Calculus

Problem 26801

A student designs a chess piece by rotating a region around the $y$ -axis. Find values of $a$ and $b$ for $C_{1}$ and the volume for $C_{2}$ .

See Solution

Problem 26802

Find the smallest value $c$ that meets the mean value theorem for $f(x)=5 x \sin (6 x)-\frac{1}{10} x^{3}$ on $[0, 3]$ .

See Solution

Problem 26803

Find the indefinite integral $\int e^{7x+5} dx$ using the substitution $u=7x+5$ , $du=7 dx$ .

See Solution

Problem 26804

Find $\mathrm{dh} / \mathrm{dt}$ when $h=3$ if the area under $y=f(x)$ from $0$ to $h$ increases at 2 sq. units/sec. $f(x)=\frac{1}{x^{4}+1}$

See Solution

Problem 26805

Find intercepts and asymptotes for $h(x)=\frac{x-6}{x^{2}-3x-28}$ . Analyze limits at vertical asymptotes.

See Solution

Problem 26806

Calculate the integral: $\int x e^{-4 x^{2}} d x=\square$

See Solution

Problem 26807

A student designs a chess piece by rotating curves $C_1$ and $C_2$ around the $y$ -axis. Find $a$ and $b$ in $x=\frac{a}{y+b}$ , then calculate the volume.

See Solution

Problem 26808

Find the degree 14 Taylor series at $x=0$ for $f(x)=\frac{d}{d x} \int_{0}^{x^{3}} \cos(t^{2}) dt$ .

See Solution

Problem 26809

Find the unique value of $x$ where the tangent lines to $6^{0.8 x}=(0.8 \ln 6) y$ and $0=(0.1 \ln 17) y-17^{0.1 x+0.5}$ are parallel.

See Solution

Problem 26810

Find the volume of the solid formed by rotating the region between $y=\mathrm{e}^{3 x}$ , $y=(2 x-1)^{4}$ , and $x=\frac{1}{2}$ about the $x$ -axis.

See Solution

Problem 26811

Evaluate the integral: $\int \frac{3 d x}{x \ln (4 x)}$

See Solution

Problem 26812

Determine if the polynomial $f(x)=2 x^{5}+10 x^{3}+7 x^{2}+9$ has a zero between -2 and -1 using the Intermediate Value Theorem.

See Solution

Problem 26813

Find the volume of the solid formed by revolving the region $R$ (bounded by $y$ -axis, $y=2$ , and $y=2 \sqrt[3]{x}$ ) around the $y$ -axis.

See Solution

Problem 26814

Find $\int x f(x) \, dx$ if $f^{\prime}(x) = -f(x)$ .

See Solution

Problem 26815

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

See Solution

Problem 26816

Find the solution to the differential equation $y^{\prime}=\ln(2x^{3}+5)$ .

See Solution

Problem 26817

Evaluate the series: $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$

See Solution

Problem 26818

Find the sum of the series $\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$ .

See Solution

Problem 26819

Calculate the integral $\int_{0}^{1} x f^{\prime \prime}(x) \, dx$ .

See Solution

Problem 26820

Quel est l'intervalle de convergence de la série $\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$ ?

See Solution

Problem 26821

Find the limit: $\lim _{x \rightarrow 0} \cot(5x)$ .

See Solution

Problem 26822

Calculate the limit: $\lim _{x \rightarrow-3}\left(\frac{x}{x+3}-\frac{18}{x^{2}-9}\right)$ .

See Solution

Problem 26823

Find the limit: $\lim _{x \rightarrow 0} \frac{\cot 4 x}{5 x}$ .

See Solution

Problem 26824

Find the limit: $\lim _{x \rightarrow 0^{+}}(x+1)^{1 / x}$ .

See Solution

Problem 26825

Find the limit: $\lim _{x \rightarrow 0^{+}}\left(x^{2} \ln \left(4 x^{2}\right)\right)$ .

See Solution

Problem 26826

Find the rate of change of a sphere's volume when its radius decreases at 3 ft/min and volume is 108 ft³. Round to three decimals.

See Solution

Problem 26827

Find the value of $\frac{d^{2}}{d x^{2}}(\sqrt{f(x)})$ at $x=2$ using the table values for $f, f', f''$ . Options: A) $\frac{3}{8}$ B) $\frac{5}{4}$ C) $-\frac{1}{32}$ D) $\frac{5}{8}$ .

See Solution

Problem 26828

Determine the radius $R$ and interval $I$ of convergence for the series $\sum_{n=1}^{\infty} \frac{x^{n+8}}{\sqrt{n}}$ .

See Solution

Problem 26829

Determine the radius $R$ and interval $I$ of convergence for the series $\sum_{n=2}^{\infty} \frac{(-1)^{n} x^{n+6}}{n+7}$ .

See Solution

Problem 26830

A right triangle has legs of 36 in and 48 in. The short leg increases by $8 \mathrm{in/sec}$ , and the long leg decreases by $8 \mathrm{in/sec}$ . Find the rate of change of the area.

See Solution

Problem 26831

A right triangle has legs 27 in and 36 in. If one leg grows at $10 \mathrm{in/sec}$ and the other shrinks at $9 \mathrm{in/sec}$ , find the area change rate.

See Solution

Problem 26832

Determine the radius $R$ and interval $I$ of convergence for the series $\sum_{n=1}^{\infty} \frac{x^{n+4}}{4 n !}$ .

See Solution

Problem 26833

A circle's radius decreases at 7 cm/min. When the radius is 3 cm, find the area change rate. Round to three decimals.

See Solution

Problem 26834

A circle's area increases by 55 sq ft/min. When the radius is 2 ft, find the radius's rate of change. Round to three decimals.

See Solution

Problem 26835

Find the radius and interval of convergence for the series $\sum_{n=3}^{\infty}(-1)^{n} \frac{n^{2} x^{n}}{5^{n}}$ . $R=\square$ ? $\vee$

See Solution

Problem 26836

A circle's area increases at 178 sq ft/sec. When the radius is 4 ft, find the radius's rate of change. Round to 3 decimal places.

See Solution

Problem 26837

Find the radius change rate when the area increases at 205 in²/min and the radius is 6 in. Round to three decimals.

See Solution

Problem 26838

Find the radius and interval of convergence for the series: $\sum_{n=3}^{\infty}(-1)^{n} \frac{n^{2} x^{n}}{5^{n}}$ . R = \square, I = ? \vee \square.

See Solution

Problem 26839

Find the radius and interval of convergence for the series $\sum_{n=1}^{\infty} \frac{x^{n}}{11^{n} n^{3}}$ .

See Solution

Problem 26840

Find the radius and interval of convergence for the series $\sum_{n=1}^{\infty} \frac{5^{n} x^{n}}{n^{4}}$ . $R=\square$ , $I=? \vee$

See Solution

Problem 26841

Find the derivative of $f(x)=3 x^{2}+2 x-14$ using first principles and confirm with shortcuts.

See Solution

Problem 26842

Find the power series for $f(x)=\frac{1}{5+x}$ centered at $x=0$ and its interval of convergence.

See Solution

Problem 26843

Find the tangent line equation to $f(x)=3 x^{2}+2 x-14$ with a slope of 4.

See Solution

Problem 26844

Find the radius and interval of convergence for the series $\sum_{n=1}^{\infty} \frac{5^{n} x^{n}}{n^{4}}$ .

See Solution

Problem 26845

Find the power series for $f(x)=\frac{4}{1-x^{3}}$ and its interval of convergence.

See Solution

Problem 26846

Find the power series for $f(x)=\frac{4}{3-x}$ centered at $x=0$ and its interval of convergence.

See Solution

Problem 26847

A water tank is an inverted pyramid with height $20 \mathrm{~m}$ and base $15 \mathrm{~m}$ . Water is pumped out at $70 \frac{\mathrm{m}^{3}}{\mathrm{~h}}$ . Find the rate of depth change when water is $4 \mathrm{~m}$ deep. Round to the nearest hundredth. Is depth increasing, decreasing, or constant?

See Solution

Problem 26848

Evaluate the integral $\int_{0.35}^{0.85} x(\sqrt{1-x^{2}}+1) \, dx$ .

See Solution

Problem 26849

Find $\lim _{x \rightarrow 2} \frac{2 x-4}{f(3 x)-1}$ using the values of $f$ and $f'$ from the table provided.

See Solution

Problem 26850

Evaluate the integral: $\int \frac{e^{x}}{e^{2 x}+3 e^{x}+2} d x$

See Solution

Problem 26851

Find the power series for $f(x)=\frac{x}{10 x^{2}+1}$ centered at $x=0$ and its interval of convergence.

See Solution

Problem 26852

Find the power series for $f(x)=\frac{4}{17+x}$ and its interval of convergence.

See Solution

Problem 26853

Find $\lim _{x \rightarrow 2} \frac{f(3 x)-2}{4-2 x}$ using the given values of $f$ and $f'$ .

See Solution

Problem 26854

Calculate the beam deflection given: $I = 5.4 \text{ in}^4$ , $E = 2.75 \times 10^{6} \text{ psi}$ , $L = 10 \text{ ft}$ , $P = 550 \text{ lb}$ .

See Solution

Problem 26855

Find the limit: $\lim _{x \rightarrow 7} \frac{f(x)-1}{x^{2}-7 x}$ .

See Solution

Problem 26856

Find the power series for $f(x)=\frac{2+x}{1-x}$ centered at $x=0$ and its interval of convergence.

See Solution

Problem 26857

Find the interval of convergence for the series:
$\sum_{n=1}^{\infty}\left[(-1)^{n+1}-\frac{1}{10^{n+1}}\right] x^{n}$

See Solution

Problem 26858

Analyze the function $f(x)$ and its second derivative $f^{\prime \prime}(x)$ based on the behavior of $f^{\prime}(x)$ given.

See Solution

Problem 26859

Find the power series for $f(x)=\ln(9-x)$ and determine the radius of convergence, $R$ .

See Solution

Problem 26860

Evaluate the integral $\int \sum_{n=0}^{\infty} \frac{t}{1-t^{5}} dt$ and find $R=\square+C$ .

See Solution

Problem 26861

Find the limit: $\lim _{x \rightarrow 7} \frac{f(x)-1}{x^{2}-7 x}$ given that $f(x)$ approaches 1 as $x$ approaches 7.

See Solution

Problem 26862

Find the Taylor series for $f$ at 5 with $f^{(n)}(5)=\frac{(-1)^{n} n !}{4^{n}(n+3)}$ . What is the radius of convergence, $R$ ?

See Solution

Problem 26863

Find the Maclaurin series for $f(x)=(1-x)^{-2}$ and its radius of convergence $R$ .

See Solution

Problem 26864

Find the Maclaurin series for $f(x) = \ln(1 + 5x)$ and its radius of convergence, $R$ .

See Solution

Problem 26865

Find the Maclaurin series for $f(x)$ given $\sum_{n=0}^{\infty} f(x)=\frac{\cos 5 x}{8}$ .

See Solution

Problem 26866

Find the Maclaurin series for $f(x) = x e^{10x}$ and determine the radius of convergence, $R$ .

See Solution

Problem 26867

Find the Maclaurin series for $f(x) = 3 e^{2x}$ and its radius of convergence $R$ .

See Solution

Problem 26868

Find the Taylor series for $f(x) = \frac{4}{x}$ centered at $a = -4$ : $\sum_{n=0}^{\infty} \square$

See Solution

Problem 26869

Find the Taylor series for $f(x) = 2 \cos x$ at $a = 7\pi$ . No need to show that $R(x) \rightarrow 0$ .

See Solution

Problem 26870

Given the equations $x=6+t^{2}$ and $y=t^{2}+t^{3}$ , find $\frac{dy}{dx}$ and $\frac{d^{2}y}{dx^{2}}$ . Also, determine when the curve is concave upward.

See Solution

Problem 26871

Problem 4: Given the series $J_{0}(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{2^{2 n}(n !)^{2}}$ , find the interval of convergence, compute $J_{0}^{\prime}(x)$ and $J_{0}^{\prime \prime}(x)$ , and explain why $x^{2} J_{0}^{\prime \prime}(x)+x J_{0}^{\prime}(x)+x^{2} J_{0}(x)=0$ .

See Solution

Problem 26872

Find the area between the x-axis and the curve defined by $x=4+e^{t}$ and $y=t-t^{2}$ .

See Solution

Problem 26873

Find the length of the curve defined by $x=6+6t^{2}$ and $y=1+4t^{3}$ for $0 \leq t \leq 3$ .

See Solution

Problem 26874

Find the length of the polar curves: $r=2 \sin(\theta)$ for $0 \leq \theta \leq \frac{\pi}{3}$ and $r=e^{7\theta}$ for $0 \leq \theta \leq \pi$ .

See Solution

Problem 26875

A cylinder's radius decreases at 2 in/sec and height increases at 5 in/sec. Find volume change rate when $r=4$ , $h=6$ . $V=\pi r^{2} h$

See Solution

Problem 26876

Find the length of the polar curve $r=2 \sin (\theta)$ for $0 \leq \theta \leq \frac{\pi}{3}$ .

See Solution

Problem 26877

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{x}-1-x}{x^{2}}$

See Solution

Problem 26878

Find the limits and derivatives:
1. $\lim _{x \rightarrow 1} \frac{x-1}{x^{2}-8 x+7}$
2. $\frac{d}{d x} \frac{\tan x}{x^{3}}$
3. $\lim _{x \rightarrow 0} \frac{7 e^{x}-7-7 x}{1-\cos x}$
4. $\lim _{x \rightarrow 0} \frac{e^{x}-1-x}{x^{2}}$
For the tangent line $y=4x+7$ at $x=-3$ , which statements are true? I. $f(0)=7$ II. $f(-3)=-5$ III. $f^{\prime}(-3)=4$ (A) I only (B) II only (C) III only (D) I and III only (E) II and III only

See Solution

Problem 26879

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{x}-1-x}{x^{2}}$ .

See Solution

Problem 26880

Find the vertical speed of an ant on the curve $x^{2}+x y+y^{2}=19$ at $(2,3)$ moving right at $3 \mathrm{~cm/sec}$ .

See Solution

Problem 26881

Find the true statements about the function $f$ given the tangent line $y=4x+7$ at $x=-3$ . Also, how fast is the area of ripples growing when radius is 3 ft?

See Solution

Problem 26882

A stone creates ripples in a pond. If the radius grows at $0.5 \mathrm{ft}/\mathrm{sec}$ , find the area growth rate at $r=3$ ft. Area: $A=\pi r^{2}$ .

See Solution

Problem 26883

Untersuche den Abbau eines Kontrastmittels:
a) Finde die Funktionsgleichung. b) Wie viel bleibt nach 60 Minuten? c) Wie lange bis 95 % abgebaut sind? d) Ist nach 24 Stunden kaum noch Kontrastmittel vorhanden?

See Solution

Problem 26884

Find the tangent line to $y=f(x)$ where $f(x)=x^{5}$ at $x=2$ with a slope of 80. Equation: $\square$ .

See Solution

Problem 26885

A 7.5 MeV proton hits ${ }_{3}^{7} \mathrm{Li}$ . Find the scattered proton energy at $\theta=90^{\circ}$ for: a) no excitation, b) excitation at 0.477 MeV.

See Solution

Problem 26886

Find the average revenue change from 1,000 to 1,050 car seats for $R(x)=24x-0.010x^{2}$ , $0 \leq x \leq 2400$ .

See Solution

Problem 26887

Bestimmen Sie die Punkte, wo der Graph von $f$ die Steigung $m$ hat, und geben Sie die Tangentengleichungen an für: a) $f(x)=\frac{1}{3} x^{3}-8 x ; m=1$ b) $f(x)=(2 x+1)^{2} ; m=8$ c) $f(x)=x^{3}-3 x^{2}+6 ; m=0$ d) $f(x)=-\frac{4}{x} ; m=1$ e) $f(x)=\frac{1}{x^{2}}+2 x ; m=\frac{9}{4}$ f) $f(x)=x^{5}+5 x^{3}+3 ; m=4$

See Solution

Problem 26888

Bestimmen Sie die Nullstellen der Tangente an den Graphen von $f(x)=-\frac{1}{2} x^{3}+\frac{1}{2} x+2$ und $f(x)=4 \sqrt{x}$ im Punkt B(1|f(1)).

See Solution

Problem 26889

Zbadaj jednostajną ciągłość i lipschitzowskość funkcji: a. $\sqrt{x}$ dla $x \geqslant 0$ ; b. $x^{2}$ dla $x \in \mathbb{R}$ .

See Solution

Problem 26890

Berechne: $\lim _{x \rightarrow \infty} \frac{3 x+2}{x}$ mit Testeinsetzung.

See Solution

Problem 26891

Differentiate these using Leibniz's Rule: a. $f(x)=\int_{1}^{x^{3}-x} \sqrt{2+t} dt$ b. $f(x)=\int_{2}^{\ln (x)} e^{-2 t} dt, \quad x>0$ c. $f(x)=\int_{e^{3 x^{2}}}^{\arctan (x)} \sin (t) dt$

See Solution

Problem 26892

Calculate these indefinite integrals: a. $\int\left(x^{\frac{1}{4}}+\frac{1}{3 x^{\frac{1}{4}}}\right) d x$ b. $\int\left(x^{-6}+6^{-x}\right) d x$ c. $\int\left(\frac{7 x+2}{x}\right) d x$ d. $\int\left(\frac{1}{\sqrt{x}}+\sqrt{e^{x}}\right) d x$

See Solution

Problem 26893

Berechnen Sie für $f(x)=x^{4}-3x^{2}-4$ : a) Liegt $Q(2,5 \mid 16,125)$ auf $f$ ? b) Achsenschnittpunkte.

See Solution

Problem 26894

Berechne die mittlere Änderungsrate (Differenzenquotient) für: a) Punkte $P(-2, 2.5)$ und $Q(1.5, 13)$ , b) im Intervall $[-2, 3]$ für $g(x)=x^{3}$ .

See Solution

Problem 26895

Berechnen Sie die mittlere Änderungsrate zwischen den Punkten $P(-2, 2,5)$ und $Q(1,5, 13)$ sowie im Intervall $[-2, 3]$ für $g(x)=x^{3}$ .

See Solution

Problem 26896

Evaluate these definite integrals: a. $\int_{0}^{1 / 2} \sin (\pi(x+1)) d x$ b. $\int_{4}^{9}\left(\frac{1+x}{\sqrt{x}}\right) d x$ c. $\int_{1}^{4}\left(\frac{3}{5 x-1}\right) d x$ d. $\int_{1}^{2}\left(\frac{1}{\sqrt{x}}+\sqrt{e^{x}}\right) d x$

See Solution

Problem 26897

Find the horizontal asymptote of $f(x)=\frac{1}{x-3}$ . Options: a) b) $y=2$ c) $y=3$ d) $y=0$

See Solution

Problem 26898

If $\int_{8}^{14} f(x) d x=10$ , find $\int_{2}^{4} f(3 x+2) d x$ . Options: A) 30 B) $\frac{10}{3}$ C) $\frac{5}{3}$ D) $\frac{20}{3}$

See Solution

Problem 26899

Calculate $\int_{-1}^{5} f(x) \, dx$ for $f(x)=\begin{cases} 3x^{2}, & x \leq 0 \\ -1, & x > 0 \end{cases}$ .

See Solution

Problem 26900

Find $\lim _{x \rightarrow-2^{+}} f(x)$ for the piecewise function $f(x)$ defined as: $3x-3$ , $x^{2}-10$ , $-2-2x$ .

See Solution

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Calculus

Problem 26801

A student designs a chess piece by rotating a region around the yyy-axis. Find values of aaa and bbb for C1C_{1}C1​ and the volume for C2C_{2}C2​.

Problem 26802

Find the smallest value ccc that meets the mean value theorem for f(x)=5xsin⁡(6x)−110x3f(x)=5 x \sin (6 x)-\frac{1}{10} x^{3}f(x)=5xsin(6x)−101​x3 on [0,3][0, 3][0,3].

Problem 26803

Find the indefinite integral ∫e7x+5dx\int e^{7x+5} dx∫e7x+5dx using the substitution u=7x+5u=7x+5u=7x+5, du=7dxdu=7 dxdu=7dx.

Problem 26804

Find dh/dt\mathrm{dh} / \mathrm{dt}dh/dt when h=3h=3h=3 if the area under y=f(x)y=f(x)y=f(x) from 000 to hhh increases at 2 sq. units/sec. f(x)=1x4+1f(x)=\frac{1}{x^{4}+1}f(x)=x4+11​

Problem 26805

Find intercepts and asymptotes for h(x)=x−6x2−3x−28h(x)=\frac{x-6}{x^{2}-3x-28}h(x)=x2−3x−28x−6​. Analyze limits at vertical asymptotes.

Problem 26806

Calculate the integral: ∫xe−4x2dx=□\int x e^{-4 x^{2}} d x=\square∫xe−4x2dx=□

Problem 26807

A student designs a chess piece by rotating curves C1C_1C1​ and C2C_2C2​ around the yyy-axis. Find aaa and bbb in x=ay+bx=\frac{a}{y+b}x=y+ba​, then calculate the volume.

Problem 26808

Find the degree 14 Taylor series at x=0x=0x=0 for f(x)=ddx∫0x3cos⁡(t2)dtf(x)=\frac{d}{d x} \int_{0}^{x^{3}} \cos(t^{2}) dtf(x)=dxd​∫0x3​cos(t2)dt.

Problem 26809

Find the unique value of xxx where the tangent lines to 60.8x=(0.8ln⁡6)y6^{0.8 x}=(0.8 \ln 6) y60.8x=(0.8ln6)y and 0=(0.1ln⁡17)y−170.1x+0.50=(0.1 \ln 17) y-17^{0.1 x+0.5}0=(0.1ln17)y−170.1x+0.5 are parallel.

Problem 26810

Find the volume of the solid formed by rotating the region between y=e3xy=\mathrm{e}^{3 x}y=e3x, y=(2x−1)4y=(2 x-1)^{4}y=(2x−1)4, and x=12x=\frac{1}{2}x=21​ about the xxx-axis.

Problem 26811

Evaluate the integral: ∫3dxxln⁡(4x)\int \frac{3 d x}{x \ln (4 x)}∫xln(4x)3dx​

Problem 26812

Determine if the polynomial f(x)=2x5+10x3+7x2+9f(x)=2 x^{5}+10 x^{3}+7 x^{2}+9f(x)=2x5+10x3+7x2+9 has a zero between -2 and -1 using the Intermediate Value Theorem.

Problem 26813

Find the volume of the solid formed by revolving the region RRR (bounded by yyy-axis, y=2y=2y=2, and y=2x3y=2 \sqrt[3]{x}y=23x​) around the yyy-axis.

Problem 26814

Find ∫xf(x) dx\int x f(x) \, dx∫xf(x)dx if f′(x)=−f(x)f^{\prime}(x) = -f(x)f′(x)=−f(x).

Problem 26815

Find the value of the series: ∑n=1∞(−1)n1n\sum_{n=1}^{\infty}(-1)^{n} \frac{1}{\sqrt{n}}n=1∑∞​(−1)nn​1​.

Problem 26816

Find the solution to the differential equation y′=ln⁡(2x3+5)y^{\prime}=\ln(2x^{3}+5)y′=ln(2x3+5).

Problem 26817

Evaluate the series: ∑n=1∞1n(n+1)\sum_{n=1}^{\infty} \frac{1}{n(n+1)}∑n=1∞​n(n+1)1​

Problem 26818

Find the sum of the series ∑n=0∞xnn!\sum_{n=0}^{\infty} \frac{x^{n}}{n !}∑n=0∞​n!xn​.

Problem 26819

Calculate the integral ∫01xf′′(x) dx\int_{0}^{1} x f^{\prime \prime}(x) \, dx∫01​xf′′(x)dx.

Problem 26820

Quel est l'intervalle de convergence de la série ∑n=0∞xnn!\sum_{n=0}^{\infty} \frac{x^{n}}{n !}∑n=0∞​n!xn​ ?

Problem 26821

Find the limit: lim⁡x→0cot⁡(5x)\lim _{x \rightarrow 0} \cot(5x)limx→0​cot(5x).

Problem 26822

Calculate the limit: lim⁡x→−3(xx+3−18x2−9)\lim _{x \rightarrow-3}\left(\frac{x}{x+3}-\frac{18}{x^{2}-9}\right)limx→−3​(x+3x​−x2−918​).

Problem 26823

Find the limit: lim⁡x→0cot⁡4x5x\lim _{x \rightarrow 0} \frac{\cot 4 x}{5 x}limx→0​5xcot4x​.

Problem 26824

Find the limit: lim⁡x→0+(x+1)1/x\lim _{x \rightarrow 0^{+}}(x+1)^{1 / x}limx→0+​(x+1)1/x.

Problem 26825

Find the limit: lim⁡x→0+(x2ln⁡(4x2))\lim _{x \rightarrow 0^{+}}\left(x^{2} \ln \left(4 x^{2}\right)\right)limx→0+​(x2ln(4x2)).

Problem 26826

Find the rate of change of a sphere's volume when its radius decreases at 3 ft/min and volume is 108 ft³. Round to three decimals.

Problem 26827

Find the value of d2dx2(f(x))\frac{d^{2}}{d x^{2}}(\sqrt{f(x)})dx2d2​(f(x)​) at x=2x=2x=2 using the table values for f,f′,f′′f, f', f''f,f′,f′′. Options: A) 38\frac{3}{8}83​ B) 54\frac{5}{4}45​ C) −132-\frac{1}{32}−321​ D) 58\frac{5}{8}85​.

Problem 26828

Determine the radius RRR and interval III of convergence for the series ∑n=1∞xn+8n\sum_{n=1}^{\infty} \frac{x^{n+8}}{\sqrt{n}}∑n=1∞​n​xn+8​.

Problem 26829

Determine the radius RRR and interval III of convergence for the series ∑n=2∞(−1)nxn+6n+7\sum_{n=2}^{\infty} \frac{(-1)^{n} x^{n+6}}{n+7}∑n=2∞​n+7(−1)nxn+6​.

Problem 26830

A right triangle has legs of 36 in and 48 in. The short leg increases by 8in/sec8 \mathrm{in/sec}8in/sec, and the long leg decreases by 8in/sec8 \mathrm{in/sec}8in/sec. Find the rate of change of the area.

Problem 26831

A right triangle has legs 27 in and 36 in. If one leg grows at 10in/sec10 \mathrm{in/sec}10in/sec and the other shrinks at 9in/sec9 \mathrm{in/sec}9in/sec, find the area change rate.

Problem 26832

Determine the radius RRR and interval III of convergence for the series ∑n=1∞xn+44n!\sum_{n=1}^{\infty} \frac{x^{n+4}}{4 n !}∑n=1∞​4n!xn+4​.

Problem 26833

A circle's radius decreases at 7 cm/min. When the radius is 3 cm, find the area change rate. Round to three decimals.

Problem 26834

A circle's area increases by 55 sq ft/min. When the radius is 2 ft, find the radius's rate of change. Round to three decimals.

Problem 26835

Find the radius and interval of convergence for the series ∑n=3∞(−1)nn2xn5n\sum_{n=3}^{\infty}(-1)^{n} \frac{n^{2} x^{n}}{5^{n}}∑n=3∞​(−1)n5nn2xn​. R=□R=\squareR=□ ? ∨\vee∨

Problem 26836

A circle's area increases at 178 sq ft/sec. When the radius is 4 ft, find the radius's rate of change. Round to 3 decimal places.

Problem 26837

Find the radius change rate when the area increases at 205 in²/min and the radius is 6 in. Round to three decimals.

Problem 26838

Find the radius and interval of convergence for the series: ∑n=3∞(−1)nn2xn5n\sum_{n=3}^{\infty}(-1)^{n} \frac{n^{2} x^{n}}{5^{n}}n=3∑∞​(−1)n5nn2xn​. R = \square, I = ? \vee \square.

Problem 26839

Find the radius and interval of convergence for the series ∑n=1∞xn11nn3\sum_{n=1}^{\infty} \frac{x^{n}}{11^{n} n^{3}}∑n=1∞​11nn3xn​.

Problem 26840

A student designs a chess piece by rotating a region around the $y$ -axis. Find values of $a$ and $b$ for $C_{1}$ and the volume for $C_{2}$ .

Find the smallest value $c$ that meets the mean value theorem for $f(x)=5 x \sin (6 x)-\frac{1}{10} x^{3}$ on $[0, 3]$ .

Find the indefinite integral $\int e^{7x+5} dx$ using the substitution $u=7x+5$ , $du=7 dx$ .

Find $\mathrm{dh} / \mathrm{dt}$ when $h=3$ if the area under $y=f(x)$ from $0$ to $h$ increases at 2 sq. units/sec. $f(x)=\frac{1}{x^{4}+1}$

Find intercepts and asymptotes for $h(x)=\frac{x-6}{x^{2}-3x-28}$ . Analyze limits at vertical asymptotes.

Calculate the integral: $\int x e^{-4 x^{2}} d x=\square$

A student designs a chess piece by rotating curves $C_1$ and $C_2$ around the $y$ -axis. Find $a$ and $b$ in $x=\frac{a}{y+b}$ , then calculate the volume.

Find the degree 14 Taylor series at $x=0$ for $f(x)=\frac{d}{d x} \int_{0}^{x^{3}} \cos(t^{2}) dt$ .

Find the unique value of $x$ where the tangent lines to $6^{0.8 x}=(0.8 \ln 6) y$ and $0=(0.1 \ln 17) y-17^{0.1 x+0.5}$ are parallel.

Find the volume of the solid formed by rotating the region between $y=\mathrm{e}^{3 x}$ , $y=(2 x-1)^{4}$ , and $x=\frac{1}{2}$ about the $x$ -axis.

Evaluate the integral: $\int \frac{3 d x}{x \ln (4 x)}$

Determine if the polynomial $f(x)=2 x^{5}+10 x^{3}+7 x^{2}+9$ has a zero between -2 and -1 using the Intermediate Value Theorem.

Find the volume of the solid formed by revolving the region $R$ (bounded by $y$ -axis, $y=2$ , and $y=2 \sqrt[3]{x}$ ) around the $y$ -axis.

Find $\int x f(x) \, dx$ if $f^{\prime}(x) = -f(x)$ .

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

Find the solution to the differential equation $y^{\prime}=\ln(2x^{3}+5)$ .

Evaluate the series: $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$

Find the sum of the series $\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$ .

Calculate the integral $\int_{0}^{1} x f^{\prime \prime}(x) \, dx$ .

Quel est l'intervalle de convergence de la série $\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$ ?

Find the limit: $\lim _{x \rightarrow 0} \cot(5x)$ .

Calculate the limit: $\lim _{x \rightarrow-3}\left(\frac{x}{x+3}-\frac{18}{x^{2}-9}\right)$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\cot 4 x}{5 x}$ .

Find the limit: $\lim _{x \rightarrow 0^{+}}(x+1)^{1 / x}$ .

Find the limit: $\lim _{x \rightarrow 0^{+}}\left(x^{2} \ln \left(4 x^{2}\right)\right)$ .

Find the value of $\frac{d^{2}}{d x^{2}}(\sqrt{f(x)})$ at $x=2$ using the table values for $f, f', f''$ . Options: A) $\frac{3}{8}$ B) $\frac{5}{4}$ C) $-\frac{1}{32}$ D) $\frac{5}{8}$ .

Determine the radius $R$ and interval $I$ of convergence for the series $\sum_{n=1}^{\infty} \frac{x^{n+8}}{\sqrt{n}}$ .

Determine the radius $R$ and interval $I$ of convergence for the series $\sum_{n=2}^{\infty} \frac{(-1)^{n} x^{n+6}}{n+7}$ .

A right triangle has legs of 36 in and 48 in. The short leg increases by $8 \mathrm{in/sec}$ , and the long leg decreases by $8 \mathrm{in/sec}$ . Find the rate of change of the area.

A right triangle has legs 27 in and 36 in. If one leg grows at $10 \mathrm{in/sec}$ and the other shrinks at $9 \mathrm{in/sec}$ , find the area change rate.

Determine the radius $R$ and interval $I$ of convergence for the series $\sum_{n=1}^{\infty} \frac{x^{n+4}}{4 n !}$ .

Find the radius and interval of convergence for the series $\sum_{n=3}^{\infty}(-1)^{n} \frac{n^{2} x^{n}}{5^{n}}$ . $R=\square$ ? $\vee$

Find the radius and interval of convergence for the series: $\sum_{n=3}^{\infty}(-1)^{n} \frac{n^{2} x^{n}}{5^{n}}$ . R = \square, I = ? \vee \square.

Find the radius and interval of convergence for the series $\sum_{n=1}^{\infty} \frac{x^{n}}{11^{n} n^{3}}$ .

Find the radius and interval of convergence for the series $\sum_{n=1}^{\infty} \frac{5^{n} x^{n}}{n^{4}}$ . $R=\square$ , $I=? \vee$

Find the derivative of $f(x)=3 x^{2}+2 x-14$ using first principles and confirm with shortcuts.

Find the power series for $f(x)=\frac{1}{5+x}$ centered at $x=0$ and its interval of convergence.

Find the tangent line equation to $f(x)=3 x^{2}+2 x-14$ with a slope of 4.

Find the radius and interval of convergence for the series $\sum_{n=1}^{\infty} \frac{5^{n} x^{n}}{n^{4}}$ .

Find the power series for $f(x)=\frac{4}{1-x^{3}}$ and its interval of convergence.

Find the power series for $f(x)=\frac{4}{3-x}$ centered at $x=0$ and its interval of convergence.

Evaluate the integral $\int_{0.35}^{0.85} x(\sqrt{1-x^{2}}+1) \, dx$ .

Find $\lim _{x \rightarrow 2} \frac{2 x-4}{f(3 x)-1}$ using the values of $f$ and $f'$ from the table provided.

Evaluate the integral: $\int \frac{e^{x}}{e^{2 x}+3 e^{x}+2} d x$

Find the power series for $f(x)=\frac{x}{10 x^{2}+1}$ centered at $x=0$ and its interval of convergence.

Find the power series for $f(x)=\frac{4}{17+x}$ and its interval of convergence.

Find $\lim _{x \rightarrow 2} \frac{f(3 x)-2}{4-2 x}$ using the given values of $f$ and $f'$ .

Calculate the beam deflection given: $I = 5.4 \text{ in}^4$ , $E = 2.75 \times 10^{6} \text{ psi}$ , $L = 10 \text{ ft}$ , $P = 550 \text{ lb}$ .

Find the limit: $\lim _{x \rightarrow 7} \frac{f(x)-1}{x^{2}-7 x}$ .

Find the power series for $f(x)=\frac{2+x}{1-x}$ centered at $x=0$ and its interval of convergence.

Find the interval of convergence for the series:
$\sum_{n=1}^{\infty}\left[(-1)^{n+1}-\frac{1}{10^{n+1}}\right] x^{n}$

Analyze the function $f(x)$ and its second derivative $f^{\prime \prime}(x)$ based on the behavior of $f^{\prime}(x)$ given.

Find the power series for $f(x)=\ln(9-x)$ and determine the radius of convergence, $R$ .

Evaluate the integral $\int \sum_{n=0}^{\infty} \frac{t}{1-t^{5}} dt$ and find $R=\square+C$ .

Find the limit: $\lim _{x \rightarrow 7} \frac{f(x)-1}{x^{2}-7 x}$ given that $f(x)$ approaches 1 as $x$ approaches 7.

Find the Taylor series for $f$ at 5 with $f^{(n)}(5)=\frac{(-1)^{n} n !}{4^{n}(n+3)}$ . What is the radius of convergence, $R$ ?

Find the Maclaurin series for $f(x)=(1-x)^{-2}$ and its radius of convergence $R$ .

Find the Maclaurin series for $f(x) = \ln(1 + 5x)$ and its radius of convergence, $R$ .

Find the Maclaurin series for $f(x)$ given $\sum_{n=0}^{\infty} f(x)=\frac{\cos 5 x}{8}$ .

Find the Maclaurin series for $f(x) = x e^{10x}$ and determine the radius of convergence, $R$ .

Find the Maclaurin series for $f(x) = 3 e^{2x}$ and its radius of convergence $R$ .

Find the Taylor series for $f(x) = \frac{4}{x}$ centered at $a = -4$ : $\sum_{n=0}^{\infty} \square$

Find the Taylor series for $f(x) = 2 \cos x$ at $a = 7\pi$ . No need to show that $R(x) \rightarrow 0$ .

Given the equations $x=6+t^{2}$ and $y=t^{2}+t^{3}$ , find $\frac{dy}{dx}$ and $\frac{d^{2}y}{dx^{2}}$ . Also, determine when the curve is concave upward.

Find the area between the x-axis and the curve defined by $x=4+e^{t}$ and $y=t-t^{2}$ .

Find the length of the curve defined by $x=6+6t^{2}$ and $y=1+4t^{3}$ for $0 \leq t \leq 3$ .

Find the length of the polar curves: $r=2 \sin(\theta)$ for $0 \leq \theta \leq \frac{\pi}{3}$ and $r=e^{7\theta}$ for $0 \leq \theta \leq \pi$ .