Calculus

Problem 14901

Berechne den Flächeninhalt unter den Graphen von $f(x)$ im Intervall [0; 2] für a) bis d): a) $f(x)=e^{x}$ , b) $f(x)=8-e^{x}$ , c) $f(x)=e^{x}+x+2$ , d) $f(x)=e^{x}-x$ .

See Solution

Problem 14902

Eine Rakete erreicht bei der zweiten Stufe 12500 m. Bestimmen Sie Höhe nach 30 s, Funktionen für Höhenzuwachs und Zuwachs von 10 s bis 30 s.

See Solution

Problem 14903

For the function $f(x)=\frac{-5 x^{2}+2 x+8}{x^{2}}$ , find: (a) domain, (b) first derivative $f^{\prime}(x)$ , (c) increase/decrease regions, (d) local extrema, (e) second derivative $f^{\prime \prime}(x)$ , (f) concavity regions, (g) inflection points.

See Solution

Problem 14904

Berechne die Flächeninhalte unter den Graphen von $f(x)$ im Intervall $[0; 2]$ für die Funktionen a) bis d).

See Solution

Problem 14905

Find the radius and height of a grain silo (cylinder + hemisphere) that minimizes material cost with volume $900 \, m^{3}$ .

See Solution

Problem 14906

Find the slope and concavity of the curve defined by $x=\cos \theta$ and $y=3 \sin \theta$ at $\theta=0$ .

See Solution

Problem 14907

Find the intervals where the function $f(x)=3 x^{2}+6 x$ is decreasing.

See Solution

Problem 14908

Find $\frac{d^{2} y}{d x^{2}}$ for the parametric equations $x=3 t^{2}-1$ and $y=\ln t$ in terms of $t$ .

See Solution

Problem 14909

Berechnen Sie die folgenden Integrale: f) $\int_{0}^{\frac{\pi}{2}} \cos (x) d x$ g) $\int_{0}^{\pi} \sin (x) d x$ h) $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{2} \cos (x) d x$ i) $\int_{-\pi}^{\pi}-\sin (x) d x$ j) $\int_{2}^{4} \frac{2}{3} e^{x} d x$

See Solution

Problem 14910

Find the slope and concavity of $y$ at $\theta=\pi$ given $x=\theta-\cos \theta$ and $y=1-\sin \theta$ .

See Solution

Problem 14911

Find the limit as $x$ approaches 5 for the function $f(x)=\frac{\sqrt{x-3}-\sqrt{2}}{x-5}$ . Show your work.

See Solution

Problem 14912

احسب $\lim_{x \rightarrow 5} \frac{\sqrt{x-3}-\sqrt{2}}{x-5}$ ووضح خطوات الحل.

See Solution

Problem 14913

Find the tangent line equation for $f(x)=x^{2} \ln (x)$ at $x=e$ .

See Solution

Problem 14914

Find the derivative of $y=\frac{1+\cos x}{1+\sin x}$ .

See Solution

Problem 14915

Find the values of $x$ where the function $g(x) = \int_{0}^{x} f(t) dt$ has a point of inflection, given $f$ has horizontal tangents at $x=2$ and $x=5$ .

See Solution

Problem 14916

Find the derivative of $y=\frac{3}{3x+7}$ .

See Solution

Problem 14917

Find the derivative of $y=\sin ^{2} x$ or $y=\sin x \cdot \sin x$ .

See Solution

Problem 14918

Berechnen Sie das Integral von $w(t)=0,001 \cdot t^{2}$ für $0 \leqq t \leqq 100$ und deuten Sie das Ergebnis.

See Solution

Problem 14919

Find the derivative of $y=\frac{3x+7}{x^{3}}$ .

See Solution

Problem 14920

Find the tangent line equation for the curve $y=x^{2} e^{-5 x}$ at $x=1$ .

See Solution

Problem 14921

Find the derivative of $y=\frac{1-x^{2}}{\sin x}$ .

See Solution

Problem 14922

Find the derivative of $y=\csc x$ .

See Solution

Problem 14923

Find the derivative of $y=\cos^{2} x$ .

See Solution

Problem 14924

Find the derivative of $y=\frac{x^{2}}{3x+7}$ .

See Solution

Problem 14925

Calculate the area under the curve from -3 to 2 for the function $f(x) = 8$ . What is $\int_{-3}^{2} 8 d x$ ?

See Solution

Problem 14926

Calculate the area under the curve: $\int_{-5}^{1} 2 x \, dx = \square$

See Solution

Problem 14927

Find the derivative of the function $y=\frac{3x^{2}+7x}{x^{2}}$ .

See Solution

Problem 14928

Find the function $f$ from the limit of Riemann sums and express it as a definite integral: $\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(x_{k}^{*}\right)^{6} \Delta x_{k} ;[2,11]$

See Solution

Problem 14929

Find the derivative of $y=\frac{3x+7}{\sqrt{x}}$ .

See Solution

Problem 14930

Find the function $f$ from the limit of Riemann sums and express it as a definite integral over $[1,2]$ : $\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n} x_{k}^{} \sin x_{k}^{} \Delta x_{k}.$

See Solution

Problem 14931

Find the function $f$ and express the limit of Riemann sums as a definite integral: $\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n} x_{k}^{} \cos x_{k}^{} \Delta x_{k} ;[1,2]$ .

See Solution

Problem 14932

Find the derivative of the function $y=\frac{2x-5}{3x+7}$ .

See Solution

Problem 14933

Find the function $f$ for the limit of Riemann sums: $\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n} x_{k}^{} \sin x_{k}^{} \Delta x_{k}$ on $[1,2]$ .

See Solution

Problem 14934

Graph a function $f$ with these properties: domain all reals, $f(-2)=3$ , $f(2)=3$ , no $x$ intercepts, $y$ intercept 5, $\lim_{x \to \infty} f(x)=0$ , $\lim_{x \to -\infty} f(x)=0$ , $f'(x)>0$ for $x<0$ , $f'(x)<0$ for $x>0$ , $f'(0)=0$ , $f''(x)<0$ for $-2<x<2$ , $f''(x)>0$ for $|x|>2$ , $f''(2)=0$ , $f''(-2)=0$ .

See Solution

Problem 14935

Find the tangent equation to the curve $x=t^2$ , $y=t-1/t$ at the point where $t=2$ .

See Solution

Problem 14936

Find the slope of $y^{2}(4-x)=x^{2}$ at the point (3, -3) by calculating the derivative with respect to x.

See Solution

Problem 14937

Evaluate $\int_{7}^{2} g(x) d x$ given $\int_{2}^{7} g(x) d x=-8$ .

See Solution

Problem 14938

Find dy/dx for the parametric equations $x=t-e^{2 t}$ and $y=e^{2 t}$ .

See Solution

Problem 14939

Find $g^{\prime}\left(\frac{5}{2}\right)$ given $g(x)-9 x \sin(g(x))=6 x^{2}-x-35$ and $g\left(\frac{5}{2}\right)=0$ .

See Solution

Problem 14940

Find $f^{\prime}(4)$ for $f(x)=(g(x))^{2}$ , $f(x)=\sqrt{h(x)}$ , and $f(x)=h(g(x))$ using given values.

See Solution

Problem 14941

Given $\int_{2}^{4} f(x) d x=-7$ , $\int_{2}^{7} f(x) d x=-3$ , and $\int_{2}^{7} g(x) d x=-8$ , find:
1. $\int_{7}^{2} g(x) d x$
2. $\int_{2}^{7} 8 g(x) d x$
3. $\int_{2}^{7}[g(x)-f(x)] d x=\square$

See Solution

Problem 14942

Find the tangent line equation to the hyperbola $3 x^{2}-3 x y-4 y^{2}+2 x=24$ at the point $(3,-3)$ .

See Solution

Problem 14943

Evaluate the integral $\int_{0}^{3 \pi / 2} x \sin x \, dx$ given areas of regions $R_{1}, R_{2}, R_{3}, R_{4}$ as $1, \pi-1, \pi+1, 2\pi-1$ .

See Solution

Problem 14944

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the circle defined by the equation $x^{2}+y^{2}=25$ .

See Solution

Problem 14945

Given $\int_{2}^{4} f(x) d x=-7$ , $\int_{2}^{7} f(x) d x=-3$ , and $\int_{2}^{7} g(x) d x=-8$ , find:
1. $\int_{7}^{2} g(x) d x$ .
2. $\int_{2}^{7} 8 g(x) d x$ .
3. $\int_{2}^{7}[g(x)-f(x)] d x$ .
4. $\int_{2}^{7}[8 g(x)-f(x)] d x=\square$ .

See Solution

Problem 14946

Given $\int_{2}^{4} f(x) d x=-7$ , $\int_{2}^{7} f(x) d x=-3$ , and $\int_{2}^{7} g(x) d x=-8$ , find:
1. $\int_{7}^{2} g(x) d x=8$ (simplify).
2. $\int_{2}^{7} 8 g(x) d x=\square$ (simplify).

See Solution

Problem 14947

Find $\frac{d y}{d x}$ for the relation $x + 3xy + y^2 = 2x$ .

See Solution

Problem 14948

Find the slope of the tangent line to the ellipse $x^{2}+4 y^{2}=16$ at the point $(4,0)$ .

See Solution

Problem 14949

Bestimme die Geschwindigkeiten des Fahrzeugs bei $t = 2$ , $5$ , $8$ und $10$ s für $f(t) = t^{2}$ .

See Solution

Problem 14950

Bestimmen Sie die Ableitung der Funktionen: a) $f(x)=2 x^{2}$ , b) $f(x)=x$ , c) $f(x)=2 x$ , d) $f(x)=5$ , e) $f(x)=-x^{2}$ , f) $f(x)=2 x+2$ , g) $f(x)=a x+b$ , h) $f(x)=a x^{2}$ , i) $f(x)=a x^{2}+b x+c$ .

See Solution

Problem 14951

Evaluate the integral $\int_{1}^{8} f(x) d x$ where $f(x)=\begin{cases} 2x & \text{if } 1 \leq x \leq 5 \\ 12-2x & \text{if } 5<x \leq 8 \end{cases}$ . Find $\int_{1}^{8} f(x) d x=\square$ .

See Solution

Problem 14952

Evaluate the integral $\int_{3}^{7}[3 f(x)-2 g(x)] d x$ given $\int_{3}^{7} f(x) d x=6$ and $\int_{3}^{7} g(x) d x=5$ .

See Solution

Problem 14953

Bestimme die Ableitungen: a) $\frac{d E}{d t}$ , b) $\frac{\mathrm{dE}}{\mathrm{da}}$ , c) $\frac{d E}{d b}$ für $E=a t+\frac{b}{2} t^{2}$ .

See Solution

Problem 14954

Find the work done by the tension in a string lifting a mass $M$ at an acceleration of $\frac{g}{4}$ over a distance $\ell$ .

See Solution

Problem 14955

Estimate the area under $f(x)=4 x^{4}$ from $x=0$ to $x=4$ using 4 equal subintervals and right endpoints.

See Solution

Problem 14956

Find the linearization $L(x)$ of the function $f(x)=\ln(x^{2})$ at the point $x=e$ .

See Solution

Problem 14957

Determine if the rectangular approximation is an underestimate or overestimate. Estimate area under $f(x)=4x^4$ from $x=0$ to $x=4$ using 4 left endpoints.

See Solution

Problem 14958

Estimate the area under $f(x)=4 x^{4}$ from $x=0$ to $x=4$ using 4 equal subintervals with left endpoints.

See Solution

Problem 14959

Hochbehälter: Gegeben ist $h(t)=\frac{1}{8} t^{2}-2 t+8$ . a) Graph zeichnen. b) Wann leer? c) Halb und $1/4$ voll? d) Durchschnittliche Sinkgeschwindigkeit?

See Solution

Problem 14960

Bestimme $\frac{d u}{d v}$ aus der Gleichung $2 \cdot u = v \cdot w^{2} - v^{3}$ (mit konstantem $w$ ).

See Solution

Problem 14961

Find the limit of $p(t)=\frac{4300 t}{t+1}$ as $t$ approaches infinity, where $p(t)$ is the rabbit population.

See Solution

Problem 14962

Evaluate the integral $\int_{-\infty}^{\infty} \frac{d x}{x^{2}+36}$ or state if it diverges.

See Solution

Problem 14963

Gegeben ist die Formel $2 \cdot u=v \cdot w^{2}-v^{3}$ . Finde a) $\frac{d u}{d v}$ (für konstantes $w$ ) und b) $\frac{d u}{d w}$ (für konstantes $v$ ).

See Solution

Problem 14964

Determine the end behavior of the function $f(x)=1536+3724 x^{3}+4 x^{6}+5248 x+6616 x^{2}+916 x^{4}+100 x^{5}$ .

See Solution

Problem 14965

Determine the truth of these statements: I. If $f^{\prime}(c)=0$ , then $f$ has a local max/min at $x=c$ . II. If $f(x)$ is increasing everywhere, it has no inflection points. III. A function $f$ can have a local min at $x=3$ with $f^{\prime}(3) \neq 0$ .

See Solution

Problem 14966

Find the limit: $\lim _{x \rightarrow-\infty} 4 x^{3}-2 x+10$ . Options: a. 0 b. $\infty$ c. $-\infty$ d. undefined

See Solution

Problem 14967

Approximate the integral $\int_{0}^{5}(-x^{2}+17x-5)dx$ using a Riemann Sum with 5 subintervals and right endpoints.

See Solution

Problem 14968

Given functions $f(x)$ , $g(x)$ , and $h(x)$ , determine their behavior at critical points $x=2$ and $x=1$ and $x=3$ .

See Solution

Problem 14969

Evaluate the integral from 8 to 27 of $\sqrt[3]{x}$ with respect to $x$ .

See Solution

Problem 14970

Evaluate the integral from -6 to 0: $\int_{-6}^{0}\left(2 x-e^{x}\right) d x$

See Solution

Problem 14971

Evaluate the integral from 0 to 1: $\int_{0}^{1} x(4 \sqrt[3]{x}+7 \sqrt[4]{x}) d x$

See Solution

Problem 14972

Calculate the sum of the series $\sum_{i=0}^{\infty}\left(\frac{1}{5}\right)^{i}$ .

See Solution

Problem 14973

Find the limit: $\lim _{x \rightarrow \infty} \frac{-2 x^{3}-4 x^{2}+10}{4 x^{3}+2 x-1}$ .

See Solution

Problem 14974

Calculate the sum of the series $\sum_{i=0}^{\infty}\left(\frac{7}{10}\right)^{i}$ .

See Solution

Problem 14975

Find the limit as $x$ approaches 2 for $\frac{x^{2}-4}{x^{2}-3x+2}$ . Options: $-1/2$ , 4, $-4$ , $-1/5$ .

See Solution

Problem 14976

Approximate the area under $f(x)=6 e^{-x^{2}}$ on $[0,6]$ using 3 equal subintervals and right endpoints. Round to 3 decimals.

See Solution

Problem 14977

Calculate the sum of the series $\sum_{i=0}^{\infty}\left(\frac{3}{5}\right)^{i}$ .

See Solution

Problem 14978

Approximate the area under $f(x)=6 e^{-x^{2}}$ on $[0,6]$ using 3 rectangles with right endpoints. Find $\Delta x$ and endpoints.

See Solution

Problem 14979

Problem 1: Start with \$1000 at 2% interest compounded yearly.
(a) Find the formula for $A(t)$ . (b) How long to double your money? (c) Compute the change rate after 10 years.

See Solution

Problem 14980

Determine if the series converges: $\sum_{k=1}^{\infty} \frac{\sqrt{9 k^{2}+4}}{k}$

See Solution

Problem 14981

Evaluate the integral from 0 to 3: $\int_{0}^{3}\left(6 e^{x}+5 \cos x\right) d x$ .

See Solution

Problem 14982

Find $\frac{d y}{d x}$ at the point $(1,0)$ for $e^{2 y}-e^{(y^{2}-y)}=x^{4}-x^{2}$ . Options: (A) 0 (B) $\frac{1}{2}$ (C) $\frac{2}{3}$ (D) 2

See Solution

Problem 14983

Find the first and second derivatives of these functions: (a) $f(x)=x^{3}+14 x^{2}+6 x+7$ , (b) $f(x)=\cos(x^{2})$ , (c) $f(x)=e^{x}+e^{-x}$ , (d) $f(x)=\ln(5x)$ , (e) $f(x)=\sin(x)\cos(x)$ .

See Solution

Problem 14984

Find the first and second derivatives for these functions: (a) $f(x)=x^{3}+14 x^{2}+6 x+7$ , (b) $f(x)=\cos(x^{2})$ , (c) $f(x)=e^{x}+e^{-x}$ , (d) $f(x)=\ln(5x)$ , (e) $f(x)=\sin(x)\cos(x)$ .

See Solution

Problem 14985

Evaluate the integral: $\int_{0}^{\pi / 4} \frac{2+5 \cos ^{2} \theta}{\cos ^{2} \theta} d \theta$ .

See Solution

Problem 14986

Aufgabe 7: a) Definiere die Ableitung einer Funktion $f$ an der Stelle $x_{0}$ . b) Unter welchen Bedingungen ist eine Funktion $g$ differenzierbar? c) Bestimme die Ableitung von $f(x)=\frac{1}{x}$ an der Stelle $x_{0}=2$ schrittweise.

See Solution

Problem 14987

Calculate $\int_{0}^{7} f(x) d x$ where $f(x) = 5$ for $x<5$ and $f(x) = x$ for $x \geq 5$ .

See Solution

Problem 14988

Calculate the integral from 8 to 27 of the cube root of x: $\int_{8}^{27} \sqrt[3]{x} dx$ .

See Solution

Problem 14989

Calculate the threshold value of infection from the function $f(t)=-t(t-21)(t+1)$ at $t=14$ days. Round to the nearest whole number.

See Solution

Problem 14990

Find the limit: As $x \rightarrow -\infty$ , what is $\arctan x$ ?

See Solution

Problem 14991

Evaluate the integral from 8 to 27 of $\sqrt[3]{x}$ with respect to $x$ .

See Solution

Problem 14992

Evaluate the integral from 4 to 5 of $(2+2y)^{2} \, dy$ .

See Solution

Problem 14993

Determine if the sequences converge or diverge. If they converge, find the limit:
1. $a_{n}=\frac{5}{n+2}$
2. $a_{n}=5 \sqrt{n+2}$
3. $a_{n}=\frac{4 n^{2}-3 n}{2 n^{2}+1}$
4. $a_{n}=\frac{4 n^{2}-3 n}{2 n+1}$
5. $a_{n}=\frac{n^{4}}{n^{3}-2 n}$
6. $a_{n}=2+(0.86)^{n}$

See Solution

Problem 14994

Evaluate the integral: $\int_{0}^{\pi / 4} \frac{2+5 \cos ^{2} \theta}{\cos ^{2} \theta} d \theta$ .

See Solution

Problem 14995

A culture of Rhodobacter sphaeroides starts with 24 bacteria and grows at $3.4657 e^{0.1386 t}$ . Find the population after 6 hours.

See Solution

Problem 14996

A bacteria colony grows at $4.0591 e^{1.1 t}$ per hour. If it starts with 40, find the population after 4 hours.

See Solution

Problem 14997

Gegeben ist die Funktion $f(x)=0,5 x^{2}$ . Bestimmen Sie den Wert von $f^{\prime}(2)$ an der Tangente im Punkt $P(2 \mid 2)$ .

See Solution

Problem 14998

Gegeben ist die Funktion $f(x)=(x+3) \cdot e^{-x}$ . Finde Nullstellen, Ableitungen, Hoch-/Tiefpunkte und skizziere den Graphen. Berechne die Tangente und Normale bei $A(0 \mid 3)$ .

See Solution

Problem 14999

Find the mass of an $881 \mathrm{~kg}$ radioactive sample after 6 days, decaying at $14\%$ per day. Round to the nearest tenth.

See Solution

Problem 15000

Gegeben ist die Funktion $f(x)=0,5 x^{2}$ . Bestimmen Sie den Wert von $f^{\prime}(2)$ anhand der Tangente im Punkt $P(2 \mid 2)$ .

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 14901

Berechne den Flächeninhalt unter den Graphen von f(x)f(x)f(x) im Intervall [0; 2] für a) bis d): a) f(x)=exf(x)=e^{x}f(x)=ex, b) f(x)=8−exf(x)=8-e^{x}f(x)=8−ex, c) f(x)=ex+x+2f(x)=e^{x}+x+2f(x)=ex+x+2, d) f(x)=ex−xf(x)=e^{x}-xf(x)=ex−x.

Problem 14902

Eine Rakete erreicht bei der zweiten Stufe 12500 m. Bestimmen Sie Höhe nach 30 s, Funktionen für Höhenzuwachs und Zuwachs von 10 s bis 30 s.

Problem 14903

Problem 14904

Berechne die Flächeninhalte unter den Graphen von f(x)f(x)f(x) im Intervall [0;2][0; 2][0;2] für die Funktionen a) bis d).

Problem 14905

Find the radius and height of a grain silo (cylinder + hemisphere) that minimizes material cost with volume 900 m3900 \, m^{3}900m3.

Problem 14906

Find the slope and concavity of the curve defined by x=cos⁡θx=\cos \thetax=cosθ and y=3sin⁡θy=3 \sin \thetay=3sinθ at θ=0\theta=0θ=0.

Problem 14907

Find the intervals where the function f(x)=3x2+6xf(x)=3 x^{2}+6 xf(x)=3x2+6x is decreasing.

Problem 14908

Find d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ for the parametric equations x=3t2−1x=3 t^{2}-1x=3t2−1 and y=ln⁡ty=\ln ty=lnt in terms of ttt.

Problem 14909

Problem 14910

Find the slope and concavity of yyy at θ=π\theta=\piθ=π given x=θ−cos⁡θx=\theta-\cos \thetax=θ−cosθ and y=1−sin⁡θy=1-\sin \thetay=1−sinθ.

Problem 14911

Find the limit as xxx approaches 5 for the function f(x)=x−3−2x−5f(x)=\frac{\sqrt{x-3}-\sqrt{2}}{x-5}f(x)=x−5x−3​−2​​. Show your work.

Problem 14912

احسب lim⁡x→5x−3−2x−5\lim_{x \rightarrow 5} \frac{\sqrt{x-3}-\sqrt{2}}{x-5}limx→5​x−5x−3​−2​​ ووضح خطوات الحل.

Problem 14913

Find the tangent line equation for f(x)=x2ln⁡(x)f(x)=x^{2} \ln (x)f(x)=x2ln(x) at x=ex=ex=e.

Problem 14914

Find the derivative of y=1+cos⁡x1+sin⁡xy=\frac{1+\cos x}{1+\sin x}y=1+sinx1+cosx​.

Problem 14915

Find the values of xxx where the function g(x)=∫0xf(t)dtg(x) = \int_{0}^{x} f(t) dtg(x)=∫0x​f(t)dt has a point of inflection, given fff has horizontal tangents at x=2x=2x=2 and x=5x=5x=5.

Problem 14916

Find the derivative of y=33x+7y=\frac{3}{3x+7}y=3x+73​.

Problem 14917

Find the derivative of y=sin⁡2xy=\sin ^{2} xy=sin2x or y=sin⁡x⋅sin⁡xy=\sin x \cdot \sin xy=sinx⋅sinx.

Problem 14918

Berechnen Sie das Integral von w(t)=0,001⋅t2w(t)=0,001 \cdot t^{2}w(t)=0,001⋅t2 für 0≦t≦1000 \leqq t \leqq 1000≦t≦100 und deuten Sie das Ergebnis.

Problem 14919

Find the derivative of y=3x+7x3y=\frac{3x+7}{x^{3}}y=x33x+7​.

Problem 14920

Find the tangent line equation for the curve y=x2e−5xy=x^{2} e^{-5 x}y=x2e−5x at x=1x=1x=1.

Problem 14921

Find the derivative of y=1−x2sin⁡xy=\frac{1-x^{2}}{\sin x}y=sinx1−x2​.

Problem 14922

Find the derivative of y=csc⁡xy=\csc xy=cscx.

Problem 14923

Find the derivative of y=cos⁡2xy=\cos^{2} xy=cos2x.

Problem 14924

Find the derivative of y=x23x+7y=\frac{x^{2}}{3x+7}y=3x+7x2​.

Problem 14925

Calculate the area under the curve from -3 to 2 for the function f(x)=8f(x) = 8f(x)=8. What is ∫−328dx\int_{-3}^{2} 8 d x∫−32​8dx?

Problem 14926

Calculate the area under the curve: ∫−512x dx=□\int_{-5}^{1} 2 x \, dx = \square∫−51​2xdx=□

Problem 14927

Find the derivative of the function y=3x2+7xx2y=\frac{3x^{2}+7x}{x^{2}}y=x23x2+7x​.

Problem 14928

Find the function fff from the limit of Riemann sums and express it as a definite integral: lim⁡Δ→0∑k=1n(xk∗)6Δxk;[2,11]\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(x_{k}^{*}\right)^{6} \Delta x_{k} ;[2,11]Δ→0lim​k=1∑n​(xk∗​)6Δxk​;[2,11]

Problem 14929

Find the derivative of y=3x+7xy=\frac{3x+7}{\sqrt{x}}y=x​3x+7​.

Problem 14930

Find the function fff from the limit of Riemann sums and express it as a definite integral over [1,2][1,2][1,2]: lim⁡Δ→0∑k=1nxk∗sin⁡xk∗Δxk.\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n} x_{k}^{*} \sin x_{k}^{*} \Delta x_{k}.Δ→0lim​k=1∑n​xk∗​sinxk∗​Δxk​.

Problem 14931

Find the function fff and express the limit of Riemann sums as a definite integral: lim⁡Δ→0∑k=1nxk∗cos⁡xk∗Δxk;[1,2]\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n} x_{k}^{*} \cos x_{k}^{*} \Delta x_{k} ;[1,2]Δ→0lim​k=1∑n​xk∗​cosxk∗​Δxk​;[1,2].

Problem 14932

Find the derivative of the function y=2x−53x+7y=\frac{2x-5}{3x+7}y=3x+72x−5​.

Problem 14933

Find the function fff for the limit of Riemann sums: lim⁡Δ→0∑k=1nxk∗sin⁡xk∗Δxk\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n} x_{k}^{*} \sin x_{k}^{*} \Delta x_{k}Δ→0lim​k=1∑n​xk∗​sinxk∗​Δxk​ on [1,2][1,2][1,2].

Problem 14934

Problem 14935

Find the tangent equation to the curve x=t2x=t^2x=t2, y=t−1/ty=t-1/ty=t−1/t at the point where t=2t=2t=2.

Problem 14936

Find the slope of y2(4−x)=x2y^{2}(4-x)=x^{2}y2(4−x)=x2 at the point (3, -3) by calculating the derivative with respect to x.

Problem 14937

Evaluate ∫72g(x)dx\int_{7}^{2} g(x) d x∫72​g(x)dx given ∫27g(x)dx=−8\int_{2}^{7} g(x) d x=-8∫27​g(x)dx=−8.

Problem 14938

Find dy/dx for the parametric equations x=t−e2tx=t-e^{2 t}x=t−e2t and y=e2ty=e^{2 t}y=e2t.

Problem 14939

Find g′(52)g^{\prime}\left(\frac{5}{2}\right)g′(25​) given g(x)−9xsin⁡(g(x))=6x2−x−35g(x)-9 x \sin(g(x))=6 x^{2}-x-35g(x)−9xsin(g(x))=6x2−x−35 and g(52)=0g\left(\frac{5}{2}\right)=0g(25​)=0.

Problem 14940

Find f′(4)f^{\prime}(4)f′(4) for f(x)=(g(x))2f(x)=(g(x))^{2}f(x)=(g(x))2, f(x)=h(x)f(x)=\sqrt{h(x)}f(x)=h(x)​, and f(x)=h(g(x))f(x)=h(g(x))f(x)=h(g(x)) using given values.

Problem 14941

Problem 14942

Berechne den Flächeninhalt unter den Graphen von $f(x)$ im Intervall [0; 2] für a) bis d): a) $f(x)=e^{x}$ , b) $f(x)=8-e^{x}$ , c) $f(x)=e^{x}+x+2$ , d) $f(x)=e^{x}-x$ .

Berechne die Flächeninhalte unter den Graphen von $f(x)$ im Intervall $[0; 2]$ für die Funktionen a) bis d).

Find the radius and height of a grain silo (cylinder + hemisphere) that minimizes material cost with volume $900 \, m^{3}$ .

Find the slope and concavity of the curve defined by $x=\cos \theta$ and $y=3 \sin \theta$ at $\theta=0$ .

Find the intervals where the function $f(x)=3 x^{2}+6 x$ is decreasing.

Find $\frac{d^{2} y}{d x^{2}}$ for the parametric equations $x=3 t^{2}-1$ and $y=\ln t$ in terms of $t$ .

Find the slope and concavity of $y$ at $\theta=\pi$ given $x=\theta-\cos \theta$ and $y=1-\sin \theta$ .

Find the limit as $x$ approaches 5 for the function $f(x)=\frac{\sqrt{x-3}-\sqrt{2}}{x-5}$ . Show your work.

احسب $\lim_{x \rightarrow 5} \frac{\sqrt{x-3}-\sqrt{2}}{x-5}$ ووضح خطوات الحل.

Find the tangent line equation for $f(x)=x^{2} \ln (x)$ at $x=e$ .

Find the derivative of $y=\frac{1+\cos x}{1+\sin x}$ .

Find the values of $x$ where the function $g(x) = \int_{0}^{x} f(t) dt$ has a point of inflection, given $f$ has horizontal tangents at $x=2$ and $x=5$ .

Find the derivative of $y=\frac{3}{3x+7}$ .

Find the derivative of $y=\sin ^{2} x$ or $y=\sin x \cdot \sin x$ .

Berechnen Sie das Integral von $w(t)=0,001 \cdot t^{2}$ für $0 \leqq t \leqq 100$ und deuten Sie das Ergebnis.

Find the derivative of $y=\frac{3x+7}{x^{3}}$ .

Find the tangent line equation for the curve $y=x^{2} e^{-5 x}$ at $x=1$ .

Find the derivative of $y=\frac{1-x^{2}}{\sin x}$ .

Find the derivative of $y=\csc x$ .

Find the derivative of $y=\cos^{2} x$ .

Find the derivative of $y=\frac{x^{2}}{3x+7}$ .

Calculate the area under the curve from -3 to 2 for the function $f(x) = 8$ . What is $\int_{-3}^{2} 8 d x$ ?

Calculate the area under the curve: $\int_{-5}^{1} 2 x \, dx = \square$

Find the derivative of the function $y=\frac{3x^{2}+7x}{x^{2}}$ .

Find the function $f$ from the limit of Riemann sums and express it as a definite integral: $\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(x_{k}^{*}\right)^{6} \Delta x_{k} ;[2,11]$

Find the derivative of $y=\frac{3x+7}{\sqrt{x}}$ .

Find the function $f$ from the limit of Riemann sums and express it as a definite integral over $[1,2]$ : $\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n} x_{k}^{} \sin x_{k}^{} \Delta x_{k}.$

Find the function $f$ and express the limit of Riemann sums as a definite integral: $\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n} x_{k}^{} \cos x_{k}^{} \Delta x_{k} ;[1,2]$ .

Find the derivative of the function $y=\frac{2x-5}{3x+7}$ .

Find the function $f$ for the limit of Riemann sums: $\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n} x_{k}^{} \sin x_{k}^{} \Delta x_{k}$ on $[1,2]$ .

Find the tangent equation to the curve $x=t^2$ , $y=t-1/t$ at the point where $t=2$ .

Find the slope of $y^{2}(4-x)=x^{2}$ at the point (3, -3) by calculating the derivative with respect to x.

Evaluate $\int_{7}^{2} g(x) d x$ given $\int_{2}^{7} g(x) d x=-8$ .

Find dy/dx for the parametric equations $x=t-e^{2 t}$ and $y=e^{2 t}$ .

Find $g^{\prime}\left(\frac{5}{2}\right)$ given $g(x)-9 x \sin(g(x))=6 x^{2}-x-35$ and $g\left(\frac{5}{2}\right)=0$ .

Find $f^{\prime}(4)$ for $f(x)=(g(x))^{2}$ , $f(x)=\sqrt{h(x)}$ , and $f(x)=h(g(x))$ using given values.

Find the tangent line equation to the hyperbola $3 x^{2}-3 x y-4 y^{2}+2 x=24$ at the point $(3,-3)$ .

Evaluate the integral $\int_{0}^{3 \pi / 2} x \sin x \, dx$ given areas of regions $R_{1}, R_{2}, R_{3}, R_{4}$ as $1, \pi-1, \pi+1, 2\pi-1$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the circle defined by the equation $x^{2}+y^{2}=25$ .

Find $\frac{d y}{d x}$ for the relation $x + 3xy + y^2 = 2x$ .

Find the slope of the tangent line to the ellipse $x^{2}+4 y^{2}=16$ at the point $(4,0)$ .

Bestimme die Geschwindigkeiten des Fahrzeugs bei $t = 2$ , $5$ , $8$ und $10$ s für $f(t) = t^{2}$ .

Evaluate the integral $\int_{3}^{7}[3 f(x)-2 g(x)] d x$ given $\int_{3}^{7} f(x) d x=6$ and $\int_{3}^{7} g(x) d x=5$ .

Bestimme die Ableitungen: a) $\frac{d E}{d t}$ , b) $\frac{\mathrm{dE}}{\mathrm{da}}$ , c) $\frac{d E}{d b}$ für $E=a t+\frac{b}{2} t^{2}$ .

Find the work done by the tension in a string lifting a mass $M$ at an acceleration of $\frac{g}{4}$ over a distance $\ell$ .

Estimate the area under $f(x)=4 x^{4}$ from $x=0$ to $x=4$ using 4 equal subintervals and right endpoints.

Find the linearization $L(x)$ of the function $f(x)=\ln(x^{2})$ at the point $x=e$ .

Determine if the rectangular approximation is an underestimate or overestimate. Estimate area under $f(x)=4x^4$ from $x=0$ to $x=4$ using 4 left endpoints.

Estimate the area under $f(x)=4 x^{4}$ from $x=0$ to $x=4$ using 4 equal subintervals with left endpoints.

Hochbehälter: Gegeben ist $h(t)=\frac{1}{8} t^{2}-2 t+8$ . a) Graph zeichnen. b) Wann leer? c) Halb und $1/4$ voll? d) Durchschnittliche Sinkgeschwindigkeit?

Bestimme $\frac{d u}{d v}$ aus der Gleichung $2 \cdot u = v \cdot w^{2} - v^{3}$ (mit konstantem $w$ ).

Find the limit of $p(t)=\frac{4300 t}{t+1}$ as $t$ approaches infinity, where $p(t)$ is the rabbit population.

Evaluate the integral $\int_{-\infty}^{\infty} \frac{d x}{x^{2}+36}$ or state if it diverges.

Gegeben ist die Formel $2 \cdot u=v \cdot w^{2}-v^{3}$ . Finde a) $\frac{d u}{d v}$ (für konstantes $w$ ) und b) $\frac{d u}{d w}$ (für konstantes $v$ ).

Determine the end behavior of the function $f(x)=1536+3724 x^{3}+4 x^{6}+5248 x+6616 x^{2}+916 x^{4}+100 x^{5}$ .

Find the limit: $\lim _{x \rightarrow-\infty} 4 x^{3}-2 x+10$ . Options: a. 0 b. $\infty$ c. $-\infty$ d. undefined

Approximate the integral $\int_{0}^{5}(-x^{2}+17x-5)dx$ using a Riemann Sum with 5 subintervals and right endpoints.

Given functions $f(x)$ , $g(x)$ , and $h(x)$ , determine their behavior at critical points $x=2$ and $x=1$ and $x=3$ .

Evaluate the integral from 8 to 27 of $\sqrt[3]{x}$ with respect to $x$ .

Evaluate the integral from -6 to 0: $\int_{-6}^{0}\left(2 x-e^{x}\right) d x$

Evaluate the integral from 0 to 1: $\int_{0}^{1} x(4 \sqrt[3]{x}+7 \sqrt[4]{x}) d x$

Calculate the sum of the series $\sum_{i=0}^{\infty}\left(\frac{1}{5}\right)^{i}$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{-2 x^{3}-4 x^{2}+10}{4 x^{3}+2 x-1}$ .

Calculate the sum of the series $\sum_{i=0}^{\infty}\left(\frac{7}{10}\right)^{i}$ .