Calculus

Problem 15501

Find the derivative of $f(x)=e^{x} \sin x$ .

See Solution

Problem 15502

Evaluate the integral: $\int \left( \frac{1}{7 x} + \frac{1}{e^{-x}} + \frac{1}{4 x^{8}} \right) d x$

See Solution

Problem 15503

Find the tangent equations to $y=2x^{3}-3x^{2}-11x+8$ at points where $y=2$ . Verify with a graphing calculator.

See Solution

Problem 15504

Explain the indeterminate form $\frac{0}{0}$ with examples for limits of functions as $x$ approaches a.

See Solution

Problem 15505

Find the area of region $R$ bounded by $y=\sinh x$ , the $x$ -axis, and line $AB$ where $AB=\frac{4}{3}$ . Show $OB=\ln 3$ and $\cosh(\ln k)=\frac{k^{2}+1}{2k}$ .

See Solution

Problem 15506

Estimate the sum of the series $\sum (-1)^n \frac{1}{x^{2}+3}$ using the alternating series theorem with error < 0.0001.

See Solution

Problem 15507

Explain how to use l'Hôpital's Rule for limits like $\frac{0}{0}$ . Choose the correct answer: A, E, E, or 2.

See Solution

Problem 15508

Convert the limit $0 \cdot \infty$ to $0 / 0$ or $\infty / \infty$ . Which statement is correct?

See Solution

Problem 15509

Evaluate the series: $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n+4}$ .

See Solution

Problem 15510

Convert the limit $\lim _{x \rightarrow 1-}(x-1) \tan \left(\frac{\pi x}{2}\right)$ from $0 \cdot \infty$ to $\frac{0}{0}$ and evaluate it.

See Solution

Problem 15511

Evaluate the limit: $\lim _{x \rightarrow-7} \frac{x^{2}+8 x+7}{-63-2 x+x^{2}}$ using I'Hôpital's Rule if needed.

See Solution

Problem 15512

Find $O B=\ln 3$ if $A B=\frac{4}{3}$ . Show $\cosh (\ln k)=\frac{k^{2}+1}{2 k}$ and area $= \frac{2}{3}$ .

See Solution

Problem 15513

Given $f(x)=x^{3} \ln (x)-x^{3}$ , find $f^{\prime}(x)$ and evaluate $f^{\prime}(e)$ to five decimal places.

See Solution

Problem 15514

Find the derivative of $y$ with respect to $x$ for $y=\sec ^{-1}\left(\frac{4 x+3}{5}\right)$ . Show your work.

See Solution

Problem 15515

2. L'azote $(\mathrm{N})$ et l'hydrogène $(\mathrm{H})$ forment l'ammoniac avec $Q(t)=100-\frac{1000}{10+t}$ .
a) Trouvez $Q(0)$ et $Q(20)$ . b) Calculez la variation de $Q$ sur $[10 \mathrm{~s}, 20 \mathrm{~s}]$ . c) Taux de variation moyen sur : (i) $[10 s, 20 s]$ (ii) $[20 s, 30 s]$ d) Évaluez $\lim _{h \rightarrow 0^{+}} \frac{Q(h+0)-Q(0)}{h}$ et interprétez. e) Trouvez la fonction du rythme de variation de $Q$ . f) Évaluez : (i) $\left.\frac{d Q}{d t}\right|_{t=10 s}$ (ii) $\left.\frac{d Q}{d t}\right|_{t=1 \min }$

See Solution

Problem 15516

Evaluate the limit: $\lim _{u \rightarrow \frac{\pi}{4}} \frac{7 \tan u-7 \cot u}{2 u-\frac{\pi}{2}}$ using l'Hôpital's Rule.

See Solution

Problem 15517

Compute $(f+g)^{\prime}(3)$ using provided values for $f$ , $g$ , $f'$ , and $g'$ . Enter DNE if not defined.

See Solution

Problem 15518

Find the derivative of $y$ with respect to $x$ for $y=\tan^{-1}(\sqrt{5x})$ . Show your work.

See Solution

Problem 15519

Find a function where $\lim _{x \rightarrow 0}[f(x)]^{2}$ exists but $\lim _{x \rightarrow 0} f(x)$ does not.

See Solution

Problem 15520

Find the derivative of $f(x) = \frac{2 \sin \sqrt{x}}{\cos (\sqrt{x})^{2} \cdot \sqrt{x}}$ .

See Solution

Problem 15521

AGNs convert energy into radiation via efficiency $\eta$ .
a) Derive luminosity $L$ for mass $M_{bh}$ and rate $\dot{M}$ . b) Does $L$ depend on $M_{bh}$ ? c) Explain need for massive black holes in galaxies. d) If $\eta=0.1$ and $L=10^{12} L_{\odot}$ , estimate $\dot{M}$ in (M⊙/yr).

See Solution

Problem 15522

Evaluate the limit:
$\lim _{x \rightarrow \frac{\pi}{2}^{-}} \frac{\tan x}{\frac{11}{2 x-\pi}}$
Use l'Hôpital's Rule and find the exact answer.

See Solution

Problem 15523

Evaluate the limit using l'Hôpital's Rule:
$\lim _{x \rightarrow \frac{\pi}{2}^{-}} \frac{\tan x}{\frac{11}{2x-\pi}}$
Find the exact answer.

See Solution

Problem 15524

Prove that for the equation $x^{2}+25 y^{2}=100$ , it follows that $\frac{d^{2} y}{d x^{2}}=-\frac{4}{25 y^{3}}$ .

See Solution

Problem 15525

Find the linearization $L(x)$ of $f(x)=\sqrt{4x+9}$ at $x=0$ . Show your work.

See Solution

Problem 15526

Find the $x$ -coordinates where the tangent to $y=\frac{1}{1-x}$ is parallel to $4x-9y=13$ . Is there a positive $x$ -coordinate?

See Solution

Problem 15527

Evaluate the limit: $\lim _{x \rightarrow-1} \frac{2 x^{3}-3 x^{2}-12 x-7}{3 x^{4}+3 x^{3}-4 x^{2}-5 x-1}$ using l'Hôpital's Rule.

See Solution

Problem 15528

Find a linear approximation of $f(x)=\sqrt{x}$ near the integer $b=9$ . Show your work.

See Solution

Problem 15529

Differentiate the function $\frac{x^{2}-18 x}{(x-9)^{2}}$ with respect to $x$ .

See Solution

Problem 15530

Find the $x$ -intercepts of the curve with slope $\frac{d y}{d x}=6 x^{2}+4 x-60$ that passes through the origin.

See Solution

Problem 15531

Find the distance covered by a car with velocity $v(t)=4+19t-5t^{2}$ after 2 hours of driving.

See Solution

Problem 15532

Find the integral $I=\int_{a}^{b} m g(x) dx$ where $g(x)$ is continuous on $[a, b]$ and $m$ is a constant.

See Solution

Problem 15533

Find and interpret the marginal revenue for the demand equation $p=21-x$ at $x=5$ , $x=15$ , and $x=11$ . What is $R^{\prime}(x)$ ?

See Solution

Problem 15534

Find the differential $d y$ for the function $y=x \sqrt{5 x+4}$ . Show your work.

See Solution

Problem 15535

Find the differential $d y$ for the function $y=x \sqrt{5 x+4}$ . Show your work.

See Solution

Problem 15536

Find the expression for $\frac{f(x+h)-f(x)}{h}$ if $f(x)=1-2 x^{2}$ . Options: A. $4 x+2 h$ B. 9 C. $2 x+h$

See Solution

Problem 15537

Find the differential $d y$ for $y=\sin(2x^2)$ . Show your work.

See Solution

Problem 15538

Find the intervals where the function $f(x)=x^{3}-5 x^{2}+10 x+1$ is increasing, decreasing, concave up, and concave down.

See Solution

Problem 15539

Find transition points for $f(x)=6 x e^{-x^{2}}$ : local max, min, inflection, increase/decrease/concavity intervals, and horizontal asymptotes.

See Solution

Problem 15540

Find transition points for $x \in [0, \frac{2 \pi}{13}]$ in $y=13x+\sin(13x)$ . Identify local minima, maxima, and inflection points.

See Solution

Problem 15541

Find the critical points of the function $f(x)=45 x^{3}-3 x^{5}$ using calculus methods. Show your work.

See Solution

Problem 15542

Find the absolute max and min of $f(x)=\sqrt[3]{x}$ on $[-3, 8]$ using the Closed Interval Method. Show your work.

See Solution

Problem 15543

Estimate the age of cave paintings with $14\%$ carbon-14 using $A=A_{0} e^{-0.000121 t}$ . Find $t$ in years.

See Solution

Problem 15544

Calculate the average rate of change of $f(x)=3x$ from $x=-2$ to $x=1$ .

See Solution

Problem 15545

Find the first and second derivatives of the logistic function $P(x)=\frac{M}{1+A e^{-k x}}$ . Determine horizontal asymptotes, intervals of increase/decrease, and inflection points.

See Solution

Problem 15546

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

See Solution

Problem 15547

Graph $H(x)=\frac{1}{(x+3)^{2}}$ to determine where the function is increasing or decreasing.

See Solution

Problem 15548

Given the logistic function $P(x)=\frac{M}{1+A e^{-k x}}$ , find $P'(x)$ , $P''(x)$ , horizontal asymptotes, intervals of increase/decrease, and inflection points.

See Solution

Problem 15549

Evaluate the integral $I=\int_{0}^{3} f^{\prime}(x) d x$ for a linear function $f$ with slope $m$ and intercept $b$ .

See Solution

Problem 15550

Analyze the function $f(x)=\frac{x^{3}+8}{x}$ . Find intervals for increasing, decreasing, concave up, and concave down. Identify asymptotes.

See Solution

Problem 15551

Find the integral $I=\int_{a}^{b} m g(x) d x$ for a continuous function $g(x)$ on $[a, b]$ and constant $m$ .

See Solution

Problem 15552

Find when the drug concentration $C(t)=\frac{4t}{0.5+t^{2}}$ is maximized. Round to two decimal places: $t=$ h.

See Solution

Problem 15553

Find the limit: $\lim _{x \rightarrow 0} \frac{S(x)}{3 x^{3}}$ where $S(x)=\int_{0}^{x} \sin(3 \pi t^{2}) dt$ .

See Solution

Problem 15554

A box has a square base $x$ and height $h$ .
(a) Find $x, h$ for volume 11 and minimum surface area.
(b) Find $x, h$ for surface area 13 and maximum volume.

See Solution

Problem 15555

Find $y(4)$ if $\frac{d y}{d x}=\frac{3}{2} x^{1/2}-5 x^{-1/2}$ and $y(1)=-7$ .

See Solution

Problem 15556

Find the point $(x_0, y_0)$ on the graph of $y=x^3+6x^2+4x-3$ where the tangent slope is minimized. Answer as $(, )$ .

See Solution

Problem 15557

Find the $x$ -intercepts of a curve with slope $\frac{d y}{d x}=3 x^{2}+2 x-30$ that passes through the origin.

See Solution

Problem 15558

Find the $y$ -intercept of the curve that passes through $(1,5)$ with slope $\frac{d y}{d x}=8 x-3$ .

See Solution

Problem 15559

Find the value of $F(1)$ where $F(x)=\frac{d}{d x}\left(\int_{0}^{x^{4}} 5 t^{3} d t\right)$ .

See Solution

Problem 15560

A cylinder's radius grows at 5 ft/min with a constant volume of 77 ft³. When the radius is 4 ft, find the height's rate of change. Use $V=\pi r^{2} h$ and round to three decimal places.

See Solution

Problem 15561

Find the resistance $R=\frac{0.234}{r^{2}}$ for $r=3.48 \mathrm{~mm}$ and its error bound. What is the max uncertainty in $r$ for $R$ < 1.2%?

See Solution

Problem 15562

Tim threw a ball to point $B$ , 10 m from shore at point $C$ , 15 m from point $A$ . Find distance $x$ where Elvis entered water to minimize time.

See Solution

Problem 15563

Find the second-order partial derivatives of the function $f(x, y)=\ln(2+x^{2}y^{2})$ .

See Solution

Problem 15564

Find the first partial derivatives of the function $f(x, y) = x^{2} + xy + y^{2} + 2x - y$ at the point $(-5, 4)$ .

See Solution

Problem 15565

Find the second-order partial derivatives of $f(x, y)=7 x \sqrt{y}+7 y \sqrt{x}$ . Show $f_{xy} = f_{yx}$ .

See Solution

Problem 15566

Maximize $2xy^2$ for a rectangle in a circle of radius $r$ . Find dimensions $x$ and $y$ in terms of $r$ . $x=$ $y=$

See Solution

Problem 15567

Estimate the average fuel consumption (mpg) from 1974 to 1987 using the Trapezoidal Rule on the points $(74, 13.2)$ to $(87, 26.6)$ . Round to two decimal places.

See Solution

Problem 15568

Find the critical point of $f(x, y)=9-9 x^{2}-6 y^{2}$ and classify it using the second derivative test.

See Solution

Problem 15569

Find the rate of change of resistance $R$ with respect to radius $r$ when $r=3.48 \mathrm{~mm}$ . Then, find expected $R$ and error bound for $r=(3.48 \pm 0.28) \mathrm{mm}$ . Lastly, determine max uncertainty in $r$ to keep uncertainty in $R$ below $1.2\%$ .

See Solution

Problem 15570

Find the critical point of $f(x, y) = x^2 + 2xy + 2y^2 - 6x + 10y + 8$ and classify it using the second derivative test.

See Solution

Problem 15571

Find the volume $V$ of water in a hemispherical container with radius $28 \mathrm{~cm}$ and water level $21 \mathrm{~cm}$ . Then estimate uncertainties in $V$ with given uncertainties in $r$ and $h$ , and find the change in water depth for $4000 \mathrm{~cm}^{3}$ removed.

See Solution

Problem 15572

When $f(x)$ is concave down on $(3,12)$ , what can we say about its rates of change in that interval? Choose the best answer: a) Increasing on $(-\infty, 3) \cup (12, \infty)$ b) Decreasing on $(-\infty, 3) \cup (12, \infty)$ c) Increasing on $(3,12)$ d) Decreasing on $(3,12)$

See Solution

Problem 15573

Find the limit of the sequence $a_{n}=\sqrt[n]{(1+\sqrt{3})^{n}+(\sqrt{5}-1)^{n}}$ as $n \to \infty$ .

See Solution

Problem 15574

Find the limit of $a_{n}=\sqrt{n^{3}-n^{2}+n}-\sqrt{n^{3}-3 n+1}$ as $n$ approaches infinity.

See Solution

Problem 15575

A cup of coffee cools from $205^{\circ} \mathrm{F}$ to $195^{\circ} \mathrm{F}$ in 10 min. When will it reach $180^{\circ} \mathrm{F}$ ? A. 15 min B. 45 min C. 27 min D. 1 hour

See Solution

Problem 15576

Find the decay constant $k$ for plutonium-240 with a half-life of 6,300 years using $Q(t)=Q_{0} e^{-k t}$ .

See Solution

Problem 15577

Earth's population model is $P(t)=\frac{64}{1+11 e^{-0.08 t}}$ . Which statements are true? A. Exponential growth B. 64 billion capacity C. 1990: 5.33 billion D. 8% annual increase.

See Solution

Problem 15578

Calculate the limit: $\lim _{n \rightarrow \infty}\left(1+\frac{\pi}{2^{n+1}}\right)^{2^{n}}$ .

See Solution

Problem 15579

Evaluate the limit as (x, y) approaches (4, 12): $\lim _{(x, y) \rightarrow(4,12)} \frac{\sqrt{x+y}-4}{x+y-16}$

See Solution

Problem 15580

Klimaanlage ausgefallen: $T(t)=\frac{200}{4+\frac{6}{0,1 t+1}}$ . a) Temperatur zu Beginn? b) Nach 60 Minuten? c) Grenztemperatur?

See Solution

Problem 15581

Evaluate the integral: $\int 4 \sin^{6}(x) \cos(x) \, dx =$

See Solution

Problem 15582

Find the limit: $\lim _{x \rightarrow+\infty} x^{2} \cdot e^{x}$ .

See Solution

Problem 15583

Evaluate the integral $\int \sec (u) \tan (u) \frac{du}{2}$ , where $u=2x$ .

See Solution

Problem 15584

Find the integral $\int x \cdot \sqrt[5]{x^{2}-6} dx$ using u-substitution: $u=\square$ , $du=\square dx$ . Rewrite as $\int \square du$ .

See Solution

Problem 15585

Find the limits:
1) $\lim_{x \rightarrow 5} \frac{x^{2}-25}{x-5}$
2) $\lim_{x \rightarrow 3} \frac{3x^{2}-27}{x-3}$
3) $\lim_{x \rightarrow 4} \frac{x^{2}-16}{x-4}$
4) $\lim_{x \rightarrow -1} \frac{x^{3}-x}{x+1}$

See Solution

Problem 15586

Check if Rolle's Theorem applies to $g(x)=x^{3}+2 x^{2}-15 x-36$ on $[-3,4]$ and find any guaranteed points.

See Solution

Problem 15587

Find the time $t$ when the drug concentration $C(t)=\frac{0.9 t}{t^{2}+18}$ is maximum, accurate to 2 decimal places. Also, find the max concentration in $\frac{\mu g}{m l}$ .

See Solution

Problem 15588

Find the time $t$ when the drug concentration $C(t)=\frac{0.2 t}{t^{2}+29}$ is maximum and the max concentration in $\frac{\mu g}{m l}$ .

See Solution

Problem 15589

Untersuchen Sie die Funktionen auf Extremalpunkte und skizzieren Sie deren Graphen: a) $f(x)=2 x^{2}+3 x-5$ b) $f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-3 x$ c) $f(x)=\frac{1}{4} x^{3}-2$

See Solution

Problem 15590

Calculate the integral $\int_{0}^{362} \sin x \, dx$ .

See Solution

Problem 15591

Check if the Mean Value Theorem applies to $f(x)=|x-5|$ on $[-5,11]$ and find the guaranteed point(s).

See Solution

Problem 15592

Find the volume change rate when the radius is 2.4 inches, expanding at 0.2 in/s. Round to the nearest hundredth.

See Solution

Problem 15593

A snowball's radius decreases at $0.3 \mathrm{~cm/min}$ . Find the volume decrease rate when the radius is $15 \mathrm{~cm}$ . $\frac{\mathrm{cm}^{3}}{\mathrm{~min}}$

See Solution

Problem 15594

A snowball's radius decreases at $0.2 \mathrm{~cm/min}$ . Find the volume decrease rate when the radius is $11 \mathrm{~cm}$ . $\frac{\mathrm{cm}^{3}}{\min }$

See Solution

Problem 15595

Check if the Mean Value Theorem applies to $f(x)=-3-x^{2}$ on $[-2,1]$ and find the guaranteed point(s).

See Solution

Problem 15596

Find $\frac{d x}{d t}$ when $103=3xy+y^{2}+x^{4}$ , $\frac{d y}{d t}=3$ , $x=-3$ , $y=-2$ .

See Solution

Problem 15597

Gegeben ist die Funktion $f$ mit $f(x)=3 x^{2}+x$ . Beantworte die Fragen: a) Ableitung von $f$ bei $-3$ . b) Finde $x_{0}$ für $f\left(x_{0}\right)=-2$ . c) Wo hat $f$ die Steigung 13? d) Wo ist die Tangente parallel zu $y=-5 x+3$ ?

See Solution

Problem 15598

a. Find dimensions of a rectangular pen with 400 m of fencing to maximize area against a barn. b. Determine dimensions of four adjacent pens (each 100 m²) to minimize fencing against a barn.

See Solution

Problem 15599

Evaluate the integral $\int \frac{x^{3}}{\left(x^{4}-4\right)^{4}} d x$ using the substitution $u=\square$ and $d u=\square d x$ .

See Solution

Problem 15600

Find $\frac{d x}{d t}$ given the equation $-4xy + x^4 + y^3 = 97$ , with $x=3$ , $y=-2$ , and $\frac{d y}{d t}=-4$ .

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 15501

Find the derivative of f(x)=exsin⁡xf(x)=e^{x} \sin xf(x)=exsinx.

Problem 15502

Evaluate the integral: ∫(17x+1e−x+14x8)dx\int \left( \frac{1}{7 x} + \frac{1}{e^{-x}} + \frac{1}{4 x^{8}} \right) d x∫(7x1​+e−x1​+4x81​)dx

Problem 15503

Find the tangent equations to y=2x3−3x2−11x+8y=2x^{3}-3x^{2}-11x+8y=2x3−3x2−11x+8 at points where y=2y=2y=2. Verify with a graphing calculator.

Problem 15504

Explain the indeterminate form 00\frac{0}{0}00​ with examples for limits of functions as xxx approaches a.

Problem 15505

Find the area of region RRR bounded by y=sinh⁡xy=\sinh xy=sinhx, the xxx-axis, and line ABABAB where AB=43AB=\frac{4}{3}AB=34​. Show OB=ln⁡3OB=\ln 3OB=ln3 and cosh⁡(ln⁡k)=k2+12k\cosh(\ln k)=\frac{k^{2}+1}{2k}cosh(lnk)=2kk2+1​.

Problem 15506

Estimate the sum of the series ∑(−1)n1x2+3\sum (-1)^n \frac{1}{x^{2}+3}∑(−1)nx2+31​ using the alternating series theorem with error < 0.0001.

Problem 15507

Explain how to use l'Hôpital's Rule for limits like 00\frac{0}{0}00​. Choose the correct answer: A, E, E, or 2.

Problem 15508

Convert the limit 0⋅∞0 \cdot \infty0⋅∞ to 0/00 / 00/0 or ∞/∞\infty / \infty∞/∞. Which statement is correct?

Problem 15509

Evaluate the series: ∑n=1∞(−1)n+11n+4\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n+4}∑n=1∞​(−1)n+1n+41​.

Problem 15510

Convert the limit lim⁡x→1−(x−1)tan⁡(πx2)\lim _{x \rightarrow 1-}(x-1) \tan \left(\frac{\pi x}{2}\right)limx→1−​(x−1)tan(2πx​) from 0⋅∞0 \cdot \infty0⋅∞ to 00\frac{0}{0}00​ and evaluate it.

Problem 15511

Evaluate the limit: lim⁡x→−7x2+8x+7−63−2x+x2\lim _{x \rightarrow-7} \frac{x^{2}+8 x+7}{-63-2 x+x^{2}}x→−7lim​−63−2x+x2x2+8x+7​ using I'Hôpital's Rule if needed.

Problem 15512

Find OB=ln⁡3O B=\ln 3OB=ln3 if AB=43A B=\frac{4}{3}AB=34​. Show cosh⁡(ln⁡k)=k2+12k\cosh (\ln k)=\frac{k^{2}+1}{2 k}cosh(lnk)=2kk2+1​ and area =23= \frac{2}{3}=32​.

Problem 15513

Given f(x)=x3ln⁡(x)−x3f(x)=x^{3} \ln (x)-x^{3}f(x)=x3ln(x)−x3, find f′(x)f^{\prime}(x)f′(x) and evaluate f′(e)f^{\prime}(e)f′(e) to five decimal places.

Problem 15514

Find the derivative of yyy with respect to xxx for y=sec⁡−1(4x+35)y=\sec ^{-1}\left(\frac{4 x+3}{5}\right)y=sec−1(54x+3​). Show your work.

Problem 15515

Problem 15516

Evaluate the limit: lim⁡u→π47tan⁡u−7cot⁡u2u−π2\lim _{u \rightarrow \frac{\pi}{4}} \frac{7 \tan u-7 \cot u}{2 u-\frac{\pi}{2}}limu→4π​​2u−2π​7tanu−7cotu​ using l'Hôpital's Rule.

Problem 15517

Compute (f+g)′(3)(f+g)^{\prime}(3)(f+g)′(3) using provided values for fff, ggg, f′f'f′, and g′g'g′. Enter DNE if not defined.

Problem 15518

Find the derivative of yyy with respect to xxx for y=tan⁡−1(5x)y=\tan^{-1}(\sqrt{5x})y=tan−1(5x​). Show your work.

Problem 15519

Find a function where lim⁡x→0[f(x)]2\lim _{x \rightarrow 0}[f(x)]^{2}limx→0​[f(x)]2 exists but lim⁡x→0f(x)\lim _{x \rightarrow 0} f(x)limx→0​f(x) does not.

Problem 15520

Find the derivative of f(x)=2sin⁡xcos⁡(x)2⋅xf(x) = \frac{2 \sin \sqrt{x}}{\cos (\sqrt{x})^{2} \cdot \sqrt{x}}f(x)=cos(x​)2⋅x​2sinx​​.

Problem 15521

Problem 15522

Evaluate the limit: lim⁡x→π2−tan⁡x112x−π \lim _{x \rightarrow \frac{\pi}{2}^{-}} \frac{\tan x}{\frac{11}{2 x-\pi}} x→2π​−lim​2x−π11​tanx​ Use l'Hôpital's Rule and find the exact answer.

Problem 15523

Evaluate the limit using l'Hôpital's Rule: lim⁡x→π2−tan⁡x112x−π \lim _{x \rightarrow \frac{\pi}{2}^{-}} \frac{\tan x}{\frac{11}{2x-\pi}} x→2π​−lim​2x−π11​tanx​ Find the exact answer.

Problem 15524

Prove that for the equation x2+25y2=100x^{2}+25 y^{2}=100x2+25y2=100, it follows that d2ydx2=−425y3\frac{d^{2} y}{d x^{2}}=-\frac{4}{25 y^{3}}dx2d2y​=−25y34​.

Problem 15525

Find the linearization L(x)L(x)L(x) of f(x)=4x+9f(x)=\sqrt{4x+9}f(x)=4x+9​ at x=0x=0x=0. Show your work.

Problem 15526

Find the xxx-coordinates where the tangent to y=11−xy=\frac{1}{1-x}y=1−x1​ is parallel to 4x−9y=134x-9y=134x−9y=13. Is there a positive xxx-coordinate?

Problem 15527

Evaluate the limit: lim⁡x→−12x3−3x2−12x−73x4+3x3−4x2−5x−1\lim _{x \rightarrow-1} \frac{2 x^{3}-3 x^{2}-12 x-7}{3 x^{4}+3 x^{3}-4 x^{2}-5 x-1}x→−1lim​3x4+3x3−4x2−5x−12x3−3x2−12x−7​ using l'Hôpital's Rule.

Problem 15528

Find a linear approximation of f(x)=xf(x)=\sqrt{x}f(x)=x​ near the integer b=9b=9b=9. Show your work.

Problem 15529

Differentiate the function x2−18x(x−9)2\frac{x^{2}-18 x}{(x-9)^{2}}(x−9)2x2−18x​ with respect to xxx.

Problem 15530

Find the xxx-intercepts of the curve with slope dydx=6x2+4x−60\frac{d y}{d x}=6 x^{2}+4 x-60dxdy​=6x2+4x−60 that passes through the origin.

Problem 15531

Find the distance covered by a car with velocity v(t)=4+19t−5t2v(t)=4+19t-5t^{2}v(t)=4+19t−5t2 after 2 hours of driving.

Problem 15532

Find the integral I=∫abmg(x)dxI=\int_{a}^{b} m g(x) dxI=∫ab​mg(x)dx where g(x)g(x)g(x) is continuous on [a,b][a, b][a,b] and mmm is a constant.

Problem 15533

Find and interpret the marginal revenue for the demand equation p=21−xp=21-xp=21−x at x=5x=5x=5, x=15x=15x=15, and x=11x=11x=11. What is R′(x)R^{\prime}(x)R′(x)?

Problem 15534

Find the differential dyd ydy for the function y=x5x+4y=x \sqrt{5 x+4}y=x5x+4​. Show your work.

Problem 15535

Find the differential dyd ydy for the function y=x5x+4y=x \sqrt{5 x+4}y=x5x+4​. Show your work.

Problem 15536

Find the expression for f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ if f(x)=1−2x2f(x)=1-2 x^{2}f(x)=1−2x2. Options: A. 4x+2h4 x+2 h4x+2h B. 9 C. 2x+h2 x+h2x+h

Problem 15537

Find the differential dyd ydy for y=sin⁡(2x2)y=\sin(2x^2)y=sin(2x2). Show your work.

Problem 15538

Find the intervals where the function f(x)=x3−5x2+10x+1f(x)=x^{3}-5 x^{2}+10 x+1f(x)=x3−5x2+10x+1 is increasing, decreasing, concave up, and concave down.

Problem 15539

Find transition points for f(x)=6xe−x2f(x)=6 x e^{-x^{2}}f(x)=6xe−x2: local max, min, inflection, increase/decrease/concavity intervals, and horizontal asymptotes.

Problem 15540

Find transition points for x∈[0,2π13]x \in [0, \frac{2 \pi}{13}]x∈[0,132π​] in y=13x+sin⁡(13x)y=13x+\sin(13x)y=13x+sin(13x). Identify local minima, maxima, and inflection points.

Problem 15541

Find the derivative of $f(x)=e^{x} \sin x$ .

Evaluate the integral: $\int \left( \frac{1}{7 x} + \frac{1}{e^{-x}} + \frac{1}{4 x^{8}} \right) d x$

Find the tangent equations to $y=2x^{3}-3x^{2}-11x+8$ at points where $y=2$ . Verify with a graphing calculator.

Explain the indeterminate form $\frac{0}{0}$ with examples for limits of functions as $x$ approaches a.

Find the area of region $R$ bounded by $y=\sinh x$ , the $x$ -axis, and line $AB$ where $AB=\frac{4}{3}$ . Show $OB=\ln 3$ and $\cosh(\ln k)=\frac{k^{2}+1}{2k}$ .

Estimate the sum of the series $\sum (-1)^n \frac{1}{x^{2}+3}$ using the alternating series theorem with error < 0.0001.

Explain how to use l'Hôpital's Rule for limits like $\frac{0}{0}$ . Choose the correct answer: A, E, E, or 2.

Convert the limit $0 \cdot \infty$ to $0 / 0$ or $\infty / \infty$ . Which statement is correct?

Evaluate the series: $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n+4}$ .

Convert the limit $\lim _{x \rightarrow 1-}(x-1) \tan \left(\frac{\pi x}{2}\right)$ from $0 \cdot \infty$ to $\frac{0}{0}$ and evaluate it.

Evaluate the limit: $\lim _{x \rightarrow-7} \frac{x^{2}+8 x+7}{-63-2 x+x^{2}}$ using I'Hôpital's Rule if needed.

Find $O B=\ln 3$ if $A B=\frac{4}{3}$ . Show $\cosh (\ln k)=\frac{k^{2}+1}{2 k}$ and area $= \frac{2}{3}$ .

Given $f(x)=x^{3} \ln (x)-x^{3}$ , find $f^{\prime}(x)$ and evaluate $f^{\prime}(e)$ to five decimal places.

Find the derivative of $y$ with respect to $x$ for $y=\sec ^{-1}\left(\frac{4 x+3}{5}\right)$ . Show your work.

Evaluate the limit: $\lim _{u \rightarrow \frac{\pi}{4}} \frac{7 \tan u-7 \cot u}{2 u-\frac{\pi}{2}}$ using l'Hôpital's Rule.

Compute $(f+g)^{\prime}(3)$ using provided values for $f$ , $g$ , $f'$ , and $g'$ . Enter DNE if not defined.

Find the derivative of $y$ with respect to $x$ for $y=\tan^{-1}(\sqrt{5x})$ . Show your work.

Find a function where $\lim _{x \rightarrow 0}[f(x)]^{2}$ exists but $\lim _{x \rightarrow 0} f(x)$ does not.

Find the derivative of $f(x) = \frac{2 \sin \sqrt{x}}{\cos (\sqrt{x})^{2} \cdot \sqrt{x}}$ .

Evaluate the limit:
$\lim _{x \rightarrow \frac{\pi}{2}^{-}} \frac{\tan x}{\frac{11}{2 x-\pi}}$
Use l'Hôpital's Rule and find the exact answer.

Evaluate the limit using l'Hôpital's Rule:
$\lim _{x \rightarrow \frac{\pi}{2}^{-}} \frac{\tan x}{\frac{11}{2x-\pi}}$
Find the exact answer.

Prove that for the equation $x^{2}+25 y^{2}=100$ , it follows that $\frac{d^{2} y}{d x^{2}}=-\frac{4}{25 y^{3}}$ .

Find the linearization $L(x)$ of $f(x)=\sqrt{4x+9}$ at $x=0$ . Show your work.

Find the $x$ -coordinates where the tangent to $y=\frac{1}{1-x}$ is parallel to $4x-9y=13$ . Is there a positive $x$ -coordinate?

Evaluate the limit: $\lim _{x \rightarrow-1} \frac{2 x^{3}-3 x^{2}-12 x-7}{3 x^{4}+3 x^{3}-4 x^{2}-5 x-1}$ using l'Hôpital's Rule.

Find a linear approximation of $f(x)=\sqrt{x}$ near the integer $b=9$ . Show your work.

Differentiate the function $\frac{x^{2}-18 x}{(x-9)^{2}}$ with respect to $x$ .

Find the $x$ -intercepts of the curve with slope $\frac{d y}{d x}=6 x^{2}+4 x-60$ that passes through the origin.

Find the distance covered by a car with velocity $v(t)=4+19t-5t^{2}$ after 2 hours of driving.

Find the integral $I=\int_{a}^{b} m g(x) dx$ where $g(x)$ is continuous on $[a, b]$ and $m$ is a constant.

Find and interpret the marginal revenue for the demand equation $p=21-x$ at $x=5$ , $x=15$ , and $x=11$ . What is $R^{\prime}(x)$ ?

Find the differential $d y$ for the function $y=x \sqrt{5 x+4}$ . Show your work.

Find the differential $d y$ for the function $y=x \sqrt{5 x+4}$ . Show your work.

Find the expression for $\frac{f(x+h)-f(x)}{h}$ if $f(x)=1-2 x^{2}$ . Options: A. $4 x+2 h$ B. 9 C. $2 x+h$

Find the differential $d y$ for $y=\sin(2x^2)$ . Show your work.

Find the intervals where the function $f(x)=x^{3}-5 x^{2}+10 x+1$ is increasing, decreasing, concave up, and concave down.

Find transition points for $f(x)=6 x e^{-x^{2}}$ : local max, min, inflection, increase/decrease/concavity intervals, and horizontal asymptotes.

Find transition points for $x \in [0, \frac{2 \pi}{13}]$ in $y=13x+\sin(13x)$ . Identify local minima, maxima, and inflection points.

Find the critical points of the function $f(x)=45 x^{3}-3 x^{5}$ using calculus methods. Show your work.

Find the absolute max and min of $f(x)=\sqrt[3]{x}$ on $[-3, 8]$ using the Closed Interval Method. Show your work.

Estimate the age of cave paintings with $14\%$ carbon-14 using $A=A_{0} e^{-0.000121 t}$ . Find $t$ in years.

Calculate the average rate of change of $f(x)=3x$ from $x=-2$ to $x=1$ .

Find the first and second derivatives of the logistic function $P(x)=\frac{M}{1+A e^{-k x}}$ . Determine horizontal asymptotes, intervals of increase/decrease, and inflection points.

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

Graph $H(x)=\frac{1}{(x+3)^{2}}$ to determine where the function is increasing or decreasing.

Given the logistic function $P(x)=\frac{M}{1+A e^{-k x}}$ , find $P'(x)$ , $P''(x)$ , horizontal asymptotes, intervals of increase/decrease, and inflection points.

Evaluate the integral $I=\int_{0}^{3} f^{\prime}(x) d x$ for a linear function $f$ with slope $m$ and intercept $b$ .

Analyze the function $f(x)=\frac{x^{3}+8}{x}$ . Find intervals for increasing, decreasing, concave up, and concave down. Identify asymptotes.

Find the integral $I=\int_{a}^{b} m g(x) d x$ for a continuous function $g(x)$ on $[a, b]$ and constant $m$ .

Find when the drug concentration $C(t)=\frac{4t}{0.5+t^{2}}$ is maximized. Round to two decimal places: $t=$ h.

Find the limit: $\lim _{x \rightarrow 0} \frac{S(x)}{3 x^{3}}$ where $S(x)=\int_{0}^{x} \sin(3 \pi t^{2}) dt$ .

A box has a square base $x$ and height $h$ .
(a) Find $x, h$ for volume 11 and minimum surface area.
(b) Find $x, h$ for surface area 13 and maximum volume.

Find $y(4)$ if $\frac{d y}{d x}=\frac{3}{2} x^{1/2}-5 x^{-1/2}$ and $y(1)=-7$ .

Find the point $(x_0, y_0)$ on the graph of $y=x^3+6x^2+4x-3$ where the tangent slope is minimized. Answer as $(, )$ .

Find the $x$ -intercepts of a curve with slope $\frac{d y}{d x}=3 x^{2}+2 x-30$ that passes through the origin.

Find the $y$ -intercept of the curve that passes through $(1,5)$ with slope $\frac{d y}{d x}=8 x-3$ .

Find the value of $F(1)$ where $F(x)=\frac{d}{d x}\left(\int_{0}^{x^{4}} 5 t^{3} d t\right)$ .

A cylinder's radius grows at 5 ft/min with a constant volume of 77 ft³. When the radius is 4 ft, find the height's rate of change. Use $V=\pi r^{2} h$ and round to three decimal places.

Find the resistance $R=\frac{0.234}{r^{2}}$ for $r=3.48 \mathrm{~mm}$ and its error bound. What is the max uncertainty in $r$ for $R$ < 1.2%?

Tim threw a ball to point $B$ , 10 m from shore at point $C$ , 15 m from point $A$ . Find distance $x$ where Elvis entered water to minimize time.

Find the second-order partial derivatives of the function $f(x, y)=\ln(2+x^{2}y^{2})$ .

Find the first partial derivatives of the function $f(x, y) = x^{2} + xy + y^{2} + 2x - y$ at the point $(-5, 4)$ .

Find the second-order partial derivatives of $f(x, y)=7 x \sqrt{y}+7 y \sqrt{x}$ . Show $f_{xy} = f_{yx}$ .

Maximize $2xy^2$ for a rectangle in a circle of radius $r$ . Find dimensions $x$ and $y$ in terms of $r$ . $x=$ $y=$

Estimate the average fuel consumption (mpg) from 1974 to 1987 using the Trapezoidal Rule on the points $(74, 13.2)$ to $(87, 26.6)$ . Round to two decimal places.

Find the critical point of $f(x, y)=9-9 x^{2}-6 y^{2}$ and classify it using the second derivative test.

Find the critical point of $f(x, y) = x^2 + 2xy + 2y^2 - 6x + 10y + 8$ and classify it using the second derivative test.

Find the limit of the sequence $a_{n}=\sqrt[n]{(1+\sqrt{3})^{n}+(\sqrt{5}-1)^{n}}$ as $n \to \infty$ .

Find the limit of $a_{n}=\sqrt{n^{3}-n^{2}+n}-\sqrt{n^{3}-3 n+1}$ as $n$ approaches infinity.