Calculus

Problem 18201

Evaluate the integral: $\int(7 x^{2}+3 x-5) \, dx = \square + C$

See Solution

Problem 18202

Check if the series $\sum_{n=2}^{\infty} \frac{1}{\sqrt[3]{n-1}}$ and $\sum_{n=1}^{\infty} \frac{1}{n 13}$ converge or diverge.

See Solution

Problem 18203

Evaluate the integral: $\int \frac{d x}{12 x^{3}}=\square+C$

See Solution

Problem 18204

Find an antiderivative $G$ for $g(z)=\frac{1}{z^{6}}$ . What is $G(z)=?$

See Solution

Problem 18205

Evaluate the integral: $\int\left(\frac{5}{\sqrt[4]{x^{3}}}-e+3 x^{2}\right) d x$

See Solution

Problem 18206

Find an antiderivative $P$ of $p(t)=\frac{1}{\sqrt[5]{t^{4}}}$ .

See Solution

Problem 18207

Calculate the integral: $\int (2 t \sqrt{t} + \frac{1}{t^{2} \sqrt{t}}) \, dt$

See Solution

Problem 18208

Find the formula for the $n$ th term of partial sums $S_{n}$ for the series $\sum_{k=1}^{\infty}(\sqrt{k+10}-\sqrt{k+9})$ and evaluate $\lim_{n \to \infty} S_{n}$ .

See Solution

Problem 18209

Find the formula for the $n$ th term of the series $\sum_{k=1}^{\infty}(\sqrt{k+5}-\sqrt{k+4})$ .

See Solution

Problem 18210

Find the $n$ th term of the partial sums $S_n$ for the series $\sum_{k=1}^{\infty}(\sqrt{k+5}-\sqrt{k+4})$ and evaluate $\lim_{n \rightarrow \infty} S_n$ .

See Solution

Problem 18211

A piston moves into a cylinder (radius $4 \mathrm{~cm}$ ) at $3 \mathrm{~cm/s}$ . Find the volume change rate when the piston is $4 \mathrm{~cm}$ from the base.

See Solution

Problem 18212

Find dy/dx for $y=\tan^{-1} \frac{2x}{3}$ .

See Solution

Problem 18213

Evaluate the series $\sum_{k=1}^{\infty}\left(\frac{6}{5^{k}}-\frac{6}{5^{k+1}}\right)$ using telescoping and geometric series methods.

See Solution

Problem 18214

Find the volume of the solid formed by rotating the area in the first quadrant bounded by $y=x^{6}$ , $y=1$ , and the $y$ -axis around the $x$ -axis. Volume =

See Solution

Problem 18215

Find the derivative of $y=\tan^{-1} \frac{2x}{3}$ .

See Solution

Problem 18216

Find the function whose derivative is $f^{\prime}(x)=e^{3 x}$ and passes through the point $P=(0, \frac{4}{3})$ .

See Solution

Problem 18217

Find the antiderivative $x(t)$ such that $\frac{d x}{d t}=5 e^{t}-6$ and $x(0)=7$ .

See Solution

Problem 18218

Find the fluid force on a submerged plate with edges at $y=x^{\frac{1}{3}}$ and $y=-x^{\frac{1}{3}}$ in water at $\sqrt[3]{6} \mathrm{ft}$ depth. Density: $62.5 \mathrm{lb} / \mathrm{ft}^{3}$ . $F=\square \mathrm{lb}$

See Solution

Problem 18219

Find the rate of change of unit revenue $R$ when unit cost $C$ changes by \ $15/unit and$ R = \ $2000$ .

See Solution

Problem 18220

Find the derivative of $y$ with respect to $x$ for $y=2 \sin^{-1}(4 x^{4})$ .

See Solution

Problem 18221

Find the rate of change of the area (dA/dt) for a $7 \mathrm{~cm} \times 4 \mathrm{~cm}$ image with length change $0.5 \mathrm{~cm/s}$ and width change $0.3 \mathrm{~cm/s}$ .

See Solution

Problem 18222

Calculate the total force on a dam with width $w=3540 \mathrm{~m}$ , height $h=85 \mathrm{~m}$ , base width $b=2910 \mathrm{~m}$ , $\rho=1000 \mathrm{~kg/m^3}$ , and $g=9.8 \mathrm{~m/s^2}$ . Use a definite integral.

See Solution

Problem 18223

Find the derivative of $y$ where $y=\tan^{-1}(\ln(3x))$ .

See Solution

Problem 18224

Evaluate the series: $\sum_{k=2}^{\infty}\left(\frac{1}{3}\right)^{3 k}$ . Does it converge or diverge?

See Solution

Problem 18225

Evaluate the series $\sum_{k=0}^{\infty}\left[2\left(\frac{5}{7}\right)^{k}-3\left(\frac{5}{8}\right)^{k}\right]$ or state if it diverges.

See Solution

Problem 18226

Evaluate the series $\sum_{k=0}^{\infty}\left[6\left(\frac{1}{7}\right)^{k}-3\left(\frac{5}{8}\right)^{k}\right]$ or state if it diverges.

See Solution

Problem 18227

Evaluate the integral: $\int_{2}^{3}(5 x+9) d x$ using the Fundamental Theorem of Calculus.

See Solution

Problem 18228

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{3}\left(\frac{x^{3}}{5}+4 x\right) d x$

See Solution

Problem 18229

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{1}(2 y^{2}+y^{4}) d y$ .

See Solution

Problem 18230

Find the derivative of $y$ where $y=\sec^{-1}\left(\frac{6x+7}{7}\right)$ .

See Solution

Problem 18231

Evaluate the integral from 3 to 5 of $\frac{5 x^{2}+4}{x^{2}}$ dx.

See Solution

Problem 18232

Evaluate the integral from 0 to 1 of $x^{3/4}$ with respect to $x$ .

See Solution

Problem 18233

Find the general antiderivative of the function: $\int\left(\frac{\sqrt{y}}{3}+\frac{6}{\sqrt{y}}\right) d y$

See Solution

Problem 18234

Find the derivative of $y$ where $y=\sec^{-1}\left(\frac{6x+7}{7}\right)$ .

See Solution

Problem 18235

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{8} t^{2} dt$ .

See Solution

Problem 18236

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{-3}^{-2} \frac{5}{r^{3}} d r$ .

See Solution

Problem 18237

Evaluate the limit using l'Hôpital's Rule: $\lim _{x \rightarrow 0} \frac{\sin 2 x}{6 x}$ .

See Solution

Problem 18238

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{1}^{3} e^{-0.6 t} dt$ .

See Solution

Problem 18239

Evaluate the area between the $x$ -axis and $f(x)=15 x^{2}+3$ from $x=-3$ to $x=10$ using the Fundamental Theorem of Calculus.

See Solution

Problem 18240

Find the general antiderivative of the function: $\int\left(\frac{1}{x^{4}}-x^{4}-\frac{1}{5}\right) d x$

See Solution

Problem 18241

Find the derivative of the inverse function at 0: $\left(f^{-1}\right)^{\prime}(0)$ where $f(x)=\int_{t}^{x} \frac{d t}{\sqrt{4+t^{4}}}$ .

See Solution

Problem 18242

Evaluate the series: $\sum_{k=2}^{\infty}\left(\frac{2}{5}\right)^{3 k}$ . Does it converge or diverge?

See Solution

Problem 18243

The company's net worth growth rate is $f'(t)=2850-11t^2$ . How does it change from 1990 to 2000? Choose A, B, C, or D. If worth \$55000 in 1990, what is it in 2000?

See Solution

Problem 18244

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{2}^{4} \frac{5}{x} d x$ .

See Solution

Problem 18245

Find the integral $\int_{a}^{b} 28 e^{0.08 t} \, dt$ for oil used from 1990 to 1998. What are $a$ and $b$ ?

See Solution

Problem 18246

Find the derivative of the inverse function at 0: $\left(f^{-1}\right)^{\prime}(0)$ where $f(x)=\int_{4}^{x} \frac{d t}{\sqrt{4+t^{4}}}$ .

See Solution

Problem 18247

Find the derivative of the integral: $\frac{d}{d x} \int_{0}^{x^{2}} \frac{t d t}{t+5}$ .

See Solution

Problem 18248

Find the derivative with respect to $\theta$ of the integral $\int_{1}^{\theta} 3 \cot (u) \, du$ .

See Solution

Problem 18249

Find the derivative of the integral $\frac{d}{d s} \int_{-7}^{s} \tan \left(\frac{1}{u^{2}+5}\right) d u$ .

See Solution

Problem 18250

Find the total cost for the first 49 units given the marginal cost function $m c(x)=\frac{13}{\sqrt{x}}$ .

See Solution

Problem 18251

Find the derivative: $\frac{d}{d x} \int_{1}^{1 / x} \cos ^{5}(t) d t$

See Solution

Problem 18252

Find the tangent line equation for $y=-4 \sin x$ at $x=\frac{\pi}{4}$ .

See Solution

Problem 18253

Find $\left(f^{-1}\right)^{\prime}(3)$ for $f(x)=\frac{x+5}{x+1}$ where $x>-1$ .

See Solution

Problem 18254

Find $\left(f^{-1}\right)^{\prime}(3)$ for the function $f(x) = \frac{x+5}{x+1}$ , where $x > -1$ .

See Solution

Problem 18255

Find the derivative: $\frac{d}{d x} \int_{\sqrt{x}}^{x^{4}} \tan (3 t) d t$

See Solution

Problem 18256

Compute the integral $\int_{-2}^{2} \frac{x^{2}+2}{x^{3}+3 x^{2}-9 x-27} d x$ and approximate it to 3 decimal places.

See Solution

Problem 18257

Evaluate the limit using I'Hôpital's Rule: $\lim _{x \rightarrow \infty} \frac{5 x+7}{8 x^{2}+5 x-8}$ .

See Solution

Problem 18258

An object's acceleration is $a(t)=7 \sin (t)$ with initial velocity $v(0)=-12 \mathrm{~m/s}$ .
a) Find $v(t)$ . b) Calculate displacement from $t=0$ to $t=3$ . c) Determine total distance traveled from $t=0$ to $t=3$ .

See Solution

Problem 18259

Differentiate the integral $\int_{-u}^{5u} \sqrt{x^{2}+2} \, dx$ with respect to $u$ .

See Solution

Problem 18260

Find an antiderivative of the function $-\frac{20}{x^{6}}$ .

See Solution

Problem 18261

Find $\left(f^{-1}\right)^{\prime}(1)$ for $f(x)=\cos(2x)$ , where $0 \leq x \leq \pi/2$ .

See Solution

Problem 18262

Find $\left(f^{-1}\right)^{\prime}(63)$ for the function $f(x) = x^{3} - 1$ .

See Solution

Problem 18263

Find a tangent line to $y=-\cos x$ with slope -1. Is there more than one solution? Explain.

See Solution

Problem 18264

Evaluate the limit using l'Hôpital's Rule: $\lim _{x \rightarrow 1} \frac{x^{3}-8 x^{2}+7}{x-1}$

See Solution

Problem 18265

Find the number of cars passing an intersection from 6 am to 7 am given $r(t)=400+1000 t-150 t^{2}$ .

See Solution

Problem 18266

Find the limit as $x$ approaches 0 from the right: $\lim _{x \rightarrow 0^{+}} x^{-3 / \ln x}$ .

See Solution

Problem 18267

Evaluate the series $\sum_{k=2}^{\infty}\left(\frac{3}{8}\right)^{4 k}$ or state if it diverges.

See Solution

Problem 18268

Solve the integral $\int_{-5}^{5} \frac{x^{3} \sin ^{2} x}{x^{4}+2 x^{2}+1} d x$ .

See Solution

Problem 18269

Complete the chart for the equation $\frac{1}{(x-5)^{2}}$ as $x \to \infty$ , $x \to -\infty$ , $x \to 5^{-}$ , and $x \to 5^{+}$ .

See Solution

Problem 18270

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(1+\frac{2}{x^{4}}\right)^{x}$ .

See Solution

Problem 18271

Find the general antiderivative of $-6 \cos t$ .

See Solution

Problem 18272

Find the limit as $n$ approaches infinity of the sum for $f(x)=\sqrt[6]{x}$ from $1$ to $18$ using right endpoints.

See Solution

Problem 18273

Evaluate the limit using l'Hôpital's Rule: $\lim _{x \rightarrow 0} \frac{\cos (5 x)-1}{x^{2}}$ .

See Solution

Problem 18274

Find $\int_{2}^{6}[f(x)-4 g(x)] d x$ given $\int_{2}^{6} f(x) d x=-2$ and $\int_{2}^{6} g(x) d x=10$ .

See Solution

Problem 18275

Estimate the area under $f(x)=x^{2}$ from $x=0$ to $x=1$ using the midpoint sum with 2 equal-width rectangles.

See Solution

Problem 18276

An object is thrown upwards at $1.8 \mathrm{~m/s}$ from a $34.5 \mathrm{~m}$ building. Find its impact velocity on the ground.

See Solution

Problem 18277

Find the derivative of $y=\int_{0}^{x} \frac{d t}{2 t+5}$ .

See Solution

Problem 18278

A rocket launched from 0 m reaches 382 m. What is its velocity when it returns to 0 m? Remember, velocity is a vector.

See Solution

Problem 18279

Find the value of $g^{\prime}(3)$ for the differentiable function $g(x)=f^{-1}(x)$ given $f(3)=15, f(6)=3, f^{\prime}(3)=-8, f^{\prime}(6)=-2$ .

See Solution

Problem 18280

Estimate the area under $f(x)=25-x^{2}$ from $x=-5$ to $x=5$ using the midpoint sum with 2 equal-width rectangles.

See Solution

Problem 18281

A sphere's radius decreases at 2 cm/s. When $r=3$ cm, find the rate of change of surface area $S=4\pi r^2$ .

See Solution

Problem 18282

Find the acceleration of a particle with velocity $v(t)=7-(1.01)^{-t^{2}}$ at time $t=3$ .

See Solution

Problem 18283

Find the average rate of change for $g(x)=\frac{1}{2}(4)^{x}$ from $x=1$ to $x=5$ .

See Solution

Problem 18284

Calculate $\left(\int_{0}^{3} e^{-z} d z\right)\left(\int_{0}^{2 \pi} 7 d \theta\right)$ .

See Solution

Problem 18285

A spacecraft moves away from Earth. Given acceleration $a = \frac{4 \times 10^{5}}{x^{2}}$ , find: (a) $v^{2}$ in terms of $x$ , (b) $v$ at $x=10000$ , (c) terminal velocity. Ans: (a) $v^{2}=21+\frac{8 \times 10^{5}}{x}$ , (b) $10 \mathrm{kms}^{-1}$ , (c) $4.58 \mathrm{kms}^{-1}$ .

See Solution

Problem 18286

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

See Solution

Problem 18287

Find $g'(1)$ if $g(x)=f^{-1}(x)$ and $f(x)=2x+e^x$ with $(0,1)$ on $f$ .

See Solution

Problem 18288

Find $\int_{3}^{3} f(x) d x$ and $\int_{4}^{2} f(x) d x$ given that $\int_{2}^{4} f(x) d x=-3$ .

See Solution

Problem 18289

Calculate the integral from 0 to 9 of $4 \sqrt{x}$ with respect to $x$ .

See Solution

Problem 18290

Calculate the area between the curve $y=\frac{3}{x^{3}}$ and the $x$ -axis for $1 \leq x \leq 3$ .

See Solution

Problem 18291

An object is thrown upward at $8.9 \mathrm{~m/s}$ from a $31.8 \mathrm{~m}$ building. Find its impact velocity on the ground.

See Solution

Problem 18292

Estimate the distance a race car travels in 8 seconds using left endpoint values for 8 intervals of length 1. Use velocities: $0, 10, 23, 19, 29, 32, 34, 12, 5$ .

See Solution

Problem 18293

Determine if the sequence $a_{n}=\frac{n^{2} \cos (n)}{6+n^{2}}$ converges or diverges. Enter DIVERGES if it diverges.

See Solution

Problem 18294

Find the average value of $f(x)=\frac{1}{x}$ on $[7,7 e]$ . What is the value of $f=\square$ ?

See Solution

Problem 18295

What is the result of integrating population growth rate from $t=a$ to $t=b$ where $b>a$ ? Choose the correct answer.

See Solution

Problem 18296

Calculate the average value of $f(x)=-\cos x$ from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ . Answer: $\square$ .

See Solution

Problem 18297

Evaluate the expression: $-\left(\frac{\theta}{2}-\frac{1}{4} \sin (2 \theta) \big|_{\theta=0}^{\pi / 2}\right)\left(\frac{\cos ^{3}(\varphi)}{3}-\cos (\varphi) \big|_{\varphi=0}^{\pi / 2}\right)\left(3 \rho^{5} \big|_{\rho=0}^{3}\right)$ .

See Solution

Problem 18298

Given $a_{n}=\frac{2 n}{6 n+1}$ , is $\{a_{n}\}$ convergent or divergent? Also, is $\sum_{n=1}^{\infty} a_{n}$ convergent or divergent?

See Solution

Problem 18299

A pizza pan cools from $450^{\circ} \mathrm{F}$ to $75^{\circ} \mathrm{F}$ . After 5 min it's $300^{\circ} \mathrm{F}$ . Find when it reaches $135^{\circ} \mathrm{F}$ and $240^{\circ} \mathrm{F}$ . What trends do you observe?

See Solution

Problem 18300

Graph the function and find the area to evaluate the integral: $\int_{-2}^{y}|x| d x$ .

See Solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337

Calculus

Problem 18201

Evaluate the integral: ∫(7x2+3x−5) dx=□+C\int(7 x^{2}+3 x-5) \, dx = \square + C∫(7x2+3x−5)dx=□+C

Problem 18202

Check if the series ∑n=2∞1n−13\sum_{n=2}^{\infty} \frac{1}{\sqrt[3]{n-1}}∑n=2∞​3n−1​1​ and ∑n=1∞1n13\sum_{n=1}^{\infty} \frac{1}{n 13}∑n=1∞​n131​ converge or diverge.

Problem 18203

Evaluate the integral: ∫dx12x3=□+C\int \frac{d x}{12 x^{3}}=\square+C∫12x3dx​=□+C

Problem 18204

Find an antiderivative GGG for g(z)=1z6g(z)=\frac{1}{z^{6}}g(z)=z61​. What is G(z)=?G(z)=?G(z)=?

Problem 18205

Evaluate the integral: ∫(5x34−e+3x2)dx\int\left(\frac{5}{\sqrt[4]{x^{3}}}-e+3 x^{2}\right) d x∫(4x3​5​−e+3x2)dx

Problem 18206

Find an antiderivative PPP of p(t)=1t45p(t)=\frac{1}{\sqrt[5]{t^{4}}}p(t)=5t4​1​.

Problem 18207

Calculate the integral: ∫(2tt+1t2t) dt\int (2 t \sqrt{t} + \frac{1}{t^{2} \sqrt{t}}) \, dt∫(2tt​+t2t​1​)dt

Problem 18208

Find the formula for the nnnth term of partial sums SnS_{n}Sn​ for the series ∑k=1∞(k+10−k+9)\sum_{k=1}^{\infty}(\sqrt{k+10}-\sqrt{k+9})∑k=1∞​(k+10​−k+9​) and evaluate lim⁡n→∞Sn\lim_{n \to \infty} S_{n}limn→∞​Sn​.

Problem 18209

Find the formula for the nnnth term of the series ∑k=1∞(k+5−k+4)\sum_{k=1}^{\infty}(\sqrt{k+5}-\sqrt{k+4})∑k=1∞​(k+5​−k+4​).

Problem 18210

Find the nnnth term of the partial sums SnS_nSn​ for the series ∑k=1∞(k+5−k+4)\sum_{k=1}^{\infty}(\sqrt{k+5}-\sqrt{k+4})∑k=1∞​(k+5​−k+4​) and evaluate lim⁡n→∞Sn\lim_{n \rightarrow \infty} S_nlimn→∞​Sn​.

Problem 18211

A piston moves into a cylinder (radius 4 cm4 \mathrm{~cm}4 cm) at 3 cm/s3 \mathrm{~cm/s}3 cm/s. Find the volume change rate when the piston is 4 cm4 \mathrm{~cm}4 cm from the base.

Problem 18212

Find dy/dx for y=tan⁡−12x3y=\tan^{-1} \frac{2x}{3}y=tan−132x​.

Problem 18213

Evaluate the series ∑k=1∞(65k−65k+1)\sum_{k=1}^{\infty}\left(\frac{6}{5^{k}}-\frac{6}{5^{k+1}}\right)∑k=1∞​(5k6​−5k+16​) using telescoping and geometric series methods.

Problem 18214

Find the volume of the solid formed by rotating the area in the first quadrant bounded by y=x6y=x^{6}y=x6, y=1y=1y=1, and the yyy-axis around the xxx-axis. Volume =

Problem 18215

Find the derivative of y=tan⁡−12x3y=\tan^{-1} \frac{2x}{3}y=tan−132x​.

Problem 18216

Find the function whose derivative is f′(x)=e3xf^{\prime}(x)=e^{3 x}f′(x)=e3x and passes through the point P=(0,43)P=(0, \frac{4}{3})P=(0,34​).

Problem 18217

Find the antiderivative x(t)x(t)x(t) such that dxdt=5et−6\frac{d x}{d t}=5 e^{t}-6dtdx​=5et−6 and x(0)=7x(0)=7x(0)=7.

Problem 18218

Find the fluid force on a submerged plate with edges at y=x13y=x^{\frac{1}{3}}y=x31​ and y=−x13y=-x^{\frac{1}{3}}y=−x31​ in water at 63ft\sqrt[3]{6} \mathrm{ft}36​ft depth. Density: 62.5lb/ft362.5 \mathrm{lb} / \mathrm{ft}^{3}62.5lb/ft3. F=□lb F=\square \mathrm{lb} F=□lb

Problem 18219

Find the rate of change of unit revenue RRR when unit cost CCC changes by \15/unitand15/unit and 15/unitandR = \200020002000.

Problem 18220

Find the derivative of yyy with respect to xxx for y=2sin⁡−1(4x4)y=2 \sin^{-1}(4 x^{4})y=2sin−1(4x4).

Problem 18221

Find the rate of change of the area (dA/dt) for a 7 cm×4 cm7 \mathrm{~cm} \times 4 \mathrm{~cm}7 cm×4 cm image with length change 0.5 cm/s0.5 \mathrm{~cm/s}0.5 cm/s and width change 0.3 cm/s0.3 \mathrm{~cm/s}0.3 cm/s.

Problem 18222

Calculate the total force on a dam with width w=3540 mw=3540 \mathrm{~m}w=3540 m, height h=85 mh=85 \mathrm{~m}h=85 m, base width b=2910 mb=2910 \mathrm{~m}b=2910 m, ρ=1000 kg/m3\rho=1000 \mathrm{~kg/m^3}ρ=1000 kg/m3, and g=9.8 m/s2g=9.8 \mathrm{~m/s^2}g=9.8 m/s2. Use a definite integral.

Problem 18223

Find the derivative of yyy where y=tan⁡−1(ln⁡(3x))y=\tan^{-1}(\ln(3x))y=tan−1(ln(3x)).

Problem 18224

Evaluate the series: ∑k=2∞(13)3k\sum_{k=2}^{\infty}\left(\frac{1}{3}\right)^{3 k}∑k=2∞​(31​)3k. Does it converge or diverge?

Problem 18225

Evaluate the series ∑k=0∞[2(57)k−3(58)k]\sum_{k=0}^{\infty}\left[2\left(\frac{5}{7}\right)^{k}-3\left(\frac{5}{8}\right)^{k}\right]k=0∑∞​[2(75​)k−3(85​)k] or state if it diverges.

Problem 18226

Evaluate the series ∑k=0∞[6(17)k−3(58)k]\sum_{k=0}^{\infty}\left[6\left(\frac{1}{7}\right)^{k}-3\left(\frac{5}{8}\right)^{k}\right]k=0∑∞​[6(71​)k−3(85​)k] or state if it diverges.

Problem 18227

Evaluate the integral: ∫23(5x+9)dx\int_{2}^{3}(5 x+9) d x∫23​(5x+9)dx using the Fundamental Theorem of Calculus.

Problem 18228

Evaluate the integral using the Fundamental Theorem of Calculus: ∫03(x35+4x)dx\int_{0}^{3}\left(\frac{x^{3}}{5}+4 x\right) d x∫03​(5x3​+4x)dx

Problem 18229

Evaluate the integral using the Fundamental Theorem of Calculus: ∫01(2y2+y4)dy\int_{0}^{1}(2 y^{2}+y^{4}) d y∫01​(2y2+y4)dy.

Problem 18230

Find the derivative of yyy where y=sec⁡−1(6x+77)y=\sec^{-1}\left(\frac{6x+7}{7}\right)y=sec−1(76x+7​).

Problem 18231

Evaluate the integral from 3 to 5 of 5x2+4x2\frac{5 x^{2}+4}{x^{2}}x25x2+4​ dx.

Problem 18232

Evaluate the integral from 0 to 1 of x3/4x^{3/4}x3/4 with respect to xxx.

Problem 18233

Find the general antiderivative of the function: ∫(y3+6y)dy\int\left(\frac{\sqrt{y}}{3}+\frac{6}{\sqrt{y}}\right) d y∫(3y​​+y​6​)dy

Problem 18234

Find the derivative of yyy where y=sec⁡−1(6x+77)y=\sec^{-1}\left(\frac{6x+7}{7}\right)y=sec−1(76x+7​).

Problem 18235

Evaluate the integral using the Fundamental Theorem of Calculus: ∫08t2dt\int_{0}^{8} t^{2} dt∫08​t2dt.

Problem 18236

Evaluate the integral using the Fundamental Theorem of Calculus: ∫−3−25r3dr\int_{-3}^{-2} \frac{5}{r^{3}} d r∫−3−2​r35​dr.

Problem 18237

Evaluate the limit using l'Hôpital's Rule: lim⁡x→0sin⁡2x6x\lim _{x \rightarrow 0} \frac{\sin 2 x}{6 x}x→0lim​6xsin2x​.

Problem 18238

Evaluate the integral using the Fundamental Theorem of Calculus: ∫13e−0.6tdt\int_{1}^{3} e^{-0.6 t} dt∫13​e−0.6tdt.

Problem 18239

Evaluate the area between the xxx-axis and f(x)=15x2+3f(x)=15 x^{2}+3f(x)=15x2+3 from x=−3x=-3x=−3 to x=10x=10x=10 using the Fundamental Theorem of Calculus.

Problem 18240

Evaluate the integral: $\int(7 x^{2}+3 x-5) \, dx = \square + C$

Check if the series $\sum_{n=2}^{\infty} \frac{1}{\sqrt[3]{n-1}}$ and $\sum_{n=1}^{\infty} \frac{1}{n 13}$ converge or diverge.

Evaluate the integral: $\int \frac{d x}{12 x^{3}}=\square+C$

Find an antiderivative $G$ for $g(z)=\frac{1}{z^{6}}$ . What is $G(z)=?$

Evaluate the integral: $\int\left(\frac{5}{\sqrt[4]{x^{3}}}-e+3 x^{2}\right) d x$

Find an antiderivative $P$ of $p(t)=\frac{1}{\sqrt[5]{t^{4}}}$ .

Calculate the integral: $\int (2 t \sqrt{t} + \frac{1}{t^{2} \sqrt{t}}) \, dt$

Find the formula for the $n$ th term of partial sums $S_{n}$ for the series $\sum_{k=1}^{\infty}(\sqrt{k+10}-\sqrt{k+9})$ and evaluate $\lim_{n \to \infty} S_{n}$ .

Find the formula for the $n$ th term of the series $\sum_{k=1}^{\infty}(\sqrt{k+5}-\sqrt{k+4})$ .

Find the $n$ th term of the partial sums $S_n$ for the series $\sum_{k=1}^{\infty}(\sqrt{k+5}-\sqrt{k+4})$ and evaluate $\lim_{n \rightarrow \infty} S_n$ .

A piston moves into a cylinder (radius $4 \mathrm{~cm}$ ) at $3 \mathrm{~cm/s}$ . Find the volume change rate when the piston is $4 \mathrm{~cm}$ from the base.

Find dy/dx for $y=\tan^{-1} \frac{2x}{3}$ .

Evaluate the series $\sum_{k=1}^{\infty}\left(\frac{6}{5^{k}}-\frac{6}{5^{k+1}}\right)$ using telescoping and geometric series methods.

Find the volume of the solid formed by rotating the area in the first quadrant bounded by $y=x^{6}$ , $y=1$ , and the $y$ -axis around the $x$ -axis. Volume =

Find the derivative of $y=\tan^{-1} \frac{2x}{3}$ .

Find the function whose derivative is $f^{\prime}(x)=e^{3 x}$ and passes through the point $P=(0, \frac{4}{3})$ .

Find the antiderivative $x(t)$ such that $\frac{d x}{d t}=5 e^{t}-6$ and $x(0)=7$ .

Find the fluid force on a submerged plate with edges at $y=x^{\frac{1}{3}}$ and $y=-x^{\frac{1}{3}}$ in water at $\sqrt[3]{6} \mathrm{ft}$ depth. Density: $62.5 \mathrm{lb} / \mathrm{ft}^{3}$ . $F=\square \mathrm{lb}$

Find the rate of change of unit revenue $R$ when unit cost $C$ changes by \ $15/unit and$ R = \ $2000$ .

Find the derivative of $y$ with respect to $x$ for $y=2 \sin^{-1}(4 x^{4})$ .

Find the rate of change of the area (dA/dt) for a $7 \mathrm{~cm} \times 4 \mathrm{~cm}$ image with length change $0.5 \mathrm{~cm/s}$ and width change $0.3 \mathrm{~cm/s}$ .

Calculate the total force on a dam with width $w=3540 \mathrm{~m}$ , height $h=85 \mathrm{~m}$ , base width $b=2910 \mathrm{~m}$ , $\rho=1000 \mathrm{~kg/m^3}$ , and $g=9.8 \mathrm{~m/s^2}$ . Use a definite integral.

Find the derivative of $y$ where $y=\tan^{-1}(\ln(3x))$ .

Evaluate the series: $\sum_{k=2}^{\infty}\left(\frac{1}{3}\right)^{3 k}$ . Does it converge or diverge?

Evaluate the series $\sum_{k=0}^{\infty}\left[2\left(\frac{5}{7}\right)^{k}-3\left(\frac{5}{8}\right)^{k}\right]$ or state if it diverges.

Evaluate the series $\sum_{k=0}^{\infty}\left[6\left(\frac{1}{7}\right)^{k}-3\left(\frac{5}{8}\right)^{k}\right]$ or state if it diverges.

Evaluate the integral: $\int_{2}^{3}(5 x+9) d x$ using the Fundamental Theorem of Calculus.

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{3}\left(\frac{x^{3}}{5}+4 x\right) d x$

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{1}(2 y^{2}+y^{4}) d y$ .

Find the derivative of $y$ where $y=\sec^{-1}\left(\frac{6x+7}{7}\right)$ .

Evaluate the integral from 3 to 5 of $\frac{5 x^{2}+4}{x^{2}}$ dx.

Evaluate the integral from 0 to 1 of $x^{3/4}$ with respect to $x$ .

Find the general antiderivative of the function: $\int\left(\frac{\sqrt{y}}{3}+\frac{6}{\sqrt{y}}\right) d y$

Find the derivative of $y$ where $y=\sec^{-1}\left(\frac{6x+7}{7}\right)$ .

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{8} t^{2} dt$ .

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{-3}^{-2} \frac{5}{r^{3}} d r$ .

Evaluate the limit using l'Hôpital's Rule: $\lim _{x \rightarrow 0} \frac{\sin 2 x}{6 x}$ .

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{1}^{3} e^{-0.6 t} dt$ .

Evaluate the area between the $x$ -axis and $f(x)=15 x^{2}+3$ from $x=-3$ to $x=10$ using the Fundamental Theorem of Calculus.

Find the general antiderivative of the function: $\int\left(\frac{1}{x^{4}}-x^{4}-\frac{1}{5}\right) d x$

Find the derivative of the inverse function at 0: $\left(f^{-1}\right)^{\prime}(0)$ where $f(x)=\int_{t}^{x} \frac{d t}{\sqrt{4+t^{4}}}$ .

Evaluate the series: $\sum_{k=2}^{\infty}\left(\frac{2}{5}\right)^{3 k}$ . Does it converge or diverge?

The company's net worth growth rate is $f'(t)=2850-11t^2$ . How does it change from 1990 to 2000? Choose A, B, C, or D. If worth \$55000 in 1990, what is it in 2000?

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{2}^{4} \frac{5}{x} d x$ .

Find the integral $\int_{a}^{b} 28 e^{0.08 t} \, dt$ for oil used from 1990 to 1998. What are $a$ and $b$ ?

Find the derivative of the inverse function at 0: $\left(f^{-1}\right)^{\prime}(0)$ where $f(x)=\int_{4}^{x} \frac{d t}{\sqrt{4+t^{4}}}$ .

Find the derivative of the integral: $\frac{d}{d x} \int_{0}^{x^{2}} \frac{t d t}{t+5}$ .

Find the derivative with respect to $\theta$ of the integral $\int_{1}^{\theta} 3 \cot (u) \, du$ .

Find the derivative of the integral $\frac{d}{d s} \int_{-7}^{s} \tan \left(\frac{1}{u^{2}+5}\right) d u$ .

Find the total cost for the first 49 units given the marginal cost function $m c(x)=\frac{13}{\sqrt{x}}$ .

Find the derivative: $\frac{d}{d x} \int_{1}^{1 / x} \cos ^{5}(t) d t$

Find the tangent line equation for $y=-4 \sin x$ at $x=\frac{\pi}{4}$ .

Find $\left(f^{-1}\right)^{\prime}(3)$ for $f(x)=\frac{x+5}{x+1}$ where $x>-1$ .

Find $\left(f^{-1}\right)^{\prime}(3)$ for the function $f(x) = \frac{x+5}{x+1}$ , where $x > -1$ .

Find the derivative: $\frac{d}{d x} \int_{\sqrt{x}}^{x^{4}} \tan (3 t) d t$

Compute the integral $\int_{-2}^{2} \frac{x^{2}+2}{x^{3}+3 x^{2}-9 x-27} d x$ and approximate it to 3 decimal places.

Evaluate the limit using I'Hôpital's Rule: $\lim _{x \rightarrow \infty} \frac{5 x+7}{8 x^{2}+5 x-8}$ .

An object's acceleration is $a(t)=7 \sin (t)$ with initial velocity $v(0)=-12 \mathrm{~m/s}$ .
a) Find $v(t)$ . b) Calculate displacement from $t=0$ to $t=3$ . c) Determine total distance traveled from $t=0$ to $t=3$ .

Differentiate the integral $\int_{-u}^{5u} \sqrt{x^{2}+2} \, dx$ with respect to $u$ .

Find an antiderivative of the function $-\frac{20}{x^{6}}$ .

Find $\left(f^{-1}\right)^{\prime}(1)$ for $f(x)=\cos(2x)$ , where $0 \leq x \leq \pi/2$ .

Find $\left(f^{-1}\right)^{\prime}(63)$ for the function $f(x) = x^{3} - 1$ .

Find a tangent line to $y=-\cos x$ with slope -1. Is there more than one solution? Explain.

Evaluate the limit using l'Hôpital's Rule: $\lim _{x \rightarrow 1} \frac{x^{3}-8 x^{2}+7}{x-1}$

Find the number of cars passing an intersection from 6 am to 7 am given $r(t)=400+1000 t-150 t^{2}$ .

Find the limit as $x$ approaches 0 from the right: $\lim _{x \rightarrow 0^{+}} x^{-3 / \ln x}$ .

Evaluate the series $\sum_{k=2}^{\infty}\left(\frac{3}{8}\right)^{4 k}$ or state if it diverges.

Solve the integral $\int_{-5}^{5} \frac{x^{3} \sin ^{2} x}{x^{4}+2 x^{2}+1} d x$ .

Complete the chart for the equation $\frac{1}{(x-5)^{2}}$ as $x \to \infty$ , $x \to -\infty$ , $x \to 5^{-}$ , and $x \to 5^{+}$ .

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(1+\frac{2}{x^{4}}\right)^{x}$ .

Find the general antiderivative of $-6 \cos t$ .

Find the limit as $n$ approaches infinity of the sum for $f(x)=\sqrt[6]{x}$ from $1$ to $18$ using right endpoints.

Evaluate the limit using l'Hôpital's Rule: $\lim _{x \rightarrow 0} \frac{\cos (5 x)-1}{x^{2}}$ .

Find $\int_{2}^{6}[f(x)-4 g(x)] d x$ given $\int_{2}^{6} f(x) d x=-2$ and $\int_{2}^{6} g(x) d x=10$ .