Calculus

Problem 17601

Find relative maxima and minima for the function $f(x)=2 x^{3}+3 x^{2}-36 x+6$ . Calculate their values.

See Solution

Problem 17602

Find the cost function $C(x)$ given the marginal cost $M C=35 x^{4/3}-14 x^{3/4}+1$ and fixed costs of \$7,000.

See Solution

Problem 17603

Gegeben ist die Funktion $f(x)=(2-a)e^{-\frac{x}{a-1}}$ mit $a \in] 1 ; 2[$ . Bestimme $a$ , sodass $I(a)=\int_{0}^{\infty} f(x) \, dx$ maximal ist. Skizziere $f(x)$ für $a \in\{1.3 ; 1.5 ; 1.7\}$ .

See Solution

Problem 17604

Ein Körper wird mit $6 \mathrm{~m/s}$ von $140 \mathrm{~m}$ Höhe abgeworfen. Berechne die Zeit $T$ bis zum Boden. $T=55.3$ Sekunden.

See Solution

Problem 17605

Bestimmen Sie die Ableitung von $f$ an $x_{0}=2$ mit dem Differenzenquotienten für $h \rightarrow 0$ für die Funktionen: a) $f(x)=x^{2}$ , b) $f(x)=\frac{2}{x}$ , c) $f(x)=2 x^{2}-3$ , d) $f(x)=x^{4}$ , e) $f(x)=x^{3}$ , f) $f(x)=4 x-x^{2}$ , g) $f(x)=-\frac{1}{x}$ , h) $f(x)=5$ .

See Solution

Problem 17606

Berechnen Sie den Grenzwert $a$ der Folgen und finden Sie $n_{\varepsilon}$ mit $\left|a_{n}-a\right|<10^{-18}$ für (a) $a_{n}=\sum_{k=0}^{n} 99\left(\frac{1}{100}\right)^{k}$ und (b) $a_{n}=\frac{n^{3}-27}{8 n^{3}}$ .

See Solution

Problem 17607

Calculate the integral $\int e^{x} \cos x \, dx$ .

See Solution

Problem 17608

Napisz równanie stycznej do funkcji f w punkcie $\mathrm{x}_{0}$ dla: a) $f(x)=\frac{8}{4+x^{2}}, x_{0}=2$ b) $f(x)=\ln x-\frac{1}{x}, x_{0}=1$ c) $f(x)=\arcsin \frac{2 x}{1+x^{2}}, x_{0}=\sqrt{3}$

See Solution

Problem 17609

Calculate the indefinite integral and include the constant of integration $C$ : $\int\left(12 e^{3 x}+8 x\right) d x$

See Solution

Problem 17610

Calculate the length of the curve $y=\ln (\cos (x))$ for $0 \leq x \leq \frac{\pi}{6}$ .

See Solution

Problem 17611

Calculate the indefinite integral and include the constant $C$ for integration: $\int e^{-3 x} d x$

See Solution

Problem 17612

Find the surface area when the curve $x=\sqrt{a^{2}-y^{2}}$ (for $0 \leq y \leq a/6$ ) is rotated about the $y$ -axis.

See Solution

Problem 17613

Calculate the indefinite integral: $\int\left(e^{5 x}-\frac{5}{x}\right) d x$ and include the constant $C$ .

See Solution

Problem 17614

Calculate the curve length for $y=\int_{1}^{x} \sqrt{t^{3}-1} dt$ from $x=16$ to $x=25$ .

See Solution

Problem 17615

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

See Solution

Problem 17616

Calculate the indefinite integral: $\int(9 \sqrt{x}-3) \, dx$ (Include constant $C$ ).

See Solution

Problem 17617

Calculate the indefinite integral and include the constant of integration $C$ for the result: $\int(24 x^{3}+8 x-5) dx$

See Solution

Problem 17618

Find the limit as $n$ approaches infinity for the sequence $a_{n}=\frac{n^{2} \cos (n)}{4+n^{2}}$ .

See Solution

Problem 17619

Find the first eight partial sums of the series $\sum_{n=1}^{\infty} \frac{5}{\sqrt[3]{n}}$ to four decimal places. Is it convergent or divergent?

See Solution

Problem 17620

Calculate the indefinite integral: $\int\left(10 \sqrt{x^{3}}-8 x\right) d x$ (Include constant $C$ ).

See Solution

Problem 17621

Calculate the indefinite integral and include the constant $C$ for integration: $\int \frac{3 x^{3}+2 x^{2}+3 x}{x} d x$

See Solution

Problem 17622

Consider the sequence $a_{n}=\frac{5 n}{3 n+1}$ . Is it convergent or divergent? Also, is $\sum_{n=1}^{\infty} a_{n}$ convergent or divergent?

See Solution

Problem 17623

Is $y=\frac{1}{2} x \sin (x)$ a solution for the equation $y^{\prime \prime}+y=\sin (x)$ ?

See Solution

Problem 17624

Find the slope of the tangent to the curve given by $x=t^{2}+2t$ and $y=2^{t}-2t$ at (24,8). Round to two decimals.

See Solution

Problem 17625

Evaluate the series $\sum_{n=4}^{\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)$ .

See Solution

Problem 17626

Find $\frac{d x}{d t}$ , $\frac{d y}{d t}$ , and $\frac{d y}{d x}$ for $x=5 t^{3}+6 t$ , $y=3 t-4 t^{2}$ .

See Solution

Problem 17627

Find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ for $x=t^{2}+3$ , $y=t^{2}+5t$ . When is the curve concave upward?

See Solution

Problem 17628

Calculate the area between the parametric curve $x=\sin^{2}(t)$ , $y=8\cos(t)$ , and the $y$ -axis using Area = $\int (x(t) * y'(t)) dt$ .

See Solution

Problem 17629

Solve the equation $\frac{d y}{d x}=x \sqrt{y}$ . Include initial conditions if provided.

See Solution

Problem 17630

Calculate the area under the curve defined by $x=5 \sin (t)$ and $y=\sin (t) \cos (t)$ for $0 \leq t \leq \frac{\pi}{2}$ .

See Solution

Problem 17631

Bestimmen Sie die Punkte, wo der Graph von $f$ eine waagerechte Tangente hat: a) $f(x)=x^{2} e^{3 x}$ , b) $f(x)=2 x(x+1) e^{x}$ , c) $f(x)=\frac{2}{3} e^{2 x-1}$ , d) $f(x)=(x+3)^{3} e^{x}$ .

See Solution

Problem 17632

Maximize production $Q=50 L^{0.6} K^{0.4}$ under $8L + 4K=400$ . Find optimal $L$ and $K$ .

See Solution

Problem 17633

Find the time $t$ (in days) when cases $N=\frac{800}{1+790 e^{-0.1 t}}$ has a point of inflection. Round to nearest integer.

See Solution

Problem 17634

Bestimme die Besucherzahl $x$ , die die Heizkosten $H(x)=55-0,001 \cdot x^{2}$ minimal macht, für $0 \leq x \leq 145$ .

See Solution

Problem 17635

أي تعبير يمكن استخدامه لحساب ميل المستقيم المماس للدالة $y=\frac{1}{x^{2}}$ عند $x=3$ ؟

See Solution

Problem 17636

A flu outbreak starts with 5 cases on day $t=0$ . The growth rate is $r(t)=14 e^{0.04 t}$ .
(a) Find the total cases $F(t)$ in the first $t$ days. (b) Calculate $F(25)$ and round to the nearest whole number.

See Solution

Problem 17637

Find the derivative $\frac{d y}{d x}$ for the function $y=(\arcsin x) \sqrt{1-x^{2}}$ .

See Solution

Problem 17638

Find the derivative $\frac{d y}{d x}$ at $x=1$ for the function $y=\ln x-\frac{1}{e^{x}}$ .

See Solution

Problem 17639

Find the instantaneous velocity of $f(t)=-t^{2}+t$ at $t=3$ seconds. Options: -5, -4, -3, -2.

See Solution

Problem 17640

Find the $\mathrm{x}$ -values where the tangent line of $f(x)=2 \sin x+\sin ^{2} x$ is horizontal for $0 \leq x<2 \pi$ . $x=$

See Solution

Problem 17641

Find the instantaneous velocity of $f(t)=t^{2}-3t$ at $t=10$ seconds. Options: 21, 17, 5, 15.

See Solution

Problem 17642

Finde die Punkte, wo der Graph der Funktionen $f(x)=2 x(x+1) e^{x}$ , $f(x)=\frac{2}{3} e^{2 x-1}$ und $f(x)=(x+3)^{3} e^{x}$ eine waagerechte Tangente hat.

See Solution

Problem 17643

Calculate the integral $I=\int_{0}^{5}|x-3| dx$ . What is the value of $I$ ?

See Solution

Problem 17644

Evaluate the integral $I=\int_{0}^{\pi / 2} \frac{\sin (x) \cos (x)}{\cos ^{2}(x)+7} d x$ . Find $I$ .

See Solution

Problem 17645

Evaluate the integral $\int_{-1}^{0}-x \ln |x| d x$ . Does it diverge or converge to $-\frac{1}{4}$ , 0, or $\frac{1}{4}$ ?

See Solution

Problem 17646

Evaluate the integral: $\int \frac{\sqrt{x}}{1+x^{3}} d x=$ . Choose the correct answer from the options provided.

See Solution

Problem 17647

Evaluate the convergence of the integral $\int_{1}^{\infty} \frac{\operatorname{sech}\left(x^{2023}\right)}{\sqrt[2023]{2024+x^{2024}}} d x$ . Choose A, B, C, or D.

See Solution

Problem 17648

找出函数的临界点，并使用二阶导数测试分类该点。 $f(x, y)=4-4 x^{2}-4 y^{2}$ 临界点 $(x, y)=(\square)$ 分类 $\quad$ 选择- $v$ 最后，确定函数的相对极值。相对最小值相对最大值

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Problem 17649

Evaluate the integral $\int_{1}^{\infty} \frac{\operatorname{sech}\left(x^{2023}\right)}{\sqrt[2023]{2024+x^{2024}}} d x$ for convergence. Options: A) Converges B) 2023 C) Improper II D) Diverges

See Solution

Problem 17650

Evaluate the integral from $\ln 2$ to $\ln 3$ of $\frac{3 \sinh x}{\cosh ^{2} x+\cosh x-2} dx$ . What is the result? A) $\ln \left(\frac{11}{26}\right)$ B) $\ln \left(\frac{169}{726}\right)$ C) $\ln \left(\frac{26}{11}\right)$ D) $\ln \left(\frac{726}{169}\right)$

See Solution

Problem 17651

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

See Solution

Problem 17652

Find critical points of $f(x, y)=x^{3}-2xy+y^{2}+3$ and classify them using the Second Derivative Test.

See Solution

Problem 17653

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

See Solution

Problem 17654

Given the function $f(x)=x-5 x^{1 / 5}$ , find critical values, intervals of increase/decrease, and local maxima.

See Solution

Problem 17655

Find the instantaneous velocity of $f(t)=t^{3}$ at $t=4$ seconds. Choices: 48, 60, 24, 36.

See Solution

Problem 17656

Find the average velocity of a car with position $y(t)=3 t^{2}-6 t$ from $t=2$ to $t=5$ .

See Solution

Problem 17657

Find the instantaneous velocity of $f(t)=-t^{2}+t$ at $t=4$ seconds. Options: $-3$ , $-1$ , $-7$ , $-5$ .

See Solution

Problem 17658

Find the instantaneous velocity of $f(t)=t^{2}-3t$ at $t=7$ seconds. Choices: 11, 8, 4, 5.

See Solution

Problem 17659

Find the average velocity of $s(t)=4 t^{2}-4 t$ from $t=0$ to $t=2$ .

See Solution

Problem 17660

Find the interval where the function $F(x)=\frac{1}{x^2}$ has a positive, decreasing slope. Options: a. $x<-6$ b. $-6<x<-2$ c. $-2<x<2$ d. $x>2$

See Solution

Problem 17661

Find the limit as $x$ approaches 0 from the right: $\lim _{x \rightarrow 0^{+}}(1+5 x)^{5 / x}$ .

See Solution

Problem 17662

Evaluate the integral: $\int \sin^{3}(y) \cos^{4}(y) \, dy$ (use $\mathrm{C}$ for the constant).

See Solution

Problem 17663

Determine the inflection points for the function $f(x)=-x^{2}-12 x-35$ . Are there any?

See Solution

Problem 17664

Solve the equation $y' - y = e^{x}$ . Include the constant if there's an initial value.

See Solution

Problem 17665

Find the critical point of $f(x, y) = xy + \ln(x) + 32y^2$ , classify it, and determine relative extrema.

See Solution

Problem 17666

Find the critical point of $f(x, y)=x^{2}-e^{y^{2}}$ and classify it using the Second Derivative Test. Then find relative extrema.

See Solution

Problem 17667

Find the average rate of change of $f(x)=-\frac{3}{x^{2}-2x}$ on the interval $\left[-1, \frac{1}{3}\right]$ .

See Solution

Problem 17668

Is the sequence $a_{n}=\frac{9 n !}{2^{n}}$ convergent or divergent? If convergent, find the limit; otherwise, state DIVERGES.

See Solution

Problem 17669

Identify why these integrals are improper: (a) $\int_{1}^{9} \frac{d x}{x-8}$ ; (b) $\int_{8}^{\infty} \frac{d x}{x^{2}-9}$ .

See Solution

Problem 17670

Find the volume of the solid formed by rotating the region between $x=1+(y-4)^{2}$ and $x=2$ around the $x$ -axis using cylindrical shells.

See Solution

Problem 17671

Find local extrema of $f(x)=\frac{x^{2}}{x^{2}+1}$ using the first derivative test.

See Solution

Problem 17672

Find the acceleration of the particle at $t=\pi$ for $x(t)=\sin(2t)-\cos(3t)$ . Options: (A) 9, (B) $1/9$ , (C) 0, (D) $-\frac{1}{9}$ , (E) -9.

See Solution

Problem 17673

Evaluate the improper integrals: (c) $\int_{0}^{1} \tan (\pi x) d x$ and (d) $\int_{-\infty}^{-1} \frac{e^{x}}{x} d x$ .

See Solution

Problem 17674

Find the slope of the tangent to $f(x)=\frac{2 x-1}{x+2}$ at $x=-3$ , rounded to 1 decimal place.

See Solution

Problem 17675

Given the function $f(x)=9 x^{2} \ln (x)$ for $x>0$ , find critical values, intervals of increase/decrease, local extrema, concavity, and inflection points.

See Solution

Problem 17676

Calculate the surface area from rotating the curve $y=\sqrt{x^{2}+1}$ around the $x$ -axis for $0 \leq x \leq 4$ .

See Solution

Problem 17677

Bestimme die Punkte, an denen der Graph von $f(x)=(x+3)^{3} e^{x}$ eine waagerechte Tangente hat.

See Solution

Problem 17678

Find the tangent slope $m_{\tan }$ using $m_{\text {tan }}=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ and the equation at given points:
6. $f(x)=x^{2}+x$ , $a=1$
7. $f(x)=1-x-x^{2}$ , $a=0$

See Solution

Problem 17679

Find the slope of the tangent to the curve $f(x)=\frac{2 x-1}{x+2}$ at $x=-3$ , rounded to 1 decimal place.

See Solution

Problem 17680

Calculate the surface area from rotating the curve $y=\sqrt{x^{2}+1}$ around the $x$ -axis for $0 \leq x \leq 2$ .

See Solution

Problem 17681

Find the indefinite integral and verify by differentiating: $\int(4 x^{11}-5 x^{14}) dx = \square$

See Solution

Problem 17682

Find the displacement of an object with velocity $v=3 t^{2}+2$ ft/s on $[0,4]$ using midpoints for $n=4$ and $n=8$ subintervals.

See Solution

Problem 17683

Find the slope of the tangent line and its equation for $f(x)=1-x-x^{2}$ at $x=0$ . Use $m_{\tan }=\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}$ .

See Solution

Problem 17684

Solve the equation: $t^{2} \frac{d y}{d t}+3 t y=\sqrt{1+t^{2}}$ .

See Solution

Problem 17685

For the functions, find the tangent slope $m_{\tan }=f^{\prime}(a)$ and the tangent line equation at $x=a$ :
1. $f(x)=x^{2}+x, a=1$
2. $f(x)=1-x-x^{2}, a=0$
3. $f(x)=\frac{7}{x}, a=3$
4. $f(x)=\sqrt{x+8}, a=1$

See Solution

Problem 17686

Find the indefinite integral and verify by differentiating:
$\int\left(5 x^{\frac{1}{3}}+4 x^{-\frac{1}{3}}+11\right) d x=\square$

See Solution

Problem 17687

A particle moves on $y=x^{2}$ with $dx/dt=8$ m/s. Find $d\theta/dt$ when $x=1$ m. Write the relation: $d\theta/dt=\square \frac{dx}{dt}$ .

See Solution

Problem 17688

Find all antiderivatives of $f(r)=\frac{15}{r^{2}+1}$ and verify by differentiating. Antiderivatives are $F(r)=\square$ .

See Solution

Problem 17689

Calculate the left and right Riemann sums for $f(x)=x+2$ on $[0,5]$ with $n=5$ .

See Solution

Problem 17690

Evaluate the double integral: $\int_{0}^{2} \int_{1}^{3} x^{2} y^{3} d y d x$ .

See Solution

Problem 17691

Find the derivative of the function $y=(6 t-1)(3 t-4)^{-1}$ .

See Solution

Problem 17692

Compute the integral $\int(2 \sec x \tan x + 7 \sec ^{2} x) \, dx$ .

See Solution

Problem 17693

Determine if the sequence $a_{n}=\frac{7 n}{2^{n}}$ converges or diverges. Find the limit if it converges.

See Solution

Problem 17694

Find the antiderivative $F$ of $f(x)=9 \sqrt{x}+4$ with $F(1)=3$ . What is $F(x)=\square$ ?

See Solution

Problem 17695

Find the derivative of $f(x)=\left(\sec (5 x)+7^{3 x}\right)^{10}$ .

See Solution

Problem 17696

Trouvez l'équation de la tangente à la courbe $f(x)=\log _{2}\left(x^{2}+4\right)$ au point $(2,3)$ .

See Solution

Problem 17697

A particle moves on $y=x^{2}$ with $dx/dt=10 \mathrm{~m/s}$ . Find $d\theta/dt$ when $x=1 \mathrm{~m}$ . Relate them: $\frac{d \theta}{d t}=\square \frac{d x}{d t}$ .

See Solution

Problem 17698

AP Calculus: Find $f^{\prime}(2)$ for $f(x)=x^{2}+9x$ using $f^{\prime}(a)=\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$ .

See Solution

Problem 17699

Find the normal line equation to the curve $f(x)=\log _2(x^2+4)$ at the point $(2,3)$ .

See Solution

Problem 17700

Find the critical points of the function $f(x)=-6 x^{2}-2 x^{3}$ .

See Solution

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Calculus

Problem 17601

Find relative maxima and minima for the function f(x)=2x3+3x2−36x+6f(x)=2 x^{3}+3 x^{2}-36 x+6f(x)=2x3+3x2−36x+6. Calculate their values.

Problem 17602

Find the cost function C(x)C(x)C(x) given the marginal cost MC=35x4/3−14x3/4+1M C=35 x^{4/3}-14 x^{3/4}+1MC=35x4/3−14x3/4+1 and fixed costs of \$7,000.

Problem 17603

Problem 17604

Ein Körper wird mit 6 m/s6 \mathrm{~m/s}6 m/s von 140 m140 \mathrm{~m}140 m Höhe abgeworfen. Berechne die Zeit TTT bis zum Boden. T=55.3T=55.3T=55.3 Sekunden.

Problem 17605

Problem 17606

Problem 17607

Calculate the integral ∫excos⁡x dx\int e^{x} \cos x \, dx∫excosxdx.

Problem 17608

Problem 17609

Calculate the indefinite integral and include the constant of integration CCC: ∫(12e3x+8x)dx\int\left(12 e^{3 x}+8 x\right) d x∫(12e3x+8x)dx

Problem 17610

Calculate the length of the curve y=ln⁡(cos⁡(x))y=\ln (\cos (x))y=ln(cos(x)) for 0≤x≤π60 \leq x \leq \frac{\pi}{6}0≤x≤6π​.

Problem 17611

Calculate the indefinite integral and include the constant CCC for integration: ∫e−3xdx\int e^{-3 x} d x∫e−3xdx

Problem 17612

Find the surface area when the curve x=a2−y2x=\sqrt{a^{2}-y^{2}}x=a2−y2​ (for 0≤y≤a/60 \leq y \leq a/60≤y≤a/6) is rotated about the yyy-axis.

Problem 17613

Calculate the indefinite integral: ∫(e5x−5x)dx\int\left(e^{5 x}-\frac{5}{x}\right) d x∫(e5x−x5​)dx and include the constant CCC.

Problem 17614

Calculate the curve length for y=∫1xt3−1dty=\int_{1}^{x} \sqrt{t^{3}-1} dty=∫1x​t3−1​dt from x=16x=16x=16 to x=25x=25x=25.

Problem 17615

Find the area of the surface formed by rotating y=7−xy=\sqrt{7-x}y=7−x​ from x=1x=1x=1 to x=7x=7x=7 about the xxx-axis.

Problem 17616

Calculate the indefinite integral: ∫(9x−3) dx\int(9 \sqrt{x}-3) \, dx∫(9x​−3)dx (Include constant CCC).

Problem 17617

Calculate the indefinite integral and include the constant of integration CCC for the result: ∫(24x3+8x−5)dx\int(24 x^{3}+8 x-5) dx∫(24x3+8x−5)dx

Problem 17618

Find the limit as nnn approaches infinity for the sequence an=n2cos⁡(n)4+n2a_{n}=\frac{n^{2} \cos (n)}{4+n^{2}}an​=4+n2n2cos(n)​.

Problem 17619

Find the first eight partial sums of the series ∑n=1∞5n3\sum_{n=1}^{\infty} \frac{5}{\sqrt[3]{n}}n=1∑∞​3n​5​ to four decimal places. Is it convergent or divergent?

Problem 17620

Calculate the indefinite integral: ∫(10x3−8x)dx\int\left(10 \sqrt{x^{3}}-8 x\right) d x∫(10x3​−8x)dx (Include constant CCC).

Problem 17621

Calculate the indefinite integral and include the constant CCC for integration: ∫3x3+2x2+3xxdx\int \frac{3 x^{3}+2 x^{2}+3 x}{x} d x∫x3x3+2x2+3x​dx

Problem 17622

Consider the sequence an=5n3n+1a_{n}=\frac{5 n}{3 n+1}an​=3n+15n​. Is it convergent or divergent? Also, is ∑n=1∞an\sum_{n=1}^{\infty} a_{n}∑n=1∞​an​ convergent or divergent?

Problem 17623

Is y=12xsin⁡(x)y=\frac{1}{2} x \sin (x)y=21​xsin(x) a solution for the equation y′′+y=sin⁡(x)y^{\prime \prime}+y=\sin (x)y′′+y=sin(x)?

Problem 17624

Find the slope of the tangent to the curve given by x=t2+2tx=t^{2}+2tx=t2+2t and y=2t−2ty=2^{t}-2ty=2t−2t at (24,8). Round to two decimals.

Problem 17625

Evaluate the series ∑n=4∞(1n−1n+1)\sum_{n=4}^{\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)∑n=4∞​(n​1​−n+1​1​).

Problem 17626

Find dxdt\frac{d x}{d t}dtdx​, dydt\frac{d y}{d t}dtdy​, and dydx\frac{d y}{d x}dxdy​ for x=5t3+6tx=5 t^{3}+6 tx=5t3+6t, y=3t−4t2y=3 t-4 t^{2}y=3t−4t2.

Problem 17627

Find dydx\frac{d y}{d x}dxdy​ and d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ for x=t2+3x=t^{2}+3x=t2+3, y=t2+5ty=t^{2}+5ty=t2+5t. When is the curve concave upward?

Problem 17628

Calculate the area between the parametric curve x=sin⁡2(t)x=\sin^{2}(t)x=sin2(t), y=8cos⁡(t)y=8\cos(t)y=8cos(t), and the yyy-axis using Area = ∫(x(t)∗y′(t))dt\int (x(t) * y'(t)) dt∫(x(t)∗y′(t))dt.

Problem 17629

Solve the equation dydx=xy\frac{d y}{d x}=x \sqrt{y}dxdy​=xy​. Include initial conditions if provided.

Problem 17630

Calculate the area under the curve defined by x=5sin⁡(t)x=5 \sin (t)x=5sin(t) and y=sin⁡(t)cos⁡(t)y=\sin (t) \cos (t)y=sin(t)cos(t) for 0≤t≤π20 \leq t \leq \frac{\pi}{2}0≤t≤2π​.

Problem 17631

Bestimmen Sie die Punkte, wo der Graph von fff eine waagerechte Tangente hat: a) f(x)=x2e3xf(x)=x^{2} e^{3 x}f(x)=x2e3x, b) f(x)=2x(x+1)exf(x)=2 x(x+1) e^{x}f(x)=2x(x+1)ex, c) f(x)=23e2x−1f(x)=\frac{2}{3} e^{2 x-1}f(x)=32​e2x−1, d) f(x)=(x+3)3exf(x)=(x+3)^{3} e^{x}f(x)=(x+3)3ex.

Problem 17632

Maximize production Q=50L0.6K0.4Q=50 L^{0.6} K^{0.4}Q=50L0.6K0.4 under 8L+4K=4008L + 4K=4008L+4K=400. Find optimal LLL and KKK.

Problem 17633

Find the time ttt (in days) when cases N=8001+790e−0.1tN=\frac{800}{1+790 e^{-0.1 t}}N=1+790e−0.1t800​ has a point of inflection. Round to nearest integer.

Problem 17634

Bestimme die Besucherzahl xxx, die die Heizkosten H(x)=55−0,001⋅x2H(x)=55-0,001 \cdot x^{2}H(x)=55−0,001⋅x2 minimal macht, für 0≤x≤1450 \leq x \leq 1450≤x≤145.

Problem 17635

أي تعبير يمكن استخدامه لحساب ميل المستقيم المماس للدالة y=1x2y=\frac{1}{x^{2}}y=x21​ عند x=3x=3x=3؟

Problem 17636

A flu outbreak starts with 5 cases on day t=0t=0t=0. The growth rate is r(t)=14e0.04tr(t)=14 e^{0.04 t}r(t)=14e0.04t. (a) Find the total cases F(t)F(t)F(t) in the first ttt days. (b) Calculate F(25)F(25)F(25) and round to the nearest whole number.

Problem 17637

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=(arcsin⁡x)1−x2y=(\arcsin x) \sqrt{1-x^{2}}y=(arcsinx)1−x2​.

Problem 17638

Find the derivative dydx\frac{d y}{d x}dxdy​ at x=1x=1x=1 for the function y=ln⁡x−1exy=\ln x-\frac{1}{e^{x}}y=lnx−ex1​.

Problem 17639

Find the instantaneous velocity of f(t)=−t2+tf(t)=-t^{2}+tf(t)=−t2+t at t=3t=3t=3 seconds. Options: -5, -4, -3, -2.

Problem 17640

Find the x\mathrm{x}x-values where the tangent line of f(x)=2sin⁡x+sin⁡2xf(x)=2 \sin x+\sin ^{2} xf(x)=2sinx+sin2x is horizontal for 0≤x<2π0 \leq x<2 \pi0≤x<2π. x= x= x=

Problem 17641

Find the instantaneous velocity of f(t)=t2−3tf(t)=t^{2}-3tf(t)=t2−3t at t=10t=10t=10 seconds. Options: 21, 17, 5, 15.

Problem 17642

Find relative maxima and minima for the function $f(x)=2 x^{3}+3 x^{2}-36 x+6$ . Calculate their values.

Find the cost function $C(x)$ given the marginal cost $M C=35 x^{4/3}-14 x^{3/4}+1$ and fixed costs of \$7,000.

Ein Körper wird mit $6 \mathrm{~m/s}$ von $140 \mathrm{~m}$ Höhe abgeworfen. Berechne die Zeit $T$ bis zum Boden. $T=55.3$ Sekunden.

Calculate the integral $\int e^{x} \cos x \, dx$ .

Calculate the indefinite integral and include the constant of integration $C$ : $\int\left(12 e^{3 x}+8 x\right) d x$

Calculate the length of the curve $y=\ln (\cos (x))$ for $0 \leq x \leq \frac{\pi}{6}$ .

Calculate the indefinite integral and include the constant $C$ for integration: $\int e^{-3 x} d x$

Find the surface area when the curve $x=\sqrt{a^{2}-y^{2}}$ (for $0 \leq y \leq a/6$ ) is rotated about the $y$ -axis.

Calculate the indefinite integral: $\int\left(e^{5 x}-\frac{5}{x}\right) d x$ and include the constant $C$ .

Calculate the curve length for $y=\int_{1}^{x} \sqrt{t^{3}-1} dt$ from $x=16$ to $x=25$ .

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

Calculate the indefinite integral: $\int(9 \sqrt{x}-3) \, dx$ (Include constant $C$ ).

Calculate the indefinite integral and include the constant of integration $C$ for the result: $\int(24 x^{3}+8 x-5) dx$

Find the limit as $n$ approaches infinity for the sequence $a_{n}=\frac{n^{2} \cos (n)}{4+n^{2}}$ .

Find the first eight partial sums of the series $\sum_{n=1}^{\infty} \frac{5}{\sqrt[3]{n}}$ to four decimal places. Is it convergent or divergent?

Calculate the indefinite integral: $\int\left(10 \sqrt{x^{3}}-8 x\right) d x$ (Include constant $C$ ).

Calculate the indefinite integral and include the constant $C$ for integration: $\int \frac{3 x^{3}+2 x^{2}+3 x}{x} d x$

Consider the sequence $a_{n}=\frac{5 n}{3 n+1}$ . Is it convergent or divergent? Also, is $\sum_{n=1}^{\infty} a_{n}$ convergent or divergent?

Is $y=\frac{1}{2} x \sin (x)$ a solution for the equation $y^{\prime \prime}+y=\sin (x)$ ?

Find the slope of the tangent to the curve given by $x=t^{2}+2t$ and $y=2^{t}-2t$ at (24,8). Round to two decimals.

Evaluate the series $\sum_{n=4}^{\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)$ .

Find $\frac{d x}{d t}$ , $\frac{d y}{d t}$ , and $\frac{d y}{d x}$ for $x=5 t^{3}+6 t$ , $y=3 t-4 t^{2}$ .

Find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ for $x=t^{2}+3$ , $y=t^{2}+5t$ . When is the curve concave upward?

Calculate the area between the parametric curve $x=\sin^{2}(t)$ , $y=8\cos(t)$ , and the $y$ -axis using Area = $\int (x(t) * y'(t)) dt$ .

Solve the equation $\frac{d y}{d x}=x \sqrt{y}$ . Include initial conditions if provided.

Calculate the area under the curve defined by $x=5 \sin (t)$ and $y=\sin (t) \cos (t)$ for $0 \leq t \leq \frac{\pi}{2}$ .

Bestimmen Sie die Punkte, wo der Graph von $f$ eine waagerechte Tangente hat: a) $f(x)=x^{2} e^{3 x}$ , b) $f(x)=2 x(x+1) e^{x}$ , c) $f(x)=\frac{2}{3} e^{2 x-1}$ , d) $f(x)=(x+3)^{3} e^{x}$ .

Maximize production $Q=50 L^{0.6} K^{0.4}$ under $8L + 4K=400$ . Find optimal $L$ and $K$ .

Find the time $t$ (in days) when cases $N=\frac{800}{1+790 e^{-0.1 t}}$ has a point of inflection. Round to nearest integer.

Bestimme die Besucherzahl $x$ , die die Heizkosten $H(x)=55-0,001 \cdot x^{2}$ minimal macht, für $0 \leq x \leq 145$ .

أي تعبير يمكن استخدامه لحساب ميل المستقيم المماس للدالة $y=\frac{1}{x^{2}}$ عند $x=3$ ؟

A flu outbreak starts with 5 cases on day $t=0$ . The growth rate is $r(t)=14 e^{0.04 t}$ .
(a) Find the total cases $F(t)$ in the first $t$ days. (b) Calculate $F(25)$ and round to the nearest whole number.

Find the derivative $\frac{d y}{d x}$ for the function $y=(\arcsin x) \sqrt{1-x^{2}}$ .

Find the derivative $\frac{d y}{d x}$ at $x=1$ for the function $y=\ln x-\frac{1}{e^{x}}$ .

Find the instantaneous velocity of $f(t)=-t^{2}+t$ at $t=3$ seconds. Options: -5, -4, -3, -2.

Find the $\mathrm{x}$ -values where the tangent line of $f(x)=2 \sin x+\sin ^{2} x$ is horizontal for $0 \leq x<2 \pi$ . $x=$

Find the instantaneous velocity of $f(t)=t^{2}-3t$ at $t=10$ seconds. Options: 21, 17, 5, 15.

Finde die Punkte, wo der Graph der Funktionen $f(x)=2 x(x+1) e^{x}$ , $f(x)=\frac{2}{3} e^{2 x-1}$ und $f(x)=(x+3)^{3} e^{x}$ eine waagerechte Tangente hat.

Calculate the integral $I=\int_{0}^{5}|x-3| dx$ . What is the value of $I$ ?

Evaluate the integral $I=\int_{0}^{\pi / 2} \frac{\sin (x) \cos (x)}{\cos ^{2}(x)+7} d x$ . Find $I$ .

Evaluate the integral $\int_{-1}^{0}-x \ln |x| d x$ . Does it diverge or converge to $-\frac{1}{4}$ , 0, or $\frac{1}{4}$ ?

Evaluate the integral: $\int \frac{\sqrt{x}}{1+x^{3}} d x=$ . Choose the correct answer from the options provided.

Evaluate the convergence of the integral $\int_{1}^{\infty} \frac{\operatorname{sech}\left(x^{2023}\right)}{\sqrt[2023]{2024+x^{2024}}} d x$ . Choose A, B, C, or D.

找出函数的临界点，并使用二阶导数测试分类该点。 $f(x, y)=4-4 x^{2}-4 y^{2}$ 临界点 $(x, y)=(\square)$ 分类 $\quad$ 选择- $v$ 最后，确定函数的相对极值。相对最小值相对最大值

Evaluate the integral $\int_{1}^{\infty} \frac{\operatorname{sech}\left(x^{2023}\right)}{\sqrt[2023]{2024+x^{2024}}} d x$ for convergence. Options: A) Converges B) 2023 C) Improper II D) Diverges

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

Find critical points of $f(x, y)=x^{3}-2xy+y^{2}+3$ and classify them using the Second Derivative Test.

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

Given the function $f(x)=x-5 x^{1 / 5}$ , find critical values, intervals of increase/decrease, and local maxima.

Find the instantaneous velocity of $f(t)=t^{3}$ at $t=4$ seconds. Choices: 48, 60, 24, 36.

Find the average velocity of a car with position $y(t)=3 t^{2}-6 t$ from $t=2$ to $t=5$ .

Find the instantaneous velocity of $f(t)=-t^{2}+t$ at $t=4$ seconds. Options: $-3$ , $-1$ , $-7$ , $-5$ .

Find the instantaneous velocity of $f(t)=t^{2}-3t$ at $t=7$ seconds. Choices: 11, 8, 4, 5.

Find the average velocity of $s(t)=4 t^{2}-4 t$ from $t=0$ to $t=2$ .

Find the interval where the function $F(x)=\frac{1}{x^2}$ has a positive, decreasing slope. Options: a. $x<-6$ b. $-6<x<-2$ c. $-2<x<2$ d. $x>2$

Find the limit as $x$ approaches 0 from the right: $\lim _{x \rightarrow 0^{+}}(1+5 x)^{5 / x}$ .

Evaluate the integral: $\int \sin^{3}(y) \cos^{4}(y) \, dy$ (use $\mathrm{C}$ for the constant).

Determine the inflection points for the function $f(x)=-x^{2}-12 x-35$ . Are there any?

Solve the equation $y' - y = e^{x}$ . Include the constant if there's an initial value.

Find the critical point of $f(x, y) = xy + \ln(x) + 32y^2$ , classify it, and determine relative extrema.

Find the critical point of $f(x, y)=x^{2}-e^{y^{2}}$ and classify it using the Second Derivative Test. Then find relative extrema.

Find the average rate of change of $f(x)=-\frac{3}{x^{2}-2x}$ on the interval $\left[-1, \frac{1}{3}\right]$ .

Is the sequence $a_{n}=\frac{9 n !}{2^{n}}$ convergent or divergent? If convergent, find the limit; otherwise, state DIVERGES.

Identify why these integrals are improper: (a) $\int_{1}^{9} \frac{d x}{x-8}$ ; (b) $\int_{8}^{\infty} \frac{d x}{x^{2}-9}$ .

Find the volume of the solid formed by rotating the region between $x=1+(y-4)^{2}$ and $x=2$ around the $x$ -axis using cylindrical shells.

Find local extrema of $f(x)=\frac{x^{2}}{x^{2}+1}$ using the first derivative test.

Find the acceleration of the particle at $t=\pi$ for $x(t)=\sin(2t)-\cos(3t)$ . Options: (A) 9, (B) $1/9$ , (C) 0, (D) $-\frac{1}{9}$ , (E) -9.

Evaluate the improper integrals: (c) $\int_{0}^{1} \tan (\pi x) d x$ and (d) $\int_{-\infty}^{-1} \frac{e^{x}}{x} d x$ .

Find the slope of the tangent to $f(x)=\frac{2 x-1}{x+2}$ at $x=-3$ , rounded to 1 decimal place.