Calculus

Problem 13301

Find $f^{\prime}(x)$ if $f(x)=\int_{x}^{13} t^{3} dt$ .

See Solution

Problem 13302

Find the area between the polar curves $r=\sin \theta$ and $r=1+\cos \theta$ . First, find their intersection points, then integrate to get the area.

See Solution

Problem 13303

Find the integral for the area of the region between $r=1+\sin \theta$ and $r=1+\cos \theta$ without evaluating it.

See Solution

Problem 13304

Given the functions $f$ and $g$ with their values and derivatives, find $(f g)^{\prime}(1)$ . Provide the exact integer result.

See Solution

Problem 13305

Compute the value of $(f / g)^{\prime}(3)$ using the given values for $f$ , $g$ , $f'$ , and $g'$ . Round to three decimal places.

See Solution

Problem 13306

Find the tangent line equation to the spiral $r=e^{\frac{\theta}{4}}$ at $\theta=8$ . Use decimal approximation.

See Solution

Problem 13307

Find global extrema of $f(x)=\frac{3 x^{2}}{x^{2}+1}$ on the interval $[-6,6]$ .

See Solution

Problem 13308

Find the integral for the area of the loop defined by $r=1-\cos 2\theta$ . No need to evaluate it.

See Solution

Problem 13309

Construct the integral to find the area of the region from the curve $r=3-3 \cos \theta$ without evaluating it.

See Solution

Problem 13310

Calculate the limit of Riemann sums for $\int_{0}^{1}(4+x^{2}) dx$ using the sum formula for squares.

See Solution

Problem 13311

Find the cross partials of $z=f(x, y)=(x \times y)^{2}-\ln (x \times y)$ . What are $f_{yx}$ and $f_{xy}$ ?

See Solution

Problem 13312

Identify the region with area given by $\text{area}=\lim_{n \to \infty} \sum_{i=1}^{n} \frac{\pi}{3n} \cos\left(\frac{i\pi}{3n}\right)$ without calculating the limit.

See Solution

Problem 13313

Estimate the area under $f(x)=4 \sin x$ from $x=0$ to $x=\frac{\pi}{4}$ using 5 rectangles with right endpoints.

See Solution

Problem 13314

Gegeben ist $f(x)=(3x^3+4)^2$ . a) Zeigen Sie, dass $f'(x) \neq 3 \cdot 2 \cdot (3x^3+4)$ . b) Mit welchem Term multiplizieren?

See Solution

Problem 13315

Find the limit: $\lim _{x \rightarrow \infty}\left(1+\frac{1}{6 x}\right)^{-4 x}$ .

See Solution

Problem 13316

Find the cross partial derivatives of $z=f(x, y)=(x \times y)^{2}-\ln (x \times y)$ . What are $f_{yx}$ and $f_{xy}$ ?

See Solution

Problem 13317

Evaluate the integral using integration by parts: $\int x e^{2x} dx$ .

See Solution

Problem 13318

You have \$141,000 to invest. Compare 11% daily compounded vs 0.94% continuously compounded after one year.

See Solution

Problem 13319

Find $\frac{d y}{d x}$ using implicit differentiation for $\sqrt[3]{x}+\sqrt[3]{y}=8$ . Choose A or B.

See Solution

Problem 13320

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x^{3}+8 y^{5}=\ln y$ .

See Solution

Problem 13321

Find the value of $I=\int_{0}^{5} f(x) d x$ given $\int_{0}^{7} f(x) d x=5$ and $\int_{5}^{7} f(x) d x=9$ .

See Solution

Problem 13322

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x^{5} \cdot y^{5}=4$ .

See Solution

Problem 13323

Calculate the integral $I_{1}=\int_{0}^{3} 3 \sqrt{9-x^{2}} \, dx$ .

See Solution

Problem 13324

Find $\frac{\mathrm{dq}}{\mathrm{dp}}$ for the demand equation $p=\frac{20}{(q+4)^{5}}$ .

See Solution

Problem 13325

Differentiate the function using logarithmic differentiation: $f(x)=(x+7)^{5}(6 x-7)^{4}$ . Find $f^{\prime}(x)=\square$ .

See Solution

Problem 13326

Differentiate the function using logarithmic differentiation: $f(x)=\frac{(x+1)(8 x+1)(7 x+1)}{\sqrt{6 x+1}}$ . Find $f^{\prime}(x)=\square$ .

See Solution

Problem 13327

Find the derivative $y^{\prime}$ of the function $y=4 x^{\sqrt{x}}$ .

See Solution

Problem 13328

Is the function $f(x)=\left\{\begin{array}{ll} x^{2}-21 x+3, & x \leq 3 \\ -15 x-6, & x>3 \end{array}\right.$ differentiable, continuous, both, or neither at $x=3$ ?

See Solution

Problem 13329

Find the derivative using logarithmic differentiation: $\frac{d}{d x}\left(1+\frac{9}{x}\right)^{x}=\square$

See Solution

Problem 13330

Given $f(5)=9$ , $f'(5)=9$ , $g(5)=4$ , $g'(5)=6$ , calculate $\left.\frac{d}{d x}(2 f(x)-3 g(x)+4)\right|_{x=5}$ .

See Solution

Problem 13331

Find the derivative of $y=(5x+2)^{x}$ with respect to $x$ . What is $\frac{dy}{dx}$ ?

See Solution

Problem 13332

Find the value of the integral $I=\int_{4}^{5}(6 f(x)-7 g(x)+5) d x$ given $\int_{4}^{5} f(x) d x=9$ and $\int_{4}^{5} g(x) d x=36$ .

See Solution

Problem 13333

Find the derivative of $y$ where $y=(9x+4)^{x}$ . What is $\frac{dy}{dx}$ ?

See Solution

Problem 13334

Calculate the integral $I=\int_{2}^{5} f(x) d x$ given that $\int_{2}^{9} f(x) d x=4$ and $\int_{5}^{9} f(x) d x=8$ .

See Solution

Problem 13335

Find the tangent line equation for $y=(x+1)(x+2)^{2}(x+3)^{2}$ at $x=2$ in slope-intercept form.

See Solution

Problem 13336

Find the integral $I=\int_{0}^{1}(4+x^{2}) dx$ using the formula for the sum of squares $1^{2}+2^{2}+\ldots+n^{2}=\frac{1}{6} n(n+1)(2 n+1)$ .

See Solution

Problem 13337

Find the tangent line equation for $y=x^{13 x}$ at $x=e$ . The equation is $\square$ .

See Solution

Problem 13338

Find the limit of $\sum_{k=1}^{n}\left(\frac{4}{n^{3}} k^{2}-\frac{3}{n^{2}} k+\frac{5}{n}\right)$ as $n \rightarrow \infty$ .

See Solution

Problem 13339

Find if the limit $\lim _{n \rightarrow \infty} \frac{1}{n^{2}}\left\{\sum_{k=1}^{n}(5+8 k)\right\}$ exists and its value.

See Solution

Problem 13340

Find the average rate of change of $f(x)=(x-3)^{3}-1$ from $x=2$ to $x=5$ .

See Solution

Problem 13341

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $7 x^{4} y^{5}-x+y^{2}=49$ .

See Solution

Problem 13342

Minimize $f(x, y, z)=x^{2}+y^{2}-z^{2}$ with constraints $x+y+z=9$ and $x+2y+3z=10$ .
Minimize $f(x, y, z)=xyz$ with constraints $x^{2}+y^{2}+z^{2}=1$ , $x \geq y$ , $y \leq 0$ , $z \geq 0$ .

See Solution

Problem 13343

Evaluate the integral of $x \ln x$ with respect to $x$ : $\int x \ln x \, dx$ .

See Solution

Problem 13344

Find $f(x)$ such that $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{3}{n}\left(6+\frac{k}{n}\right)^{2}=\int_{0}^{1} f(x) dx$ .

See Solution

Problem 13345

Find $\frac{d y}{d x}$ using implicit differentiation for $(1+e^{2 x})^{2}=5+\ln (x+y)$ , $y \neq -x$ .

See Solution

Problem 13346

Find the instantaneous rate of change of $f(x)=2 x^{2}+4 x-3$ at $x=1$ .

See Solution

Problem 13347

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

See Solution

Problem 13348

Express the limit as a definite integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(5 x_{i}^{} \sin x_{i}^{}\right) \Delta x$ for $[1,4]$ .

See Solution

Problem 13349

Estimate the area under $f(x)=\sin x$ from $x=0$ to $x=\frac{\pi}{4}$ using 5 rectangles with right endpoints.

See Solution

Problem 13350

Find the value of the definite integral $I=\int_{a}^{b} f(x) d x$ given the Riemann sum approximation $\sum_{i=1}^{n} f(x_{i}^{*}) \Delta x_{i}=\frac{5 n^{2}-6 n+3}{n^{2}}$ .

See Solution

Problem 13351

How many days does it take Earth to orbit the Sun, given Sun's mass $=1.989 \times 10^{30} \mathrm{~kg}$ and orbit radius $=1.496 \times 10^{8}$ ?

See Solution

Problem 13352

Calculate the area between the curves $y=3 x^{4}+2 x^{2}$ and $y=3 x^{4}+64-2 x^{2}$ at their intersection points.

See Solution

Problem 13353

Determine where the function $f(x)$ is increasing given its derivative $f^{\prime}(x)=(x+1)(x-2)^{2}(x-4)^{3}$ .

See Solution

Problem 13354

Determine if $f(x)=3 e^{x^{2}}$ is increasing or decreasing on $[0,1]$ and estimate the signed area using $L_{4}$ or $R_{4}$ .

See Solution

Problem 13355

An airplane flies at 400 km/h and 500 m high. Find the distance change rate 0.600 min after passing an observer. Answer: 397 km/h

See Solution

Problem 13356

Determine the intervals where the function $f(x)=\frac{x^{2}}{2}-2 x+(3+2 x) \ln x+c$ is concave up.

See Solution

Problem 13357

Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for these functions: 1- $f(x, y)=\mathrm{e}^{x y^{3}+x}$ 2- $f(x, y)=\operatorname{Ln}(\mathrm{x}+\mathrm{y})$ 3- $f(x, y)=(3 y+x y)^{4}$ 4- $f(x, y)=y \ln (x y)$ 5- $f(x, y)=y \cos (x y)$ .

See Solution

Problem 13358

Find the limit: $\lim _{x \rightarrow 0^{+}}(f(x)+1)^{1 / \sin (2 x)}$ given $\lim _{x \rightarrow 0^{+}} f(x)=0$ and $\lim _{x \rightarrow 0^{+}} f^{\prime}(x)=2$ .

See Solution

Problem 13359

Is the numerical derivative of a function always an approximation of the actual derivative? Explain or provide a counterexample.

See Solution

Problem 13360

Given $f(4)=5$ and $f^{\prime}(4)=8$ , find $(f h)^{\prime}(4)$ where $h(x)=\sqrt{x}$ . Round to three decimals.

See Solution

Problem 13361

A right triangle has legs of 9 in and 12 in. The short leg grows by 9 in/sec, and the long leg shrinks by 2 in/sec. Find the area change rate.

See Solution

Problem 13362

Find the integral of $x^{3} e^{2 x}$ with respect to $x$ .

See Solution

Problem 13363

Given values for functions $f$ , $g$ and their derivatives, find $(f g)^{\prime}(1)$ . Provide an integer answer.

See Solution

Problem 13364

Find the derivative of $f(x)=\left(\frac{x^{5}+4}{x^{2}-5}\right)^{\frac{1}{5}}$ using the chain rule.

See Solution

Problem 13365

A right triangle has legs 30 in and 40 in. Short leg increases by 8 in/sec, long leg decreases by 2 in/sec. Find hypotenuse's rate of change.

See Solution

Problem 13366

Find the derivative of $f(x) = \frac{\sqrt[5]{x^2 - 3}}{-x - 5}$ using the chain rule.

See Solution

Problem 13367

Find the rate of change of the radius when the sphere's volume is 1585 m³ and decreasing at 4027 m³/min. Use $V=\frac{4}{3} \pi r^{3}$ . Round to three decimal places.

See Solution

Problem 13368

A cone's radius decreases at 1 m/min, volume at 61 m³/min. Find height change rate when radius is 2 m, volume 41 m³. Use $V=\frac{1}{3} \pi r^{2} h$ . Round to three decimals.

See Solution

Problem 13369

Find the derivative of $y=\sec(2x^{4})$ using the chain rule.

See Solution

Problem 13370

If $f$ and $g$ are continuous with $f(x) \geq 0$ , which statements are true? I. $\int_{a}^{b} f(x) g(x) dx = \left(\int_{a}^{b} f(x) dx\right)\left(\int_{a}^{b} g(x) dx\right)$ II. $\int_{a}^{b}\{f(x)+g(x)\} dx = \int_{a}^{b} f(x) dx+\int_{a}^{b} g(x) dx$ III. $\int_{a}^{b} \sqrt{f(x)} dx = \sqrt{\int_{a}^{b} f(x) dx}$

See Solution

Problem 13371

Find the inflection point(s) of $f(x)=5 x e^{-4 x}$ .

See Solution

Problem 13372

Find $(f h)'(4)$ given $f(4)=10$ , $f'(4)=9$ , and $h(x)=\sqrt{x}$ . Round your answer to three decimal places.

See Solution

Problem 13373

Calculate the integral $\int x^{3} \ln (x) \, dx$ .

See Solution

Problem 13374

Compute the value of $(f g)'(1)$ using the given values for $f(1)$ , $f'(1)$ , $g(1)$ , and $g'(1)$ .

See Solution

Problem 13375

Find the value of $(f / g)^{\prime}(3)$ using the given values. Round your answer to three decimal places.

See Solution

Problem 13376

Find $h'(1)$ for $h(x) = x f(x) \cos(x)$ given $f(1)=7$ and $f'(1)=6$ . Round to three decimal places.

See Solution

Problem 13377

Find $g^{\prime}(8)$ for $g(x)=\frac{f(x)}{1+f(x)}$ given $f(8)=2$ and $f^{\prime}(8)=2$ . Round to three decimal places.

See Solution

Problem 13378

Find the derivatives: $f^{\prime}(2)$ and $f^{\prime}(4)$ for the function $f(x)=-3 x^{1/6}$ .

See Solution

Problem 13379

Find the derivative of $y=-3 x^{8}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 13380

Find the derivative of the function $f(x)=\frac{-6}{x^{2}}$ at $x=3$ : $f^{\prime}(3)=$

See Solution

Problem 13381

Find the derivatives $f^{\prime}(2)$ and $f^{\prime}(4)$ for the function $f(x)=-3 x^{1/6}$ .

See Solution

Problem 13382

A bank account grows exponentially. At $\$ 4000$ , it grows at $\$ 100$ per year. How fast is it growing at $\$ 5000$ ?

See Solution

Problem 13383

Find two positive numbers with a product of 8. Minimize their sum and verify using the second derivative test.

See Solution

Problem 13384

Differentiate the function $\frac{-x^{2}+64}{(x^{2}+64)^{2}}$ with respect to $x$ .

See Solution

Problem 13385

Find the area $A$ that maximizes savings $S=360 A-0.10 A^{3}$ and calculate the maximum savings.

See Solution

Problem 13386

Find the horizontal asymptote of $f(x)=\frac{4(x+3)(4x-1)}{(8-x)(8x+2)}$ . What is $y=$ ?

See Solution

Problem 13387

Find the inflection points of the function $f(x)=\frac{x}{x^{2}+64}$ .

See Solution

Problem 13388

Find the conclusion from the Taylor polynomial $T_{3}(x)=3-(x-\pi)-(x-\pi)^{2}+4(x-\pi)^{3}$ .

See Solution

Problem 13389

Find the slope of the secant line for $f(x)=2 x^{2}-20 x+36$ on the interval $[6, x_{2}]$ for $x_{2}$ values: 7, 6.1, 6.01, 6.001.

See Solution

Problem 13390

Find the slope of the secant line for $f(x)=2 x^{2}-4 x-5$ on intervals $[2, x_{2}]$ for $x_{2}=3, 2.1, 2.01, 2.001$ .

See Solution

Problem 13391

Is $\frac{x^{n+1}}{n+1}$ an antiderivative of $x^{n}$ for real $n$ ? Choose true/false options regarding $n$ .

See Solution

Problem 13392

A radioactive spill of 2000 tons has a half-life of 36 days. When will it drop below 100 tons for swimming safety?

See Solution

Problem 13393

Find the integral $I_{1}=\int_{0}^{4} 3 \sqrt{16-x^{2}} d x$ using known areas.

See Solution

Problem 13394

Find $f(x)$ such that $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{4}{n} \left(2+\frac{k}{n}\right)^{2} = \int_{0}^{1} f(x) \, dx$ .

See Solution

Problem 13395

How old is a bone that has lost about $30.2\%$ of its carbon-14, given the half-life is 5750 years?

See Solution

Problem 13396

How long for bacteria to double at a 3.4% growth rate per minute? Use $A=A_{0} e^{r t}$ and round to the nearest whole number.

See Solution

Problem 13397

Find the Riemann sums using 6 rectangles for $f(x)=9-x$ on $[2,4]$ and $g(x)=2x^2-x-1$ on $[2,5]$ . Also, find it for $g(x)=x^2+1$ on $[1,3]$ with 8 rectangles and for $f(x)=\cos x$ on $\left[0, \frac{\pi}{2}\right]$ with 4 rectangles.

See Solution

Problem 13398

A satellite orbits Earth at an altitude of $2.50 \times 10^{6} \mathrm{~m}$ . Find its speed in $\mathrm{km/s}$ and acceleration in $\mathrm{m/s}^{2}$ . Use $G = 6.674 \times 10^{-11} \mathrm{N(m/kg)^2}$ and $M = 5.972 \times 10^{24} \mathrm{kg}$ .

See Solution

Problem 13399

Find the inflection points of the function $f(x)=\cos\left(3x-\frac{\pi}{2}\right)$ in the interval $(-2\pi, 0)$ .

See Solution

Problem 13400

Find the derivative $f'(x)$ of $f(x)=\int_{5}^{x} t^{3} dt$ and evaluate it at $x = -3$ .

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 13301

Find f′(x)f^{\prime}(x)f′(x) if f(x)=∫x13t3dtf(x)=\int_{x}^{13} t^{3} dtf(x)=∫x13​t3dt.

Problem 13302

Find the area between the polar curves r=sin⁡θr=\sin \thetar=sinθ and r=1+cos⁡θr=1+\cos \thetar=1+cosθ. First, find their intersection points, then integrate to get the area.

Problem 13303

Find the integral for the area of the region between r=1+sin⁡θr=1+\sin \thetar=1+sinθ and r=1+cos⁡θr=1+\cos \thetar=1+cosθ without evaluating it.

Problem 13304

Given the functions fff and ggg with their values and derivatives, find (fg)′(1)(f g)^{\prime}(1)(fg)′(1). Provide the exact integer result.

Problem 13305

Compute the value of (f/g)′(3)(f / g)^{\prime}(3)(f/g)′(3) using the given values for fff, ggg, f′f'f′, and g′g'g′. Round to three decimal places.

Problem 13306

Find the tangent line equation to the spiral r=eθ4r=e^{\frac{\theta}{4}}r=e4θ​ at θ=8\theta=8θ=8. Use decimal approximation.

Problem 13307

Find global extrema of f(x)=3x2x2+1f(x)=\frac{3 x^{2}}{x^{2}+1}f(x)=x2+13x2​ on the interval [−6,6][-6,6][−6,6].

Problem 13308

Find the integral for the area of the loop defined by r=1−cos⁡2θr=1-\cos 2\thetar=1−cos2θ. No need to evaluate it.

Problem 13309

Construct the integral to find the area of the region from the curve r=3−3cos⁡θr=3-3 \cos \thetar=3−3cosθ without evaluating it.

Problem 13310

Calculate the limit of Riemann sums for ∫01(4+x2)dx\int_{0}^{1}(4+x^{2}) dx∫01​(4+x2)dx using the sum formula for squares.

Problem 13311

Find the cross partials of z=f(x,y)=(x×y)2−ln⁡(x×y)z=f(x, y)=(x \times y)^{2}-\ln (x \times y)z=f(x,y)=(x×y)2−ln(x×y). What are fyxf_{yx}fyx​ and fxyf_{xy}fxy​?

Problem 13312

Identify the region with area given by area=lim⁡n→∞∑i=1nπ3ncos⁡(iπ3n)\text{area}=\lim_{n \to \infty} \sum_{i=1}^{n} \frac{\pi}{3n} \cos\left(\frac{i\pi}{3n}\right)area=n→∞lim​i=1∑n​3nπ​cos(3niπ​) without calculating the limit.

Problem 13313

Estimate the area under f(x)=4sin⁡xf(x)=4 \sin xf(x)=4sinx from x=0x=0x=0 to x=π4x=\frac{\pi}{4}x=4π​ using 5 rectangles with right endpoints.

Problem 13314

Gegeben ist f(x)=(3x3+4)2f(x)=(3x^3+4)^2f(x)=(3x3+4)2. a) Zeigen Sie, dass f′(x)≠3⋅2⋅(3x3+4)f'(x) \neq 3 \cdot 2 \cdot (3x^3+4)f′(x)=3⋅2⋅(3x3+4). b) Mit welchem Term multiplizieren?

Problem 13315

Find the limit: lim⁡x→∞(1+16x)−4x\lim _{x \rightarrow \infty}\left(1+\frac{1}{6 x}\right)^{-4 x}limx→∞​(1+6x1​)−4x.

Problem 13316

Find the cross partial derivatives of z=f(x,y)=(x×y)2−ln⁡(x×y)z=f(x, y)=(x \times y)^{2}-\ln (x \times y)z=f(x,y)=(x×y)2−ln(x×y). What are fyxf_{yx}fyx​ and fxyf_{xy}fxy​?

Problem 13317

Evaluate the integral using integration by parts: ∫xe2xdx\int x e^{2x} dx∫xe2xdx.

Problem 13318

You have \$141,000 to invest. Compare 11% daily compounded vs 0.94% continuously compounded after one year.

Problem 13319

Find dydx\frac{d y}{d x}dxdy​ using implicit differentiation for x3+y3=8\sqrt[3]{x}+\sqrt[3]{y}=83x​+3y​=8. Choose A or B.

Problem 13320

Find dydx\frac{d y}{d x}dxdy​ using implicit differentiation for the equation x3+8y5=ln⁡yx^{3}+8 y^{5}=\ln yx3+8y5=lny.

Problem 13321

Find the value of I=∫05f(x)dxI=\int_{0}^{5} f(x) d xI=∫05​f(x)dx given ∫07f(x)dx=5\int_{0}^{7} f(x) d x=5∫07​f(x)dx=5 and ∫57f(x)dx=9\int_{5}^{7} f(x) d x=9∫57​f(x)dx=9.

Problem 13322

Find dydx\frac{d y}{d x}dxdy​ using implicit differentiation for the equation x5⋅y5=4x^{5} \cdot y^{5}=4x5⋅y5=4.

Problem 13323

Calculate the integral I1=∫0339−x2 dxI_{1}=\int_{0}^{3} 3 \sqrt{9-x^{2}} \, dxI1​=∫03​39−x2​dx.

Problem 13324

Find dqdp\frac{\mathrm{dq}}{\mathrm{dp}}dpdq​ for the demand equation p=20(q+4)5p=\frac{20}{(q+4)^{5}}p=(q+4)520​.

Problem 13325

Differentiate the function using logarithmic differentiation: f(x)=(x+7)5(6x−7)4f(x)=(x+7)^{5}(6 x-7)^{4}f(x)=(x+7)5(6x−7)4. Find f′(x)=□f^{\prime}(x)=\squaref′(x)=□.

Problem 13326

Differentiate the function using logarithmic differentiation: f(x)=(x+1)(8x+1)(7x+1)6x+1f(x)=\frac{(x+1)(8 x+1)(7 x+1)}{\sqrt{6 x+1}}f(x)=6x+1​(x+1)(8x+1)(7x+1)​. Find f′(x)=□f^{\prime}(x)=\squaref′(x)=□.

Problem 13327

Find the derivative y′y^{\prime}y′ of the function y=4xxy=4 x^{\sqrt{x}}y=4xx​.

Problem 13328

Is the function f(x)={x2−21x+3,x≤3−15x−6,x>3f(x)=\left\{\begin{array}{ll} x^{2}-21 x+3, & x \leq 3 \\ -15 x-6, & x>3 \end{array}\right.f(x)={x2−21x+3,−15x−6,​x≤3x>3​ differentiable, continuous, both, or neither at x=3x=3x=3?

Problem 13329

Find the derivative using logarithmic differentiation: ddx(1+9x)x=□\frac{d}{d x}\left(1+\frac{9}{x}\right)^{x}=\squaredxd​(1+x9​)x=□

Problem 13330

Given f(5)=9f(5)=9f(5)=9, f′(5)=9f'(5)=9f′(5)=9, g(5)=4g(5)=4g(5)=4, g′(5)=6g'(5)=6g′(5)=6, calculate ddx(2f(x)−3g(x)+4)∣x=5\left.\frac{d}{d x}(2 f(x)-3 g(x)+4)\right|_{x=5}dxd​(2f(x)−3g(x)+4)∣∣​x=5​.

Problem 13331

Find the derivative of y=(5x+2)xy=(5x+2)^{x}y=(5x+2)x with respect to xxx. What is dydx\frac{dy}{dx}dxdy​?

Problem 13332

Find the value of the integral I=∫45(6f(x)−7g(x)+5)dxI=\int_{4}^{5}(6 f(x)-7 g(x)+5) d xI=∫45​(6f(x)−7g(x)+5)dx given ∫45f(x)dx=9\int_{4}^{5} f(x) d x=9∫45​f(x)dx=9 and ∫45g(x)dx=36\int_{4}^{5} g(x) d x=36∫45​g(x)dx=36.

Problem 13333

Find the derivative of yyy where y=(9x+4)xy=(9x+4)^{x}y=(9x+4)x. What is dydx\frac{dy}{dx}dxdy​?

Problem 13334

Calculate the integral I=∫25f(x)dxI=\int_{2}^{5} f(x) d xI=∫25​f(x)dx given that ∫29f(x)dx=4\int_{2}^{9} f(x) d x=4∫29​f(x)dx=4 and ∫59f(x)dx=8\int_{5}^{9} f(x) d x=8∫59​f(x)dx=8.

Problem 13335

Find the tangent line equation for y=(x+1)(x+2)2(x+3)2y=(x+1)(x+2)^{2}(x+3)^{2}y=(x+1)(x+2)2(x+3)2 at x=2x=2x=2 in slope-intercept form.

Problem 13336

Find the integral I=∫01(4+x2)dxI=\int_{0}^{1}(4+x^{2}) dxI=∫01​(4+x2)dx using the formula for the sum of squares 12+22+…+n2=16n(n+1)(2n+1)1^{2}+2^{2}+\ldots+n^{2}=\frac{1}{6} n(n+1)(2 n+1)12+22+…+n2=61​n(n+1)(2n+1).

Problem 13337

Find the tangent line equation for y=x13xy=x^{13 x}y=x13x at x=ex=ex=e. The equation is □\square□.

Problem 13338

Find the limit of ∑k=1n(4n3k2−3n2k+5n)\sum_{k=1}^{n}\left(\frac{4}{n^{3}} k^{2}-\frac{3}{n^{2}} k+\frac{5}{n}\right)∑k=1n​(n34​k2−n23​k+n5​) as n→∞n \rightarrow \inftyn→∞.

Problem 13339

Find if the limit lim⁡n→∞1n2{∑k=1n(5+8k)}\lim _{n \rightarrow \infty} \frac{1}{n^{2}}\left\{\sum_{k=1}^{n}(5+8 k)\right\}limn→∞​n21​{∑k=1n​(5+8k)} exists and its value.

Problem 13340

Find $f^{\prime}(x)$ if $f(x)=\int_{x}^{13} t^{3} dt$ .

Find the area between the polar curves $r=\sin \theta$ and $r=1+\cos \theta$ . First, find their intersection points, then integrate to get the area.

Find the integral for the area of the region between $r=1+\sin \theta$ and $r=1+\cos \theta$ without evaluating it.

Given the functions $f$ and $g$ with their values and derivatives, find $(f g)^{\prime}(1)$ . Provide the exact integer result.

Compute the value of $(f / g)^{\prime}(3)$ using the given values for $f$ , $g$ , $f'$ , and $g'$ . Round to three decimal places.

Find the tangent line equation to the spiral $r=e^{\frac{\theta}{4}}$ at $\theta=8$ . Use decimal approximation.

Find global extrema of $f(x)=\frac{3 x^{2}}{x^{2}+1}$ on the interval $[-6,6]$ .

Find the integral for the area of the loop defined by $r=1-\cos 2\theta$ . No need to evaluate it.

Construct the integral to find the area of the region from the curve $r=3-3 \cos \theta$ without evaluating it.

Calculate the limit of Riemann sums for $\int_{0}^{1}(4+x^{2}) dx$ using the sum formula for squares.

Find the cross partials of $z=f(x, y)=(x \times y)^{2}-\ln (x \times y)$ . What are $f_{yx}$ and $f_{xy}$ ?

Identify the region with area given by $\text{area}=\lim_{n \to \infty} \sum_{i=1}^{n} \frac{\pi}{3n} \cos\left(\frac{i\pi}{3n}\right)$ without calculating the limit.

Estimate the area under $f(x)=4 \sin x$ from $x=0$ to $x=\frac{\pi}{4}$ using 5 rectangles with right endpoints.

Gegeben ist $f(x)=(3x^3+4)^2$ . a) Zeigen Sie, dass $f'(x) \neq 3 \cdot 2 \cdot (3x^3+4)$ . b) Mit welchem Term multiplizieren?

Find the limit: $\lim _{x \rightarrow \infty}\left(1+\frac{1}{6 x}\right)^{-4 x}$ .

Find the cross partial derivatives of $z=f(x, y)=(x \times y)^{2}-\ln (x \times y)$ . What are $f_{yx}$ and $f_{xy}$ ?

Evaluate the integral using integration by parts: $\int x e^{2x} dx$ .

Find $\frac{d y}{d x}$ using implicit differentiation for $\sqrt[3]{x}+\sqrt[3]{y}=8$ . Choose A or B.

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x^{3}+8 y^{5}=\ln y$ .

Find the value of $I=\int_{0}^{5} f(x) d x$ given $\int_{0}^{7} f(x) d x=5$ and $\int_{5}^{7} f(x) d x=9$ .

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x^{5} \cdot y^{5}=4$ .

Calculate the integral $I_{1}=\int_{0}^{3} 3 \sqrt{9-x^{2}} \, dx$ .

Find $\frac{\mathrm{dq}}{\mathrm{dp}}$ for the demand equation $p=\frac{20}{(q+4)^{5}}$ .

Differentiate the function using logarithmic differentiation: $f(x)=(x+7)^{5}(6 x-7)^{4}$ . Find $f^{\prime}(x)=\square$ .

Differentiate the function using logarithmic differentiation: $f(x)=\frac{(x+1)(8 x+1)(7 x+1)}{\sqrt{6 x+1}}$ . Find $f^{\prime}(x)=\square$ .

Find the derivative $y^{\prime}$ of the function $y=4 x^{\sqrt{x}}$ .

Is the function $f(x)=\left\{\begin{array}{ll} x^{2}-21 x+3, & x \leq 3 \\ -15 x-6, & x>3 \end{array}\right.$ differentiable, continuous, both, or neither at $x=3$ ?

Find the derivative using logarithmic differentiation: $\frac{d}{d x}\left(1+\frac{9}{x}\right)^{x}=\square$

Given $f(5)=9$ , $f'(5)=9$ , $g(5)=4$ , $g'(5)=6$ , calculate $\left.\frac{d}{d x}(2 f(x)-3 g(x)+4)\right|_{x=5}$ .

Find the derivative of $y=(5x+2)^{x}$ with respect to $x$ . What is $\frac{dy}{dx}$ ?

Find the value of the integral $I=\int_{4}^{5}(6 f(x)-7 g(x)+5) d x$ given $\int_{4}^{5} f(x) d x=9$ and $\int_{4}^{5} g(x) d x=36$ .

Find the derivative of $y$ where $y=(9x+4)^{x}$ . What is $\frac{dy}{dx}$ ?

Calculate the integral $I=\int_{2}^{5} f(x) d x$ given that $\int_{2}^{9} f(x) d x=4$ and $\int_{5}^{9} f(x) d x=8$ .

Find the tangent line equation for $y=(x+1)(x+2)^{2}(x+3)^{2}$ at $x=2$ in slope-intercept form.

Find the integral $I=\int_{0}^{1}(4+x^{2}) dx$ using the formula for the sum of squares $1^{2}+2^{2}+\ldots+n^{2}=\frac{1}{6} n(n+1)(2 n+1)$ .

Find the tangent line equation for $y=x^{13 x}$ at $x=e$ . The equation is $\square$ .

Find the limit of $\sum_{k=1}^{n}\left(\frac{4}{n^{3}} k^{2}-\frac{3}{n^{2}} k+\frac{5}{n}\right)$ as $n \rightarrow \infty$ .

Find if the limit $\lim _{n \rightarrow \infty} \frac{1}{n^{2}}\left\{\sum_{k=1}^{n}(5+8 k)\right\}$ exists and its value.

Find the average rate of change of $f(x)=(x-3)^{3}-1$ from $x=2$ to $x=5$ .

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $7 x^{4} y^{5}-x+y^{2}=49$ .

Evaluate the integral of $x \ln x$ with respect to $x$ : $\int x \ln x \, dx$ .

Find $f(x)$ such that $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{3}{n}\left(6+\frac{k}{n}\right)^{2}=\int_{0}^{1} f(x) dx$ .

Find $\frac{d y}{d x}$ using implicit differentiation for $(1+e^{2 x})^{2}=5+\ln (x+y)$ , $y \neq -x$ .

Find the instantaneous rate of change of $f(x)=2 x^{2}+4 x-3$ at $x=1$ .

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

Express the limit as a definite integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(5 x_{i}^{} \sin x_{i}^{}\right) \Delta x$ for $[1,4]$ .

Estimate the area under $f(x)=\sin x$ from $x=0$ to $x=\frac{\pi}{4}$ using 5 rectangles with right endpoints.

Find the value of the definite integral $I=\int_{a}^{b} f(x) d x$ given the Riemann sum approximation $\sum_{i=1}^{n} f(x_{i}^{*}) \Delta x_{i}=\frac{5 n^{2}-6 n+3}{n^{2}}$ .

How many days does it take Earth to orbit the Sun, given Sun's mass $=1.989 \times 10^{30} \mathrm{~kg}$ and orbit radius $=1.496 \times 10^{8}$ ?

Calculate the area between the curves $y=3 x^{4}+2 x^{2}$ and $y=3 x^{4}+64-2 x^{2}$ at their intersection points.

Determine where the function $f(x)$ is increasing given its derivative $f^{\prime}(x)=(x+1)(x-2)^{2}(x-4)^{3}$ .

Determine if $f(x)=3 e^{x^{2}}$ is increasing or decreasing on $[0,1]$ and estimate the signed area using $L_{4}$ or $R_{4}$ .

Determine the intervals where the function $f(x)=\frac{x^{2}}{2}-2 x+(3+2 x) \ln x+c$ is concave up.

Find the limit: $\lim _{x \rightarrow 0^{+}}(f(x)+1)^{1 / \sin (2 x)}$ given $\lim _{x \rightarrow 0^{+}} f(x)=0$ and $\lim _{x \rightarrow 0^{+}} f^{\prime}(x)=2$ .

Given $f(4)=5$ and $f^{\prime}(4)=8$ , find $(f h)^{\prime}(4)$ where $h(x)=\sqrt{x}$ . Round to three decimals.

Find the integral of $x^{3} e^{2 x}$ with respect to $x$ .

Given values for functions $f$ , $g$ and their derivatives, find $(f g)^{\prime}(1)$ . Provide an integer answer.

Find the derivative of $f(x)=\left(\frac{x^{5}+4}{x^{2}-5}\right)^{\frac{1}{5}}$ using the chain rule.

Find the derivative of $f(x) = \frac{\sqrt[5]{x^2 - 3}}{-x - 5}$ using the chain rule.

Find the rate of change of the radius when the sphere's volume is 1585 m³ and decreasing at 4027 m³/min. Use $V=\frac{4}{3} \pi r^{3}$ . Round to three decimal places.

A cone's radius decreases at 1 m/min, volume at 61 m³/min. Find height change rate when radius is 2 m, volume 41 m³. Use $V=\frac{1}{3} \pi r^{2} h$ . Round to three decimals.

Find the derivative of $y=\sec(2x^{4})$ using the chain rule.

Find the inflection point(s) of $f(x)=5 x e^{-4 x}$ .

Find $(f h)'(4)$ given $f(4)=10$ , $f'(4)=9$ , and $h(x)=\sqrt{x}$ . Round your answer to three decimal places.

Calculate the integral $\int x^{3} \ln (x) \, dx$ .

Compute the value of $(f g)'(1)$ using the given values for $f(1)$ , $f'(1)$ , $g(1)$ , and $g'(1)$ .