Calculus

Problem 16801

Evaluate the integral $\int \frac{1}{7 x^{7}} d x=\square$ (Provide the exact answer.)

See Solution

Problem 16802

Test if the series converge: 1) $\sum_{n=1}^{\infty} \frac{n^{n}}{n !}$ , 2) $\sum_{n=1}^{\infty}\left(\frac{2 n+3}{3 n+2}\right)^{n}$ .

See Solution

Problem 16803

Evaluate the definite integral $\int_{0}^{3}(7-x^{2})dx$ using its definition.

See Solution

Problem 16804

Evaluate the integral from 0 to 3 of the function $7 - x^{2}$ : $\int_{0}^{3}\left(7-x^{2}\right) d x$ .

See Solution

Problem 16805

Determine convergence or divergence of the series and if convergent, classify as conditional or absolute: $\sum_{n=1}^{\infty} \frac{n \ln n}{(-2)^{n}}$

See Solution

Problem 16806

Check if the series $\sum_{n=1}^{\infty} \frac{n \ln n}{(-2)^{n}}$ converges or diverges.

See Solution

Problem 16807

Find the indefinite integral: $\int \frac{4}{81 w^{2}+81} d w$ and verify by differentiating your result.

See Solution

Problem 16808

Determine if these statements are true or false, and explain your reasoning: a) If $F(x)$ is an antiderivative of $f(x)$ , then $\frac{1}{a} F(a x)$ is an antiderivative of $f(a x)$ . b) Each antiderivative of an $n^{\text{th}}$ degree polynomial is an $(n+1)^{\text{st}}$ degree polynomial. c) If $f$ and $g$ are continuous with $f(x) \geq g(x) \geq 0$ for $a \leq x \leq b$ , then $\int_{a}^{b} f(x) dx \geq \int_{a}^{b} g(x) dx$ .

See Solution

Problem 16809

Find the antiderivative $F$ of $f(u)=7 e^{u}+13$ with $F(0)=-4$ . What is $F(u)$ ?

See Solution

Problem 16810

Find the indefinite integral and verify by differentiating: $\int \frac{8}{81 v^{2}+81} d v=\square$

See Solution

Problem 16811

Determine convergence or divergence of the series. If it converges, classify as conditional or absolute: $\sum_{n=1}^{\infty} \frac{n 2^{n}(n+1) !}{3^{n} n !}$

See Solution

Problem 16812

Determine the position and velocity of an object with acceleration $a(t)=-34$ , initial velocity $v(0)=50$ , and initial position $s(0)=10$ . Find $v(t)=\square$ .

See Solution

Problem 16813

Find the velocity function $v(t)$ given $a(t)=4 t^{2}+7$ and $v(0)=3$ .

See Solution

Problem 16814

Find the rate of change of the area $A$ of a circle with radius $r$ when $r=4$ and $\frac{d r}{d t}=5$ .

See Solution

Problem 16815

Is the function $f(x)=\sum_{n=0}^{\infty} x e^{-n x^{2}}$ continuous at $x=0$ ? Justify your response.

See Solution

Problem 16816

Evaluate the integral: $\int \frac{1}{12 x^{6}} d x=\square($ Type an exact answer.)

See Solution

Problem 16817

Evaluate the integral: $\int \frac{1}{5 x^{5}} d x=\square$ (Type an exact answer.)

See Solution

Problem 16818

Find the first four nonzero terms of the Taylor series for $f(x)=\sin (-4 x)$ around $x=3$ .

See Solution

Problem 16819

Is it true or false that if $\sum_{n \in \mathbb{N}} a_{n}$ is conditionally convergent, then $\sum_{n \in \mathbb{N}} n^{k} a_{n}$ diverges for each $k \in(1,+\infty)$ ? Prove or provide a counterexample.

See Solution

Problem 16820

Find the particular solution of $f'(x)=-8x$ with the initial condition $f(1)=-2$ . What is $f(x)$ ?

See Solution

Problem 16821

Prove that the function $T(x)=\sum_{n=2}^{\infty}\left(\frac{1}{x-n \pi}+\frac{1}{n \pi}\right)$ is defined for $x \in [0, \pi]$ .

See Solution

Problem 16822

Find the rate of change of $f(x)=x^{2}+2x$ from $x=3$ to $x=5$ . The answer is 10. Explain this in context of the graph.

See Solution

Problem 16823

Considérez la fonction $f(x)=x-\sin (2 x)$ sur $[0, \frac{\pi}{2}]$ . Trouvez $f'(x)$ , $f''(x)$ et le tableau de variation.

See Solution

Problem 16824

Calculate the infinite series: $\sum_{n=1}^{\infty} \frac{n 2^{n}(n+1) !}{3^{n} n !}$ .

See Solution

Problem 16825

Find $F^{\prime}(x)$ using the Second Fundamental Theorem of Calculus for $F(x)=\int_{0}^{x} t^{4} \sqrt{1+t^{5}} dt$ .

See Solution

Problem 16826

Find the average height change per foot for a 25 ft drop from 0 ft to -100 ft. Options: $4 \mathrm{ft}$ , $-\frac{1}{4} \mathrm{ft}$ , $-4 \mathrm{ft}$ , $\frac{1}{4} \mathrm{ft}$ .

See Solution

Problem 16827

Find the sum of the series $-((1 / 2))-\frac{((1 / 2))^{2}}{2}-\frac{((1 / 2))^{3}}{3}-\cdots$ as a Taylor series.

See Solution

Problem 16828

Calculate the sum of the series: $6 - \frac{6^3}{3!} + \frac{6^5}{5!} - \frac{6^7}{7!} + \cdots$

See Solution

Problem 16829

Find the first few coefficients $c_n$ of the power series for $f(x)=\ln(5-x)=\sum_{n=0}^{\infty} c_{n} x^{n}$ .

See Solution

Problem 16830

Find local maxima, minima, and saddle points of $f(x, y)=x^{2}+7xy+y^{2}$ .

See Solution

Problem 16831

Find local maxima, minima, and saddle points of $f(x, y)=x^{2}+14 x+y^{2}-4 y+3$ . Options include points like $(-7,2)$ and $(7,-2)$ .

See Solution

Problem 16832

Find local maxima, minima, and saddle points of $f(x, y)=10xy(x+y)+4$ . Identify points from the options given.

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Problem 16833

Find local maxima, minima, and saddle points of $f(x, y)=2-x^{4} y^{4}$ . Identify points from options A-G.

See Solution

Problem 16834

Find local maxima, minima, and saddle points for $f(x, y)=x^{2}+2x+y^{2}+12y+3$ .

See Solution

Problem 16835

Find local maxima, minima, and saddle points of $f(x, y)=x^{2}+y x+y^{2}$ . Options: (A) (B) (C) (D) (E) (F) (G).

See Solution

Problem 16836

Find the degree 4 Taylor polynomial for $f(x)=\ln(x^{8})$ at $a=1$ . Compute derivatives and evaluate them.

See Solution

Problem 16837

Find the absolute max and min of $f(x, y)=6x+7y$ in the triangle with vertices $(0,0),(1,0),(0,1)$ .

See Solution

Problem 16838

Find the absolute max and min of $f(x, y)=x^{2}+xy+y^{2}$ on $-8 \leq x, y \leq 8$ .

See Solution

Problem 16839

$\lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\sin x}\right)$ limitinin sonucu nedir? A) -1 B) $-\frac{1}{2}$ C) 0 D) $\frac{1}{2}$ E) 1

See Solution

Problem 16840

Find the concavity intervals for the function $m(x)=\frac{x^{2}+5}{x-5}$ .

See Solution

Problem 16841

Given $f(t)=t^{2}+t-2$ , find: (a) average rate of change from $t=1$ to $t=2$ ; (b) instantaneous rate at $t=1$ .

See Solution

Problem 16842

An inverted pyramid with a square top (sides $2 \mathrm{~cm}$ ) is filled with water at $50 \mathrm{~cm}^3/\mathrm{s}$ . Find the water level rise rate when the level is $2 \mathrm{~cm}$ . $\mathrm{cm} / \mathrm{sec}$

See Solution

Problem 16843

Find local maxima of the function $h$ from its graph with a peak at (-2,2). (a) Local maximum locations: $\square$ (b) Local maximum values: $\square$

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Problem 16844

Find the extrema and inflection points of the function $g(t)=t^{2}(2t+5)$ .

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Problem 16845

F(x) = { x^2 \cos(1/x) for x ≠ 0; 0 for x = 0. Bu fonksiyonun sürekli olduğunu gösterin.

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Problem 16846

Gegeben sind $f(x)=-x^{2}+4$ und $g(x)=x^{2}-5x+6$ . Bestimme $f'$ , $g'$ , Steigungsstelle, Extremstelle, Tangentengleichung und Schnittwinkel.

See Solution

Problem 16847

$|a|<1$ için dizinin $1+a+a^{\wedge} 2+\ldots+a^{\wedge} n$ monoton artan ve sınırlı olduğunu gösterin.

See Solution

Problem 16848

Given $f(x)=x(x+5)^{2}$ , find the intercepts, increasing/decreasing intervals, extremum points, concavity, and inflection points. Sketch the graph.

See Solution

Problem 16849

Aufgabe 4: Gegeben ist $f_{t}(x)=x^{3}-\frac{3}{2} t x^{2}$ mit $t>0$ .
a) Einfluss von $t$ auf den Graphen. b) Nullstellen in Abhängigkeit von $t$ berechnen. c) Lokale Extrempunkte und $t$ für Tiefpunkt 4 Längeneinheiten unter der x-Achse finden. d) Tangentengleichung am Punkt $\left(\frac{1}{2} t, -\frac{1}{4} t^{3}\right)$ bestimmen.

See Solution

Problem 16850

Finden Sie die Stammfunktion für die folgenden Funktionen: a) $f(x)=x^{4}$ , b) $f(x)=x^{3}-x^{2}$ , c) $f(x)=x^{n}$ , d) $f(x)=x^{-2}$ , e) $f(x)=\frac{1}{5} x^{-4}$ , f) $f(x)=\frac{1}{x}$ .

See Solution

Problem 16851

Calculate the integral: $\int x(2x^{2}+1) \, dx$

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Problem 16852

Evaluate the integral: $\int\left(\frac{12 \sqrt{3}}{3 \pi^{2}+x^{2}}\right)$ .

See Solution

Problem 16853

Calculate the integral $\int \frac{\ln ^{3} x}{x\left(\ln ^{4} x+5\right)} d x$ .

See Solution

Problem 16854

Calculate the integral: $\int\left(3 \sqrt{x}+\frac{1}{\sqrt{x}}\right) d x$ .

See Solution

Problem 16855

Bestimme die Ableitungsfunktion für die folgenden Funktionen: a) $f(x)=a x^{2}+b x+c$ , b) $f(x)=a x+c$ , c) $f(x)=x^{c+1}$ , d) $f(t)=t^{2}+3 t$ , e) $f(x)=x-t$ , f) $f(t)=x-t$ .

See Solution

Problem 16856

Gjeni integralet e mëposhtme: a) $\int(2x + 3x^5) \, dx$ b) $\int(x^{-4} - 4x^7) \, dx$ c) $\int(3\sqrt{x} + \frac{1}{\sqrt{x}}) \, dx$

See Solution

Problem 16857

Untersuchen Sie das Verhalten der Funktion $f(x)=\frac{1}{2} x^{3}-2 x^{2}$ für $x \rightarrow \infty$ und $x \rightarrow -\infty$ . Ändern Sie die Koeffizienten, damit $g$ folgende Grenzen hat: I: $\lim_{x \rightarrow -\infty} g(x)=\infty$ , II: $\lim_{x \rightarrow \pm \infty} g(x)=-\infty$ , III: $\lim_{x \rightarrow \pm \infty} g(x)=\infty$ .

See Solution

Problem 16858

Gjeni intervalet e $x$ ku $y=x^{3}+5 x^{2}-8 x+4$ është monoton zbritës. Zgjidhni integralet: $\int(2 x+3 x^{5}) dx$ , $\int(x^{-4}-4 x^{3}) dx$ , $\int(3 \sqrt{x}+\frac{1}{\sqrt{x}}) dx$ .

See Solution

Problem 16859

Bestimmen Sie die Ableitung $f^{\prime}$ für die folgenden Funktionen: a) $f(x)=x(5-x)$ , b) $f(x)=(x+x^{2})x$ , c) $f(x)=x^{2}(x+2)5$ , d) $f(x)=(x+2)^{2}$ , e) $f(x)=2(x-2)^{2}$ , f) $f(x)=(x-7)(x+7)$ .

See Solution

Problem 16860

Find the derivatives of these functions:
1. $g(x)=x^{-3}$
2. $g(x)=c+\sqrt{x}$
3. $g(x)=\sqrt{x} \cdot 0.2$
4. $g(x)=\sqrt{x}-1.4 \cdot x^{4}$
5. $g(x)=-1.5 \cdot \cos (x)+x^{7}$
6. $k(x)=x^{1}$
7. $h(x)=6-x^{-3}$
8. $k(x)=\sin (x)-3$
9. $h(x)=-b \cdot \cos (x)$
10. $k(x)=7 \cdot x^{-3}$
11. $h(x)=7 \cdot x^{-3}-2 \cdot x^{2}$
12. $k(x)=\sin (x)+2+c$
13. $h(x)=0.1-\sin (x)-0.5$
14. $k(x)=-2 \cdot \cos (x)+1.3$
15. $h(x)=0.1-\sin (x)-0.5$

See Solution

Problem 16861

Find the limit of $f(x)=\frac{x^{2}-1}{x-1}$ as $x$ approaches 1. Options: 1, 2, Undefined, 0.

See Solution

Problem 16862

Calculate the derivative of the function $f(x)=\sin (2 x)+\cos (3 x)$ .

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Problem 16863

Find the derivative of $f(x)=(x^{2}+1)x^{3/2}$ . What is $f'(x)$ ?

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Problem 16864

Find the derivative of $f(x)=e^{2 x^{2}-3 x+4}$ . What is $f'(x)$ ?

See Solution

Problem 16865

Find the derivative of $f(x)=\ln(2x^{3}-5x+1)$ . What is $f'(x)$ ?

See Solution

Problem 16866

Find the volume $V$ using cylindrical shells for the region bounded by $y=\frac{7}{x}$ , $y=0$ , $x=1$ , $x=4$ . $V=\square$

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Problem 16867

Find the average value of $f(x)=-2 x^{3}-x-3$ on $[1,3]$ . Provide the exact answer as a fraction if necessary.

See Solution

Problem 16868

Determine if the series $\frac{1}{4}-\frac{2}{5}+\frac{3}{6}-\frac{4}{7}+\frac{5}{8}-\ldots$ is convergent or divergent.

See Solution

Problem 16869

Find the derivative of $f(x)=(2x+1)^{5}(3x-2)^{3}$ . What is $f'(x)$ ?

See Solution

Problem 16870

Find the volume $V$ of the solid formed by rotating the area between $x=1+y^{2}$ , $x=0$ , $y=1$ , and $y=3$ around the $x$ -axis using cylindrical shells. $v=$

See Solution

Problem 16871

Test the series for convergence or divergence: $\sum_{n=1}^{\infty}(-1)^{n} \frac{7 n-1}{3 n+1}$ . What is the limit as $n \to \infty$ ?

See Solution

Problem 16872

Calculate the average value of $f(x)=-\frac{3}{\sqrt[7]{x}}$ on the interval $[0,2]$ . Use exact fractions/roots.

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Problem 16873

Find $F(2)$ given $F(0)=-3$ and $F^{\prime}(x)=2x$ .

See Solution

Problem 16874

Prove the series converges and find how many terms are needed for an error less than 0.00005 in $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{9}}$ .

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Problem 16875

Untersuchen Sie die Folgen auf Monotonie und Beschränktheit:
(a) $a_{n}=\frac{n+3}{2 n}$ , (b) $a_{n}=n \cos (\pi(n+1))$ , (c) $a_{n}=1+\left(-\frac{3}{4}\right)^{n}$ , (d) $a_{n}=\cos \left(\frac{\pi}{6}\right) \sin \left(\frac{\pi n}{2}\right)$ .

See Solution

Problem 16876

Approximate the series sum to four decimal places: $\sum_{n=1}^{\infty} \frac{(-1)^{n-1} n^{2}}{10^{n}}$ .

See Solution

Problem 16877

Find integrals for the surface area of the curve $y=x^{6}$ from $0 \leq x \leq 1$ rotated about the $x$ -axis and $y$ -axis.

See Solution

Problem 16878

Determine the values of $p$ for which the series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{p+3}}$ converges.

See Solution

Problem 16879

Find the value of $\int_{-7}^{-2} F'(x) \, dx$ for the points (-8,2),(-7,-4),(-3,3),(-2,4),(0,4),(8,2).

See Solution

Problem 16880

Find the total distance a particle travels from $t=0$ to $t=2$ seconds with velocity $v(t)=2 t^{2}-2$ m/s.

See Solution

Problem 16881

Find the area of the surface formed by rotating the curve $9x = y^2 + 45$ for $5 \leq x \leq 9$ about the $x$ -axis.

See Solution

Problem 16882

Find the area of the surface formed by rotating the curve $9x = y^2 + 45$ from $x = 5$ to $x = 9$ around the $x$ -axis.

See Solution

Problem 16883

Find the total distance a particle travels from $t=0$ to $t=3$ given its velocity $v(t)=4t-6$ m/s. Enter an exact answer.

See Solution

Problem 16884

Find the total distance a particle travels from $t=0$ to $t=4$ seconds with velocity $v(t)=-t^{2}+2t+3$ .

See Solution

Problem 16885

Find $\int_{-3}^{3} F^{\prime}(x) \, dx$ for the linear function $F(x)$ defined by the points (-8,-7),(-3,-4),...,(8,6).

See Solution

Problem 16886

Find the integral of $\csc^{3}(3x) \cot^{5}(3x) \, dx$ .

See Solution

Problem 16887

Find the area of the surface formed by rotating the curve $y=\sqrt{1+5x}$ from $x=1$ to $x=7$ about the $x$ -axis.

See Solution

Problem 16888

A spring needs 5 J to stretch from 32 cm to 50 cm.
(a) Find work (in J) to stretch from 34 cm to 42 cm.
(b) How far (in cm) will 45 N stretch it?

See Solution

Problem 16889

Find the value of $F(1)$ given $F(0)=-1$ and the points of $F'(x)$ : $(-2,8),(-1,4),(0,0),(1,-4),(2,-8)$ .

See Solution

Problem 16890

According to Hooke's Law, the force to stretch a spring $x$ m is $f(x)=k x$ . Given $3 \mathrm{~J}$ to stretch from $26 \mathrm{~cm}$ to $39 \mathrm{~cm}$ , find $k$ in $\mathrm{N} / \mathrm{m}$ .
(a) Work to stretch from $30 \mathrm{~cm}$ to $34 \mathrm{~cm}$ ? (Round to two decimals.)
(b) Distance for $30 \mathrm{~N}$ force? (Round to one decimal.) $\mathrm{cm}$

See Solution

Problem 16891

Given $f(x)=\frac{2}{3} x^{3}-6 x^{2}+10$ , find: a) $y$ -intercept, b) critical points, c) increasing/decreasing intervals, d) concavity, e) sketch the graph.

See Solution

Problem 16892

In 2012, a city's population was 5.71M with a growth rate of $2.12\%$ . Find the growth function, population in 2018, time to reach 9M, and doubling time.

See Solution

Problem 16893

Evaluate the integral $\int \frac{\tan ^{2} \theta}{\sec \theta} d \theta - \int \frac{\tan \theta}{\sec \theta} d \theta$ .

See Solution

Problem 16894

Calculate the time for \$ 6,200 to double at a 3\% annual interest rate, compounded continuously. Round to 4 decimal places.

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Problem 16895

Hochbehälter: Gegeben ist $h(t)=\frac{1}{8} t^{2}-2 t+8$ . a) Graph zeichnen. b) Wann leer? c) Halb und 1/4 voll? d) Durchschnittliche Sinkgeschwindigkeit?

See Solution

Problem 16896

Evaluate the integral: $\int\left(\sec ^{2} \theta-1\right) \cos \theta d \theta$

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Problem 16897

A bacteria culture starts with 640 and grows proportionally. After 4 hours, there are 2560 bacteria.
(a) Find $P(t)$ as a function of $t$ . $P(t)=$ (b) What is the population after 7 hours? bacteria (c) How long to reach 1790 bacteria? hours

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Problem 16898

Ein Gartenpool wird mit einem Schlauch gefüllt. Bestimme die Füllhöhe nach 10 Minuten mit $v(t)=0,1 t^{2}-\frac{1}{t^{3}+1}+40$ .

See Solution

Problem 16899

Find the total mass of a 9 m rod with linear density $\rho(x)=7x+6$ kg/m.

See Solution

Problem 16900

Find the population in 1998 if the rate of change is given by $r(t)=36-3t^{2}$ and the population was 3000 in 1991.

See Solution

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Calculus

Problem 16801

Evaluate the integral ∫17x7dx=□\int \frac{1}{7 x^{7}} d x=\square∫7x71​dx=□ (Provide the exact answer.)

Problem 16802

Test if the series converge: 1) ∑n=1∞nnn!\sum_{n=1}^{\infty} \frac{n^{n}}{n !}∑n=1∞​n!nn​, 2) ∑n=1∞(2n+33n+2)n\sum_{n=1}^{\infty}\left(\frac{2 n+3}{3 n+2}\right)^{n}∑n=1∞​(3n+22n+3​)n.

Problem 16803

Evaluate the definite integral ∫03(7−x2)dx\int_{0}^{3}(7-x^{2})dx∫03​(7−x2)dx using its definition.

Problem 16804

Evaluate the integral from 0 to 3 of the function 7−x27 - x^{2}7−x2: ∫03(7−x2)dx\int_{0}^{3}\left(7-x^{2}\right) d x∫03​(7−x2)dx.

Problem 16805

Determine convergence or divergence of the series and if convergent, classify as conditional or absolute: ∑n=1∞nln⁡n(−2)n\sum_{n=1}^{\infty} \frac{n \ln n}{(-2)^{n}}n=1∑∞​(−2)nnlnn​

Problem 16806

Check if the series ∑n=1∞nln⁡n(−2)n\sum_{n=1}^{\infty} \frac{n \ln n}{(-2)^{n}}∑n=1∞​(−2)nnlnn​ converges or diverges.

Problem 16807

Find the indefinite integral: ∫481w2+81dw\int \frac{4}{81 w^{2}+81} d w∫81w2+814​dw and verify by differentiating your result.

Problem 16808

Problem 16809

Find the antiderivative FFF of f(u)=7eu+13f(u)=7 e^{u}+13f(u)=7eu+13 with F(0)=−4F(0)=-4F(0)=−4. What is F(u)F(u)F(u)?

Problem 16810

Find the indefinite integral and verify by differentiating: ∫881v2+81dv=□\int \frac{8}{81 v^{2}+81} d v=\square∫81v2+818​dv=□

Problem 16811

Determine convergence or divergence of the series. If it converges, classify as conditional or absolute: ∑n=1∞n2n(n+1)!3nn!\sum_{n=1}^{\infty} \frac{n 2^{n}(n+1) !}{3^{n} n !}n=1∑∞​3nn!n2n(n+1)!​

Problem 16812

Determine the position and velocity of an object with acceleration a(t)=−34a(t)=-34a(t)=−34, initial velocity v(0)=50v(0)=50v(0)=50, and initial position s(0)=10s(0)=10s(0)=10. Find v(t)=□v(t)=\squarev(t)=□.

Problem 16813

Find the velocity function v(t)v(t)v(t) given a(t)=4t2+7a(t)=4 t^{2}+7a(t)=4t2+7 and v(0)=3v(0)=3v(0)=3.

Problem 16814

Find the rate of change of the area AAA of a circle with radius rrr when r=4r=4r=4 and drdt=5\frac{d r}{d t}=5dtdr​=5.

Problem 16815

Is the function f(x)=∑n=0∞xe−nx2f(x)=\sum_{n=0}^{\infty} x e^{-n x^{2}}f(x)=∑n=0∞​xe−nx2 continuous at x=0x=0x=0? Justify your response.

Problem 16816

Evaluate the integral: ∫112x6dx=□(\int \frac{1}{12 x^{6}} d x=\square(∫12x61​dx=□( Type an exact answer.)

Problem 16817

Evaluate the integral: ∫15x5dx=□\int \frac{1}{5 x^{5}} d x=\square∫5x51​dx=□ (Type an exact answer.)

Problem 16818

Find the first four nonzero terms of the Taylor series for f(x)=sin⁡(−4x)f(x)=\sin (-4 x)f(x)=sin(−4x) around x=3x=3x=3.

Problem 16819

Is it true or false that if ∑n∈Nan\sum_{n \in \mathbb{N}} a_{n}∑n∈N​an​ is conditionally convergent, then ∑n∈Nnkan\sum_{n \in \mathbb{N}} n^{k} a_{n}∑n∈N​nkan​ diverges for each k∈(1,+∞)k \in(1,+\infty)k∈(1,+∞)? Prove or provide a counterexample.

Problem 16820

Find the particular solution of f′(x)=−8xf'(x)=-8xf′(x)=−8x with the initial condition f(1)=−2f(1)=-2f(1)=−2. What is f(x)f(x)f(x)?

Problem 16821

Prove that the function T(x)=∑n=2∞(1x−nπ+1nπ)T(x)=\sum_{n=2}^{\infty}\left(\frac{1}{x-n \pi}+\frac{1}{n \pi}\right)T(x)=∑n=2∞​(x−nπ1​+nπ1​) is defined for x∈[0,π]x \in [0, \pi]x∈[0,π].

Problem 16822

Find the rate of change of f(x)=x2+2xf(x)=x^{2}+2xf(x)=x2+2x from x=3x=3x=3 to x=5x=5x=5. The answer is 10. Explain this in context of the graph.

Problem 16823

Considérez la fonction f(x)=x−sin⁡(2x)f(x)=x-\sin (2 x)f(x)=x−sin(2x) sur [0,π2][0, \frac{\pi}{2}][0,2π​]. Trouvez f′(x)f'(x)f′(x), f′′(x)f''(x)f′′(x) et le tableau de variation.

Problem 16824

Calculate the infinite series: ∑n=1∞n2n(n+1)!3nn!\sum_{n=1}^{\infty} \frac{n 2^{n}(n+1) !}{3^{n} n !}∑n=1∞​3nn!n2n(n+1)!​.

Problem 16825

Find F′(x)F^{\prime}(x)F′(x) using the Second Fundamental Theorem of Calculus for F(x)=∫0xt41+t5dtF(x)=\int_{0}^{x} t^{4} \sqrt{1+t^{5}} dtF(x)=∫0x​t41+t5​dt.

Problem 16826

Find the average height change per foot for a 25 ft drop from 0 ft to -100 ft. Options: 4ft4 \mathrm{ft}4ft, −14ft-\frac{1}{4} \mathrm{ft}−41​ft, −4ft-4 \mathrm{ft}−4ft, 14ft\frac{1}{4} \mathrm{ft}41​ft.

Problem 16827

Find the sum of the series −((1/2))−((1/2))22−((1/2))33−⋯-((1 / 2))-\frac{((1 / 2))^{2}}{2}-\frac{((1 / 2))^{3}}{3}-\cdots−((1/2))−2((1/2))2​−3((1/2))3​−⋯ as a Taylor series.

Problem 16828

Calculate the sum of the series: 6−633!+655!−677!+⋯6 - \frac{6^3}{3!} + \frac{6^5}{5!} - \frac{6^7}{7!} + \cdots6−3!63​+5!65​−7!67​+⋯

Problem 16829

Find the first few coefficients cnc_ncn​ of the power series for f(x)=ln⁡(5−x)=∑n=0∞cnxnf(x)=\ln(5-x)=\sum_{n=0}^{\infty} c_{n} x^{n}f(x)=ln(5−x)=∑n=0∞​cn​xn.

Problem 16830

Find local maxima, minima, and saddle points of f(x,y)=x2+7xy+y2f(x, y)=x^{2}+7xy+y^{2}f(x,y)=x2+7xy+y2.

Problem 16831

Find local maxima, minima, and saddle points of f(x,y)=x2+14x+y2−4y+3f(x, y)=x^{2}+14 x+y^{2}-4 y+3f(x,y)=x2+14x+y2−4y+3. Options include points like (−7,2)(-7,2)(−7,2) and (7,−2)(7,-2)(7,−2).

Problem 16832

Find local maxima, minima, and saddle points of f(x,y)=10xy(x+y)+4f(x, y)=10xy(x+y)+4f(x,y)=10xy(x+y)+4. Identify points from the options given.

Problem 16833

Find local maxima, minima, and saddle points of f(x,y)=2−x4y4f(x, y)=2-x^{4} y^{4}f(x,y)=2−x4y4. Identify points from options A-G.

Problem 16834

Find local maxima, minima, and saddle points for f(x,y)=x2+2x+y2+12y+3f(x, y)=x^{2}+2x+y^{2}+12y+3f(x,y)=x2+2x+y2+12y+3.

Problem 16835

Find local maxima, minima, and saddle points of f(x,y)=x2+yx+y2f(x, y)=x^{2}+y x+y^{2}f(x,y)=x2+yx+y2. Options: (A) (B) (C) (D) (E) (F) (G).

Problem 16836

Find the degree 4 Taylor polynomial for f(x)=ln⁡(x8)f(x)=\ln(x^{8})f(x)=ln(x8) at a=1a=1a=1. Compute derivatives and evaluate them.

Problem 16837

Find the absolute max and min of f(x,y)=6x+7yf(x, y)=6x+7yf(x,y)=6x+7y in the triangle with vertices (0,0),(1,0),(0,1)(0,0),(1,0),(0,1)(0,0),(1,0),(0,1).

Problem 16838

Find the absolute max and min of f(x,y)=x2+xy+y2f(x, y)=x^{2}+xy+y^{2}f(x,y)=x2+xy+y2 on −8≤x,y≤8-8 \leq x, y \leq 8−8≤x,y≤8.

Problem 16839

lim⁡x→0(1x−1sin⁡x)\lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\sin x}\right)limx→0​(x1​−sinx1​) limitinin sonucu nedir? A) -1 B) −12-\frac{1}{2}−21​ C) 0 D) 12\frac{1}{2}21​ E) 1

Problem 16840

Find the concavity intervals for the function m(x)=x2+5x−5m(x)=\frac{x^{2}+5}{x-5}m(x)=x−5x2+5​.

Evaluate the integral $\int \frac{1}{7 x^{7}} d x=\square$ (Provide the exact answer.)

Test if the series converge: 1) $\sum_{n=1}^{\infty} \frac{n^{n}}{n !}$ , 2) $\sum_{n=1}^{\infty}\left(\frac{2 n+3}{3 n+2}\right)^{n}$ .

Evaluate the definite integral $\int_{0}^{3}(7-x^{2})dx$ using its definition.

Evaluate the integral from 0 to 3 of the function $7 - x^{2}$ : $\int_{0}^{3}\left(7-x^{2}\right) d x$ .

Determine convergence or divergence of the series and if convergent, classify as conditional or absolute: $\sum_{n=1}^{\infty} \frac{n \ln n}{(-2)^{n}}$

Check if the series $\sum_{n=1}^{\infty} \frac{n \ln n}{(-2)^{n}}$ converges or diverges.

Find the indefinite integral: $\int \frac{4}{81 w^{2}+81} d w$ and verify by differentiating your result.

Find the antiderivative $F$ of $f(u)=7 e^{u}+13$ with $F(0)=-4$ . What is $F(u)$ ?

Find the indefinite integral and verify by differentiating: $\int \frac{8}{81 v^{2}+81} d v=\square$

Determine convergence or divergence of the series. If it converges, classify as conditional or absolute: $\sum_{n=1}^{\infty} \frac{n 2^{n}(n+1) !}{3^{n} n !}$

Determine the position and velocity of an object with acceleration $a(t)=-34$ , initial velocity $v(0)=50$ , and initial position $s(0)=10$ . Find $v(t)=\square$ .

Find the velocity function $v(t)$ given $a(t)=4 t^{2}+7$ and $v(0)=3$ .

Find the rate of change of the area $A$ of a circle with radius $r$ when $r=4$ and $\frac{d r}{d t}=5$ .

Is the function $f(x)=\sum_{n=0}^{\infty} x e^{-n x^{2}}$ continuous at $x=0$ ? Justify your response.

Evaluate the integral: $\int \frac{1}{12 x^{6}} d x=\square($ Type an exact answer.)

Evaluate the integral: $\int \frac{1}{5 x^{5}} d x=\square$ (Type an exact answer.)

Find the first four nonzero terms of the Taylor series for $f(x)=\sin (-4 x)$ around $x=3$ .

Is it true or false that if $\sum_{n \in \mathbb{N}} a_{n}$ is conditionally convergent, then $\sum_{n \in \mathbb{N}} n^{k} a_{n}$ diverges for each $k \in(1,+\infty)$ ? Prove or provide a counterexample.

Find the particular solution of $f'(x)=-8x$ with the initial condition $f(1)=-2$ . What is $f(x)$ ?

Prove that the function $T(x)=\sum_{n=2}^{\infty}\left(\frac{1}{x-n \pi}+\frac{1}{n \pi}\right)$ is defined for $x \in [0, \pi]$ .

Find the rate of change of $f(x)=x^{2}+2x$ from $x=3$ to $x=5$ . The answer is 10. Explain this in context of the graph.

Considérez la fonction $f(x)=x-\sin (2 x)$ sur $[0, \frac{\pi}{2}]$ . Trouvez $f'(x)$ , $f''(x)$ et le tableau de variation.

Calculate the infinite series: $\sum_{n=1}^{\infty} \frac{n 2^{n}(n+1) !}{3^{n} n !}$ .

Find $F^{\prime}(x)$ using the Second Fundamental Theorem of Calculus for $F(x)=\int_{0}^{x} t^{4} \sqrt{1+t^{5}} dt$ .

Find the average height change per foot for a 25 ft drop from 0 ft to -100 ft. Options: $4 \mathrm{ft}$ , $-\frac{1}{4} \mathrm{ft}$ , $-4 \mathrm{ft}$ , $\frac{1}{4} \mathrm{ft}$ .

Find the sum of the series $-((1 / 2))-\frac{((1 / 2))^{2}}{2}-\frac{((1 / 2))^{3}}{3}-\cdots$ as a Taylor series.

Calculate the sum of the series: $6 - \frac{6^3}{3!} + \frac{6^5}{5!} - \frac{6^7}{7!} + \cdots$

Find the first few coefficients $c_n$ of the power series for $f(x)=\ln(5-x)=\sum_{n=0}^{\infty} c_{n} x^{n}$ .

Find local maxima, minima, and saddle points of $f(x, y)=x^{2}+7xy+y^{2}$ .

Find local maxima, minima, and saddle points of $f(x, y)=x^{2}+14 x+y^{2}-4 y+3$ . Options include points like $(-7,2)$ and $(7,-2)$ .

Find local maxima, minima, and saddle points of $f(x, y)=10xy(x+y)+4$ . Identify points from the options given.

Find local maxima, minima, and saddle points of $f(x, y)=2-x^{4} y^{4}$ . Identify points from options A-G.

Find local maxima, minima, and saddle points for $f(x, y)=x^{2}+2x+y^{2}+12y+3$ .

Find local maxima, minima, and saddle points of $f(x, y)=x^{2}+y x+y^{2}$ . Options: (A) (B) (C) (D) (E) (F) (G).

Find the degree 4 Taylor polynomial for $f(x)=\ln(x^{8})$ at $a=1$ . Compute derivatives and evaluate them.

Find the absolute max and min of $f(x, y)=6x+7y$ in the triangle with vertices $(0,0),(1,0),(0,1)$ .

Find the absolute max and min of $f(x, y)=x^{2}+xy+y^{2}$ on $-8 \leq x, y \leq 8$ .

$\lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\sin x}\right)$ limitinin sonucu nedir? A) -1 B) $-\frac{1}{2}$ C) 0 D) $\frac{1}{2}$ E) 1

Find the concavity intervals for the function $m(x)=\frac{x^{2}+5}{x-5}$ .

Given $f(t)=t^{2}+t-2$ , find: (a) average rate of change from $t=1$ to $t=2$ ; (b) instantaneous rate at $t=1$ .

An inverted pyramid with a square top (sides $2 \mathrm{~cm}$ ) is filled with water at $50 \mathrm{~cm}^3/\mathrm{s}$ . Find the water level rise rate when the level is $2 \mathrm{~cm}$ . $\mathrm{cm} / \mathrm{sec}$

Find local maxima of the function $h$ from its graph with a peak at (-2,2). (a) Local maximum locations: $\square$ (b) Local maximum values: $\square$

Find the extrema and inflection points of the function $g(t)=t^{2}(2t+5)$ .

Gegeben sind $f(x)=-x^{2}+4$ und $g(x)=x^{2}-5x+6$ . Bestimme $f'$ , $g'$ , Steigungsstelle, Extremstelle, Tangentengleichung und Schnittwinkel.

$|a|<1$ için dizinin $1+a+a^{\wedge} 2+\ldots+a^{\wedge} n$ monoton artan ve sınırlı olduğunu gösterin.

Given $f(x)=x(x+5)^{2}$ , find the intercepts, increasing/decreasing intervals, extremum points, concavity, and inflection points. Sketch the graph.

Finden Sie die Stammfunktion für die folgenden Funktionen: a) $f(x)=x^{4}$ , b) $f(x)=x^{3}-x^{2}$ , c) $f(x)=x^{n}$ , d) $f(x)=x^{-2}$ , e) $f(x)=\frac{1}{5} x^{-4}$ , f) $f(x)=\frac{1}{x}$ .

Calculate the integral: $\int x(2x^{2}+1) \, dx$

Evaluate the integral: $\int\left(\frac{12 \sqrt{3}}{3 \pi^{2}+x^{2}}\right)$ .

Calculate the integral $\int \frac{\ln ^{3} x}{x\left(\ln ^{4} x+5\right)} d x$ .

Calculate the integral: $\int\left(3 \sqrt{x}+\frac{1}{\sqrt{x}}\right) d x$ .

Bestimme die Ableitungsfunktion für die folgenden Funktionen: a) $f(x)=a x^{2}+b x+c$ , b) $f(x)=a x+c$ , c) $f(x)=x^{c+1}$ , d) $f(t)=t^{2}+3 t$ , e) $f(x)=x-t$ , f) $f(t)=x-t$ .

Gjeni integralet e mëposhtme: a) $\int(2x + 3x^5) \, dx$ b) $\int(x^{-4} - 4x^7) \, dx$ c) $\int(3\sqrt{x} + \frac{1}{\sqrt{x}}) \, dx$

Find the limit of $f(x)=\frac{x^{2}-1}{x-1}$ as $x$ approaches 1. Options: 1, 2, Undefined, 0.

Calculate the derivative of the function $f(x)=\sin (2 x)+\cos (3 x)$ .

Find the derivative of $f(x)=(x^{2}+1)x^{3/2}$ . What is $f'(x)$ ?

Find the derivative of $f(x)=e^{2 x^{2}-3 x+4}$ . What is $f'(x)$ ?

Find the derivative of $f(x)=\ln(2x^{3}-5x+1)$ . What is $f'(x)$ ?

Find the volume $V$ using cylindrical shells for the region bounded by $y=\frac{7}{x}$ , $y=0$ , $x=1$ , $x=4$ . $V=\square$

Find the average value of $f(x)=-2 x^{3}-x-3$ on $[1,3]$ . Provide the exact answer as a fraction if necessary.

Determine if the series $\frac{1}{4}-\frac{2}{5}+\frac{3}{6}-\frac{4}{7}+\frac{5}{8}-\ldots$ is convergent or divergent.

Find the derivative of $f(x)=(2x+1)^{5}(3x-2)^{3}$ . What is $f'(x)$ ?

Find the volume $V$ of the solid formed by rotating the area between $x=1+y^{2}$ , $x=0$ , $y=1$ , and $y=3$ around the $x$ -axis using cylindrical shells. $v=$

Test the series for convergence or divergence: $\sum_{n=1}^{\infty}(-1)^{n} \frac{7 n-1}{3 n+1}$ . What is the limit as $n \to \infty$ ?

Calculate the average value of $f(x)=-\frac{3}{\sqrt[7]{x}}$ on the interval $[0,2]$ . Use exact fractions/roots.

Find $F(2)$ given $F(0)=-3$ and $F^{\prime}(x)=2x$ .

Prove the series converges and find how many terms are needed for an error less than 0.00005 in $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{9}}$ .