Calculus

Problem 21001

A conical tank is 24 ft wide and 6 ft deep. Water flows in at 9 ft³/sec. Find the depth change rate when water is 2 ft deep. Rate = ft/sec.

See Solution

Problem 21002

Calculate the indefinite integral: $\int \frac{x^{2}}{x^{3}+1} d x$ (include absolute values and use $C$ for the constant).

See Solution

Problem 21003

Find the derivative of $y=\log _{3} e^{4 x}$ . What is $\frac{d y}{d x}=\square$ ?

See Solution

Problem 21004

Find the derivative of $y$ where $y=\log _{6} x+\log _{6} x^{4}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 21005

A conical tank is 6 ft wide and 9 ft deep. Water flows in at $5 \mathrm{ft}^{3}/\mathrm{sec}$ . Find the depth change rate when water is 3 ft deep. Rate of change = $\square \mathrm{ft}/\mathrm{sec}$ .

See Solution

Problem 21006

Compare US Personal Income equality in 2020. Shade area for Gini index using $G = 2 \times \text{Area}$ from $\int_0^1 (x - Y) \, dx$ .

See Solution

Problem 21007

Find the derivative of $y$ with respect to $t$ using logarithmic differentiation, where $y=(\sqrt[12]{t})^{t}$ . $\frac{\mathrm{dy}}{\mathrm{dt}}=\square$

See Solution

Problem 21008

Determine if the series $\sum_{n=2}^{\infty} \frac{n^{2}}{n^{3}+1}$ converges or diverges using the integral test. Evaluate $\int_{2}^{\infty} \frac{x^{2}}{x^{3}+1} dx$ .

See Solution

Problem 21009

Find the limit as $h$ approaches 0: $\lim _{h \rightarrow 0} \frac{\sqrt{13 h+1}-1}{h}$ .

See Solution

Problem 21010

Growth model: (a) How many fruit flies were initially in the bottle? (b) Find the number of fruit flies after 20 days. (c) What does $P(t)$ approach as $t \rightarrow \infty$ ?

See Solution

Problem 21011

Find the area change rate of a circle with radius $r$ at $r=7 \mathrm{~cm}$ .

See Solution

Problem 21012

Use the Integral Test to check if the series $\sum_{n=1}^{\infty} n^{-5}$ converges. Evaluate $\int_{1}^{\infty} x^{-5} dx$ .

See Solution

Problem 21013

Determine if the series $\sum_{n=1}^{\infty} n^{4} e^{-n^{5}}$ converges or diverges using the Integral Test. Evaluate $\int_{1}^{\infty} x^{4} e^{-x^{5}} d x$ .

See Solution

Problem 21014

Find the surface area when the curve $y=\frac{1}{3} x^{3/2}$ , for $0 \leq x \leq 12$ , is rotated about the $y$ -axis.

See Solution

Problem 21015

Find the derivative of the function $f(x)=2 x^{2}-x+2$ using $f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$ .

See Solution

Problem 21016

Is the series $\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{14}+\cdots$ convergent or divergent?

See Solution

Problem 21017

Calculate the integral: $\int \cos ^{3} 4 \theta \sin ^{-2} 4 \theta d \theta = \square$

See Solution

Problem 21018

Check if the Mean Value Theorem applies to $f(x)=6 x^{2}+11 x+11$ on $[9,12]$ . If yes, find $c$ values; else, enter DNE.

See Solution

Problem 21019

Evaluate if the sequence $a_{n}=\frac{2 n}{9 n+1}$ is convergent and if $\sum_{n=1}^{\infty} a_{n}$ is convergent.

See Solution

Problem 21020

Evaluate the integral: $\int \cos^{3} 4\theta \sin^{-2} 4\theta d\theta$ . What is the result?

See Solution

Problem 21021

Find the limit as $x$ approaches -1 from the right: $\lim _{x \rightarrow-1^{+}} \cos ^{-1} x = \square$ .

See Solution

Problem 21022

Is the series $\sum_{n=1}^{\infty} \frac{4}{n^{2}+n^{3}}$ convergent or divergent?

See Solution

Problem 21023

Find the limit: $\lim _{x \rightarrow \infty} \tan ^{-1} x = \square$ (Type the exact answer.)

See Solution

Problem 21024

Find the radius and interval of convergence for the series $\sum_{n=2}^{\infty}(-1)^{n} \frac{x^{n}}{6^{n} \ln (n)}$ .

See Solution

Problem 21025

Find the derivative of $s=\frac{6}{7 \pi} \sin (7 t)+\frac{6}{5 \pi} \cos (5 t)$ . What is $\frac{\mathrm{ds}}{\mathrm{dt}}=\square$ ?

See Solution

Problem 21026

Is the series $\sum_{k=1}^{\infty} k e^{-k^{2}}$ convergent or divergent?

See Solution

Problem 21027

Evaluate the integral: $\int_{\frac{\pi}{8}}^{\frac{\pi}{4}} 6 \sin ^{2} 4 x \cos ^{3} 4 x d x$ .

See Solution

Problem 21028

Calculate the Gini index using the Lorenz curve $Y=5.078x^5-9.479x^4+6.151x^3-1.058x^2+0.308x$ .

See Solution

Problem 21029

Is the series $\sum_{n=1}^{\infty} \frac{5}{\pi^{n}}$ convergent or divergent? If convergent, find the sum.

See Solution

Problem 21030

Evaluate the integral from $\frac{\pi}{8}$ to $\frac{\pi}{4}$ of $6 \sin^{2} 4x \cos^{3} 4x \, dx$ .

See Solution

Problem 21031

Find the derivative of $y$ where $y=6 x^{2} \arcsin(6 x^{2})+\sqrt{1-36 x^{4}}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 21032

Find the derivative of $y=\arccos(\sqrt{6} t)$ with respect to $t$ : $\frac{\mathrm{dy}}{\mathrm{dt}}=\square$ .

See Solution

Problem 21033

Find the radius and interval of convergence for the series $\sum_{n=0}^{\infty} \frac{(x-6)^{n}}{n^{3}+1}$ .

See Solution

Problem 21034

Differentiate $\tan y = x$ to find $\frac{d y}{d x}$ , leading to $\frac{d y}{d x}=\frac{1}{1+x^{2}}$ .

See Solution

Problem 21035

Calculate the integral of the function: $\int(6 x^{3}+7 x^{2}) \, dx$

See Solution

Problem 21036

Find the radius of convergence, $R$ , and interval, $I$ , for the series $\sum_{n=0}^{\infty}(-1)^{n} \frac{(x-8)^{n}}{6 n+1}$ .

See Solution

Problem 21037

Find the number of cars passing an intersection from 6 am to 9 am using $r(t)=400+600 t-180 t^{2}$ .

See Solution

Problem 21038

Calculate the integral from 0 to 1 of the function $-5.078 x^{5}+9.479 x^{4}-5.093 x^{2}-0.308 x+x$ .

See Solution

Problem 21039

Find the rate of change of the surface area of a cylinder with height $16 \text{ cm}$ when $r=5 \text{ cm}$ and $\frac{dr}{dt}=5 \text{ cm/s}$ .

See Solution

Problem 21040

Find the radius of convergence, $R$ , for the series $\sum_{n=1}^{\infty} \frac{n}{5^{n}}(x+2)^{n}$ .

See Solution

Problem 21041

Calculate the integral of $\left(\frac{6}{x^{4}}+7 x+3\right)$ with respect to $x$ .

See Solution

Problem 21042

A cube with edge length $x$ decreases at $5 \mathrm{~m/min}$ . Find the rates of change for surface area and volume when $x=3 \mathrm{~m}$ .

See Solution

Problem 21043

Given $s=\sqrt{2 x^{2}+6 y^{2}}$ , find $\frac{\mathrm{ds}}{\mathrm{dt}}$ in terms of $\frac{\mathrm{dx}}{\mathrm{dt}}$ when $y$ is constant. $\frac{\mathrm{ds}}{\mathrm{dt}}=\square$

See Solution

Problem 21044

Find $\frac{d y}{d t}$ when $y=x^{2}+4$ , $\frac{d x}{d t}=2$ , and $x=5$ . Simplify your answer.

See Solution

Problem 21045

A balloon inflates at $1200$ cm³/s. Find the radius increase rate when $r = 15$ cm. Use $V = \frac{4}{3} \pi r^3$ .

See Solution

Problem 21046

Given $s=\sqrt{2 x^{2}+6 y^{2}}$ , find $\frac{\mathrm{ds}}{\mathrm{dt}}$ for constant $y$ and for variable $x$ and $y$ .

See Solution

Problem 21047

Find the radius of convergence $R$ for the series $\sum_{n=1}^{\infty} \frac{n}{b^{n}}(x-a)^{n}, b>0$ . Then, determine the interval of convergence $I$ in interval notation.

See Solution

Problem 21048

Calculate the integral of $(x+3)(x-5)$ with respect to $x$ .

See Solution

Problem 21049

Evaluate the integral: $\int \sin^{2}\left(\theta-\frac{\pi}{15}\right) d\theta = \square$

See Solution

Problem 21050

Let $x$ and $y$ be functions of $t$ . Find $\frac{ds}{dt}$ for constant $y$ , both $x$ and $y$ , and relate $\frac{dx}{dt}$ to $\frac{dy}{dt}$ for constant $s$ .

See Solution

Problem 21051

Evaluate the integral: $\int \frac{x^{2}-11}{x^{3}-2 x^{2}+x} dx$ and find its partial fraction decomposition.

See Solution

Problem 21052

Find the derivative of $g(w)=\frac{w+4}{8 \sqrt{w}+1}$ . What is $g'(w)$ ?

See Solution

Problem 21053

A circle's radius increases at $3 \mathrm{~cm/s}$ . Find the area increase rate when the radius is $8 \mathrm{~cm}$ .
$\text { Rate }=\square \mathrm{cm}^{2} / \mathrm{sec}$

See Solution

Problem 21054

Find the radius of convergence, $R$ , for the series $\sum_{n=1}^{\infty} n !(6 x-1)^{n}$ . What is $R$ ?

See Solution

Problem 21055

Find the area increase rate of a heated circular plate with radius growing at $0.03 \mathrm{~cm/min}$ when $r = 50 \mathrm{~cm}$ .

See Solution

Problem 21056

A cube's edge grows at $5 \mathrm{~cm/s}$ . Find the volume's growth rate when the edge is $10 \mathrm{~cm}$ . Rate = $\square \mathrm{cm}^{3}/\mathrm{sec}$ .

See Solution

Problem 21057

Find the total cost for producing the first 64 units, given the marginal cost function $c(x)=\frac{11}{\sqrt{x}}$ .

See Solution

Problem 21058

Find the total cost function $M(x)$ for drilling an oil well with fixed cost \ $10,000 and marginal cost$ M^{\prime}(x)=1000+58 x $.$ $M(x)=\square$ $

See Solution

Problem 21059

Calculate the value of an annuity after 266 monthly payments of \$550 at an 8% annual interest rate compounded monthly.

See Solution

Problem 21060

a. Show that $\int f^{-1}(x) d x = \int y f^{\prime}(y) d y$ . b. Prove $\int f^{-1}(x) d x = y f(y) - \int f(y) d y$ using $\int x f^{\prime}(x) d x = x f(x) - \int f(x) d x$ . c. Find $\int \ln x d x$ in terms of $x$ . d. Find $\int \sin^{-1} x d x$ in terms of $x$ . e. Find $\int \tan^{-1} x d x$ in terms of $x$ .

See Solution

Problem 21061

Evaluate the integral: $\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} 8 \sqrt{\sec ^{2} \theta-1} d \theta$ . Type the exact answer.

See Solution

Problem 21062

Find $\frac{d r}{d \theta}$ using implicit differentiation for $\sin(r \theta^{3})=\frac{1}{2}$ .

See Solution

Problem 21063

Evaluate the integral: $\int_{-\frac{\pi}{2}}^{\frac{\pi}{3}} 8 \sqrt{\sec ^{2} \theta-1} d \theta$ .

See Solution

Problem 21064

Evaluate the integral: $\int_{-\frac{\pi}{2}}^{\frac{\pi}{3}} 8 \sqrt{\sec ^{2} \theta-1} d \theta = \square$ (Exact answer.)

See Solution

Problem 21065

Find the total words memorized in the first 10 minutes using $M^{\prime}(t)=-0.009 t^{2}+0.7 t$ . Round to the nearest word.

See Solution

Problem 21066

Find the mean value of $f(x)=\sin x$ on $[0, \pi]$ and the value of $c$ for the Mean Value Theorem.

See Solution

Problem 21067

Evaluate the integral: $\int \frac{6 x^{2}+x+7}{(x+1)(x^{2}+1)} d x$ . What is the result? $\int \frac{6 x^{2}+x+7}{(x+1)(x^{2}+1)} d x=\square$

See Solution

Problem 21068

Calculate the integral: $\int \sin^{3} x \cos x \, dx$

See Solution

Problem 21069

Calculate the integral from 0 to 7 of the function $x+\frac{1}{2}$ .

See Solution

Problem 21070

Calculate the curve length of $y=\ln (4 \sec x)$ from $x=0$ to $x=\frac{\pi}{3}$ using the appropriate integral.

See Solution

Problem 21071

A ball is thrown from $8.2 \mathrm{~m}$ with an initial speed of $58 \mathrm{~m/s}$ . Find $v(t)$ and $h(t)$ after $t$ seconds.

See Solution

Problem 21072

Evaluate the integral from 2 to 3: $\int_{2}^{3}-\frac{5}{x^{2}} d x$ .

See Solution

Problem 21073

Find the integral of $(3x)^{-3}$ multiplied by $x^x$ .

See Solution

Problem 21074

Evaluate the integral: $\int_{-1}^{0}\left(3 e^{3 t}+7 t\right) d t$

See Solution

Problem 21075

Evaluate the limit: $\lim _{x \rightarrow-19} \frac{x+19}{361-x^{2}}$ .

See Solution

Problem 21076

Evaluate the limit: $\lim _{x \rightarrow 7}\left(\frac{x^{2}}{7}+\frac{7}{x^{2}}\right)$ .

See Solution

Problem 21077

Find the maximum height and time for an object fired from a 200 ft tower with height $h(t)=-16 t^{2}+80 t+200$ .

See Solution

Problem 21078

Use substitution to simplify $\int(\sec^2 x) \ln(\tan x + 9) dx$ and find the integral's value.

See Solution

Problem 21079

Calculate the integral from 3 to 15 of $6 x^{-1}$ .

See Solution

Problem 21080

Approximate the area under $f(x)=\frac{1}{2} x^{2}$ from 1 to 3 using 10 left rectangles. Then find the exact area using integrals.

See Solution

Problem 21081

Find $f(5)-f(2)$ if $f^{\prime}(x)=2 x+6$ . $f(5)-f(2)=\square$

See Solution

Problem 21082

Find the distance a particle travels from the 2nd hour to the 5th hour given $v(t)=3 t^{2}+4 t$ . Calculate from $t=2$ to $t=5$ .

See Solution

Problem 21083

A balloon's radius increases at 5 ft/s. Find the volume's rate of change when the radius is 6 ft, using $v=\frac{4}{3}\pi r^3$ .

See Solution

Problem 21084

Using the Gompertz model with $d=A=1$ , show that as $N$ approaches 0, $R(N)=0$ . Complete the table for $N=0.1, 0.01, 0.001$ .

See Solution

Problem 21085

Determine the radius of convergence $R$ and the interval of convergence $I$ for the series $\sum_{n=1}^{\infty}(-1)^{n} \frac{n^{3} x^{n}}{2^{n}}$ .

See Solution

Problem 21086

Find the radius of convergence $R$ and the interval of convergence $I$ for the series $\sum_{n=1}^{\infty} \frac{x^{n+2}}{3n!}$ .

See Solution

Problem 21087

Evaluate the integral from 1 to 4: $\int_{1}^{4}(9 \sqrt{t}+4 t) d t$ .

See Solution

Problem 21088

Determine the area change rate of a circle as its radius $r$ changes, specifically when $r=2 \mathrm{~cm}$ .

See Solution

Problem 21089

Find the limit using graphs and tables: $\lim _{x \rightarrow 2^{-}} \frac{4}{x-2}$

See Solution

Problem 21090

Find the radius $R$ and interval $I$ of convergence for the series $\sum_{n=1}^{\infty} \frac{n}{5^{n}}(x+3)^{n}$ .

See Solution

Problem 21091

Calculate the area under the curve from 0 to 2 for the function given by $\sqrt{4-x^{2}}$ .

See Solution

Problem 21092

What is the best choice for $d v$ in $\int x^{n} e^{a x} d x$ using integration by parts?
A. $e^{a x}$ B. $x^{n} d x$ C. $x^{n}$ D. $e^{a x} d x$

See Solution

Problem 21093

Find the limit: $\lim _{t \rightarrow 13 e} \ln \left(\frac{13}{t}\right)$ .

See Solution

Problem 21094

Sketch the graph of $y=-5 \sin x$ for $0<x<2\pi$ and find its absolute extreme values. Options for extreme values are given.

See Solution

Problem 21095

Evaluate the integral from 1 to 2: $\int_{1}^{2}\left(\frac{6-7 x^{4}}{x^{7}}\right) d x$ .

See Solution

Problem 21096

Calculate the curve length of $y=\ln (2 \sec x)$ for $0 \leq x \leq \frac{\pi}{4}$ . The length is $\square$ .

See Solution

Problem 21097

Evaluate the integral: $\int_{0}^{\ln 6} \frac{e^{x}+e^{-x}}{9} d x$ .

See Solution

Problem 21098

Find the sky diver's velocity $v(t)=56(1-e^{-0.15t})$ after 3s and 5s. Round to the nearest whole number.

See Solution

Problem 21099

Compute $f(5)-f(2)$ given that $f^{\prime}(t)=0.4 t-e^{-5 t}$ .

See Solution

Problem 21100

Find the radius of convergence $R$ and interval $I$ for the series $\sum_{n=1}^{\infty} n !(6 x-1)^{n}$ .

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 21001

A conical tank is 24 ft wide and 6 ft deep. Water flows in at 9 ft³/sec. Find the depth change rate when water is 2 ft deep. Rate = ft/sec.

Problem 21002

Calculate the indefinite integral: ∫x2x3+1dx\int \frac{x^{2}}{x^{3}+1} d x∫x3+1x2​dx (include absolute values and use CCC for the constant).

Problem 21003

Find the derivative of y=log⁡3e4xy=\log _{3} e^{4 x}y=log3​e4x. What is dydx=□\frac{d y}{d x}=\squaredxdy​=□?

Problem 21004

Find the derivative of yyy where y=log⁡6x+log⁡6x4y=\log _{6} x+\log _{6} x^{4}y=log6​x+log6​x4. What is dydx\frac{d y}{d x}dxdy​?

Problem 21005

A conical tank is 6 ft wide and 9 ft deep. Water flows in at 5ft3/sec5 \mathrm{ft}^{3}/\mathrm{sec}5ft3/sec. Find the depth change rate when water is 3 ft deep. Rate of change = □ft/sec\square \mathrm{ft}/\mathrm{sec}□ft/sec.

Problem 21006

Compare US Personal Income equality in 2020. Shade area for Gini index using G=2×AreaG = 2 \times \text{Area}G=2×Area from ∫01(x−Y) dx\int_0^1 (x - Y) \, dx∫01​(x−Y)dx.

Problem 21007

Find the derivative of yyy with respect to ttt using logarithmic differentiation, where y=(t12)ty=(\sqrt[12]{t})^{t}y=(12t​)t. dydt=□\frac{\mathrm{dy}}{\mathrm{dt}}=\squaredtdy​=□

Problem 21008

Determine if the series ∑n=2∞n2n3+1\sum_{n=2}^{\infty} \frac{n^{2}}{n^{3}+1}∑n=2∞​n3+1n2​ converges or diverges using the integral test. Evaluate ∫2∞x2x3+1dx\int_{2}^{\infty} \frac{x^{2}}{x^{3}+1} dx∫2∞​x3+1x2​dx.

Problem 21009

Find the limit as hhh approaches 0: lim⁡h→013h+1−1h\lim _{h \rightarrow 0} \frac{\sqrt{13 h+1}-1}{h}limh→0​h13h+1​−1​.

Problem 21010

Growth model: (a) How many fruit flies were initially in the bottle? (b) Find the number of fruit flies after 20 days. (c) What does P(t)P(t)P(t) approach as t→∞t \rightarrow \inftyt→∞?

Problem 21011

Find the area change rate of a circle with radius rrr at r=7 cmr=7 \mathrm{~cm}r=7 cm.

Problem 21012

Use the Integral Test to check if the series ∑n=1∞n−5\sum_{n=1}^{\infty} n^{-5}∑n=1∞​n−5 converges. Evaluate ∫1∞x−5dx\int_{1}^{\infty} x^{-5} dx∫1∞​x−5dx.

Problem 21013

Determine if the series ∑n=1∞n4e−n5\sum_{n=1}^{\infty} n^{4} e^{-n^{5}}∑n=1∞​n4e−n5 converges or diverges using the Integral Test. Evaluate ∫1∞x4e−x5dx\int_{1}^{\infty} x^{4} e^{-x^{5}} d x∫1∞​x4e−x5dx.

Problem 21014

Find the surface area when the curve y=13x3/2y=\frac{1}{3} x^{3/2}y=31​x3/2, for 0≤x≤120 \leq x \leq 120≤x≤12, is rotated about the yyy-axis.

Problem 21015

Find the derivative of the function f(x)=2x2−x+2f(x)=2 x^{2}-x+2f(x)=2x2−x+2 using f′(x)=lim⁡z→xf(z)−f(x)z−xf^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}f′(x)=limz→x​z−xf(z)−f(x)​.

Problem 21016

Is the series 12+15+18+111+114+⋯\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{14}+\cdots21​+51​+81​+111​+141​+⋯ convergent or divergent?

Problem 21017

Calculate the integral: ∫cos⁡34θsin⁡−24θdθ=□\int \cos ^{3} 4 \theta \sin ^{-2} 4 \theta d \theta = \square∫cos34θsin−24θdθ=□

Problem 21018

Check if the Mean Value Theorem applies to f(x)=6x2+11x+11f(x)=6 x^{2}+11 x+11f(x)=6x2+11x+11 on [9,12][9,12][9,12]. If yes, find ccc values; else, enter DNE.

Problem 21019

Evaluate if the sequence an=2n9n+1a_{n}=\frac{2 n}{9 n+1}an​=9n+12n​ is convergent and if ∑n=1∞an\sum_{n=1}^{\infty} a_{n}∑n=1∞​an​ is convergent.

Problem 21020

Evaluate the integral: ∫cos⁡34θsin⁡−24θdθ\int \cos^{3} 4\theta \sin^{-2} 4\theta d\theta∫cos34θsin−24θdθ. What is the result?

Problem 21021

Find the limit as xxx approaches -1 from the right: lim⁡x→−1+cos⁡−1x=□\lim _{x \rightarrow-1^{+}} \cos ^{-1} x = \squarelimx→−1+​cos−1x=□.

Problem 21022

Is the series ∑n=1∞4n2+n3\sum_{n=1}^{\infty} \frac{4}{n^{2}+n^{3}}n=1∑∞​n2+n34​ convergent or divergent?

Problem 21023

Find the limit: lim⁡x→∞tan⁡−1x=□\lim _{x \rightarrow \infty} \tan ^{-1} x = \squarex→∞lim​tan−1x=□ (Type the exact answer.)

Problem 21024

Find the radius and interval of convergence for the series ∑n=2∞(−1)nxn6nln⁡(n)\sum_{n=2}^{\infty}(-1)^{n} \frac{x^{n}}{6^{n} \ln (n)}∑n=2∞​(−1)n6nln(n)xn​.

Problem 21025

Find the derivative of s=67πsin⁡(7t)+65πcos⁡(5t)s=\frac{6}{7 \pi} \sin (7 t)+\frac{6}{5 \pi} \cos (5 t)s=7π6​sin(7t)+5π6​cos(5t). What is dsdt=□\frac{\mathrm{ds}}{\mathrm{dt}}=\squaredtds​=□?

Problem 21026

Is the series ∑k=1∞ke−k2\sum_{k=1}^{\infty} k e^{-k^{2}}k=1∑∞​ke−k2 convergent or divergent?

Problem 21027

Evaluate the integral: ∫π8π46sin⁡24xcos⁡34xdx\int_{\frac{\pi}{8}}^{\frac{\pi}{4}} 6 \sin ^{2} 4 x \cos ^{3} 4 x d x∫8π​4π​​6sin24xcos34xdx.

Problem 21028

Calculate the Gini index using the Lorenz curve Y=5.078x5−9.479x4+6.151x3−1.058x2+0.308xY=5.078x^5-9.479x^4+6.151x^3-1.058x^2+0.308xY=5.078x5−9.479x4+6.151x3−1.058x2+0.308x.

Problem 21029

Is the series ∑n=1∞5πn\sum_{n=1}^{\infty} \frac{5}{\pi^{n}}n=1∑∞​πn5​ convergent or divergent? If convergent, find the sum.

Problem 21030

Evaluate the integral from π8\frac{\pi}{8}8π​ to π4\frac{\pi}{4}4π​ of 6sin⁡24xcos⁡34x dx6 \sin^{2} 4x \cos^{3} 4x \, dx6sin24xcos34xdx.

Problem 21031

Find the derivative of yyy where y=6x2arcsin⁡(6x2)+1−36x4y=6 x^{2} \arcsin(6 x^{2})+\sqrt{1-36 x^{4}}y=6x2arcsin(6x2)+1−36x4​. What is dydx\frac{d y}{d x}dxdy​?

Problem 21032

Find the derivative of y=arccos⁡(6t)y=\arccos(\sqrt{6} t)y=arccos(6​t) with respect to ttt: dydt=□\frac{\mathrm{dy}}{\mathrm{dt}}=\squaredtdy​=□.

Problem 21033

Find the radius and interval of convergence for the series ∑n=0∞(x−6)nn3+1 \sum_{n=0}^{\infty} \frac{(x-6)^{n}}{n^{3}+1} ∑n=0∞​n3+1(x−6)n​.

Problem 21034

Differentiate tan⁡y=x\tan y = xtany=x to find dydx\frac{d y}{d x}dxdy​, leading to dydx=11+x2\frac{d y}{d x}=\frac{1}{1+x^{2}}dxdy​=1+x21​.

Problem 21035

Calculate the integral of the function: ∫(6x3+7x2) dx\int(6 x^{3}+7 x^{2}) \, dx∫(6x3+7x2)dx

Problem 21036

Find the radius of convergence, RRR, and interval, III, for the series ∑n=0∞(−1)n(x−8)n6n+1\sum_{n=0}^{\infty}(-1)^{n} \frac{(x-8)^{n}}{6 n+1}∑n=0∞​(−1)n6n+1(x−8)n​.

Problem 21037

Find the number of cars passing an intersection from 6 am to 9 am using r(t)=400+600t−180t2r(t)=400+600 t-180 t^{2}r(t)=400+600t−180t2.

Problem 21038

Calculate the integral from 0 to 1 of the function −5.078x5+9.479x4−5.093x2−0.308x+x-5.078 x^{5}+9.479 x^{4}-5.093 x^{2}-0.308 x+x−5.078x5+9.479x4−5.093x2−0.308x+x.

Problem 21039

Find the rate of change of the surface area of a cylinder with height 16 cm16 \text{ cm}16 cm when r=5 cmr=5 \text{ cm}r=5 cm and drdt=5 cm/s\frac{dr}{dt}=5 \text{ cm/s}dtdr​=5 cm/s.

Problem 21040

Calculate the indefinite integral: $\int \frac{x^{2}}{x^{3}+1} d x$ (include absolute values and use $C$ for the constant).

Find the derivative of $y=\log _{3} e^{4 x}$ . What is $\frac{d y}{d x}=\square$ ?

Find the derivative of $y$ where $y=\log _{6} x+\log _{6} x^{4}$ . What is $\frac{d y}{d x}$ ?

A conical tank is 6 ft wide and 9 ft deep. Water flows in at $5 \mathrm{ft}^{3}/\mathrm{sec}$ . Find the depth change rate when water is 3 ft deep. Rate of change = $\square \mathrm{ft}/\mathrm{sec}$ .

Compare US Personal Income equality in 2020. Shade area for Gini index using $G = 2 \times \text{Area}$ from $\int_0^1 (x - Y) \, dx$ .

Find the derivative of $y$ with respect to $t$ using logarithmic differentiation, where $y=(\sqrt[12]{t})^{t}$ . $\frac{\mathrm{dy}}{\mathrm{dt}}=\square$

Determine if the series $\sum_{n=2}^{\infty} \frac{n^{2}}{n^{3}+1}$ converges or diverges using the integral test. Evaluate $\int_{2}^{\infty} \frac{x^{2}}{x^{3}+1} dx$ .

Find the limit as $h$ approaches 0: $\lim _{h \rightarrow 0} \frac{\sqrt{13 h+1}-1}{h}$ .

Growth model: (a) How many fruit flies were initially in the bottle? (b) Find the number of fruit flies after 20 days. (c) What does $P(t)$ approach as $t \rightarrow \infty$ ?

Find the area change rate of a circle with radius $r$ at $r=7 \mathrm{~cm}$ .

Use the Integral Test to check if the series $\sum_{n=1}^{\infty} n^{-5}$ converges. Evaluate $\int_{1}^{\infty} x^{-5} dx$ .

Determine if the series $\sum_{n=1}^{\infty} n^{4} e^{-n^{5}}$ converges or diverges using the Integral Test. Evaluate $\int_{1}^{\infty} x^{4} e^{-x^{5}} d x$ .

Find the surface area when the curve $y=\frac{1}{3} x^{3/2}$ , for $0 \leq x \leq 12$ , is rotated about the $y$ -axis.

Find the derivative of the function $f(x)=2 x^{2}-x+2$ using $f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$ .

Is the series $\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{14}+\cdots$ convergent or divergent?

Calculate the integral: $\int \cos ^{3} 4 \theta \sin ^{-2} 4 \theta d \theta = \square$

Check if the Mean Value Theorem applies to $f(x)=6 x^{2}+11 x+11$ on $[9,12]$ . If yes, find $c$ values; else, enter DNE.

Evaluate if the sequence $a_{n}=\frac{2 n}{9 n+1}$ is convergent and if $\sum_{n=1}^{\infty} a_{n}$ is convergent.

Evaluate the integral: $\int \cos^{3} 4\theta \sin^{-2} 4\theta d\theta$ . What is the result?

Find the limit as $x$ approaches -1 from the right: $\lim _{x \rightarrow-1^{+}} \cos ^{-1} x = \square$ .

Is the series $\sum_{n=1}^{\infty} \frac{4}{n^{2}+n^{3}}$ convergent or divergent?

Find the limit: $\lim _{x \rightarrow \infty} \tan ^{-1} x = \square$ (Type the exact answer.)

Find the radius and interval of convergence for the series $\sum_{n=2}^{\infty}(-1)^{n} \frac{x^{n}}{6^{n} \ln (n)}$ .

Find the derivative of $s=\frac{6}{7 \pi} \sin (7 t)+\frac{6}{5 \pi} \cos (5 t)$ . What is $\frac{\mathrm{ds}}{\mathrm{dt}}=\square$ ?

Is the series $\sum_{k=1}^{\infty} k e^{-k^{2}}$ convergent or divergent?

Evaluate the integral: $\int_{\frac{\pi}{8}}^{\frac{\pi}{4}} 6 \sin ^{2} 4 x \cos ^{3} 4 x d x$ .

Calculate the Gini index using the Lorenz curve $Y=5.078x^5-9.479x^4+6.151x^3-1.058x^2+0.308x$ .

Is the series $\sum_{n=1}^{\infty} \frac{5}{\pi^{n}}$ convergent or divergent? If convergent, find the sum.

Evaluate the integral from $\frac{\pi}{8}$ to $\frac{\pi}{4}$ of $6 \sin^{2} 4x \cos^{3} 4x \, dx$ .

Find the derivative of $y$ where $y=6 x^{2} \arcsin(6 x^{2})+\sqrt{1-36 x^{4}}$ . What is $\frac{d y}{d x}$ ?

Find the derivative of $y=\arccos(\sqrt{6} t)$ with respect to $t$ : $\frac{\mathrm{dy}}{\mathrm{dt}}=\square$ .

Find the radius and interval of convergence for the series $\sum_{n=0}^{\infty} \frac{(x-6)^{n}}{n^{3}+1}$ .

Differentiate $\tan y = x$ to find $\frac{d y}{d x}$ , leading to $\frac{d y}{d x}=\frac{1}{1+x^{2}}$ .

Calculate the integral of the function: $\int(6 x^{3}+7 x^{2}) \, dx$

Find the radius of convergence, $R$ , and interval, $I$ , for the series $\sum_{n=0}^{\infty}(-1)^{n} \frac{(x-8)^{n}}{6 n+1}$ .

Find the number of cars passing an intersection from 6 am to 9 am using $r(t)=400+600 t-180 t^{2}$ .

Calculate the integral from 0 to 1 of the function $-5.078 x^{5}+9.479 x^{4}-5.093 x^{2}-0.308 x+x$ .

Find the rate of change of the surface area of a cylinder with height $16 \text{ cm}$ when $r=5 \text{ cm}$ and $\frac{dr}{dt}=5 \text{ cm/s}$ .

Find the radius of convergence, $R$ , for the series $\sum_{n=1}^{\infty} \frac{n}{5^{n}}(x+2)^{n}$ .

Calculate the integral of $\left(\frac{6}{x^{4}}+7 x+3\right)$ with respect to $x$ .

A cube with edge length $x$ decreases at $5 \mathrm{~m/min}$ . Find the rates of change for surface area and volume when $x=3 \mathrm{~m}$ .

Given $s=\sqrt{2 x^{2}+6 y^{2}}$ , find $\frac{\mathrm{ds}}{\mathrm{dt}}$ in terms of $\frac{\mathrm{dx}}{\mathrm{dt}}$ when $y$ is constant. $\frac{\mathrm{ds}}{\mathrm{dt}}=\square$

Find $\frac{d y}{d t}$ when $y=x^{2}+4$ , $\frac{d x}{d t}=2$ , and $x=5$ . Simplify your answer.

A balloon inflates at $1200$ cm³/s. Find the radius increase rate when $r = 15$ cm. Use $V = \frac{4}{3} \pi r^3$ .

Given $s=\sqrt{2 x^{2}+6 y^{2}}$ , find $\frac{\mathrm{ds}}{\mathrm{dt}}$ for constant $y$ and for variable $x$ and $y$ .

Find the radius of convergence $R$ for the series $\sum_{n=1}^{\infty} \frac{n}{b^{n}}(x-a)^{n}, b>0$ . Then, determine the interval of convergence $I$ in interval notation.

Calculate the integral of $(x+3)(x-5)$ with respect to $x$ .

Evaluate the integral: $\int \sin^{2}\left(\theta-\frac{\pi}{15}\right) d\theta = \square$

Let $x$ and $y$ be functions of $t$ . Find $\frac{ds}{dt}$ for constant $y$ , both $x$ and $y$ , and relate $\frac{dx}{dt}$ to $\frac{dy}{dt}$ for constant $s$ .

Evaluate the integral: $\int \frac{x^{2}-11}{x^{3}-2 x^{2}+x} dx$ and find its partial fraction decomposition.

Find the derivative of $g(w)=\frac{w+4}{8 \sqrt{w}+1}$ . What is $g'(w)$ ?

A circle's radius increases at $3 \mathrm{~cm/s}$ . Find the area increase rate when the radius is $8 \mathrm{~cm}$ .
$\text { Rate }=\square \mathrm{cm}^{2} / \mathrm{sec}$

Find the radius of convergence, $R$ , for the series $\sum_{n=1}^{\infty} n !(6 x-1)^{n}$ . What is $R$ ?

Find the area increase rate of a heated circular plate with radius growing at $0.03 \mathrm{~cm/min}$ when $r = 50 \mathrm{~cm}$ .

A cube's edge grows at $5 \mathrm{~cm/s}$ . Find the volume's growth rate when the edge is $10 \mathrm{~cm}$ . Rate = $\square \mathrm{cm}^{3}/\mathrm{sec}$ .

Find the total cost for producing the first 64 units, given the marginal cost function $c(x)=\frac{11}{\sqrt{x}}$ .

Find the total cost function $M(x)$ for drilling an oil well with fixed cost \ $10,000 and marginal cost$ M^{\prime}(x)=1000+58 x $.$ $M(x)=\square$ $

Evaluate the integral: $\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} 8 \sqrt{\sec ^{2} \theta-1} d \theta$ . Type the exact answer.

Find $\frac{d r}{d \theta}$ using implicit differentiation for $\sin(r \theta^{3})=\frac{1}{2}$ .

Evaluate the integral: $\int_{-\frac{\pi}{2}}^{\frac{\pi}{3}} 8 \sqrt{\sec ^{2} \theta-1} d \theta$ .

Evaluate the integral: $\int_{-\frac{\pi}{2}}^{\frac{\pi}{3}} 8 \sqrt{\sec ^{2} \theta-1} d \theta = \square$ (Exact answer.)

Find the total words memorized in the first 10 minutes using $M^{\prime}(t)=-0.009 t^{2}+0.7 t$ . Round to the nearest word.

Find the mean value of $f(x)=\sin x$ on $[0, \pi]$ and the value of $c$ for the Mean Value Theorem.

Evaluate the integral: $\int \frac{6 x^{2}+x+7}{(x+1)(x^{2}+1)} d x$ . What is the result? $\int \frac{6 x^{2}+x+7}{(x+1)(x^{2}+1)} d x=\square$

Calculate the integral: $\int \sin^{3} x \cos x \, dx$

Calculate the integral from 0 to 7 of the function $x+\frac{1}{2}$ .

Calculate the curve length of $y=\ln (4 \sec x)$ from $x=0$ to $x=\frac{\pi}{3}$ using the appropriate integral.

A ball is thrown from $8.2 \mathrm{~m}$ with an initial speed of $58 \mathrm{~m/s}$ . Find $v(t)$ and $h(t)$ after $t$ seconds.

Evaluate the integral from 2 to 3: $\int_{2}^{3}-\frac{5}{x^{2}} d x$ .

Find the integral of $(3x)^{-3}$ multiplied by $x^x$ .

Evaluate the integral: $\int_{-1}^{0}\left(3 e^{3 t}+7 t\right) d t$