Calculus

Problem 11101

Find the average value of $f(x)=\frac{1}{\sqrt{4x-3}}$ from $x=3$ to $x=21$ , rounded to four decimal places.

See Solution

Problem 11102

Find the inflection point of the polynomial $f(x)$ where $f^{\prime \prime}(x)=3x+5$ . Options: $x=-\frac{5}{3}, x=\frac{3}{5}, x=-\frac{3}{5}, x=\frac{5}{3}$ .

See Solution

Problem 11103

Calculate the average value of $f(x)=\frac{1}{\sqrt{4 x-3}}$ over the interval from $x=3$ to $x=21$ .

See Solution

Problem 11104

Use a calculator to estimate the value of $e^{1.1}$ .

See Solution

Problem 11105

Invest \$2700 at 3.9% APR compounded continuously.
a. Find the value after 5 years: \$2700e^{(5 \cdot 0.039)}.
b. Define function $f(t) = 2700e^{(0.039t)}$ .
c. What is the annual percent change (APY)?

See Solution

Problem 11106

Bestimme die Ableitung von $f$ an $x_{0}=2$ mit dem Differenzenquotienten für kleine $h$ für die Funktionen: a) $f(x)=x^{2}$ , b) $f(x)=x^{3}$ , c) $f(x)=2 x^{2}-3$ , d) $f(x)=x^{4}$ , e) $f(x)=\frac{2}{x}$ , f) $f(x)=4 x-x^{2}$ , g) $f(x)=\sqrt{x}$ , h) $f(x)=5$ .

See Solution

Problem 11107

Find $x$ such that $|T_{5}(x) - f(x)| < 0.001938$ for $|x| \leq 1$ , where $T_{5}(x)=1-x^{2}/2+x^{4}/4!$ .

See Solution

Problem 11108

Find the first and second derivatives of $f(x)=\frac{1}{4}x^4 + x$ .

See Solution

Problem 11109

Find all values of $c$ for which the function $f(x)=\frac{c}{x}+x^{2}$ has a relative minimum but no relative maximum.

See Solution

Problem 11110

Calculate the integral: $\int_{\pi / 2}^{\pi} \frac{\sin 2 x}{2 \sin x} d x$ .

See Solution

Problem 11111

Design a rectangular crate with a square base and volume $5632 \mathrm{ft}^{3}$ . Minimize material costs: top/sides at \$4/ft², bottom at \$7/ft². Find dimensions.

See Solution

Problem 11112

Find the differential of the function $y=\sqrt{3+\cos (\theta)}$ . What is $d y$ ?

See Solution

Problem 11113

Find the differential of the function $y=\ln (\sin (8 \theta))$ . What is $d y$ ?

See Solution

Problem 11114

Determine if $f'(x)$ is positive (+) or negative (-) for $x=\frac{1}{3}$ and $x>\frac{1}{3}$ for $f(x)=3x^2-2x+1$ .

See Solution

Problem 11115

Find the differential of $y=\ln (\sin (8 \theta))$ . What is $d y$ ?

See Solution

Problem 11116

Find the linear approximation $L(x)$ for the following functions near given points:
1. $f(x)=x+x^{4}, a=0$
2. $f(x)=\frac{1}{x}, a=2$
3. $f(x)=\tan x, a=\frac{\pi}{4}$
4. $f(x)=\sin x, a=\frac{\pi}{2}$
5. $f(x)=x \sin x, a=2 \pi$
6. $f(x)=\sin ^{2} x, a=0$

See Solution

Problem 11117

The function $f(x)=x^{5}-3$ has a critical point at $x=0$ . Is it a local max, min, or no extrema?

See Solution

Problem 11118

Find the tangent line equation for $f(x)=\frac{1}{6+x^{2}}$ at $x=3$ with slope $-\frac{2}{75}$ .

See Solution

Problem 11119

Find the derivative $f^{\prime}(x)$ of the constant function $f(x)=-6$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 11120

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{-2}$ . What is $\frac{d y}{d x}=$ ?

See Solution

Problem 11121

Find the derivative $f^{\prime}(x)$ of the function $f(x)=4 x^{8}$ .

See Solution

Problem 11122

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

See Solution

Problem 11123

Find the derivative $\frac{d y}{d x}$ for $y=\frac{1}{x^{11}}$ . What is $\frac{d y}{d x}=\square$ ?

See Solution

Problem 11124

Find the derivative $h^{\prime}(t)$ for the function $h(t)=7.5+7.4 t+0.7 t^{3}$ .

See Solution

Problem 11125

Find the derivative $G^{\prime}(w)$ for the function $G(w)=\frac{6}{7 w^{6}}+4 \sqrt[3]{w}$ . What is $G^{\prime}(w)=\square$ ?

See Solution

Problem 11126

Find the derivative $y^{\prime}$ of the function $y=6 x^{-5}-2 x^{-1}$ .

See Solution

Problem 11127

Finde die Tangentengleichung für $f(x)=\frac{1}{3} x^{3}-2 x+3$ am Punkt $(x_0, f(x_0))$ .

See Solution

Problem 11128

Find the tangent line equation for $f(x)=16 e^{x}+7 x$ at $x=0$ . $y=\square$

See Solution

Problem 11129

Differentiate the function $f(x)=10-4 x+4 e^{x}$ . Find $f^{\prime}(x)=\square$ .

See Solution

Problem 11130

Find the rate of learning after 10 hours using $N(t)=4+7 \ln t$ . What is $N'(10)$ in words/minute per hour?

See Solution

Problem 11131

Find the critical points of the function $y=256 \sqrt{x}-x^{2}$ . Provide exact values for $x$ .

See Solution

Problem 11132

Bestimmen Sie die Tangenten- und Normalengleichung von $f$ an den Punkten B: a) $f(x)=x^{2}-x ; B(-2 \mid 6)$ , b) $f(x)=\frac{4}{x}+2 ; B(4 \mid 3)$ .

See Solution

Problem 11133

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-5 \ln x + 11 x^{2} - 2$ .

See Solution

Problem 11134

Find the velocity of a particle at $t=4$ seconds, given $s(t)=9 t^{2}+29 t$ .

See Solution

Problem 11135

Find the velocity of a particle at $t=8$ seconds given $s(t)=9 t^{2}+21 t$ .

See Solution

Problem 11136

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

See Solution

Problem 11137

Evaluate the integral $\int_{\sqrt{2}}^{1}\left(\frac{u^{7}}{2}-\frac{1}{u^{5}}\right) du$ .

See Solution

Problem 11138

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(2 x+5)(7 x-6)$ .

See Solution

Problem 11139

Check if the Mean Value Theorem applies to $f(x)=\frac{1}{x}$ on $[1,3]$ and find $c$ (round to 3 decimal places).

See Solution

Problem 11140

Differentiate the function $y=\log_{7}(x \log_{9}(x))$ and find $y'=\square$ .

See Solution

Problem 11141

Find the derivative of $g(x)=\sec^{-1}(9 e^{x})$ . What is $g'(x)$ ?

See Solution

Problem 11142

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{3 x-7}{2 x+3}$ .

See Solution

Problem 11143

Find the $y$ -intercept of the tangent line to $f(x)=(3x^2-1)(x^2+2)$ at $x=1$ .

See Solution

Problem 11144

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

See Solution

Problem 11145

Compute the following values using linear approximation within 0.01: (2.001)^{6}, \sin(0.02), \cos(0.03), (15.99)^{1/4}, \frac{1}{0.98}, \sin(3.14).

See Solution

Problem 11146

Find the slope of the tangent line to the curve $y=\frac{4}{x}$ at the point $(1,4)$ .

See Solution

Problem 11147

Differentiate the function $h(x)=e^{x^{7}+\ln (x)}$ and find $h^{\prime}(x)$ .

See Solution

Problem 11148

Calculate $\int_{1}^{4} (f(x)-g(x)) dx$ for $f(x)=-x^2+6x-4$ and $g(x)=x$ .

See Solution

Problem 11149

Find the velocity of the object at $t=3$ given its position $s(t)=-3t+t^{2}$ in metres.

See Solution

Problem 11150

Find the missing expression in the equation: $\frac{d}{d x}\left(9-2 x^{2}\right)^{3}=3\left(9-2 x^{2}\right)^{2} ?$

See Solution

Problem 11151

Find the point where the tangent to $y=(x-2)^{2}(x+3)$ is horizontal. Options: $(-1,18)$ , $(1,4)$ , $(-2,16)$ , $(2,0)$ .

See Solution

Problem 11152

Find the expression for ? to make the equation valid: $\frac{\mathrm{d}}{\mathrm{dx}} e^{\mathrm{x}^{7}+6}=e^{\mathrm{x}^{7}+6} ?$

See Solution

Problem 11153

Find the velocity of the object at time $t=1$ for the position function $s(t)=5 t(t-2)^{2}$ .

See Solution

Problem 11154

Find the derivative $f^{\prime}(x)$ for the function $f(x)=3 x^{4} \ln x$ .

See Solution

Problem 11155

Find the instantaneous pulse rate change for a person $x$ inches tall, given $y=590 x^{-1/2}$ for heights 35 and 65 inches.

See Solution

Problem 11156

Find the learning rate for $y=40\sqrt{x}$ at $x=1$ hour and $x=16$ hours. What is the rate at 1 hour?

See Solution

Problem 11157

Find the slope of the tangent to $y=\sqrt{1-x}$ at the point $(-8,3)$ .

See Solution

Problem 11158

A couple invested a \$134,000 inheritance at 7.7\% interest for 14 years. What will be their half-year payments for 20 years?

See Solution

Problem 11159

Find the learning rate at 1 hour and 16 hours for $y=30 \sqrt{x}, 0 \leq x \leq 16$ . Rate at 1 hour is $\square$ .

See Solution

Problem 11160

Find the gradient of $g(x)=\frac{1}{4} \tan x-3 \cos x-x^{3}$ at $x=\frac{\pi}{6}$ .

See Solution

Problem 11161

Find the learning rate using $y=30 \sqrt{x}$ at 1 hour and 9 hours. What is the rate at 1 hour?

See Solution

Problem 11162

Finde die unbekannte Größe, sodass die folgenden Integrale gleich null sind:
a) $\int_{-1}^{2}(x^{2}-6) dx=0$ , b) $\int_{0}^{b}(x^{2}-8x) dx=0$ , c) $\int(\frac{1}{3} x^{4}-3 x^{2}) dx=0$ .

See Solution

Problem 11163

Find the learning rate at 1 hour and 16 hours given $y=50 \sqrt{x}, 0 \leq x \leq 16$ . Rate at 1 hour: $\square$ .

See Solution

Problem 11164

Find $x$ in $[0, 2\pi)$ such that the derivative $h'(x)=0$ for $h(x)=\sin x+\cos x$ .

See Solution

Problem 11165

A radioactive sample halves in 7 years. How long to decay to $\frac{5}{6}$ of its original amount? (3 decimal places)

See Solution

Problem 11166

Find the learning rate for $y = 50 \sqrt{x}$ at $x = 1$ and $x = 4$ . Answer: $\square$ items per hour.

See Solution

Problem 11167

A substance decays to $\frac{1}{3}$ in 10 hours. What is its half-life? Round to 3 decimal places.

See Solution

Problem 11168

Find the instantaneous rate of change of pulse rate $y=598 x^{-1/2}$ for heights 31 and 75 inches. Round to the nearest hundredth.

See Solution

Problem 11169

Find the second derivative of the function $f(x)=-7 x^{3}-4 x^{2}+2 x$ .

See Solution

Problem 11170

Find the instantaneous rate of change of pulse rate $y=598 x^{-1/2}$ for heights 31 inches and 75 inches.

See Solution

Problem 11171

Find the first time $t>0$ when the acceleration $a(t)=0$ for the particle with velocity $v(t)=e^{t^{2}-\sqrt{t}}-\sin t$ .

See Solution

Problem 11172

Find $\left(f^{-1}\right)^{\prime}(1)$ for the function $f(x)=\frac{3 x-1}{x+3}$ .

See Solution

Problem 11173

Find the value of $a$ such that the tangent line to $f(x)=\ln(x^{2}+1)$ at $x=2$ is parallel to $g(x)=x^{5}+x^{3}$ at $x=a$ .

See Solution

Problem 11174

Find $\left(f^{-1}\right)^{\prime}(6)$ using the values of $f(x)$ and $f^{\prime}(x)$ given in the table.

See Solution

Problem 11175

Find $P^{\prime}(t)$ when $P(t)$ satisfies $y^{\prime}=3y$ and $P(0)=600$ , given $P(t)=1000$ .

See Solution

Problem 11176

Find $\left(f^{-1}\right)^{\prime}(6)$ using $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$ . Also, write the tangent line equation for $y=g^{-1}(x)$ at $x=2$ .

See Solution

Problem 11177

Find the elasticity of demand for $D(p) = 202 - p^{2}$ at $p = 12$ and classify it as elastic, inelastic, or unit elasticity.

See Solution

Problem 11178

Find the integral of the function: $\int\left(2 x^{3}-4 x\right) d x$ .

See Solution

Problem 11179

Find the elasticity of demand for $D(p)=202-p^{2}$ at $p=12$ and classify it as elastic, inelastic, or unit elastic.

See Solution

Problem 11180

Find the production level $x$ that minimizes the average cost given $c(x)=x^{3}-22x^{2}+20,000x$ .

See Solution

Problem 11181

Find the production level $x$ that minimizes the average cost given $c(x)=x^{3}-16 x^{2}+10,000 x$ . Options: 7, 8, 9, 10 items.

See Solution

Problem 11182

Calculate the integral from 0 to 3 of the function $\frac{1}{3} x^{2}-\frac{2}{3} x+\frac{4}{3}$ .

See Solution

Problem 11183

Evaluate the integral from 0 to 1: $\int_{0}^{1}(2 x-2)^{2} d x$ .

See Solution

Problem 11184

Find $h'(-1)$ for $h(x)=f(x)+3g(x)$ , where $f'(x)=2g'(x)$ and $\lim_{x\to-1}\frac{f(x)f(-1)}{x+1}=-6$ .

See Solution

Problem 11185

Find the elasticity of demand $E=\frac{5 p}{2 q+5 p}$ for $q^2 + 5pq = 75$ and check if demand is elastic at $p=2$ .

See Solution

Problem 11186

Find the production level $x$ that minimizes the average cost given $c(x)=x^{3}-22x^{2}+20,000x$ . Options: 10, 11, 12, 13 items.

See Solution

Problem 11187

An airline's ticket price $p$ affects demand $q=324-0.01 p^{2}$ . Find elasticity $E(p)$ and analyze revenue behavior.

See Solution

Problem 11188

An airline's ticket price $p$ affects demand $q=432-0.03 p^{2}$ . Find elasticity $E(p)$ and revenue behavior for $p$ values.

See Solution

Problem 11189

Find the production level $x$ that minimizes the average cost given $c(x)=x^{3}-22 x^{2}+10,000 x$ . Options: 10, 11, 12, 13 items.

See Solution

Problem 11190

Find the production level $x$ that minimizes the average cost given $c(x)=x^{3}-18x^{2}+30,000x$ . Options: 8, 9, 10, 11 items.

See Solution

Problem 11191

Check if the functions $C_{1}(t)=\frac{7.6 t}{t^{2}+0.2}$ and $C_{2}(t)=\frac{7.2 t}{t^{2}+0.15}$ are increasing or decreasing.

See Solution

Problem 11192

Find the first and second derivatives of the function $f(x)=e^{11 x}$ . a. $f^{\prime}(x)=$ b. $f^{\prime \prime}(x)=$

See Solution

Problem 11193

Find $g^{\prime}(3)$ if $g^{\prime}(x)=-4x^{2}-3x-7$ .

See Solution

Problem 11194

Estimate $\Delta f=f(7.01)-f(7)$ for $f(x)=x^{4}$ using Linear Approximation. Round to two decimal places.

See Solution

Problem 11195

Find constants $K, m,$ and $n$ for the integral $\int x^{2} \sqrt{16 - x^{2}} dx$ using $x = 4 \sin(u)$ .

See Solution

Problem 11196

Find the indefinite integral: $\int \frac{6 x^{3}+4 x^{2}-45 x-33}{x^{2}-9} d x$ and decompose it as $a x+b+\frac{c}{x-3}+\frac{d}{x+3}$ .

See Solution

Problem 11197

Find the limit using I'Hospital's Rule if needed: $\lim _{x \rightarrow 0} \frac{x}{\tan ^{-1}(9 x)}$

See Solution

Problem 11198

Bestimme, wann die Bevölkerung von 60 Millionen auf 120 Millionen wächst und welche Sättigungsgrenze $L$ erreicht wird. Gegeben ist $N'(t) = a e^{bt}$ .

See Solution

Problem 11199

Find the position of a particle at $t=28 \, \text{s}$ given $x_{0}=19 \, \text{m}$ , $v(0)=20 \, \text{m/s}$ , $v(28)=5 \, \text{m/s}$ .

See Solution

Problem 11200

Find the $x$ -values of relative extrema for $f(x)=x^{4}-72 x^{2}+7$ and their corresponding values. Derivative: $f^{\prime}(x)=\square$ .

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 11101

Find the average value of f(x)=14x−3f(x)=\frac{1}{\sqrt{4x-3}}f(x)=4x−3​1​ from x=3x=3x=3 to x=21x=21x=21, rounded to four decimal places.

Problem 11102

Find the inflection point of the polynomial f(x)f(x)f(x) where f′′(x)=3x+5f^{\prime \prime}(x)=3x+5f′′(x)=3x+5. Options: x=−53,x=35,x=−35,x=53x=-\frac{5}{3}, x=\frac{3}{5}, x=-\frac{3}{5}, x=\frac{5}{3}x=−35​,x=53​,x=−53​,x=35​.

Problem 11103

Calculate the average value of f(x)=14x−3f(x)=\frac{1}{\sqrt{4 x-3}}f(x)=4x−3​1​ over the interval from x=3x=3x=3 to x=21x=21x=21.

Problem 11104

Use a calculator to estimate the value of e1.1e^{1.1}e1.1.

Problem 11105

Invest \$2700 at 3.9% APR compounded continuously. a. Find the value after 5 years: \$2700e^{(5 \cdot 0.039)}. b. Define function f(t)=2700e(0.039t)f(t) = 2700e^{(0.039t)}f(t)=2700e(0.039t). c. What is the annual percent change (APY)?

Problem 11106

Problem 11107

Find xxx such that ∣T5(x)−f(x)∣<0.001938|T_{5}(x) - f(x)| < 0.001938∣T5​(x)−f(x)∣<0.001938 for ∣x∣≤1|x| \leq 1∣x∣≤1, where T5(x)=1−x2/2+x4/4!T_{5}(x)=1-x^{2}/2+x^{4}/4!T5​(x)=1−x2/2+x4/4!.

Problem 11108

Find the first and second derivatives of f(x)=14x4+xf(x)=\frac{1}{4}x^4 + xf(x)=41​x4+x.

Problem 11109

Find all values of ccc for which the function f(x)=cx+x2f(x)=\frac{c}{x}+x^{2}f(x)=xc​+x2 has a relative minimum but no relative maximum.

Problem 11110

Calculate the integral: ∫π/2πsin⁡2x2sin⁡xdx\int_{\pi / 2}^{\pi} \frac{\sin 2 x}{2 \sin x} d x∫π/2π​2sinxsin2x​dx.

Problem 11111

Design a rectangular crate with a square base and volume 5632ft35632 \mathrm{ft}^{3}5632ft3. Minimize material costs: top/sides at \$4/ft², bottom at \$7/ft². Find dimensions.

Problem 11112

Find the differential of the function y=3+cos⁡(θ)y=\sqrt{3+\cos (\theta)}y=3+cos(θ)​. What is dyd ydy?

Problem 11113

Find the differential of the function y=ln⁡(sin⁡(8θ))y=\ln (\sin (8 \theta))y=ln(sin(8θ)). What is dyd ydy?

Problem 11114

Determine if f′(x)f'(x)f′(x) is positive (+) or negative (-) for x=13x=\frac{1}{3}x=31​ and x>13x>\frac{1}{3}x>31​ for f(x)=3x2−2x+1f(x)=3x^2-2x+1f(x)=3x2−2x+1.

Problem 11115

Find the differential of y=ln⁡(sin⁡(8θ))y=\ln (\sin (8 \theta))y=ln(sin(8θ)). What is dyd ydy?

Problem 11116

Problem 11117

The function f(x)=x5−3f(x)=x^{5}-3f(x)=x5−3 has a critical point at x=0x=0x=0. Is it a local max, min, or no extrema?

Problem 11118

Find the tangent line equation for f(x)=16+x2f(x)=\frac{1}{6+x^{2}}f(x)=6+x21​ at x=3x=3x=3 with slope −275-\frac{2}{75}−752​.

Problem 11119

Find the derivative f′(x)f^{\prime}(x)f′(x) of the constant function f(x)=−6f(x)=-6f(x)=−6. What is f′(x)f^{\prime}(x)f′(x)?

Problem 11120

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=x−2y=x^{-2}y=x−2. What is dydx=\frac{d y}{d x}=dxdy​=?

Problem 11121

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=4x8f(x)=4 x^{8}f(x)=4x8.

Problem 11122

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=−7x6f(x)=-7 x^{6}f(x)=−7x6.

Problem 11123

Find the derivative dydx\frac{d y}{d x}dxdy​ for y=1x11y=\frac{1}{x^{11}}y=x111​. What is dydx=□\frac{d y}{d x}=\squaredxdy​=□?

Problem 11124

Find the derivative h′(t)h^{\prime}(t)h′(t) for the function h(t)=7.5+7.4t+0.7t3h(t)=7.5+7.4 t+0.7 t^{3}h(t)=7.5+7.4t+0.7t3.

Problem 11125

Find the derivative G′(w)G^{\prime}(w)G′(w) for the function G(w)=67w6+4w3G(w)=\frac{6}{7 w^{6}}+4 \sqrt[3]{w}G(w)=7w66​+43w​. What is G′(w)=□G^{\prime}(w)=\squareG′(w)=□?

Problem 11126

Find the derivative y′y^{\prime}y′ of the function y=6x−5−2x−1y=6 x^{-5}-2 x^{-1}y=6x−5−2x−1.

Problem 11127

Finde die Tangentengleichung für f(x)=13x3−2x+3f(x)=\frac{1}{3} x^{3}-2 x+3f(x)=31​x3−2x+3 am Punkt (x0,f(x0))(x_0, f(x_0))(x0​,f(x0​)).

Problem 11128

Find the tangent line equation for f(x)=16ex+7xf(x)=16 e^{x}+7 xf(x)=16ex+7x at x=0x=0x=0. y=□y=\squarey=□

Problem 11129

Differentiate the function f(x)=10−4x+4exf(x)=10-4 x+4 e^{x}f(x)=10−4x+4ex. Find f′(x)=□f^{\prime}(x)=\squaref′(x)=□.

Problem 11130

Find the rate of learning after 10 hours using N(t)=4+7ln⁡tN(t)=4+7 \ln tN(t)=4+7lnt. What is N′(10)N'(10)N′(10) in words/minute per hour?

Problem 11131

Find the critical points of the function y=256x−x2y=256 \sqrt{x}-x^{2}y=256x​−x2. Provide exact values for xxx.

Problem 11132

Bestimmen Sie die Tangenten- und Normalengleichung von fff an den Punkten B: a) f(x)=x2−x;B(−2∣6)f(x)=x^{2}-x ; B(-2 \mid 6)f(x)=x2−x;B(−2∣6), b) f(x)=4x+2;B(4∣3)f(x)=\frac{4}{x}+2 ; B(4 \mid 3)f(x)=x4​+2;B(4∣3).

Problem 11133

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=−5ln⁡x+11x2−2f(x)=-5 \ln x + 11 x^{2} - 2f(x)=−5lnx+11x2−2.

Problem 11134

Find the velocity of a particle at t=4t=4t=4 seconds, given s(t)=9t2+29ts(t)=9 t^{2}+29 ts(t)=9t2+29t.

Problem 11135

Find the velocity of a particle at t=8t=8t=8 seconds given s(t)=9t2+21ts(t)=9 t^{2}+21 ts(t)=9t2+21t.

Problem 11136

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=ex+2x−ln⁡xf(x)=e^{x}+2 x-\ln xf(x)=ex+2x−lnx.

Problem 11137

Evaluate the integral ∫21(u72−1u5)du\int_{\sqrt{2}}^{1}\left(\frac{u^{7}}{2}-\frac{1}{u^{5}}\right) du∫2​1​(2u7​−u51​)du.

Problem 11138

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=(2x+5)(7x−6)f(x)=(2 x+5)(7 x-6)f(x)=(2x+5)(7x−6).

Problem 11139

Check if the Mean Value Theorem applies to f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ on [1,3][1,3][1,3] and find ccc (round to 3 decimal places).

Problem 11140

Differentiate the function y=log⁡7(xlog⁡9(x))y=\log_{7}(x \log_{9}(x))y=log7​(xlog9​(x)) and find y′=□y'=\squarey′=□.

Problem 11141

Find the average value of $f(x)=\frac{1}{\sqrt{4x-3}}$ from $x=3$ to $x=21$ , rounded to four decimal places.

Find the inflection point of the polynomial $f(x)$ where $f^{\prime \prime}(x)=3x+5$ . Options: $x=-\frac{5}{3}, x=\frac{3}{5}, x=-\frac{3}{5}, x=\frac{5}{3}$ .

Calculate the average value of $f(x)=\frac{1}{\sqrt{4 x-3}}$ over the interval from $x=3$ to $x=21$ .

Use a calculator to estimate the value of $e^{1.1}$ .

Invest \$2700 at 3.9% APR compounded continuously.
a. Find the value after 5 years: \$2700e^{(5 \cdot 0.039)}.
b. Define function $f(t) = 2700e^{(0.039t)}$ .
c. What is the annual percent change (APY)?

Find $x$ such that $|T_{5}(x) - f(x)| < 0.001938$ for $|x| \leq 1$ , where $T_{5}(x)=1-x^{2}/2+x^{4}/4!$ .

Find the first and second derivatives of $f(x)=\frac{1}{4}x^4 + x$ .

Find all values of $c$ for which the function $f(x)=\frac{c}{x}+x^{2}$ has a relative minimum but no relative maximum.

Calculate the integral: $\int_{\pi / 2}^{\pi} \frac{\sin 2 x}{2 \sin x} d x$ .

Design a rectangular crate with a square base and volume $5632 \mathrm{ft}^{3}$ . Minimize material costs: top/sides at \$4/ft², bottom at \$7/ft². Find dimensions.

Find the differential of the function $y=\sqrt{3+\cos (\theta)}$ . What is $d y$ ?

Find the differential of the function $y=\ln (\sin (8 \theta))$ . What is $d y$ ?

Determine if $f'(x)$ is positive (+) or negative (-) for $x=\frac{1}{3}$ and $x>\frac{1}{3}$ for $f(x)=3x^2-2x+1$ .

Find the differential of $y=\ln (\sin (8 \theta))$ . What is $d y$ ?

The function $f(x)=x^{5}-3$ has a critical point at $x=0$ . Is it a local max, min, or no extrema?

Find the tangent line equation for $f(x)=\frac{1}{6+x^{2}}$ at $x=3$ with slope $-\frac{2}{75}$ .

Find the derivative $f^{\prime}(x)$ of the constant function $f(x)=-6$ . What is $f^{\prime}(x)$ ?

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{-2}$ . What is $\frac{d y}{d x}=$ ?

Find the derivative $f^{\prime}(x)$ of the function $f(x)=4 x^{8}$ .

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

Find the derivative $\frac{d y}{d x}$ for $y=\frac{1}{x^{11}}$ . What is $\frac{d y}{d x}=\square$ ?

Find the derivative $h^{\prime}(t)$ for the function $h(t)=7.5+7.4 t+0.7 t^{3}$ .

Find the derivative $G^{\prime}(w)$ for the function $G(w)=\frac{6}{7 w^{6}}+4 \sqrt[3]{w}$ . What is $G^{\prime}(w)=\square$ ?

Find the derivative $y^{\prime}$ of the function $y=6 x^{-5}-2 x^{-1}$ .

Finde die Tangentengleichung für $f(x)=\frac{1}{3} x^{3}-2 x+3$ am Punkt $(x_0, f(x_0))$ .

Find the tangent line equation for $f(x)=16 e^{x}+7 x$ at $x=0$ . $y=\square$

Differentiate the function $f(x)=10-4 x+4 e^{x}$ . Find $f^{\prime}(x)=\square$ .

Find the rate of learning after 10 hours using $N(t)=4+7 \ln t$ . What is $N'(10)$ in words/minute per hour?

Find the critical points of the function $y=256 \sqrt{x}-x^{2}$ . Provide exact values for $x$ .

Bestimmen Sie die Tangenten- und Normalengleichung von $f$ an den Punkten B: a) $f(x)=x^{2}-x ; B(-2 \mid 6)$ , b) $f(x)=\frac{4}{x}+2 ; B(4 \mid 3)$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-5 \ln x + 11 x^{2} - 2$ .

Find the velocity of a particle at $t=4$ seconds, given $s(t)=9 t^{2}+29 t$ .

Find the velocity of a particle at $t=8$ seconds given $s(t)=9 t^{2}+21 t$ .

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

Evaluate the integral $\int_{\sqrt{2}}^{1}\left(\frac{u^{7}}{2}-\frac{1}{u^{5}}\right) du$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(2 x+5)(7 x-6)$ .

Check if the Mean Value Theorem applies to $f(x)=\frac{1}{x}$ on $[1,3]$ and find $c$ (round to 3 decimal places).

Differentiate the function $y=\log_{7}(x \log_{9}(x))$ and find $y'=\square$ .

Find the derivative of $g(x)=\sec^{-1}(9 e^{x})$ . What is $g'(x)$ ?

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{3 x-7}{2 x+3}$ .

Find the $y$ -intercept of the tangent line to $f(x)=(3x^2-1)(x^2+2)$ at $x=1$ .

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

Find the slope of the tangent line to the curve $y=\frac{4}{x}$ at the point $(1,4)$ .

Differentiate the function $h(x)=e^{x^{7}+\ln (x)}$ and find $h^{\prime}(x)$ .

Calculate $\int_{1}^{4} (f(x)-g(x)) dx$ for $f(x)=-x^2+6x-4$ and $g(x)=x$ .

Find the velocity of the object at $t=3$ given its position $s(t)=-3t+t^{2}$ in metres.

Find the missing expression in the equation: $\frac{d}{d x}\left(9-2 x^{2}\right)^{3}=3\left(9-2 x^{2}\right)^{2} ?$

Find the point where the tangent to $y=(x-2)^{2}(x+3)$ is horizontal. Options: $(-1,18)$ , $(1,4)$ , $(-2,16)$ , $(2,0)$ .

Find the expression for ? to make the equation valid: $\frac{\mathrm{d}}{\mathrm{dx}} e^{\mathrm{x}^{7}+6}=e^{\mathrm{x}^{7}+6} ?$

Find the velocity of the object at time $t=1$ for the position function $s(t)=5 t(t-2)^{2}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=3 x^{4} \ln x$ .

Find the instantaneous pulse rate change for a person $x$ inches tall, given $y=590 x^{-1/2}$ for heights 35 and 65 inches.

Find the learning rate for $y=40\sqrt{x}$ at $x=1$ hour and $x=16$ hours. What is the rate at 1 hour?

Find the slope of the tangent to $y=\sqrt{1-x}$ at the point $(-8,3)$ .

Find the learning rate at 1 hour and 16 hours for $y=30 \sqrt{x}, 0 \leq x \leq 16$ . Rate at 1 hour is $\square$ .

Find the gradient of $g(x)=\frac{1}{4} \tan x-3 \cos x-x^{3}$ at $x=\frac{\pi}{6}$ .

Find the learning rate using $y=30 \sqrt{x}$ at 1 hour and 9 hours. What is the rate at 1 hour?

Find the learning rate at 1 hour and 16 hours given $y=50 \sqrt{x}, 0 \leq x \leq 16$ . Rate at 1 hour: $\square$ .

Find $x$ in $[0, 2\pi)$ such that the derivative $h'(x)=0$ for $h(x)=\sin x+\cos x$ .

A radioactive sample halves in 7 years. How long to decay to $\frac{5}{6}$ of its original amount? (3 decimal places)

Find the learning rate for $y = 50 \sqrt{x}$ at $x = 1$ and $x = 4$ . Answer: $\square$ items per hour.

A substance decays to $\frac{1}{3}$ in 10 hours. What is its half-life? Round to 3 decimal places.

Find the instantaneous rate of change of pulse rate $y=598 x^{-1/2}$ for heights 31 and 75 inches. Round to the nearest hundredth.

Find the second derivative of the function $f(x)=-7 x^{3}-4 x^{2}+2 x$ .

Find the instantaneous rate of change of pulse rate $y=598 x^{-1/2}$ for heights 31 inches and 75 inches.

Find the first time $t>0$ when the acceleration $a(t)=0$ for the particle with velocity $v(t)=e^{t^{2}-\sqrt{t}}-\sin t$ .

Find $\left(f^{-1}\right)^{\prime}(1)$ for the function $f(x)=\frac{3 x-1}{x+3}$ .

Find the value of $a$ such that the tangent line to $f(x)=\ln(x^{2}+1)$ at $x=2$ is parallel to $g(x)=x^{5}+x^{3}$ at $x=a$ .

Find $\left(f^{-1}\right)^{\prime}(6)$ using the values of $f(x)$ and $f^{\prime}(x)$ given in the table.