Calculus

Problem 29301

Evaluate the integral $\int_{-1}^{0} \frac{x^{5}}{\sqrt{12 x^{2}+4}} d x$ .

See Solution

Problem 29302

Calculate the indefinite integral and include the constant $C$ :
$\int \frac{x+4}{\sqrt{x^{2}+8 x-6}} d x$

See Solution

Problem 29303

Differentiate the function $4x^{5}-\frac{4}{x}+3\sqrt{x}$ .

See Solution

Problem 29304

Find $\frac{dv}{dt}$ for the function $3 \tan(2t) - 4e^{3t} + \ln(5)$ .

See Solution

Problem 29305

Find $u$ and $d v$ to solve the integral $\int x e^{237 x} d x$ where $d v = {}^{d v} d x$ .

See Solution

Problem 29306

Differentiate $f(\theta)=(2 \theta+1) \ln (1+2 \theta)$ with respect to $\theta$ .

See Solution

Problem 29307

Find the derivative of $y=\frac{4}{\sqrt{3x^{2}+1}}$ with respect to $x$ , simplified.

See Solution

Problem 29308

Evaluate the integral $\int_{\pi / 6}^{\pi / 2} 3^{\csc x} 9^{\csc x} \csc x \cot x \, dx$ .

See Solution

Problem 29309

Find the derivative of the integral: $\frac{d}{d x} \int_{4}^{\sqrt{3 x}} \cos (4 t) d t$ .

See Solution

Problem 29310

Find the average rate of change of $g(x)=4x^{2}+\frac{3}{x^{3}}$ over the interval $[-2,1]$ .

See Solution

Problem 29311

Find the derivative of the function $f(x)=\int_{0}^{3 x} \sin(t^{2}) dt$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 29312

Evaluate the integral: $\int \frac{x^{2}-4 x+4}{x^{3}+1} d x$

See Solution

Problem 29313

Find the derivative of the function $3 \sin (5 x) + 7$ .

See Solution

Problem 29314

Find the derivative of $y=\left(x^{3}-9\right)^{4}$ with respect to $x$ : $\frac{d y}{d x}$ .

See Solution

Problem 29315

Evaluate the integral: $\int \frac{11-2 x}{\sqrt{1-8 x-x^{2}}} d x$

See Solution

Problem 29316

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{x}{x-2}$ .

See Solution

Problem 29317

Find $f^{\prime \prime}\left(\frac{\pi}{2}\right)$ for the function $f(x)=\sin(3x)$ .

See Solution

Problem 29318

Evaluate the integral: $\int \frac{11-2 x}{\sqrt{1-8 x-x^{2}}} \, dx$

See Solution

Problem 29319

Find $\frac{d y}{d x}$ if $x^{3}+y^{3}=3 x y$ .

See Solution

Problem 29320

Find the value of $k$ if $y=x e^{k x}$ and $\frac{d^{2} y}{d x^{2}}=6$ at $x=0$ .

See Solution

Problem 29321

Find the derivative of $\tan(3x) + e^{3y}$ with respect to $x$ .

See Solution

Problem 29322

Find the derivative of the integral $\int_{-2}^{x^{5}} \sqrt{t^{2}-t} d t$ with respect to $x$ .

See Solution

Problem 29323

Find the limit as $x$ approaches 0 from the left: $\lim _{x \rightarrow 0^{-}} \frac{\sin x}{x}$ .

See Solution

Problem 29324

Find $f^{\prime}(x)$ for the function $f(x)=\int_{-1}^{3 x} \cos(t^{2}) dt$ .

See Solution

Problem 29325

Find the acceleration of a particle at time $t=4$ if its velocity is $v(t)=3+4.1 \cos (0.9 t)$ .

See Solution

Problem 29326

Find $F^{\prime}(x)$ for the function $F(x)=\int_{4}^{x^{3}} \sin(t^{2}+2t) dt$ .

See Solution

Problem 29327

Find $g'(2)$ where $g(x)=\int_{-1}^{x} f(t) dt$ and $f$ is the derivative of $g$ .

See Solution

Problem 29328

Find the value of $\int_{3}^{-4}(f(x)-1) d x$ given $\int_{3}^{5} f(x) d x=2$ and $\int_{-4}^{5} f(x) d x=-4$ .

See Solution

Problem 29329

Find the average value of the function $f(x)=-x^{2}+4$ over the interval $[-1,1]$ in simplest form.

See Solution

Problem 29330

Calculate the integral: $\int\left(\frac{10 x+2}{x^{2}+25}\right) d x$

See Solution

Problem 29331

Explore Taylor polynomials up to degree five for $f(x)=e^{x}$ at $a=1$ . Find $T_0(x)$ to $T_5(x)$ and coefficients.

See Solution

Problem 29332

Calculate the integral: $\int \frac{5x}{49x^4 + 16} \, dx$

See Solution

Problem 29333

Calculate the integral: $\int\left(\frac{4 x+5}{\sqrt{81-x^{2}}}\right) dx$

See Solution

Problem 29334

Evaluate the integral: $\int \frac{e^{x}}{\sqrt{1-81 e^{2 x}}} d x$

See Solution

Problem 29335

Find the total cost for 30 units using $C(x)=14x+1500$ , the marginal cost, and the cost to increase production to 31 units.

See Solution

Problem 29336

Calculate the integral: $\int \frac{5 \sin (x) \cos (x)}{\sqrt{10+\sin ^{2}(x)}} \, dx$

See Solution

Problem 29337

Calculate the integral: $\int\left(\frac{4}{x-3}-\frac{3}{(x+4)^{2}}-\frac{6}{x^{2}+16}\right) dx$

See Solution

Problem 29338

Evaluate the integral: $\int_{0}^{a} 6 x \sqrt{a^{2}-x^{2}} \, dx$ , with $a$ being a constant.

See Solution

Problem 29339

Calculate the integral of $\left(\frac{1}{2} f(x)-g(x)\right)$ with respect to $x$ .

See Solution

Problem 29340

Find the limit as $x$ approaches 1 from the left: $\lim _{x \rightarrow 1^{-}} \cosh (\sqrt{1-x})$ .

See Solution

Problem 29341

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan ^{2}(3 x)}{x \sin (7 x)}$

See Solution

Problem 29342

Find the limit as $\theta$ approaches $\frac{\pi}{2}$ for the expression $5 \theta \sin \theta$ .

See Solution

Problem 29343

Differentiate $y=\left[9^{\ln x}+\sin ^{2}\left(x^{e}\right)\right]^{\sin ^{-1} x}$ using logarithmic differentiation.

See Solution

Problem 29344

Find the definite integral of the piecewise function with points (2,3), (4,-3), (7,6) and determine its starting point.

See Solution

Problem 29345

Find the area under the curve $y=f(x)$ from $x=2$ to $x=7$ using geometric formulas and definite integral properties.

See Solution

Problem 29346

Find the limit: $L = \lim _{h \rightarrow 0} \frac{\sqrt{4(a+h)}-\sqrt{4 a}}{h}$ .

See Solution

Problem 29347

Find the point of inflection and the slope of the tangent for the function $y = x^3 + 3x^2$ .

See Solution

Problem 29348

Calculate the integral from 3 to 4 of the function $\frac{5}{v}$ .

See Solution

Problem 29349

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{2}-2 x+1}{x^{4}-1}$ .

See Solution

Problem 29350

A 5 kg book falls and lands at 5.7 m/s. Calculate the height it fell from. A. 3.3 m B. 1.7 m C. 16.7 m D. 0.9 m

See Solution

Problem 29351

Compare the vertical distance a projectile falls below a straight-line path with the distance it would fall from rest in the same time.

See Solution

Problem 29352

Bestimmen Sie die Fläche zwischen $f(x) = \frac{1}{4}x^3$ und der x-Achse von $x=0$ bis $x=2$ .

See Solution

Problem 29353

Bestimmen Sie die Steigung von $f$ bei $x_{0}=\pi$ und die Tangentengleichung an $f$ im Punkt $P$ .

See Solution

Problem 29354

Bestimmen Sie die Steigung von $f$ bei $x_{0}=\pi$ für a) $f(x)=-9 \sin (x)$ , b) $f(x)=5+\cos (x)$ , c) $f(x)=5 x-\cos (x)$ . Finden Sie die Tangentengleichung für a) $f(x)=\cos (x)$ bei $P\left(\frac{7}{4} \pi, \square\right)$ , b) $f(x)=3 \sin (x)$ bei $P\left(\frac{5 \pi}{3}, \square\right)$ , c) $f(x)=x+2 \sin (x)$ bei $P\left(\frac{\pi}{4}, \square\right)$ .

See Solution

Problem 29355

Bestimmen Sie die Tangentengleichungen für die Funktionen an den Punkten: a) $f(x)=\cos (x)$ bei $P\left(\frac{7}{4} \pi, \square\right)$ b) $f(x)=3 \sin (x)$ bei $P\left(\frac{5 \pi}{3}, \square\right)$ c) $f(x)=x+2 \sin (x)$ bei $P\left(\frac{\pi}{4}, \square\right)$

See Solution

Problem 29356

Verify if $y=\varphi(x)=(1-\sin(x))^{-1/2}$ is a solution of $2y' = y^3 \cos(x)$ . Calculate $2y'$ and $y^3 \cos(x)$ .

See Solution

Problem 29357

$x^{\frac{2}{3}}+y^{\frac{2}{3}}=4$ denklemi için $x=8$ 'de $\frac{d}{dx}y^x$ 'in değerini bulun.

See Solution

Problem 29358

Bestimme die Ableitung der Funktionen: a) $f(x)=e^{0,1 x+5}$ , b) $f(t)=3,5 \cdot e^{-4 t-9}$ , c) $f(x)=1,5 x-1,5 \cdot e^{\frac{2}{3} x}$ , d) $f(x)=e^{-(x+2)}$ .

See Solution

Problem 29359

Find the volumes of solids formed by revolving regions bounded by the given equations around specified lines.

See Solution

Problem 29360

Find the volume of the solid formed by revolving the area between these curves around the line $y=4$ : 17. $y=x, y=3, x=0$ ; 18. $y=\frac{1}{2} x^{3}, y=4, x=0$ ; 19. $y=\frac{3}{1+x}, y=0, x=0, x=3$ .

See Solution

Problem 29361

Find the volume of the solid formed by revolving the area between these curves around the line $y=4$ :
1. $y=x$ , $y=3$ , $x=0$
2. $y=\frac{1}{2} x^{3}$ , $y=4$ , $x=0$
3. $y=\frac{3}{1+x}$ , $y=0$ , $x=0$ , $x=3$

See Solution

Problem 29362

Berechnen Sie das Integral $\int_{-2}^{4} 3 \, dx$ und zeichnen Sie den Graphen der Funktion.

See Solution

Problem 29363

Bestimmen Sie die Nullstellen der Funktion $f(x)=x^{3}-4x$ und die Tangente im Punkt $W$ .

See Solution

Problem 29364

How far does an object fall during the third second after being dropped? A. $4.9 \mathrm{~m}$ B. $9.8 \mathrm{~m}$ C. $15 \mathrm{~m}$ D. $25 \mathrm{~m}$

See Solution

Problem 29365

Find the speed of a particle at $t=2$ sec given its position function $x(t)=5 t^{2}-2 t-4$ .

See Solution

Problem 29366

Find the limits as $x$ approaches infinity: 1) $-2x^3 - 2x$ , 2) $2x^4 + 6x^3 - 2x$ , 3) $-9x^5 - 6x^3 - x$ .

See Solution

Problem 29367

How far does an object fall during the third second after being released from a building? Use $s = \frac{1}{2}gt^2$ .

See Solution

Problem 29368

Bestimmen Sie die Konzentration $f(1)$ , wann $f(x)=500$ und erklären Sie $f'(10)=0$ , $f(10)=1190$ . Ordnen Sie Ableitungen zu.

See Solution

Problem 29369

Determine if the following limit statements are true or false:
1. $\lim _{x \rightarrow-\infty}(-2 x^{3}-2 x)=\infty$
2. $\lim _{x \rightarrow \infty}(2 x^{4}+6 x^{3}-2 x)=\infty$
3. $\lim _{x \rightarrow \infty}(-9 x^{5}-6 x^{3}-x)=-\infty$

See Solution

Problem 29370

Gegeben ist die Funktion $f(x)=3-\frac{1}{2} \cdot e^{-x}$ .
a) Finde die Schnittpunkte von $\mathrm{K}_{f}$ mit den Achsen und die Asymptote. b) Beschreibe die Entstehung von $\mathrm{K}_{f}$ aus $e^{x}$ und zeige, dass $f$ monoton wächst. c) Erkläre, wie man den Flächeninhalt zwischen $f$ , $y=3$ , der $y$ -Achse und $x=4$ berechnet. d) Bestimme die Koordinaten des Berührpunkts B der Tangente $y=\frac{1}{2} x+\frac{5}{2}$ . e) Beurteile die Aussage über ganzrationale Funktionen vierten Grades mit drei Wendepunkten.

See Solution

Problem 29371

Find the limit of the function and determine if the statements are True or False:
1. $\lim _{x \rightarrow-\infty}\left(-2 x^{3}-2 x\right)=\infty$
2. $\lim _{x \rightarrow \infty}\left(2 x^{4}+6 x^{3}-2 x\right)=\infty$
3. $\lim _{x \rightarrow \infty}\left(-9 x^{5}-6 x^{3}-x\right)=-\infty$

See Solution

Problem 29372

Berechne den Steigungswinkel und die Steigung in Prozent für die Punkte $B_{1}(10, h(10))$ , $B_{2}(30, h(30))$ , $B_{3}(50, h(50))$ und $B_{4}(60, h(60))$ der Funktion $h(x)=-0,00001 \cdot x^{4}-0,0006 \cdot x^{3}+0,1 \cdot x^{2}$ .

See Solution

Problem 29373

Berechne die Fläche zwischen dem Graphen von $f$ , der Tangente in $P$ und der $x$ -Achse für $f(x)=\frac{1}{2} x^{2}$ und $P(3 \mid 4,5)$ .

See Solution

Problem 29374

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x^{2}}{2+2^{x}}$

See Solution

Problem 29375

Soru 4. Aşağıdaki limitleri hesaplayın. (a) $\lim _{x \rightarrow \infty} x(\sin (5 / x))$ (b) $\lim _{x \rightarrow 0} \frac{1}{\sin x}-\cot x$

See Solution

Problem 29376

A spring stretches from $9.0 \, \mathrm{cm}$ to $12.0 \, \mathrm{cm}$ . Calculate the work done in this stretch.

See Solution

Problem 29377

Untersuchen Sie das Grenzverhalten von $f: x \rightarrow \frac{7}{x+2}-4$ für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ .

See Solution

Problem 29378

Oblicz sumę szeregu $\sum_{n=0}^{\infty} \frac{(-1)^{n+1}}{2^{2 n}}$ .

See Solution

Problem 29379

A patient takes 600 mg of aspirin.
a) Write the equation for aspirin amount after $t$ minutes. b) How many half-lives in 3 hours? c) Amount left after 5 hours?

See Solution

Problem 29380

Find $\lim _{x \rightarrow 2} f(x)$ where $f(x)=\ln x$ for $0<x \leq 2$ and $f(x)=x^{2} \ln 2$ for $2<x \leq 4$ .

See Solution

Problem 29381

Berechne die Steigung von $f(x)=2 x^{2}+1$ bei $x_{0}=3$ mit der h-Methode oder der $(x \rightarrow x_{0})$ -Methode.

See Solution

Problem 29382

Estimate the rate of change in shirts sold at 4 weeks for $S(t)=-x^{2}+5x+2$ using a centered interval.

See Solution

Problem 29383

Find the limit as $x$ approaches 2 for the expression $\frac{x^{2}-4}{x^{2}-5x+6}$ .

See Solution

Problem 29384

Find the tangent line equation to $y=x \tan (x)$ at $x=\frac{\pi}{4}$ with slope $y^{\prime}\left(\frac{\pi}{4}\right)=1+\frac{\pi}{2}$ .

See Solution

Problem 29385

Given the table of values for $f(x)$ at different $x$ , which limit conclusion is correct as $x$ approaches 3? A. $\lim _{x \rightarrow 3} f(x)=0$ B. $\lim _{x \rightarrow 3} f(x)=3$ C. $\lim _{x \rightarrow 3} f(x)=10$ D. $\lim _{x \rightarrow 3} f(x)$ does not exist

See Solution

Problem 29386

Estimate the average rate of change of Marie's college fund $A(t)=100(1.04)^{t}$ from age 17 to 18, rounded to the nearest tenth.

See Solution

Problem 29387

Find the tangent line equation at point $(1,0)$ for the curve $x^{2} \sin y+(2 x+y)^{3}=8$ .

See Solution

Problem 29388

Find stationary points of $y=3 x^{4}-16 x^{3}+18 x^{2}+6$ and values of $c$ for four real roots in $3 x^{4}-16 x^{3}+18 x^{2}+6=c$ .

See Solution

Problem 29389

Find the velocity and acceleration of a train at $t=5 \mathrm{~s}$ given $s(t)=200(t+1)^{-2}$ . Is it speeding up or slowing down?

See Solution

Problem 29390

Estimate $31^{4 / 5}$ using a linear approximation.

See Solution

Problem 29391

Find the rate of change of depth $D(t)=7 \sin \left(\frac{\pi}{6} t-\frac{7 \pi}{6}\right)+4$ at 6 a.m. (in ft/hr).

See Solution

Problem 29392

Une boule est lâchée d'une hauteur de 553,2 m. Quelle est sa vitesse juste avant de toucher le sol?

See Solution

Problem 29393

Evaluate these integrals: a) $\int \sin \theta d \theta$ , b) $\int \frac{x+\sqrt{x}+\sqrt{x^{3}}}{x^{2}} d x$ , c) $\int_{0}^{1}(x+1)^{2} d x$ .

See Solution

Problem 29394

Find the derivative of $y=(6 x-7)^{95}$ , that is, calculate $y^{\prime}=$ .

See Solution

Problem 29395

Find the velocity of the object at time $t=\frac{\pi}{4}$ given $S(t)=12+7 \cos t$ .

See Solution

Problem 29396

A quantity starts at 2300 and grows at $45\%$ per minute. Find its value after 0.2 hours, rounded to two decimal places.

See Solution

Problem 29397

Arianys invested \ $61,000 at$ 7 \frac{1}{8} \% $continuous interest. Yusuf invested \$ 61,000 at$ 7 \frac{5}{8} \%$ monthly. After 14 years, how much more does Yusuf have than Arianys?

See Solution

Problem 29398

Approximate $\int_{0}^{4} x^{3} d x$ with a left Riemann sum for $n=8$ , then find the exact value using the Fundamental Theorem.

See Solution

Problem 29399

Find the derivative of $\ln(5x^{2}+4)$ with respect to $x$ .

See Solution

Problem 29400

For the function $y=x^{3}(x^{2}+1)$ , find: a) $\left.\frac{d y}{d x}\right|_{x=-1}=m$ , b) the point at $x=-1$ , c) the tangent line equation at $x=-1$ .

See Solution

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Calculus

Problem 29301

Evaluate the integral ∫−10x512x2+4dx\int_{-1}^{0} \frac{x^{5}}{\sqrt{12 x^{2}+4}} d x∫−10​12x2+4​x5​dx.

Problem 29302

Calculate the indefinite integral and include the constant CCC: ∫x+4x2+8x−6dx \int \frac{x+4}{\sqrt{x^{2}+8 x-6}} d x ∫x2+8x−6​x+4​dx

Problem 29303

Differentiate the function 4x5−4x+3x4x^{5}-\frac{4}{x}+3\sqrt{x}4x5−x4​+3x​.

Problem 29304

Find dvdt\frac{dv}{dt}dtdv​ for the function 3tan⁡(2t)−4e3t+ln⁡(5)3 \tan(2t) - 4e^{3t} + \ln(5)3tan(2t)−4e3t+ln(5).

Problem 29305

Find uuu and dvd vdv to solve the integral ∫xe237xdx\int x e^{237 x} d x∫xe237xdx where dv=dvdxd v = {}^{d v} d xdv=dvdx.

Problem 29306

Differentiate f(θ)=(2θ+1)ln⁡(1+2θ)f(\theta)=(2 \theta+1) \ln (1+2 \theta)f(θ)=(2θ+1)ln(1+2θ) with respect to θ\thetaθ.

Problem 29307

Find the derivative of y=43x2+1y=\frac{4}{\sqrt{3x^{2}+1}}y=3x2+1​4​ with respect to xxx, simplified.

Problem 29308

Evaluate the integral ∫π/6π/23csc⁡x9csc⁡xcsc⁡xcot⁡x dx\int_{\pi / 6}^{\pi / 2} 3^{\csc x} 9^{\csc x} \csc x \cot x \, dx∫π/6π/2​3cscx9cscxcscxcotxdx.

Problem 29309

Find the derivative of the integral: ddx∫43xcos⁡(4t)dt\frac{d}{d x} \int_{4}^{\sqrt{3 x}} \cos (4 t) d tdxd​∫43x​​cos(4t)dt.

Problem 29310

Find the average rate of change of g(x)=4x2+3x3g(x)=4x^{2}+\frac{3}{x^{3}}g(x)=4x2+x33​ over the interval [−2,1][-2,1][−2,1].

Problem 29311

Find the derivative of the function f(x)=∫03xsin⁡(t2)dtf(x)=\int_{0}^{3 x} \sin(t^{2}) dtf(x)=∫03x​sin(t2)dt. What is f′(x)f^{\prime}(x)f′(x)?

Problem 29312

Evaluate the integral: ∫x2−4x+4x3+1dx\int \frac{x^{2}-4 x+4}{x^{3}+1} d x∫x3+1x2−4x+4​dx

Problem 29313

Find the derivative of the function 3sin⁡(5x)+73 \sin (5 x) + 73sin(5x)+7.

Problem 29314

Find the derivative of y=(x3−9)4y=\left(x^{3}-9\right)^{4}y=(x3−9)4 with respect to xxx: dydx\frac{d y}{d x}dxdy​.

Problem 29315

Evaluate the integral: ∫11−2x1−8x−x2dx\int \frac{11-2 x}{\sqrt{1-8 x-x^{2}}} d x∫1−8x−x2​11−2x​dx

Problem 29316

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=xx−2y=\frac{x}{x-2}y=x−2x​.

Problem 29317

Find f′′(π2)f^{\prime \prime}\left(\frac{\pi}{2}\right)f′′(2π​) for the function f(x)=sin⁡(3x)f(x)=\sin(3x)f(x)=sin(3x).

Problem 29318

Evaluate the integral: ∫11−2x1−8x−x2 dx\int \frac{11-2 x}{\sqrt{1-8 x-x^{2}}} \, dx∫1−8x−x2​11−2x​dx

Problem 29319

Find dydx\frac{d y}{d x}dxdy​ if x3+y3=3xyx^{3}+y^{3}=3 x yx3+y3=3xy.

Problem 29320

Find the value of kkk if y=xekxy=x e^{k x}y=xekx and d2ydx2=6\frac{d^{2} y}{d x^{2}}=6dx2d2y​=6 at x=0x=0x=0.

Problem 29321

Find the derivative of tan⁡(3x)+e3y\tan(3x) + e^{3y}tan(3x)+e3y with respect to xxx.

Problem 29322

Find the derivative of the integral ∫−2x5t2−tdt\int_{-2}^{x^{5}} \sqrt{t^{2}-t} d t∫−2x5​t2−t​dt with respect to xxx.

Problem 29323

Find the limit as xxx approaches 0 from the left: lim⁡x→0−sin⁡xx\lim _{x \rightarrow 0^{-}} \frac{\sin x}{x}limx→0−​xsinx​.

Problem 29324

Find f′(x)f^{\prime}(x)f′(x) for the function f(x)=∫−13xcos⁡(t2)dtf(x)=\int_{-1}^{3 x} \cos(t^{2}) dtf(x)=∫−13x​cos(t2)dt.

Problem 29325

Find the acceleration of a particle at time t=4t=4t=4 if its velocity is v(t)=3+4.1cos⁡(0.9t)v(t)=3+4.1 \cos (0.9 t)v(t)=3+4.1cos(0.9t).

Problem 29326

Find F′(x)F^{\prime}(x)F′(x) for the function F(x)=∫4x3sin⁡(t2+2t)dtF(x)=\int_{4}^{x^{3}} \sin(t^{2}+2t) dtF(x)=∫4x3​sin(t2+2t)dt.

Problem 29327

Find g′(2)g'(2)g′(2) where g(x)=∫−1xf(t)dtg(x)=\int_{-1}^{x} f(t) dtg(x)=∫−1x​f(t)dt and fff is the derivative of ggg.

Problem 29328

Find the value of ∫3−4(f(x)−1)dx\int_{3}^{-4}(f(x)-1) d x∫3−4​(f(x)−1)dx given ∫35f(x)dx=2\int_{3}^{5} f(x) d x=2∫35​f(x)dx=2 and ∫−45f(x)dx=−4\int_{-4}^{5} f(x) d x=-4∫−45​f(x)dx=−4.

Problem 29329

Find the average value of the function f(x)=−x2+4f(x)=-x^{2}+4f(x)=−x2+4 over the interval [−1,1][-1,1][−1,1] in simplest form.

Problem 29330

Calculate the integral: ∫(10x+2x2+25)dx\int\left(\frac{10 x+2}{x^{2}+25}\right) d x∫(x2+2510x+2​)dx

Problem 29331

Explore Taylor polynomials up to degree five for f(x)=exf(x)=e^{x}f(x)=ex at a=1a=1a=1. Find T0(x)T_0(x)T0​(x) to T5(x)T_5(x)T5​(x) and coefficients.

Problem 29332

Calculate the integral: ∫5x49x4+16 dx\int \frac{5x}{49x^4 + 16} \, dx∫49x4+165x​dx

Problem 29333

Calculate the integral: ∫(4x+581−x2)dx\int\left(\frac{4 x+5}{\sqrt{81-x^{2}}}\right) dx∫(81−x2​4x+5​)dx

Problem 29334

Evaluate the integral: ∫ex1−81e2xdx\int \frac{e^{x}}{\sqrt{1-81 e^{2 x}}} d x∫1−81e2x​ex​dx

Problem 29335

Find the total cost for 30 units using C(x)=14x+1500C(x)=14x+1500C(x)=14x+1500, the marginal cost, and the cost to increase production to 31 units.

Problem 29336

Calculate the integral: ∫5sin⁡(x)cos⁡(x)10+sin⁡2(x) dx\int \frac{5 \sin (x) \cos (x)}{\sqrt{10+\sin ^{2}(x)}} \, dx∫10+sin2(x)​5sin(x)cos(x)​dx

Problem 29337

Calculate the integral: ∫(4x−3−3(x+4)2−6x2+16)dx\int\left(\frac{4}{x-3}-\frac{3}{(x+4)^{2}}-\frac{6}{x^{2}+16}\right) dx∫(x−34​−(x+4)23​−x2+166​)dx

Problem 29338

Evaluate the integral: ∫0a6xa2−x2 dx\int_{0}^{a} 6 x \sqrt{a^{2}-x^{2}} \, dx∫0a​6xa2−x2​dx, with aaa being a constant.

Problem 29339

Calculate the integral of (12f(x)−g(x))\left(\frac{1}{2} f(x)-g(x)\right)(21​f(x)−g(x)) with respect to xxx.

Problem 29340

Evaluate the integral $\int_{-1}^{0} \frac{x^{5}}{\sqrt{12 x^{2}+4}} d x$ .

Calculate the indefinite integral and include the constant $C$ :
$\int \frac{x+4}{\sqrt{x^{2}+8 x-6}} d x$

Differentiate the function $4x^{5}-\frac{4}{x}+3\sqrt{x}$ .

Find $\frac{dv}{dt}$ for the function $3 \tan(2t) - 4e^{3t} + \ln(5)$ .

Find $u$ and $d v$ to solve the integral $\int x e^{237 x} d x$ where $d v = {}^{d v} d x$ .

Differentiate $f(\theta)=(2 \theta+1) \ln (1+2 \theta)$ with respect to $\theta$ .

Find the derivative of $y=\frac{4}{\sqrt{3x^{2}+1}}$ with respect to $x$ , simplified.

Evaluate the integral $\int_{\pi / 6}^{\pi / 2} 3^{\csc x} 9^{\csc x} \csc x \cot x \, dx$ .

Find the derivative of the integral: $\frac{d}{d x} \int_{4}^{\sqrt{3 x}} \cos (4 t) d t$ .

Find the average rate of change of $g(x)=4x^{2}+\frac{3}{x^{3}}$ over the interval $[-2,1]$ .

Find the derivative of the function $f(x)=\int_{0}^{3 x} \sin(t^{2}) dt$ . What is $f^{\prime}(x)$ ?

Evaluate the integral: $\int \frac{x^{2}-4 x+4}{x^{3}+1} d x$

Find the derivative of the function $3 \sin (5 x) + 7$ .

Find the derivative of $y=\left(x^{3}-9\right)^{4}$ with respect to $x$ : $\frac{d y}{d x}$ .

Evaluate the integral: $\int \frac{11-2 x}{\sqrt{1-8 x-x^{2}}} d x$

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{x}{x-2}$ .

Find $f^{\prime \prime}\left(\frac{\pi}{2}\right)$ for the function $f(x)=\sin(3x)$ .

Evaluate the integral: $\int \frac{11-2 x}{\sqrt{1-8 x-x^{2}}} \, dx$

Find $\frac{d y}{d x}$ if $x^{3}+y^{3}=3 x y$ .

Find the value of $k$ if $y=x e^{k x}$ and $\frac{d^{2} y}{d x^{2}}=6$ at $x=0$ .

Find the derivative of $\tan(3x) + e^{3y}$ with respect to $x$ .

Find the derivative of the integral $\int_{-2}^{x^{5}} \sqrt{t^{2}-t} d t$ with respect to $x$ .

Find the limit as $x$ approaches 0 from the left: $\lim _{x \rightarrow 0^{-}} \frac{\sin x}{x}$ .

Find $f^{\prime}(x)$ for the function $f(x)=\int_{-1}^{3 x} \cos(t^{2}) dt$ .

Find the acceleration of a particle at time $t=4$ if its velocity is $v(t)=3+4.1 \cos (0.9 t)$ .

Find $F^{\prime}(x)$ for the function $F(x)=\int_{4}^{x^{3}} \sin(t^{2}+2t) dt$ .

Find $g'(2)$ where $g(x)=\int_{-1}^{x} f(t) dt$ and $f$ is the derivative of $g$ .

Find the value of $\int_{3}^{-4}(f(x)-1) d x$ given $\int_{3}^{5} f(x) d x=2$ and $\int_{-4}^{5} f(x) d x=-4$ .

Find the average value of the function $f(x)=-x^{2}+4$ over the interval $[-1,1]$ in simplest form.

Calculate the integral: $\int\left(\frac{10 x+2}{x^{2}+25}\right) d x$

Explore Taylor polynomials up to degree five for $f(x)=e^{x}$ at $a=1$ . Find $T_0(x)$ to $T_5(x)$ and coefficients.

Calculate the integral: $\int \frac{5x}{49x^4 + 16} \, dx$

Calculate the integral: $\int\left(\frac{4 x+5}{\sqrt{81-x^{2}}}\right) dx$

Evaluate the integral: $\int \frac{e^{x}}{\sqrt{1-81 e^{2 x}}} d x$

Find the total cost for 30 units using $C(x)=14x+1500$ , the marginal cost, and the cost to increase production to 31 units.

Calculate the integral: $\int \frac{5 \sin (x) \cos (x)}{\sqrt{10+\sin ^{2}(x)}} \, dx$

Calculate the integral: $\int\left(\frac{4}{x-3}-\frac{3}{(x+4)^{2}}-\frac{6}{x^{2}+16}\right) dx$

Evaluate the integral: $\int_{0}^{a} 6 x \sqrt{a^{2}-x^{2}} \, dx$ , with $a$ being a constant.

Calculate the integral of $\left(\frac{1}{2} f(x)-g(x)\right)$ with respect to $x$ .

Find the limit as $x$ approaches 1 from the left: $\lim _{x \rightarrow 1^{-}} \cosh (\sqrt{1-x})$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan ^{2}(3 x)}{x \sin (7 x)}$

Find the limit as $\theta$ approaches $\frac{\pi}{2}$ for the expression $5 \theta \sin \theta$ .

Differentiate $y=\left[9^{\ln x}+\sin ^{2}\left(x^{e}\right)\right]^{\sin ^{-1} x}$ using logarithmic differentiation.

Find the area under the curve $y=f(x)$ from $x=2$ to $x=7$ using geometric formulas and definite integral properties.

Find the limit: $L = \lim _{h \rightarrow 0} \frac{\sqrt{4(a+h)}-\sqrt{4 a}}{h}$ .

Find the point of inflection and the slope of the tangent for the function $y = x^3 + 3x^2$ .

Calculate the integral from 3 to 4 of the function $\frac{5}{v}$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{2}-2 x+1}{x^{4}-1}$ .

Bestimmen Sie die Fläche zwischen $f(x) = \frac{1}{4}x^3$ und der x-Achse von $x=0$ bis $x=2$ .

Bestimmen Sie die Steigung von $f$ bei $x_{0}=\pi$ und die Tangentengleichung an $f$ im Punkt $P$ .

Verify if $y=\varphi(x)=(1-\sin(x))^{-1/2}$ is a solution of $2y' = y^3 \cos(x)$ . Calculate $2y'$ and $y^3 \cos(x)$ .

$x^{\frac{2}{3}}+y^{\frac{2}{3}}=4$ denklemi için $x=8$ 'de $\frac{d}{dx}y^x$ 'in değerini bulun.

Bestimme die Ableitung der Funktionen: a) $f(x)=e^{0,1 x+5}$ , b) $f(t)=3,5 \cdot e^{-4 t-9}$ , c) $f(x)=1,5 x-1,5 \cdot e^{\frac{2}{3} x}$ , d) $f(x)=e^{-(x+2)}$ .

Find the volume of the solid formed by revolving the area between these curves around the line $y=4$ : 17. $y=x, y=3, x=0$ ; 18. $y=\frac{1}{2} x^{3}, y=4, x=0$ ; 19. $y=\frac{3}{1+x}, y=0, x=0, x=3$ .

Find the volume of the solid formed by revolving the area between these curves around the line $y=4$ :
1. $y=x$ , $y=3$ , $x=0$
2. $y=\frac{1}{2} x^{3}$ , $y=4$ , $x=0$
3. $y=\frac{3}{1+x}$ , $y=0$ , $x=0$ , $x=3$

Berechnen Sie das Integral $\int_{-2}^{4} 3 \, dx$ und zeichnen Sie den Graphen der Funktion.

Bestimmen Sie die Nullstellen der Funktion $f(x)=x^{3}-4x$ und die Tangente im Punkt $W$ .

How far does an object fall during the third second after being dropped? A. $4.9 \mathrm{~m}$ B. $9.8 \mathrm{~m}$ C. $15 \mathrm{~m}$ D. $25 \mathrm{~m}$

Find the speed of a particle at $t=2$ sec given its position function $x(t)=5 t^{2}-2 t-4$ .

Find the limits as $x$ approaches infinity: 1) $-2x^3 - 2x$ , 2) $2x^4 + 6x^3 - 2x$ , 3) $-9x^5 - 6x^3 - x$ .

How far does an object fall during the third second after being released from a building? Use $s = \frac{1}{2}gt^2$ .

Bestimmen Sie die Konzentration $f(1)$ , wann $f(x)=500$ und erklären Sie $f'(10)=0$ , $f(10)=1190$ . Ordnen Sie Ableitungen zu.

Berechne die Fläche zwischen dem Graphen von $f$ , der Tangente in $P$ und der $x$ -Achse für $f(x)=\frac{1}{2} x^{2}$ und $P(3 \mid 4,5)$ .

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x^{2}}{2+2^{x}}$