Calculus

Problem 27101

Find the derivatives of the following functions: a. $y=2 e^{x^{3}}$ , b. $y=x e^{3 x}$ , c. $f(x)=\frac{e^{-x^{3}}}{x}$ , d. $f(x)=\sqrt{x} e^{x}$ , e. $h(t)=e t^{2}+3 e^{-t}$ , f. $g(t)=\frac{e^{2 t}}{1+e^{2 t}}$ .

See Solution

Problem 27102

Find the derivative of $f(x)=\frac{x^{2}+5}{3 \sqrt{x}}$ and show your work.

See Solution

Problem 27103

For the polynomial $g$ , if the rate of change increases for $x<2$ and decreases for $x>2$ , what is true about $g$ at $x=2$ ?

See Solution

Problem 27104

Find the average rate of change of $y=x^{3}$ from $x=1$ to $x=4$ . Options: 13, 21, 28, 14.

See Solution

Problem 27105

Find the average rate of change (AROC) of $f(x)=\frac{2 x-3}{x+3}$ from $x=1$ to $x=6$ .

See Solution

Problem 27106

Find the derivative $f^{\prime}(4)$ for the function $f(x)=6^{\sin x}$ . Options: A. -3.922 B. -302 C. 3.922 D. 4.545 E. does not exist.

See Solution

Problem 27107

Find the slope of the tangent to the particle's path at $t=2$ given $x(t) = \frac{1}{3}t - 2$ , $y(t) = -2t + 1$ . Options: (A) $\frac{9}{4}$ , (B) $-\frac{1}{6}$ , (C) $-2$ , (D) $-6$ .

See Solution

Problem 27108

Find the population of a town after 10 years if it grows at $1.4\%$ annually from a current population of 2000.

See Solution

Problem 27109

Find the average rate of change of the ball's height $s(t)=-5 t^{2}+20 t$ between $t=0$ and $t=0.9$ .

See Solution

Problem 27110

Find the value of $\int_{1}^{2} x f''(x) dx$ given $f(1)=-2$ , $f(2)=1$ , $f(3)=6$ , $f(4)=3$ . Choices: (A) 3, (B) 4, (C) 7, (D) 9.

See Solution

Problem 27111

Find $\left(f^{-1}\right)^{\prime}(4)$ for the function $f(x)=x^{3}-2 x^{2}+2 x$ given the point $(2,4)$ .

See Solution

Problem 27112

Find the IROC of the temperature model $T(t)=\frac{110 t+700}{t+35}$ at $t=95$ seconds. Options: 0.91, 0.19 °C/min, °C/sec.

See Solution

Problem 27113

Find $\frac{d^{2} y}{d t^{2}}$ if $y=b \sin (a t)$ . Choose from: A. $-b \sin (a t)$ B. $-a^{2} b \sin (a t)$ C. $)^{2} b \sin (a t)$ D. $a b \cos (a t)$ E. $-\sin (a t)$ .

See Solution

Problem 27114

Find $f^{\prime}\left(\frac{1}{4}\right)$ if $f(x)=\arccos(2x)$ . Choices: A) $\frac{4}{\sqrt{3}}$ , B) $-2\sqrt{2}$ , C) $2\sqrt{2}$ , D) $-\frac{8}{\sqrt{3}}$ , E) $-\frac{4}{\sqrt{3}}$ .

See Solution

Problem 27115

A rock is thrown up with an initial speed of $80 \mathrm{ft} / \mathrm{sec}$ . What is its maximum height? (A) $200 ft$ (B) $259 ft$ (C) $100 ft$ (D) $295 ft$

See Solution

Problem 27116

Find $\frac{d y}{d x}$ if $y=\arctan \left(2 e^{x}\right)$ . Options: A. $\frac{1}{1+4 e^{2 x}}$ , B. $\frac{2 e^{x}}{1+2 e^{2 x}}$ , C. $\frac{1}{1+2 e^{2 x}}$ , D. $\frac{2 e^{x}}{1+4 e^{2 x}}$ , E. $\frac{2 e^{x}}{1+2 e^{x^{2}}}$ .

See Solution

Problem 27117

Find the derivative $f'(x)$ for $f(x) = x + p(q(x))$ using $2 f'(x) = 1 + p'(q(x)) q'(x)$ . Use the table for values.

See Solution

Problem 27118

Find the derivative $f^{\prime}(1)$ for the function $f(x)=3^{x}$ . A. 3 B. $\ln 3$ C. $\ln 27$ D. $\frac{3}{\ln 3}$ E) $\ln 9$

See Solution

Problem 27119

Find the limit as $v$ approaches $4$ from the right: $\lim _{v \rightarrow 4^{+}} \frac{4-v}{|4-v|}$ .

See Solution

Problem 27120

Find the limit: $\lim _{x \rightarrow 0^{+}} \arctan \left(\frac{1}{x}\right)$ .

See Solution

Problem 27121

Find the derivative of the function: $8x^{2} + 2x^{3} + 4 + 17x$ .

See Solution

Problem 27122

Two cars leave an intersection; one goes north at $20 \mathrm{hm/h}$ , the other east at $50 \mathrm{km/h}$ . Find the distance change rate after 2 hours.

See Solution

Problem 27123

Given the demand function $p=1200-3q$ , find the Revenue Function $R(q)$ , the quantity for max revenue, and the max revenue amount.

See Solution

Problem 27124

Elena cuts squares of side length $x$ from a 11 in by 17 in paper to form a box. Find $x$ for max volume $V(x)=x(11-2x)(17-2x)$ .

See Solution

Problem 27125

Mr. Rooley's class studies bacteria growth with $n(t)=975 e^{0.46 t}$ . Find the growth rate and initial population.

See Solution

Problem 27126

Mr. Rooley's class measures bacteria growth: $n(t)=975 e^{0.46 t}$ . What is the relative growth rate in %?

See Solution

Problem 27127

Bestimmen Sie die Hoch- und Tiefpunkte der Funktionen $f$ durch die zweite Ableitung: a) $f(x)=x^{2}-2 x+2$ , b) $f(x)=x^{3}-3 x^{2}+2$ , c) $f(x)=\frac{1}{3} x^{3}-\frac{5}{2} x^{2}+6 x$ .

See Solution

Problem 27128

Mr. Rooley's class swabs sinks for bacteria growth modeled by $n(t)=975 e^{0.46 t}$ . Find growth rate, initial pop, and $n(4)$ .

See Solution

Problem 27129

Trova i valori di $\alpha \in \mathbb{R}$ per cui la funzione $f$ è derivabile in zero.

See Solution

Problem 27130

Trova l'espressione esplicita di $f^{\prime}(x)$ per $f(x)=\int_{3 x^{2}}^{7 x} e^{-t^{2}} \mathrm{~d} t$ .

See Solution

Problem 27131

Mr. Rooley's students swab sinks to measure bacteria growth with $n(t)=930 e^{0.47 t}$ . Find:
(a) Growth Rate = $\square \%$
(b) Initial Population = $\square$ cells
(c) Population after 6 hours = $\square$ cells.

See Solution

Problem 27132

Finde die Hoch- und Tiefpunkte der Funktionen a) $f(x)=8 x^{3}-6 x^{2}$ , b) $f(x)=2 x^{4}-6$ , c) $f(x)=-\frac{1}{8} x^{4}-\frac{1}{3} x^{3}+1$ mit der zweiten Ableitung.

See Solution

Problem 27133

Mr. Rooley's students swab sinks for bacteria. Given $n(t)=905 e^{0.34 t}$ , find: (a) growth rate, (b) initial population, (c) population at $t=3$ .

See Solution

Problem 27134

Mr. Rooley's class measures bacteria growth with $n(t)=1000 e^{0.21 t}$ .
(a) Find the growth rate: Growth Rate = $\square \%$ (b) Initial population at $t=0$ = cells (c) Bacteria count after 5 hours =

See Solution

Problem 27135

Solve the equation: $(x+y) dx + (x+y^2) dy = 0$ using exact methods.

See Solution

Problem 27136

Find the limit: $\lim _{x \rightarrow 0} \frac{\frac{1}{\sqrt{x+1}}-1}{x}$

See Solution

Problem 27137

Calculez la limite suivante : $\lim _{x \rightarrow 3} \ln \left(\frac{x^{2}-9}{4 x-12}\right)$

See Solution

Problem 27138

Find the units $x$ (in hundreds) to maximize revenue given $R(x)=40 x^{2} e^{-0.4 x}+30$ with a max of 800 units.

See Solution

Problem 27139

Find the number of items $x$ that maximize profit given $P(x)=10^{6}\left[1+(x-1) e^{-0.001 x}\right]$ . Max 2000 or 500 items.

See Solution

Problem 27140

Trova la derivata della funzione composta $g(f)$ con $f(x)=\log(x)$ e $g(y)=\arctan(y)$ . Qual è la risposta corretta?

See Solution

Problem 27141

Given a velocity-time graph with $v_i = 4 \, \text{m/s}$ at $t_i = 0$ s and $v_f = 10 \, \text{m/s}$ at $t_f = 12$ s, find the displacement.

See Solution

Problem 27142

Find $\frac{d y}{d x}$ for these functions: a. $y=\sin 2 x$ , b. $y=2 \cos 3 x$ , c. $y=\sin (x^{3}-2 x+4)$ , f. $y=2^{x}+2 \sin x-2 \cos x$ , g. $y=\sin (e^{x})$ , h. $y=3 \sin (3 x+2 \pi)$ .

See Solution

Problem 27143

Calcola la derivata della funzione composta $g(f)$ , dove $f(x)=\log(x)$ e $g(y)=\arctan(y)$ .

See Solution

Problem 27144

Bestimme die Stammfunktion von $f_{u}(x)=(k x+2) \cdot e^{u x}$ .

See Solution

Problem 27145

Find the tangent equations for these functions at the specified $x$ -coordinates: a. $f(x)=\sin x, x=\frac{\pi}{3}$ b. $f(x)=x+\sin x, x=0$ c. $f(x)=\cos(4x), x=\frac{\pi}{4}$ d. $f(x)=\sin(2x)+\cos x, x=\frac{\pi}{2}$ e. $f(x)=\cos(2x+\frac{\pi}{3}), x=\frac{\pi}{4}$ f. $f(x)=2\sin x \cos x, x=\frac{\pi}{2}$

See Solution

Problem 27146

Bestimme die Stammfunktion von $f_{a}(x)=(a x+2) \cdot e^{a x}$ .

See Solution

Problem 27147

Find the convexity/concavity intervals and inflection points of $y=x^{4}-2x^{3}-7$ .

See Solution

Problem 27148

Find the derivative $f^{\prime}\left(\frac{\pi}{6}\right)$ for $f(x)=\ln (\csc x)$ and simplify.

See Solution

Problem 27149

Find where $f(x)=6x+\frac{6}{x}$ is increasing, decreasing, and the $x$ -coordinates of relative maxima and minima.

See Solution

Problem 27150

Estimez $\sqrt{98}$ en utilisant les différentielles.

See Solution

Problem 27151

A roller coaster starts at height $12\,m$ and descends to $4\,m$ . Find its speed at point $C$ assuming no friction.

See Solution

Problem 27152

Find the derivative $\frac{d y}{d x}$ for the function $y=\sin^{-1}(\ln x)$ .

See Solution

Problem 27153

Find the decay constant for Pu-239 (half-life 24,110 years), remaining amount after 5,000 years from 20g, and time to decay to 1g.

See Solution

Problem 27154

Evaluate the expression $e^{-y} \int_{0}^{\infty} \frac{1}{y} e^{-\frac{x}{y}} d x$ .

See Solution

Problem 27155

Trouvez l'équation de la tangente à $y=\sin (\pi x / 27)$ en $x=9$ .

See Solution

Problem 27156

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

See Solution

Problem 27157

Find the derivative of the function given by $y=\left(\arcsin \left(1-x^{2}\right)\right)$ .

See Solution

Problem 27158

Calculate the limit: $\lim _{x \rightarrow 0} \frac{2}{\sin (2 x)}$ .

See Solution

Problem 27159

¿Cuánto tiempo tarda un bolígrafo en alcanzar $19.62 \frac{\mathrm{m}}{\mathrm{s}}$ al caer sin resistencia del aire?

See Solution

Problem 27160

1. For the curve $x y^{2}-x^{3} y=6$ : a. Prove $\frac{d y}{d x}=\frac{3 x^{2} y-y^{2}}{2 x y-x^{3}}$ . b. Find points with $x=1$ and tangent line equations. c. Determine $x$ -coordinates where tangent lines are vertical.

See Solution

Problem 27161

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

See Solution

Problem 27162

16. Use Newton's cooling law $T=T_{s}+(T_{0}-T_{s}) e^{-kt}$ to find cooling times for coffee at 140°F in two scenarios.

See Solution

Problem 27163

Approximate the integral of $y=0.5 x^{3}-3 x^{2}+4 x+12$ from $x=0$ to $x=8$ using the Trapezium rule with step size $2$ .

See Solution

Problem 27164

Find the potential difference between points A and B if an electron speeds up from $6 \times 10^{5} \mathrm{~m/s}$ to $12 \times 10^{5} \mathrm{~m/s}$ .

See Solution

Problem 27165

Is $h(x)=\frac{5}{x^{3}}$ concave down at $x=4$ ? Use calculus to explain your reasoning.

See Solution

Problem 27166

Find critical points of the function $t(x)=\frac{6}{\sqrt{2x-2}}$ .

See Solution

Problem 27167

Approximate the area under $y=0.5 x^{3}-3 x^{2}+4 x+12$ from $x=0$ to $x=8$ using the Trapezium rule with step size 1.

See Solution

Problem 27168

Differentiate $y=\frac{6}{(x+4)}$ to find $\frac{d y}{d x}=\frac{a}{(x^{2}+b x+c)}$ . Find values of $a$ , $b$ , and $c$ .

See Solution

Problem 27169

Differentiate $y=\frac{x^{2}}{(4 x-6)}$ and express as $\frac{d y}{d x}=\frac{(a x^{2}+b x)}{(c x^{2}+d x+e)}$ . Find $a$ , $b$ , $c$ , $d$ , and $e$ .

See Solution

Problem 27170

Differentiate and simplify $y=\frac{4 x^{2}+3}{3 x-5}$ to find $a$ , $b$ , $c$ , $d$ , and $e$ in $\frac{d y}{d x}=\frac{(a x^{2}+b x+c)}{(d x^{2}+e x+f)}$ .

See Solution

Problem 27171

Find the derivative $f'(x)$ of the function $f(x) = \frac{x^2}{4x - 6}$ and evaluate it at $x = 2$ .

See Solution

Problem 27172

Find the derivative of $y = 4x^2 + \frac{3}{3x-5}$ at $x=2$ and $x=-2$ .

See Solution

Problem 27173

Find the derivative values at $x=2$ and $x=-2$ for the function $y=\frac{4x^2+3}{3x-5}$ .

See Solution

Problem 27174

Find the derivative of $y = \frac{4x^2 + 3}{3x - 5}$ at $x = 2$ and $x = -2$ .

See Solution

Problem 27175

Approximate the area under $y=x^{4}-4 x^{3}+6 x+13$ from $x=-2$ to $x=4$ using Mid-Ordinate rule (step size 2).

See Solution

Problem 27176

Find the critical points of the function $f(x) = x^{\frac{1}{3}}(x - 6)^2$ by solving $f'(x) = 0$ or when $f'(x)$ is undefined.

See Solution

Problem 27177

Approximate the area under $y=x^{4}-4 x^{3}+6 x+13$ from $x=-2$ to $x=4$ using Mid-Ordinate and Simpson's rules.

See Solution

Problem 27178

Differentiate $y=\frac{6}{(x+4)}$ and express as $\frac{d y}{d x}=\frac{a}{(x^{2}+b x+c)}$ . Find $a$ , $b$ , and $c$ .

See Solution

Problem 27179

Approximate the area under $y=x^{4}-4x^{3}+6x+13$ from $x=-2$ to $x=4$ using the Mid-Ordinate rule with step size 2.

See Solution

Problem 27180

Approximate the area under $y=0.5 x^{3}-3 x^{2}+4 x+12$ from $x=0$ to $x=8$ using the Trapezium rule with step size 2.

See Solution

Problem 27181

A well produces 50,000 liters/week and drops by $5\%$ per year. How many liters can it produce before going dry?

See Solution

Problem 27182

Approximate the area under $y=0.5 x^{3}-3 x^{2}+5 x+13$ from $x=0$ to $x=8$ using the Trapezium rule with step size 2.

See Solution

Problem 27183

Approximate the area under $y=0.5 x^{3}-3 x^{2}+5 x+13$ from $x=0$ to $x=8$ using the Trapezium rule with step size 2. What is the integral's value?

See Solution

Problem 27184

Approximate the area under $y=0.5 x^{3}-3 x^{2}+5 x+13$ from $x=0$ to $x=8$ using the Trapezium rule with step size 2. What is the integral value? (1 decimal place)
Then, do the same with step size 1. What is the integral value? (1 decimal place)

See Solution

Problem 27185

Find the average value of the function $Y=x^4-4x^3+9x+12$ from $x=-2$ to $x=4$ using integration.

See Solution

Problem 27186

Approximate the area under $f(x) = x^4 - 4x^3 + 9x + 12$ from $x = -2$ to $x = 4$ using Simpson's rule with step size 1.

See Solution

Problem 27187

Integrate $f(x) = x^4 - 4x^3 + 9x + 12$ to find the area from $x = -2$ to $x = 4$ . What is the area?

See Solution

Problem 27188

Approximate the area under $y=0.5 x^{3}-3 x^{2}+4 x+9$ from $x=0$ to $x=8$ using the Trapezium rule with step size 2.

See Solution

Problem 27189

Given the curve $y+\cos y=x+1$ for $0 \leq y \leq 2 \pi$ : a. Find $\frac{d y}{d x}$ in terms of $y$ . b. Find equations for vertical tangent lines. c. Find $\frac{d^{2} y}{d x^{2}}$ in terms of $y$ .

See Solution

Problem 27190

Approximate the area under $y=x^{4}-4 x^{3}+9 x+12$ from $x=-2$ to $x=4$ using the Mid-Ordinate rule with step size $2$ .

See Solution

Problem 27191

Find the average of $f(x) = x^4 - 4x^3 + 9x + 12$ from $x = -2$ to $x = 4$ .

See Solution

Problem 27192

Find the average value of the function $y=0.5x^3-3x^2+5x+13$ from $x=0$ to $x=8$ .

See Solution

Problem 27193

Approximate the area under $y=x^{4}-4 x^{3}+9 x+12$ from $x=-2$ to $x=4$ using the Mid-Ordinate rule (step size 1).

See Solution

Problem 27194

Find the rate of change of revenue $R=1200x-x^2$ when 210 units are sold, with sales increasing at 18 units/day.

See Solution

Problem 27195

Approximate the area under $y=0.5 x^{3}-3 x^{2}+4 x+9$ from $x=0$ to $x=8$ using the Trapezium rule with step size 1. What is the integral value? (1 decimal place)

See Solution

Problem 27196

Find critical points, increasing/decreasing intervals, relative extrema, concavity, and inflection points for $y=5 x^{3}+2 x^{2}-3 x$ .

See Solution

Problem 27197

Find the average value of $y = 0.5x^3 - 3x^2 + 4x + 9$ from $x = 0$ to $x = 8$ .

See Solution

Problem 27198

Use the Trapezium rule with step size 1 to estimate the area under $f(x) = 0.5x^3 - 3x^2 + 4x + 9$ from $x=0$ to $x=8$ . What is the approximate integral value?

See Solution

Problem 27199

Integrate $f(x) = 0.5x^3 - 3x^2 + 4x + 9$ to find the area from $x=0$ to $x=8$ . What is this area?

See Solution

Problem 27200

Find the average value of the function $y = 0.5x^3 - 3x^2 + 4x + 9$ from $x=0$ to $x=8$ .

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 27101

Problem 27102

Find the derivative of f(x)=x2+53xf(x)=\frac{x^{2}+5}{3 \sqrt{x}}f(x)=3x​x2+5​ and show your work.

Problem 27103

For the polynomial ggg, if the rate of change increases for x<2x<2x<2 and decreases for x>2x>2x>2, what is true about ggg at x=2x=2x=2?

Problem 27104

Find the average rate of change of y=x3y=x^{3}y=x3 from x=1x=1x=1 to x=4x=4x=4. Options: 13, 21, 28, 14.

Problem 27105

Find the average rate of change (AROC) of f(x)=2x−3x+3f(x)=\frac{2 x-3}{x+3}f(x)=x+32x−3​ from x=1x=1x=1 to x=6x=6x=6.

Problem 27106

Find the derivative f′(4)f^{\prime}(4)f′(4) for the function f(x)=6sin⁡xf(x)=6^{\sin x}f(x)=6sinx. Options: A. -3.922 B. -302 C. 3.922 D. 4.545 E. does not exist.

Problem 27107

Find the slope of the tangent to the particle's path at t=2t=2t=2 given x(t)=13t−2x(t) = \frac{1}{3}t - 2x(t)=31​t−2, y(t)=−2t+1y(t) = -2t + 1y(t)=−2t+1. Options: (A) 94\frac{9}{4}49​, (B) −16-\frac{1}{6}−61​, (C) −2-2−2, (D) −6-6−6.

Problem 27108

Find the population of a town after 10 years if it grows at 1.4%1.4\%1.4% annually from a current population of 2000.

Problem 27109

Find the average rate of change of the ball's height s(t)=−5t2+20ts(t)=-5 t^{2}+20 ts(t)=−5t2+20t between t=0t=0t=0 and t=0.9t=0.9t=0.9.

Problem 27110

Find the value of ∫12xf′′(x)dx\int_{1}^{2} x f''(x) dx∫12​xf′′(x)dx given f(1)=−2f(1)=-2f(1)=−2, f(2)=1f(2)=1f(2)=1, f(3)=6f(3)=6f(3)=6, f(4)=3f(4)=3f(4)=3. Choices: (A) 3, (B) 4, (C) 7, (D) 9.

Problem 27111

Find (f−1)′(4)\left(f^{-1}\right)^{\prime}(4)(f−1)′(4) for the function f(x)=x3−2x2+2xf(x)=x^{3}-2 x^{2}+2 xf(x)=x3−2x2+2x given the point (2,4)(2,4)(2,4).

Problem 27112

Find the IROC of the temperature model T(t)=110t+700t+35T(t)=\frac{110 t+700}{t+35}T(t)=t+35110t+700​ at t=95t=95t=95 seconds. Options: 0.91, 0.19 °C/min, °C/sec.

Problem 27113

Problem 27114

Find f′(14)f^{\prime}\left(\frac{1}{4}\right)f′(41​) if f(x)=arccos⁡(2x)f(x)=\arccos(2x)f(x)=arccos(2x). Choices: A) 43\frac{4}{\sqrt{3}}3​4​, B) −22-2\sqrt{2}−22​, C) 222\sqrt{2}22​, D) −83-\frac{8}{\sqrt{3}}−3​8​, E) −43-\frac{4}{\sqrt{3}}−3​4​.

Problem 27115

A rock is thrown up with an initial speed of 80ft/sec80 \mathrm{ft} / \mathrm{sec}80ft/sec. What is its maximum height? (A) 200ft200 ft200ft (B) 259ft259 ft259ft (C) 100ft100 ft100ft (D) 295ft295 ft295ft

Problem 27116

Problem 27117

Find the derivative f′(x)f'(x)f′(x) for f(x)=x+p(q(x))f(x) = x + p(q(x))f(x)=x+p(q(x)) using 2f′(x)=1+p′(q(x))q′(x)2 f'(x) = 1 + p'(q(x)) q'(x)2f′(x)=1+p′(q(x))q′(x). Use the table for values.

Problem 27118

Find the derivative f′(1)f^{\prime}(1)f′(1) for the function f(x)=3xf(x)=3^{x}f(x)=3x. A. 3 B. ln⁡3\ln 3ln3 C. ln⁡27\ln 27ln27 D. 3ln⁡3\frac{3}{\ln 3}ln33​ E) ln⁡9\ln 9ln9

Problem 27119

Find the limit as vvv approaches 444 from the right: lim⁡v→4+4−v∣4−v∣\lim _{v \rightarrow 4^{+}} \frac{4-v}{|4-v|}limv→4+​∣4−v∣4−v​.

Problem 27120

Find the limit: lim⁡x→0+arctan⁡(1x)\lim _{x \rightarrow 0^{+}} \arctan \left(\frac{1}{x}\right)limx→0+​arctan(x1​).

Problem 27121

Find the derivative of the function: 8x2+2x3+4+17x8x^{2} + 2x^{3} + 4 + 17x8x2+2x3+4+17x.

Problem 27122

Two cars leave an intersection; one goes north at 20hm/h20 \mathrm{hm/h}20hm/h, the other east at 50km/h50 \mathrm{km/h}50km/h. Find the distance change rate after 2 hours.

Problem 27123

Given the demand function p=1200−3qp=1200-3qp=1200−3q, find the Revenue Function R(q)R(q)R(q), the quantity for max revenue, and the max revenue amount.

Problem 27124

Elena cuts squares of side length xxx from a 11 in by 17 in paper to form a box. Find xxx for max volume V(x)=x(11−2x)(17−2x)V(x)=x(11-2x)(17-2x)V(x)=x(11−2x)(17−2x).

Problem 27125

Mr. Rooley's class studies bacteria growth with n(t)=975e0.46tn(t)=975 e^{0.46 t}n(t)=975e0.46t. Find the growth rate and initial population.

Problem 27126

Mr. Rooley's class measures bacteria growth: n(t)=975e0.46tn(t)=975 e^{0.46 t}n(t)=975e0.46t. What is the relative growth rate in %?

Problem 27127

Bestimmen Sie die Hoch- und Tiefpunkte der Funktionen fff durch die zweite Ableitung: a) f(x)=x2−2x+2f(x)=x^{2}-2 x+2f(x)=x2−2x+2, b) f(x)=x3−3x2+2f(x)=x^{3}-3 x^{2}+2f(x)=x3−3x2+2, c) f(x)=13x3−52x2+6xf(x)=\frac{1}{3} x^{3}-\frac{5}{2} x^{2}+6 xf(x)=31​x3−25​x2+6x.

Problem 27128

Mr. Rooley's class swabs sinks for bacteria growth modeled by n(t)=975e0.46tn(t)=975 e^{0.46 t}n(t)=975e0.46t. Find growth rate, initial pop, and n(4)n(4)n(4).

Problem 27129

Trova i valori di α∈R\alpha \in \mathbb{R}α∈R per cui la funzione fff è derivabile in zero.

Problem 27130

Trova l'espressione esplicita di f′(x)f^{\prime}(x)f′(x) per f(x)=∫3x27xe−t2 dtf(x)=\int_{3 x^{2}}^{7 x} e^{-t^{2}} \mathrm{~d} tf(x)=∫3x27x​e−t2 dt.

Problem 27131

Mr. Rooley's students swab sinks to measure bacteria growth with n(t)=930e0.47tn(t)=930 e^{0.47 t}n(t)=930e0.47t. Find: (a) Growth Rate = □%\square \%□% (b) Initial Population = □\square□ cells (c) Population after 6 hours = □\square□ cells.

Problem 27132

Finde die Hoch- und Tiefpunkte der Funktionen a) f(x)=8x3−6x2f(x)=8 x^{3}-6 x^{2}f(x)=8x3−6x2, b) f(x)=2x4−6f(x)=2 x^{4}-6f(x)=2x4−6, c) f(x)=−18x4−13x3+1f(x)=-\frac{1}{8} x^{4}-\frac{1}{3} x^{3}+1f(x)=−81​x4−31​x3+1 mit der zweiten Ableitung.

Problem 27133

Mr. Rooley's students swab sinks for bacteria. Given n(t)=905e0.34tn(t)=905 e^{0.34 t}n(t)=905e0.34t, find: (a) growth rate, (b) initial population, (c) population at t=3t=3t=3.

Problem 27134

Mr. Rooley's class measures bacteria growth with n(t)=1000e0.21tn(t)=1000 e^{0.21 t}n(t)=1000e0.21t. (a) Find the growth rate: Growth Rate = □%\square \%□% (b) Initial population at t=0t=0t=0 = cells (c) Bacteria count after 5 hours =

Problem 27135

Solve the equation: (x+y)dx+(x+y2)dy=0(x+y) dx + (x+y^2) dy = 0(x+y)dx+(x+y2)dy=0 using exact methods.

Problem 27136

Find the limit: lim⁡x→01x+1−1x\lim _{x \rightarrow 0} \frac{\frac{1}{\sqrt{x+1}}-1}{x}limx→0​xx+1​1​−1​

Problem 27137

Calculez la limite suivante : lim⁡x→3ln⁡(x2−94x−12) \lim _{x \rightarrow 3} \ln \left(\frac{x^{2}-9}{4 x-12}\right) x→3lim​ln(4x−12x2−9​)

Problem 27138

Find the units xxx (in hundreds) to maximize revenue given R(x)=40x2e−0.4x+30R(x)=40 x^{2} e^{-0.4 x}+30R(x)=40x2e−0.4x+30 with a max of 800 units.

Problem 27139

Find the number of items xxx that maximize profit given P(x)=106[1+(x−1)e−0.001x]P(x)=10^{6}\left[1+(x-1) e^{-0.001 x}\right]P(x)=106[1+(x−1)e−0.001x]. Max 2000 or 500 items.

Problem 27140

Trova la derivata della funzione composta g(f)g(f)g(f) con f(x)=log⁡(x)f(x)=\log(x)f(x)=log(x) e g(y)=arctan⁡(y)g(y)=\arctan(y)g(y)=arctan(y). Qual è la risposta corretta?

Problem 27141

Given a velocity-time graph with vi=4 m/sv_i = 4 \, \text{m/s}vi​=4m/s at ti=0t_i = 0ti​=0 s and vf=10 m/sv_f = 10 \, \text{m/s}vf​=10m/s at tf=12t_f = 12tf​=12 s, find the displacement.

Find the derivative of $f(x)=\frac{x^{2}+5}{3 \sqrt{x}}$ and show your work.

For the polynomial $g$ , if the rate of change increases for $x<2$ and decreases for $x>2$ , what is true about $g$ at $x=2$ ?

Find the average rate of change of $y=x^{3}$ from $x=1$ to $x=4$ . Options: 13, 21, 28, 14.

Find the average rate of change (AROC) of $f(x)=\frac{2 x-3}{x+3}$ from $x=1$ to $x=6$ .

Find the derivative $f^{\prime}(4)$ for the function $f(x)=6^{\sin x}$ . Options: A. -3.922 B. -302 C. 3.922 D. 4.545 E. does not exist.

Find the slope of the tangent to the particle's path at $t=2$ given $x(t) = \frac{1}{3}t - 2$ , $y(t) = -2t + 1$ . Options: (A) $\frac{9}{4}$ , (B) $-\frac{1}{6}$ , (C) $-2$ , (D) $-6$ .

Find the population of a town after 10 years if it grows at $1.4\%$ annually from a current population of 2000.

Find the average rate of change of the ball's height $s(t)=-5 t^{2}+20 t$ between $t=0$ and $t=0.9$ .

Find the value of $\int_{1}^{2} x f''(x) dx$ given $f(1)=-2$ , $f(2)=1$ , $f(3)=6$ , $f(4)=3$ . Choices: (A) 3, (B) 4, (C) 7, (D) 9.

Find $\left(f^{-1}\right)^{\prime}(4)$ for the function $f(x)=x^{3}-2 x^{2}+2 x$ given the point $(2,4)$ .

Find the IROC of the temperature model $T(t)=\frac{110 t+700}{t+35}$ at $t=95$ seconds. Options: 0.91, 0.19 °C/min, °C/sec.

Find $f^{\prime}\left(\frac{1}{4}\right)$ if $f(x)=\arccos(2x)$ . Choices: A) $\frac{4}{\sqrt{3}}$ , B) $-2\sqrt{2}$ , C) $2\sqrt{2}$ , D) $-\frac{8}{\sqrt{3}}$ , E) $-\frac{4}{\sqrt{3}}$ .

A rock is thrown up with an initial speed of $80 \mathrm{ft} / \mathrm{sec}$ . What is its maximum height? (A) $200 ft$ (B) $259 ft$ (C) $100 ft$ (D) $295 ft$

Find the derivative $f'(x)$ for $f(x) = x + p(q(x))$ using $2 f'(x) = 1 + p'(q(x)) q'(x)$ . Use the table for values.

Find the derivative $f^{\prime}(1)$ for the function $f(x)=3^{x}$ . A. 3 B. $\ln 3$ C. $\ln 27$ D. $\frac{3}{\ln 3}$ E) $\ln 9$

Find the limit as $v$ approaches $4$ from the right: $\lim _{v \rightarrow 4^{+}} \frac{4-v}{|4-v|}$ .

Find the limit: $\lim _{x \rightarrow 0^{+}} \arctan \left(\frac{1}{x}\right)$ .

Find the derivative of the function: $8x^{2} + 2x^{3} + 4 + 17x$ .

Two cars leave an intersection; one goes north at $20 \mathrm{hm/h}$ , the other east at $50 \mathrm{km/h}$ . Find the distance change rate after 2 hours.

Given the demand function $p=1200-3q$ , find the Revenue Function $R(q)$ , the quantity for max revenue, and the max revenue amount.

Elena cuts squares of side length $x$ from a 11 in by 17 in paper to form a box. Find $x$ for max volume $V(x)=x(11-2x)(17-2x)$ .

Mr. Rooley's class studies bacteria growth with $n(t)=975 e^{0.46 t}$ . Find the growth rate and initial population.

Mr. Rooley's class measures bacteria growth: $n(t)=975 e^{0.46 t}$ . What is the relative growth rate in %?

Bestimmen Sie die Hoch- und Tiefpunkte der Funktionen $f$ durch die zweite Ableitung: a) $f(x)=x^{2}-2 x+2$ , b) $f(x)=x^{3}-3 x^{2}+2$ , c) $f(x)=\frac{1}{3} x^{3}-\frac{5}{2} x^{2}+6 x$ .

Mr. Rooley's class swabs sinks for bacteria growth modeled by $n(t)=975 e^{0.46 t}$ . Find growth rate, initial pop, and $n(4)$ .

Trova i valori di $\alpha \in \mathbb{R}$ per cui la funzione $f$ è derivabile in zero.

Trova l'espressione esplicita di $f^{\prime}(x)$ per $f(x)=\int_{3 x^{2}}^{7 x} e^{-t^{2}} \mathrm{~d} t$ .

Mr. Rooley's students swab sinks to measure bacteria growth with $n(t)=930 e^{0.47 t}$ . Find:
(a) Growth Rate = $\square \%$
(b) Initial Population = $\square$ cells
(c) Population after 6 hours = $\square$ cells.

Finde die Hoch- und Tiefpunkte der Funktionen a) $f(x)=8 x^{3}-6 x^{2}$ , b) $f(x)=2 x^{4}-6$ , c) $f(x)=-\frac{1}{8} x^{4}-\frac{1}{3} x^{3}+1$ mit der zweiten Ableitung.

Mr. Rooley's students swab sinks for bacteria. Given $n(t)=905 e^{0.34 t}$ , find: (a) growth rate, (b) initial population, (c) population at $t=3$ .

Mr. Rooley's class measures bacteria growth with $n(t)=1000 e^{0.21 t}$ .
(a) Find the growth rate: Growth Rate = $\square \%$ (b) Initial population at $t=0$ = cells (c) Bacteria count after 5 hours =

Solve the equation: $(x+y) dx + (x+y^2) dy = 0$ using exact methods.

Find the limit: $\lim _{x \rightarrow 0} \frac{\frac{1}{\sqrt{x+1}}-1}{x}$

Calculez la limite suivante : $\lim _{x \rightarrow 3} \ln \left(\frac{x^{2}-9}{4 x-12}\right)$

Find the units $x$ (in hundreds) to maximize revenue given $R(x)=40 x^{2} e^{-0.4 x}+30$ with a max of 800 units.

Find the number of items $x$ that maximize profit given $P(x)=10^{6}\left[1+(x-1) e^{-0.001 x}\right]$ . Max 2000 or 500 items.

Trova la derivata della funzione composta $g(f)$ con $f(x)=\log(x)$ e $g(y)=\arctan(y)$ . Qual è la risposta corretta?

Given a velocity-time graph with $v_i = 4 \, \text{m/s}$ at $t_i = 0$ s and $v_f = 10 \, \text{m/s}$ at $t_f = 12$ s, find the displacement.

Calcola la derivata della funzione composta $g(f)$ , dove $f(x)=\log(x)$ e $g(y)=\arctan(y)$ .

Bestimme die Stammfunktion von $f_{u}(x)=(k x+2) \cdot e^{u x}$ .

Bestimme die Stammfunktion von $f_{a}(x)=(a x+2) \cdot e^{a x}$ .

Find the convexity/concavity intervals and inflection points of $y=x^{4}-2x^{3}-7$ .

Find the derivative $f^{\prime}\left(\frac{\pi}{6}\right)$ for $f(x)=\ln (\csc x)$ and simplify.

Find where $f(x)=6x+\frac{6}{x}$ is increasing, decreasing, and the $x$ -coordinates of relative maxima and minima.

Estimez $\sqrt{98}$ en utilisant les différentielles.

A roller coaster starts at height $12\,m$ and descends to $4\,m$ . Find its speed at point $C$ assuming no friction.

Find the derivative $\frac{d y}{d x}$ for the function $y=\sin^{-1}(\ln x)$ .

Evaluate the expression $e^{-y} \int_{0}^{\infty} \frac{1}{y} e^{-\frac{x}{y}} d x$ .

Trouvez l'équation de la tangente à $y=\sin (\pi x / 27)$ en $x=9$ .

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

Find the derivative of the function given by $y=\left(\arcsin \left(1-x^{2}\right)\right)$ .

Calculate the limit: $\lim _{x \rightarrow 0} \frac{2}{\sin (2 x)}$ .

¿Cuánto tiempo tarda un bolígrafo en alcanzar $19.62 \frac{\mathrm{m}}{\mathrm{s}}$ al caer sin resistencia del aire?

1. For the curve $x y^{2}-x^{3} y=6$ : a. Prove $\frac{d y}{d x}=\frac{3 x^{2} y-y^{2}}{2 x y-x^{3}}$ . b. Find points with $x=1$ and tangent line equations. c. Determine $x$ -coordinates where tangent lines are vertical.

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

16. Use Newton's cooling law $T=T_{s}+(T_{0}-T_{s}) e^{-kt}$ to find cooling times for coffee at 140°F in two scenarios.

Approximate the integral of $y=0.5 x^{3}-3 x^{2}+4 x+12$ from $x=0$ to $x=8$ using the Trapezium rule with step size $2$ .

Find the potential difference between points A and B if an electron speeds up from $6 \times 10^{5} \mathrm{~m/s}$ to $12 \times 10^{5} \mathrm{~m/s}$ .

Is $h(x)=\frac{5}{x^{3}}$ concave down at $x=4$ ? Use calculus to explain your reasoning.

Find critical points of the function $t(x)=\frac{6}{\sqrt{2x-2}}$ .

Approximate the area under $y=0.5 x^{3}-3 x^{2}+4 x+12$ from $x=0$ to $x=8$ using the Trapezium rule with step size 1.

Differentiate $y=\frac{6}{(x+4)}$ to find $\frac{d y}{d x}=\frac{a}{(x^{2}+b x+c)}$ . Find values of $a$ , $b$ , and $c$ .

Differentiate $y=\frac{x^{2}}{(4 x-6)}$ and express as $\frac{d y}{d x}=\frac{(a x^{2}+b x)}{(c x^{2}+d x+e)}$ . Find $a$ , $b$ , $c$ , $d$ , and $e$ .

Differentiate and simplify $y=\frac{4 x^{2}+3}{3 x-5}$ to find $a$ , $b$ , $c$ , $d$ , and $e$ in $\frac{d y}{d x}=\frac{(a x^{2}+b x+c)}{(d x^{2}+e x+f)}$ .

Find the derivative $f'(x)$ of the function $f(x) = \frac{x^2}{4x - 6}$ and evaluate it at $x = 2$ .

Find the derivative of $y = 4x^2 + \frac{3}{3x-5}$ at $x=2$ and $x=-2$ .

Find the derivative values at $x=2$ and $x=-2$ for the function $y=\frac{4x^2+3}{3x-5}$ .

Find the derivative of $y = \frac{4x^2 + 3}{3x - 5}$ at $x = 2$ and $x = -2$ .