Calculus

Problem 17701

Find the limit: $\lim _{n \rightarrow \infty} \sqrt{\frac{n+(-1)^{n} \sqrt[n]{n}}{(3 n+\sqrt{2 n})^{2}}}$ .

See Solution

Problem 17702

Find the limit: $\lim _{n \rightarrow \infty} \frac{n^{2}+(-1)^{n} \cdot n}{(n+\sqrt{3})^{2}}$ .

See Solution

Problem 17703

Soit $f(x)=\log _{2}(x^{2}+4)$ . Trouvez $f^{\prime}(x)$ et calculez $f^{\prime}(2)$ .

See Solution

Problem 17704

Find the limit: $\lim _{n \rightarrow \infty} \sqrt{\frac{n+(-1)^{n} \sqrt[4]{n}}{(3 n+\sqrt{2 n})^{2}}}$ .

See Solution

Problem 17705

Find the limit as $n$ approaches infinity: $\lim _{n \rightarrow \infty} \sqrt{n^{2}+5 n}-\sqrt{n^{2}-n}$ .

See Solution

Problem 17706

Find the limit: $\lim _{n \rightarrow \infty} \sqrt{5 n^{2}-3 n}-n \sqrt{5}+8$ .

See Solution

Problem 17707

Find the electric potential $V$ for a charged disk when $D \gg R$ : $V=2 \pi k_{e} \sigma\left(\sqrt{D^{2}+R^{2}}-D\right)$ .

See Solution

Problem 17708

Find a formula for the change in volume $V=\pi r^{2} h$ of a cylinder when the radius changes from $r_{0}$ to $r_{0}+dr$ and height is constant.

See Solution

Problem 17709

Estimate the change in volume $V=2 x^{3}$ when edge lengths change from $x_{0}$ to $x_{0}+d x$ .

See Solution

Problem 17710

Find the point $(x, y)$ in rectangle $R$ with corners $(0,0),(1,0),(0,2),(1,2)$ where land price $p(x, y)=210-15\left(x-\frac{1}{5}\right)^{2}-20\left(y-\frac{3}{2}\right)^{2}$ is max.

See Solution

Problem 17711

Find the total profit from selling the first 60 tickets, given the marginal profit function $P'(x) = 8x - 1176$ .

See Solution

Problem 17712

Approximate the area under $f(x)=0.04 x^{4}-2.25 x^{2}+82$ from $x=3$ to $x=11$ using 4 left endpoints.

See Solution

Problem 17713

Find the total cost of roasting 160 lb of coffee if the marginal cost is $C'(x)=-0.019x+4.75$ for $x \leq 200$ . Total cost is \$\square.

See Solution

Problem 17714

Find intervals where $f(x)=x^{3} \ln x$ is concave up and down for $x>0$ . Concave up: $(0,1/e^{5/6}) \cup (1/e^{5/6}, \infty)$ ; down: $(0,1)$ .

See Solution

Problem 17715

Find where the function $f(x)=x^{3} \ln x$ is concave up and down for $x>0$ .

See Solution

Problem 17716

Find intervals of concavity for $f(x)=x^{3} \ln x$ and identify inflection points and critical numbers.

See Solution

Problem 17717

Find the critical point of $f(x, y) = x^2 + 2xy + 2y^2 - 6x + 10y + 8$ . Classify it using the second derivative test. Also, find relative extrema: min value 351, max value DNE.

See Solution

Problem 17718

Find the critical point of $f(x, y)=x^{2}+2xy+2y^{2}-6x+10y+8$ and classify it using the second derivative test. Determine relative extrema.

See Solution

Problem 17719

Find the volume of the solid formed by rotating the region between $x=2+(y-4)^{2}$ and $x=3$ around the $x$ -axis using cylindrical shells.

See Solution

Problem 17720

Determine if the sequence $a_{n}=\frac{5 n !}{2^{n}}$ converges or diverges. If it converges, find the limit.

See Solution

Problem 17721

Determine why the integral $\int_{1}^{6} \frac{d x}{x-5}$ is improper due to discontinuities or infinite intervals.

See Solution

Problem 17722

Evaluate the improper integral $\int_{8}^{\infty} \frac{d x}{x^{2}-9}$ and identify its type.

See Solution

Problem 17723

Calculate the limit: $\lim _{n \rightarrow \infty} \sqrt[3]{n(n+1)^{2}} - \sqrt[3]{n(n-1)^{2}}$ .

See Solution

Problem 17724

Find the average rate of change of $f(x)=3$ from $x=0$ to $x=3$ . Options: -3, 3, 0, 9.

See Solution

Problem 17725

Evaluate the improper integral $\int_{0}^{1} \tan (\pi x) d x$ considering its discontinuities.

See Solution

Problem 17726

Given that $f(x)$ is increasing at a decreasing rate, what are the signs of $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ ?

See Solution

Problem 17727

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

See Solution

Problem 17728

Determine if $x=22$ is a relative max or min for $f(x)$ given $f'(x)<0$ for $x<22$ and $f'(x)>0$ for $x>22$ .

See Solution

Problem 17729

Find the first derivative of $f(x)=\frac{x}{2}$ . Options: 2, 0, $\frac{1}{2}$ , 1.

See Solution

Problem 17730

Find the partial derivative $f_{x}(x, y)$ for $f(x, y)=4 x^{2}+5 x^{2} y+12 x y^{2}$ and evaluate at $x=12$ , $y=6$ .

See Solution

Problem 17731

Find $x$ where the tangent line of $f(x)=2 x^{3}+9 x^{2}-60 x+4$ is horizontal. Answer the smaller and larger values.

See Solution

Problem 17732

Is it true or false that if $f(x)$ has a min at $x=100$ , then $f^{\prime}(95)$ is negative?

See Solution

Problem 17733

Find the derivative of $y=\ln(4x+7)$ . Options: $1/(4x+7)$ , $4/(4x+7)$ , $1/4x$ , $4x+7$ .

See Solution

Problem 17734

Soit $f(x)=\tan (x / 10)$ pour $2 \pi \leq x \leq 3 \pi$ . Estimez $\int_{2 \pi}^{3 \pi} f(x) d x$ par la méthode du point milieu avec 5 sous-intervalles.
a) Longueur commune $\Delta x$ ? b) Points milieux $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ ? c) Valeurs $f(m_{1}), f(m_{2}), f(m_{3}), f(m_{4}), f(m_{5})$ ? d) Approximation de $\int_{2 \pi}^{3 \pi} f(x) d x$ ?

See Solution

Problem 17735

Considérez $f(x)=\tan (x / 10)$ pour $2 \pi \leq x \leq 3 \pi$ . Estimez $\int_{2 \pi}^{3 \pi} f(x) d x$ avec les étapes suivantes : a) Calculez $\Delta x$ . b) Trouvez les points milieux $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ . c) Listez $f\left(m_{1}\right), f\left(m_{2}\right), f\left(m_{3}\right), f\left(m_{4}\right), f\left(m_{5}\right)$ .

See Solution

Problem 17736

Find critical points, domain endpoints, and local extremes for the function $y=x^{4/5}(x+2)$ .

See Solution

Problem 17737

Find the intervals where $f(x)=-x^{4}-9 x^{3}+3 x-2$ is concave up and down, and the $x$ -coordinates of inflection points.

See Solution

Problem 17738

See Solution

Problem 17739

Calculez l'intégrale $\int \frac{x^{3}}{\sqrt{x^{2}+16}} d x$ avec substitution trigonométrique. Quelle identitié utiliser ? Quel est $A$ ?

See Solution

Problem 17740

Find the intervals where $f(x)=\frac{1}{4 x^{2}+5}$ is concave up, concave down, and the inflection points at $x=$ .

See Solution

Problem 17741

Soit $f(x)=\sqrt{x^{2}-4 x}$ pour $4 \leq x \leq 7$ . Estimez $\int_{4}^{7} f(x) d x$ avec la méthode de Simpson en 6 intervalles. Trouvez $\Delta x$ et l'approximation à 3 décimales.

See Solution

Problem 17742

Estimez $\int_{2}^{5} f(x) d x$ par 3 méthodes (points médians, trapèzes, Simpson) en utilisant les valeurs du tableau.

See Solution

Problem 17743

Estimez $\int_{2 \pi}^{3 \pi} \tan(x/10) dx$ par la méthode du point milieu avec 5 sous-intervalles. Donnez la réponse à 3 décimales.

See Solution

Problem 17744

Estimez $\int_{2}^{5} f(x) d x$ en utilisant les méthodes suivantes : (a) points médians, (b) trapèzes, (c) Simpson.

See Solution

Problem 17745

Determine why the integral $\int_{1}^{8} \frac{d x}{x-7}$ is improper.

See Solution

Problem 17746

Find the volume of the solid formed by rotating the area between $x=4+(y-5)^{2}$ and $x=5$ around the $x$ -axis using cylindrical shells.

See Solution

Problem 17747

Evaluate the improper integral $\int_{8}^{\infty} \frac{d x}{x^{2}-9}$ considering its discontinuities.

See Solution

Problem 17748

Find the derivative $f^{\prime}(10)$ for the function $f(x)=5 x^{\ln (x)}$ .

See Solution

Problem 17749

Calculate the integral $\int_{8}^{\infty} \frac{d x}{x^{2}-9}$ .

See Solution

Problem 17750

Find the derivative of $f(x)=7 \sec^{-1}(2x)$ and evaluate $f^{\prime}(5)$ .

See Solution

Problem 17751

Determine if $\sum_{n=1}^{\infty} \ln(a_{n})$ converges or diverges given that $\sum_{n=1}^{\infty} a_{n}$ converges.

See Solution

Problem 17752

Given a convergent series $\sum_{n=1}^{\infty} a_{n}$ , check if these series converge or diverge: a) $\sum_{n=1}^{\infty} \ln(a_{n})$ , b) $\sum_{n=1}^{\infty} \ln(\frac{1}{a_{n}})$ .

See Solution

Problem 17753

Find the derivative of $f(x)=\cot^{-1}(x)$ and calculate $f'(0.5)$ .

See Solution

Problem 17754

Find the length of the curve $y=\ln(1-x^{2})$ for $0 \leq x \leq \frac{1}{6}$ .

See Solution

Problem 17755

Given $x = 4 \tan(Q)$ , find $\sin(Q)$ , $\csc(Q)$ , $\cos(Q)$ , $\tan(Q)$ , $\cot(Q)$ and calculate $\frac{dx}{dQ}$ . Then, simplify $\int \frac{x^3}{\sqrt{x^2 + 16}} dx = \int g(Q) dQ$ . Evaluate $\int g(Q) dQ$ and rewrite the result in terms of $x$ .

See Solution

Problem 17756

Find the volume of the solid formed by rotating the region between $x=4+(y-5)^{2}$ and $x=5$ about the $x$ -axis using cylindrical shells.

See Solution

Problem 17757

Find the absolute max and min of $f(x)=x^{3}-6 x^{2}-63 x+11$ on these intervals:
(A) $[-4,0]$
(B) $[-1,8]$
(C) $[-4,8]$
Use -1000 for non-existent extrema.

See Solution

Problem 17758

Determine if the sequence $a_{n}=\frac{9 n !}{2^{n}}$ converges or diverges. If it converges, find the limit.

See Solution

Problem 17759

Find the max and min of $f(x)=-\left(x^{2}+x\right)^{\frac{2}{3}}$ on $[-3,4]$ . Max at $x=\square$ , min at $x=\square$ .

See Solution

Problem 17760

Find the critical numbers of $f(x)=-4 x^{5}-3 x^{3}+1$ and sketch the variation table for max and min values.

See Solution

Problem 17761

Fonction $f(x)=x^{2}-2 \ln x$ : 1. Trouvez le domaine. 2. Asymptotes. 3. Calculez $f^{\prime}(x)$ et critiques. 4. Calculez $f^{\prime \prime}(x)$ et critiques. 5. Tableau de variations.

See Solution

Problem 17762

Find critical points of $f(x) = x e^{-x^{2}}$ and classify them using the Second Derivative Test.

See Solution

Problem 17763

Given $xy=4$ , find $dy/dt$ when $x=6$ , $dx/dt=14$ and $dx/dt$ when $x=1$ , $dy/dt=-8$ .

See Solution

Problem 17764

Given $y=\sqrt{x}$ , find $d y / d t$ when $x=1$ , $d x / d t=4$ , and $d x / d t$ when $x=25$ , $d y / d t=3$ .

See Solution

Problem 17765

Find the derivative of y=(6t-1)/(3t-4} with respect to $t$ .

See Solution

Problem 17766

A worker pulls a 5-meter plank up a wall at 0.15 m/s. Find the speed of the plank's end on the ground when it's 2.1 m from the wall. Round to two decimal places. $\mathrm{m} / \mathrm{sec}$

See Solution

Problem 17767

Find $d y / d t$ and $d x / d t$ for $x^2 + y^2 = 100$ .
(a) Given $x=6, y=8, d x / d t=5$ , find $d y / d t$ .
(b) Given $x=8, y=6, d y / d t=-4$ , find $d x / d t$ .

See Solution

Problem 17768

Two planes are flying towards a point at right angles. One is 150 miles away at 600 mph, the other is 200 miles away at 800 mph.
(a) Find the rate of distance change in $\mathrm{mph}$ .
(b) Determine the time available to redirect one plane in $h$ .

See Solution

Problem 17769

A cube's edges expand at 7 cm/s. Find volume change rates when edges are 5 cm and 11 cm: $\mathrm{cm}^{3} / \mathrm{sec}$ .

See Solution

Problem 17770

A 25-ft ladder leans against a wall. The base moves away at 2 ft/s. Find the top's velocity and area/angle rates at 7 ft.

See Solution

Problem 17771

A 25 ft ladder leans against a wall. The base moves away at 2 ft/sec. Find rates for top height, area, and angle at 7 ft from wall.

See Solution

Problem 17772

Find $d y$ for the equation $y=\cos(7x^2)dx a$ .

See Solution

Problem 17773

Evaluate the integral from 4 to infinity: $\int_{4}^{\infty} 11 e^{-y / 2} d y$ .

See Solution

Problem 17774

Evaluate the integral $\int_{-\infty}^{16} r e^{r / 4} d r$ .

See Solution

Problem 17775

Evaluate the integral $\int_{-\infty}^{16} r e^{r / 4} d r$ and determine if it is convergent or divergent.

See Solution

Problem 17776

Identify critical points, endpoints, and extreme values for the function $y=x^{\frac{4}{7}}(x^{2}-4)$ .

See Solution

Problem 17777

Find the general antiderivative of $f(x)=4 e^{x}+3 \sec ^{2}(x)$ .

See Solution

Problem 17778

Find the general antiderivative of $g(x)=\sqrt[6]{x^{5}}+\sqrt[5]{x^{6}}$ .

See Solution

Problem 17779

Find values of $c$ for the equation $\frac{f(1)-f(-2)}{1-(-2)}=f^{\prime}(c)$ with $f(x)=3x^{2}+5x-2$ .

See Solution

Problem 17780

Find the general antiderivative of $f(x)=7 x^{6}-8 x^{3}+9 x^{2}$ .

See Solution

Problem 17781

Find the general antiderivative of $f(x)=\frac{6 \sqrt{5}-4 x+x^{3}}{\sqrt{x}}$ .

See Solution

Problem 17782

Find the antiderivative of the function $f(x)=\frac{6-4 x+x^{3}}{\sqrt{x}}$ .

See Solution

Problem 17783

Find the limit: $\lim _{x \rightarrow 0}\left(x+e^{x / 7}\right)^{7 / x}$ .

See Solution

Problem 17784

Find the limit as $x$ approaches 1 from the left for $x^{\frac{6}{1-x}}$ .

See Solution

Problem 17785

Find the function $f$ given that $f^{\prime \prime}(x)=\sin x+\cos x$ , $f^{\prime}(0)=9$ , and $f(0)=9$ .

See Solution

Problem 17786

Find the limit: $\lim _{x \rightarrow 1^{+}} x^{\frac{6}{1-x}}$ .

See Solution

Problem 17787

Find the particle's position at time $t=9$ given $a(t)=36t+10$ , $s(0)=15$ , and $v(0)=15$ .

See Solution

Problem 17788

Find $f$ given $f'(t)=3 \csc(t)(\csc(t)+5 \cot(t)), 0<t<\pi$ , and $f\left(\frac{\pi}{4}\right)=-15 \sqrt{2}$ .

See Solution

Problem 17789

A warship traveled 154 sea miles in 22 hours. Show its speed exceeded 6.6 knots using the Mean Value Theorem.

See Solution

Problem 17790

Find the absolute maximum of $f(x)=2 x^{2} \ln \left(\frac{1}{x}\right)$ and its location.

See Solution

Problem 17791

Gegeben ist die Funktion $f(x)=2 e^{3 x}+1$ . Finde die Stelle, wo $f(x)=4$ und die Stelle mit der Steigung 3.

See Solution

Problem 17792

Untersuchen Sie die Funktion $f_{1}(t)=-5 t^{2}+130 t$ auf Monotonie und Krümmungsverhalten im Intervall $[0 ; 12]$ .

See Solution

Problem 17793

Bestimme die Werte für $a$ und $b$ , damit die Funktionen $f(x)=x^{2}-2x+2$ und $g(x)=a(x-4)^{2}+b$ an $P(2;2)$ tangential sind.

See Solution

Problem 17794

Calculate the integral $\int_{0}^{2}(x^{2}-2x+1) \, dx$ .

See Solution

Problem 17795

Calculate the integral $\int_{1}^{3}(0.5 x-1)^{2} dx$ .

See Solution

Problem 17796

Calculate the integral $\int_{0}^{2}(x-1)^{3} dx$ .

See Solution

Problem 17797

Find the elasticity of the function $p=D(x)=163 e^{-0.025 x}$ .

See Solution

Problem 17798

Calculate the integral $\int_{0}^{2}(x-2)(x+2) dx$ .

See Solution

Problem 17799

Find the elasticity of the function $p=D(x)=238 e^{-0.125 x}$ .

See Solution

Problem 17800

Berechne den Gesamtflächeninhalt des markierten Bereichs für $f(x)= \frac{1}{2}x^2 - \frac{5}{4}x$ von $x=2$ bis $x=6$ .

See Solution

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Calculus

Problem 17701

Find the limit: lim⁡n→∞n+(−1)nnn(3n+2n)2\lim _{n \rightarrow \infty} \sqrt{\frac{n+(-1)^{n} \sqrt[n]{n}}{(3 n+\sqrt{2 n})^{2}}}limn→∞​(3n+2n​)2n+(−1)nnn​​​.

Problem 17702

Find the limit: lim⁡n→∞n2+(−1)n⋅n(n+3)2\lim _{n \rightarrow \infty} \frac{n^{2}+(-1)^{n} \cdot n}{(n+\sqrt{3})^{2}}limn→∞​(n+3​)2n2+(−1)n⋅n​.

Problem 17703

Soit f(x)=log⁡2(x2+4)f(x)=\log _{2}(x^{2}+4)f(x)=log2​(x2+4). Trouvez f′(x)f^{\prime}(x)f′(x) et calculez f′(2)f^{\prime}(2)f′(2).

Problem 17704

Find the limit: lim⁡n→∞n+(−1)nn4(3n+2n)2\lim _{n \rightarrow \infty} \sqrt{\frac{n+(-1)^{n} \sqrt[4]{n}}{(3 n+\sqrt{2 n})^{2}}}limn→∞​(3n+2n​)2n+(−1)n4n​​​.

Problem 17705

Find the limit as nnn approaches infinity: lim⁡n→∞n2+5n−n2−n\lim _{n \rightarrow \infty} \sqrt{n^{2}+5 n}-\sqrt{n^{2}-n}limn→∞​n2+5n​−n2−n​.

Problem 17706

Find the limit: lim⁡n→∞5n2−3n−n5+8\lim _{n \rightarrow \infty} \sqrt{5 n^{2}-3 n}-n \sqrt{5}+8limn→∞​5n2−3n​−n5​+8.

Problem 17707

Find the electric potential VVV for a charged disk when D≫RD \gg RD≫R: V=2πkeσ(D2+R2−D)V=2 \pi k_{e} \sigma\left(\sqrt{D^{2}+R^{2}}-D\right)V=2πke​σ(D2+R2​−D).

Problem 17708

Find a formula for the change in volume V=πr2hV=\pi r^{2} hV=πr2h of a cylinder when the radius changes from r0r_{0}r0​ to r0+drr_{0}+drr0​+dr and height is constant.

Problem 17709

Estimate the change in volume V=2x3V=2 x^{3}V=2x3 when edge lengths change from x0x_{0}x0​ to x0+dxx_{0}+d xx0​+dx.

Problem 17710

Problem 17711

Find the total profit from selling the first 60 tickets, given the marginal profit function P′(x)=8x−1176P'(x) = 8x - 1176P′(x)=8x−1176.

Problem 17712

Approximate the area under f(x)=0.04x4−2.25x2+82f(x)=0.04 x^{4}-2.25 x^{2}+82f(x)=0.04x4−2.25x2+82 from x=3x=3x=3 to x=11x=11x=11 using 4 left endpoints.

Problem 17713

Find the total cost of roasting 160 lb of coffee if the marginal cost is C′(x)=−0.019x+4.75C'(x)=-0.019x+4.75C′(x)=−0.019x+4.75 for x≤200x \leq 200x≤200. Total cost is \$\square.

Problem 17714

Find intervals where f(x)=x3ln⁡xf(x)=x^{3} \ln xf(x)=x3lnx is concave up and down for x>0x>0x>0. Concave up: (0,1/e5/6)∪(1/e5/6,∞)(0,1/e^{5/6}) \cup (1/e^{5/6}, \infty)(0,1/e5/6)∪(1/e5/6,∞); down: (0,1)(0,1)(0,1).

Problem 17715

Find where the function f(x)=x3ln⁡xf(x)=x^{3} \ln xf(x)=x3lnx is concave up and down for x>0x>0x>0.

Problem 17716

Find intervals of concavity for f(x)=x3ln⁡xf(x)=x^{3} \ln xf(x)=x3lnx and identify inflection points and critical numbers.

Problem 17717

Find the critical point of f(x,y)=x2+2xy+2y2−6x+10y+8f(x, y) = x^2 + 2xy + 2y^2 - 6x + 10y + 8f(x,y)=x2+2xy+2y2−6x+10y+8. Classify it using the second derivative test. Also, find relative extrema: min value 351, max value DNE.

Problem 17718

Find the critical point of f(x,y)=x2+2xy+2y2−6x+10y+8f(x, y)=x^{2}+2xy+2y^{2}-6x+10y+8f(x,y)=x2+2xy+2y2−6x+10y+8 and classify it using the second derivative test. Determine relative extrema.

Problem 17719

Find the volume of the solid formed by rotating the region between x=2+(y−4)2x=2+(y-4)^{2}x=2+(y−4)2 and x=3x=3x=3 around the xxx-axis using cylindrical shells.

Problem 17720

Determine if the sequence an=5n!2na_{n}=\frac{5 n !}{2^{n}}an​=2n5n!​ converges or diverges. If it converges, find the limit.

Problem 17721

Determine why the integral ∫16dxx−5\int_{1}^{6} \frac{d x}{x-5}∫16​x−5dx​ is improper due to discontinuities or infinite intervals.

Problem 17722

Evaluate the improper integral ∫8∞dxx2−9\int_{8}^{\infty} \frac{d x}{x^{2}-9}∫8∞​x2−9dx​ and identify its type.

Problem 17723

Calculate the limit: lim⁡n→∞n(n+1)23−n(n−1)23\lim _{n \rightarrow \infty} \sqrt[3]{n(n+1)^{2}} - \sqrt[3]{n(n-1)^{2}}limn→∞​3n(n+1)2​−3n(n−1)2​.

Problem 17724

Find the average rate of change of f(x)=3f(x)=3f(x)=3 from x=0x=0x=0 to x=3x=3x=3. Options: -3, 3, 0, 9.

Problem 17725

Evaluate the improper integral ∫01tan⁡(πx)dx\int_{0}^{1} \tan (\pi x) d x∫01​tan(πx)dx considering its discontinuities.

Problem 17726

Given that f(x)f(x)f(x) is increasing at a decreasing rate, what are the signs of f′(x)f^{\prime}(x)f′(x) and f′′(x)f^{\prime \prime}(x)f′′(x)?

Problem 17727

Find the partial derivative fx(x,y)f_{x}(x, y)fx​(x,y) for f(x,y)=4x2y2−16x2+4yf(x, y)=4 x^{2} y^{2}-16 x^{2}+4 yf(x,y)=4x2y2−16x2+4y.

Problem 17728

Determine if x=22x=22x=22 is a relative max or min for f(x)f(x)f(x) given f′(x)<0f'(x)<0f′(x)<0 for x<22x<22x<22 and f′(x)>0f'(x)>0f′(x)>0 for x>22x>22x>22.

Problem 17729

Find the first derivative of f(x)=x2f(x)=\frac{x}{2}f(x)=2x​. Options: 2, 0, 12\frac{1}{2}21​, 1.

Problem 17730

Find the partial derivative fx(x,y)f_{x}(x, y)fx​(x,y) for f(x,y)=4x2+5x2y+12xy2f(x, y)=4 x^{2}+5 x^{2} y+12 x y^{2}f(x,y)=4x2+5x2y+12xy2 and evaluate at x=12x=12x=12, y=6y=6y=6.

Problem 17731

Find xxx where the tangent line of f(x)=2x3+9x2−60x+4f(x)=2 x^{3}+9 x^{2}-60 x+4f(x)=2x3+9x2−60x+4 is horizontal. Answer the smaller and larger values.

Problem 17732

Is it true or false that if f(x)f(x)f(x) has a min at x=100x=100x=100, then f′(95)f^{\prime}(95)f′(95) is negative?

Problem 17733

Find the derivative of y=ln⁡(4x+7)y=\ln(4x+7)y=ln(4x+7). Options: 1/(4x+7)1/(4x+7)1/(4x+7), 4/(4x+7)4/(4x+7)4/(4x+7), 1/4x1/4x1/4x, 4x+74x+74x+7.

Problem 17734

Problem 17735

Problem 17736

Find critical points, domain endpoints, and local extremes for the function y=x4/5(x+2)y=x^{4/5}(x+2)y=x4/5(x+2).

Problem 17737

Find the intervals where f(x)=−x4−9x3+3x−2f(x)=-x^{4}-9 x^{3}+3 x-2f(x)=−x4−9x3+3x−2 is concave up and down, and the xxx-coordinates of inflection points.

Problem 17738

Problem 17739

Calculez l'intégrale ∫x3x2+16dx\int \frac{x^{3}}{\sqrt{x^{2}+16}} d x∫x2+16​x3​dx avec substitution trigonométrique. Quelle identitié utiliser ? Quel est AAA ?

Problem 17740

Find the intervals where f(x)=14x2+5f(x)=\frac{1}{4 x^{2}+5}f(x)=4x2+51​ is concave up, concave down, and the inflection points at x=x=x=.

Problem 17741

Soit f(x)=x2−4xf(x)=\sqrt{x^{2}-4 x}f(x)=x2−4x​ pour 4≤x≤74 \leq x \leq 74≤x≤7. Estimez ∫47f(x)dx\int_{4}^{7} f(x) d x∫47​f(x)dx avec la méthode de Simpson en 6 intervalles. Trouvez Δx\Delta xΔx et l'approximation à 3 décimales.

Problem 17742

Find the limit: $\lim _{n \rightarrow \infty} \sqrt{\frac{n+(-1)^{n} \sqrt[n]{n}}{(3 n+\sqrt{2 n})^{2}}}$ .

Find the limit: $\lim _{n \rightarrow \infty} \frac{n^{2}+(-1)^{n} \cdot n}{(n+\sqrt{3})^{2}}$ .

Soit $f(x)=\log _{2}(x^{2}+4)$ . Trouvez $f^{\prime}(x)$ et calculez $f^{\prime}(2)$ .

Find the limit: $\lim _{n \rightarrow \infty} \sqrt{\frac{n+(-1)^{n} \sqrt[4]{n}}{(3 n+\sqrt{2 n})^{2}}}$ .

Find the limit as $n$ approaches infinity: $\lim _{n \rightarrow \infty} \sqrt{n^{2}+5 n}-\sqrt{n^{2}-n}$ .

Find the limit: $\lim _{n \rightarrow \infty} \sqrt{5 n^{2}-3 n}-n \sqrt{5}+8$ .

Find the electric potential $V$ for a charged disk when $D \gg R$ : $V=2 \pi k_{e} \sigma\left(\sqrt{D^{2}+R^{2}}-D\right)$ .

Find a formula for the change in volume $V=\pi r^{2} h$ of a cylinder when the radius changes from $r_{0}$ to $r_{0}+dr$ and height is constant.

Estimate the change in volume $V=2 x^{3}$ when edge lengths change from $x_{0}$ to $x_{0}+d x$ .

Find the total profit from selling the first 60 tickets, given the marginal profit function $P'(x) = 8x - 1176$ .

Approximate the area under $f(x)=0.04 x^{4}-2.25 x^{2}+82$ from $x=3$ to $x=11$ using 4 left endpoints.

Find the total cost of roasting 160 lb of coffee if the marginal cost is $C'(x)=-0.019x+4.75$ for $x \leq 200$ . Total cost is \$\square.

Find intervals where $f(x)=x^{3} \ln x$ is concave up and down for $x>0$ . Concave up: $(0,1/e^{5/6}) \cup (1/e^{5/6}, \infty)$ ; down: $(0,1)$ .

Find where the function $f(x)=x^{3} \ln x$ is concave up and down for $x>0$ .

Find intervals of concavity for $f(x)=x^{3} \ln x$ and identify inflection points and critical numbers.

Find the critical point of $f(x, y) = x^2 + 2xy + 2y^2 - 6x + 10y + 8$ . Classify it using the second derivative test. Also, find relative extrema: min value 351, max value DNE.

Find the critical point of $f(x, y)=x^{2}+2xy+2y^{2}-6x+10y+8$ and classify it using the second derivative test. Determine relative extrema.

Find the volume of the solid formed by rotating the region between $x=2+(y-4)^{2}$ and $x=3$ around the $x$ -axis using cylindrical shells.

Determine if the sequence $a_{n}=\frac{5 n !}{2^{n}}$ converges or diverges. If it converges, find the limit.

Determine why the integral $\int_{1}^{6} \frac{d x}{x-5}$ is improper due to discontinuities or infinite intervals.

Evaluate the improper integral $\int_{8}^{\infty} \frac{d x}{x^{2}-9}$ and identify its type.

Calculate the limit: $\lim _{n \rightarrow \infty} \sqrt[3]{n(n+1)^{2}} - \sqrt[3]{n(n-1)^{2}}$ .

Find the average rate of change of $f(x)=3$ from $x=0$ to $x=3$ . Options: -3, 3, 0, 9.

Evaluate the improper integral $\int_{0}^{1} \tan (\pi x) d x$ considering its discontinuities.

Given that $f(x)$ is increasing at a decreasing rate, what are the signs of $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ ?

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

Determine if $x=22$ is a relative max or min for $f(x)$ given $f'(x)<0$ for $x<22$ and $f'(x)>0$ for $x>22$ .

Find the first derivative of $f(x)=\frac{x}{2}$ . Options: 2, 0, $\frac{1}{2}$ , 1.

Find the partial derivative $f_{x}(x, y)$ for $f(x, y)=4 x^{2}+5 x^{2} y+12 x y^{2}$ and evaluate at $x=12$ , $y=6$ .

Find $x$ where the tangent line of $f(x)=2 x^{3}+9 x^{2}-60 x+4$ is horizontal. Answer the smaller and larger values.

Is it true or false that if $f(x)$ has a min at $x=100$ , then $f^{\prime}(95)$ is negative?

Find the derivative of $y=\ln(4x+7)$ . Options: $1/(4x+7)$ , $4/(4x+7)$ , $1/4x$ , $4x+7$ .

Find critical points, domain endpoints, and local extremes for the function $y=x^{4/5}(x+2)$ .

Find the intervals where $f(x)=-x^{4}-9 x^{3}+3 x-2$ is concave up and down, and the $x$ -coordinates of inflection points.

Calculez l'intégrale $\int \frac{x^{3}}{\sqrt{x^{2}+16}} d x$ avec substitution trigonométrique. Quelle identitié utiliser ? Quel est $A$ ?

Find the intervals where $f(x)=\frac{1}{4 x^{2}+5}$ is concave up, concave down, and the inflection points at $x=$ .

Soit $f(x)=\sqrt{x^{2}-4 x}$ pour $4 \leq x \leq 7$ . Estimez $\int_{4}^{7} f(x) d x$ avec la méthode de Simpson en 6 intervalles. Trouvez $\Delta x$ et l'approximation à 3 décimales.

Estimez $\int_{2}^{5} f(x) d x$ par 3 méthodes (points médians, trapèzes, Simpson) en utilisant les valeurs du tableau.

Estimez $\int_{2 \pi}^{3 \pi} \tan(x/10) dx$ par la méthode du point milieu avec 5 sous-intervalles. Donnez la réponse à 3 décimales.

Estimez $\int_{2}^{5} f(x) d x$ en utilisant les méthodes suivantes : (a) points médians, (b) trapèzes, (c) Simpson.

Determine why the integral $\int_{1}^{8} \frac{d x}{x-7}$ is improper.

Find the volume of the solid formed by rotating the area between $x=4+(y-5)^{2}$ and $x=5$ around the $x$ -axis using cylindrical shells.

Evaluate the improper integral $\int_{8}^{\infty} \frac{d x}{x^{2}-9}$ considering its discontinuities.

Find the derivative $f^{\prime}(10)$ for the function $f(x)=5 x^{\ln (x)}$ .

Calculate the integral $\int_{8}^{\infty} \frac{d x}{x^{2}-9}$ .

Find the derivative of $f(x)=7 \sec^{-1}(2x)$ and evaluate $f^{\prime}(5)$ .

Determine if $\sum_{n=1}^{\infty} \ln(a_{n})$ converges or diverges given that $\sum_{n=1}^{\infty} a_{n}$ converges.

Given a convergent series $\sum_{n=1}^{\infty} a_{n}$ , check if these series converge or diverge: a) $\sum_{n=1}^{\infty} \ln(a_{n})$ , b) $\sum_{n=1}^{\infty} \ln(\frac{1}{a_{n}})$ .

Find the derivative of $f(x)=\cot^{-1}(x)$ and calculate $f'(0.5)$ .

Find the length of the curve $y=\ln(1-x^{2})$ for $0 \leq x \leq \frac{1}{6}$ .

Find the volume of the solid formed by rotating the region between $x=4+(y-5)^{2}$ and $x=5$ about the $x$ -axis using cylindrical shells.

Find the absolute max and min of $f(x)=x^{3}-6 x^{2}-63 x+11$ on these intervals:
(A) $[-4,0]$
(B) $[-1,8]$
(C) $[-4,8]$
Use -1000 for non-existent extrema.

Determine if the sequence $a_{n}=\frac{9 n !}{2^{n}}$ converges or diverges. If it converges, find the limit.

Find the max and min of $f(x)=-\left(x^{2}+x\right)^{\frac{2}{3}}$ on $[-3,4]$ . Max at $x=\square$ , min at $x=\square$ .

Find the critical numbers of $f(x)=-4 x^{5}-3 x^{3}+1$ and sketch the variation table for max and min values.

Fonction $f(x)=x^{2}-2 \ln x$ : 1. Trouvez le domaine. 2. Asymptotes. 3. Calculez $f^{\prime}(x)$ et critiques. 4. Calculez $f^{\prime \prime}(x)$ et critiques. 5. Tableau de variations.

Find critical points of $f(x) = x e^{-x^{2}}$ and classify them using the Second Derivative Test.

Given $xy=4$ , find $dy/dt$ when $x=6$ , $dx/dt=14$ and $dx/dt$ when $x=1$ , $dy/dt=-8$ .

Given $y=\sqrt{x}$ , find $d y / d t$ when $x=1$ , $d x / d t=4$ , and $d x / d t$ when $x=25$ , $d y / d t=3$ .

Find the derivative of y=(6t-1)/(3t-4} with respect to $t$ .

A worker pulls a 5-meter plank up a wall at 0.15 m/s. Find the speed of the plank's end on the ground when it's 2.1 m from the wall. Round to two decimal places. $\mathrm{m} / \mathrm{sec}$

Find $d y / d t$ and $d x / d t$ for $x^2 + y^2 = 100$ .
(a) Given $x=6, y=8, d x / d t=5$ , find $d y / d t$ .
(b) Given $x=8, y=6, d y / d t=-4$ , find $d x / d t$ .

Two planes are flying towards a point at right angles. One is 150 miles away at 600 mph, the other is 200 miles away at 800 mph.
(a) Find the rate of distance change in $\mathrm{mph}$ .
(b) Determine the time available to redirect one plane in $h$ .

A cube's edges expand at 7 cm/s. Find volume change rates when edges are 5 cm and 11 cm: $\mathrm{cm}^{3} / \mathrm{sec}$ .

Find $d y$ for the equation $y=\cos(7x^2)dx a$ .

Evaluate the integral from 4 to infinity: $\int_{4}^{\infty} 11 e^{-y / 2} d y$ .

Evaluate the integral $\int_{-\infty}^{16} r e^{r / 4} d r$ .