Calculus

Problem 17501

Calculate the average value $f_{\text {ave }}$ of the function $f(x)=3 x^{2}+4 x$ over the interval $[-1,5]$ .

See Solution

Problem 17502

Quelle est la forme de la solution particulière $y_{p}$ pour l'équation $y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=4 x e^{2 x} \cos (3 x)$ ?

See Solution

Problem 17503

Calculate the average value $g_{\text {ave }}$ of the function $g(t)=\frac{3}{1+t^{2}}$ over the interval $[0,5]$ .

See Solution

Problem 17504

Calculate the average value $h_{\text{ave}}$ of the function $h(x)=7 \cos^{4}(x) \sin(x)$ over the interval $[0, \pi]$ .

See Solution

Problem 17505

Evaluate the integral: $\int \frac{\sin (28 x)}{1+\cos ^{2}(28 x)} d x$ (Use $C$ for the constant).

See Solution

Problem 17506

Déterminez le domaine de la fonction $f(x, y)=\ln(2x^{2}+3y^{2})$ .

See Solution

Problem 17507

Trouver la forme de la solution particulière $y_{p}$ pour l'équation $y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=r(x)$ avec $r(x)=4 x e^{2 x} \cos (3 x)$ .

See Solution

Problem 17508

Trouver la forme de la solution particulière $y_{p}$ pour l'équation $y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=r(x)$ .

See Solution

Problem 17509

Evaluate the integral: $\int_{0}^{16} \frac{x}{\sqrt{1+5 x}} d x$

See Solution

Problem 17510

Équation différentielle avec polynôme $p(\lambda)=(\lambda-1)(\lambda+1)^{2}(\lambda^{2}+9)$ , trouver $y_{p}$ .

See Solution

Problem 17511

Trouvez $\alpha$ , $\beta$ , $C_{1}$ et $C_{2}$ pour que $y=e^{\alpha x}(C_{1} \cos (\beta x)+C_{2} \sin (\beta x))$ soit solution de $y^{\prime \prime}+16 y=0$ avec $y(\pi / 16)=-2 \sqrt{2}$ et $y^{\prime}(\pi / 16)=12 \sqrt{2}$ .

See Solution

Problem 17512

Quelle forme de solution particulière $y_{p}$ utiliser pour l'équation $y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_{0} y = r(x)$ ?

See Solution

Problem 17513

Trouver la forme de la solution particulière $y_{p}$ pour l'équation différentielle donnée avec $r(x)=e^{x}+e^{-x}(\cos(3x)-2\sin(3x))$ .

See Solution

Problem 17514

Identifiez les solutions d'une équation différentielle avec racine double 1 et racines simples $-2 \pm i$ .

See Solution

Problem 17515

Evaluate the integral: $\int \frac{5 t-3}{t+1} d t$ (include absolute values and use constant $C$ ).

See Solution

Problem 17516

Evaluate the integral from 0 to 2: $\int_{0}^{2} \frac{x^{2}+x+1}{(x+1)^{2}(x+2)} d x$

See Solution

Problem 17517

A 2500-kg rocket's velocity $v(m)$ as mass $m$ changes. Find $v(1849)$ if $\frac{d v}{d m}=-60 \mathrm{~m}^{-1/2}$ and $v(2500)=0$ .

See Solution

Problem 17518

Find $f^{\prime}$ and $f$ given $f^{\prime \prime}(x)=\cos(x)$ , $f^{\prime}\left(\frac{\pi}{2}\right)=13$ , $f\left(\frac{\pi}{2}\right)=14$ .

See Solution

Problem 17519

Quel est le domaine de la fonction $f(x, y)=\ln (x+y)$ ? Choisissez parmi les options suivantes.

See Solution

Problem 17520

Quelle est la forme de la solution particulière $y_{p}$ pour l'équation $y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=r(x)$ avec $r(x)=4 x e^{2 x} \cos (3 x)$ ?

See Solution

Problem 17521

Trouver les solutions d'une équation différentielle avec racines $1$ (double) et $-2 \pm i$ (simples) parmi les fonctions données.

See Solution

Problem 17522

Find the particular solution form $y_{p}$ for the equation $y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=r(x)$ with $r(x)=e^{x}+e^{-x}(\cos(3x)-2\sin(3x))$ . Which option fits?

See Solution

Problem 17523

Calculate the integral $\int_{\frac{\sqrt{3}}{3}}^{\sqrt{3}} \frac{d x}{x^{2}+1}$ .

See Solution

Problem 17524

Evaluate the integral $\int_{2}^{\sqrt{x}} \frac{1}{4 t+1} d t$ .

See Solution

Problem 17525

Trouver la forme de la solution particulière $y_{p}$ pour l'équation $y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=r(x)$ avec $r(x)=e^{x}+e^{-x}(\cos(3x)-2\sin(3x))$ .

See Solution

Problem 17526

Trouver $\alpha$ , $\beta$ , $C_{1}$ et $C_{2}$ pour que $y=e^{\alpha x}(C_{1} \cos (\beta x)+C_{2} \sin (\beta x))$ soit solution de $y^{\prime \prime}+16 y=0$ avec $y(\pi / 16)=2 \sqrt{2}$ et $y^{\prime}(\pi / 16)=12 \sqrt{2}$ .

See Solution

Problem 17527

Write the integral $I=\int_{0}^{6}\left|x^{2}-4\right| dx$ as a sum of integrals and evaluate.

See Solution

Problem 17528

Differentiate the integral $\int_{-u}^{8 u} \sqrt{x^{2}+3} \, dx$ with respect to $u$ .

See Solution

Problem 17529

Find the area under $f(x)=12 \cos (x)$ from $0$ to $\frac{\pi}{2}$ using the Fundamental Theorem of Calculus, Part I.

See Solution

Problem 17530

Rewrite the integral without absolute values and evaluate: $\int_{\pi / 13}^{\pi}|\cos (x)| d x \approx$

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Problem 17531

Find the area under the curve of $f(x)=4 x^{2}$ from $x=0$ to $x=5$ using the Fundamental Theorem of Calculus.

See Solution

Problem 17532

Find the differential of $y=(-2+2 r)^{-4}$ . What is $\mathrm{d} y$ in terms of $\mathrm{d} r$ ?

See Solution

Problem 17533

Find the differential $d y$ for the function $y=x^{9}-4 \sqrt{x}$ in terms of $x$ and $dx$ .

See Solution

Problem 17534

Find the differential of $y=\sqrt{2+t^{2}}$ . What is $\mathrm{d} y$ in terms of $\mathrm{d} t$ ?

See Solution

Problem 17535

Find the derivative $d y / d x$ using implicit differentiation for $x^{2} y^{2}+x \sin y=4$ .

See Solution

Problem 17536

Find $d y$ for $y=\tan(3x+6)$ at $x=1$ with $d x=0.3$ and $d x=0.6$ .

See Solution

Problem 17537

Find the tangent line equation $y=m x+b$ to the curve $y=11 x e^{x}$ at $(0,0$ . Determine $m$ and $b$ .

See Solution

Problem 17538

Approximate $f^{\prime}(-4.3)$ using $f(-4.3)=-5.2$ and $f(-3.8)=3.7$ . $f^{\prime}(-4.3) \approx$

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Problem 17539

Find the linearization $L(x)$ of $f(x)=\ln(x)$ at the point $a=1$ . What is $L(x)$ ?

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Problem 17540

Find the linearization of $f(x)=\sqrt{2+7 x}$ at $a=0$ . Determine $A$ and $B$ in $L(x)=A+B x$ .

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Problem 17541

Find the differential $d y$ for $y=\ln(3+x^{8})$ in terms of $x$ and $dx$ . Enter $dx$ as $dx$ . $d y=$

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Problem 17542

Approximate $\sqrt{81.3}$ using local linearization of $f(x)=\sqrt{x}$ at $x=81$ : find $L_{81}(x)$ and compute $L_{81}(81.3)$ .

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Problem 17543

Find the critical point $c$ of $f(x)=x^{2}-10x+3$ , compute $f(c)$ , and determine min/max on $[0,10]$ and $[0,1]$ .

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Problem 17544

Déterminez l'ensemble image de la fonction $f(x, y)=1-\ln(1-x^{2}-y^{2})$ . Choisissez la bonne réponse parmi : A, B, C, D, E.

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Problem 17545

Find the three critical numbers of the function $F(x)=x^{4/5}(8x-32)^{2}$ in increasing order.

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Problem 17546

Find $d y$ for $y=2 x^{2}+2 x+3$ at $x=3$ for $d x=0.1$ and $d x=0.2$ .

See Solution

Problem 17547

Find $f^{\prime}(1)$ if $2 + f(x) + x^{2}(f(x))^{3} = 0$ and $f(1) = -1$ .

See Solution

Problem 17548

Déterminez l'ensemble image de la fonction $f(x, y)=1-\ln \left(1-x^{2}-y^{2}\right)$ .

See Solution

Problem 17549

Find the number $c$ that meets the Mean Value Theorem for $f(x)=2x^{3}+2x-14$ on the interval $[0,2]$ .

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Problem 17550

Find the tangent line equation to the hyperbola $4 x^{2}+4 x y-y^{2}-2 x=154$ at the point $(5,4)$ .

See Solution

Problem 17551

Identifiez l'équation du plan tangent à $z=4 x^{2} y$ au point $(-1,-3,-12)$ parmi les options suivantes.

See Solution

Problem 17552

Find the number $c$ that meets the Mean Value Theorem for $f(x)=e^{-9x}$ on the interval $[0,3]$ .

See Solution

Problem 17553

Les racines du polynôme caractéristique d'une équation différentielle d'ordre 2 sont $\lambda_{1}=7$ et $\lambda_{2}=0$ . Quelle équation parmi les suivantes peut avoir une solution particulière de la forme $y_{p}=A x$ ? A. $y^{\prime \prime}-y^{\prime}=3 x$ B. $y^{\prime \prime}-y^{\prime}=3$ C. $y^{\prime \prime}-7 y^{\prime}=3$ D. $y^{\prime \prime}-7 y^{\prime}=3 x$

See Solution

Problem 17554

Déterminez l'ensemble image de la fonction $f(x, y)=1-\ln(1-x^{2}-y^{2})$ parmi les choix suivants : A. $]0,1]$ , B. $]1,+\infty[$ , C. $[1,+\infty[$ , D. $-\infty, 1]$ , E. Aucun.

See Solution

Problem 17555

Quelle est la forme d'une solution particulière de l'équation $y^{\prime \prime}-6 y^{\prime}+13 y=2 x e^{3 x} \cos (2 x)$ ?

See Solution

Problem 17556

Déterminez l'ensemble image de la fonction $f(x, y)=1-\ln(1-x^{2}-y^{2})$ parmi les choix suivants : A. $\bigcirc 0,1]$ , B. $\bigcirc 1,+\infty[$ , C. $\bigcirc[1,+\infty[$ , D. $]-\infty, 1]$ , E. Aucun.

See Solution

Problem 17557

Les racines du polynôme caractéristique sont $\lambda_{1}=7$ et $\lambda_{2}=0$ . Quelle équation différentielle pourrait avoir $y_{p}=A x$ ? A. $7 y^{\prime \prime}-y^{\prime}=3 x$ B. $y^{\prime \prime}-y^{\prime}=3$ C. $y^{\prime \prime}-7 y^{\prime}=3$ D. $y^{\prime \prime}-7 y^{\prime}=3 x$

See Solution

Problem 17558

Identifiez l'équation du plan tangent à $z=4 x^{2} y$ au point $(-1,-3,-12)$ parmi les choix suivants.

See Solution

Problem 17559

Trouver la solution particulière d'une équation différentielle avec $r(x)=3 \mathrm{e}^{x}+x^{2}$ et $P(\lambda)=\left(\lambda^{2}-1\right) \lambda^{2}(\lambda+1)^{2}$ . Choisissez une réponse : A, B, C, D, E, F, G.

See Solution

Problem 17560

Trouvez l'équation du plan tangent à $z=4 x^{2} y$ au point $(-1,-3,-12)$ .

See Solution

Problem 17561

Find the position of a particle, $s(t)$ , given $a(t)=2t+7$ , $s(0)=7$ , and $v(0)=-5$ .

See Solution

Problem 17562

Trouver la valeur de $a$ pour l'équation différentielle $y^{(7)}-a y=4 e^{3 x}$ sans solution particulière de la forme $y=A e^{3 x}$ .

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Problem 17563

Trouver la solution particulière d'une équation différentielle avec $r(x)=3 e^{x}+x^{2}$ et $P(\Lambda)=\left(\lambda^{2}-1\right) \Lambda^{2}(\lambda+1)^{2}$ .

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Problem 17564

A stone is dropped from a 550 m tower.
(a) Initial velocity $v(0)=0$ m/s and distance $s(0)=550$ m. Find $v(t)$ and $s(t)$ .
(b) Time to reach ground? Round to 2 decimal places.
(c) Impact velocity? Round to 1 decimal place.
(d) If thrown down at 4 m/s, time to reach ground? Round to 2 decimal places.

See Solution

Problem 17565

Determine if the series $\sum_{n=1}^{\infty} \frac{(-2)^{n}}{e^{n+1}}$ converges or diverges and find its sum if it converges.

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Problem 17566

Determine convergence/divergence of the series and find the sum if possible: $\sum_{n=1}^{\infty}\left(\frac{4}{5}\right)^{n}$

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Problem 17567

Differentiate the function $y=\ln \left(x+\sqrt{11+x^{2}}\right)$ after simplifying it using logarithm properties.

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Problem 17568

Find the percentage of a brand of computer chips expected to be usable after 4 years using $P(t)=100(1-e^{-0.07 t})$ .

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Problem 17569

Berechnen Sie alle positiven Werte von $k$ für die Gleichung $\int_{0}^{k}(0,25 x+2) d x=4,5$ .

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Problem 17570

A child flies a kite at a height of $a$ ft, moving horizontally at $30 \mathrm{ft/sec}$ . How fast to let out string when kite is $150 \mathrm{ft}$ away?

See Solution

Problem 17571

Find the rate of decrease of ice thickness and outer surface area when the ice is 5 inches thick, melting at 14 in³/min.

See Solution

Problem 17572

Calculate the total amount after 6 years for an investment of \$2500 at a continuous compound interest rate of 4\% per year.

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Problem 17573

Find the limit as $x$ approaches $\pm \infty$ for the expression $\frac{8-x-x^{2}}{x^{2}-x-6}$ .

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Problem 17574

Coffee is draining from a cone into a cylinder at $5 \, \text{in}^3/\text{min}$ . The cone's base diameter is 4 inches.
a. Find the pot's rising level speed when the cone's coffee is 5 inches deep.
b. Find the cone's falling level speed when the coffee is 5 inches deep.

See Solution

Problem 17575

Calculate the derivative of the function $y=\frac{7+3x-6\sqrt{x}}{x}$ . What is $y'=\square$ ?

See Solution

Problem 17576

Find the derivative of $y=(5 t-1)(4 t-3)^{-1}$ . What is $\frac{\mathrm{dy}}{\mathrm{dt}}$ ?

See Solution

Problem 17577

Gegeben ist die Funktion $f(x)=\frac{1}{x^{2}-9}-2$ .
a) Zeigen Sie $f(-x)=f(x)$ und interpretieren Sie. b) Zeigen Sie die Nullstellen $x_{1}=-\sqrt{9,5}$ , $x_{2}=\sqrt{9,5}$ und dass $P\left(0 \mid-2 \frac{1}{9}\right)$ auf $G_{f}$ liegt. c) Bestimmen Sie $\lim _{x \rightarrow-\infty} f(x)$ und $\lim _{x \rightarrow+\infty} f(x)$ . d) Bestimmen Sie die einseitigen Grenzwerte bei $x=-3$ und $x=3$ . e) Welcher Graph gehört zu $f$ ?

See Solution

Problem 17578

Find the derivative of $y=\frac{7+2x-6\sqrt{x}}{x}$ . What is $y'=\square$ ?

See Solution

Problem 17579

Find all derivatives of the function $y=(x-4)(x+2)(x+4)$ . What is $y^{\prime}$ ?

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Problem 17580

Find the first and second derivatives of $y=\frac{4 x^{3}+5}{x}$ . What is $y^{\prime}$ ?

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Problem 17581

Find the integral of $\sin(3x) \cos^2(x)$ with respect to $x$ : $\int \sin(3x) \cos^2(x) \, dx$ .

See Solution

Problem 17582

Find the body's velocity, speed, acceleration, and jerk at $t=\frac{\pi}{4}$ sec for $s=9 \sin t + 14 \cos t$ .

See Solution

Problem 17583

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{-2(x+\Delta x)^{2}+2 x^{2}}{\Delta x}$ .

See Solution

Problem 17584

Find all derivatives of the function $y=(x-2)(x+1)(x+2)$ . What is $y^{\prime}=\square$ ?

See Solution

Problem 17585

Find the derivative $\frac{d y}{d x}$ for the function $y=9 x^{5} \sin x \cos x$ .

See Solution

Problem 17586

Find the derivative of $y=(\csc x+\cot x)^{-1}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 17587

Find critical numbers, max and min of $f(x)=2 x^{3}+3 x^{2}-72 x$ on $[-5,4]$ .

See Solution

Problem 17588

Find the limit: $\lim _{x \rightarrow 1} \frac{1-x}{\sqrt{x+3}-2 x}$ .

See Solution

Problem 17589

Find the limit as $x$ approaches -5 for the expression $\frac{2 x^{2}-50}{x+5}$ .

See Solution

Problem 17590

Untersuchen Sie die Folgen auf Konvergenz, Grenzwert, Häufungspunkte, Limes superior und Limes inferior:
(a) $a_{n}=\frac{36 n^{3}-9}{24 n^{2}+12 \sqrt{n}}$ (b) $a_{n}=(-1)^{n} \sqrt[n]{\frac{2 n}{3^{n^{2}}}}+\frac{\cos (n)}{n}$ (c) $a_{n}=\operatorname{Re}\left(2^{-n}(-1-\mathrm{i} \sqrt{3})^{n}\right)$ (d) $a_{n}=\sum_{k=0}^{n} \frac{2^{k}+(-1)^{k}}{5^{k}}$

See Solution

Problem 17591

Find the limit as $x$ approaches -2 for the expression $\frac{\frac{2}{x}+\frac{1}{x+3}}{x+2}$ .

See Solution

Problem 17592

Given $y = x^2 + 5$ and $\frac{dx}{dt} = 4$ when $x = 5$ , find $\frac{dy}{dt}$ at $x = 5$ .

See Solution

Problem 17593

Find the limit: $\lim _{x \rightarrow 2} \frac{\sqrt{6-x}-x}{x-2}$ .

See Solution

Problem 17594

Given $y=x^{2}+5$ and $\frac{d x}{d t}=4$ at $x=5$ , find $\frac{d y}{d t}$ when $x=5$ .

See Solution

Problem 17595

Find the relationship between $\frac{d A}{d t}$ and $\frac{d x}{d t}$ if $A=11 x^{2}$ . $\frac{d A}{d t}=\square \frac{d x}{d t}$

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Problem 17596

A rectangle's length $\ell$ decreases at $2 \mathrm{~cm/s}$ and width $w$ increases at $2 \mathrm{~cm/s}$ . Find the rates of change of area, perimeter, and diagonal lengths when $\ell=15 \mathrm{~cm}$ and $w=8 \mathrm{~cm}$ .

See Solution

Problem 17597

A balloon rises at $5 \mathrm{ft/sec}$ and a bike moves at $16 \mathrm{ft/sec}$ . Find how fast distance $s(t)$ increases after 6 sec.

See Solution

Problem 17598

Coffee drains into a cylindrical pot at $6 \, \text{in.}^{3}/\text{min}$ with radius $4 \, \text{in.}$ . Find the rise rate when the cone is $7 \, \text{in.}$ deep. Round to three decimal places.

See Solution

Problem 17599

A rectangle's length $\ell$ increases at $2 \mathrm{~cm/sec}$ and width $w$ decreases at $2 \mathrm{~cm/sec}$ . Find the rates of change of area, perimeter, and diagonals when $\ell=24 \mathrm{~cm}$ and $w=7 \mathrm{~cm}$ .

See Solution

Problem 17600

Coffee drains from a conical filter into a cylindrical pot at 6 in. $^{3}$ /min. Both have a radius of 4 in.
(a) Find the rise rate in the pot when the cone is 7 in. deep. Answer: $0.119$ in./min.
(b) Find the fall rate in the cone.

See Solution

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Calculus

Problem 17501

Calculate the average value fave f_{\text {ave }}fave ​ of the function f(x)=3x2+4xf(x)=3 x^{2}+4 xf(x)=3x2+4x over the interval [−1,5][-1,5][−1,5].

Problem 17502

Quelle est la forme de la solution particulière ypy_{p}yp​ pour l'équation y(n)+an−1y(n−1)+⋯+a0y=4xe2xcos⁡(3x)y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=4 x e^{2 x} \cos (3 x)y(n)+an−1​y(n−1)+⋯+a0​y=4xe2xcos(3x)?

Problem 17503

Calculate the average value gave g_{\text {ave }}gave ​ of the function g(t)=31+t2g(t)=\frac{3}{1+t^{2}}g(t)=1+t23​ over the interval [0,5][0,5][0,5].

Problem 17504

Calculate the average value haveh_{\text{ave}}have​ of the function h(x)=7cos⁡4(x)sin⁡(x)h(x)=7 \cos^{4}(x) \sin(x)h(x)=7cos4(x)sin(x) over the interval [0,π][0, \pi][0,π].

Problem 17505

Evaluate the integral: ∫sin⁡(28x)1+cos⁡2(28x)dx\int \frac{\sin (28 x)}{1+\cos ^{2}(28 x)} d x∫1+cos2(28x)sin(28x)​dx (Use CCC for the constant).

Problem 17506

Déterminez le domaine de la fonction f(x,y)=ln⁡(2x2+3y2)f(x, y)=\ln(2x^{2}+3y^{2})f(x,y)=ln(2x2+3y2).

Problem 17507

Trouver la forme de la solution particulière ypy_{p}yp​ pour l'équation y(n)+an−1y(n−1)+⋯+a0y=r(x)y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=r(x)y(n)+an−1​y(n−1)+⋯+a0​y=r(x) avec r(x)=4xe2xcos⁡(3x)r(x)=4 x e^{2 x} \cos (3 x)r(x)=4xe2xcos(3x).

Problem 17508

Trouver la forme de la solution particulière ypy_{p}yp​ pour l'équation y(n)+an−1y(n−1)+⋯+a0y=r(x)y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=r(x)y(n)+an−1​y(n−1)+⋯+a0​y=r(x).

Problem 17509

Evaluate the integral: ∫016x1+5xdx\int_{0}^{16} \frac{x}{\sqrt{1+5 x}} d x∫016​1+5x​x​dx

Problem 17510

Équation différentielle avec polynôme p(λ)=(λ−1)(λ+1)2(λ2+9)p(\lambda)=(\lambda-1)(\lambda+1)^{2}(\lambda^{2}+9)p(λ)=(λ−1)(λ+1)2(λ2+9), trouver ypy_{p}yp​.

Problem 17511

Problem 17512

Quelle forme de solution particulière ypy_{p}yp​ utiliser pour l'équation y(n)+an−1y(n−1)+⋯+a0y=r(x)y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_{0} y = r(x)y(n)+an−1​y(n−1)+⋯+a0​y=r(x)?

Problem 17513

Trouver la forme de la solution particulière ypy_{p}yp​ pour l'équation différentielle donnée avec r(x)=ex+e−x(cos⁡(3x)−2sin⁡(3x))r(x)=e^{x}+e^{-x}(\cos(3x)-2\sin(3x))r(x)=ex+e−x(cos(3x)−2sin(3x)).

Problem 17514

Identifiez les solutions d'une équation différentielle avec racine double 1 et racines simples −2±i-2 \pm i−2±i.

Problem 17515

Evaluate the integral: ∫5t−3t+1dt\int \frac{5 t-3}{t+1} d t∫t+15t−3​dt (include absolute values and use constant CCC).

Problem 17516

Evaluate the integral from 0 to 2: ∫02x2+x+1(x+1)2(x+2)dx\int_{0}^{2} \frac{x^{2}+x+1}{(x+1)^{2}(x+2)} d x∫02​(x+1)2(x+2)x2+x+1​dx

Problem 17517

A 2500-kg rocket's velocity v(m)v(m)v(m) as mass mmm changes. Find v(1849)v(1849)v(1849) if dvdm=−60 m−1/2\frac{d v}{d m}=-60 \mathrm{~m}^{-1/2}dmdv​=−60 m−1/2 and v(2500)=0v(2500)=0v(2500)=0.

Problem 17518

Find f′f^{\prime}f′ and fff given f′′(x)=cos⁡(x)f^{\prime \prime}(x)=\cos(x)f′′(x)=cos(x), f′(π2)=13f^{\prime}\left(\frac{\pi}{2}\right)=13f′(2π​)=13, f(π2)=14f\left(\frac{\pi}{2}\right)=14f(2π​)=14.

Problem 17519

Quel est le domaine de la fonction f(x,y)=ln⁡(x+y)f(x, y)=\ln (x+y)f(x,y)=ln(x+y) ? Choisissez parmi les options suivantes.

Problem 17520

Quelle est la forme de la solution particulière ypy_{p}yp​ pour l'équation y(n)+an−1y(n−1)+⋯+a0y=r(x)y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=r(x)y(n)+an−1​y(n−1)+⋯+a0​y=r(x) avec r(x)=4xe2xcos⁡(3x)r(x)=4 x e^{2 x} \cos (3 x)r(x)=4xe2xcos(3x) ?

Problem 17521

Trouver les solutions d'une équation différentielle avec racines 111 (double) et −2±i-2 \pm i−2±i (simples) parmi les fonctions données.

Problem 17522

Problem 17523

Calculate the integral ∫333dxx2+1\int_{\frac{\sqrt{3}}{3}}^{\sqrt{3}} \frac{d x}{x^{2}+1}∫33​​3​​x2+1dx​.

Problem 17524

Evaluate the integral ∫2x14t+1dt\int_{2}^{\sqrt{x}} \frac{1}{4 t+1} d t∫2x​​4t+11​dt.

Problem 17525

Problem 17526

Problem 17527

Write the integral I=∫06∣x2−4∣dxI=\int_{0}^{6}\left|x^{2}-4\right| dxI=∫06​∣∣​x2−4∣∣​dx as a sum of integrals and evaluate.

Problem 17528

Differentiate the integral ∫−u8ux2+3 dx\int_{-u}^{8 u} \sqrt{x^{2}+3} \, dx∫−u8u​x2+3​dx with respect to uuu.

Problem 17529

Find the area under f(x)=12cos⁡(x)f(x)=12 \cos (x)f(x)=12cos(x) from 000 to π2\frac{\pi}{2}2π​ using the Fundamental Theorem of Calculus, Part I.

Problem 17530

Rewrite the integral without absolute values and evaluate: ∫π/13π∣cos⁡(x)∣dx≈\int_{\pi / 13}^{\pi}|\cos (x)| d x \approx∫π/13π​∣cos(x)∣dx≈

Problem 17531

Find the area under the curve of f(x)=4x2f(x)=4 x^{2}f(x)=4x2 from x=0x=0x=0 to x=5x=5x=5 using the Fundamental Theorem of Calculus.

Problem 17532

Find the differential of y=(−2+2r)−4y=(-2+2 r)^{-4}y=(−2+2r)−4. What is dy\mathrm{d} ydy in terms of dr\mathrm{d} rdr?

Problem 17533

Find the differential dyd ydy for the function y=x9−4xy=x^{9}-4 \sqrt{x}y=x9−4x​ in terms of xxx and dxdxdx.

Problem 17534

Find the differential of y=2+t2y=\sqrt{2+t^{2}}y=2+t2​. What is dy\mathrm{d} ydy in terms of dt\mathrm{d} tdt?

Problem 17535

Find the derivative dy/dxd y / d xdy/dx using implicit differentiation for x2y2+xsin⁡y=4x^{2} y^{2}+x \sin y=4x2y2+xsiny=4.

Problem 17536

Find dyd ydy for y=tan⁡(3x+6)y=\tan(3x+6)y=tan(3x+6) at x=1x=1x=1 with dx=0.3d x=0.3dx=0.3 and dx=0.6d x=0.6dx=0.6.

Problem 17537

Find the tangent line equation y=mx+by=m x+by=mx+b to the curve y=11xexy=11 x e^{x}y=11xex at (0,0(0,0(0,0. Determine mmm and bbb.

Problem 17538

Approximate f′(−4.3)f^{\prime}(-4.3)f′(−4.3) using f(−4.3)=−5.2f(-4.3)=-5.2f(−4.3)=−5.2 and f(−3.8)=3.7f(-3.8)=3.7f(−3.8)=3.7. f′(−4.3)≈f^{\prime}(-4.3) \approxf′(−4.3)≈

Problem 17539

Find the linearization L(x)L(x)L(x) of f(x)=ln⁡(x)f(x)=\ln(x)f(x)=ln(x) at the point a=1a=1a=1. What is L(x)L(x)L(x)?

Problem 17540

Find the linearization of f(x)=2+7xf(x)=\sqrt{2+7 x}f(x)=2+7x​ at a=0a=0a=0. Determine AAA and BBB in L(x)=A+BxL(x)=A+B xL(x)=A+Bx.

Problem 17541

Find the differential dyd ydy for y=ln⁡(3+x8)y=\ln(3+x^{8})y=ln(3+x8) in terms of xxx and dxdxdx. Enter dxdxdx as dxdxdx. dy= d y= dy=

Problem 17542

Calculate the average value $f_{\text {ave }}$ of the function $f(x)=3 x^{2}+4 x$ over the interval $[-1,5]$ .

Quelle est la forme de la solution particulière $y_{p}$ pour l'équation $y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=4 x e^{2 x} \cos (3 x)$ ?

Calculate the average value $g_{\text {ave }}$ of the function $g(t)=\frac{3}{1+t^{2}}$ over the interval $[0,5]$ .

Calculate the average value $h_{\text{ave}}$ of the function $h(x)=7 \cos^{4}(x) \sin(x)$ over the interval $[0, \pi]$ .

Evaluate the integral: $\int \frac{\sin (28 x)}{1+\cos ^{2}(28 x)} d x$ (Use $C$ for the constant).

Déterminez le domaine de la fonction $f(x, y)=\ln(2x^{2}+3y^{2})$ .

Trouver la forme de la solution particulière $y_{p}$ pour l'équation $y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=r(x)$ avec $r(x)=4 x e^{2 x} \cos (3 x)$ .

Trouver la forme de la solution particulière $y_{p}$ pour l'équation $y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=r(x)$ .

Evaluate the integral: $\int_{0}^{16} \frac{x}{\sqrt{1+5 x}} d x$

Équation différentielle avec polynôme $p(\lambda)=(\lambda-1)(\lambda+1)^{2}(\lambda^{2}+9)$ , trouver $y_{p}$ .

Quelle forme de solution particulière $y_{p}$ utiliser pour l'équation $y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_{0} y = r(x)$ ?

Trouver la forme de la solution particulière $y_{p}$ pour l'équation différentielle donnée avec $r(x)=e^{x}+e^{-x}(\cos(3x)-2\sin(3x))$ .

Identifiez les solutions d'une équation différentielle avec racine double 1 et racines simples $-2 \pm i$ .

Evaluate the integral: $\int \frac{5 t-3}{t+1} d t$ (include absolute values and use constant $C$ ).

Evaluate the integral from 0 to 2: $\int_{0}^{2} \frac{x^{2}+x+1}{(x+1)^{2}(x+2)} d x$

A 2500-kg rocket's velocity $v(m)$ as mass $m$ changes. Find $v(1849)$ if $\frac{d v}{d m}=-60 \mathrm{~m}^{-1/2}$ and $v(2500)=0$ .

Find $f^{\prime}$ and $f$ given $f^{\prime \prime}(x)=\cos(x)$ , $f^{\prime}\left(\frac{\pi}{2}\right)=13$ , $f\left(\frac{\pi}{2}\right)=14$ .

Quel est le domaine de la fonction $f(x, y)=\ln (x+y)$ ? Choisissez parmi les options suivantes.

Quelle est la forme de la solution particulière $y_{p}$ pour l'équation $y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=r(x)$ avec $r(x)=4 x e^{2 x} \cos (3 x)$ ?

Trouver les solutions d'une équation différentielle avec racines $1$ (double) et $-2 \pm i$ (simples) parmi les fonctions données.

Calculate the integral $\int_{\frac{\sqrt{3}}{3}}^{\sqrt{3}} \frac{d x}{x^{2}+1}$ .

Evaluate the integral $\int_{2}^{\sqrt{x}} \frac{1}{4 t+1} d t$ .

Write the integral $I=\int_{0}^{6}\left|x^{2}-4\right| dx$ as a sum of integrals and evaluate.

Differentiate the integral $\int_{-u}^{8 u} \sqrt{x^{2}+3} \, dx$ with respect to $u$ .

Find the area under $f(x)=12 \cos (x)$ from $0$ to $\frac{\pi}{2}$ using the Fundamental Theorem of Calculus, Part I.

Rewrite the integral without absolute values and evaluate: $\int_{\pi / 13}^{\pi}|\cos (x)| d x \approx$

Find the area under the curve of $f(x)=4 x^{2}$ from $x=0$ to $x=5$ using the Fundamental Theorem of Calculus.

Find the differential of $y=(-2+2 r)^{-4}$ . What is $\mathrm{d} y$ in terms of $\mathrm{d} r$ ?

Find the differential $d y$ for the function $y=x^{9}-4 \sqrt{x}$ in terms of $x$ and $dx$ .

Find the differential of $y=\sqrt{2+t^{2}}$ . What is $\mathrm{d} y$ in terms of $\mathrm{d} t$ ?

Find the derivative $d y / d x$ using implicit differentiation for $x^{2} y^{2}+x \sin y=4$ .

Find $d y$ for $y=\tan(3x+6)$ at $x=1$ with $d x=0.3$ and $d x=0.6$ .

Find the tangent line equation $y=m x+b$ to the curve $y=11 x e^{x}$ at $(0,0$ . Determine $m$ and $b$ .

Approximate $f^{\prime}(-4.3)$ using $f(-4.3)=-5.2$ and $f(-3.8)=3.7$ . $f^{\prime}(-4.3) \approx$

Find the linearization $L(x)$ of $f(x)=\ln(x)$ at the point $a=1$ . What is $L(x)$ ?

Find the linearization of $f(x)=\sqrt{2+7 x}$ at $a=0$ . Determine $A$ and $B$ in $L(x)=A+B x$ .

Find the differential $d y$ for $y=\ln(3+x^{8})$ in terms of $x$ and $dx$ . Enter $dx$ as $dx$ . $d y=$

Approximate $\sqrt{81.3}$ using local linearization of $f(x)=\sqrt{x}$ at $x=81$ : find $L_{81}(x)$ and compute $L_{81}(81.3)$ .

Find the critical point $c$ of $f(x)=x^{2}-10x+3$ , compute $f(c)$ , and determine min/max on $[0,10]$ and $[0,1]$ .

Déterminez l'ensemble image de la fonction $f(x, y)=1-\ln(1-x^{2}-y^{2})$ . Choisissez la bonne réponse parmi : A, B, C, D, E.

Find the three critical numbers of the function $F(x)=x^{4/5}(8x-32)^{2}$ in increasing order.

Find $d y$ for $y=2 x^{2}+2 x+3$ at $x=3$ for $d x=0.1$ and $d x=0.2$ .

Find $f^{\prime}(1)$ if $2 + f(x) + x^{2}(f(x))^{3} = 0$ and $f(1) = -1$ .

Déterminez l'ensemble image de la fonction $f(x, y)=1-\ln \left(1-x^{2}-y^{2}\right)$ .

Find the number $c$ that meets the Mean Value Theorem for $f(x)=2x^{3}+2x-14$ on the interval $[0,2]$ .

Find the tangent line equation to the hyperbola $4 x^{2}+4 x y-y^{2}-2 x=154$ at the point $(5,4)$ .

Identifiez l'équation du plan tangent à $z=4 x^{2} y$ au point $(-1,-3,-12)$ parmi les options suivantes.

Find the number $c$ that meets the Mean Value Theorem for $f(x)=e^{-9x}$ on the interval $[0,3]$ .

Déterminez l'ensemble image de la fonction $f(x, y)=1-\ln(1-x^{2}-y^{2})$ parmi les choix suivants : A. $]0,1]$ , B. $]1,+\infty[$ , C. $[1,+\infty[$ , D. $-\infty, 1]$ , E. Aucun.

Quelle est la forme d'une solution particulière de l'équation $y^{\prime \prime}-6 y^{\prime}+13 y=2 x e^{3 x} \cos (2 x)$ ?

Identifiez l'équation du plan tangent à $z=4 x^{2} y$ au point $(-1,-3,-12)$ parmi les choix suivants.

Trouver la solution particulière d'une équation différentielle avec $r(x)=3 \mathrm{e}^{x}+x^{2}$ et $P(\lambda)=\left(\lambda^{2}-1\right) \lambda^{2}(\lambda+1)^{2}$ . Choisissez une réponse : A, B, C, D, E, F, G.

Trouvez l'équation du plan tangent à $z=4 x^{2} y$ au point $(-1,-3,-12)$ .

Find the position of a particle, $s(t)$ , given $a(t)=2t+7$ , $s(0)=7$ , and $v(0)=-5$ .

Trouver la valeur de $a$ pour l'équation différentielle $y^{(7)}-a y=4 e^{3 x}$ sans solution particulière de la forme $y=A e^{3 x}$ .

Trouver la solution particulière d'une équation différentielle avec $r(x)=3 e^{x}+x^{2}$ et $P(\Lambda)=\left(\lambda^{2}-1\right) \Lambda^{2}(\lambda+1)^{2}$ .

Determine if the series $\sum_{n=1}^{\infty} \frac{(-2)^{n}}{e^{n+1}}$ converges or diverges and find its sum if it converges.

Determine convergence/divergence of the series and find the sum if possible: $\sum_{n=1}^{\infty}\left(\frac{4}{5}\right)^{n}$

Differentiate the function $y=\ln \left(x+\sqrt{11+x^{2}}\right)$ after simplifying it using logarithm properties.

Find the percentage of a brand of computer chips expected to be usable after 4 years using $P(t)=100(1-e^{-0.07 t})$ .

Berechnen Sie alle positiven Werte von $k$ für die Gleichung $\int_{0}^{k}(0,25 x+2) d x=4,5$ .

A child flies a kite at a height of $a$ ft, moving horizontally at $30 \mathrm{ft/sec}$ . How fast to let out string when kite is $150 \mathrm{ft}$ away?

Find the limit as $x$ approaches $\pm \infty$ for the expression $\frac{8-x-x^{2}}{x^{2}-x-6}$ .

Coffee is draining from a cone into a cylinder at $5 \, \text{in}^3/\text{min}$ . The cone's base diameter is 4 inches.
a. Find the pot's rising level speed when the cone's coffee is 5 inches deep.
b. Find the cone's falling level speed when the coffee is 5 inches deep.