Calculus

Problem 8301

Find the derivative $\frac{d y}{d x}$ from the equation $y^{\prime} \cos(y+x+x^{2}) - y \sin(y+x+x^{2})(y^{\prime}+1+2x) = 3x^{2}$ at point $P=(0, \frac{\pi}{2})$ .

See Solution

Problem 8302

Determine where the function $k(x)=5 x^{4}+20 x^{3}$ is increasing or decreasing. Provide your answer in interval notation.

See Solution

Problem 8303

Calculate the partial derivative of $2 x^{2}+3 \sqrt{y}=-4 x+y$ with respect to $x$ at the point $(-2,9)$ .

See Solution

Problem 8304

Find the implicit derivative of $2x^{2}+3\sqrt{y}=-4x+y$ .

See Solution

Problem 8305

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for $4 x^{2}-5 y^{2}=11$ using implicit differentiation.

See Solution

Problem 8306

Find the second derivative of $y$ with respect to $x$ for $4 x^{2}-5 y^{2}=11$ using implicit differentiation.

See Solution

Problem 8307

Find horizontal tangent lines for the curve $y - 3y^{2} = 2x - x^{2}$ . If none, explain why.

See Solution

Problem 8308

Find the integral $\int x^{x}(1+\ln x) \, dx$ given that $f'(x)=x^{x}(1+\ln x)$ . Explain your reasoning.

See Solution

Problem 8309

Find $\lim _{x \rightarrow 4} f(x)$ for $f(x)=8x+3$ . Prove $|f(x)-35|=8|x-4|$ and show $\delta=\frac{\epsilon}{8}$ .

See Solution

Problem 8310

A ball's position is given by $s(t)=-16 t^{2}+128 t+5$ . Find: a. velocity at $t=1$ , b. acceleration at $t=5$ , c. time for velocity -64 ft/s, d. time at height 117 ft.

See Solution

Problem 8311

Find $\delta$ for $\lim _{x \rightarrow 2} f(x)=7$ with $f(x)=4x-1$ and $|f(x)-7|<0.01$ .

See Solution

Problem 8312

Bestimme die Ableitung von $2x \cdot (4x - 1)$ .

See Solution

Problem 8313

A stone is thrown up at $128 \mathrm{ft} / \mathrm{sec}$ . Find: a. max height, b. time to hit ground, c. speed when it hits.

See Solution

Problem 8314

A stone is thrown up at $128 \mathrm{ft/sec}$ . Find: a. max height, b. time to hit ground, c. speed on impact using integrals.

See Solution

Problem 8315

Bestimmen Sie die Koordinaten des Punktes $P$ , wo der Graph $G_{f}$ von $f(x)=-\frac{1}{4} x^{2}+x$ die Steigung -2 hat.

See Solution

Problem 8316

Bestimmen Sie die Ableitung von $f_{a}(x)=\frac{1}{a} x^{a}$ .

See Solution

Problem 8317

Evaluate the integral $\int_{12}^{31} \frac{5 x^{2}}{\left(x^{3}+4\right)^{\frac{1}{2}}} d x$ using $u=x^{3}+4$ . Round to three decimal places.

See Solution

Problem 8318

Given $y=e^{3 x} \operatorname{Sin} 5 x$ , find (a) (i) $\frac{d y}{d x}$ , (ii) $\frac{d^{2} y}{d x^{2}}$ . (b) Show that $\frac{d^{2} y}{d x^{2}}+3 \frac{d y}{d x}-20 y=9 e^{3 x}[5 \operatorname{Cos} 5 x-3 \operatorname{Sin} 5 x]$ .

See Solution

Problem 8319

Untersuchen Sie die Intervalle, in denen $f(x)=\frac{1}{8} x^{3}-\frac{3}{4} x^{2}-2,5$ linksgekrümmt ist. Bestimmen Sie die Extrempunkte von $f(x)=-\frac{1}{6} x^{3}+2 x$ .

See Solution

Problem 8320

Find the horizontal asymptotes of $y=\frac{2x}{\sqrt{x^2-1}}$ . Options: (A) $y=0$ (B) $y=1$ (C) $y=2$ (D) $y=-2$ and $y=2$ (E) $y=-1$ and $y=1$ .

See Solution

Problem 8321

Cell growth rate $x$ at time $t$ increases proportionally to $(20-x)$ .
(a) Write the differential equation. (b) If initially there are 10 cells and after 1.5 hours there are 18, find the cell count after 2 hours (2 sig. figs.).

See Solution

Problem 8322

Berechnen Sie den Flächeninhalt eines gleichschenkligen Dreiecks und bestimmen Sie die Intervalle, in denen $f(x)=\frac{2}{3} x^{3}+x^{2}-12 x+1$ monoton wachsend ist. Untersuchen Sie auch die Krümmung von $f(x)=\frac{1}{8} x^{3}-\frac{3}{4} x^{2}-2,5$ .

See Solution

Problem 8323

A car's position is given by $s(t)=(4 t^{2}-3)e^{-0.5 t}$ . How far does it travel from $t=0$ to when it stops?

See Solution

Problem 8324

The function $g$ is continuous on $[2,6]$ with values: $g(2)=7$ , $g(3)=4$ , $g(4)=1$ , $g(5)=4$ , $g(6)=7$ . True statement?

See Solution

Problem 8325

Find the limit: $\lim_{x \rightarrow \infty} \frac{2x + 1}{x}$ .

See Solution

Problem 8326

Gegeben ist die Funktion $f(x)=x^{2}$ und der Punkt $T(1 /-3)$ . Bestimme die Tangentengleichung in $P(u / f(u))$ .

See Solution

Problem 8327

Find the maximum slope of the tangent line to the graph of $f(x)=3x^2-2x^3$ .

See Solution

Problem 8328

Gegeben ist die Funktion $f(x)=(2 x+10) \cdot e^{5 x+3}$ . Finde die Nullstelle und den $y$ -Achsen-Schnittpunkt. Bestimme die Steigung bei $x=-2$ .

See Solution

Problem 8329

Berechne die Ableitung von $g(x)=\frac{x^{2}+2x}{3x-4}$ und finde die Tangenten bei $P(3/g(3))$ . Untersuche auch Asymptoten.

See Solution

Problem 8330

Bestimmen Sie die Ableitungsfunktion für die folgenden Funktionen: a) $f(x)=0,5 x$ , b) $f(x)=2 x+3$ , c) $f(x)=x^{2}+5$ , d) $f(x)=3 x^{2}$ , e) $f(x)=-x^{2}+7 x$ , f) $f(x)=3 x^{2}-2 x+4$ .

See Solution

Problem 8331

Find the curve where the slope of the tangent line at $(x, y)$ is $3 \sqrt{x}$ and it passes through $(9, 4)$ .

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Problem 8332

Find the hydrostatic force on one side of an isosceles triangle (base 12 ft, sides 10 ft) submerged 2 ft below water. Set up Riemann Sum and integral without evaluating.

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Problem 8333

Untersuchen Sie, ob die Funktion $g(x)=f(x) \cdot \cos (x)$ im Punkt $P(0 \mid 1)$ auch eine waagerechte Tangente hat.

See Solution

Problem 8334

Leiten Sie die Funktion $g(x)=c \cdot f(x)$ mit der Produktregel ab. Welche Regel ergibt sich?

See Solution

Problem 8335

Bestimmen Sie die Summe des Funktionsterms $f(x)$ und leiten Sie $f$ einmal ab für: a) $f(x)=\frac{x^{4}-x^{3}}{x^{2}}$ , b) $f(x)=\frac{x^{12}+x^{8}+1}{x^{10}}$ , c) $f(x)=\frac{x^{4}+2 x^{2}}{x^{5}}$ .

See Solution

Problem 8336

Bestimme die Ableitung der Funktion $f(x)=(5x^4-2x)^{-3}$ mit der Kettenregel.

See Solution

Problem 8337

Find the limit of $f(x)=\frac{\sin(x)}{x}$ as $x$ approaches $0$ from the left, right, and both sides.

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Problem 8338

Calculate the average rate of change for $f(x)=x^{2}+8x$ from $x=2$ to $x=3$ . Simplify your answer.

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Problem 8339

Calculate the average rate of change of $f(x)=\sqrt{x}$ from $x_{1}=16$ to $x_{2}=81$ . Simplify your answer.

See Solution

Problem 8340

Find the average rate of change of $f(x)=\sqrt{x}$ from $x_{1}=16$ to $x_{2}=81$ . Simplify your answer.

See Solution

Problem 8341

Analyze the series $S = \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \ldots + \frac{1}{\sqrt{n}}$ for convergence or divergence.

See Solution

Problem 8342

Find the max and saddle points of $f(x, y)=-x^{3}+8xy-4y^{2}+3$ .

See Solution

Problem 8343

Bestimme die Fußpunkte des Hügels für die Funktion $f(x)=-\frac{1}{2} x^{2}+4 x-6$ und den Steigungswinkel am westlichen Fußpunkt.

See Solution

Problem 8344

Differentiate the function $f(x) = x^{3} - 4x^{2} - 3x$ .

See Solution

Problem 8345

Eine Kleinstadt hat 2006 Neubaugebiete.
a) Bestimme, wann die Einwohnerzahl mit $f(x)=1000 \cdot x^{2} \cdot e^{-x}$ am stärksten zunimmt. b) Berechne die Veränderung der Einwohnerzahl von 2006 bis 2014. c) Finde den Durchschnitt der jährlichen Zunahme von 2006 bis 2014.

See Solution

Problem 8346

Find the value of $\int_{0}^{b} x f\left(x^{2}\right) d x$ given that $\int_{0}^{b^{2}} f(x) d x=214.2$ . Round to two decimal places.

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Problem 8347

If $h(a)=h(b)$ , find the value of $\int_{a}^{b} g(h(x)) h^{\prime}(x) d x$ .

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Problem 8348

Find $\int_{0}^{a} f(3 x) d x$ given that $\int_{0}^{3 a} f(x) d x=13.8$ . Round to two decimal places.

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Problem 8349

A balloon rises at 3 m/s from a point 30 m away. Find the angle of elevation's rate of change when the balloon is 30 m high.

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Problem 8350

Evaluate the integral $\int_{0}^{\infty} e^{\frac{-x^{2}}{3.5}} d x$ using $\int_{0}^{\infty} e^{-x^{2}} d x=\frac{1}{2} \sqrt{\pi}$ . Round to five decimal places.

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Problem 8351

Find the derivative $\vec{r}^{\prime}(t)$ of the vector function $\vec{r}(t)=\langle-\sin(t), 2e^{-2t}, 3t^{2}-2\rangle$ .

See Solution

Problem 8352

Find the derivative of the function $7 \ln x + \ln 2$ .

See Solution

Problem 8353

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle 2 \ln (4 t), 3 e^{4 t},-3 \sqrt{4 t}\rangle$ .

See Solution

Problem 8354

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle 5 \sin (t-2),-4 t e^{5 t}, 2 t \ln (-4 t)\rangle$ .

See Solution

Problem 8355

Find the derivative $\vec{r}^{\prime}(t)$ of the vector function $\vec{r}(t)=\left\langle 2 e^{-2 t}, 4 \ln (5 t),-2 \sqrt{2 t}\right\rangle$ .

See Solution

Problem 8356

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\left\langle\frac{-3 t^{2}}{-t^{3}+2}, \frac{6 t}{t^{2}-2}, \frac{5}{2 t-1}\right\rangle$ .

See Solution

Problem 8357

Find the second derivative of the function $f(x)=-4 x^{3}-30 x^{2}-48 x-36$ .

See Solution

Problem 8358

Find $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle 5 \sin(t-2), -4t e^{5t}, 2t \ln(-4t) \rangle$ .

See Solution

Problem 8359

Find local max/min points of $f(x)=2 x^{3}+6 x^{2}-18 x+12$ using the Second Derivative Test. Enter as ordered pairs.

See Solution

Problem 8360

Find the derivative of $f(x)=-4 x^{3}+6 x^{2}+240 x-12$ .

See Solution

Problem 8361

Find the second derivative of the vector function $\vec{r}(t)=(3t+7\sin(t))\vec{i}+(5t+7\cos(t))\vec{j}$ .

See Solution

Problem 8362

Find $\lim _{x \rightarrow \frac{4}{3}} f(x)$ for the function $f(x)=\frac{(3 x-4)(x^{2}-x+2)}{(3 x-4)(x-3)}$ . Options: (A) $\infty$ , (B) $-\frac{22}{15}$ , (C) 0, (D) does not exist.

See Solution

Problem 8363

Find the second derivative of the function $f(x) = x^{3} - 6 \sqrt{x} - 5$ .

See Solution

Problem 8364

Find the tangent line $(L)$ to the curve $\vec{r}(t)=\langle 3t^{2}+5, t^{3}-4t^{4},-3t^{3}\rangle$ at the point $(8,-3,-3)$ .

See Solution

Problem 8365

Find the tangent line $(L)$ to the vector function $\vec{r}(t)=\langle-5 t-5,-4 e^{-2 t}, 3 e^{5 t}\rangle$ at $t=-3$ .

See Solution

Problem 8366

Find the local max and min of $f(x)=5 x^{2}-20 x$ on $[0,8]$ as ordered pairs.

See Solution

Problem 8367

Find the local min and max of $f(x)=2x^3−150x$ in the interval $[-7,8]$ .

See Solution

Problem 8368

Find the absolute extrema of $f(x)=5x^{2}-20x$ on $[0,7]$ . Provide $(x, f(x))$ as an ordered pair.

See Solution

Problem 8369

Berechnen Sie die Integrale: a) $\int \sqrt{1-x^{2}} d x$ und b) $\int \sqrt{1+x^{2}} d x$ .

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Problem 8370

Find $\lim _{x \rightarrow 2 \pi} \frac{\sin(2 \pi)-\sin(x)}{x-2 \pi}$ . What is the limit?

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Problem 8371

Найдите предел $a_{n}=\frac{4 n-3}{2 n+5}$ при $n \to \infty$ и значение $a$ при $n = 2$ .

See Solution

Problem 8372

Find $f^{\prime}(0)$ for the function $f(x)=2 \cos x+3 e^{x}$ . Choose from (A) -3, (B) 0, (C) 3, (D) 5.

See Solution

Problem 8373

Find $\lim _{x \rightarrow 2} \frac{\ln 2 - \ln x}{x - 2}$ . Choose from (A) $-\ln 2$ , (B) $-\frac{1}{2}$ , (C) $\frac{1}{2}$ , (D) $\ln 2$ .

See Solution

Problem 8374

Find critical numbers of $f(x)=2x^{3}-24x$ in the interval $[-7,3]$ .

See Solution

Problem 8375

Find local extrema of $f(x)=-6 x^{3}-72 x^{2}-270 x$ on $[-7,7]$ . Provide your answer as an ordered pair.

See Solution

Problem 8376

Find the tangent line equation for $f(x)=(x^{2}+31)^{\frac{4}{5}}$ at $x=1$ .

See Solution

Problem 8377

A 13-foot ladder leans against a wall. If Jack pulls it away at 0.5 ft/s, how fast is the top sliding down when the foot is 5 ft away?

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Problem 8378

Find the tangent line equation for $f(x)=x(1-x)^{4}$ at $x=2$ . What is $y=$ ?

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Problem 8379

Find the derivative of $f(g(x))$ where $f(x)=x^{3}$ and $g(x)=3 x-8$ .

See Solution

Problem 8380

Find $g^{\prime}(3)$ where $g$ is the inverse of $f(x)=\sqrt{e^{x}+3x+8}$ and $f(0)=3$ .

See Solution

Problem 8381

Find the tangent line equation for $f(x)=4 e^{x}-1$ at $x=0$ .

See Solution

Problem 8382

Find the limit: $\lim _{x \rightarrow-2} \frac{3 x^{2}+5 x-2}{4 x^{3}+12 x^{2}-7 x-30}$

See Solution

Problem 8383

Find the derivative of $f(x)=x^{2} \cdot \arctan (5 x)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 8384

Find the derivative $\frac{d y}{d x}$ for $y=x^{4} \sqrt{x+1}$ . Choose the correct option among the given choices.

See Solution

Problem 8385

Find the derivative $f'(x)$ for $f(x)=\arcsin x-2x$ and the tangent line at $x=0$ .

See Solution

Problem 8386

Two planes fly at the same height: A at 310 mi/h east and B at 260 mi/h north. Find the rate of distance change to the airport.

See Solution

Problem 8387

Zeigen Sie, dass die Wendepunkte der Funktion $f_{t}(x)=\frac{2}{t^{2}} x^{3}-\frac{12}{t} x^{2}-3$ auf einer Geraden liegen.

See Solution

Problem 8388

A cylinder has a radius of 9 ft and height of 22 ft. If the surface area increases at $320 \mathrm{ft}^{2}/\mathrm{sec}$ and height decreases at $-4 \mathrm{ft}/\mathrm{sec}$ , find the rate of change of the radius.

See Solution

Problem 8389

A penny is dropped from a height of $373 \mathrm{~m}$ . What is its velocity upon hitting the ground? (Ignore air resistance.)

See Solution

Problem 8390

Write the composite function $f(g(x))$ for $y=\cos(\sin(x))$ with $f(u)=\cos(u)$ and $g(x)=\sin(x)$ . Find $\frac{d y}{d x}$ .

See Solution

Problem 8391

A conical tank is 6 m wide and 8 m deep. If water height decreases at 13.5 m/min, find volume change when height is 4 m.

See Solution

Problem 8392

Ein Turmspringer springt waagerecht ab. Gegeben ist $h(t)=10-5 t^{2}$ .
a) Durchschnittsgeschwindigkeit in der ersten Sekunde? b) Momentangeschwindigkeit nach 1 Sekunde? c) Momentangeschwindigkeit beim Eintauchen?

See Solution

Problem 8393

Calculer la dérivée de $f(x)=\frac{(10 x^{8}+7)^{4}}{(7 x^{8}-6)^{4}}$ . Trouver $f^{\prime}(x)$ .

See Solution

Problem 8394

Find the derivative of $f(x)=(2x-3)^{3} \cdot 3x$ .

See Solution

Problem 8395

Bestimme den Differenzenquotienten von $f(x)=\frac{1}{2} x^{2}-4$ in den Intervallen $[0; 2]$ und $[-1; 11]$ .

See Solution

Problem 8396

Find the derivative or integral of the function $g(x)=\frac{x^{2}}{x+1}$ or solve for a specific x-value.

See Solution

Problem 8397

Bestimme die Tangentengleichung der Funktion $f(x)=\sqrt{x}$ im Punkt $B(4, f(4))$ .

See Solution

Problem 8398

Bestimme den Differenzenquotienten von $f(x)=\frac{1}{2} x^{2}-4$ für die Intervalle: a) $[0 ; 2]$ , b) $[-1 ; 1]$ , c) $[-2 ;-1,9]$ .

See Solution

Problem 8399

Gegeben ist die Funktion $f_{a}(x)$ . Bestimme die Ableitung und die Steigung bei $x=0$ . Für welches $a$ ist die Steigung 1? a) $f_{a}(x)=-x^{2}+a x$ b) $f_{a}(x)=a x^{3}-3 a x$ c) $f_{a}(x)=a x^{4}-4 x^{3}+a^{2} x$

See Solution

Problem 8400

Find the derivative of $f(x)=\frac{1+x e^{x}}{1+x}$ . What is $f'(x)$ ?

See Solution

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Calculus

Problem 8301

Find the derivative dydx\frac{d y}{d x}dxdy​ from the equation y′cos⁡(y+x+x2)−ysin⁡(y+x+x2)(y′+1+2x)=3x2y^{\prime} \cos(y+x+x^{2}) - y \sin(y+x+x^{2})(y^{\prime}+1+2x) = 3x^{2}y′cos(y+x+x2)−ysin(y+x+x2)(y′+1+2x)=3x2 at point P=(0,π2)P=(0, \frac{\pi}{2})P=(0,2π​).

Problem 8302

Determine where the function k(x)=5x4+20x3k(x)=5 x^{4}+20 x^{3}k(x)=5x4+20x3 is increasing or decreasing. Provide your answer in interval notation.

Problem 8303

Calculate the partial derivative of 2x2+3y=−4x+y2 x^{2}+3 \sqrt{y}=-4 x+y2x2+3y​=−4x+y with respect to xxx at the point (−2,9)(-2,9)(−2,9).

Problem 8304

Find the implicit derivative of 2x2+3y=−4x+y2x^{2}+3\sqrt{y}=-4x+y2x2+3y​=−4x+y.

Problem 8305

Find the second derivative d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ for 4x2−5y2=114 x^{2}-5 y^{2}=114x2−5y2=11 using implicit differentiation.

Problem 8306

Find the second derivative of yyy with respect to xxx for 4x2−5y2=114 x^{2}-5 y^{2}=114x2−5y2=11 using implicit differentiation.

Problem 8307

Find horizontal tangent lines for the curve y−3y2=2x−x2y - 3y^{2} = 2x - x^{2}y−3y2=2x−x2. If none, explain why.

Problem 8308

Find the integral ∫xx(1+ln⁡x) dx\int x^{x}(1+\ln x) \, dx∫xx(1+lnx)dx given that f′(x)=xx(1+ln⁡x)f'(x)=x^{x}(1+\ln x)f′(x)=xx(1+lnx). Explain your reasoning.

Problem 8309

Find lim⁡x→4f(x)\lim _{x \rightarrow 4} f(x)limx→4​f(x) for f(x)=8x+3f(x)=8x+3f(x)=8x+3. Prove ∣f(x)−35∣=8∣x−4∣|f(x)-35|=8|x-4|∣f(x)−35∣=8∣x−4∣ and show δ=ϵ8\delta=\frac{\epsilon}{8}δ=8ϵ​.

Problem 8310

A ball's position is given by s(t)=−16t2+128t+5s(t)=-16 t^{2}+128 t+5s(t)=−16t2+128t+5. Find: a. velocity at t=1t=1t=1, b. acceleration at t=5t=5t=5, c. time for velocity -64 ft/s, d. time at height 117 ft.

Problem 8311

Find δ\deltaδ for lim⁡x→2f(x)=7\lim _{x \rightarrow 2} f(x)=7limx→2​f(x)=7 with f(x)=4x−1f(x)=4x-1f(x)=4x−1 and ∣f(x)−7∣<0.01|f(x)-7|<0.01∣f(x)−7∣<0.01.

Problem 8312

Bestimme die Ableitung von 2x⋅(4x−1)2x \cdot (4x - 1)2x⋅(4x−1).

Problem 8313

A stone is thrown up at 128ft/sec128 \mathrm{ft} / \mathrm{sec}128ft/sec. Find: a. max height, b. time to hit ground, c. speed when it hits.

Problem 8314

A stone is thrown up at 128ft/sec128 \mathrm{ft/sec}128ft/sec. Find: a. max height, b. time to hit ground, c. speed on impact using integrals.

Problem 8315

Bestimmen Sie die Koordinaten des Punktes PPP, wo der Graph GfG_{f}Gf​ von f(x)=−14x2+xf(x)=-\frac{1}{4} x^{2}+xf(x)=−41​x2+x die Steigung -2 hat.

Problem 8316

Bestimmen Sie die Ableitung von fa(x)=1axaf_{a}(x)=\frac{1}{a} x^{a}fa​(x)=a1​xa.

Problem 8317

Evaluate the integral ∫12315x2(x3+4)12dx\int_{12}^{31} \frac{5 x^{2}}{\left(x^{3}+4\right)^{\frac{1}{2}}} d x∫1231​(x3+4)21​5x2​dx using u=x3+4u=x^{3}+4u=x3+4. Round to three decimal places.

Problem 8318

Problem 8319

Untersuchen Sie die Intervalle, in denen f(x)=18x3−34x2−2,5f(x)=\frac{1}{8} x^{3}-\frac{3}{4} x^{2}-2,5f(x)=81​x3−43​x2−2,5 linksgekrümmt ist. Bestimmen Sie die Extrempunkte von f(x)=−16x3+2xf(x)=-\frac{1}{6} x^{3}+2 xf(x)=−61​x3+2x.

Problem 8320

Find the horizontal asymptotes of y=2xx2−1y=\frac{2x}{\sqrt{x^2-1}}y=x2−1​2x​. Options: (A) y=0y=0y=0 (B) y=1y=1y=1 (C) y=2y=2y=2 (D) y=−2y=-2y=−2 and y=2y=2y=2 (E) y=−1y=-1y=−1 and y=1y=1y=1.

Problem 8321

Cell growth rate xxx at time ttt increases proportionally to (20−x)(20-x)(20−x). (a) Write the differential equation. (b) If initially there are 10 cells and after 1.5 hours there are 18, find the cell count after 2 hours (2 sig. figs.).

Problem 8322

Problem 8323

A car's position is given by s(t)=(4t2−3)e−0.5ts(t)=(4 t^{2}-3)e^{-0.5 t}s(t)=(4t2−3)e−0.5t. How far does it travel from t=0t=0t=0 to when it stops?

Problem 8324

The function ggg is continuous on [2,6][2,6][2,6] with values: g(2)=7g(2)=7g(2)=7, g(3)=4g(3)=4g(3)=4, g(4)=1g(4)=1g(4)=1, g(5)=4g(5)=4g(5)=4, g(6)=7g(6)=7g(6)=7. True statement?

Problem 8325

Find the limit: lim⁡x→∞2x+1x\lim_{x \rightarrow \infty} \frac{2x + 1}{x}limx→∞​x2x+1​.

Problem 8326

Gegeben ist die Funktion f(x)=x2f(x)=x^{2}f(x)=x2 und der Punkt T(1/−3)T(1 /-3)T(1/−3). Bestimme die Tangentengleichung in P(u/f(u))P(u / f(u))P(u/f(u)).

Problem 8327

Find the maximum slope of the tangent line to the graph of f(x)=3x2−2x3f(x)=3x^2-2x^3f(x)=3x2−2x3.

Problem 8328

Gegeben ist die Funktion f(x)=(2x+10)⋅e5x+3f(x)=(2 x+10) \cdot e^{5 x+3}f(x)=(2x+10)⋅e5x+3. Finde die Nullstelle und den yyy-Achsen-Schnittpunkt. Bestimme die Steigung bei x=−2x=-2x=−2.

Problem 8329

Berechne die Ableitung von g(x)=x2+2x3x−4g(x)=\frac{x^{2}+2x}{3x-4}g(x)=3x−4x2+2x​ und finde die Tangenten bei P(3/g(3))P(3/g(3))P(3/g(3)). Untersuche auch Asymptoten.

Problem 8330

Problem 8331

Find the curve where the slope of the tangent line at (x,y)(x, y)(x,y) is 3x3 \sqrt{x}3x​ and it passes through (9,4)(9, 4)(9,4).

Problem 8332

Find the hydrostatic force on one side of an isosceles triangle (base 12 ft, sides 10 ft) submerged 2 ft below water. Set up Riemann Sum and integral without evaluating.

Problem 8333

Untersuchen Sie, ob die Funktion g(x)=f(x)⋅cos⁡(x)g(x)=f(x) \cdot \cos (x)g(x)=f(x)⋅cos(x) im Punkt P(0∣1)P(0 \mid 1)P(0∣1) auch eine waagerechte Tangente hat.

Problem 8334

Leiten Sie die Funktion g(x)=c⋅f(x)g(x)=c \cdot f(x)g(x)=c⋅f(x) mit der Produktregel ab. Welche Regel ergibt sich?

Problem 8335

Bestimmen Sie die Summe des Funktionsterms f(x)f(x)f(x) und leiten Sie fff einmal ab für: a) f(x)=x4−x3x2f(x)=\frac{x^{4}-x^{3}}{x^{2}}f(x)=x2x4−x3​, b) f(x)=x12+x8+1x10f(x)=\frac{x^{12}+x^{8}+1}{x^{10}}f(x)=x10x12+x8+1​, c) f(x)=x4+2x2x5f(x)=\frac{x^{4}+2 x^{2}}{x^{5}}f(x)=x5x4+2x2​.

Problem 8336

Bestimme die Ableitung der Funktion f(x)=(5x4−2x)−3f(x)=(5x^4-2x)^{-3}f(x)=(5x4−2x)−3 mit der Kettenregel.

Problem 8337

Find the limit of f(x)=sin⁡(x)xf(x)=\frac{\sin(x)}{x}f(x)=xsin(x)​ as xxx approaches 000 from the left, right, and both sides.

Problem 8338

Calculate the average rate of change for f(x)=x2+8xf(x)=x^{2}+8xf(x)=x2+8x from x=2x=2x=2 to x=3x=3x=3. Simplify your answer.

Problem 8339

Calculate the average rate of change of f(x)=xf(x)=\sqrt{x}f(x)=x​ from x1=16x_{1}=16x1​=16 to x2=81x_{2}=81x2​=81. Simplify your answer.

Problem 8340

Find the average rate of change of f(x)=xf(x)=\sqrt{x}f(x)=x​ from x1=16x_{1}=16x1​=16 to x2=81x_{2}=81x2​=81. Simplify your answer.

Problem 8341

Analyze the series S=11+12+13+…+1nS = \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \ldots + \frac{1}{\sqrt{n}}S=1​1​+2​1​+3​1​+…+n​1​ for convergence or divergence.

Find the derivative $\frac{d y}{d x}$ from the equation $y^{\prime} \cos(y+x+x^{2}) - y \sin(y+x+x^{2})(y^{\prime}+1+2x) = 3x^{2}$ at point $P=(0, \frac{\pi}{2})$ .

Determine where the function $k(x)=5 x^{4}+20 x^{3}$ is increasing or decreasing. Provide your answer in interval notation.

Calculate the partial derivative of $2 x^{2}+3 \sqrt{y}=-4 x+y$ with respect to $x$ at the point $(-2,9)$ .

Find the implicit derivative of $2x^{2}+3\sqrt{y}=-4x+y$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for $4 x^{2}-5 y^{2}=11$ using implicit differentiation.

Find the second derivative of $y$ with respect to $x$ for $4 x^{2}-5 y^{2}=11$ using implicit differentiation.

Find horizontal tangent lines for the curve $y - 3y^{2} = 2x - x^{2}$ . If none, explain why.

Find the integral $\int x^{x}(1+\ln x) \, dx$ given that $f'(x)=x^{x}(1+\ln x)$ . Explain your reasoning.

Find $\lim _{x \rightarrow 4} f(x)$ for $f(x)=8x+3$ . Prove $|f(x)-35|=8|x-4|$ and show $\delta=\frac{\epsilon}{8}$ .

A ball's position is given by $s(t)=-16 t^{2}+128 t+5$ . Find: a. velocity at $t=1$ , b. acceleration at $t=5$ , c. time for velocity -64 ft/s, d. time at height 117 ft.

Find $\delta$ for $\lim _{x \rightarrow 2} f(x)=7$ with $f(x)=4x-1$ and $|f(x)-7|<0.01$ .

Bestimme die Ableitung von $2x \cdot (4x - 1)$ .

A stone is thrown up at $128 \mathrm{ft} / \mathrm{sec}$ . Find: a. max height, b. time to hit ground, c. speed when it hits.

A stone is thrown up at $128 \mathrm{ft/sec}$ . Find: a. max height, b. time to hit ground, c. speed on impact using integrals.

Bestimmen Sie die Koordinaten des Punktes $P$ , wo der Graph $G_{f}$ von $f(x)=-\frac{1}{4} x^{2}+x$ die Steigung -2 hat.

Bestimmen Sie die Ableitung von $f_{a}(x)=\frac{1}{a} x^{a}$ .

Evaluate the integral $\int_{12}^{31} \frac{5 x^{2}}{\left(x^{3}+4\right)^{\frac{1}{2}}} d x$ using $u=x^{3}+4$ . Round to three decimal places.

Untersuchen Sie die Intervalle, in denen $f(x)=\frac{1}{8} x^{3}-\frac{3}{4} x^{2}-2,5$ linksgekrümmt ist. Bestimmen Sie die Extrempunkte von $f(x)=-\frac{1}{6} x^{3}+2 x$ .

Find the horizontal asymptotes of $y=\frac{2x}{\sqrt{x^2-1}}$ . Options: (A) $y=0$ (B) $y=1$ (C) $y=2$ (D) $y=-2$ and $y=2$ (E) $y=-1$ and $y=1$ .

Cell growth rate $x$ at time $t$ increases proportionally to $(20-x)$ .
(a) Write the differential equation. (b) If initially there are 10 cells and after 1.5 hours there are 18, find the cell count after 2 hours (2 sig. figs.).

A car's position is given by $s(t)=(4 t^{2}-3)e^{-0.5 t}$ . How far does it travel from $t=0$ to when it stops?

The function $g$ is continuous on $[2,6]$ with values: $g(2)=7$ , $g(3)=4$ , $g(4)=1$ , $g(5)=4$ , $g(6)=7$ . True statement?

Find the limit: $\lim_{x \rightarrow \infty} \frac{2x + 1}{x}$ .

Gegeben ist die Funktion $f(x)=x^{2}$ und der Punkt $T(1 /-3)$ . Bestimme die Tangentengleichung in $P(u / f(u))$ .

Find the maximum slope of the tangent line to the graph of $f(x)=3x^2-2x^3$ .

Gegeben ist die Funktion $f(x)=(2 x+10) \cdot e^{5 x+3}$ . Finde die Nullstelle und den $y$ -Achsen-Schnittpunkt. Bestimme die Steigung bei $x=-2$ .

Berechne die Ableitung von $g(x)=\frac{x^{2}+2x}{3x-4}$ und finde die Tangenten bei $P(3/g(3))$ . Untersuche auch Asymptoten.

Find the curve where the slope of the tangent line at $(x, y)$ is $3 \sqrt{x}$ and it passes through $(9, 4)$ .

Untersuchen Sie, ob die Funktion $g(x)=f(x) \cdot \cos (x)$ im Punkt $P(0 \mid 1)$ auch eine waagerechte Tangente hat.

Leiten Sie die Funktion $g(x)=c \cdot f(x)$ mit der Produktregel ab. Welche Regel ergibt sich?

Bestimmen Sie die Summe des Funktionsterms $f(x)$ und leiten Sie $f$ einmal ab für: a) $f(x)=\frac{x^{4}-x^{3}}{x^{2}}$ , b) $f(x)=\frac{x^{12}+x^{8}+1}{x^{10}}$ , c) $f(x)=\frac{x^{4}+2 x^{2}}{x^{5}}$ .

Bestimme die Ableitung der Funktion $f(x)=(5x^4-2x)^{-3}$ mit der Kettenregel.

Find the limit of $f(x)=\frac{\sin(x)}{x}$ as $x$ approaches $0$ from the left, right, and both sides.

Calculate the average rate of change for $f(x)=x^{2}+8x$ from $x=2$ to $x=3$ . Simplify your answer.

Calculate the average rate of change of $f(x)=\sqrt{x}$ from $x_{1}=16$ to $x_{2}=81$ . Simplify your answer.

Find the average rate of change of $f(x)=\sqrt{x}$ from $x_{1}=16$ to $x_{2}=81$ . Simplify your answer.

Analyze the series $S = \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \ldots + \frac{1}{\sqrt{n}}$ for convergence or divergence.

Find the max and saddle points of $f(x, y)=-x^{3}+8xy-4y^{2}+3$ .

Bestimme die Fußpunkte des Hügels für die Funktion $f(x)=-\frac{1}{2} x^{2}+4 x-6$ und den Steigungswinkel am westlichen Fußpunkt.

Differentiate the function $f(x) = x^{3} - 4x^{2} - 3x$ .

Find the value of $\int_{0}^{b} x f\left(x^{2}\right) d x$ given that $\int_{0}^{b^{2}} f(x) d x=214.2$ . Round to two decimal places.

If $h(a)=h(b)$ , find the value of $\int_{a}^{b} g(h(x)) h^{\prime}(x) d x$ .

Find $\int_{0}^{a} f(3 x) d x$ given that $\int_{0}^{3 a} f(x) d x=13.8$ . Round to two decimal places.

Evaluate the integral $\int_{0}^{\infty} e^{\frac{-x^{2}}{3.5}} d x$ using $\int_{0}^{\infty} e^{-x^{2}} d x=\frac{1}{2} \sqrt{\pi}$ . Round to five decimal places.

Find the derivative $\vec{r}^{\prime}(t)$ of the vector function $\vec{r}(t)=\langle-\sin(t), 2e^{-2t}, 3t^{2}-2\rangle$ .

Find the derivative of the function $7 \ln x + \ln 2$ .

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle 2 \ln (4 t), 3 e^{4 t},-3 \sqrt{4 t}\rangle$ .

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle 5 \sin (t-2),-4 t e^{5 t}, 2 t \ln (-4 t)\rangle$ .

Find the derivative $\vec{r}^{\prime}(t)$ of the vector function $\vec{r}(t)=\left\langle 2 e^{-2 t}, 4 \ln (5 t),-2 \sqrt{2 t}\right\rangle$ .

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\left\langle\frac{-3 t^{2}}{-t^{3}+2}, \frac{6 t}{t^{2}-2}, \frac{5}{2 t-1}\right\rangle$ .

Find the second derivative of the function $f(x)=-4 x^{3}-30 x^{2}-48 x-36$ .

Find $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle 5 \sin(t-2), -4t e^{5t}, 2t \ln(-4t) \rangle$ .

Find local max/min points of $f(x)=2 x^{3}+6 x^{2}-18 x+12$ using the Second Derivative Test. Enter as ordered pairs.

Find the derivative of $f(x)=-4 x^{3}+6 x^{2}+240 x-12$ .

Find the second derivative of the vector function $\vec{r}(t)=(3t+7\sin(t))\vec{i}+(5t+7\cos(t))\vec{j}$ .

Find the second derivative of the function $f(x) = x^{3} - 6 \sqrt{x} - 5$ .

Find the tangent line $(L)$ to the curve $\vec{r}(t)=\langle 3t^{2}+5, t^{3}-4t^{4},-3t^{3}\rangle$ at the point $(8,-3,-3)$ .

Find the tangent line $(L)$ to the vector function $\vec{r}(t)=\langle-5 t-5,-4 e^{-2 t}, 3 e^{5 t}\rangle$ at $t=-3$ .

Find the local max and min of $f(x)=5 x^{2}-20 x$ on $[0,8]$ as ordered pairs.

Find the local min and max of $f(x)=2x^3−150x$ in the interval $[-7,8]$ .

Find the absolute extrema of $f(x)=5x^{2}-20x$ on $[0,7]$ . Provide $(x, f(x))$ as an ordered pair.

Berechnen Sie die Integrale: a) $\int \sqrt{1-x^{2}} d x$ und b) $\int \sqrt{1+x^{2}} d x$ .

Find $\lim _{x \rightarrow 2 \pi} \frac{\sin(2 \pi)-\sin(x)}{x-2 \pi}$ . What is the limit?

Найдите предел $a_{n}=\frac{4 n-3}{2 n+5}$ при $n \to \infty$ и значение $a$ при $n = 2$ .

Find $f^{\prime}(0)$ for the function $f(x)=2 \cos x+3 e^{x}$ . Choose from (A) -3, (B) 0, (C) 3, (D) 5.

Find $\lim _{x \rightarrow 2} \frac{\ln 2 - \ln x}{x - 2}$ . Choose from (A) $-\ln 2$ , (B) $-\frac{1}{2}$ , (C) $\frac{1}{2}$ , (D) $\ln 2$ .

Find critical numbers of $f(x)=2x^{3}-24x$ in the interval $[-7,3]$ .