Calculus

Problem 29701

Maximize $z=4xy^{2}$ with the constraint $8x+4y=16$ . Find $x$ , $y$ , and the extreme value of $z$ .

See Solution

Problem 29702

Berechne die Ableitungen für $f(x)=x^{2}+1$ bei $x=1$ und $f(x)=2 x^{3}-x^{2}+3 x-5$ bei $x=2$ .

See Solution

Problem 29703

A firm expects a 20% annual profit growth over 5 years starting from £12m. Calculate total profits using continuous compounding.

See Solution

Problem 29704

Determine where $f$ is concave up or down, and find any points of inflection. Support your answers.

See Solution

Problem 29705

Bestimme den Differenzenquotienten für die Funktionen im angegebenen Intervall:
a) $f(x)=x^{2}+1$ , I=[0;2]
b) $f(x)=\sqrt{x}$ , I=[0;4]
c) $f(x)=\frac{1}{x}$ , I=[0,5;1]
d) $f(x)=\sqrt{x+3}-x$ , I=[1;6]

See Solution

Problem 29706

Calculate the limit as $x$ approaches -5 for the function $e^{-x}$ .

See Solution

Problem 29707

Calculate $f(x) = \frac{\sin(4x)}{x}$ for $x = -0.1, -0.01, -0.001, 0.001, 0.01, 0.1$ and round to six decimal places.

See Solution

Problem 29708

Find the limit: $\lim _{x \rightarrow 2} \frac{\sqrt{x+7}-3}{x-2}$ .

See Solution

Problem 29709

Find $\lim _{x \rightarrow 8} f(x)$ using the Squeeze Theorem, given $2-|x-8| \leq f(x) \leq 2+|x-8|$ .

See Solution

Problem 29710

Find $\lim _{x \rightarrow-3} F(x)$ for the piecewise function $F(x)=\left\{\begin{array}{cc}(x+3)^{3}+3 & x<-3 \\ e^{x}-3 & -3 \leq x\end{array}\right.$ .

See Solution

Problem 29711

Find the limit as $x$ approaches 0: $\lim_{x \rightarrow 0} \frac{\sqrt{x+5}-\sqrt{5}}{x}$ .

See Solution

Problem 29712

Find the limit: $\lim _{x \rightarrow-3^{+}} F(x)$ where $F(x)=\sin (x+3)$ for $x<-3$ and $(x+3)^{3}+2$ for $-3 \leq x$ .

See Solution

Problem 29713

Evaluate the following integrals: a) $\int(\sin x-\cos x)^{2} dx$ b) $\int_{4}^{9} \frac{1}{\sqrt{x}+x} dx$

See Solution

Problem 29714

Find numbers for the inequality: $-1 - 1 < x + 2 < -1 + 1$ when $\varepsilon = 1$ .

See Solution

Problem 29715

Prove that $\lim _{x \rightarrow-3}(x+2)=-1$ .

See Solution

Problem 29716

Find the unit price $p$ for maximum revenue from $R(p)=\frac{-1}{2} p^{2}+1900 p$ and calculate the maximum revenue.

See Solution

Problem 29717

Find appropriate values of $\delta$ for the limit $\lim _{x \rightarrow 2}(x^{2}-3 x+5)=3$ in terms of $\varepsilon$ .

See Solution

Problem 29718

Evaluate the following integrals using integration by parts: a) $\int_{0}^{1} x e^{-5 x} d x$ , b) $\int(5 x+1) \cos x d x$ .

See Solution

Problem 29719

Evaluate the following trigonometric integrals: a. $\int \sin ^{4} x \cos ^{5} x d x$ b. $\int_{0}^{\frac{\pi}{4}} \tan ^{3} x \sec ^{3} x d x$

See Solution

Problem 29720

Evaluate the integrals using partial fractions:
1. $\int \frac{11 x+7}{2 x^{2}+7 x-4} d x$
2. $\int_{0}^{2} \frac{x^{2}}{x^{2}+2 x+1} d x$

See Solution

Problem 29721

Evaluate the following integrals using trigonometric substitution: a. $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sqrt{4-x^{2}} \, dx$ b. $\int^{4} \frac{dx}{\sqrt{(x^{2}-1)^{3}}}$

See Solution

Problem 29722

Evaluate the following integrals using partial fractions: a. $\int \frac{11 x+7}{2 x^{2}+7 x-4} d x$ b. $\int_{0}^{2} \frac{x^{2}}{x^{2}+2 x+1} d x$

See Solution

Problem 29723

Approximate the integral $\int_{1}^{2} \frac{1}{x} dx$ using (a) the Trapezoidal Rule and (b) the Midpoint Rule with $n=5$ .

See Solution

Problem 29724

Evaluate the improper integrals:
1. $\int_{0}^{\infty} x e^{-2 x} d x$
2. $\int_{-\infty}^{\infty} \frac{2}{x^{2}+1} d x$

See Solution

Problem 29725

Une pierre tombe d'une tour en 1,2 s. Quel est sa vitesse au sol ? $(-12 \mathrm{~m} / \mathrm{s})$

See Solution

Problem 29726

Calculate the definite integral from 1 to 6: $\int_{1}^{6} 2^{x} \, dx$ .

See Solution

Problem 29727

Evaluate the integral: $\int_{1}^{e^{5}} \frac{2}{x} \, dx$

See Solution

Problem 29728

Evaluate the integral: $\int_{0}^{\sqrt{3} / 2} \frac{2}{\sqrt{1-x^{2}}} d x$ exactly.

See Solution

Problem 29729

Find the exact value of the integral $\int_{0}^{\pi / 4} \frac{3+\cos ^{3}(x)}{\cos ^{2}(x)} dx$ .

See Solution

Problem 29730

Exercice 8: Soit $f(x)=x^{3}-3x+1$ . Trouvez les limites en $-\infty$ et $+\infty$ , étudiez les variations et montrez que $f(x)=0$ a trois solutions.

See Solution

Problem 29731

Find $\frac{d y}{d x}$ for the equation $y^{2}=x^{2}+\sin x \cdot y$ .

See Solution

Problem 29732

Exercice 10 : Soit $f(x)=\left(-5 x^{2}+5\right) e^{x}$ , calculez les limites en $-\infty$ et $+\infty$ , étudiez les variations et trouvez la tangente en $x=0$ .

See Solution

Problem 29733

Find the initial velocity and acceleration of an auto with $x(t)=27t-4.0t^{3}$ at $t=0$ . Choices are A-E.

See Solution

Problem 29734

Find the limit: $\lim _{x \rightarrow-1^{-}} e^{\frac{-1}{1-x^{2}}}$ .

See Solution

Problem 29735

Find the radius of convergence for the series $\sum_{n=1}^{\infty} \frac{6^{n} n !}{n^{n}} x^{n}$ .

See Solution

Problem 29736

Zadanie 1. Pokaż, że jeśli $\lim _{n \rightarrow \infty} b_{n}=0$ , to szereg $\sum_{n=0}^{\infty}(b_{n}-b_{n+p})$ jest zbieżny i znajdź jego sumę w zależności od $b_{n}$ i $p$ .

See Solution

Problem 29737

Bestimme die Ableitung von $f(x)=2 x^{2}+1$ bei $x_{0}=2$ und allgemein bei $x_{0}$ . Auch für $g(x)=a \cdot x^{2}+c$ .

See Solution

Problem 29738

Is the function $f(x)=\left\{\begin{array}{ll} 3 \ln (1+x) & x \geq 0 \\ x^{5}+x & x<0 \end{array}\right.$ continuous on $\mathbb{R}$ ?

See Solution

Problem 29739

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{\tan (7 x)}$ .

See Solution

Problem 29740

1. Determine if the series $\sum_{n=0}^{\infty} \frac{n^{50}}{3^{n}}$ converges or diverges.
2. Determine if the series $\sum_{n=1}^{\infty} \frac{\arctan (n)}{n^{2}}$ converges or diverges.
3. Find the radius of convergence for $\sum_{n=1}^{\infty} \frac{6^{n} n !}{n^{n}} x^{n}$ .
4. Evaluate the limit $\lim _{x \rightarrow-\infty} e^{x} \sin \left(x^{2}\right)$ or state it does not exist.
5. Evaluate the limit $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{\tan (7 x)}$ or state it does not exist.

See Solution

Problem 29741

Find the limit: $\lim _{x \rightarrow-\infty} e^{x} \sin \left(x^{2}\right)$ , or state if it does not exist.

See Solution

Problem 29742

Aufgaben: Finde die Stammfunktionen für die folgenden Funktionen: a) $f(x)=4 x^{3}+2 x^{2}-1$ , b) $f(x)=x^{5}-4 x^{3}$ , c) $f(x)=-x^{3}-2 x+7$ , d) $f(x)=\frac{3}{4} x^{2}+\frac{8}{7} x^{3}$ , e) $f(x)=0,24 x^{7}-5,4 x$ , f) $f(x)=-25 x^{4}+33 x^{2}$ , g) $f(x)=2 x^{3}-15 x^{2}$ , h) $f(x)=162 x^{8}-54 x^{5}$ .

See Solution

Problem 29743

Bestimmen Sie die ersten drei Ableitungen von $f$ für die Funktionen: a) $(2x-3)e^x$ , b) $(x^2+1)e^x$ , c) $(5x^2+1)e^{-x}$ , d) $5x^2e^{-\frac{1}{4}x}$ . Erklären Sie den Vorteil des Ausklammerns.

See Solution

Problem 29744

1. Find the limit: $\lim _{x \rightarrow -1^{-}} e^{\frac{-1}{1-x^{2}}}$ .
2. Is the function $f(x)=\begin{cases} 3 \ln (1+x) & x \geq 0 \\ x^{5}+x & x<0 \end{cases}$ continuous on $\mathbb{R}$ ? (YES / NO)
3. Does the function from question 2 have an inverse? (YES / NO)
4. For $f(x)=\frac{x^{2}}{1+x+x^{2}}$ on $(-2, \infty)$ : (a) Does $f$ have a maximum? (YES / NO) (b) Does $f$ have a minimum? (YES / NO) (c) For which $x$ is $f$ decreasing?
5. For $f(x)=\sum_{n=0}^{\infty} e^{-2 n x}$ : (a) For which $x \in \mathbb{R}$ is $f(x)$ defined? (b) Calculate $f^{\prime}(x)$ .

See Solution

Problem 29745

Find the limit as $x$ approaches $-\infty$ for the expression $e^{x} \sin(x^{2})$ .

See Solution

Problem 29746

Bewerte die Aussagen: a) $f^{\prime}(x_{0})=0 \Rightarrow f$ hat Extremstelle bei $x_{0}$ ; b) 2. Grades Funktion hat immer Extremstelle; c) globales Maximum ist lokal.

See Solution

Problem 29747

Find the limits: 1) $\lim _{x \rightarrow-\infty} e^{x} \sin \left(x^{2}\right)$ and 2) $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{\tan (7 x)}$ .

See Solution

Problem 29748

Find the domain of $f(x) = \sum_{n=0}^{\infty} e^{-2 n x}$ and compute $f^{\prime}(x)$ .

See Solution

Problem 29749

Find $\lim _{n \rightarrow \infty} \frac{(e-1) \sum_{k=0}^{\infty} \frac{1}{e^{k}}}{\left(1+\frac{2}{n}\right)^{n}}$ .

See Solution

Problem 29750

Determine if the series $\sum_{n=1}^{\infty} \frac{n ! \sin (1 / n)}{n^{n}}$ converges or diverges.

See Solution

Problem 29751

Gegeben sind die Funktionen $f(x)=0,5 x^{2}$ und $g(x)=3 x^{3}+1$ . Bestimme den Differenzenquotienten für die Intervalle: a) $I=[0 ; 2]$ b) $I=[-1 ; 3]$ c) $I=[-1 ; 1]$ d) $I=[-2 ;-1]$

See Solution

Problem 29752

Determine the radius of convergence for the series $\sum_{n=1}^{\infty} 2^{n}\left(n+\frac{1}{n}\right) x^{n}$ .

See Solution

Problem 29753

Evaluate the integral $\int \frac{1}{x^{2}} \cos^{2}\left(\frac{1}{x}\right) dx$ using the substitution $u=-\frac{1}{x}$ .

See Solution

Problem 29754

Evaluate the integral $\int \frac{9 r^{2} d r}{\sqrt{1-r^{3}}}$ using the substitution $u=1-r^{3}$ .

See Solution

Problem 29755

Finde die Tangentengleichung der Funktion $f(x)=\cos(0,5 x)$ bei $x_{1}=2 \pi$ und $x_{2}=\frac{\pi}{3}$ .

See Solution

Problem 29756

Bestimme die Tangentengleichung $\mathrm{t}: \mathrm{y}=\square \mathrm{x}+$ für $f(x)=x^{2}-3$ im Punkt $P(1|-2)$ .

See Solution

Problem 29757

Evaluate the integral: $\int x^{1 / 2} \sin \left(x^{3 / 2}+1\right) d x$

See Solution

Problem 29758

Bestimme die Tangentengleichung an $f(x)=x^{2}-3$ im Punkt $P(1|-2)$ . Tangentengleichung: $y=\square x+$

See Solution

Problem 29759

Bestimme den Ausdruck $\frac{d f}{d a}+\frac{d f}{d b}-\frac{d f}{d c}$ für $f(a, b, c)=3 a^{2} b-3 b^{2} c+3 a c^{2}$ .

See Solution

Problem 29760

Bestimme die Tangentengleichungen an $f(x)=x^{2}-3$ bei $P(1|-2)$ und an $f(x)=\frac{1}{x}$ bei $P(2|\frac{1}{2})$ .

See Solution

Problem 29761

Evaluate the integral: $\int 3 y \sqrt{7-3 y^{2}} d y$

See Solution

Problem 29762

Evaluate the integral: $\int x \sqrt{4-x} \, dx$

See Solution

Problem 29763

Find the limit: $\lim _{x \rightarrow \infty} \frac{e^{-x}+1}{e^{-x}-1}$ . What is the value?

See Solution

Problem 29764

Find the limit: $\lim _{x \rightarrow 1^{-}} \frac{x^{2}+2 x-3}{|x-1|}$ or state if it doesn't exist.

See Solution

Problem 29765

Υπολογίστε την παράγωγο της συνάρτησης $f(x)=\frac{1}{e^{x}+1}$ για $x \in \mathbb{R}$ .

See Solution

Problem 29766

Bestimme die Integrale: a) $\int_{2}^{5} x \, dx$ b) $\int_{-1}^{1}(2x+1) \, dx$ mithilfe von Flächen.

See Solution

Problem 29767

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sin \left(\frac{1}{x}\right)}{x}$ . What is the result?

See Solution

Problem 29768

Find the limit: $\lim _{x \rightarrow \frac{\pi}{6}} \frac{\cos 3 x}{1-4 \sin ^{2} x}$ .

See Solution

Problem 29769

Find the radius of convergence for the series $\sum_{n=1}^{\infty} 2^{n}\left(n+\frac{1}{n}\right) x^{n}$ .

See Solution

Problem 29770

Is the function $f(x)=\begin{cases} 3 \ln (1+x) & x \geq 0 \\ x^{5}+x & x<0 \end{cases}$ continuous on $\mathbb{R}$ ? (YES / NO) Does it have an inverse? (YES / NO)

See Solution

Problem 29771

Is the function $f(x)=\begin{cases} 3 \ln (1+x) & x \geq 0 \\ x^{5}+x & x<0 \end{cases}$ continuous on $\mathbb{R}$ ? (YES / NO)

See Solution

Problem 29772

Find the change in momentum from $t=0$ to $t$ given $\frac{dp}{dt} = 4t \hat{i} + 3t^2 \hat{j}$ . Options: a. $t^3 \hat{i} + 2t^2 \hat{j}$ b. $4 \hat{i} + 6 \hat{j}$ c. $2t^2 \hat{i} + t^3 \hat{j}$ d. $2t^3 \hat{i} + t^2 \hat{j}$

See Solution

Problem 29773

Find the force on a $3 \mathrm{~kg}$ particle system at $t=1$ sec, given $\vec{v}=2 t^{2} \hat{i}+3 t \hat{j}$ . Choices: a. $16 \hat{i}+12 \hat{j}$ b. $8 \hat{i}+6 \hat{j}$ c. $8 \hat{i}+12 \hat{j}$ d. $12 \hat{i}+9 \hat{j}$ .

See Solution

Problem 29774

Find critical points and determine concavity for the function $y=\frac{5 x+7}{5 x}$ , $x \neq 0$ , using derivatives.

See Solution

Problem 29775

In Dagoberts Geldspeicher ist die Höhe $h'(t) = e^{-0,05t} \cdot t$ m/d. Bestimme die Höhenfunktion $h(t)$ .

See Solution

Problem 29776

Find the change in momentum of a particle with $\frac{dp}{dt} = 4t \hat{i} + 3t^{2} \hat{j}$ from $t=0$ to $t$ .

See Solution

Problem 29777

Find points of inflection for $f(x)=3x+(x+2)^{\frac{2}{5}}$ and identify concave up/down intervals.

See Solution

Problem 29778

Gjeni pikën $(a, b)$ ku $y=\frac{2 x^{2}+1}{2}$ dhe $y=\frac{1+4 x}{x}$ janë tangjente dhe pingule. Gjeni koeficientin këndor dhe ekuacionin e drejtëzës.

See Solution

Problem 29779

Një vijë kubike ka ekuacionin $y=x^{3}+x^{2}+2x+1$ . Gjeni tangjentën në $x=0$ dhe koordinatat e pikave B, C, dhe sipërfaqen e trekëndëshit $\mathrm{BCO}$ .

See Solution

Problem 29780

Find where the tangent line to $y=f(x)=x^{3} \ln (x)$ is horizontal for $x \in (0, \infty)$ .

See Solution

Problem 29781

Find the extremum points of the line $y = x^3 - 6x^2$ . Determine their coordinates and nature using the second derivative.

See Solution

Problem 29782

Is the function $f(x)=\left\{\begin{array}{ll}(x-1) \sin \left(\frac{1}{x-1}\right) & x \neq 1 \\ -1 & x=1\end{array}\right.$ continuous on $\mathbb{R}$ ?

See Solution

Problem 29783

Find the extremum points of $y=2x^3+30x^2+1$ and explain their characteristics.

See Solution

Problem 29784

Find the extremum points of $y=2x+30x^{2}+1$ and describe their nature.

See Solution

Problem 29785

Find the inverse transform of $\mathcal{L}^{-1}\left\{\frac{10 s+2}{s^{2}-6 s+5}\right\}$ .

See Solution

Problem 29786

Lancez une bille $B_{1}$ de masse $m$ pour atteindre $B_{2}$ . Trouvez $v_{0}$ et $v$ avec $g=10 \mathrm{~m/s^2}$ , $\alpha=30^\circ$ , $D=8 \mathrm{~m}$ , $h=1 \mathrm{~m}$ .

See Solution

Problem 29787

Bestimme die erste Ableitung von $f(x)=5 \sqrt{x}$ .

See Solution

Problem 29788

Find the derivative of $f(x)=2 x^{\wedge} 3-9 x^{\wedge} 2+12 x+7$ . Identify the coordinates and nature of the two extremum points.

See Solution

Problem 29789

Bestimme die Punkte, an denen die Funktion $f$ eine Steigung von -2 hat: a) $f(x)=x^{2}+3 x+5$ , b) $f(x)=2 \sqrt{x^{3}}-8 x$ .

See Solution

Problem 29790

Find the combined profit/loss function from supplier $-4x^2 + 20x$ and distributor $x^2 - 40x + 100$ , then differentiate to find the optimum value.

See Solution

Problem 29791

Find $f^{\prime}(3)$ for the function $f(x) = -0.5x^2 + 4x$ using the limit definition of the derivative.

See Solution

Problem 29792

Berechnen Sie die Fläche zwischen dem Graphen von $f(x)=\frac{1}{2} x^{2}$ , der Tangente in $P(3 \mid 4,5)$ und der $x$ -Achse.

See Solution

Problem 29793

Find the limit: $\lim _{x \rightarrow 0} \frac{-\sqrt{3}+\sqrt{x+3}}{x}$ (2 marks)

See Solution

Problem 29794

Bestimmen Sie die Ableitung von $f(t) = \frac{2 \cdot \sin t}{2 + \cos t}$ .

See Solution

Problem 29795

Find the derivative of $y=e^{2 x} \sec 3 x$ given that $\frac{d(\sec x)}{d x}=\sec x \tan x$ .

See Solution

Problem 29796

Find the limit of $\frac{j(x+h)-j(x)}{h}$ for $j(x)=4^{x-2}$ .

See Solution

Problem 29797

Determine if the limit exists: $\lim _{h \rightarrow 0} \frac{\frac{6}{7+h}-\frac{6}{7}}{h}$ . If yes, find its value.

See Solution

Problem 29798

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{9 x^{3}+4 x-4}{6 x^{4}-8 x^{3}-5}$ . What is its value?

See Solution

Problem 29799

Find constants $a$ and $b$ such that for curve $C$ : $\frac{dy}{dx}=\frac{1}{a\sqrt{b-x^2}}$ .

See Solution

Problem 29800

Choose the correct limit: If $\lim_{x \rightarrow 5^{-}} f(x)=8$ and $\lim_{x \rightarrow 5^{+}} f(x)=2$ , then $\lim_{x \rightarrow 5} f(x)$ is? Options: does not exist, $\infty$ , 8, -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 29701

Maximize z=4xy2z=4xy^{2}z=4xy2 with the constraint 8x+4y=168x+4y=168x+4y=16. Find xxx, yyy, and the extreme value of zzz.

Problem 29702

Berechne die Ableitungen für f(x)=x2+1f(x)=x^{2}+1f(x)=x2+1 bei x=1x=1x=1 und f(x)=2x3−x2+3x−5f(x)=2 x^{3}-x^{2}+3 x-5f(x)=2x3−x2+3x−5 bei x=2x=2x=2.

Problem 29703

A firm expects a 20% annual profit growth over 5 years starting from £12m. Calculate total profits using continuous compounding.

Problem 29704

Determine where f f f is concave up or down, and find any points of inflection. Support your answers.

Problem 29705

Bestimme den Differenzenquotienten für die Funktionen im angegebenen Intervall:a) f(x)=x2+1f(x)=x^{2}+1f(x)=x2+1, I=[0;2]b) f(x)=xf(x)=\sqrt{x}f(x)=x​, I=[0;4]c) f(x)=1xf(x)=\frac{1}{x}f(x)=x1​, I=[0,5;1]d) f(x)=x+3−xf(x)=\sqrt{x+3}-xf(x)=x+3​−x, I=[1;6]

Problem 29706

Calculate the limit as xxx approaches -5 for the function e−xe^{-x}e−x.

Problem 29707

Calculate f(x)=sin⁡(4x)xf(x) = \frac{\sin(4x)}{x}f(x)=xsin(4x)​ for x=−0.1,−0.01,−0.001,0.001,0.01,0.1x = -0.1, -0.01, -0.001, 0.001, 0.01, 0.1x=−0.1,−0.01,−0.001,0.001,0.01,0.1 and round to six decimal places.

Problem 29708

Find the limit: lim⁡x→2x+7−3x−2\lim _{x \rightarrow 2} \frac{\sqrt{x+7}-3}{x-2}limx→2​x−2x+7​−3​.

Problem 29709

Find lim⁡x→8f(x)\lim _{x \rightarrow 8} f(x)limx→8​f(x) using the Squeeze Theorem, given 2−∣x−8∣≤f(x)≤2+∣x−8∣2-|x-8| \leq f(x) \leq 2+|x-8|2−∣x−8∣≤f(x)≤2+∣x−8∣.

Problem 29710

Find lim⁡x→−3F(x)\lim _{x \rightarrow-3} F(x)limx→−3​F(x) for the piecewise function F(x)={(x+3)3+3x<−3ex−3−3≤xF(x)=\left\{\begin{array}{cc}(x+3)^{3}+3 & x<-3 \\ e^{x}-3 & -3 \leq x\end{array}\right.F(x)={(x+3)3+3ex−3​x<−3−3≤x​.

Problem 29711

Find the limit as xxx approaches 0: lim⁡x→0x+5−5x\lim_{x \rightarrow 0} \frac{\sqrt{x+5}-\sqrt{5}}{x}limx→0​xx+5​−5​​.

Problem 29712

Find the limit: lim⁡x→−3+F(x)\lim _{x \rightarrow-3^{+}} F(x)limx→−3+​F(x) where F(x)=sin⁡(x+3)F(x)=\sin (x+3)F(x)=sin(x+3) for x<−3x<-3x<−3 and (x+3)3+2(x+3)^{3}+2(x+3)3+2 for −3≤x-3 \leq x−3≤x.

Problem 29713

Evaluate the following integrals: a) ∫(sin⁡x−cos⁡x)2dx\int(\sin x-\cos x)^{2} dx∫(sinx−cosx)2dx b) ∫491x+xdx\int_{4}^{9} \frac{1}{\sqrt{x}+x} dx∫49​x​+x1​dx

Problem 29714

Find numbers for the inequality: −1−1<x+2<−1+1-1 - 1 < x + 2 < -1 + 1−1−1<x+2<−1+1 when ε=1\varepsilon = 1ε=1.

Problem 29715

Prove that lim⁡x→−3(x+2)=−1\lim _{x \rightarrow-3}(x+2)=-1limx→−3​(x+2)=−1.

Problem 29716

Find the unit price ppp for maximum revenue from R(p)=−12p2+1900pR(p)=\frac{-1}{2} p^{2}+1900 pR(p)=2−1​p2+1900p and calculate the maximum revenue.

Problem 29717

Find appropriate values of δ\deltaδ for the limit lim⁡x→2(x2−3x+5)=3\lim _{x \rightarrow 2}(x^{2}-3 x+5)=3limx→2​(x2−3x+5)=3 in terms of ε\varepsilonε.

Problem 29718

Evaluate the following integrals using integration by parts: a) ∫01xe−5xdx\int_{0}^{1} x e^{-5 x} d x∫01​xe−5xdx, b) ∫(5x+1)cos⁡xdx\int(5 x+1) \cos x d x∫(5x+1)cosxdx.

Problem 29719

Evaluate the following trigonometric integrals: a. ∫sin⁡4xcos⁡5xdx\int \sin ^{4} x \cos ^{5} x d x∫sin4xcos5xdx b. ∫0π4tan⁡3xsec⁡3xdx\int_{0}^{\frac{\pi}{4}} \tan ^{3} x \sec ^{3} x d x∫04π​​tan3xsec3xdx

Problem 29720

Evaluate the integrals using partial fractions: 1. ∫11x+72x2+7x−4dx\int \frac{11 x+7}{2 x^{2}+7 x-4} d x∫2x2+7x−411x+7​dx 2. ∫02x2x2+2x+1dx\int_{0}^{2} \frac{x^{2}}{x^{2}+2 x+1} d x∫02​x2+2x+1x2​dx

Problem 29721

Evaluate the following integrals using trigonometric substitution: a. ∫−π4π44−x2 dx\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sqrt{4-x^{2}} \, dx∫−4π​4π​​4−x2​dx b. ∫4dx(x2−1)3\int^{4} \frac{dx}{\sqrt{(x^{2}-1)^{3}}}∫4(x2−1)3​dx​

Problem 29722

Evaluate the following integrals using partial fractions: a. ∫11x+72x2+7x−4dx\int \frac{11 x+7}{2 x^{2}+7 x-4} d x∫2x2+7x−411x+7​dx b. ∫02x2x2+2x+1dx\int_{0}^{2} \frac{x^{2}}{x^{2}+2 x+1} d x∫02​x2+2x+1x2​dx

Problem 29723

Approximate the integral ∫121xdx\int_{1}^{2} \frac{1}{x} dx∫12​x1​dx using (a) the Trapezoidal Rule and (b) the Midpoint Rule with n=5n=5n=5.

Problem 29724

Evaluate the improper integrals: 1. ∫0∞xe−2xdx\int_{0}^{\infty} x e^{-2 x} d x∫0∞​xe−2xdx 2. ∫−∞∞2x2+1dx\int_{-\infty}^{\infty} \frac{2}{x^{2}+1} d x∫−∞∞​x2+12​dx

Problem 29725

Une pierre tombe d'une tour en 1,2 s. Quel est sa vitesse au sol ? (−12 m/s)(-12 \mathrm{~m} / \mathrm{s})(−12 m/s)

Problem 29726

Calculate the definite integral from 1 to 6: ∫162x dx\int_{1}^{6} 2^{x} \, dx∫16​2xdx.

Problem 29727

Evaluate the integral: ∫1e52x dx\int_{1}^{e^{5}} \frac{2}{x} \, dx∫1e5​x2​dx

Problem 29728

Evaluate the integral: ∫03/221−x2dx\int_{0}^{\sqrt{3} / 2} \frac{2}{\sqrt{1-x^{2}}} d x∫03​/2​1−x2​2​dx exactly.

Problem 29729

Find the exact value of the integral ∫0π/43+cos⁡3(x)cos⁡2(x)dx\int_{0}^{\pi / 4} \frac{3+\cos ^{3}(x)}{\cos ^{2}(x)} dx∫0π/4​cos2(x)3+cos3(x)​dx.

Problem 29730

Exercice 8: Soit f(x)=x3−3x+1f(x)=x^{3}-3x+1f(x)=x3−3x+1. Trouvez les limites en −∞-\infty−∞ et +∞+\infty+∞, étudiez les variations et montrez que f(x)=0f(x)=0f(x)=0 a trois solutions.

Problem 29731

Find dydx\frac{d y}{d x}dxdy​ for the equation y2=x2+sin⁡x⋅yy^{2}=x^{2}+\sin x \cdot yy2=x2+sinx⋅y.

Problem 29732

Exercice 10 : Soit f(x)=(−5x2+5)exf(x)=\left(-5 x^{2}+5\right) e^{x}f(x)=(−5x2+5)ex, calculez les limites en −∞-\infty−∞ et +∞+\infty+∞, étudiez les variations et trouvez la tangente en x=0x=0x=0.

Problem 29733

Find the initial velocity and acceleration of an auto with x(t)=27t−4.0t3x(t)=27t-4.0t^{3}x(t)=27t−4.0t3 at t=0t=0t=0. Choices are A-E.

Problem 29734

Find the limit: lim⁡x→−1−e−11−x2\lim _{x \rightarrow-1^{-}} e^{\frac{-1}{1-x^{2}}}limx→−1−​e1−x2−1​.

Problem 29735

Find the radius of convergence for the series ∑n=1∞6nn!nnxn\sum_{n=1}^{\infty} \frac{6^{n} n !}{n^{n}} x^{n}∑n=1∞​nn6nn!​xn.

Problem 29736

Zadanie 1. Pokaż, że jeśli lim⁡n→∞bn=0\lim _{n \rightarrow \infty} b_{n}=0limn→∞​bn​=0, to szereg ∑n=0∞(bn−bn+p)\sum_{n=0}^{\infty}(b_{n}-b_{n+p})∑n=0∞​(bn​−bn+p​) jest zbieżny i znajdź jego sumę w zależności od bnb_{n}bn​ i ppp.

Problem 29737

Bestimme die Ableitung von f(x)=2x2+1f(x)=2 x^{2}+1f(x)=2x2+1 bei x0=2x_{0}=2x0​=2 und allgemein bei x0x_{0}x0​. Auch für g(x)=a⋅x2+cg(x)=a \cdot x^{2}+cg(x)=a⋅x2+c.

Problem 29738

Is the function f(x)={3ln⁡(1+x)x≥0x5+xx<0f(x)=\left\{\begin{array}{ll} 3 \ln (1+x) & x \geq 0 \\ x^{5}+x & x<0 \end{array}\right.f(x)={3ln(1+x)x5+x​x≥0x<0​ continuous on R\mathbb{R}R?

Problem 29739

Find the limit: lim⁡x→0sin⁡(3x)tan⁡(7x)\lim _{x \rightarrow 0} \frac{\sin (3 x)}{\tan (7 x)}limx→0​tan(7x)sin(3x)​.

Problem 29740

Maximize $z=4xy^{2}$ with the constraint $8x+4y=16$ . Find $x$ , $y$ , and the extreme value of $z$ .

Berechne die Ableitungen für $f(x)=x^{2}+1$ bei $x=1$ und $f(x)=2 x^{3}-x^{2}+3 x-5$ bei $x=2$ .

Determine where $f$ is concave up or down, and find any points of inflection. Support your answers.

Bestimme den Differenzenquotienten für die Funktionen im angegebenen Intervall:
a) $f(x)=x^{2}+1$ , I=[0;2]
b) $f(x)=\sqrt{x}$ , I=[0;4]
c) $f(x)=\frac{1}{x}$ , I=[0,5;1]
d) $f(x)=\sqrt{x+3}-x$ , I=[1;6]

Calculate the limit as $x$ approaches -5 for the function $e^{-x}$ .

Calculate $f(x) = \frac{\sin(4x)}{x}$ for $x = -0.1, -0.01, -0.001, 0.001, 0.01, 0.1$ and round to six decimal places.

Find the limit: $\lim _{x \rightarrow 2} \frac{\sqrt{x+7}-3}{x-2}$ .

Find $\lim _{x \rightarrow 8} f(x)$ using the Squeeze Theorem, given $2-|x-8| \leq f(x) \leq 2+|x-8|$ .

Find $\lim _{x \rightarrow-3} F(x)$ for the piecewise function $F(x)=\left\{\begin{array}{cc}(x+3)^{3}+3 & x<-3 \\ e^{x}-3 & -3 \leq x\end{array}\right.$ .

Find the limit as $x$ approaches 0: $\lim_{x \rightarrow 0} \frac{\sqrt{x+5}-\sqrt{5}}{x}$ .

Find the limit: $\lim _{x \rightarrow-3^{+}} F(x)$ where $F(x)=\sin (x+3)$ for $x<-3$ and $(x+3)^{3}+2$ for $-3 \leq x$ .

Evaluate the following integrals: a) $\int(\sin x-\cos x)^{2} dx$ b) $\int_{4}^{9} \frac{1}{\sqrt{x}+x} dx$

Find numbers for the inequality: $-1 - 1 < x + 2 < -1 + 1$ when $\varepsilon = 1$ .

Prove that $\lim _{x \rightarrow-3}(x+2)=-1$ .

Find the unit price $p$ for maximum revenue from $R(p)=\frac{-1}{2} p^{2}+1900 p$ and calculate the maximum revenue.

Find appropriate values of $\delta$ for the limit $\lim _{x \rightarrow 2}(x^{2}-3 x+5)=3$ in terms of $\varepsilon$ .

Evaluate the following integrals using integration by parts: a) $\int_{0}^{1} x e^{-5 x} d x$ , b) $\int(5 x+1) \cos x d x$ .

Evaluate the following trigonometric integrals: a. $\int \sin ^{4} x \cos ^{5} x d x$ b. $\int_{0}^{\frac{\pi}{4}} \tan ^{3} x \sec ^{3} x d x$

Evaluate the integrals using partial fractions:
1. $\int \frac{11 x+7}{2 x^{2}+7 x-4} d x$
2. $\int_{0}^{2} \frac{x^{2}}{x^{2}+2 x+1} d x$

Evaluate the following integrals using trigonometric substitution: a. $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sqrt{4-x^{2}} \, dx$ b. $\int^{4} \frac{dx}{\sqrt{(x^{2}-1)^{3}}}$

Evaluate the following integrals using partial fractions: a. $\int \frac{11 x+7}{2 x^{2}+7 x-4} d x$ b. $\int_{0}^{2} \frac{x^{2}}{x^{2}+2 x+1} d x$

Approximate the integral $\int_{1}^{2} \frac{1}{x} dx$ using (a) the Trapezoidal Rule and (b) the Midpoint Rule with $n=5$ .

Evaluate the improper integrals:
1. $\int_{0}^{\infty} x e^{-2 x} d x$
2. $\int_{-\infty}^{\infty} \frac{2}{x^{2}+1} d x$

Une pierre tombe d'une tour en 1,2 s. Quel est sa vitesse au sol ? $(-12 \mathrm{~m} / \mathrm{s})$

Calculate the definite integral from 1 to 6: $\int_{1}^{6} 2^{x} \, dx$ .

Evaluate the integral: $\int_{1}^{e^{5}} \frac{2}{x} \, dx$

Evaluate the integral: $\int_{0}^{\sqrt{3} / 2} \frac{2}{\sqrt{1-x^{2}}} d x$ exactly.

Find the exact value of the integral $\int_{0}^{\pi / 4} \frac{3+\cos ^{3}(x)}{\cos ^{2}(x)} dx$ .

Exercice 8: Soit $f(x)=x^{3}-3x+1$ . Trouvez les limites en $-\infty$ et $+\infty$ , étudiez les variations et montrez que $f(x)=0$ a trois solutions.

Find $\frac{d y}{d x}$ for the equation $y^{2}=x^{2}+\sin x \cdot y$ .

Exercice 10 : Soit $f(x)=\left(-5 x^{2}+5\right) e^{x}$ , calculez les limites en $-\infty$ et $+\infty$ , étudiez les variations et trouvez la tangente en $x=0$ .

Find the initial velocity and acceleration of an auto with $x(t)=27t-4.0t^{3}$ at $t=0$ . Choices are A-E.

Find the limit: $\lim _{x \rightarrow-1^{-}} e^{\frac{-1}{1-x^{2}}}$ .

Find the radius of convergence for the series $\sum_{n=1}^{\infty} \frac{6^{n} n !}{n^{n}} x^{n}$ .

Zadanie 1. Pokaż, że jeśli $\lim _{n \rightarrow \infty} b_{n}=0$ , to szereg $\sum_{n=0}^{\infty}(b_{n}-b_{n+p})$ jest zbieżny i znajdź jego sumę w zależności od $b_{n}$ i $p$ .

Bestimme die Ableitung von $f(x)=2 x^{2}+1$ bei $x_{0}=2$ und allgemein bei $x_{0}$ . Auch für $g(x)=a \cdot x^{2}+c$ .

Is the function $f(x)=\left\{\begin{array}{ll} 3 \ln (1+x) & x \geq 0 \\ x^{5}+x & x<0 \end{array}\right.$ continuous on $\mathbb{R}$ ?

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{\tan (7 x)}$ .

Find the limit: $\lim _{x \rightarrow-\infty} e^{x} \sin \left(x^{2}\right)$ , or state if it does not exist.

Bestimmen Sie die ersten drei Ableitungen von $f$ für die Funktionen: a) $(2x-3)e^x$ , b) $(x^2+1)e^x$ , c) $(5x^2+1)e^{-x}$ , d) $5x^2e^{-\frac{1}{4}x}$ . Erklären Sie den Vorteil des Ausklammerns.

Find the limit as $x$ approaches $-\infty$ for the expression $e^{x} \sin(x^{2})$ .

Bewerte die Aussagen: a) $f^{\prime}(x_{0})=0 \Rightarrow f$ hat Extremstelle bei $x_{0}$ ; b) 2. Grades Funktion hat immer Extremstelle; c) globales Maximum ist lokal.

Find the limits: 1) $\lim _{x \rightarrow-\infty} e^{x} \sin \left(x^{2}\right)$ and 2) $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{\tan (7 x)}$ .

Find the domain of $f(x) = \sum_{n=0}^{\infty} e^{-2 n x}$ and compute $f^{\prime}(x)$ .

Find $\lim _{n \rightarrow \infty} \frac{(e-1) \sum_{k=0}^{\infty} \frac{1}{e^{k}}}{\left(1+\frac{2}{n}\right)^{n}}$ .

Determine if the series $\sum_{n=1}^{\infty} \frac{n ! \sin (1 / n)}{n^{n}}$ converges or diverges.

Gegeben sind die Funktionen $f(x)=0,5 x^{2}$ und $g(x)=3 x^{3}+1$ . Bestimme den Differenzenquotienten für die Intervalle: a) $I=[0 ; 2]$ b) $I=[-1 ; 3]$ c) $I=[-1 ; 1]$ d) $I=[-2 ;-1]$

Determine the radius of convergence for the series $\sum_{n=1}^{\infty} 2^{n}\left(n+\frac{1}{n}\right) x^{n}$ .

Evaluate the integral $\int \frac{1}{x^{2}} \cos^{2}\left(\frac{1}{x}\right) dx$ using the substitution $u=-\frac{1}{x}$ .

Evaluate the integral $\int \frac{9 r^{2} d r}{\sqrt{1-r^{3}}}$ using the substitution $u=1-r^{3}$ .

Finde die Tangentengleichung der Funktion $f(x)=\cos(0,5 x)$ bei $x_{1}=2 \pi$ und $x_{2}=\frac{\pi}{3}$ .

Bestimme die Tangentengleichung $\mathrm{t}: \mathrm{y}=\square \mathrm{x}+$ für $f(x)=x^{2}-3$ im Punkt $P(1|-2)$ .

Evaluate the integral: $\int x^{1 / 2} \sin \left(x^{3 / 2}+1\right) d x$

Bestimme die Tangentengleichung an $f(x)=x^{2}-3$ im Punkt $P(1|-2)$ . Tangentengleichung: $y=\square x+$

Bestimme den Ausdruck $\frac{d f}{d a}+\frac{d f}{d b}-\frac{d f}{d c}$ für $f(a, b, c)=3 a^{2} b-3 b^{2} c+3 a c^{2}$ .

Bestimme die Tangentengleichungen an $f(x)=x^{2}-3$ bei $P(1|-2)$ und an $f(x)=\frac{1}{x}$ bei $P(2|\frac{1}{2})$ .

Evaluate the integral: $\int 3 y \sqrt{7-3 y^{2}} d y$

Evaluate the integral: $\int x \sqrt{4-x} \, dx$

Find the limit: $\lim _{x \rightarrow \infty} \frac{e^{-x}+1}{e^{-x}-1}$ . What is the value?

Find the limit: $\lim _{x \rightarrow 1^{-}} \frac{x^{2}+2 x-3}{|x-1|}$ or state if it doesn't exist.

Υπολογίστε την παράγωγο της συνάρτησης $f(x)=\frac{1}{e^{x}+1}$ για $x \in \mathbb{R}$ .

Bestimme die Integrale: a) $\int_{2}^{5} x \, dx$ b) $\int_{-1}^{1}(2x+1) \, dx$ mithilfe von Flächen.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sin \left(\frac{1}{x}\right)}{x}$ . What is the result?

Find the limit: $\lim _{x \rightarrow \frac{\pi}{6}} \frac{\cos 3 x}{1-4 \sin ^{2} x}$ .

Find the radius of convergence for the series $\sum_{n=1}^{\infty} 2^{n}\left(n+\frac{1}{n}\right) x^{n}$ .

Is the function $f(x)=\begin{cases} 3 \ln (1+x) & x \geq 0 \\ x^{5}+x & x<0 \end{cases}$ continuous on $\mathbb{R}$ ? (YES / NO) Does it have an inverse? (YES / NO)

Is the function $f(x)=\begin{cases} 3 \ln (1+x) & x \geq 0 \\ x^{5}+x & x<0 \end{cases}$ continuous on $\mathbb{R}$ ? (YES / NO)

Find critical points and determine concavity for the function $y=\frac{5 x+7}{5 x}$ , $x \neq 0$ , using derivatives.