Calculus

Problem 25501

Find the concavity regions for the function $y=\frac{2 x+7}{2 x}$ , where $x \neq 0$ .

See Solution

Problem 25502

A child swings at a max speed of $5.20 \mathrm{~m/s}$ and is $0.860 \mathrm{~m}$ high at the lowest point. Find his max height.

See Solution

Problem 25503

Find $\lim _{h \rightarrow 0} \frac{3(x+h)^{2}+7(x+h)-3 x^{2}-7 x}{h}$ treating $x$ as constant.

See Solution

Problem 25504

Find the limit $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ for functions 37-42:
37. $f(x)=4-x$ , 38. $f(x)=2x+3$ , 39. $f(x)=x^{2}-3$ , 40. $f(x)=x^{2}+x+1$ , 41. $f(x)=x^{3}-4x^{2}$ , 42. $f(x)=5-2x-3x^{2}$ .

See Solution

Problem 25505

Find the octopus population change over the first 4 weeks given $\frac{d}{d t} P=125(2 t^{2}-2 t+1)^{1.2}(2 t-1)$ . Round to the nearest whole number.

See Solution

Problem 25506

Finde die Ableitung von $f(x) = \frac{4}{3 x^{3}}$ .

See Solution

Problem 25507

Find inflection points and concavity regions for $f(x)=8 e^{-9 x^{2}}, x \geq 0$ .

See Solution

Problem 25508

Which statement is TRUE by the Comparison Test? Analyze integrals involving $\ln(x)$ and $\cos(x)$ for convergence.

See Solution

Problem 25509

Estimate the temperature change at 4.5 meters from the heater after 7 minutes, given the rates of change.

See Solution

Problem 25510

Find the inflection points of $f(x)=8 e^{-9 x^{2}}$ for $x \geq 0$ .

See Solution

Problem 25511

Is the piecewise function $f(x)=\begin{cases} 2x-3, & -1 \leq x < 0 \\ x-3, & 0 \leq x \leq 4 \end{cases}$ differentiable?

See Solution

Problem 25512

Find the derivative of $f(x)=\cos^{2}(6x^{2})-7$ .

See Solution

Problem 25513

Find the smallest growth rate of the fungus given by $L(t)=2.8 t+0.6 \cos \left(\frac{\pi t}{12}\right)$ .

See Solution

Problem 25514

A cargo falls from a crane, taking $1.2 \mathrm{~s}$ to fall halfway. Find the total fall time to the ground.

See Solution

Problem 25515

Find the distance $r$ that minimizes the Leonard-Jones potential $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ , where $A>0$ .

See Solution

Problem 25516

Gegeben ist $f(x)=e^{x}+1$ . a) Berechne $\int_{0}^{1} f(x) d x$ . b) Bestimme $g(x)$ , das aus $f$ durch Spiegeln und Verschieben entsteht.

See Solution

Problem 25517

Find the derivative of the function given by $g^{\prime}(x)=\left(x^{2}\left(2^{x}\right)\right)^{\prime}$ .

See Solution

Problem 25518

Gitt $f(x)=\frac{x^{2}-x-2}{x^{2}+x-2}$ , finn nullpunkter, derivert, polynomdivisjon, asymptoter og skisser grafen.

See Solution

Problem 25519

Berechnen Sie die mittlere Änderungsrate zwischen den Punkten $P(3, 2.5)$ und $Q(6.5, 13)$ sowie im Intervall $[1, 4]$ für $g(x)=x^{3}$ . Untersuchen Sie die Grenzwerte: $\lim_{x \to \infty} \frac{3x+2}{x}$ und $\lim_{x \to \infty} \frac{3+x^{2}}{2x^{2}}$ . Finden Sie die Nullstellen von $f(x)=x^{2}-9$ und $g(x)=2x^{4}-4x^{2}-16$ . Analysieren Sie die Funktion $f(x)=x^{3}-3x^{2}$ : Grad, Nullstellen, Symmetrie, Punkt $Q(2.5, -3.25)$ , Schnittpunkte mit $g(x)=x$ , und zeichnen Sie $f(x)$ und $g(x)$ im Intervall $[-1, 3.5]$ . Berechnen Sie die Temperatur $T(x)=63 \cdot 0.93^{x}+19$ beim Einschenken und nach 10 Minuten, die durchschnittliche Abnahme in den ersten 6 Minuten, und die langfristige Temperatur.

See Solution

Problem 25520

Berechnen Sie $f^{\prime}(-2)$ für die Funktion $f(x)=x^{2}-2$ mit dem Differenzenquotienten für $h \rightarrow 0$ .

See Solution

Problem 25521

Bestimmen Sie die Stammfunktion von $-0,03 s^{3}+0,9 s^{2}-1,7 s+0,1$ .

See Solution

Problem 25522

Bestimme die Tangentengleichung $t(x)$ bei Punkt P durch Ableitung für $f(x)=x^{3}+x^{2}+x$ und $f(x)=-x^{3}+2x^{2}$ .

See Solution

Problem 25523

Approximate the negative root of $e^{x}=9-x^{2}$ using Newton's method to six decimal places.

See Solution

Problem 25524

Evaluate the integral $\int_{2}^{6} \frac{d x}{\sqrt[3]{x-2}}$ or state if it diverges. Show your work.

See Solution

Problem 25525

Find the limit: $\lim _{x \rightarrow 2}(2 x+3)(x^{2}+1)$ .

See Solution

Problem 25526

Find the limit: $\lim _{x \rightarrow 2} e\left(x^{2}+1\right)$ .

See Solution

Problem 25527

Find the total change in price of an ice cream cone from 2000 to 2005 given the rate $18 e^{0.04 t}$ cents/year.

See Solution

Problem 25528

Find the limits: a) $\lim _{x \rightarrow 3} \frac{x-2}{x-8}$ , b) $\lim _{x \rightarrow 2} \frac{x^{2}-5}{1-x}$ .

See Solution

Problem 25529

Find the limit as $x$ approaches 2 for the expression $\sqrt{(2x + 3)}$ .

See Solution

Problem 25530

Find the limits: b) $\lim _{x \rightarrow 3}(3 x+7)$ and c) $\lim _{x \rightarrow 3} \frac{(5 x-4)}{(3 x+7)}$ .

See Solution

Problem 25531

Find the limits: a) $\lim _{x \rightarrow 3} \frac{x}{x-3}$ and b) $\lim _{x \rightarrow 3} \frac{4 x}{2 x-6}$ .

See Solution

Problem 25532

Find the limits: a) $\lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4}$ b) $\lim _{x \rightarrow 2} \frac{3 x^{2}-7 x+2}{x-4}$

See Solution

Problem 25533

Calculate the sum of the integrals: $\int_{-1}^{0}(x^{2}-1) dx + \int_{-2}^{-1}(x^{2}-1) dx$ .

See Solution

Problem 25534

Berechne die Ableitung der Funktion $f(x)=\frac{1}{x^{2}}$ .

See Solution

Problem 25535

Find the limits as $x$ approaches infinity: $\lim_{x \rightarrow \infty}(3x^2 + 1)$ and $\lim_{x \rightarrow \infty}(2x^5 + x^4 - 1000)$ .

See Solution

Problem 25536

Find the limits: a) $\lim_{x \rightarrow \infty} \frac{1}{2x^4}$ , b) $\lim_{x \rightarrow \infty} \frac{5}{2x}$ , c) $\lim_{x \rightarrow \infty} \frac{1}{x^3}$ .

See Solution

Problem 25537

Find the limits: a) $\lim _{x \rightarrow \infty} \pi$ b) $\lim _{x \rightarrow -\infty} 2.718$

See Solution

Problem 25538

Calculate $-\int_{-1}^{0}(x^{2}-1) dx + \int_{-2}^{-1}(x^{2}-1) dx$ .

See Solution

Problem 25539

Find the limits: a) $\lim _{x \rightarrow \infty} \frac{2 x^{4}+x^{2}-3}{x^{3}+x+2}$ , b) $\lim _{x \rightarrow \infty} \frac{2 x^{3}+x^{2}-3}{x^{3}+x+2}$ , c) $\lim _{x \rightarrow \infty} \frac{x+1}{2 x^{2}+5 x}$ .

See Solution

Problem 25540

Beweisen Sie die Volumenformeln für (a) Kegel, (b) Kugel, (c) Kegelstumpf und erläutern Sie (d) den Hinweis zu (b).

See Solution

Problem 25541

Find the limits: $\lim _{x \rightarrow 4} \frac{x-4}{x+2}$ and $\lim _{x \rightarrow 2} \frac{2 x-4}{3}$ .

See Solution

Problem 25542

Bestem stammfunksjonen for $A=\int_{0}^{2,356} \sin (x)-0,3 x \, dx$ og beregn integralet.

See Solution

Problem 25543

Find the radius change rate when the circle area is $25 \pi$ in², decreasing at 205 in²/min. Round to 3 decimals.

See Solution

Problem 25544

Berechnen Sie die Integrale: a) $\int_{0}^{2}(x^{3}-24 x^{2}+60 x-32) dx$ b) $\int_{1}^{2}(7+60 x-\frac{3}{x}) dx$ c) $\int_{0}^{3}(2 e^{t}) dt$ d) $\int_{a}^{b} x dx$

See Solution

Problem 25545

A cylinder's radius increases at 8 cm/min, with a constant volume of 174 cm³. Find the height's change rate when height is 7 cm. Use $V=\pi r^{2} h$ and round to three decimal places.

See Solution

Problem 25546

Bestimmen Sie alle Häufungspunkte der Folgen $a_n$ und geben Sie jeweils $\lim_{n \rightarrow \infty} a_n$ an. a) $a_n=\frac{1}{n} \cos \left(\frac{n \pi}{4}\right)$ b) $a_n=\sin \left(\frac{n \pi}{2}\right) \frac{n+2}{2 n}$

See Solution

Problem 25547

A cylinder's height decreases at 4 in/s and volume increases at 370 in³/s. Find the radius change rate when h=7 in, V=110 in³. Use $V=\pi r^{2} h$ and round to three decimal places.

See Solution

Problem 25548

A sphere's radius decreases at 6 ft/s. When its volume is 163 ft³, find the volume's rate of change using $V=\frac{4}{3} \pi r^{3}$ . Round to three decimal places.

See Solution

Problem 25549

If $|R(x)| \rightarrow \infty$ as $x$ approaches $c$ , then $x=c$ is a vertical asymptote of the graph of $R$ .

See Solution

Problem 25550

A spherical balloon's volume increases at $100 \mathrm{~cm}^{3}/\mathrm{s}$ . Find the radius growth rate when the diameter is $50 \mathrm{~cm}$ .

See Solution

Problem 25551

Untersuche die Funktion $f(x)=x-\cos (x)$ auf Nullstellen, Extremstellen und Wendestellen.

See Solution

Problem 25552

Untersuchen Sie die Funktion $f(x)=x-\cos(x)$ auf dem Intervall $[0; 2\pi]$ hinsichtlich Nullstellen, Extremstellen und Wendepunkten.

See Solution

Problem 25553

A 10 ft ladder leans against a wall. If the base moves away at 1 ft/s, how fast does the top slide down when it's 6 ft from the wall?

See Solution

Problem 25554

Find the rate at which the water level is rising in a cone-shaped tank with base radius $2 \mathrm{~m}$ and height $4 \mathrm{~m}$ , when water is being pumped in at $2 \mathrm{~m}^{3}/\mathrm{min}$ and the water is $3 \mathrm{~m}$ deep.

See Solution

Problem 25555

A sphere's radius decreases at 2 m/min. When the radius is 7 m, find the volume change rate using $V=\frac{4}{3} \pi r^{3}$ . Round to three decimal places.

See Solution

Problem 25556

A sphere's volume increases at 1146 m³/s. When $V=426$ m³, find the rate of change of the radius using $V=\frac{4}{3} \pi r^{3}$ . Round to three decimal places.

See Solution

Problem 25557

Find intervals where the graph of $f(x)=(4x^{2}-2x)e^{-2x}$ is concave down.

See Solution

Problem 25558

Determine where the graph of $f(x)=\left(-x^{2}-1\right) e^{x}$ is concave down.

See Solution

Problem 25559

Determine the intervals where the function $f(x)=-3 x^{2} e^{-3 x+2}$ is decreasing.

See Solution

Problem 25560

A man walks at $4 \mathrm{ft/s}$ , $20 \mathrm{ft}$ from a searchlight. Find the rotation rate when he is $15 \mathrm{ft}$ away.

See Solution

Problem 25561

A 10 m ladder leans against a wall. If the foot moves away at 0.2 m/s, find the speed of the top when it's 6 m from the wall.

See Solution

Problem 25562

A milk can's volume is 250 cm³. Find $r$ and $h$ relationship. Cost is $C(A)=0.0023A+\frac{400\pi}{A^2}$ .
1. Find $\frac{dC}{dA}$ .
2. Minimize cost with respect to $A$ .
3. Profit $P(A)$ is based on sales $N(A)=11300+20\sqrt{A}$ .
4. Maximize profit using a calculator.
5. Determine can dimensions for max profit.

See Solution

Problem 25563

Find intervals where the function $f(x)=x^{3}+3 x^{2}-24 x$ is decreasing and concave down.

See Solution

Problem 25564

Find when $N(t) = 5000$ for $N(t) = 4000 e^{0.0374 t}$ and when it doubles. Answers: 6 hours, 18.5 hours.

See Solution

Problem 25565

Bestimme das Volumen des Körpers, der durch die Rotation von $f(x)=x^{2}+1$ im Intervall $[1 ; 2]$ entsteht.

See Solution

Problem 25566

Find the Jacobian for the transformation $x=u v-u w$ , $y=v w+u v$ , $z=2 w-u v$ .

See Solution

Problem 25567

Evaluate the line integral using Green's Theorem for the curve defined by $C$ : $\int_{C}\left(y-6 x^{3} \tan ^{-1} x\right) d x+\left(y^{2} \sec ^{3} y-x\right) d y$ .

See Solution

Problem 25568

Find the half-life of a drug with an initial dose of 80 mg, where $A(t)=80(0.6)^{t}$ . Round to two decimal places.

See Solution

Problem 25569

Find the divergence of the vector field $\mathbf{H}=y^{2} z \cos x \mathbf{i}+y^{2} z \sin x \mathbf{j}+y^{2} z \sin (x) \mathbf{k}$ .

See Solution

Problem 25570

Find the derivative of $f(x)=e^{x-3}$ at $x=-6$ using the limit: $\lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}$ .

See Solution

Problem 25571

Model the drug concentration in blood after one pill using the equation: $\frac{d c}{d t}=-k_{1} c, c(0)=c_{0}$ .

See Solution

Problem 25572

Find the indefinite integral: $\int(\csc w+3)^{7} \csc w \cot w \, d w$ using a variable change.

See Solution

Problem 25573

Evaluate the integral: $\int(\csc w+5)^{8} \csc w \cot w \, dw$

See Solution

Problem 25574

Use $u=4 x^{2}+5 x$ to evaluate the integral $\int(8 x+5) \sqrt{4 x^{2}+5 x} d x$ . Find the result.

See Solution

Problem 25575

Calculate the integral $\int_{5}^{\infty} \frac{d x}{(x-3)^{4}}$ .

See Solution

Problem 25576

Model the drug concentration in blood after one pill using the equation $\frac{d c}{d t}=-k_{1} c, \quad c(0)=c_{0}$ . Why?

See Solution

Problem 25577

Find the interval of convergence for the series $\sum_{n=0}^{\infty} \frac{(x-3)^{n}}{5^{n}}$ .

See Solution

Problem 25578

Find the absolute extreme values of $f(x)=4 x^{\frac{3}{4}}-x$ on $[0,256]$ . What is the absolute maximum?

See Solution

Problem 25579

Solve the equation $y^{\prime}(t)=-\frac{1}{8} y$ with initial condition $y_{0}=100$ . What is $y(t)$ ?

See Solution

Problem 25580

Evaluate the integral: $\int_{0}^{9} \frac{v^{2}+12}{\sqrt{v^{3}+36 v+4}} d v=\square$ (Provide the exact answer with radicals.)

See Solution

Problem 25581

Find the derivative of the function $f(x) = x + 2x^2 - \sqrt{3}x^3$ . What is $f'(x)$ ?

See Solution

Problem 25582

Evaluate the integral $\int_{0}^{1} \frac{64 q}{\left(5 q^{2}+3\right)^{3}} d q$ using a change of variables. Find the value: $\square$ .

See Solution

Problem 25583

Bestimme die Stammfunktionen zu $f(x)=x^{2}$ mit $\int x^{2} d x$ . Lösungen sind unbestimmte Integrale wie $F(x)+C$ .

See Solution

Problem 25584

Find the relative extrema coordinates of the function $f(x) = x^{\frac{1}{5}}(x+3)^{2}$ .

See Solution

Problem 25585

Finde die zweite Ableitung von $f(x)=x+2x^{2}-\sqrt{3}x^{3}$ .

See Solution

Problem 25586

Find the second derivative of the function: $-x^{8}+\sqrt{7} x^{5}-2$ .

See Solution

Problem 25587

Calculate the sum of the series: $\sum_{n=1}^{\infty} \frac{5}{3 n^{n}}$ .

See Solution

Problem 25588

Find the tangent lines to the curve defined by $-3y = 5x^2 + xy$ at the point where $x = 2$ .

See Solution

Problem 25589

Find the derivative of the function defined by the integral $F(x)=\int_{x^{3}}^{e^{\cos x}} t^{2} dt$ .

See Solution

Problem 25590

Find the tangent line to $y = 1 - 5x + \frac{10}{x}$ at the point $(2, -4)$ .

See Solution

Problem 25591

Find the tangent line equation for $y = x(2x + 1)$ at the point $(-3, 15)$ using the derivative.

See Solution

Problem 25592

Given the piecewise function $f(x)$ , determine:
(a) The value of $k$ for continuity at $x=-2$ . (b) The type of discontinuity at $x=0$ and why. (c) All horizontal asymptotes of $f$ with justification.

See Solution

Problem 25593

Berechne den Differenzenquotienten von $f(x)=-7 x^{2}+1$ für die Intervalle $[0; \frac{1}{2}]$ , $[0; \frac{1}{4}]$ , $[0; \frac{1}{16}]$ . Vermute den Wert für $[0; \frac{1}{n}]$ bei großen $n$ .

See Solution

Problem 25594

Find the tangent line equation to the curve $y = \sqrt{x} + \frac{2}{\sqrt{x}}$ at the point $(4,3)$ .

See Solution

Problem 25595

Find the sum of the series: $\frac{3 \cdot 2}{5^{2}}+\frac{3 \cdot 2^{2}}{5^{3}}+\frac{3 \cdot 2^{3}}{5^{4}}+\cdots$

See Solution

Problem 25596

Find the rate of change of price $p$ when demand $q=2500$ and increasing at 100 pounds/month in the equation $p q^{1.5}=70000$ .

See Solution

Problem 25597

How fast did Archimedes' tub drain if it took 5 minutes to drain 360 liters? Choose: (A) 72 L/min (B) 74 L/min (C) 76 L/min (D) 78 L/min.

See Solution

Problem 25598

Bestimme die Ableitungen und klammere aus für:
a) $f(x)=3x+2e^{x}$
b) $f(x)=\frac{1}{3}e^{3x}$
c) $f(x)=3e^{4x-3}+5$
d) $f(x)=xe^{x-4}$
e) $f(x)=(2x-1)e^{-0,5x}$
f) $f(x)=x^{2}e^{4x+2}-6x^{3}$
Für $f(x)=e^{-2x+1}+1$ :
a) Berechne $f\left(\frac{1}{2}\right)$ und $f^{\prime}\left(\frac{1}{2}\right)$ .
b) Finde die Tangentengleichung bei $x=\frac{1}{2}$ und berechne den Flächeninhalt des Dreiecks.
c) Löse $5=e^{-2x+1}+1$ und interpretiere die Lösung.
Skizziere den Graphen von $f$ für:
a) $f(x)=e^{x}-2$
b) $f(x)=e^{x-2}$
c) $f(x)=-e^{-x}$
d) $f(x)=e^{-x-1}+1$
Für $f(x)=\frac{1}{12}x^{4}-\frac{1}{3}x^{3}+1$ :
a) Bestimme Extrem- und Wendepunkte.
b) Beurteile:
1. Eine Funktion fünften Grades hat immer vier Extremstellen.
2. Eine Funktion dritten Grades hat genau eine Wendestelle.

See Solution

Problem 25599

Given $f$ with $f(0)=3$ and $f^{\prime}(x)=\cos (\pi x)+x^{4}+6$ :
(a) Find $f^{\prime \prime}(-3)$ . (b) Write the tangent line equation for $y=(f(x))^{2}$ at $x=0$ . (c) Find $g^{\prime}(3)$ for $g(x)=f(\sqrt{2 x^{2}+7})$ . (d) Find $h^{\prime}(3)$ where $h$ is the inverse of $f$ .

See Solution

Problem 25600

Find the rate of increase in daily operating budget $y$ if $P=10 x^{0.1} y^{0.9}$ and $P=5000$ , $x=160$ , hiring 10 workers/year.

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 25501

Find the concavity regions for the function y=2x+72xy=\frac{2 x+7}{2 x}y=2x2x+7​, where x≠0x \neq 0x=0.

Problem 25502

A child swings at a max speed of 5.20 m/s5.20 \mathrm{~m/s}5.20 m/s and is 0.860 m0.860 \mathrm{~m}0.860 m high at the lowest point. Find his max height.

Problem 25503

Find lim⁡h→03(x+h)2+7(x+h)−3x2−7xh\lim _{h \rightarrow 0} \frac{3(x+h)^{2}+7(x+h)-3 x^{2}-7 x}{h}limh→0​h3(x+h)2+7(x+h)−3x2−7x​ treating xxx as constant.

Problem 25504

Problem 25505

Find the octopus population change over the first 4 weeks given ddtP=125(2t2−2t+1)1.2(2t−1)\frac{d}{d t} P=125(2 t^{2}-2 t+1)^{1.2}(2 t-1)dtd​P=125(2t2−2t+1)1.2(2t−1). Round to the nearest whole number.

Problem 25506

Finde die Ableitung von f(x)=43x3f(x) = \frac{4}{3 x^{3}}f(x)=3x34​.

Problem 25507

Find inflection points and concavity regions for f(x)=8e−9x2,x≥0f(x)=8 e^{-9 x^{2}}, x \geq 0f(x)=8e−9x2,x≥0.

Problem 25508

Which statement is TRUE by the Comparison Test? Analyze integrals involving ln⁡(x)\ln(x)ln(x) and cos⁡(x)\cos(x)cos(x) for convergence.

Problem 25509

Estimate the temperature change at 4.5 meters from the heater after 7 minutes, given the rates of change.

Problem 25510

Find the inflection points of f(x)=8e−9x2f(x)=8 e^{-9 x^{2}}f(x)=8e−9x2 for x≥0x \geq 0x≥0.

Problem 25511

Is the piecewise function f(x)={2x−3,−1≤x<0x−3,0≤x≤4f(x)=\begin{cases} 2x-3, & -1 \leq x < 0 \\ x-3, & 0 \leq x \leq 4 \end{cases}f(x)={2x−3,x−3,​−1≤x<00≤x≤4​ differentiable?

Problem 25512

Find the derivative of f(x)=cos⁡2(6x2)−7f(x)=\cos^{2}(6x^{2})-7f(x)=cos2(6x2)−7.

Problem 25513

Find the smallest growth rate of the fungus given by L(t)=2.8t+0.6cos⁡(πt12)L(t)=2.8 t+0.6 \cos \left(\frac{\pi t}{12}\right)L(t)=2.8t+0.6cos(12πt​).

Problem 25514

A cargo falls from a crane, taking 1.2 s1.2 \mathrm{~s}1.2 s to fall halfway. Find the total fall time to the ground.

Problem 25515

Find the distance rrr that minimizes the Leonard-Jones potential V(r)=1r6−Ar3V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}V(r)=r61​−r3A​, where A>0A>0A>0.

Problem 25516

Gegeben ist f(x)=ex+1f(x)=e^{x}+1f(x)=ex+1. a) Berechne ∫01f(x)dx\int_{0}^{1} f(x) d x∫01​f(x)dx. b) Bestimme g(x)g(x)g(x), das aus fff durch Spiegeln und Verschieben entsteht.

Problem 25517

Find the derivative of the function given by g′(x)=(x2(2x))′g^{\prime}(x)=\left(x^{2}\left(2^{x}\right)\right)^{\prime}g′(x)=(x2(2x))′.

Problem 25518

Gitt f(x)=x2−x−2x2+x−2f(x)=\frac{x^{2}-x-2}{x^{2}+x-2}f(x)=x2+x−2x2−x−2​, finn nullpunkter, derivert, polynomdivisjon, asymptoter og skisser grafen.

Problem 25519

Problem 25520

Berechnen Sie f′(−2)f^{\prime}(-2)f′(−2) für die Funktion f(x)=x2−2f(x)=x^{2}-2f(x)=x2−2 mit dem Differenzenquotienten für h→0h \rightarrow 0h→0.

Problem 25521

Bestimmen Sie die Stammfunktion von −0,03s3+0,9s2−1,7s+0,1-0,03 s^{3}+0,9 s^{2}-1,7 s+0,1−0,03s3+0,9s2−1,7s+0,1.

Problem 25522

Bestimme die Tangentengleichung t(x)t(x)t(x) bei Punkt P durch Ableitung für f(x)=x3+x2+xf(x)=x^{3}+x^{2}+xf(x)=x3+x2+x und f(x)=−x3+2x2f(x)=-x^{3}+2x^{2}f(x)=−x3+2x2.

Problem 25523

Approximate the negative root of ex=9−x2e^{x}=9-x^{2}ex=9−x2 using Newton's method to six decimal places.

Problem 25524

Evaluate the integral ∫26dxx−23\int_{2}^{6} \frac{d x}{\sqrt[3]{x-2}}∫26​3x−2​dx​ or state if it diverges. Show your work.

Problem 25525

Find the limit: lim⁡x→2(2x+3)(x2+1)\lim _{x \rightarrow 2}(2 x+3)(x^{2}+1)limx→2​(2x+3)(x2+1).

Problem 25526

Find the limit: lim⁡x→2e(x2+1)\lim _{x \rightarrow 2} e\left(x^{2}+1\right)limx→2​e(x2+1).

Problem 25527

Find the total change in price of an ice cream cone from 2000 to 2005 given the rate 18e0.04t18 e^{0.04 t}18e0.04t cents/year.

Problem 25528

Find the limits: a) lim⁡x→3x−2x−8\lim _{x \rightarrow 3} \frac{x-2}{x-8}limx→3​x−8x−2​, b) lim⁡x→2x2−51−x\lim _{x \rightarrow 2} \frac{x^{2}-5}{1-x}limx→2​1−xx2−5​.

Problem 25529

Find the limit as xxx approaches 2 for the expression (2x+3)\sqrt{(2x + 3)}(2x+3)​.

Problem 25530

Find the limits: b) lim⁡x→3(3x+7)\lim _{x \rightarrow 3}(3 x+7)limx→3​(3x+7) and c) lim⁡x→3(5x−4)(3x+7)\lim _{x \rightarrow 3} \frac{(5 x-4)}{(3 x+7)}limx→3​(3x+7)(5x−4)​.

Problem 25531

Find the limits: a) lim⁡x→3xx−3\lim _{x \rightarrow 3} \frac{x}{x-3}limx→3​x−3x​ and b) lim⁡x→34x2x−6\lim _{x \rightarrow 3} \frac{4 x}{2 x-6}limx→3​2x−64x​.

Problem 25532

Find the limits: a) lim⁡x→2x−2x2−4\lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4}limx→2​x2−4x−2​ b) lim⁡x→23x2−7x+2x−4\lim _{x \rightarrow 2} \frac{3 x^{2}-7 x+2}{x-4}limx→2​x−43x2−7x+2​

Problem 25533

Calculate the sum of the integrals: ∫−10(x2−1)dx+∫−2−1(x2−1)dx\int_{-1}^{0}(x^{2}-1) dx + \int_{-2}^{-1}(x^{2}-1) dx∫−10​(x2−1)dx+∫−2−1​(x2−1)dx.

Problem 25534

Berechne die Ableitung der Funktion f(x)=1x2f(x)=\frac{1}{x^{2}}f(x)=x21​.

Problem 25535

Find the limits as xxx approaches infinity: lim⁡x→∞(3x2+1)\lim_{x \rightarrow \infty}(3x^2 + 1)limx→∞​(3x2+1) and lim⁡x→∞(2x5+x4−1000)\lim_{x \rightarrow \infty}(2x^5 + x^4 - 1000)limx→∞​(2x5+x4−1000).

Problem 25536

Find the limits: a) lim⁡x→∞12x4\lim_{x \rightarrow \infty} \frac{1}{2x^4}limx→∞​2x41​, b) lim⁡x→∞52x\lim_{x \rightarrow \infty} \frac{5}{2x}limx→∞​2x5​, c) lim⁡x→∞1x3\lim_{x \rightarrow \infty} \frac{1}{x^3}limx→∞​x31​.

Problem 25537

Find the limits: a) lim⁡x→∞π\lim _{x \rightarrow \infty} \pilimx→∞​π b) lim⁡x→−∞2.718\lim _{x \rightarrow -\infty} 2.718limx→−∞​2.718

Problem 25538

Calculate −∫−10(x2−1)dx+∫−2−1(x2−1)dx-\int_{-1}^{0}(x^{2}-1) dx + \int_{-2}^{-1}(x^{2}-1) dx−∫−10​(x2−1)dx+∫−2−1​(x2−1)dx.

Problem 25539

Problem 25540

Beweisen Sie die Volumenformeln für (a) Kegel, (b) Kugel, (c) Kegelstumpf und erläutern Sie (d) den Hinweis zu (b).

Problem 25541

Find the limits: lim⁡x→4x−4x+2\lim _{x \rightarrow 4} \frac{x-4}{x+2}limx→4​x+2x−4​ and lim⁡x→22x−43\lim _{x \rightarrow 2} \frac{2 x-4}{3}limx→2​32x−4​.

Find the concavity regions for the function $y=\frac{2 x+7}{2 x}$ , where $x \neq 0$ .

A child swings at a max speed of $5.20 \mathrm{~m/s}$ and is $0.860 \mathrm{~m}$ high at the lowest point. Find his max height.

Find $\lim _{h \rightarrow 0} \frac{3(x+h)^{2}+7(x+h)-3 x^{2}-7 x}{h}$ treating $x$ as constant.

Find the octopus population change over the first 4 weeks given $\frac{d}{d t} P=125(2 t^{2}-2 t+1)^{1.2}(2 t-1)$ . Round to the nearest whole number.

Finde die Ableitung von $f(x) = \frac{4}{3 x^{3}}$ .

Find inflection points and concavity regions for $f(x)=8 e^{-9 x^{2}}, x \geq 0$ .

Which statement is TRUE by the Comparison Test? Analyze integrals involving $\ln(x)$ and $\cos(x)$ for convergence.

Find the inflection points of $f(x)=8 e^{-9 x^{2}}$ for $x \geq 0$ .

Is the piecewise function $f(x)=\begin{cases} 2x-3, & -1 \leq x < 0 \\ x-3, & 0 \leq x \leq 4 \end{cases}$ differentiable?

Find the derivative of $f(x)=\cos^{2}(6x^{2})-7$ .

Find the smallest growth rate of the fungus given by $L(t)=2.8 t+0.6 \cos \left(\frac{\pi t}{12}\right)$ .

A cargo falls from a crane, taking $1.2 \mathrm{~s}$ to fall halfway. Find the total fall time to the ground.

Find the distance $r$ that minimizes the Leonard-Jones potential $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ , where $A>0$ .

Gegeben ist $f(x)=e^{x}+1$ . a) Berechne $\int_{0}^{1} f(x) d x$ . b) Bestimme $g(x)$ , das aus $f$ durch Spiegeln und Verschieben entsteht.

Find the derivative of the function given by $g^{\prime}(x)=\left(x^{2}\left(2^{x}\right)\right)^{\prime}$ .

Gitt $f(x)=\frac{x^{2}-x-2}{x^{2}+x-2}$ , finn nullpunkter, derivert, polynomdivisjon, asymptoter og skisser grafen.

Berechnen Sie $f^{\prime}(-2)$ für die Funktion $f(x)=x^{2}-2$ mit dem Differenzenquotienten für $h \rightarrow 0$ .

Bestimmen Sie die Stammfunktion von $-0,03 s^{3}+0,9 s^{2}-1,7 s+0,1$ .

Bestimme die Tangentengleichung $t(x)$ bei Punkt P durch Ableitung für $f(x)=x^{3}+x^{2}+x$ und $f(x)=-x^{3}+2x^{2}$ .

Approximate the negative root of $e^{x}=9-x^{2}$ using Newton's method to six decimal places.

Evaluate the integral $\int_{2}^{6} \frac{d x}{\sqrt[3]{x-2}}$ or state if it diverges. Show your work.

Find the limit: $\lim _{x \rightarrow 2}(2 x+3)(x^{2}+1)$ .

Find the limit: $\lim _{x \rightarrow 2} e\left(x^{2}+1\right)$ .

Find the total change in price of an ice cream cone from 2000 to 2005 given the rate $18 e^{0.04 t}$ cents/year.

Find the limits: a) $\lim _{x \rightarrow 3} \frac{x-2}{x-8}$ , b) $\lim _{x \rightarrow 2} \frac{x^{2}-5}{1-x}$ .

Find the limit as $x$ approaches 2 for the expression $\sqrt{(2x + 3)}$ .

Find the limits: b) $\lim _{x \rightarrow 3}(3 x+7)$ and c) $\lim _{x \rightarrow 3} \frac{(5 x-4)}{(3 x+7)}$ .

Find the limits: a) $\lim _{x \rightarrow 3} \frac{x}{x-3}$ and b) $\lim _{x \rightarrow 3} \frac{4 x}{2 x-6}$ .

Find the limits: a) $\lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4}$ b) $\lim _{x \rightarrow 2} \frac{3 x^{2}-7 x+2}{x-4}$

Calculate the sum of the integrals: $\int_{-1}^{0}(x^{2}-1) dx + \int_{-2}^{-1}(x^{2}-1) dx$ .

Berechne die Ableitung der Funktion $f(x)=\frac{1}{x^{2}}$ .

Find the limits as $x$ approaches infinity: $\lim_{x \rightarrow \infty}(3x^2 + 1)$ and $\lim_{x \rightarrow \infty}(2x^5 + x^4 - 1000)$ .

Find the limits: a) $\lim_{x \rightarrow \infty} \frac{1}{2x^4}$ , b) $\lim_{x \rightarrow \infty} \frac{5}{2x}$ , c) $\lim_{x \rightarrow \infty} \frac{1}{x^3}$ .

Find the limits: a) $\lim _{x \rightarrow \infty} \pi$ b) $\lim _{x \rightarrow -\infty} 2.718$

Calculate $-\int_{-1}^{0}(x^{2}-1) dx + \int_{-2}^{-1}(x^{2}-1) dx$ .

Find the limits: $\lim _{x \rightarrow 4} \frac{x-4}{x+2}$ and $\lim _{x \rightarrow 2} \frac{2 x-4}{3}$ .

Bestem stammfunksjonen for $A=\int_{0}^{2,356} \sin (x)-0,3 x \, dx$ og beregn integralet.

Find the radius change rate when the circle area is $25 \pi$ in², decreasing at 205 in²/min. Round to 3 decimals.

A cylinder's radius increases at 8 cm/min, with a constant volume of 174 cm³. Find the height's change rate when height is 7 cm. Use $V=\pi r^{2} h$ and round to three decimal places.

A cylinder's height decreases at 4 in/s and volume increases at 370 in³/s. Find the radius change rate when h=7 in, V=110 in³. Use $V=\pi r^{2} h$ and round to three decimal places.

A sphere's radius decreases at 6 ft/s. When its volume is 163 ft³, find the volume's rate of change using $V=\frac{4}{3} \pi r^{3}$ . Round to three decimal places.

If $|R(x)| \rightarrow \infty$ as $x$ approaches $c$ , then $x=c$ is a vertical asymptote of the graph of $R$ .

A spherical balloon's volume increases at $100 \mathrm{~cm}^{3}/\mathrm{s}$ . Find the radius growth rate when the diameter is $50 \mathrm{~cm}$ .

Untersuche die Funktion $f(x)=x-\cos (x)$ auf Nullstellen, Extremstellen und Wendestellen.

Untersuchen Sie die Funktion $f(x)=x-\cos(x)$ auf dem Intervall $[0; 2\pi]$ hinsichtlich Nullstellen, Extremstellen und Wendepunkten.

Find the rate at which the water level is rising in a cone-shaped tank with base radius $2 \mathrm{~m}$ and height $4 \mathrm{~m}$ , when water is being pumped in at $2 \mathrm{~m}^{3}/\mathrm{min}$ and the water is $3 \mathrm{~m}$ deep.

A sphere's radius decreases at 2 m/min. When the radius is 7 m, find the volume change rate using $V=\frac{4}{3} \pi r^{3}$ . Round to three decimal places.

A sphere's volume increases at 1146 m³/s. When $V=426$ m³, find the rate of change of the radius using $V=\frac{4}{3} \pi r^{3}$ . Round to three decimal places.

Find intervals where the graph of $f(x)=(4x^{2}-2x)e^{-2x}$ is concave down.

Determine where the graph of $f(x)=\left(-x^{2}-1\right) e^{x}$ is concave down.

Determine the intervals where the function $f(x)=-3 x^{2} e^{-3 x+2}$ is decreasing.

A man walks at $4 \mathrm{ft/s}$ , $20 \mathrm{ft}$ from a searchlight. Find the rotation rate when he is $15 \mathrm{ft}$ away.

Find intervals where the function $f(x)=x^{3}+3 x^{2}-24 x$ is decreasing and concave down.

Find when $N(t) = 5000$ for $N(t) = 4000 e^{0.0374 t}$ and when it doubles. Answers: 6 hours, 18.5 hours.

Bestimme das Volumen des Körpers, der durch die Rotation von $f(x)=x^{2}+1$ im Intervall $[1 ; 2]$ entsteht.

Find the Jacobian for the transformation $x=u v-u w$ , $y=v w+u v$ , $z=2 w-u v$ .

Evaluate the line integral using Green's Theorem for the curve defined by $C$ : $\int_{C}\left(y-6 x^{3} \tan ^{-1} x\right) d x+\left(y^{2} \sec ^{3} y-x\right) d y$ .

Find the half-life of a drug with an initial dose of 80 mg, where $A(t)=80(0.6)^{t}$ . Round to two decimal places.

Find the divergence of the vector field $\mathbf{H}=y^{2} z \cos x \mathbf{i}+y^{2} z \sin x \mathbf{j}+y^{2} z \sin (x) \mathbf{k}$ .

Find the derivative of $f(x)=e^{x-3}$ at $x=-6$ using the limit: $\lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}$ .

Model the drug concentration in blood after one pill using the equation: $\frac{d c}{d t}=-k_{1} c, c(0)=c_{0}$ .

Find the indefinite integral: $\int(\csc w+3)^{7} \csc w \cot w \, d w$ using a variable change.

Evaluate the integral: $\int(\csc w+5)^{8} \csc w \cot w \, dw$

Use $u=4 x^{2}+5 x$ to evaluate the integral $\int(8 x+5) \sqrt{4 x^{2}+5 x} d x$ . Find the result.