Calculus

Problem 31601

Find the derivative of $\frac{18 x+20}{x}$ . What is $\frac{d}{d x} \frac{18 x+20}{x}=\square$ ?

See Solution

Problem 31602

Evaluate the series: $\sum_{n=1}^{\infty}\left(\tan ^{-1}(n)-\tan ^{-1}(n+1)\right)$ .

See Solution

Problem 31603

Differentiate $\frac{9 x^{3}}{5}-\frac{3}{7 x^{3}}$ with respect to $x$ . Find the result: $\frac{d}{d x}\left(\frac{9 x^{3}}{5}-\frac{3}{7 x^{3}}\right)=\square$

See Solution

Problem 31604

Find the limits: a) $\lim _{x \rightarrow-\infty} \frac{x^{5}-x^{2}}{x^{3}-2 x}$ , b) $\lim _{x \rightarrow \infty} \frac{9 x^{2}-x+8}{2 x^{4}+x^{3}-7}$ .

See Solution

Problem 31605

Find the derivative $h^{\prime}(t)$ for the function $h(t)=14.1+0.9 t-2.6 t^{2}$ .

See Solution

Problem 31606

Compute $\lim _{t \rightarrow+\infty} \frac{P(t)}{P(t)+C(t)+J(t)}$ for $P(t)=1.745t+29.84$ , $C(t)=1.097t+10.65$ , $J(t)=1.915t+12.36$ . Interpret the result.

See Solution

Problem 31607

Find the derivative $y^{\prime}$ of the function $y=6 x^{-5}+3 x^{-1}$ . What is $y^{\prime}$ ?

See Solution

Problem 31608

Find the derivative $\frac{d y}{d x}$ for $y=\frac{1}{x^{6}}$ . What is $\frac{d y}{d x}=\square$ ?

See Solution

Problem 31609

Find the derivative of $f(x)=4 x^{2}+3 x+2$ using the limit definition: $f^{\prime}(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ . Simplify $f(x+h)-f(x)$ .

See Solution

Problem 31610

Find $f(x)=2 x^{4}-4 x^{2}+8$ : (A) $f^{\prime}(x)$ , (B) slope at $x=3$ , (C) tangent line at $x=3$ , (D) horizontal tangent points.

See Solution

Problem 31611

Find the derivative of the function: $\frac{9 x^{3}}{5} - \frac{3}{7 x^{3}}$ .

See Solution

Problem 31612

Find $\lim_{x \rightarrow+\infty} S(x)$ for $S(x)=573-33 e^{-0.0131 x}$ and interpret the result for high incomes.

See Solution

Problem 31613

Determine the marginal cost function from $C(x)=240+8.7 x-0.04 x^{2}$ . Find $C^{\prime}(x)$ .

See Solution

Problem 31614

Find the limits: 1) $\lim _{x \rightarrow-\infty}\left[12-\left(\frac{6}{5}\right)^{x}\right]$ and 2) $\lim _{x \rightarrow \infty} \frac{10 x^{2}}{\sqrt{4 x^{4}+1}}$ .

See Solution

Problem 31615

Given sales function $S(t)=0.02 t^{3}+0.6 t^{2}+5 t+4$ , find $S^{\prime}(t)$ , $S(3)$ , and interpret $S(8)$ and $S^{\prime}(8)$ .

See Solution

Problem 31616

Find $f(x)=2 x^{4}-4 x^{2}+8$ : (A) $f^{\prime}(x)$ , (B) slope at $x=3$ , (C) tangent line at $x=3$ , (D) horizontal tangents.

See Solution

Problem 31617

Find the limits as $x \to -\infty$ for two expressions and determine asymptotes and discontinuities for given functions.

See Solution

Problem 31618

Invest \ $15,000 at either$ 10\% $monthly or$ 9.87\%$ continuously. Which gives more in 5 years?

See Solution

Problem 31619

Find the marginal revenue function for $R(x)=x(14-0.09 x)$ . What is $R^{\prime}(x)$ ?

See Solution

Problem 31620

Find the arc length of the curve $y=\frac{x^{4}}{4}+\frac{1}{8 x^{2}}$ on $[1,3]$ by integrating with respect to $x$ . Length: $\square$ .

See Solution

Problem 31621

Find $\lim_{t \to +\infty} F(t)$ and $\lim_{t \to +\infty} \frac{F(t)}{Y(t)}$ for Facebook and YouTube models. Interpret results.

See Solution

Problem 31622

Given the cost function $C(x)=20,000+500x$ , find: A) average cost for 200 frames, B) marginal average cost at 200, C) estimate for 201 frames.

See Solution

Problem 31623

Find $\lim _{t \rightarrow+\infty}[p(t)-q(t)]$ where $p(t)$ and $q(t)$ are percentages of children's speech abilities. Interpret the result.

See Solution

Problem 31624

Calculate the limit as $t$ approaches infinity for $p(t)=100\left(1-\frac{13,200}{t^{4.5}}\right)$ . What does it imply?

See Solution

Problem 31625

Calculate the limits: $\lim _{t \rightarrow+\infty} I(t)$ and $\lim _{t \rightarrow+\infty} \frac{I(t)}{E(t)}$ for imports $I(t)$ and exports $E(t)$ . Interpret results.

See Solution

Problem 31626

Find the cost of producing the 61st food processor using $C(x)=1800+40x-0.1x^2$ . Exact and marginal cost needed.

See Solution

Problem 31627

Find the limit: $\lim _{x \rightarrow 4} \frac{\left[\frac{1}{x+1}\right]-\left(\frac{1}{5}\right)}{x-4}$ . What is the approximate value?

See Solution

Problem 31628

Estimate the limit as $x$ approaches 4 for $\frac{[1 /(x+1)]-(1 / 5)}{x-4}$ and round to four decimal places.

See Solution

Problem 31629

Water flows into a conical tank at $9 \mathrm{ft}^3/\min$ . With height $10 \mathrm{ft}$ and radius $5 \mathrm{ft}$ , find the rise rate when water is $6 \mathrm{ft}$ deep.

See Solution

Problem 31630

Water fills a conical tank at $9 \mathrm{ft}^{3}/\mathrm{min}$ . With height $10 \mathrm{ft}$ and radius $5 \mathrm{ft}$ , find the rise rate when water is $6 \mathrm{ft}$ deep.

See Solution

Problem 31631

Complete the table for $f(x) = \frac{6 \sin (x)}{x}$ as $x \to 0$ . Estimate the limit: $\lim _{x \rightarrow 0} \frac{6 \sin (x)}{x}$ .

See Solution

Problem 31632

Calculate the curve length for $x=\int_{0}^{y} \sqrt{2 \sec ^{4} t-1} d t$ from $y=-\frac{\pi}{4}$ to $y=\frac{\pi}{4}$ .

See Solution

Problem 31633

Find the limit as $x$ approaches $-\infty$ for $\frac{2x^{2}-1}{x+9}$ . If it doesn't exist, enter DNE.

See Solution

Problem 31634

Find the integral for the length of the curve $x=2 \sin y$ from $y=\frac{\pi}{10}$ to $y=\frac{9 \pi}{10}$ .

See Solution

Problem 31635

Find the slope of the tangent line for $f(x)=1-3x$ using the four-step process. Simplify your answers completely.

See Solution

Problem 31636

Find the limit: $\lim _{x \rightarrow-3}\left(\frac{x^{2}-9}{x+3}\right)$ .

See Solution

Problem 31637

Determine the marginal cost function for the cost function $C(x)=190+4.4 x-0.01 x^{2}$ .

See Solution

Problem 31638

Find the limit as $x$ approaches 0 for the expression $\frac{15 x^{2}-4 x}{x}$ .

See Solution

Problem 31639

Calculate the limit: $\lim _{x \rightarrow 1^{+}}\left(\frac{x+8}{x+1}\right)$ . If it doesn't exist, write DNE.

See Solution

Problem 31640

Calculate the limit as $x$ approaches 8 for the expression $(x^{2}+10)(x^{2}-64)$ .

See Solution

Problem 31641

Find the limit as $x$ approaches 0 from the left of $\frac{1}{x}$ .

See Solution

Problem 31642

Calculate the one-sided limit: $\lim _{x \rightarrow 1^{-}}(3 x-6)$ .

See Solution

Problem 31643

Find the limits: $\lim _{x \rightarrow \infty} (x^3-x)$ and $\lim _{x \rightarrow -\infty} (x^3-x)$ .

See Solution

Problem 31644

Calculate the limit as $t$ approaches 6 for the expression $10 t^{2}-8 t+1$ .

See Solution

Problem 31645

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(\frac{8 x^{2}+6 x+14}{x^{4}-x^{2}}\right)$ .

See Solution

Problem 31646

Find the limit as $x$ approaches -172.5 for $\sqrt[3]{2x + 2}$ .

See Solution

Problem 31647

Calculate the limit: $\lim _{x \rightarrow 4^{+}}(4 \sqrt{x-4})$ .

See Solution

Problem 31648

Find the one-sided limit as $x$ approaches 0 from the left: $\lim _{x \rightarrow 0^{-}}\left(\frac{1}{x}\right)$ .

See Solution

Problem 31649

Find the limit as x approaches 11: $\lim _{x \rightarrow 11}\left(\frac{x+11}{x-11}\right)$ .

See Solution

Problem 31650

Evaluate the telescoping series: $\sum_{n=1}^{\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)$ .

See Solution

Problem 31651

Set up an integral for the curve length of $y^{2}+3y=x+1$ from $(-3,-1)$ to $(17,3)$ . Find $L=\int \square d y$ .

See Solution

Problem 31652

Find the sum of the series $\sum_{n=2}^{\infty} \frac{6}{n(n+3)}$ .

See Solution

Problem 31653

Show that the series converges using the Limit Comparison Test with $\frac{1}{n^{1/2}}$ and evaluate the limit: $\lim_{n \to \infty} \frac{\frac{1}{n^{3/2}}}{\frac{1}{\sqrt{n^3}+2}}$

See Solution

Problem 31654

Find the value of the series $\sum_{n=1}^{\infty} \frac{-1-(-2)^{n}}{3^{n}}$ . Choose a, b, c, or d.

See Solution

Problem 31655

Find the sum of the series $\sum_{n=1}^{\infty}\left(\frac{9}{n^{2}}-\frac{9}{(n+1)^{2}}\right)$ .

See Solution

Problem 31656

Determine the convergence of the series $\sum_{n=0}^{\infty} \frac{(-5)^{n+1}}{3^{n}}$ and find its sum.

See Solution

Problem 31657

Find the limit as $x$ approaches 1 for $\frac{x^{4}-1}{x^{6}-1}$ . Create a value table and estimate the limit.

See Solution

Problem 31658

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} dt$ from $x=0$ to $x=\frac{\pi}{4}$ . Set up the integral.

See Solution

Problem 31659

Determine if the series $\sum_{n=1}^{\infty} \frac{3^{n}-2^{n}}{5^{n}}$ converges or diverges, and find its sum.

See Solution

Problem 31660

Determine the convergence of the series $\sum_{n=2}^{\infty} \frac{6}{n(n+3)}$ : a. $\frac{13}{6}$ b. $\frac{5}{3}$ c. diverges d. $\frac{11}{3}$ e. $\frac{13}{2}$ .

See Solution

Problem 31661

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{4}-1}{x^{6}-1}$ . Use a table and graph to estimate it.

See Solution

Problem 31662

Evaluate the integral: $\int \sin^{2} \beta \cos^{2} \beta \, d\beta$

See Solution

Problem 31663

Find the limit as $x$ approaches 1 for $\frac{x^{4}-1}{x^{6}-1}$ and create a value table rounded to four decimals.

See Solution

Problem 31664

Find the derivative of $f(x)=3 x^{8}-3 x^{5}-2 x^{3}+2 x$ and then calculate $f^{\prime}(5)$ .

See Solution

Problem 31665

Find the derivative of the function $f(x)=4 x^{2}-7 x-5$ , i.e., calculate $f^{\prime}(x)$ .

See Solution

Problem 31666

Find the limit as $x$ approaches 1 for $\frac{x^{4}-1}{x^{6}-1}$ . Estimate using a table and graph.

See Solution

Problem 31667

Find the derivative of the function $f(t) = 9 \sin t - 9 \pi \cos t$ , i.e., calculate $f^{\prime}(t)$ .

See Solution

Problem 31668

Complete the table and estimate the limit:
$\lim _{x \rightarrow 4} \frac{[1 /(x+1)]-(1 / 5)}{x-4}$
Estimate: $\approx -0.04$ . Use a graphing tool to confirm.

See Solution

Problem 31669

Set up an integral for the surface area of the curve $x^{1/2} + y^{1/2} = 9$ revolved around the x-axis.

See Solution

Problem 31670

Find the derivative using logarithmic differentiation for $f(x)=\left(\frac{1+2 x^{2}}{9-4 x}\right)^{4}$ .

See Solution

Problem 31671

A colony of 90 bacteria grows at $3.5\%$ per hour. How long until the population doubles?

See Solution

Problem 31672

Find the derivative $f'(t)$ of the function $f(t)=\frac{\sqrt{4}}{t^{7}}$ and evaluate $f'(2)$ .

See Solution

Problem 31673

Complete the table to find $\lim _{x \rightarrow 0} \frac{6 \sin (x)}{x}$ and estimate the limit. Round to five decimal places.

See Solution

Problem 31674

Find limits for the function $f(x)=\begin{cases} 2x+1, & x \leq -1 \\ 3x, & -1<x<0 \\ \sqrt{x}, & x \geq 0 \end{cases}$ for: a. $\lim_{x \to -3} f(x)$ , b. $\lim_{x \to -1^+} f(x)$ , c. $\lim_{x \to -1^-} f(x)$ , d. $\lim_{x \to -1} f(x)$ , e. $\lim_{x \to 0} f(x)$ , f. $\lim_{x \to 2} f(x)$ .

See Solution

Problem 31675

Find the derivative $f'(x)$ if $f(x) = 4 + \frac{4}{x} + \frac{4}{x^2}$ .

See Solution

Problem 31676

Find $\frac{d y}{d x}$ using logarithmic differentiation for $y = x^{\log _{5}(x)}$ .

See Solution

Problem 31677

Find the derivative of the function $f(x)=(x-9)(7x+9)$ , i.e., compute $f^{\prime}(x)$ .

See Solution

Problem 31678

Find the derivative of the function $f(t)=4 t^{-\frac{3}{4}}$ . What is $f^{\prime}(t)$ ? $f^{\prime}(t)=$

See Solution

Problem 31679

Complete the table and estimate the limit: $\lim _{x \rightarrow 0} \frac{6 \sin (x)}{x} \approx 6$ . Round answers to five decimal places.

See Solution

Problem 31680

Find the derivative of the function $f(t)=4 t^{-\frac{3}{4}}$ . What is $f^{\prime}(t)$ ?

See Solution

Problem 31681

Find the limits of the piecewise function:
$f(x)=\left\{\begin{array}{lr} 2 x+1, & x \leq-1 \\ 3 x, & -1<x<0 \\ \sqrt{x}, & x \geq 0 \end{array}\right.$
a. $\lim _{x \rightarrow-3} f(x)$ , b. $\lim _{x \rightarrow-1^{+}} f(x)$ , c. $\lim _{x \rightarrow-1^{-}} f(x)$ , d. $\lim _{x \rightarrow-1} f(x)$ , e. $\lim _{x \rightarrow 0} f(x)$ , f. $\lim _{x \rightarrow 2} f(x)$ .

See Solution

Problem 31682

Differentiate the function $V(r)=\frac{4}{3} \pi r^{3}$ .

See Solution

Problem 31683

Find the marginal cost function for $C(x)=210+5.1 x-0.01 x^{2}$ . What is $C^{\prime}(x)$ ?

See Solution

Problem 31684

Determine the marginal revenue function from $R(x)=x(21-0.07 x)$ . Find $R^{\prime}(x)$ .

See Solution

Problem 31685

Find the derivative $f^{\prime}(t)$ for the function $f(t)=-4 t^{2}-6 t+7$ .

See Solution

Problem 31686

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{1}{x^{9}}$ .

See Solution

Problem 31687

Find the derivative of the function $f(x)=8 \sqrt{x}(x-2)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 31688

Differentiate $\frac{0.2}{\sqrt{x}}-4.7 x^{-2}+4 x$ with respect to $x$ .

See Solution

Problem 31689

Find the derivative of the function $\frac{18 x+39}{x}$ . What is $\frac{d}{d x} \frac{18 x+39}{x}=\square$ ?

See Solution

Problem 31690

Find the derivative $f^{\prime}(x)$ for the function $f(x)=7 x^{5}$ .

See Solution

Problem 31691

Find $f^{\prime}(0)$ for $f(x)=5 \tan (x)-\tan (3 x)$ and $g^{\prime}\left(\frac{\pi}{4}\right)$ for $g(x)=\csc (x)$ .

See Solution

Problem 31692

Find the relative minimum point of $f(x)=\frac{x^{2}-2}{x^{2}-1}$ using $f'(x)$ and $f''(x)$ .

See Solution

Problem 31693

Find the derivative $r^{\prime}(x)$ for $r(x)=6-4x^{2}$ and calculate $r^{\prime}(1)$ , $r^{\prime}(2)$ , $r^{\prime}(3)$ .

See Solution

Problem 31694

Find the inflection point using $f(x)=\frac{x^{2}-3}{x-2}$ , $f^{\prime}(x)=\frac{x^{2}-4x+3}{(x-2)^{2}}$ , and $f^{\prime \prime}(x)=\frac{2}{(x-2)^{3}}$ .

See Solution

Problem 31695

Given $f(x)=2 x^{3}$ on $[1,2]$ , find the Riemann sum $R_{N}$ , simplify it, and compute the area as $N \to \infty$ .

See Solution

Problem 31696

Find the derivative $f^{\prime}(x)$ for $f(x)=2x^{2}+3x-2$ and calculate $f^{\prime}(-2)$ , $f^{\prime}(6)$ , $f^{\prime}(8)$ .

See Solution

Problem 31697

Find the marginal revenue function for $R(x)=6 x-0.08 x^{2}$ , and calculate $R^{\prime}(x)=\square$ .

See Solution

Problem 31698

For the function $y=f(x)=7-x^{2}$ , find: (A) $\frac{f(-1)-f(-2)}{1}$ , (B) $\frac{f(-2+h)-f(-2)}{h}$ , (C) $\lim _{h \rightarrow 0} \frac{f(-2+h)-f(-2)}{h}$ .

See Solution

Problem 31699

The cost to make $x$ frames is $C(x)=90,000+500 x$ . Find: (A) avg cost for 50 frames, (B) marginal avg cost at 50, (C) estimate avg cost for 51 frames.

See Solution

Problem 31700

Calculate the value of $V=\left.x\left(-\frac{14 y^{3}}{3}+35 y\right)\right|_{-\sqrt{5} / \sqrt{2}} ^{\sqrt{5} / \sqrt{2}}$ .

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 31601

Find the derivative of 18x+20x\frac{18 x+20}{x}x18x+20​. What is ddx18x+20x=□\frac{d}{d x} \frac{18 x+20}{x}=\squaredxd​x18x+20​=□?

Problem 31602

Evaluate the series: ∑n=1∞(tan⁡−1(n)−tan⁡−1(n+1))\sum_{n=1}^{\infty}\left(\tan ^{-1}(n)-\tan ^{-1}(n+1)\right)∑n=1∞​(tan−1(n)−tan−1(n+1)).

Problem 31603

Differentiate 9x35−37x3\frac{9 x^{3}}{5}-\frac{3}{7 x^{3}}59x3​−7x33​ with respect to xxx. Find the result: ddx(9x35−37x3)=□\frac{d}{d x}\left(\frac{9 x^{3}}{5}-\frac{3}{7 x^{3}}\right)=\squaredxd​(59x3​−7x33​)=□

Problem 31604

Find the limits: a) lim⁡x→−∞x5−x2x3−2x\lim _{x \rightarrow-\infty} \frac{x^{5}-x^{2}}{x^{3}-2 x}limx→−∞​x3−2xx5−x2​, b) lim⁡x→∞9x2−x+82x4+x3−7\lim _{x \rightarrow \infty} \frac{9 x^{2}-x+8}{2 x^{4}+x^{3}-7}limx→∞​2x4+x3−79x2−x+8​.

Problem 31605

Find the derivative h′(t)h^{\prime}(t)h′(t) for the function h(t)=14.1+0.9t−2.6t2h(t)=14.1+0.9 t-2.6 t^{2}h(t)=14.1+0.9t−2.6t2.

Problem 31606

Problem 31607

Find the derivative y′y^{\prime}y′ of the function y=6x−5+3x−1y=6 x^{-5}+3 x^{-1}y=6x−5+3x−1. What is y′y^{\prime}y′?

Problem 31608

Find the derivative dydx\frac{d y}{d x}dxdy​ for y=1x6y=\frac{1}{x^{6}}y=x61​. What is dydx=□\frac{d y}{d x}=\squaredxdy​=□?

Problem 31609

Find the derivative of f(x)=4x2+3x+2f(x)=4 x^{2}+3 x+2f(x)=4x2+3x+2 using the limit definition: f′(x)=lim⁡h→0f(x+h)−f(x)hf^{\prime}(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}f′(x)=limh→0​hf(x+h)−f(x)​. Simplify f(x+h)−f(x)f(x+h)-f(x)f(x+h)−f(x).

Problem 31610

Find f(x)=2x4−4x2+8f(x)=2 x^{4}-4 x^{2}+8f(x)=2x4−4x2+8: (A) f′(x)f^{\prime}(x)f′(x), (B) slope at x=3x=3x=3, (C) tangent line at x=3x=3x=3, (D) horizontal tangent points.

Problem 31611

Find the derivative of the function: 9x35−37x3\frac{9 x^{3}}{5} - \frac{3}{7 x^{3}}59x3​−7x33​.

Problem 31612

Find lim⁡x→+∞S(x)\lim_{x \rightarrow+\infty} S(x)limx→+∞​S(x) for S(x)=573−33e−0.0131xS(x)=573-33 e^{-0.0131 x}S(x)=573−33e−0.0131x and interpret the result for high incomes.

Problem 31613

Determine the marginal cost function from C(x)=240+8.7x−0.04x2C(x)=240+8.7 x-0.04 x^{2}C(x)=240+8.7x−0.04x2. Find C′(x)C^{\prime}(x)C′(x).

Problem 31614

Find the limits: 1) lim⁡x→−∞[12−(65)x]\lim _{x \rightarrow-\infty}\left[12-\left(\frac{6}{5}\right)^{x}\right]limx→−∞​[12−(56​)x] and 2) lim⁡x→∞10x24x4+1\lim _{x \rightarrow \infty} \frac{10 x^{2}}{\sqrt{4 x^{4}+1}}limx→∞​4x4+1​10x2​.

Problem 31615

Given sales function S(t)=0.02t3+0.6t2+5t+4S(t)=0.02 t^{3}+0.6 t^{2}+5 t+4S(t)=0.02t3+0.6t2+5t+4, find S′(t)S^{\prime}(t)S′(t), S(3)S(3)S(3), and interpret S(8)S(8)S(8) and S′(8)S^{\prime}(8)S′(8).

Problem 31616

Find f(x)=2x4−4x2+8f(x)=2 x^{4}-4 x^{2}+8f(x)=2x4−4x2+8: (A) f′(x)f^{\prime}(x)f′(x), (B) slope at x=3x=3x=3, (C) tangent line at x=3x=3x=3, (D) horizontal tangents.

Problem 31617

Find the limits as x→−∞x \to -\inftyx→−∞ for two expressions and determine asymptotes and discontinuities for given functions.

Problem 31618

Invest \15,000ateither15,000 at either 15,000ateither10\%monthlyor monthly or monthlyor9.87\%$ continuously. Which gives more in 5 years?

Problem 31619

Find the marginal revenue function for R(x)=x(14−0.09x)R(x)=x(14-0.09 x)R(x)=x(14−0.09x). What is R′(x)R^{\prime}(x)R′(x)?

Problem 31620

Find the arc length of the curve y=x44+18x2y=\frac{x^{4}}{4}+\frac{1}{8 x^{2}}y=4x4​+8x21​ on [1,3][1,3][1,3] by integrating with respect to xxx. Length: □\square□.

Problem 31621

Find lim⁡t→+∞F(t)\lim_{t \to +\infty} F(t)limt→+∞​F(t) and lim⁡t→+∞F(t)Y(t)\lim_{t \to +\infty} \frac{F(t)}{Y(t)}limt→+∞​Y(t)F(t)​ for Facebook and YouTube models. Interpret results.

Problem 31622

Given the cost function C(x)=20,000+500xC(x)=20,000+500xC(x)=20,000+500x, find: A) average cost for 200 frames, B) marginal average cost at 200, C) estimate for 201 frames.

Problem 31623

Find lim⁡t→+∞[p(t)−q(t)]\lim _{t \rightarrow+\infty}[p(t)-q(t)]limt→+∞​[p(t)−q(t)] where p(t)p(t)p(t) and q(t)q(t)q(t) are percentages of children's speech abilities. Interpret the result.

Problem 31624

Calculate the limit as ttt approaches infinity for p(t)=100(1−13,200t4.5)p(t)=100\left(1-\frac{13,200}{t^{4.5}}\right)p(t)=100(1−t4.513,200​). What does it imply?

Problem 31625

Calculate the limits: lim⁡t→+∞I(t)\lim _{t \rightarrow+\infty} I(t)t→+∞lim​I(t) and lim⁡t→+∞I(t)E(t)\lim _{t \rightarrow+\infty} \frac{I(t)}{E(t)}t→+∞lim​E(t)I(t)​ for imports I(t)I(t)I(t) and exports E(t)E(t)E(t). Interpret results.

Problem 31626

Find the cost of producing the 61st food processor using C(x)=1800+40x−0.1x2C(x)=1800+40x-0.1x^2C(x)=1800+40x−0.1x2. Exact and marginal cost needed.

Problem 31627

Find the limit: lim⁡x→4[1x+1]−(15)x−4\lim _{x \rightarrow 4} \frac{\left[\frac{1}{x+1}\right]-\left(\frac{1}{5}\right)}{x-4}x→4lim​x−4[x+11​]−(51​)​. What is the approximate value?

Problem 31628

Estimate the limit as xxx approaches 4 for [1/(x+1)]−(1/5)x−4\frac{[1 /(x+1)]-(1 / 5)}{x-4}x−4[1/(x+1)]−(1/5)​ and round to four decimal places.

Problem 31629

Water flows into a conical tank at 9ft3/min⁡9 \mathrm{ft}^3/\min9ft3/min. With height 10ft10 \mathrm{ft}10ft and radius 5ft5 \mathrm{ft}5ft, find the rise rate when water is 6ft6 \mathrm{ft}6ft deep.

Problem 31630

Water fills a conical tank at 9ft3/min9 \mathrm{ft}^{3}/\mathrm{min}9ft3/min. With height 10ft10 \mathrm{ft}10ft and radius 5ft5 \mathrm{ft}5ft, find the rise rate when water is 6ft6 \mathrm{ft}6ft deep.

Problem 31631

Complete the table for f(x)=6sin⁡(x)xf(x) = \frac{6 \sin (x)}{x}f(x)=x6sin(x)​ as x→0x \to 0x→0. Estimate the limit: lim⁡x→06sin⁡(x)x\lim _{x \rightarrow 0} \frac{6 \sin (x)}{x}x→0lim​x6sin(x)​.

Problem 31632

Calculate the curve length for x=∫0y2sec⁡4t−1dtx=\int_{0}^{y} \sqrt{2 \sec ^{4} t-1} d tx=∫0y​2sec4t−1​dt from y=−π4y=-\frac{\pi}{4}y=−4π​ to y=π4y=\frac{\pi}{4}y=4π​.

Problem 31633

Find the limit as xxx approaches −∞-\infty−∞ for 2x2−1x+9\frac{2x^{2}-1}{x+9}x+92x2−1​. If it doesn't exist, enter DNE.

Problem 31634

Find the integral for the length of the curve x=2sin⁡yx=2 \sin yx=2siny from y=π10y=\frac{\pi}{10}y=10π​ to y=9π10y=\frac{9 \pi}{10}y=109π​.

Problem 31635

Find the slope of the tangent line for f(x)=1−3xf(x)=1-3xf(x)=1−3x using the four-step process. Simplify your answers completely.

Problem 31636

Find the limit: lim⁡x→−3(x2−9x+3)\lim _{x \rightarrow-3}\left(\frac{x^{2}-9}{x+3}\right)limx→−3​(x+3x2−9​).

Problem 31637

Determine the marginal cost function for the cost function C(x)=190+4.4x−0.01x2C(x)=190+4.4 x-0.01 x^{2}C(x)=190+4.4x−0.01x2.

Problem 31638

Find the limit as xxx approaches 0 for the expression 15x2−4xx\frac{15 x^{2}-4 x}{x}x15x2−4x​.

Problem 31639

Calculate the limit: lim⁡x→1+(x+8x+1)\lim _{x \rightarrow 1^{+}}\left(\frac{x+8}{x+1}\right)x→1+lim​(x+1x+8​). If it doesn't exist, write DNE.

Problem 31640

Calculate the limit as xxx approaches 8 for the expression (x2+10)(x2−64)(x^{2}+10)(x^{2}-64)(x2+10)(x2−64).

Find the derivative of $\frac{18 x+20}{x}$ . What is $\frac{d}{d x} \frac{18 x+20}{x}=\square$ ?

Evaluate the series: $\sum_{n=1}^{\infty}\left(\tan ^{-1}(n)-\tan ^{-1}(n+1)\right)$ .

Differentiate $\frac{9 x^{3}}{5}-\frac{3}{7 x^{3}}$ with respect to $x$ . Find the result: $\frac{d}{d x}\left(\frac{9 x^{3}}{5}-\frac{3}{7 x^{3}}\right)=\square$

Find the limits: a) $\lim _{x \rightarrow-\infty} \frac{x^{5}-x^{2}}{x^{3}-2 x}$ , b) $\lim _{x \rightarrow \infty} \frac{9 x^{2}-x+8}{2 x^{4}+x^{3}-7}$ .

Find the derivative $h^{\prime}(t)$ for the function $h(t)=14.1+0.9 t-2.6 t^{2}$ .

Find the derivative $y^{\prime}$ of the function $y=6 x^{-5}+3 x^{-1}$ . What is $y^{\prime}$ ?

Find the derivative $\frac{d y}{d x}$ for $y=\frac{1}{x^{6}}$ . What is $\frac{d y}{d x}=\square$ ?

Find the derivative of $f(x)=4 x^{2}+3 x+2$ using the limit definition: $f^{\prime}(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ . Simplify $f(x+h)-f(x)$ .

Find $f(x)=2 x^{4}-4 x^{2}+8$ : (A) $f^{\prime}(x)$ , (B) slope at $x=3$ , (C) tangent line at $x=3$ , (D) horizontal tangent points.

Find the derivative of the function: $\frac{9 x^{3}}{5} - \frac{3}{7 x^{3}}$ .

Find $\lim_{x \rightarrow+\infty} S(x)$ for $S(x)=573-33 e^{-0.0131 x}$ and interpret the result for high incomes.

Determine the marginal cost function from $C(x)=240+8.7 x-0.04 x^{2}$ . Find $C^{\prime}(x)$ .

Find the limits: 1) $\lim _{x \rightarrow-\infty}\left[12-\left(\frac{6}{5}\right)^{x}\right]$ and 2) $\lim _{x \rightarrow \infty} \frac{10 x^{2}}{\sqrt{4 x^{4}+1}}$ .

Given sales function $S(t)=0.02 t^{3}+0.6 t^{2}+5 t+4$ , find $S^{\prime}(t)$ , $S(3)$ , and interpret $S(8)$ and $S^{\prime}(8)$ .

Find $f(x)=2 x^{4}-4 x^{2}+8$ : (A) $f^{\prime}(x)$ , (B) slope at $x=3$ , (C) tangent line at $x=3$ , (D) horizontal tangents.

Find the limits as $x \to -\infty$ for two expressions and determine asymptotes and discontinuities for given functions.

Invest \ $15,000 at either$ 10\% $monthly or$ 9.87\%$ continuously. Which gives more in 5 years?

Find the marginal revenue function for $R(x)=x(14-0.09 x)$ . What is $R^{\prime}(x)$ ?

Find the arc length of the curve $y=\frac{x^{4}}{4}+\frac{1}{8 x^{2}}$ on $[1,3]$ by integrating with respect to $x$ . Length: $\square$ .

Find $\lim_{t \to +\infty} F(t)$ and $\lim_{t \to +\infty} \frac{F(t)}{Y(t)}$ for Facebook and YouTube models. Interpret results.

Given the cost function $C(x)=20,000+500x$ , find: A) average cost for 200 frames, B) marginal average cost at 200, C) estimate for 201 frames.

Find $\lim _{t \rightarrow+\infty}[p(t)-q(t)]$ where $p(t)$ and $q(t)$ are percentages of children's speech abilities. Interpret the result.

Calculate the limit as $t$ approaches infinity for $p(t)=100\left(1-\frac{13,200}{t^{4.5}}\right)$ . What does it imply?

Calculate the limits: $\lim _{t \rightarrow+\infty} I(t)$ and $\lim _{t \rightarrow+\infty} \frac{I(t)}{E(t)}$ for imports $I(t)$ and exports $E(t)$ . Interpret results.

Find the cost of producing the 61st food processor using $C(x)=1800+40x-0.1x^2$ . Exact and marginal cost needed.

Find the limit: $\lim _{x \rightarrow 4} \frac{\left[\frac{1}{x+1}\right]-\left(\frac{1}{5}\right)}{x-4}$ . What is the approximate value?

Estimate the limit as $x$ approaches 4 for $\frac{[1 /(x+1)]-(1 / 5)}{x-4}$ and round to four decimal places.

Water flows into a conical tank at $9 \mathrm{ft}^3/\min$ . With height $10 \mathrm{ft}$ and radius $5 \mathrm{ft}$ , find the rise rate when water is $6 \mathrm{ft}$ deep.

Water fills a conical tank at $9 \mathrm{ft}^{3}/\mathrm{min}$ . With height $10 \mathrm{ft}$ and radius $5 \mathrm{ft}$ , find the rise rate when water is $6 \mathrm{ft}$ deep.

Complete the table for $f(x) = \frac{6 \sin (x)}{x}$ as $x \to 0$ . Estimate the limit: $\lim _{x \rightarrow 0} \frac{6 \sin (x)}{x}$ .

Calculate the curve length for $x=\int_{0}^{y} \sqrt{2 \sec ^{4} t-1} d t$ from $y=-\frac{\pi}{4}$ to $y=\frac{\pi}{4}$ .

Find the limit as $x$ approaches $-\infty$ for $\frac{2x^{2}-1}{x+9}$ . If it doesn't exist, enter DNE.

Find the integral for the length of the curve $x=2 \sin y$ from $y=\frac{\pi}{10}$ to $y=\frac{9 \pi}{10}$ .

Find the slope of the tangent line for $f(x)=1-3x$ using the four-step process. Simplify your answers completely.

Find the limit: $\lim _{x \rightarrow-3}\left(\frac{x^{2}-9}{x+3}\right)$ .

Determine the marginal cost function for the cost function $C(x)=190+4.4 x-0.01 x^{2}$ .

Find the limit as $x$ approaches 0 for the expression $\frac{15 x^{2}-4 x}{x}$ .

Calculate the limit: $\lim _{x \rightarrow 1^{+}}\left(\frac{x+8}{x+1}\right)$ . If it doesn't exist, write DNE.

Calculate the limit as $x$ approaches 8 for the expression $(x^{2}+10)(x^{2}-64)$ .

Find the limit as $x$ approaches 0 from the left of $\frac{1}{x}$ .

Calculate the one-sided limit: $\lim _{x \rightarrow 1^{-}}(3 x-6)$ .

Find the limits: $\lim _{x \rightarrow \infty} (x^3-x)$ and $\lim _{x \rightarrow -\infty} (x^3-x)$ .

Calculate the limit as $t$ approaches 6 for the expression $10 t^{2}-8 t+1$ .

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(\frac{8 x^{2}+6 x+14}{x^{4}-x^{2}}\right)$ .

Find the limit as $x$ approaches -172.5 for $\sqrt[3]{2x + 2}$ .

Calculate the limit: $\lim _{x \rightarrow 4^{+}}(4 \sqrt{x-4})$ .

Find the one-sided limit as $x$ approaches 0 from the left: $\lim _{x \rightarrow 0^{-}}\left(\frac{1}{x}\right)$ .

Find the limit as x approaches 11: $\lim _{x \rightarrow 11}\left(\frac{x+11}{x-11}\right)$ .

Evaluate the telescoping series: $\sum_{n=1}^{\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)$ .

Set up an integral for the curve length of $y^{2}+3y=x+1$ from $(-3,-1)$ to $(17,3)$ . Find $L=\int \square d y$ .

Find the sum of the series $\sum_{n=2}^{\infty} \frac{6}{n(n+3)}$ .

Show that the series converges using the Limit Comparison Test with $\frac{1}{n^{1/2}}$ and evaluate the limit: $\lim_{n \to \infty} \frac{\frac{1}{n^{3/2}}}{\frac{1}{\sqrt{n^3}+2}}$

Find the value of the series $\sum_{n=1}^{\infty} \frac{-1-(-2)^{n}}{3^{n}}$ . Choose a, b, c, or d.

Find the sum of the series $\sum_{n=1}^{\infty}\left(\frac{9}{n^{2}}-\frac{9}{(n+1)^{2}}\right)$ .

Determine the convergence of the series $\sum_{n=0}^{\infty} \frac{(-5)^{n+1}}{3^{n}}$ and find its sum.

Find the limit as $x$ approaches 1 for $\frac{x^{4}-1}{x^{6}-1}$ . Create a value table and estimate the limit.

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} dt$ from $x=0$ to $x=\frac{\pi}{4}$ . Set up the integral.

Determine if the series $\sum_{n=1}^{\infty} \frac{3^{n}-2^{n}}{5^{n}}$ converges or diverges, and find its sum.

Determine the convergence of the series $\sum_{n=2}^{\infty} \frac{6}{n(n+3)}$ : a. $\frac{13}{6}$ b. $\frac{5}{3}$ c. diverges d. $\frac{11}{3}$ e. $\frac{13}{2}$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{4}-1}{x^{6}-1}$ . Use a table and graph to estimate it.

Evaluate the integral: $\int \sin^{2} \beta \cos^{2} \beta \, d\beta$

Find the limit as $x$ approaches 1 for $\frac{x^{4}-1}{x^{6}-1}$ and create a value table rounded to four decimals.

Find the derivative of $f(x)=3 x^{8}-3 x^{5}-2 x^{3}+2 x$ and then calculate $f^{\prime}(5)$ .

Find the derivative of the function $f(x)=4 x^{2}-7 x-5$ , i.e., calculate $f^{\prime}(x)$ .

Find the limit as $x$ approaches 1 for $\frac{x^{4}-1}{x^{6}-1}$ . Estimate using a table and graph.

Find the derivative of the function $f(t) = 9 \sin t - 9 \pi \cos t$ , i.e., calculate $f^{\prime}(t)$ .

Complete the table and estimate the limit:
$\lim _{x \rightarrow 4} \frac{[1 /(x+1)]-(1 / 5)}{x-4}$
Estimate: $\approx -0.04$ . Use a graphing tool to confirm.

Set up an integral for the surface area of the curve $x^{1/2} + y^{1/2} = 9$ revolved around the x-axis.

Find the derivative using logarithmic differentiation for $f(x)=\left(\frac{1+2 x^{2}}{9-4 x}\right)^{4}$ .

A colony of 90 bacteria grows at $3.5\%$ per hour. How long until the population doubles?

Find the derivative $f'(t)$ of the function $f(t)=\frac{\sqrt{4}}{t^{7}}$ and evaluate $f'(2)$ .

Complete the table to find $\lim _{x \rightarrow 0} \frac{6 \sin (x)}{x}$ and estimate the limit. Round to five decimal places.