentiate the function $4x^5 - \frac{4}{x} + 3\sqrt{x}$ .

See Solution

Problem 29111

Find the derivative $dv/dt$ of the function $3 \tan (2 t) - 4 e^{3 t} + \ln 5$ .

See Solution

Problem 29112

Annabelle invested $\$5,900$ at $2\%$ interest compounded continuously. How long until the account reaches $\$7,440$ ?

See Solution

Problem 29113

Find $f^{\prime}(x)$ and $f^{\prime}(-2) \cdot (15-20)$ for:
15. $f(x)=3x-10$
16. $f(x)=x^{2}-3x+e$
17. $f(x)=-x^{3}-2x^{2}+3x-5$
18. $f(x)=-4x\left(x^{3}-\frac{1}{x^{3}}\right)$
19. $f(x)=\frac{3x^{3}-2x^{2}+4x}{2x}$
20. $f(x)=x^{3}+8\left(\frac{1}{x}-\frac{1}{x^{2}}\right)$

See Solution

Problem 29114

Curve $C$ has polar equation $r^{2}=\frac{1}{\theta^{2}+1}$ for $0 \leq \theta \leq \pi$ .
(a) Sketch $C$ and find the furthest point's polar coordinates.
Calculate the area enclosed by $C$ , the initial line, and $\theta=\pi$ .
Show that at the furthest point, $(\theta+\frac{1}{\theta}) \cot \theta-1=0$ and verify a root exists between 1.1 and 1.2.

See Solution

Problem 29115

Calculate the left endpoint Riemann sum for $f(x)=\frac{x^{2}}{9}$ on $[2,6]$ using 8 rectangles. What is the sum?

See Solution

Problem 29116

Bestimme die 1. und 2. Ableitung der Funktion $f(x)=5 \cdot e^{-3 x+2}$ .

See Solution

Problem 29117

Find the surface area of the curve $y=2 \sqrt{x+1}$ , for $0 \leq x \leq 5$ , revolved around the $x$ -axis.

See Solution

Problem 29118

Bestimmen Sie die Ableitungsfunktion von $f(x) = 4x + 1,871$ , also $f'(x)$ .

See Solution

Problem 29119

Find the length of the curve $x=\tan (y)$ from $y=\frac{\pi}{6}$ to $y=\frac{\pi}{3}$ .

See Solution

Problem 29120

Find the surface area of the curve $y=2 \sqrt{x+1}$ from $0 \leq x \leq 1$ revolved around the $x$ -axis.

See Solution

Problem 29121

Find the length of the curve $y=\frac{1}{3}(x^{2}+3)^{3/2}$ for $1 \leq x \leq 2$ .

See Solution

Problem 29122

Find the surface area from revolving $x=\frac{2}{\pi} \sqrt{3y+1}$ , $0 \leq y \leq 1$ about the $y$ -axis.

See Solution

Problem 29123

Find the surface area from revolving $x=\frac{2}{\pi} \sqrt{5y+1}$ , $0 \leq y \leq 1$ about the $y$ -axis.

See Solution

Problem 29124

Bestimme die Intervallgrenze $u$ für die Funktion $f(x)=\frac{1}{x^{2}}$ im Intervall $I=[u ; 6]$ , wenn der Flächeninhalt $A=\frac{5}{6}$ ist.

See Solution

Problem 29125

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}^{+}}\left(-1+5^{\tan x}\right)=?$ Choose: a. 0, b. 1, c. $\infty$ , d. 2, e. -2, f. -1, 9. Else

See Solution

Problem 29126

Find the length of the curve $y=\frac{1}{3}(x^{2}+6)^{3/2}$ for $1 \leq x \leq 2$ .

See Solution

Problem 29127

Bestimmen Sie die Stammfunktionen für die folgenden Funktionen: (1) $f(x)=1+e^{3-2 x}$ , (2) $f(x)=\frac{1}{4} e^{2 x}-3 e^{-x}+5$ , (3) $f(x)=e^{5-2 x}+\frac{7}{e^{3 x}}$ .

See Solution

Problem 29128

Gegeben ist die Funktion $g(x)=-x(x-2)^{2}(x-4)$ .
a) Beschreiben Sie 4 Merkmale des Graphen.
b) Berechnen Sie den Flächeninhalt $A$ zwischen $g(x)$ und der $x$ -Achse, wenn $\int_{0}^{2} g(x) \, dx = \frac{64}{15}$ .
Aufgabe A4: Bestimmen Sie je eine Stammfunktion für die Funktionen:
1) $f(x)=1+e^{3-2 x}$ , 2) $f(x)=\frac{1}{4} e^{2 x}-3 e^{-x}+5$ , 3) $f(x)=e^{5-2 x}+\frac{7}{e^{3 x}}$ .

See Solution

Problem 29129

Calculate the area in the first quadrant under the curve $y=\frac{\ln ^{4} x}{x}$ from $x=1$ to $x=8$ .

See Solution

Problem 29130

Calculate the area in the first quadrant under $y=\frac{3 \ln ^{2} x}{x}$ from $x=1$ to $x=8$ .

See Solution

Problem 29131

Calculate the area in the first quadrant under $y=\frac{4 \ln ^{3} x}{x}$ from $x=1$ to $x=5$ .

See Solution

Problem 29132

Calculate the sum: $\sum_{n=1}^{\infty} n \ln \left(1+\frac{1}{n^{2}}\right)$ .

See Solution

Problem 29133

An object on a 30° incline accelerates at 12 ft/s² from 4 ft/s. How long to reach the end?

See Solution

Problem 29134

Zeige, dass $F(x) = x \ln(x) - x$ die Stammfunktion von $f(x) = \ln(x)$ ist.

See Solution

Problem 29135

An object is shot down at 160 ft/s from 9600 ft. When does it hit the ground? Solve for $t$ in $9600 = 160t + \frac{1}{2}gt^2$ .

See Solution

Problem 29136

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

See Solution

Problem 29137

Finde die Stammfunktionen für: $f(x) = x$ , $f(x) = \sqrt{x}$ , $f(x) = \frac{1}{x}$ , $f(x) = \frac{1}{x^{2}}$ .

See Solution

Problem 29138

Gegeben sind die Angebots- und Nachfragefunktionen $p_{N}(x)=10 e^{-0,5 \cdot x+2}$ und $p_{A}(x)=10 e^{0,75 \cdot x-3}$ für $0 \leq x \leq 9$ .
1) Finde das Marktgleichgewicht. 2) Berechne $\int_{0}^{4} p_{N}(x) \, dx$ und zeige es grafisch. 3) Berechne $\int_{0}^{4} 10 \, dx$ und zeige es grafisch. 4) Analysiere die Differenz der Integrale $\int_{0}^{4} p_{N}(x) \, dx - \int_{0}^{4} 10 \, dx$ .

See Solution

Problem 29139

1. Finde die Ableitung für: a) $f(x)=3 x^{2}$ b) $f(x)=-7 x^{4}$ c) $f(x)=5 x$ d) $f(x)=-\frac{1}{5} x^{10}$ e) $f(x)=\frac{4}{x}$ f) $f(x)=-\frac{2}{5} x^{-5}$ g) $f(x)=\frac{2}{3} x^{-6}$ h) $f(x)=4 \sqrt{x}$ 2. Bestimme $f^{\prime}(x)$ für: a) $f(x)=2 x^{3}+5 x^{2}$ b) $f(x)=4 x^{5}-3 x$ c) $f(x)=2 x^{7}-3 x^{4}$ d) $f(x)=-2 x^{3}+x^{2}+4$ e) $f(x)=2 x^{3}(x^{2}+x-1)$ f) $f(u)=u(u+3)^{2}$ 3. Finde die ersten drei Ableitungen für: a) $f(x)=-\frac{3}{2} x^{2}+\frac{5}{4} x+2$ b) $f(x)=0,2 x^{3}-0,5 x^{2}+0,8$ c) $f(x)=(x-2)(x+2)(x-5)$ d) $f(t)=0,5(2 t-3)^{2}$ e) $f(x)=2 x^{-1}-4 x^{-2}$ f) $f(x)=\frac{3}{x}+4 \sqrt{x}-2$ 4. Berechne die Steigung von $f$ bei $A(2 \mid f(2))$ für: a) $f(x)=\frac{3}{2} x^{2}$ b) $f(t)=\frac{1}{4} t^{4}-\frac{5}{3} t^{3}$ c) $f(z)=\frac{3}{4} z^{2}-3 z^{-1}$ d) $f(x)=-x-\frac{2}{x^{4}}$

See Solution

Problem 29140

Bestimme die Stammfunktion von $f(x)=\frac{x^{6}}{2}+\frac{2}{x^{\frac{1}{9}}}$ .

See Solution

Problem 29141

Bestimme die allgemeine Stammfunktion von $f(x)=\frac{x^{3}}{4}-\frac{4}{7 x^{3}}$ .

See Solution

Problem 29142

Finde die allgemeine Stammfunktion von $f(x)=\frac{7}{3 x^{7}}$ .

See Solution

Problem 29143

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

See Solution

Problem 29144

Find the derivative of the function $f(x) = x^{2x}$ .

See Solution

Problem 29145

Finde die allgemeine Stammfunktion von $f(x)=2 x^{4}(3-x^{5})$ .

See Solution

Problem 29146

Find the limit of the series $\sum_{n=1}^{\infty}(\sqrt{n}-\sqrt{n-1})$ .

See Solution

Problem 29147

Find $b$ if the average value of $f(x)=3 x^{2}-6 x$ on $[0, b]$ is 4.

See Solution

Problem 29148

Calculate $\int_{2}^{0} f(2 x) \mathrm{d} x$ given that $\int_{0}^{4} f(x) \mathrm{d} x=-12$ .

See Solution

Problem 29149

Find the height $m_{2}$ rises before stopping if $m_{1}=3.7 \mathrm{~kg}$ , $m_{2}=4.1 \mathrm{~kg}$ , and initial speed is $0.20 \mathrm{~m/s}$ .

See Solution

Problem 29150

Calculate the volume of the solid formed by rotating the area between $y=\sqrt{x}$ , $y=0$ , $x=0$ , and $x=3$ around the x-axis.

See Solution

Problem 29151

Find the surface area from revolving $y=4 \sqrt{\sin x}$ , for $\frac{\pi}{4} \leq x \leq \frac{\pi}{2}$ , about the $x$ -axis.

See Solution

Problem 29152

Prove the convergence of $\sum_{n=1}^{\infty} \frac{4}{3^{n}+2 n}$ using one of the following tests: a. Limit Comparison with $\sum \frac{1}{3^{n}}$ b. Limit Comparison with $\sum \frac{2}{n}$ c. Alternating series d. Divergence Test.

See Solution

Problem 29153

Identify the conditionally convergent series among: a. $\sum_{n=20}^{\infty} \frac{(-1)^{n+1}}{n^{3 / 4}}$ , b. $\sum_{n=3}^{\infty} \frac{(-1)^{n+1}}{n^{5}}$ , d. $\sum_{n=7}^{\infty} \frac{e^{n}+2}{e^{n}}$ .

See Solution

Problem 29154

Data una successione convergente $\{a_n\}$ con $a_n > 0$ , analizza il comportamento di $\{b_n\}$ , $\{c_n\}$ e $\{d_n\}$ quando $n \to \infty$ .

See Solution

Problem 29155

Prove the convergence of the series $\sum_{n=1}^{\infty} \frac{4}{3^{n}+2 n}$ .

See Solution

Problem 29156

Find the surface area from revolving $x=\frac{2}{\pi} \sqrt{3 y+1}$ , $0 \leq y \leq 1$ about the $y$ axis.

See Solution

Problem 29157

Find the sum of the series $\sum_{n=1}^{\infty} \frac{(-2)^{n-1}}{2^{2 n}}$ .

See Solution

Problem 29158

Find $f(x)$ if $\int_{1}^{x} \frac{f(t)}{e^{t}} d t = x^{7} - 6x$ .

See Solution

Problem 29159

Is the series $\sum_{n=1}^{\infty} \frac{1}{n^{\sqrt{2}}}$ convergent?
Select one: True False

See Solution

Problem 29160

Calcola il limite: $\lim _{x \rightarrow+\infty}\left(1+\frac{1}{2 x^{2}}\right)^{3 x^{2}}$ . Qual è il risultato?

See Solution

Problem 29161

Calcola la derivata di $f(x)=\frac{\tan (\arccos (x))}{\sqrt{1-x^{2}}}$ per $x \in(0,1)$ . Quale affermazione è vera?

See Solution

Problem 29162

Calcola l'integrale $1=\int_{0}^{1} \frac{2 x^{2}-3}{1+x^{2}} d x$ .

See Solution

Problem 29163

Consider the function $f(x)$ defined as:
$f(x) = e^x$ if $x \geq 0$ , and $f(x) = 1$ if $x > 0$ .
Determine its continuity and differentiability at $x = 0$ .

See Solution

Problem 29164

Trova l'equazione della retta tangente al grafico di $f(x)=\int_{1}^{e^{2}} \frac{t}{t^{2}+1} d t$ in $x=1$ tra le opzioni: (a) $y=x-1$ , (b) $y=x+1$ , (c) $y=2 x$ , (d) Non esiste, (e) Nessuna delle precedenti.

See Solution

Problem 29165

Match the trigonometric substitution for each integral: (i) $\int \frac{x^{2}}{\sqrt{4-x^{2}}} d x$ (ii) $\int \frac{d x}{x^{2} \sqrt{x^{2}-9}}$ (iii) $\int \frac{x^{2}}{\sqrt{16+x^{2}}} d x$

See Solution

Problem 29166

Calcola il limite $L:=\lim _{x \rightarrow+\infty}\left(\left(x^{3}-2 x^{2}\right)^{1 / 3}-\left(x^{3}+x\right)^{1 / 3}\right)$ . Qual è il valore di $L$ ? (a) $L=-2 / 3$ (b) $L=3 / 2$ (c) $L=+\infty$ (d) $L=-\infty$ (e) Nessuna delle precedenti.

See Solution

Problem 29167

Calcola l'integrale indefinito $I:=\int \frac{4^{x}}{2^{x}-1} d x$ e scegli la risposta corretta tra le opzioni.

See Solution

Problem 29168

Find $\int_{2}^{0} f(2 x) \mathrm{d} x$ if $\int_{0}^{4} f(x) \mathrm{d} x=-12$ . Options: a. 4, b. -6, c. -4, d. 6.

See Solution

Problem 29169

Find the limit: $\lim _{x \rightarrow 10} \sqrt{20}$ .

See Solution

Problem 29170

Trova $a>0$ tale che $\int_{0}^{a} e^{2t} \,dx = 1$ . Opzioni: (a) $a=2 \log (2)$ , (b) non esiste, (c) $a=\log (\sqrt{3})$ , (d) $a=e^{3/2}$ , (e) nessuna delle precedenti.

See Solution

Problem 29171

Calcola il limite $L=\lim _{x \rightarrow 0} \frac{f(x)}{x^{3}}$ con $f(x)=\int_{x}^{2 x} \sin(t^{2}) \mathrm{d} t$ . Qual è $L$ ?

See Solution

Problem 29172

Calcola il limite $L=\lim _{x \rightarrow 0} \frac{f(x)}{x^{3}}$ per $f(x)=\int_{x}^{2 x} \sin \left(t^{2}\right) d t$ . Qual è il valore di $L$ ? (a) non esiste, (b) $7/3$ , (c) $-2/3$ , (d) $0$ , (e) nessuna delle precedenti.

See Solution

Problem 29173

Trova l'equazione della retta tangente a $f(x)=\frac{1}{\tan x}$ in $x=\frac{\pi}{4}$ . Qual è?

See Solution

Problem 29174

Find the limit: $\operatorname{Lt}_{x \rightarrow \infty} e^{-x^{2}} = ?$ Options: a) 0 b) 1 c) $\underset{x \rightarrow \infty}{\operatorname{Lt}} e^{-x}$ d) $\infty$

See Solution

Problem 29175

Un échantillon de $20 \mathrm{mg}$ de thorium 233 a diminué à $17 \mathrm{mg}$ en $5 \mathrm{min}$ . Trouve la demi-vie.

See Solution

Problem 29176

Calcola il limite $L=\lim _{x \rightarrow+\infty} \frac{x(2 x+1)^{x}}{(2 x)^{x+1}}$ . Qual è il valore di $L$ ?

See Solution

Problem 29177

Calcola il limite $L=\lim _{x \rightarrow+\infty} x^{2}\left(x-\left(1+x^{3}\right)^{1 / 3}\right)$ . Qual è il valore di $L$ ?

See Solution

Problem 29178

Trova l'integrale generale $I$ della funzione $f(x)=\frac{x(1+\arctan(x^{2}))}{1+x^{4}}$ . Quale opzione è corretta?

See Solution

Problem 29179

Trova $a>0$ tale che: (a) $a=2 \log (2)$ e $\int_{0}^{a} e^{2 x} d x=1$ ; (b) Non esiste; (c) $a=\log (\sqrt{3})$ ; (d) $a=e^{3 / 2}$ ; (e) Nessuna delle precedenti.

See Solution

Problem 29180

Berechne die Ableitung der Funktion $f(x)=2,4 \cdot 10^{-4} x^{3}+0,04 x^{2}-3,2$ . Was ist $f'(x)$ ?

See Solution

Problem 29181

Find the line integral of $f(x, y) = x y$ from $(1,1)$ to $(4,5)$ .

See Solution

Problem 29182

A player spikes the ball at 10 ft with an initial velocity of -55 ft/s. How much time do opponents have to return it?

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Problem 29183

Approximate the area between the $x$ -axis and $f(x)$ from $x=-1$ to $x=3$ using a trapezoidal sum with 4 subdivisions.

See Solution

Problem 29184

Evaluate the line integral of $\mathbf{F}(x, y)=x^{2} \mathbf{i}+x \mathbf{j}$ from $(0,0)$ to $(1,1)$ along $y^{2}=x$ .

See Solution

Problem 29185

Un échantillon de $20 \mathrm{mg}$ de thorium 233 s'est désintégré en $17 \mathrm{mg}$ après $5 \mathrm{~min}$ . Trouve la demi-vie et le temps pour atteindre $1 \mathrm{mg}$ .

See Solution

Problem 29186

Approximate the area under $f(x)$ from $x=0$ to $x=9$ using a trapezoidal sum with 3 subdivisions. Given points are: $(0, 2)$ , $(3, 5)$ , $(6, 7)$ , $(9, 11)$ .

See Solution

Problem 29187

Find the derivative of $\frac{1}{\sqrt{3 x+2}}$ . Choose from: a) $\frac{3}{2(3 x+2)^{\frac{3}{2}}}$ b) $\frac{-3}{2(3 x+2)^{\frac{3}{2}}}$ c) $\frac{3}{2(3 x-2)^{\frac{3}{2}}}$ d) $\frac{-3}{2(3 x-2)^{\frac{3}{2}}}$

See Solution

Problem 29188

Find an approximation for $h^{\prime}(10)$ using the given depth data, and explain its meaning regarding pool depth.

See Solution

Problem 29189

Find the derivative of $\frac{\sin x+\cos x}{\sqrt{1+\sin 2 x}}$ . What is it? a) 0 b) 1 c) 2 d) 3

See Solution

Problem 29190

Approximate the area under $g(x)=x^{2}$ from $x=1$ to $x=4$ using a trapezoidal sum with 3 subdivisions.

See Solution

Problem 29191

Approximate the area under $f(x)=\frac{2}{x}$ from $x=0.5$ to $x=2$ using a trapezoidal sum with 3 subdivisions.

See Solution

Problem 29192

A quantity starts at 840 and decays at $5\%$ daily. Find its value after 10 weeks, rounded to the nearest hundredth.

See Solution

Problem 29193

A quantity starts at 510 and decays at $65\%$ per month. Find its value after 0.25 years, rounded to two decimal places.

See Solution

Problem 29194

Approximate the area under $f(x)$ from $x=0$ to $x=9$ using a trapezoidal sum with 3 equal parts.

See Solution

Problem 29195

Approximate the area under $f(x)=\frac{2}{x}$ from $x=0.5$ to $x=2$ using a trapezoidal sum with 3 subdivisions.

See Solution

Problem 29196

Approximate the area under $f(x)=\frac{2}{x}$ from $x=0.5$ to $x=2$ using a trapezoidal sum with 3 subdivisions.

See Solution

Problem 29197

Approximate the area under $f(x)=(x-3)^{2}$ from $x=0$ to $x=6$ using a midpoint Riemann sum with 3 subdivisions.

See Solution

Problem 29198

Evaluate $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ for each function: a. $f(x)=\frac{2 x+3}{x-1}$ , b. $f(x)=\frac{5 x^{2}-3}{x^{2}+2}$ , c. $f(x)=\frac{-5 x^{2}+3 x}{2 x^{2}-5}$ , d. $f(x)=\frac{2 x^{5}-3 x^{2}+5}{3 x^{4}+5 x-4}$ .

See Solution

Problem 29199

Find the volume when the curve $y=x^{2}+2$ from $x=1$ to $x=2$ is rotated $2\pi$ radians around the $x$ -axis.

See Solution

Problem 29200

Find the limit of the sequence $a_n = 7 + \frac{1}{\sqrt{n}}$ without using Theorems 3.4.3 or 3.4.6. Prove your answer.

See Solution

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Calculus

Problem 29101

Find the sum of the series: ∑k=1∞1(k+1)(k+2)\sum_{k=1}^{\infty} \frac{1}{(k+1)(k+2)}∑k=1∞​(k+1)(k+2)1​.

Problem 29102

Find the average rate of change of the function h(x)=−x2+x+8h(x)=-x^{2}+x+8h(x)=−x2+x+8 over the interval [−2,5][-2,5][−2,5].

Problem 29103

Bestimme die Asymptoten, Nullstellen und den Extrempunkt der Funktion f(x)=25(x2−4)2−1f(x)=\frac{25}{\left(x^{2}-4\right)^{2}}-1f(x)=(x2−4)225​−1. Zeichne den Graphen für −6<x<6-6<x<6−6<x<6.

Problem 29104

Find ∑r=1∞1(kr+1)(kr−k+1)\sum_{r=1}^{\infty} \frac{1}{(k r+1)(k r-k+1)}∑r=1∞​(kr+1)(kr−k+1)1​ and ∑r=nn21(kr+1)(kr−k+1)\sum_{r=n}^{n^{2}} \frac{1}{(k r+1)(k r-k+1)}∑r=nn2​(kr+1)(kr−k+1)1​.

Problem 29105

Find the derivative of f(x)=4x3sin⁡(x)+ln⁡(x)f(x)=\frac{4 \sqrt[3]{x}}{\sin (x)+\ln (x)}f(x)=sin(x)+ln(x)43x​​ for x>0x > 0x>0. Does fff have an inverse?

Problem 29106

How long will it take for \$440 to grow to \$610 at a continuous interest rate of 2%? Round to the nearest tenth of a year.

Problem 29107

Problem 29108

Find the left and right endpoint Riemann sums for f(x)=13xf(x)=\frac{13}{x}f(x)=x13​ on [2,4][2, 4][2,4] using 4 rectangles.

Problem 29109

Find the tangent line to f(x)=ln⁡(x2+ex)f(x)=\ln(x^{2}+e^{x})f(x)=ln(x2+ex) at the point where x=1x=1x=1.

Problem 29110

Differentiate the function 4x5−4x+3x4x^5 - \frac{4}{x} + 3\sqrt{x}4x5−x4​+3x​.

Problem 29111

Find the derivative dv/dtdv/dtdv/dt of the function 3tan⁡(2t)−4e3t+ln⁡53 \tan (2 t) - 4 e^{3 t} + \ln 53tan(2t)−4e3t+ln5.

Problem 29112

Annabelle invested $5,900\$5,900$5,900 at 2%2\%2% interest compounded continuously. How long until the account reaches $7,440\$7,440$7,440?

Problem 29113

Problem 29114

Problem 29115

Calculate the left endpoint Riemann sum for f(x)=x29f(x)=\frac{x^{2}}{9}f(x)=9x2​ on [2,6][2,6][2,6] using 8 rectangles. What is the sum?

Problem 29116

Bestimme die 1. und 2. Ableitung der Funktion f(x)=5⋅e−3x+2f(x)=5 \cdot e^{-3 x+2}f(x)=5⋅e−3x+2.

Problem 29117

Find the surface area of the curve y=2x+1y=2 \sqrt{x+1}y=2x+1​, for 0≤x≤50 \leq x \leq 50≤x≤5, revolved around the xxx-axis.

Problem 29118

Bestimmen Sie die Ableitungsfunktion von f(x)=4x+1,871f(x) = 4x + 1,871f(x)=4x+1,871, also f′(x)f'(x)f′(x).

Problem 29119

Find the length of the curve x=tan⁡(y)x=\tan (y)x=tan(y) from y=π6y=\frac{\pi}{6}y=6π​ to y=π3y=\frac{\pi}{3}y=3π​.

Problem 29120

Find the surface area of the curve y=2x+1y=2 \sqrt{x+1}y=2x+1​ from 0≤x≤10 \leq x \leq 10≤x≤1 revolved around the xxx-axis.

Problem 29121

Find the length of the curve y=13(x2+3)3/2y=\frac{1}{3}(x^{2}+3)^{3/2}y=31​(x2+3)3/2 for 1≤x≤21 \leq x \leq 21≤x≤2.

Problem 29122

Find the surface area from revolving x=2π3y+1x=\frac{2}{\pi} \sqrt{3y+1}x=π2​3y+1​, 0≤y≤10 \leq y \leq 10≤y≤1 about the yyy-axis.

Problem 29123

Find the surface area from revolving x=2π5y+1x=\frac{2}{\pi} \sqrt{5y+1}x=π2​5y+1​, 0≤y≤10 \leq y \leq 10≤y≤1 about the yyy-axis.

Problem 29124

Bestimme die Intervallgrenze uuu für die Funktion f(x)=1x2f(x)=\frac{1}{x^{2}}f(x)=x21​ im Intervall I=[u;6]I=[u ; 6]I=[u;6], wenn der Flächeninhalt A=56A=\frac{5}{6}A=65​ ist.

Problem 29125

Find the limit: lim⁡x→π2+(−1+5tan⁡x)=?\lim _{x \rightarrow \frac{\pi}{2}^{+}}\left(-1+5^{\tan x}\right)=?limx→2π​+​(−1+5tanx)=? Choose: a. 0, b. 1, c. ∞\infty∞, d. 2, e. -2, f. -1, 9. Else

Problem 29126

Find the length of the curve y=13(x2+6)3/2y=\frac{1}{3}(x^{2}+6)^{3/2}y=31​(x2+6)3/2 for 1≤x≤21 \leq x \leq 21≤x≤2.

Problem 29127

Bestimmen Sie die Stammfunktionen für die folgenden Funktionen: (1) f(x)=1+e3−2xf(x)=1+e^{3-2 x}f(x)=1+e3−2x, (2) f(x)=14e2x−3e−x+5f(x)=\frac{1}{4} e^{2 x}-3 e^{-x}+5f(x)=41​e2x−3e−x+5, (3) f(x)=e5−2x+7e3xf(x)=e^{5-2 x}+\frac{7}{e^{3 x}}f(x)=e5−2x+e3x7​.

Problem 29128

Problem 29129

Calculate the area in the first quadrant under the curve y=ln⁡4xxy=\frac{\ln ^{4} x}{x}y=xln4x​ from x=1x=1x=1 to x=8x=8x=8.

Problem 29130

Calculate the area in the first quadrant under y=3ln⁡2xxy=\frac{3 \ln ^{2} x}{x}y=x3ln2x​ from x=1x=1x=1 to x=8x=8x=8.

Problem 29131

Calculate the area in the first quadrant under y=4ln⁡3xxy=\frac{4 \ln ^{3} x}{x}y=x4ln3x​ from x=1x=1x=1 to x=5x=5x=5.

Problem 29132

Calculate the sum: ∑n=1∞nln⁡(1+1n2)\sum_{n=1}^{\infty} n \ln \left(1+\frac{1}{n^{2}}\right)∑n=1∞​nln(1+n21​).

Problem 29133

An object on a 30° incline accelerates at 12 ft/s² from 4 ft/s. How long to reach the end?

Problem 29134

Zeige, dass F(x)=xln⁡(x)−xF(x) = x \ln(x) - xF(x)=xln(x)−x die Stammfunktion von f(x)=ln⁡(x)f(x) = \ln(x)f(x)=ln(x) ist.

Problem 29135

An object is shot down at 160 ft/s from 9600 ft. When does it hit the ground? Solve for ttt in 9600=160t+12gt29600 = 160t + \frac{1}{2}gt^29600=160t+21​gt2.

Problem 29136

Problem 29137

Finde die Stammfunktionen für: f(x)=xf(x) = xf(x)=x, f(x)=xf(x) = \sqrt{x}f(x)=x​, f(x)=1xf(x) = \frac{1}{x}f(x)=x1​, f(x)=1x2f(x) = \frac{1}{x^{2}}f(x)=x21​.

Problem 29138

Problem 29139

Problem 29140

Bestimme die Stammfunktion von f(x)=x62+2x19f(x)=\frac{x^{6}}{2}+\frac{2}{x^{\frac{1}{9}}}f(x)=2x6​+x91​2​.

Problem 29141

Bestimme die allgemeine Stammfunktion von f(x)=x34−47x3f(x)=\frac{x^{3}}{4}-\frac{4}{7 x^{3}}f(x)=4x3​−7x34​.

Problem 29142

Finde die allgemeine Stammfunktion von f(x)=73x7f(x)=\frac{7}{3 x^{7}}f(x)=3x77​.

Problem 29143

Problem 29144

Find the sum of the series: $\sum_{k=1}^{\infty} \frac{1}{(k+1)(k+2)}$ .

Find the average rate of change of the function $h(x)=-x^{2}+x+8$ over the interval $[-2,5]$ .

Bestimme die Asymptoten, Nullstellen und den Extrempunkt der Funktion $f(x)=\frac{25}{\left(x^{2}-4\right)^{2}}-1$ . Zeichne den Graphen für $-6<x<6$ .

Find $\sum_{r=1}^{\infty} \frac{1}{(k r+1)(k r-k+1)}$ and $\sum_{r=n}^{n^{2}} \frac{1}{(k r+1)(k r-k+1)}$ .

Find the derivative of $f(x)=\frac{4 \sqrt[3]{x}}{\sin (x)+\ln (x)}$ for $x > 0$ . Does $f$ have an inverse?

Find the left and right endpoint Riemann sums for $f(x)=\frac{13}{x}$ on $[2, 4]$ using 4 rectangles.

Find the tangent line to $f(x)=\ln(x^{2}+e^{x})$ at the point where $x=1$ .

Differentiate the function $4x^5 - \frac{4}{x} + 3\sqrt{x}$ .

Find the derivative $dv/dt$ of the function $3 \tan (2 t) - 4 e^{3 t} + \ln 5$ .

Annabelle invested $\$5,900$ at $2\%$ interest compounded continuously. How long until the account reaches $\$7,440$ ?

Calculate the left endpoint Riemann sum for $f(x)=\frac{x^{2}}{9}$ on $[2,6]$ using 8 rectangles. What is the sum?

Bestimme die 1. und 2. Ableitung der Funktion $f(x)=5 \cdot e^{-3 x+2}$ .

Find the surface area of the curve $y=2 \sqrt{x+1}$ , for $0 \leq x \leq 5$ , revolved around the $x$ -axis.

Bestimmen Sie die Ableitungsfunktion von $f(x) = 4x + 1,871$ , also $f'(x)$ .

Find the length of the curve $x=\tan (y)$ from $y=\frac{\pi}{6}$ to $y=\frac{\pi}{3}$ .

Find the surface area of the curve $y=2 \sqrt{x+1}$ from $0 \leq x \leq 1$ revolved around the $x$ -axis.

Find the length of the curve $y=\frac{1}{3}(x^{2}+3)^{3/2}$ for $1 \leq x \leq 2$ .

Find the surface area from revolving $x=\frac{2}{\pi} \sqrt{3y+1}$ , $0 \leq y \leq 1$ about the $y$ -axis.

Find the surface area from revolving $x=\frac{2}{\pi} \sqrt{5y+1}$ , $0 \leq y \leq 1$ about the $y$ -axis.

Bestimme die Intervallgrenze $u$ für die Funktion $f(x)=\frac{1}{x^{2}}$ im Intervall $I=[u ; 6]$ , wenn der Flächeninhalt $A=\frac{5}{6}$ ist.

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}^{+}}\left(-1+5^{\tan x}\right)=?$ Choose: a. 0, b. 1, c. $\infty$ , d. 2, e. -2, f. -1, 9. Else

Find the length of the curve $y=\frac{1}{3}(x^{2}+6)^{3/2}$ for $1 \leq x \leq 2$ .

Bestimmen Sie die Stammfunktionen für die folgenden Funktionen: (1) $f(x)=1+e^{3-2 x}$ , (2) $f(x)=\frac{1}{4} e^{2 x}-3 e^{-x}+5$ , (3) $f(x)=e^{5-2 x}+\frac{7}{e^{3 x}}$ .

Calculate the area in the first quadrant under the curve $y=\frac{\ln ^{4} x}{x}$ from $x=1$ to $x=8$ .

Calculate the area in the first quadrant under $y=\frac{3 \ln ^{2} x}{x}$ from $x=1$ to $x=8$ .

Calculate the area in the first quadrant under $y=\frac{4 \ln ^{3} x}{x}$ from $x=1$ to $x=5$ .

Calculate the sum: $\sum_{n=1}^{\infty} n \ln \left(1+\frac{1}{n^{2}}\right)$ .

Zeige, dass $F(x) = x \ln(x) - x$ die Stammfunktion von $f(x) = \ln(x)$ ist.

An object is shot down at 160 ft/s from 9600 ft. When does it hit the ground? Solve for $t$ in $9600 = 160t + \frac{1}{2}gt^2$ .

Finde die Stammfunktionen für: $f(x) = x$ , $f(x) = \sqrt{x}$ , $f(x) = \frac{1}{x}$ , $f(x) = \frac{1}{x^{2}}$ .

Bestimme die Stammfunktion von $f(x)=\frac{x^{6}}{2}+\frac{2}{x^{\frac{1}{9}}}$ .

Bestimme die allgemeine Stammfunktion von $f(x)=\frac{x^{3}}{4}-\frac{4}{7 x^{3}}$ .

Finde die allgemeine Stammfunktion von $f(x)=\frac{7}{3 x^{7}}$ .

Find the derivative of the function $f(x) = x^{2x}$ .

Finde die allgemeine Stammfunktion von $f(x)=2 x^{4}(3-x^{5})$ .

Find the limit of the series $\sum_{n=1}^{\infty}(\sqrt{n}-\sqrt{n-1})$ .

Find $b$ if the average value of $f(x)=3 x^{2}-6 x$ on $[0, b]$ is 4.

Calculate $\int_{2}^{0} f(2 x) \mathrm{d} x$ given that $\int_{0}^{4} f(x) \mathrm{d} x=-12$ .

Find the height $m_{2}$ rises before stopping if $m_{1}=3.7 \mathrm{~kg}$ , $m_{2}=4.1 \mathrm{~kg}$ , and initial speed is $0.20 \mathrm{~m/s}$ .

Calculate the volume of the solid formed by rotating the area between $y=\sqrt{x}$ , $y=0$ , $x=0$ , and $x=3$ around the x-axis.

Find the surface area from revolving $y=4 \sqrt{\sin x}$ , for $\frac{\pi}{4} \leq x \leq \frac{\pi}{2}$ , about the $x$ -axis.

Prove the convergence of $\sum_{n=1}^{\infty} \frac{4}{3^{n}+2 n}$ using one of the following tests: a. Limit Comparison with $\sum \frac{1}{3^{n}}$ b. Limit Comparison with $\sum \frac{2}{n}$ c. Alternating series d. Divergence Test.

Data una successione convergente $\{a_n\}$ con $a_n > 0$ , analizza il comportamento di $\{b_n\}$ , $\{c_n\}$ e $\{d_n\}$ quando $n \to \infty$ .

Prove the convergence of the series $\sum_{n=1}^{\infty} \frac{4}{3^{n}+2 n}$ .

Find the surface area from revolving $x=\frac{2}{\pi} \sqrt{3 y+1}$ , $0 \leq y \leq 1$ about the $y$ axis.

Find the sum of the series $\sum_{n=1}^{\infty} \frac{(-2)^{n-1}}{2^{2 n}}$ .

Find $f(x)$ if $\int_{1}^{x} \frac{f(t)}{e^{t}} d t = x^{7} - 6x$ .

Is the series $\sum_{n=1}^{\infty} \frac{1}{n^{\sqrt{2}}}$ convergent?
Select one: True False

Calcola il limite: $\lim _{x \rightarrow+\infty}\left(1+\frac{1}{2 x^{2}}\right)^{3 x^{2}}$ . Qual è il risultato?

Calcola la derivata di $f(x)=\frac{\tan (\arccos (x))}{\sqrt{1-x^{2}}}$ per $x \in(0,1)$ . Quale affermazione è vera?

Calcola l'integrale $1=\int_{0}^{1} \frac{2 x^{2}-3}{1+x^{2}} d x$ .

Consider the function $f(x)$ defined as:
$f(x) = e^x$ if $x \geq 0$ , and $f(x) = 1$ if $x > 0$ .
Determine its continuity and differentiability at $x = 0$ .

Trova l'equazione della retta tangente al grafico di $f(x)=\int_{1}^{e^{2}} \frac{t}{t^{2}+1} d t$ in $x=1$ tra le opzioni: (a) $y=x-1$ , (b) $y=x+1$ , (c) $y=2 x$ , (d) Non esiste, (e) Nessuna delle precedenti.

Match the trigonometric substitution for each integral: (i) $\int \frac{x^{2}}{\sqrt{4-x^{2}}} d x$ (ii) $\int \frac{d x}{x^{2} \sqrt{x^{2}-9}}$ (iii) $\int \frac{x^{2}}{\sqrt{16+x^{2}}} d x$

Calcola l'integrale indefinito $I:=\int \frac{4^{x}}{2^{x}-1} d x$ e scegli la risposta corretta tra le opzioni.

Find $\int_{2}^{0} f(2 x) \mathrm{d} x$ if $\int_{0}^{4} f(x) \mathrm{d} x=-12$ . Options: a. 4, b. -6, c. -4, d. 6.

Find the limit: $\lim _{x \rightarrow 10} \sqrt{20}$ .

Trova $a>0$ tale che $\int_{0}^{a} e^{2t} \,dx = 1$ . Opzioni: (a) $a=2 \log (2)$ , (b) non esiste, (c) $a=\log (\sqrt{3})$ , (d) $a=e^{3/2}$ , (e) nessuna delle precedenti.

Calcola il limite $L=\lim _{x \rightarrow 0} \frac{f(x)}{x^{3}}$ con $f(x)=\int_{x}^{2 x} \sin(t^{2}) \mathrm{d} t$ . Qual è $L$ ?

Trova l'equazione della retta tangente a $f(x)=\frac{1}{\tan x}$ in $x=\frac{\pi}{4}$ . Qual è?

Find the limit: $\operatorname{Lt}_{x \rightarrow \infty} e^{-x^{2}} = ?$ Options: a) 0 b) 1 c) $\underset{x \rightarrow \infty}{\operatorname{Lt}} e^{-x}$ d) $\infty$