Calculus

Problem 1301

Find the limits: 1. $\lim _{x \rightarrow 3} \frac{3-x}{x^{2}-9}$ and 2. $\lim _{x \rightarrow 4} \frac{x^{2}-5 x+4}{x^{2}-2 x-8}$ .

See Solution

Problem 1302

Find the limit as x approaches 1: $\lim_{{x \to 1}} \frac{\sqrt{2 x^{3}+x+1}-2 x}{x-1}$

See Solution

Problem 1303

Find the limits: 1. $\lim _{x \rightarrow 4} \frac{\sqrt{x+5}-3}{x-4}$ , 2. $\lim _{x \rightarrow \frac{\pi}{4}} \frac{1-\tan x}{\sin x-\cos x}$ . Also, find $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ for $f(x)=x^{2}-x$ and $f(x)=\frac{1}{x+3}$ .

See Solution

Problem 1304

Find the limit as $h$ approaches 0 for the difference quotient of: 1. $f(x)=x^{2}-x$ , 2. $f(x)=\frac{1}{x+3}$ .

See Solution

Problem 1305

מה הגבול של $\left(\sqrt[3]{x^{3}+x^{2}}-x\right)$ כש- $x$ מתקרב למינוס אינסוף?

See Solution

Problem 1306

Evaluate the piecewise function $f$ at $x=3$ to check if it's continuous and differentiable. What is true about $f$ at $x=3$ ?

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Problem 1307

Given that $f(x)$ is continuous at $x=-9$ with $f(-9)=3$ and $f^{\prime}(-9)=-1$ , which statements are true?
1. $f(x)$ is differentiable at $x=-9$
2. $\lim _{x \rightarrow 3} f(x)=-9$
3. $\lim _{x \rightarrow -9} f(x)=3$
4. None of the above.

See Solution

Problem 1308

Given $g(x)$ is differentiable at $x=-3$ with $g(-3)=9$ and $g'(-3)=0$ , which statements are true?

See Solution

Problem 1309

Given the piecewise function $f$ , determine if it is continuous or differentiable at $x=-3$ . Options: 1) neither, 2) continuous only, 3) both.

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Problem 1310

Is the function $f(x)=\left\{\begin{array}{cc}\sin (x) & x<\frac{\pi}{2} \\ \frac{2 x}{\pi} & x \geq \frac{\pi}{2}\end{array}\right.$ continuous at $x=\frac{\pi}{2}$ ? Explain.

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Problem 1311

Find the acceleration due to gravity at 6989 m, the horizontal asymptote of $g(h)$ , and solve $g(h)=0$ .

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Problem 1312

When can average speed equal instantaneous speed?

See Solution

Problem 1313

True or false: The integral from -2 to 2 of $\frac{1}{x^{2}}$ equals $-1$ .

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Problem 1314

True or false: For any continuous function $f$ , is it true that $\int_{3}^{3} f(x) dx = f(3)$ ?

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Problem 1315

Find the function $y(x)=7+\int_{2}^{x} \sin(t^{2}) dt$ given $\frac{d y}{d x}=\sin(x^{2})$ and $y(2)=7$ .

See Solution

Problem 1316

Determine if the integral $\int_{0}^{1} \frac{1}{\sqrt{x}} d x$ converges or diverges.

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Problem 1317

Determine if the integral $\int_{0}^{1} \frac{1}{\sqrt{x}} d x$ converges or diverges.

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Problem 1318

True or false: A continuous function has an antiderivative.

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Problem 1319

True or false: $\int \frac{1}{1+x^{2}} d x=\ln \left(\left|1+x^{2}\right|\right)+C$

See Solution

Problem 1320

True or false: The integral of $\sin(x)$ is $\cos(x) + C$ .

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Problem 1321

True or false: If $f$ is continuous on $[a, b]$ , then $\left[\int_{a}^{b} f(x) dx\right]^{2}=\int_{a}^{b}[f(x)]^{2} dx$ ?

See Solution

Problem 1322

True or false: For any continuous function $f$ , is it true that $\frac{d}{d x} \int_{x}^{0} f(t) d t=-f(x)$ ?

See Solution

Problem 1323

True or false: For any continuous function $f$ , does $\int_{a}^{b} f(x) dx + \int_{b}^{a} f(x) dx = 0$ when $a < b$ ?

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Problem 1324

True or false: If $0 \leq f(x) \leq g(x)$ and $\int_{3}^{\infty} g(x) dx$ diverges, does $\int_{3}^{\infty} f(x) dx$ also diverge?

See Solution

Problem 1325

True or false: For differentiable functions $f$ and $g$ , is it true that $\int f(x) g^{\prime}(x) dx = f(x) g(x) - \int f^{\prime}(x) g(x) dx$ ?

See Solution

Problem 1326

Find the volume of a solid with base $\mathrm{R}$ (bounded by $y=x(1-x)^{1/2}$ and the $x$ -axis) and square cross-sections.

See Solution

Problem 1327

Find the value of the integral $\int_{1}^{4} \frac{f(\sqrt{x})}{\sqrt{x}} d x$ given that $\int_{1}^{2} f(x) d x=6$ .

See Solution

Problem 1328

Given $f$ is continuous with $f(x) \geq 0$ for $a \leq x \leq b$ , and the area from $a$ to $b$ is 8. If $F(a)=6$ , find $F(b)$ .

See Solution

Problem 1329

A moped slows down from $31 \mathrm{ft} / \mathrm{s}$ at $3 \mathrm{ft} / \mathrm{s}^{2}$ . How far does it travel before stopping?

See Solution

Problem 1330

Find $F^{\prime}(3)$ where $F(x)=\int_{x}^{x^{2}} f(t) dt$ , given $f(3)=3$ and $f(9)=7$ .

See Solution

Problem 1331

Find the value of $\int_{2}^{5}[2 f(x)-3 g(x)+9] d x$ given that $\int_{2}^{5} f(x) d x=8$ and $\int_{2}^{5} g(x) d x=4$ .

See Solution

Problem 1332

Find the exact value of $y(2)$ given the differential equation $\frac{d y}{d x}=3 x^{2}+2$ and $y(1)=6$ .

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Problem 1333

Evaluate the integral $\int_{1}^{4} x f^{\prime}(x) d x$ given that the area under $y=f(x)$ from $x=1$ to $x=4$ is 9.

See Solution

Problem 1334

A moped slows down from $31 \mathrm{ft} / \mathrm{s}$ at $3 \mathrm{ft} / \mathrm{s}^{2}$ . How far does it travel before stopping? Provide a numerical answer.

See Solution

Problem 1335

Estimate the area under $f(x)$ from $x=0$ to $x=8$ using 4 rectangles and left endpoints. Calculate $L_{4}$ .

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Problem 1336

Find the average rate of change of $f(x)=x^{2}+1$ from $x=-10$ to $x=10$ . A. 0 B. 0.05 C. 5.05 D. 101

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Problem 1337

Given the Beverton-Holt model $R(N_t)=\frac{R_0}{1+a N_t}$ with $a=0.05$ , $R_0=5$ , find $N_t$ for $t=1,2,\ldots,5$ and $\lim_{t \to \infty} N_t$ for $N_0=2$ .

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Problem 1338

Use the Beverton-Holt model $R(N_{t})=\frac{R_{0}}{1+a N_{t}}$ with $a=0.05$ , $R_{0}=5$ . Find $N_{t}$ for $t=1,2,\ldots,5$ and $\lim_{t \to \infty} N_{t}$ for $N_{0}=2$ .

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Problem 1339

Solve the equation $\frac{d y}{d x}=-4(1+y^{2}) x^{7}$ using separation of variables. Let $C$ be the constant. $y=$

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Problem 1340

Solve the equation $(10+x^{4}) \frac{dy}{dx}=\frac{x^{3}}{y}$ using separation of variables. Find $y^{2}=$ .

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Problem 1341

Solve the equation $e^{-y} \sin (x)+\frac{d y}{d x}=0$ using separation of variables. Find $y=C$ .

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Problem 1342

Solve the initial value problem using separation of variables: $\frac{d y}{d x}-x^{2} e^{y}=9 e^{y}$ , $y(0)=3$ . Find $y=$

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Problem 1343

Solve the equation $\frac{d y}{d t}=5 y$ with the initial condition $y(2)=2$ . Find $y=$

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Problem 1344

Solve the equation $\frac{d u}{d t}=e^{3 u+4 t}$ with $u(0)=13$ . Find $u(t)$ .

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Problem 1345

Solve the equation $\frac{d y}{d x}=\frac{-0.4}{\cos (y)}$ with initial condition $y(0)=\frac{\pi}{3}$ . Find $y(x)$ .

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Problem 1346

Solve the differential equation $y' = x^4 \tan(y)$ for the general solution, using the constant $\mathrm{C}$ .

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Problem 1347

A rocket starts from rest with mass $m_0$ and burns fuel at rate $k$ . Find $v(t)$ from $m = m_0 - kt$ and $m \frac{dv}{dt} = ck - mg$ .
(a) $v(t) = \, \mathrm{m/sec}$
(b) If fuel is 80% of $m_0$ and lasts 110s, find $v(110)$ with $g=9.8 \, \mathrm{m/s}^2$ and $c=2500 \, \mathrm{m/s}$ .
$v(110) = \, \mathrm{m/sec}$ [Round to nearest whole number]

See Solution

Problem 1348

Find the derivative of $e^{x} \cos x$ .

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Problem 1349

Find the horizontal asymptote of the drug concentration function $C(t)=\frac{t}{7t^{2}+8}$ . What is $C=\square$ ?

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Problem 1350

Find the horizontal asymptote of $C(t)=\frac{t}{7 t^{2}+8}$ , identify the graph, and determine when concentration is highest.

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Problem 1351

Find the displacement of a mass on a spring from $t=0$ to $t=\pi$ given $v(t)=6 \sin(t)-6 \cos(t)$ . Evaluate $\int_{0}^{\pi}(6 \sin(t)-6 \cos(t)) dt$ .

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Problem 1352

Calculate the work done by the force $F(x)=x^{-1}+4x$ from $x=3$ to $x=5$ : $W=\int_{3}^{5}(x^{-1}+4x)dx$

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Problem 1353

Find the distance a wave travels from $t=1$ to $t=4$ given $v=\sqrt{\frac{8}{x}}$ . Evaluate the integral $\int_{1}^{4} \sqrt{\frac{8}{x}} d x$ .

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Problem 1354

A car's velocity is given by $v(t)=2 t^{1/2}+5$ . Find the displacement (in m) from $t=2$ to $t=8$ .

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Problem 1355

Find the displacement and total distance traveled by a particle with velocity $v(t)=4-2t$ from $t=0$ to $t=6$ .

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Problem 1356

A stone is thrown from a $120 \mathrm{ft}$ cliff at $148 \mathrm{ft/s}$ . Find: (a) time to highest point, (b) max height, (c) time to beach, (d) impact velocity.

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Problem 1357

Find the derivative of $f(x)=4 \cos 2 x+\log x+x$ .

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Problem 1358

Find the position of a particle at time $t=7$ given $a(t)=18t+2$ , $s(0)=16$ , and $v(0)=7$ .

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Problem 1359

A particle has acceleration $a(t)=t-5/2$ m/s² for $0 \leq t \leq 9$ . Find $v(t)$ , $d(t)$ , and total distance traveled.

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Problem 1360

Find the water flow in liters from a tank in the first 16 minutes, given $r(t)=200-4t$ , for $0 \leq t \leq 50$ .

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Problem 1361

Calculate the total oil leaked in the first 5 hours after the tanker breaks apart using $R(t)=\frac{0.9}{1+t^{2}}$ .

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Problem 1362

Find the displacement and total distance traveled by a particle with velocity $v(t)=4-2t$ from $t=0$ to $t=6$ .

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Problem 1363

A stone thrown from a 120 ft cliff at 148 ft/s, with gravity -32 ft/s². Find time to highest point, max height, time to beach, and impact velocity.

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Problem 1364

Find $f$ for the integral $\int \frac{x}{\sqrt{3 x^{2}+2}} d x$ using the substitution $u=3 x^{2}+2$ .

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Problem 1365

Find $f$ for the integral $\int x \sqrt{x+4} dx$ using the substitution $u=x+4$ , so $\int f(u) du$ . $f(u)=$

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Problem 1366

Evaluate the integral using substitution: $\int \frac{x^{6}}{\sqrt{x^{7}-4}} d x = C$ .

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Problem 1367

Evaluate the integral using substitution: $\int 7 x^{3} \cos(4 x^{4}) \, dx = C$ .

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Problem 1368

Find $\frac{\partial f}{\partial x}$ , $\frac{\partial f}{\partial y}$ , and evaluate at (1,-1) for $f(x, y)=6 x e^{5 x y}$ .

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Problem 1369

Evaluate the integral using substitution: $\int -x(5-8x)^{5} dx =$

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Problem 1370

Evaluate the integral: $\int \frac{x^{6}}{(x^{7}+5)^{7}} \, dx =$

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Problem 1371

Evaluate the integral from 0 to 1: $\int_{0}^{1} 6 x^{4}\left(1-x^{5}\right)^{3} d x=$

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Problem 1372

Evaluate the integral: $\int -3 \sin(x) \sqrt{\cos(x)} \, dx =$

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Problem 1373

Evaluate the integral using substitution: $\int \cos (x)(13-\cos (x))^{8} \sin (x) d x =$

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Problem 1374

Find the second partial derivatives $f_{xx}(x, y), f_{yy}(x, y), f_{xy}(x, y), f_{yx}(x, y)$ for $f(x, y)=7x^2y$ and evaluate at $(1,-1)$ .

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Problem 1375

Evaluate the integral: $\int_{-\pi / 2}^{\pi / 2} \sin ^{6}(x) \cos (x) d x.$

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Problem 1376

Evaluate the integral using substitution: $\int 7 \sin ^{2}(4 x) \cos ^{3}(4 x) d x.$

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Problem 1377

Evaluate the integral using substitution: $\int 5 t \sin(t^{2}) \cos(t^{2}) \, dt =$

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Problem 1378

Find the partial derivatives $\frac{\partial f}{\partial x}$ , $\frac{\partial f}{\partial y}$ , and evaluate at $(1,-1)$ for $f(x, y)=13,000-30 x+20 y+8 x y$ .

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Problem 1379

Find the general antiderivative of $\frac{d y}{d x}=7 e^{x}+4$ . Antiderivative $=$ $+C$ .

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Problem 1380

Find the antiderivative for $\frac{d x}{d t}=2 e^{t}-3$ with the condition $x(0)=4$ . What is $x$ ?

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Problem 1381

Find the antiderivatives of $\frac{d x}{d t}=2 t^{-1}+5$ . What is $x$ ? Include the constant $C$ .

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Problem 1382

Find the function $f(x)$ given $f^{\prime \prime \prime}(x)=e^{x}$ , $f^{\prime \prime}(0)=6$ , $f^{\prime}(0)=10$ .

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Problem 1383

Find an antiderivative of $f(x)=\frac{2}{x}-10 e^{x}$ .

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Problem 1384

Find the function $f(x)$ where $f'(x)=9^{x}$ and $f(4)=-5$ . What is $f(x)$ ?

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Problem 1385

Evaluate the integral $\int_{-2}^{8} f(x) dx$ where $f(x)=x$ for $x<1$ and $f(x)=\frac{1}{x}$ for $x \geq 1$ .

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Problem 1386

Calculate the area between $f(x)=3^{-x}$ and $f(x)=0$ from $x=2$ to $x=7$ using integration. Provide the exact answer.

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Problem 1387

Calculate the area between $f(x)=3^{-x}$ and $f(x)=0$ for $x$ in $[2,7]$ using integration. Provide the exact value.

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Problem 1388

Evaluate the integral of $y=7xe^{-x}$ from $x=2$ to $x$ .

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Problem 1389

Calculate the volume $V$ of the solid formed by rotating the area under $y=\frac{4}{x^{2}}$ from $x=1$ to $x=\infty$ around the $\mathrm{x}$ -axis. $V=$

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Problem 1390

Find the volume $V$ of the solid formed by revolving the area under $y=10 e^{-x}$ in the first quadrant around the $y$ -axis.

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Problem 1391

Calculate the area under the curve $y=7 x e^{-x}$ for $x$ values starting from 2.

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Problem 1392

Evaluate the integral from 4 to 3 of $\frac{\cos (\arctan (5 t))}{1+(5 t)^{2}} dt$ .

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Problem 1393

Find $f(x)$ given that $f'(x)=\frac{4}{\sqrt{1-x^{2}}}$ and $f\left(\frac{1}{2}\right)=8$ .

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Problem 1394

Evaluate the integral for $x>0$ : $\int \frac{1}{x \sqrt{64 x^{2}-1}} d x=$

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Problem 1395

Evaluate the integral for $x>0$ : $\int \frac{1}{x \sqrt{64 x^{2}-1}} d x = C$ .

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Problem 1396

Tad drops a cherry pit from $0.8 \mathrm{~m}$ at $18 \mathrm{~m/s}$ . How far does it land from the drop point?

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Problem 1397

Choose $u$ for integration by parts in $\int x^{4} \tan^{-1}(x) \, dx$ . Recall: $\int u \, dv = uv - \int v \, du$ .

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Problem 1398

Use the Beverton-Holt model $R(N_{t})=\frac{R_{0}}{1+a N_{t}}$ with $a=0.03$ , $R_{0}=3$ . Find $N_{t}$ for $t=1,2,\ldots,5$ and $\lim_{t \to \infty} N_{t}$ for $N_{0}=2$ .

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Problem 1399

Calculate the integral $\int \ln \left(x^{2}\right) d x$ . Use "C" for the constant in your solution.

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Problem 1400

Calculate the indefinite integral $\int \ln \left(x^{2}\right) d x$ . Use "C" for the constant.

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Calculus

Problem 1301

Find the limits: 1. lim⁡x→33−xx2−9\lim _{x \rightarrow 3} \frac{3-x}{x^{2}-9}limx→3​x2−93−x​ and 2. lim⁡x→4x2−5x+4x2−2x−8\lim _{x \rightarrow 4} \frac{x^{2}-5 x+4}{x^{2}-2 x-8}limx→4​x2−2x−8x2−5x+4​.

Problem 1302

Find the limit as x approaches 1: lim⁡x→12x3+x+1−2xx−1\lim_{{x \to 1}} \frac{\sqrt{2 x^{3}+x+1}-2 x}{x-1}limx→1​x−12x3+x+1​−2x​

Problem 1303

Problem 1304

Find the limit as hhh approaches 0 for the difference quotient of: 1. f(x)=x2−xf(x)=x^{2}-xf(x)=x2−x, 2. f(x)=1x+3f(x)=\frac{1}{x+3}f(x)=x+31​.

Problem 1305

מה הגבול של (x3+x23−x)\left(\sqrt[3]{x^{3}+x^{2}}-x\right)(3x3+x2​−x) כש-xxx מתקרב למינוס אינסוף?

Problem 1306

Evaluate the piecewise function fff at x=3x=3x=3 to check if it's continuous and differentiable. What is true about fff at x=3x=3x=3?

Problem 1307

Problem 1308

Given g(x)g(x)g(x) is differentiable at x=−3x=-3x=−3 with g(−3)=9g(-3)=9g(−3)=9 and g′(−3)=0g'(-3)=0g′(−3)=0, which statements are true?

Problem 1309

Given the piecewise function fff, determine if it is continuous or differentiable at x=−3x=-3x=−3. Options: 1) neither, 2) continuous only, 3) both.

Problem 1310

Is the function f(x)={sin⁡(x)x<π22xπx≥π2f(x)=\left\{\begin{array}{cc}\sin (x) & x<\frac{\pi}{2} \\ \frac{2 x}{\pi} & x \geq \frac{\pi}{2}\end{array}\right.f(x)={sin(x)π2x​​x<2π​x≥2π​​ continuous at x=π2x=\frac{\pi}{2}x=2π​? Explain.

Problem 1311

Find the acceleration due to gravity at 6989 m, the horizontal asymptote of g(h)g(h)g(h), and solve g(h)=0g(h)=0g(h)=0.

Problem 1312

When can average speed equal instantaneous speed?

Problem 1313

True or false: The integral from -2 to 2 of 1x2\frac{1}{x^{2}}x21​ equals −1-1−1.

Problem 1314

True or false: For any continuous function fff, is it true that ∫33f(x)dx=f(3)\int_{3}^{3} f(x) dx = f(3)∫33​f(x)dx=f(3)?

Problem 1315

Find the function y(x)=7+∫2xsin⁡(t2)dty(x)=7+\int_{2}^{x} \sin(t^{2}) dty(x)=7+∫2x​sin(t2)dt given dydx=sin⁡(x2)\frac{d y}{d x}=\sin(x^{2})dxdy​=sin(x2) and y(2)=7y(2)=7y(2)=7.

Problem 1316

Determine if the integral ∫011xdx\int_{0}^{1} \frac{1}{\sqrt{x}} d x∫01​x​1​dx converges or diverges.

Problem 1317

Determine if the integral ∫011xdx\int_{0}^{1} \frac{1}{\sqrt{x}} d x∫01​x​1​dx converges or diverges.

Problem 1318

True or false: A continuous function has an antiderivative.

Problem 1319

True or false: ∫11+x2dx=ln⁡(∣1+x2∣)+C\int \frac{1}{1+x^{2}} d x=\ln \left(\left|1+x^{2}\right|\right)+C∫1+x21​dx=ln(∣∣​1+x2∣∣​)+C

Problem 1320

True or false: The integral of sin⁡(x)\sin(x)sin(x) is cos⁡(x)+C\cos(x) + Ccos(x)+C.

Problem 1321

True or false: If fff is continuous on [a,b][a, b][a,b], then [∫abf(x)dx]2=∫ab[f(x)]2dx\left[\int_{a}^{b} f(x) dx\right]^{2}=\int_{a}^{b}[f(x)]^{2} dx[∫ab​f(x)dx]2=∫ab​[f(x)]2dx?

Problem 1322

True or false: For any continuous function fff, is it true that ddx∫x0f(t)dt=−f(x)\frac{d}{d x} \int_{x}^{0} f(t) d t=-f(x)dxd​∫x0​f(t)dt=−f(x)?

Problem 1323

True or false: For any continuous function fff, does ∫abf(x)dx+∫baf(x)dx=0\int_{a}^{b} f(x) dx + \int_{b}^{a} f(x) dx = 0∫ab​f(x)dx+∫ba​f(x)dx=0 when a<ba < ba<b?

Problem 1324

True or false: If 0≤f(x)≤g(x)0 \leq f(x) \leq g(x)0≤f(x)≤g(x) and ∫3∞g(x)dx\int_{3}^{\infty} g(x) dx∫3∞​g(x)dx diverges, does ∫3∞f(x)dx\int_{3}^{\infty} f(x) dx∫3∞​f(x)dx also diverge?

Problem 1325

True or false: For differentiable functions fff and ggg, is it true that ∫f(x)g′(x)dx=f(x)g(x)−∫f′(x)g(x)dx\int f(x) g^{\prime}(x) dx = f(x) g(x) - \int f^{\prime}(x) g(x) dx∫f(x)g′(x)dx=f(x)g(x)−∫f′(x)g(x)dx?

Problem 1326

Find the volume of a solid with base R\mathrm{R}R (bounded by y=x(1−x)1/2y=x(1-x)^{1/2}y=x(1−x)1/2 and the xxx-axis) and square cross-sections.

Problem 1327

Find the value of the integral ∫14f(x)xdx\int_{1}^{4} \frac{f(\sqrt{x})}{\sqrt{x}} d x∫14​x​f(x​)​dx given that ∫12f(x)dx=6\int_{1}^{2} f(x) d x=6∫12​f(x)dx=6.

Problem 1328

Given fff is continuous with f(x)≥0f(x) \geq 0f(x)≥0 for a≤x≤ba \leq x \leq ba≤x≤b, and the area from aaa to bbb is 8. If F(a)=6F(a)=6F(a)=6, find F(b)F(b)F(b).

Problem 1329

A moped slows down from 31ft/s31 \mathrm{ft} / \mathrm{s}31ft/s at 3ft/s23 \mathrm{ft} / \mathrm{s}^{2}3ft/s2. How far does it travel before stopping?

Problem 1330

Find F′(3)F^{\prime}(3)F′(3) where F(x)=∫xx2f(t)dtF(x)=\int_{x}^{x^{2}} f(t) dtF(x)=∫xx2​f(t)dt, given f(3)=3f(3)=3f(3)=3 and f(9)=7f(9)=7f(9)=7.

Problem 1331

Find the value of ∫25[2f(x)−3g(x)+9]dx\int_{2}^{5}[2 f(x)-3 g(x)+9] d x∫25​[2f(x)−3g(x)+9]dx given that ∫25f(x)dx=8\int_{2}^{5} f(x) d x=8∫25​f(x)dx=8 and ∫25g(x)dx=4\int_{2}^{5} g(x) d x=4∫25​g(x)dx=4.

Problem 1332

Find the exact value of y(2)y(2)y(2) given the differential equation dydx=3x2+2\frac{d y}{d x}=3 x^{2}+2dxdy​=3x2+2 and y(1)=6y(1)=6y(1)=6.

Problem 1333

Evaluate the integral ∫14xf′(x)dx\int_{1}^{4} x f^{\prime}(x) d x∫14​xf′(x)dx given that the area under y=f(x)y=f(x)y=f(x) from x=1x=1x=1 to x=4x=4x=4 is 9.

Problem 1334

A moped slows down from 31ft/s31 \mathrm{ft} / \mathrm{s}31ft/s at 3ft/s23 \mathrm{ft} / \mathrm{s}^{2}3ft/s2. How far does it travel before stopping? Provide a numerical answer.

Problem 1335

Estimate the area under f(x)f(x)f(x) from x=0x=0x=0 to x=8x=8x=8 using 4 rectangles and left endpoints. Calculate L4L_{4}L4​.

Problem 1336

Find the average rate of change of f(x)=x2+1f(x)=x^{2}+1f(x)=x2+1 from x=−10x=-10x=−10 to x=10x=10x=10. A. 0 B. 0.05 C. 5.05 D. 101

Problem 1337

Given the Beverton-Holt model R(Nt)=R01+aNtR(N_t)=\frac{R_0}{1+a N_t}R(Nt​)=1+aNt​R0​​ with a=0.05a=0.05a=0.05, R0=5R_0=5R0​=5, find NtN_tNt​ for t=1,2,…,5t=1,2,\ldots,5t=1,2,…,5 and lim⁡t→∞Nt\lim_{t \to \infty} N_tlimt→∞​Nt​ for N0=2N_0=2N0​=2.

Problem 1338

Use the Beverton-Holt model R(Nt)=R01+aNtR(N_{t})=\frac{R_{0}}{1+a N_{t}}R(Nt​)=1+aNt​R0​​ with a=0.05a=0.05a=0.05, R0=5R_{0}=5R0​=5. Find NtN_{t}Nt​ for t=1,2,…,5t=1,2,\ldots,5t=1,2,…,5 and lim⁡t→∞Nt\lim_{t \to \infty} N_{t}limt→∞​Nt​ for N0=2N_{0}=2N0​=2.

Problem 1339

Solve the equation dydx=−4(1+y2)x7\frac{d y}{d x}=-4(1+y^{2}) x^{7}dxdy​=−4(1+y2)x7 using separation of variables. Let CCC be the constant. y=y=y=

Problem 1340

Solve the equation (10+x4)dydx=x3y(10+x^{4}) \frac{dy}{dx}=\frac{x^{3}}{y}(10+x4)dxdy​=yx3​ using separation of variables. Find y2=y^{2}=y2=.

Problem 1341

Find the limits: 1. $\lim _{x \rightarrow 3} \frac{3-x}{x^{2}-9}$ and 2. $\lim _{x \rightarrow 4} \frac{x^{2}-5 x+4}{x^{2}-2 x-8}$ .

Find the limit as x approaches 1: $\lim_{{x \to 1}} \frac{\sqrt{2 x^{3}+x+1}-2 x}{x-1}$

Find the limit as $h$ approaches 0 for the difference quotient of: 1. $f(x)=x^{2}-x$ , 2. $f(x)=\frac{1}{x+3}$ .

מה הגבול של $\left(\sqrt[3]{x^{3}+x^{2}}-x\right)$ כש- $x$ מתקרב למינוס אינסוף?

Evaluate the piecewise function $f$ at $x=3$ to check if it's continuous and differentiable. What is true about $f$ at $x=3$ ?

Given $g(x)$ is differentiable at $x=-3$ with $g(-3)=9$ and $g'(-3)=0$ , which statements are true?

Given the piecewise function $f$ , determine if it is continuous or differentiable at $x=-3$ . Options: 1) neither, 2) continuous only, 3) both.

Is the function $f(x)=\left\{\begin{array}{cc}\sin (x) & x<\frac{\pi}{2} \\ \frac{2 x}{\pi} & x \geq \frac{\pi}{2}\end{array}\right.$ continuous at $x=\frac{\pi}{2}$ ? Explain.

Find the acceleration due to gravity at 6989 m, the horizontal asymptote of $g(h)$ , and solve $g(h)=0$ .

True or false: The integral from -2 to 2 of $\frac{1}{x^{2}}$ equals $-1$ .

True or false: For any continuous function $f$ , is it true that $\int_{3}^{3} f(x) dx = f(3)$ ?

Find the function $y(x)=7+\int_{2}^{x} \sin(t^{2}) dt$ given $\frac{d y}{d x}=\sin(x^{2})$ and $y(2)=7$ .

Determine if the integral $\int_{0}^{1} \frac{1}{\sqrt{x}} d x$ converges or diverges.

Determine if the integral $\int_{0}^{1} \frac{1}{\sqrt{x}} d x$ converges or diverges.

True or false: $\int \frac{1}{1+x^{2}} d x=\ln \left(\left|1+x^{2}\right|\right)+C$

True or false: The integral of $\sin(x)$ is $\cos(x) + C$ .

True or false: If $f$ is continuous on $[a, b]$ , then $\left[\int_{a}^{b} f(x) dx\right]^{2}=\int_{a}^{b}[f(x)]^{2} dx$ ?

True or false: For any continuous function $f$ , is it true that $\frac{d}{d x} \int_{x}^{0} f(t) d t=-f(x)$ ?

True or false: For any continuous function $f$ , does $\int_{a}^{b} f(x) dx + \int_{b}^{a} f(x) dx = 0$ when $a < b$ ?

True or false: If $0 \leq f(x) \leq g(x)$ and $\int_{3}^{\infty} g(x) dx$ diverges, does $\int_{3}^{\infty} f(x) dx$ also diverge?

True or false: For differentiable functions $f$ and $g$ , is it true that $\int f(x) g^{\prime}(x) dx = f(x) g(x) - \int f^{\prime}(x) g(x) dx$ ?

Find the volume of a solid with base $\mathrm{R}$ (bounded by $y=x(1-x)^{1/2}$ and the $x$ -axis) and square cross-sections.

Find the value of the integral $\int_{1}^{4} \frac{f(\sqrt{x})}{\sqrt{x}} d x$ given that $\int_{1}^{2} f(x) d x=6$ .

Given $f$ is continuous with $f(x) \geq 0$ for $a \leq x \leq b$ , and the area from $a$ to $b$ is 8. If $F(a)=6$ , find $F(b)$ .

A moped slows down from $31 \mathrm{ft} / \mathrm{s}$ at $3 \mathrm{ft} / \mathrm{s}^{2}$ . How far does it travel before stopping?

Find $F^{\prime}(3)$ where $F(x)=\int_{x}^{x^{2}} f(t) dt$ , given $f(3)=3$ and $f(9)=7$ .

Find the value of $\int_{2}^{5}[2 f(x)-3 g(x)+9] d x$ given that $\int_{2}^{5} f(x) d x=8$ and $\int_{2}^{5} g(x) d x=4$ .

Find the exact value of $y(2)$ given the differential equation $\frac{d y}{d x}=3 x^{2}+2$ and $y(1)=6$ .

Evaluate the integral $\int_{1}^{4} x f^{\prime}(x) d x$ given that the area under $y=f(x)$ from $x=1$ to $x=4$ is 9.

A moped slows down from $31 \mathrm{ft} / \mathrm{s}$ at $3 \mathrm{ft} / \mathrm{s}^{2}$ . How far does it travel before stopping? Provide a numerical answer.

Estimate the area under $f(x)$ from $x=0$ to $x=8$ using 4 rectangles and left endpoints. Calculate $L_{4}$ .

Find the average rate of change of $f(x)=x^{2}+1$ from $x=-10$ to $x=10$ . A. 0 B. 0.05 C. 5.05 D. 101

Given the Beverton-Holt model $R(N_t)=\frac{R_0}{1+a N_t}$ with $a=0.05$ , $R_0=5$ , find $N_t$ for $t=1,2,\ldots,5$ and $\lim_{t \to \infty} N_t$ for $N_0=2$ .

Use the Beverton-Holt model $R(N_{t})=\frac{R_{0}}{1+a N_{t}}$ with $a=0.05$ , $R_{0}=5$ . Find $N_{t}$ for $t=1,2,\ldots,5$ and $\lim_{t \to \infty} N_{t}$ for $N_{0}=2$ .

Solve the equation $\frac{d y}{d x}=-4(1+y^{2}) x^{7}$ using separation of variables. Let $C$ be the constant. $y=$

Solve the equation $(10+x^{4}) \frac{dy}{dx}=\frac{x^{3}}{y}$ using separation of variables. Find $y^{2}=$ .

Solve the equation $e^{-y} \sin (x)+\frac{d y}{d x}=0$ using separation of variables. Find $y=C$ .

Solve the initial value problem using separation of variables: $\frac{d y}{d x}-x^{2} e^{y}=9 e^{y}$ , $y(0)=3$ . Find $y=$

Solve the equation $\frac{d y}{d t}=5 y$ with the initial condition $y(2)=2$ . Find $y=$

Solve the equation $\frac{d u}{d t}=e^{3 u+4 t}$ with $u(0)=13$ . Find $u(t)$ .

Solve the equation $\frac{d y}{d x}=\frac{-0.4}{\cos (y)}$ with initial condition $y(0)=\frac{\pi}{3}$ . Find $y(x)$ .

Solve the differential equation $y' = x^4 \tan(y)$ for the general solution, using the constant $\mathrm{C}$ .

Find the derivative of $e^{x} \cos x$ .

Find the horizontal asymptote of the drug concentration function $C(t)=\frac{t}{7t^{2}+8}$ . What is $C=\square$ ?

Find the horizontal asymptote of $C(t)=\frac{t}{7 t^{2}+8}$ , identify the graph, and determine when concentration is highest.

Find the displacement of a mass on a spring from $t=0$ to $t=\pi$ given $v(t)=6 \sin(t)-6 \cos(t)$ . Evaluate $\int_{0}^{\pi}(6 \sin(t)-6 \cos(t)) dt$ .

Calculate the work done by the force $F(x)=x^{-1}+4x$ from $x=3$ to $x=5$ : $W=\int_{3}^{5}(x^{-1}+4x)dx$

Find the distance a wave travels from $t=1$ to $t=4$ given $v=\sqrt{\frac{8}{x}}$ . Evaluate the integral $\int_{1}^{4} \sqrt{\frac{8}{x}} d x$ .

A car's velocity is given by $v(t)=2 t^{1/2}+5$ . Find the displacement (in m) from $t=2$ to $t=8$ .

Find the displacement and total distance traveled by a particle with velocity $v(t)=4-2t$ from $t=0$ to $t=6$ .

A stone is thrown from a $120 \mathrm{ft}$ cliff at $148 \mathrm{ft/s}$ . Find: (a) time to highest point, (b) max height, (c) time to beach, (d) impact velocity.

Find the derivative of $f(x)=4 \cos 2 x+\log x+x$ .

Find the position of a particle at time $t=7$ given $a(t)=18t+2$ , $s(0)=16$ , and $v(0)=7$ .

A particle has acceleration $a(t)=t-5/2$ m/s² for $0 \leq t \leq 9$ . Find $v(t)$ , $d(t)$ , and total distance traveled.

Find the water flow in liters from a tank in the first 16 minutes, given $r(t)=200-4t$ , for $0 \leq t \leq 50$ .

Calculate the total oil leaked in the first 5 hours after the tanker breaks apart using $R(t)=\frac{0.9}{1+t^{2}}$ .

Find the displacement and total distance traveled by a particle with velocity $v(t)=4-2t$ from $t=0$ to $t=6$ .

Find $f$ for the integral $\int \frac{x}{\sqrt{3 x^{2}+2}} d x$ using the substitution $u=3 x^{2}+2$ .

Find $f$ for the integral $\int x \sqrt{x+4} dx$ using the substitution $u=x+4$ , so $\int f(u) du$ . $f(u)=$

Evaluate the integral using substitution: $\int \frac{x^{6}}{\sqrt{x^{7}-4}} d x = C$ .

Evaluate the integral using substitution: $\int 7 x^{3} \cos(4 x^{4}) \, dx = C$ .

Find $\frac{\partial f}{\partial x}$ , $\frac{\partial f}{\partial y}$ , and evaluate at (1,-1) for $f(x, y)=6 x e^{5 x y}$ .

Evaluate the integral using substitution: $\int -x(5-8x)^{5} dx =$

Evaluate the integral: $\int \frac{x^{6}}{(x^{7}+5)^{7}} \, dx =$

Evaluate the integral from 0 to 1: $\int_{0}^{1} 6 x^{4}\left(1-x^{5}\right)^{3} d x=$

Evaluate the integral: $\int -3 \sin(x) \sqrt{\cos(x)} \, dx =$

Evaluate the integral using substitution: $\int \cos (x)(13-\cos (x))^{8} \sin (x) d x =$

Find the second partial derivatives $f_{xx}(x, y), f_{yy}(x, y), f_{xy}(x, y), f_{yx}(x, y)$ for $f(x, y)=7x^2y$ and evaluate at $(1,-1)$ .