Calculus

Problem 17101

Calculate the integral: $\int \frac{\operatorname{sen}^{3} x}{\sqrt{\cos x}} d x$

See Solution

Problem 17102

When the stock price $\mathrm{P}(\mathrm{t})$ is at its highest, what are the signs of $\mathrm{P}^{\prime}(\mathrm{t})$ and $\mathrm{P}^{\prime \prime}(\mathrm{t})$ ? Explain.

See Solution

Problem 17103

Find $f^{\prime}(x)$ for $f(x)=3 e^{x^{2}-9 x+4}$ at $x=0$ and where the tangent is horizontal.

See Solution

Problem 17104

Find the increase in cost from producing 0 to 900 bikes using the marginal cost function $C^{\prime}(x)=200-\frac{x}{3}$ . Set up and evaluate the integral. The increase in cost is \$ \square (round to the nearest dollar).

See Solution

Problem 17105

Fish Population Dynamics: Given $p(N)=\frac{r}{1+0.1 N}$ , find equilibria for $N_{t+1}=\left(\frac{r}{1+0.1 N_{t}}\right) N_{t}$ with $r=2.5$ . Then check stability using the derivative test.

See Solution

Problem 17106

Determine where the graph of $f(x) = x^{12} + 9x^{2}$ is concave up, concave down, and find inflection points.

See Solution

Problem 17107

Bestimmen Sie die Ableitung der folgenden Funktionen: a) $f(x)=3 x^{3}+2 x^{2}$ , b) $f(t)=t^{4}-t+1$ , c) $f(x)=(3+x)(3-x)$ , d) $f(x)=3 x^{3} \cdot 2 x^{2}$ , e) $f(x)=3 x^{2}-4 a^{2}$ , f) $f(a)=3 x^{2}-4 a^{2}$ , g) $f(x)=2(x-a)^{2}$ , h) $f(t)=t^{2}-(2-t)^{2}$ .

See Solution

Problem 17108

Find the limit: $\lim _{n \rightarrow \infty} \prod_{k=2}^{n}\left(1+\frac{1}{k}\right)=$

See Solution

Problem 17109

Find $f^{\prime}(x)$ and the tangent line at $x=2$ for $f(x)=\frac{9 x}{2^{x}}$ .

See Solution

Problem 17110

Find the derivative of $5 - x^{2} = \sin(x \cdot y^{2})$ with respect to $x$ using implicit differentiation.

See Solution

Problem 17111

Find the indefinite integral: $\int(r-7)(r+7) \, dr$ (use $C$ for the constant).

See Solution

Problem 17112

Calculate the indefinite integral and use $C$ for the constant: $\int\left(\frac{8}{z^{5}}+\frac{3}{\sqrt{z}}\right) dz$

See Solution

Problem 17113

Calculate the indefinite integral: $\int\left(5 \sqrt[3]{t^{2}}+\frac{1}{\sqrt[3]{t^{2}}}\right) d t$ . Use constant $C$ .

See Solution

Problem 17114

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{9 x}{2^{x}}$ .

See Solution

Problem 17115

Find $f^{\prime}(x)$ and the tangent line at $x=2$ for $f(x)=x(1-x)^{2}$ .

See Solution

Problem 17116

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{e^{x}}{7 x^{2}+1}$ .

See Solution

Problem 17117

Identify which of the following series statements are implied by the $\boldsymbol{n}$ -th term test. Select all that apply:
1. $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}$ converges.
2. $\sum_{n=0}^{\infty} \frac{n-1}{n+1}$ diverges.
3. $\sum_{n=0}^{\infty} e^{-2 n^{2}+3}$ converges.
4. $\sum_{n=0}^{\infty} \frac{1}{n^{2}+3 n+2}$ converges.
5. $\sum_{n=1}^{\infty}(-1)^{n} \frac{n+1}{3 n+2}$ diverges.
6. $\sum^{\infty} \frac{1}{n}$ diverges.

See Solution

Problem 17118

Find the antiderivative of $\frac{d x}{d t}=8 e^{t}-3$ with the condition $x(0)=5$ . What is $x(t)=\square$ ?

See Solution

Problem 17119

Find the limit: $\lim _{n \rightarrow \infty} \sqrt{n^{2}+3 n}-n$ .

See Solution

Problem 17120

Bestimmen Sie die Ableitungen von $f(x)$ für: a) $e^{-4 x}$ , b) $e^{x^{2}}$ , c) $e^{2 x+1}$ , d) $e^{-\sqrt{x}}$ , e) $2 \cdot e^{0,5 x}$ , f) $x-e^{x^{3}}$ , g) $(1-x) \cdot e^{x}$ , h) $x^{2} \cdot e^{-x}$ .

See Solution

Problem 17121

A patient has a temperature of 104°F changing at $t^{2}-3t$ °/hr for $0 \leq t \leq 3$ .
(a) Find $T(t)$ . (b) Calculate $T(2)$ and round to one decimal place.

See Solution

Problem 17122

Bestimmen Sie die Stammfunktion für die folgenden Funktionen: a) $f(x)=\frac{1}{4} x^{-2}$ , b) $f(x)=\frac{3}{x^{4}}$ , c) $f(x)=0,3 x^{-10}$ , d) $f(x)=0,15 x^{-6}$ , e) $f(x)=\frac{-5}{x^{6}}$ , f) $f(x)=\frac{x^{2}+2 x}{x^{5}}$ , g) $f(x)=\frac{x(x+1)}{x^{4}}$ , h) $f(x)=\frac{6 x^{2}-21 x}{3 x^{7}}$ .

See Solution

Problem 17123

Der DAX sinkt jede Woche um $2.9\%$ . Nach 100 Wochen, was ist sein Wert in Dezimalform? Geben Sie das Ergebnis gerundet an!

See Solution

Problem 17124

Explain when the minimum of a function $f(x, y)$ is higher than its maximum. What type of extrema is this? Provide an example and graph.

See Solution

Problem 17125

Find the derivatives of these functions:
1. $f(x)=-2 x^{4}+3 x^{2}-4 x+2$ , $f(x)=\frac{1}{4} x^{4}+\frac{3}{8} x^{3}-\frac{4}{5} x^{2}$ , $f(x)=-x^{4}-6 x^{2}-2.25$ , $f(x)=-(x-6)^{2}(x+1)$ , $f(x)=\frac{1}{16}(x^{5}+x^{3}-1)$ , $I(t)=0.125 t^{4}-1.5 t^{2}-3$ , $f(x)=a x^{2}+b x+c+\frac{d}{x}$ .
2. $f(x)=6 e^{x}+7$ , $f(x)=\frac{1}{2} x^{2}+4 x+5 e^{x}$ , $f(x)=e^{x}(e^{-x}-3)$ .
3. $f(x)=2 x-3 \sin (x)+4$ , $f(x)=4 \cos (x)-8 \sin (x)$ .
4. Find the instantaneous rate of change for: $f(x)=1.25(\cos (x)+2 e^{x}-1)$ , $f(x)=\frac{a}{2}(x-3)+a e^{x}$ , $f(u)=7 \sin (u)-3 e^{u}+\sin (0.5)+3 e^{2}$ .

See Solution

Problem 17126

Find the limit infimum: $\varliminf_{n \rightarrow+\infty}\left(\frac{(3 n-1) \cos (n \pi)}{n}+7\right)=$

See Solution

Problem 17127

Find the rate of change of the bacteria population, given by $N(x)=\frac{12100}{1+99 e^{-0.25 x}}$ , at $x=4$ hours. Round to the nearest integer.

See Solution

Problem 17128

Find the derivatives of these functions: a) $f(x)=6 e^{x}+7$ , c) $f(x)=\frac{1}{2} x^{2}+4 x+5 e^{x}$ , e) $f(x)=e^{x}(e^{-x}-3)$ .

See Solution

Problem 17129

Differentiate these functions: a) $f(x)=2x-3\sin(x)+4$ b) $f(x)=4\cos(x)-8\sin(x)$

See Solution

Problem 17130

A pyramid's volume is $V=\frac{x^{2} h}{3}$ . If $h=150$ m and $x=60$ m, find the volume's decrease rate with $dx/dt=-0.004$ m/yr.

See Solution

Problem 17131

Find the rate of change of the bacteria population, $N(x)=\frac{12000}{1+99 e^{-0.25 x}}$ , at $x=1$ hour. Round to the nearest integer.

See Solution

Problem 17132

Find the rate of change of the bacteria population at $x=1$ hour using $N(x)=\frac{12000}{1+99 e^{-0.25 x}}$ .

See Solution

Problem 17133

Find the rate of change of the bacteria population, given $N(x)=\frac{13000}{1+99 e^{-0.3 x}}$ , at $x=3$ .

See Solution

Problem 17134

Berechnen Sie den Grenzwert der Reihe: $\lim _{N \rightarrow+\infty} \sum_{k=2}^{N} \frac{4 \cdot 3^{k}}{2^{2 k}}$

See Solution

Problem 17135

What are the units and meaning of $\left(g^{-1}\right)^{\prime}(20)$ for the average daily emails $g(t)$ in millions?

See Solution

Problem 17136

Find the derivatives of these functions:
a) $f(x)=-2 x^{4}+3 x^{2}-4 x+2$
c) $f(x)=\frac{1}{4} x^{4}+\frac{3}{8} x^{3}-\frac{4}{5} x^{2$
e) $f(x)=-x^{4}-6 x^{2}-2.25$
g) $f(x)=-(x-6)^{2}(x+1)$
i) $f(x)=\frac{1}{16}(x^{5}+x^{3}-1)$
k) $I(t)=0.125 t^{4}-1.5 t^{2}-3$
m) $f(x)=a x^{2}+b x+c+\frac{d}{x}$

See Solution

Problem 17137

Calculate $3 \sum_{k=1}^{\infty} \frac{1}{(k+1)^{2}}$ .

See Solution

Problem 17138

Find the intervals where the function $g(x)=7(7 e^{-2 x}+3)^{2}$ is increasing or decreasing using $g^{\prime}(x)$ .

See Solution

Problem 17139

Find the integral of $\frac{2 x+5}{\sqrt{x^{2}+5 x+7}}$ with respect to $x$ .

See Solution

Problem 17140

Design a closed rectangular crate with a square base and volume $5184 \mathrm{ft}^{3}$ . Minimize material cost: top/sides at \$2/ft², bottom at \$10/ft². What are the dimensions?

See Solution

Problem 17141

Find the absolute extrema of $f(x)=\ln(8x^{2}-2x-7)$ on $[6.4,18.9]$ . Round answers to two decimal places.

See Solution

Problem 17142

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

See Solution

Problem 17143

Find the integral of $\frac{1}{x \ln x}$ with respect to $x$ .

See Solution

Problem 17144

Leite die Funktion $f(x)$ für die folgenden Fälle ab: a) $2 \cdot e^{x}$ , b) $e^{x}+x^{2}$ , c) $\frac{1}{4} \cdot e^{x}$ , d) $-e^{x}$ , e) $e^{x}+e^{x}$ , f) $1-e^{x}$ , g) $-1,5 \cdot e^{x}$ , h) $3^{2} \cdot e^{x}$ .

See Solution

Problem 17145

Evaluate the following limits: a. $\lim _{x \rightarrow-1}(9-x^{2})$ b. $\lim _{x \rightarrow 0} \sqrt{\frac{x+20}{2 x+5}}$ c. $\lim _{x \rightarrow 5} \sqrt{x-1}$

See Solution

Problem 17146

Evaluate the following limits and explain if any do not exist: a. $\lim _{x \rightarrow 0^{+}} x^{4}$ b. $\lim _{x \rightarrow 2^{-}}(x^{2}-4)$ c. $\lim _{x \rightarrow 3^{-}}(x^{2}-4)$ d. $\lim _{x \rightarrow 1^{+}} \frac{1}{x-3}$ e. $\lim _{x \rightarrow 3^{+}} \frac{1}{x+2}$ f. $\lim _{x \rightarrow 3} \frac{1}{x-3}$

See Solution

Problem 17147

Find the integral of $\ln \left(x^{7}\right) d x$ .

See Solution

Problem 17148

Evaluate the integral $\int 3x e^{3x} dx$ .

See Solution

Problem 17149

Find the time $t$ (from 6 A.M.) when the speed $f(t)=20t-40\sqrt{t}+50$ is lowest, and determine that speed.

See Solution

Problem 17150

Find the integral of $x\left(x^{2}+3\right)^{\frac{5}{4}}$ with respect to $x$ .

See Solution

Problem 17151

Evaluate the integral: $\int \ln \left(x^{3}\right) d x$

See Solution

Problem 17152

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

See Solution

Problem 17153

Minimize average cost for $C(x)=0.2(0.01 x^{2}+120)$ , where $x$ is units produced. What is the optimal production level?

See Solution

Problem 17154

Evaluate the integral of $\frac{\ln x}{x}$ with respect to $x$ : $\int \frac{\ln x}{x} d x$ .

See Solution

Problem 17155

Bestimmen Sie die Ableitungen von $f(x)=-0,5 x^{4}+4 x^{2}-x+1$ , $g(x)=-0,5 x^{4}+4 x^{2}-x+8$ , $h(x)=-0,5 x^{4}+4 x^{2}-x-2$ und vergleichen Sie diese.

See Solution

Problem 17156

Maximize revenue: Show that if $R(x)$ is concave down ( $R''(x)<0$ ), then $\bar{R}(x)=R'(x)$ gives max average revenue.

See Solution

Problem 17157

Maximize profit for the Pulsar TV with demand $p=-0.05 x+600$ and cost $C(x)=0.000002 x^{3}-0.03 x^{2}+400 x+80,000$ . Find optimal $x$ .

See Solution

Problem 17158

Die Funktion $s(t)=\frac{1}{6} t^{3}+\frac{1}{2} t^{2}$ beschreibt die Bewegung eines Objekts. Berechne die mittlere und Momentangeschwindigkeit nach 3 Sekunden und erkläre den Unterschied.

See Solution

Problem 17159

Find the production level $x$ for maximum profit given $P^{\prime}(x)=-0.004 x+20$ and fixed costs of \$16,000/month. What is the max profit?

See Solution

Problem 17160

Bestimme $a$ und $b$ für $f(x)=x^{3}+a x^{2}+x+b$ mit $f(0)=-2$ und $f'(1)=0$ . Berechne dann $\int_{0}^{2} f(x) d x$ und das Volumen $V$ für $x_{1}=0$ und $x_{2}=2$ .

See Solution

Problem 17161

Calculate the integral from 2 to 4 of the constant function 3: $\int_{2}^{4} 3 d x$ .

See Solution

Problem 17162

Find the integral $\int \frac{x^{4}}{1-x^{5}} d x$ .

See Solution

Problem 17163

Find the derivative of $f(x)=x^{4 x}$ using logarithmic differentiation. What is $f^{\prime}(x)=?$

See Solution

Problem 17164

Find the life expectancy function $g(t)$ given $g'(t)=\frac{5.45218}{(1+1.09 t)^{0.9}}$ and $g(0)=50.02$ . What is $g(100)$ ?

See Solution

Problem 17165

Find the slope of the tangent line to $2 x^{6}+9 x y-8 y^{3}=-1780$ at $(-1,6)$ . The slope is $\square$ .

See Solution

Problem 17166

Find the first partial derivative of $f(x, y)=(7 x^{2}+3 y)^{3}$ with respect to $x$ .

See Solution

Problem 17167

Find the first partial derivative of $f(x, y)=\ln \left(\sqrt{3 x+y^{2}}\right)$ with respect to $y$ .

See Solution

Problem 17168

Identify which statements about series are implied by the $n$ -th term test. Select all that apply.

See Solution

Problem 17169

Gegeben sind die Funktionen $f(x)=0,75 x^{3}-1,25 x^{2}-1$ und $g(x)=x^{2}-1$ . a) Zeichne die Graphen von $f$ und $g$ . b) Berechne $\int_{0}^{3}(g(x)-f(x)) d x$ und erkläre das Ergebnis. c) Bestimme die Bogenlänge von $f$ für $a=0$ und $b=3$ mit $L=\int_{a}^{b} \sqrt{1+\left(f^{\prime}(x)\right)^{2}} \mathrm{~d} x$ .

See Solution

Problem 17170

Find the first derivative of $\frac{x^{3}-5 x^{2}}{2 x \sqrt{x}}$ with respect to $x$ . Which option is correct?

See Solution

Problem 17171

Find the second partial derivatives $f_{xx}, f_{yy}$ , and $f_{xy}$ for the function $f=\ln (5 x)+3 x \ln (y)$ .

See Solution

Problem 17172

Find the first derivative with respect to $x$ of the expression $\frac{x^{3}-5 x^{2}}{2 x \sqrt{x}}$ .

See Solution

Problem 17173

Find the first derivative of $\frac{x^{3}-5 x^{2}}{2 x \sqrt{x}}$ with respect to $x$ . Choose from options A-F.

See Solution

Problem 17174

What is the increase in satisfaction from consuming one more unit of good $x$ when $x=20$ and $y=30$ for $U(x, y)=2 \ln x+\ln y$ ?

See Solution

Problem 17175

Find the first derivative of $\frac{x^{3}-5 x^{2}}{2 x \sqrt{x}}$ .

See Solution

Problem 17176

Find the first derivative of $\frac{x^{3}-5 x^{2}}{2 x \sqrt{x}}$ with respect to $x$ . Choose the correct option.

See Solution

Problem 17177

Encuentra la expresión de la primera derivada respecto a $x$ de $\frac{x^{3}-5 x^{2}}{2 x \sqrt{x}}$ .

See Solution

Problem 17178

Find the first derivative of $\frac{x^{3}-5 x^{2}}{2 x \sqrt{x}}$ with respect to $x$ . Which option is correct?

See Solution

Problem 17179

Find the infinite sum $S=\sum_{n=1}^{\infty} a_{n}$ given $S_{T}=\frac{2 T-3}{T+5}$ . Options: 0, 1, 2, 3, 4, 5.

See Solution

Problem 17180

Find the derivative of the function $f(x)=5 \sin^{-1}(x^{2})$ . What is $f'(x)=\square$ ?

See Solution

Problem 17181

Find the derivative of $g(x)=\ln (\sqrt[5]{x})$ and set it equal to $I$ .

See Solution

Problem 17182

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

See Solution

Problem 17183

Find the limit: $\lim _{x \rightarrow \infty} \frac{5 x^{2}-7 x}{1-3 x^{2}}$

See Solution

Problem 17184

Find the derivative of the function $f(x)=x^{2} \tan^{-1}(3x)$ . What is $f'(x)$ ?

See Solution

Problem 17185

Calculate the limit: $\lim _{n \rightarrow \infty} n\left(\sqrt[3]{n^{3}+n+2}-n\right)$ .

See Solution

Problem 17186

A rodeo rider is thrown off a bronco from $1.6 \mathrm{~m}$ high with an initial speed of $4.0 \mathrm{~m/s}$ . Find: (a) maximum height reached, (b) speed upon hitting the ground.

See Solution

Problem 17187

Calculate the indefinite integral and include the constant $C$ : $\int\left(\frac{8}{z^{5}}+\frac{3}{\sqrt{z}}\right) d z$

See Solution

Problem 17188

Calculate the indefinite integral and include the constant $C$ for the result:
$\int(24 x^{3}+8 x-5) \, dx$

See Solution

Problem 17189

Calculate the indefinite integral: $\int(9 \sqrt{x}-3) \, dx$ and include the constant $C$ .

See Solution

Problem 17190

Calculate the integral $\int_{0}^{3} x^{2} d x$ .

See Solution

Problem 17191

Differentiate the function $x e^{\ln x^{2}}$ with respect to $x$ .

See Solution

Problem 17192

If $f$ and $g$ are twice differentiable with $g(x)=c^{f(x)}$ and $g^{\prime}(x)=h(x) c^{\prime(x)}$ , find $h(x)$ .

See Solution

Problem 17193

Find $\frac{d}{d x}[g(f^{2}(x))]$ at $x=16$ using the values for $f$ and $g$ provided in the table.

See Solution

Problem 17194

Find the demand elasticity $E$ at price $p=6$ and the price $p$ where elasticity is unitary.
(a) $E(6)=\square$ (b) $p=\square$

See Solution

Problem 17195

Find the cost, average cost, and marginal cost at production level 1650 for $C(x)=78400+400x+x^{2}$ . Also, determine the production level that minimizes average cost and the minimal average cost.

See Solution

Problem 17196

Find the profit from selling 26 answering machines using $P(x)=20x-0.9x^2-240$ . Also, approximate using marginal profit.

See Solution

Problem 17197

Given $f(x)=2 x \sqrt{4 x^{2}+1}$ , find critical values and intervals for increasing and decreasing behavior.

See Solution

Problem 17198

Find the demand elasticity for $Q=47 \sqrt{32-p^{2}}$ at $p=2$ and determine when it is unitary.

See Solution

Problem 17199

Evaluate the integral $\int_{0}^{2} x\left(x^{2}-1\right)^{3} d x$ .

See Solution

Problem 17200

Find the mass of a radioactive material at $t=0$ and after 20 years using $m(t)=225 e^{-0.045 t}$ .

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 17101

Calculate the integral: ∫sen⁡3xcos⁡xdx\int \frac{\operatorname{sen}^{3} x}{\sqrt{\cos x}} d x∫cosx​sen3x​dx

Problem 17102

When the stock price P(t)\mathrm{P}(\mathrm{t})P(t) is at its highest, what are the signs of P′(t)\mathrm{P}^{\prime}(\mathrm{t})P′(t) and P′′(t)\mathrm{P}^{\prime \prime}(\mathrm{t})P′′(t)? Explain.

Problem 17103

Find f′(x)f^{\prime}(x)f′(x) for f(x)=3ex2−9x+4f(x)=3 e^{x^{2}-9 x+4}f(x)=3ex2−9x+4 at x=0x=0x=0 and where the tangent is horizontal.

Problem 17104

Find the increase in cost from producing 0 to 900 bikes using the marginal cost function C′(x)=200−x3C^{\prime}(x)=200-\frac{x}{3}C′(x)=200−3x​. Set up and evaluate the integral. The increase in cost is \$ \square (round to the nearest dollar).

Problem 17105

Fish Population Dynamics: Given p(N)=r1+0.1Np(N)=\frac{r}{1+0.1 N}p(N)=1+0.1Nr​, find equilibria for Nt+1=(r1+0.1Nt)NtN_{t+1}=\left(\frac{r}{1+0.1 N_{t}}\right) N_{t}Nt+1​=(1+0.1Nt​r​)Nt​ with r=2.5r=2.5r=2.5. Then check stability using the derivative test.

Problem 17106

Determine where the graph of f(x)=x12+9x2f(x) = x^{12} + 9x^{2}f(x)=x12+9x2 is concave up, concave down, and find inflection points.

Problem 17107

Problem 17108

Find the limit: lim⁡n→∞∏k=2n(1+1k)=\lim _{n \rightarrow \infty} \prod_{k=2}^{n}\left(1+\frac{1}{k}\right)=limn→∞​∏k=2n​(1+k1​)=

Problem 17109

Find f′(x)f^{\prime}(x)f′(x) and the tangent line at x=2x=2x=2 for f(x)=9x2xf(x)=\frac{9 x}{2^{x}}f(x)=2x9x​.

Problem 17110

Find the derivative of 5−x2=sin⁡(x⋅y2)5 - x^{2} = \sin(x \cdot y^{2})5−x2=sin(x⋅y2) with respect to xxx using implicit differentiation.

Problem 17111

Find the indefinite integral: ∫(r−7)(r+7) dr\int(r-7)(r+7) \, dr∫(r−7)(r+7)dr (use CCC for the constant).

Problem 17112

Calculate the indefinite integral and use CCC for the constant: ∫(8z5+3z)dz\int\left(\frac{8}{z^{5}}+\frac{3}{\sqrt{z}}\right) dz∫(z58​+z​3​)dz

Problem 17113

Calculate the indefinite integral: ∫(5t23+1t23)dt\int\left(5 \sqrt[3]{t^{2}}+\frac{1}{\sqrt[3]{t^{2}}}\right) d t∫(53t2​+3t2​1​)dt. Use constant CCC.

Problem 17114

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=9x2xf(x)=\frac{9 x}{2^{x}}f(x)=2x9x​.

Problem 17115

Find f′(x)f^{\prime}(x)f′(x) and the tangent line at x=2x=2x=2 for f(x)=x(1−x)2f(x)=x(1-x)^{2}f(x)=x(1−x)2.

Problem 17116

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=ex7x2+1f(x)=\frac{e^{x}}{7 x^{2}+1}f(x)=7x2+1ex​.

Problem 17117

Problem 17118

Find the antiderivative of dxdt=8et−3\frac{d x}{d t}=8 e^{t}-3dtdx​=8et−3 with the condition x(0)=5x(0)=5x(0)=5. What is x(t)=□x(t)=\squarex(t)=□?

Problem 17119

Find the limit: lim⁡n→∞n2+3n−n\lim _{n \rightarrow \infty} \sqrt{n^{2}+3 n}-nlimn→∞​n2+3n​−n.

Problem 17120

Problem 17121

A patient has a temperature of 104°F changing at t2−3tt^{2}-3tt2−3t °/hr for 0≤t≤30 \leq t \leq 30≤t≤3. (a) Find T(t)T(t)T(t). (b) Calculate T(2)T(2)T(2) and round to one decimal place.

Problem 17122

Problem 17123

Der DAX sinkt jede Woche um 2.9%2.9\%2.9%. Nach 100 Wochen, was ist sein Wert in Dezimalform? Geben Sie das Ergebnis gerundet an!

Problem 17124

Explain when the minimum of a function f(x,y)f(x, y)f(x,y) is higher than its maximum. What type of extrema is this? Provide an example and graph.

Problem 17125

Problem 17126

Find the limit infimum: lim‾⁡n→+∞((3n−1)cos⁡(nπ)n+7)=\varliminf_{n \rightarrow+\infty}\left(\frac{(3 n-1) \cos (n \pi)}{n}+7\right)=lim​n→+∞​(n(3n−1)cos(nπ)​+7)=

Problem 17127

Find the rate of change of the bacteria population, given by N(x)=121001+99e−0.25xN(x)=\frac{12100}{1+99 e^{-0.25 x}}N(x)=1+99e−0.25x12100​, at x=4x=4x=4 hours. Round to the nearest integer.

Problem 17128

Find the derivatives of these functions: a) f(x)=6ex+7f(x)=6 e^{x}+7f(x)=6ex+7, c) f(x)=12x2+4x+5exf(x)=\frac{1}{2} x^{2}+4 x+5 e^{x}f(x)=21​x2+4x+5ex, e) f(x)=ex(e−x−3)f(x)=e^{x}(e^{-x}-3)f(x)=ex(e−x−3).

Problem 17129

Differentiate these functions: a) f(x)=2x−3sin⁡(x)+4f(x)=2x-3\sin(x)+4f(x)=2x−3sin(x)+4 b) f(x)=4cos⁡(x)−8sin⁡(x)f(x)=4\cos(x)-8\sin(x)f(x)=4cos(x)−8sin(x)

Problem 17130

A pyramid's volume is V=x2h3V=\frac{x^{2} h}{3}V=3x2h​. If h=150h=150h=150 m and x=60x=60x=60 m, find the volume's decrease rate with dx/dt=−0.004dx/dt=-0.004dx/dt=−0.004 m/yr.

Problem 17131

Find the rate of change of the bacteria population, N(x)=120001+99e−0.25xN(x)=\frac{12000}{1+99 e^{-0.25 x}}N(x)=1+99e−0.25x12000​, at x=1x=1x=1 hour. Round to the nearest integer.

Problem 17132

Find the rate of change of the bacteria population at x=1x=1x=1 hour using N(x)=120001+99e−0.25xN(x)=\frac{12000}{1+99 e^{-0.25 x}}N(x)=1+99e−0.25x12000​.

Problem 17133

Find the rate of change of the bacteria population, given N(x)=130001+99e−0.3xN(x)=\frac{13000}{1+99 e^{-0.3 x}}N(x)=1+99e−0.3x13000​, at x=3x=3x=3.

Problem 17134

Berechnen Sie den Grenzwert der Reihe: lim⁡N→+∞∑k=2N4⋅3k22k\lim _{N \rightarrow+\infty} \sum_{k=2}^{N} \frac{4 \cdot 3^{k}}{2^{2 k}}N→+∞lim​k=2∑N​22k4⋅3k​

Problem 17135

What are the units and meaning of (g−1)′(20)\left(g^{-1}\right)^{\prime}(20)(g−1)′(20) for the average daily emails g(t)g(t)g(t) in millions?

Problem 17136

Problem 17137

Calculate 3∑k=1∞1(k+1)23 \sum_{k=1}^{\infty} \frac{1}{(k+1)^{2}}3∑k=1∞​(k+1)21​.

Problem 17138

Find the intervals where the function g(x)=7(7e−2x+3)2g(x)=7(7 e^{-2 x}+3)^{2}g(x)=7(7e−2x+3)2 is increasing or decreasing using g′(x)g^{\prime}(x)g′(x).

Problem 17139

Find the integral of 2x+5x2+5x+7\frac{2 x+5}{\sqrt{x^{2}+5 x+7}}x2+5x+7​2x+5​ with respect to xxx.

Problem 17140

Design a closed rectangular crate with a square base and volume 5184ft35184 \mathrm{ft}^{3}5184ft3. Minimize material cost: top/sides at \$2/ft², bottom at \$10/ft². What are the dimensions?

Problem 17141

Find the absolute extrema of f(x)=ln⁡(8x2−2x−7)f(x)=\ln(8x^{2}-2x-7)f(x)=ln(8x2−2x−7) on [6.4,18.9][6.4,18.9][6.4,18.9]. Round answers to two decimal places.

Problem 17142

Find the integral of xexx e^{x}xex with respect to xxx.

Problem 17143

Calculate the integral: $\int \frac{\operatorname{sen}^{3} x}{\sqrt{\cos x}} d x$

When the stock price $\mathrm{P}(\mathrm{t})$ is at its highest, what are the signs of $\mathrm{P}^{\prime}(\mathrm{t})$ and $\mathrm{P}^{\prime \prime}(\mathrm{t})$ ? Explain.

Find $f^{\prime}(x)$ for $f(x)=3 e^{x^{2}-9 x+4}$ at $x=0$ and where the tangent is horizontal.

Find the increase in cost from producing 0 to 900 bikes using the marginal cost function $C^{\prime}(x)=200-\frac{x}{3}$ . Set up and evaluate the integral. The increase in cost is \$ \square (round to the nearest dollar).

Fish Population Dynamics: Given $p(N)=\frac{r}{1+0.1 N}$ , find equilibria for $N_{t+1}=\left(\frac{r}{1+0.1 N_{t}}\right) N_{t}$ with $r=2.5$ . Then check stability using the derivative test.

Determine where the graph of $f(x) = x^{12} + 9x^{2}$ is concave up, concave down, and find inflection points.

Find the limit: $\lim _{n \rightarrow \infty} \prod_{k=2}^{n}\left(1+\frac{1}{k}\right)=$

Find $f^{\prime}(x)$ and the tangent line at $x=2$ for $f(x)=\frac{9 x}{2^{x}}$ .

Find the derivative of $5 - x^{2} = \sin(x \cdot y^{2})$ with respect to $x$ using implicit differentiation.

Find the indefinite integral: $\int(r-7)(r+7) \, dr$ (use $C$ for the constant).

Calculate the indefinite integral and use $C$ for the constant: $\int\left(\frac{8}{z^{5}}+\frac{3}{\sqrt{z}}\right) dz$

Calculate the indefinite integral: $\int\left(5 \sqrt[3]{t^{2}}+\frac{1}{\sqrt[3]{t^{2}}}\right) d t$ . Use constant $C$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{9 x}{2^{x}}$ .

Find $f^{\prime}(x)$ and the tangent line at $x=2$ for $f(x)=x(1-x)^{2}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{e^{x}}{7 x^{2}+1}$ .

Find the antiderivative of $\frac{d x}{d t}=8 e^{t}-3$ with the condition $x(0)=5$ . What is $x(t)=\square$ ?

Find the limit: $\lim _{n \rightarrow \infty} \sqrt{n^{2}+3 n}-n$ .

A patient has a temperature of 104°F changing at $t^{2}-3t$ °/hr for $0 \leq t \leq 3$ .
(a) Find $T(t)$ . (b) Calculate $T(2)$ and round to one decimal place.

Der DAX sinkt jede Woche um $2.9\%$ . Nach 100 Wochen, was ist sein Wert in Dezimalform? Geben Sie das Ergebnis gerundet an!

Explain when the minimum of a function $f(x, y)$ is higher than its maximum. What type of extrema is this? Provide an example and graph.

Find the limit infimum: $\varliminf_{n \rightarrow+\infty}\left(\frac{(3 n-1) \cos (n \pi)}{n}+7\right)=$

Find the rate of change of the bacteria population, given by $N(x)=\frac{12100}{1+99 e^{-0.25 x}}$ , at $x=4$ hours. Round to the nearest integer.

Find the derivatives of these functions: a) $f(x)=6 e^{x}+7$ , c) $f(x)=\frac{1}{2} x^{2}+4 x+5 e^{x}$ , e) $f(x)=e^{x}(e^{-x}-3)$ .

Differentiate these functions: a) $f(x)=2x-3\sin(x)+4$ b) $f(x)=4\cos(x)-8\sin(x)$

A pyramid's volume is $V=\frac{x^{2} h}{3}$ . If $h=150$ m and $x=60$ m, find the volume's decrease rate with $dx/dt=-0.004$ m/yr.

Find the rate of change of the bacteria population, $N(x)=\frac{12000}{1+99 e^{-0.25 x}}$ , at $x=1$ hour. Round to the nearest integer.

Find the rate of change of the bacteria population at $x=1$ hour using $N(x)=\frac{12000}{1+99 e^{-0.25 x}}$ .

Find the rate of change of the bacteria population, given $N(x)=\frac{13000}{1+99 e^{-0.3 x}}$ , at $x=3$ .

Berechnen Sie den Grenzwert der Reihe: $\lim _{N \rightarrow+\infty} \sum_{k=2}^{N} \frac{4 \cdot 3^{k}}{2^{2 k}}$

What are the units and meaning of $\left(g^{-1}\right)^{\prime}(20)$ for the average daily emails $g(t)$ in millions?

Calculate $3 \sum_{k=1}^{\infty} \frac{1}{(k+1)^{2}}$ .

Find the intervals where the function $g(x)=7(7 e^{-2 x}+3)^{2}$ is increasing or decreasing using $g^{\prime}(x)$ .

Find the integral of $\frac{2 x+5}{\sqrt{x^{2}+5 x+7}}$ with respect to $x$ .

Design a closed rectangular crate with a square base and volume $5184 \mathrm{ft}^{3}$ . Minimize material cost: top/sides at \$2/ft², bottom at \$10/ft². What are the dimensions?

Find the absolute extrema of $f(x)=\ln(8x^{2}-2x-7)$ on $[6.4,18.9]$ . Round answers to two decimal places.

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

Find the integral of $\frac{1}{x \ln x}$ with respect to $x$ .

Evaluate the following limits: a. $\lim _{x \rightarrow-1}(9-x^{2})$ b. $\lim _{x \rightarrow 0} \sqrt{\frac{x+20}{2 x+5}}$ c. $\lim _{x \rightarrow 5} \sqrt{x-1}$

Find the integral of $\ln \left(x^{7}\right) d x$ .

Evaluate the integral $\int 3x e^{3x} dx$ .

Find the time $t$ (from 6 A.M.) when the speed $f(t)=20t-40\sqrt{t}+50$ is lowest, and determine that speed.

Find the integral of $x\left(x^{2}+3\right)^{\frac{5}{4}}$ with respect to $x$ .

Evaluate the integral: $\int \ln \left(x^{3}\right) d x$

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

Minimize average cost for $C(x)=0.2(0.01 x^{2}+120)$ , where $x$ is units produced. What is the optimal production level?

Evaluate the integral of $\frac{\ln x}{x}$ with respect to $x$ : $\int \frac{\ln x}{x} d x$ .

Bestimmen Sie die Ableitungen von $f(x)=-0,5 x^{4}+4 x^{2}-x+1$ , $g(x)=-0,5 x^{4}+4 x^{2}-x+8$ , $h(x)=-0,5 x^{4}+4 x^{2}-x-2$ und vergleichen Sie diese.

Maximize revenue: Show that if $R(x)$ is concave down ( $R''(x)<0$ ), then $\bar{R}(x)=R'(x)$ gives max average revenue.

Maximize profit for the Pulsar TV with demand $p=-0.05 x+600$ and cost $C(x)=0.000002 x^{3}-0.03 x^{2}+400 x+80,000$ . Find optimal $x$ .

Die Funktion $s(t)=\frac{1}{6} t^{3}+\frac{1}{2} t^{2}$ beschreibt die Bewegung eines Objekts. Berechne die mittlere und Momentangeschwindigkeit nach 3 Sekunden und erkläre den Unterschied.

Find the production level $x$ for maximum profit given $P^{\prime}(x)=-0.004 x+20$ and fixed costs of \$16,000/month. What is the max profit?

Bestimme $a$ und $b$ für $f(x)=x^{3}+a x^{2}+x+b$ mit $f(0)=-2$ und $f'(1)=0$ . Berechne dann $\int_{0}^{2} f(x) d x$ und das Volumen $V$ für $x_{1}=0$ und $x_{2}=2$ .

Calculate the integral from 2 to 4 of the constant function 3: $\int_{2}^{4} 3 d x$ .

Find the integral $\int \frac{x^{4}}{1-x^{5}} d x$ .

Find the derivative of $f(x)=x^{4 x}$ using logarithmic differentiation. What is $f^{\prime}(x)=?$

Find the life expectancy function $g(t)$ given $g'(t)=\frac{5.45218}{(1+1.09 t)^{0.9}}$ and $g(0)=50.02$ . What is $g(100)$ ?

Find the slope of the tangent line to $2 x^{6}+9 x y-8 y^{3}=-1780$ at $(-1,6)$ . The slope is $\square$ .

Find the first partial derivative of $f(x, y)=(7 x^{2}+3 y)^{3}$ with respect to $x$ .

Find the first partial derivative of $f(x, y)=\ln \left(\sqrt{3 x+y^{2}}\right)$ with respect to $y$ .

Identify which statements about series are implied by the $n$ -th term test. Select all that apply.

Find the first derivative of $\frac{x^{3}-5 x^{2}}{2 x \sqrt{x}}$ with respect to $x$ . Which option is correct?

Find the second partial derivatives $f_{xx}, f_{yy}$ , and $f_{xy}$ for the function $f=\ln (5 x)+3 x \ln (y)$ .

Find the first derivative with respect to $x$ of the expression $\frac{x^{3}-5 x^{2}}{2 x \sqrt{x}}$ .

Find the first derivative of $\frac{x^{3}-5 x^{2}}{2 x \sqrt{x}}$ with respect to $x$ . Choose from options A-F.

What is the increase in satisfaction from consuming one more unit of good $x$ when $x=20$ and $y=30$ for $U(x, y)=2 \ln x+\ln y$ ?