Calculus

Problem 5201

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

See Solution

Problem 5202

Find the limit: $\lim _{y \rightarrow 2} \frac{y^{2}-y-2}{y^{2}-4 y+4}$ .

See Solution

Problem 5203

Find the limit of $g(x)=\frac{|x|}{x}$ as $x$ approaches 0 from the left, right, and directly.

See Solution

Problem 5204

Find the limits of the piecewise function $f(x)=\left\{\begin{array}{ll}1-x^{2} & x \geq 0 \\ x^{2}-1 & x<0\end{array}\right.$ :
1. $\lim_{{x \to 0^+}} f(x)$
2. $\lim_{{x \to 0^-}} f(x)$
3. $\lim_{{x \to 0}} f(x)$ or explain if it doesn’t exist.

See Solution

Problem 5205

Find the limit: $\lim _{x \rightarrow 25} \frac{\sqrt{x}-5}{x-25}$ .

See Solution

Problem 5206

Find the limit of $h(x)=\sqrt{x}$ for $x \geq 0$ and $h(x)=\sqrt{-x}$ for $x<0$ as $x$ approaches $0$ .

See Solution

Problem 5207

Find $\lim _{x \rightarrow 4} 2 f(x) g(x)$ if $\lim _{x \rightarrow 4} f(x) = 5$ and $\lim _{x \rightarrow 4} g(x) = -3$ .

See Solution

Problem 5208

Find the limit: $\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{x+3}-2}$

See Solution

Problem 5209

Find the limit: $\lim _{x \rightarrow 4} \frac{4-x}{5-\sqrt{x^{2}+9}}$

See Solution

Problem 5210

Find the interval of convergence for the series $1+\frac{2}{3} x+\frac{4}{9} x^{2}+\ldots$ . Choices: (a) $-\frac{3}{2}<x<\frac{3}{2}$ , (b) $-3<x<3$ , (c) $-9<x<9$ , (d) None.

See Solution

Problem 5211

Rewrite the function $f(x)=x(x-3)$ and find its derivative $f'(x)$ .

See Solution

Problem 5212

Find the derivative of $f(x)=\frac{5}{x^{3}}-\frac{x^{3}}{5}$ .

See Solution

Problem 5213

Rewrite $f(x)=(x+4)^{2}$ and find its derivative $f'(x)$ .

See Solution

Problem 5214

Forme den Funktionsterm $f(x)=\sqrt[4]{x}+\frac{1}{x^{2}}$ um und leite ihn anschließend ab.

See Solution

Problem 5215

Formen Sie die Funktion $f(x)=x \cdot\left(x+\frac{1}{x}\right)$ um und bestimmen Sie die Ableitung.

See Solution

Problem 5216

Rewrite $f(x) = \frac{2}{\sqrt{x}} - \frac{3}{x^{5}}$ and find $f'(x)$ .

See Solution

Problem 5217

Find the intervals where the function $f(x)=3 x^{3}+9 x^{2}-72 x$ is increasing or decreasing. Use $\varnothing$ for empty intervals.

See Solution

Problem 5218

Bestimme die erste und zweite Ableitung der Funktionen: 1. $f(x)=3x \cos(x)$ , 2. $g(x)=(x^{2}-3) \sin(x)$ , 3. $h(x)=\sqrt{x} \cos(x)$ , 4. $j(x)=(\sqrt{x}-1)x^{4}$ .

See Solution

Problem 5219

Find the limit $\lim _{x \rightarrow 0^{-}} f(x)$ where $f(x)=\lfloor x+4\rfloor$ for $x<0$ , $x^{2}$ for $0 \leq x<7$ , and $2-x$ for $x \geq 7$ . Options: a) 3 b) 4 c) 5 d) None.

See Solution

Problem 5220

Find if the statement is true or false: $\lim _{x \rightarrow 0.79^{-}} f(x)=0$ . a) True b) False

See Solution

Problem 5221

Überprüfen Sie die Integrationsregeln für $k=2$ , $f(x)=x^{2}$ und $g(x)=-x+1$ . Prüfen Sie (3) und (4).

See Solution

Problem 5222

Determine if the statement is true or false:
$\lim _{x \rightarrow 3^{-}} f(x)=4$
a) True b) False

See Solution

Problem 5223

Find the intervals where revenue $R(x)=214.956 x-0.009 x^{2}$ is increasing or decreasing for $0 \leq x \leq 18000$ .

See Solution

Problem 5224

Find the derivative of $y=x(5^{-3x})$ . What is $\frac{d y}{d x}=?$

See Solution

Problem 5225

Find the derivative of $P=t^{6} \ln(t+6)$ with respect to $t$ .

See Solution

Problem 5226

Find the derivative of $f(t)=t e^{3+5 t^{3}}$ with respect to $t$ .

See Solution

Problem 5227

Find the intervals where the function $f(x)=x^{2}-6 x+10$ is increasing or decreasing. Use $\varnothing$ for empty intervals.

See Solution

Problem 5228

Find the derivative of $f(z)=(\sqrt{6 z+9}) e^{6 z}$ with respect to $z$ .

See Solution

Problem 5229

Find the total number of ants on the paraboloid surface $\mathrm{z}=x^{2}+y^{2}$ using the density $f(x, y, z)=32\left(x^{2}+y^{2}\right)$ . Use polar coordinates for integration.

See Solution

Problem 5230

Find values of $x$ for horizontal tangents of $f(x)=2x^{2}-12x+14$ . Select "None" if none exist.

See Solution

Problem 5231

Find where the function $f(x)=-2 x^{3}-6 x^{2}+90 x$ is increasing or decreasing. Use $\varnothing$ for empty intervals.

See Solution

Problem 5232

Gegeben ist die Funktion $f(x)=\frac{1}{x} \cdot \sin x$ . a) Vervollständige die Wertetabelle. b) Bestimme $\lim _{x \rightarrow-\infty} f(x)$ und $\lim _{x \rightarrow+\infty} f(x)$ .

See Solution

Problem 5233

Find $\frac{d A}{d t}$ for a circle's area $A$ with radius $r$ as it expands, in terms of $\frac{d r}{d t}$ .

See Solution

Problem 5234

(a) Find $\frac{d A}{d t}$ for a circle's area $A$ with radius $r$ expanding over time.
(b) If the radius of an oil spill increases at $2 \mathrm{~m/s}$ , how fast is the area increasing when $r = 38 \mathrm{~m}$ ?

See Solution

Problem 5235

Bestimme $\lim _{x \rightarrow \pm \infty} \frac{1}{x}$ und $\lim _{x \rightarrow \pm \infty} \sin x$ .

See Solution

Problem 5236

How fast is the area of an oil spill increasing at a radius of 38 m if the radius grows at 2 m/s?

See Solution

Problem 5237

(a) Find $\frac{d A}{d t}$ for a circle area $A$ with radius $r$ expanding at rate $\frac{d r}{d t}$ . (b) If the radius of an oil spill increases at $2 \mathrm{~m/s}$ , how fast is the area increasing when $r=38 \mathrm{~m}$ ? $\mathrm{m}^{2}/\mathrm{s}$

See Solution

Problem 5238

Find the integral of $\frac{3}{x^{2}}$ with respect to $x$ .

See Solution

Problem 5239

Bestimmen Sie die Funktion $f(x)$ mit $f'(0)=3$ , $f(0)=0$ , Wendestelle bei $x=2$ mit Steigung -1.

See Solution

Problem 5240

Find the derivative $\frac{d y}{d t}$ for $y=(2.6 t-3 t^{2})(4 t+1.4)$ .

See Solution

Problem 5241

Find the derivative of the vector function $\mathbf{r}(t)=t^{3} \mathbf{i}-\sqrt{t} \mathbf{j}$ .

See Solution

Problem 5242

Find the square size $x$ (in inches) to cut from corners to maximize the volume $V(x)=x(14-2x)(12-2x)$ of a box.

See Solution

Problem 5243

Integrate: $\int \frac{\left(6 x^{2}-4\right) \ln \left(x^{3}-2 x\right)}{x^{3}-2 x} d x$

See Solution

Problem 5244

Given $4 x^{2}+16 y^{2}=100$ , find $\frac{d x}{d t}$ for (a) $x=3$ , $y=2$ , $\frac{d y}{d t}=\frac{1}{4}$ and (b) $\frac{d x}{d t}=4$ , $x=-3$ , $y=2$ .

See Solution

Problem 5245

Find the derivative $f^{\prime}(x)$ and the values of $x$ where $f^{\prime}(x)=0$ for $f(x)=\frac{x}{x^{2}+4}$ .

See Solution

Problem 5246

Gegeben ist die Funktion $f(x)=\frac{1}{9}(3 x+2)^{3}$ . Bestimme die Steigung in $P(2, f(2))$ , Punkte mit waagerechter Tangente und wo die Tangente die Steigung 1 hat.

See Solution

Problem 5247

Find the derivative of the vector function $\mathbf{q}(t)=\left\langle e^{t^{2}}, t \sin (2 t), \frac{t}{1-4 t^{3}}\right\rangle$ .

See Solution

Problem 5248

A potato is launched from a 50-ft building with an initial speed of $75 \mathrm{ft/s}$ . Find: a. Velocity after $0.5 \mathrm{s}$ and $2.75 \mathrm{s}$ . b. Speed after $0.5 \mathrm{s}$ and $2.75 \mathrm{s}$ . c. Acceleration after $0.5 \mathrm{s}$ and $2.75 \mathrm{s}$ . d. Total air time. e. Impact velocity.

See Solution

Problem 5249

A potato is launched from a 50-ft building with an initial velocity of $75 \mathrm{ft/s}$ .
a. Find its velocity at $0.5 \mathrm{s}$ and $2.75 \mathrm{s}$ . b. Find its speed at $0.5 \mathrm{s}$ and $2.75 \mathrm{s}$ . c. Find its acceleration at $0.5 \mathrm{s}$ and $2.75 \mathrm{s}$ . d. How long is the potato in the air? e. What is the velocity when it hits the ground?

See Solution

Problem 5250

The AROC of $f(x)=3x+b$ over $[2,2+h]$ is positive for any $h>0$ and constant $b$ . True or False?

See Solution

Problem 5251

Finde den Fehler in den Ableitungen der Funktionen a) bis d): a) $f(x)=2 x^{3}-4 x^{2}+5$ $f'(x)=6 x^{2}+8 x+5$ b) $f(x)=4 x^{2}+\frac{1}{x}+2$ $f'(x)=8 x+\frac{1}{x^{2}}$ c) $f(x)=x^{2}+c^{3}+2 x+3$ $f'(x)=2 x+3 c^{2}+2$ d) $f(x)=\sqrt{x}+x^{3}+\frac{1}{x}$ $f'(x)=\frac{1}{\sqrt{x}}+3 x^{2}+\frac{1}{x^{2}}$

See Solution

Problem 5252

Find $f^{\prime}(x)$ , the tangent line at $x=0$ for $f(x)=3 e^{x^{2}-7 x+5}$ , and where the tangent is horizontal.

See Solution

Problem 5253

Find the derivative $g^{\prime}(x)$ for the function $g(x)=9 x e^{5 x}$ .

See Solution

Problem 5254

Find $\frac{d y}{d x}$ using the chain rule for $y=\sin u$ and $u=3 x-6$ .

See Solution

Problem 5255

Berechnen Sie die Integrale und zeichnen Sie die Graphen von $\mathrm{f}$ für die Intervalle: a) $\int_{1}^{2}(x^{2}+1)dx$ , b) $\int_{-1}^{2}(x-2)dx$ , c) $\int_{0}^{3}(2-\frac{1}{2}x^{2})dx$ , d) $\int_{0}^{4}\sqrt{x}dx$ , e) $\int_{-1}^{1}x^{3}dx$ , f) $\int_{-1}^{2}(x^{3}-4x)dx$ .

See Solution

Problem 5256

Bestimme die Bedeutung von $f(t)=(60-2 t)^{3}$ und $f^{\prime}(2)$ für einen Würfel mit Kantenlängen, die sich verringern.

See Solution

Problem 5257

Ein Pkw beschleunigt. Skizziere den Geschwindigkeitsgraphen v, untersuche $v'$ und interpretiere $v(2)=3$ , $v'(12)=1$ .

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Problem 5258

Is it true or false that the AROC of $f(x)=P(0.2)^{x}$ on $[3,3+h]$ is positive for any $h>0$ ?

See Solution

Problem 5259

Bestimme $f(t)=(60-2t)^{3}$ und $f^{\prime}(2)$ für ein Würfelmodell mit Kantenlängen, die sich pro Minute verkürzen.

See Solution

Problem 5260

Bestimmen Sie die Stammfunktion von $f(x)=(x+3)(x+2)$ .

See Solution

Problem 5261

Find $\lim _{t \rightarrow 31} Y(t)$ where $Y(t)$ is the year at time $t$ days from Dec 1, 2022. Options: 2022, 2023, limit doesn't exist, not enough info.

See Solution

Problem 5262

Bestimmen Sie die Stammfunktion von $f(x)=\frac{x^{2}+2 x+1}{x}$ nach Vereinfachung des Ausdrucks.

See Solution

Problem 5263

Analyze how $C^{\prime}(t)$ changes in these scenarios:
1. 22152 cases on April 10, 2020.
2. Cases surged in December 2021.
3. Fewer cases now than January 2022.
4. Cases rose by 500 on June 12, 2020.
5. Cases stabilized in April 2021.
Consider the graph of $C(t)$ and its tangent line.

See Solution

Problem 5264

Find values of $a$ and $b$ so that the function $f(x)=\begin{cases} x^{2}+4x+5 & \text{if } x \leq 3 \\ ax+b & \text{if } x>3 \end{cases}$ is differentiable everywhere. $a=$ , $b=$ .

See Solution

Problem 5265

Finde eine Stammfunktion von $f(x)=(3 x+4)^{3}$ und erkläre, warum es verschiedene Ergebnisse geben kann.

See Solution

Problem 5266

Calculate the integral of the function: $\int(3 x^{4}-6 x+8) \, dx$ .

See Solution

Problem 5267

Find the derivative of $y=6 \cot n \theta$ .

See Solution

Problem 5268

Find the slope of the tangent line to $y(b)$ at $b = 6$ , where $y(b) = b^{7} - b^{5} + 4b^{2}$ .

See Solution

Problem 5269

Calculate the integral of the function: $\int 2 x^{3} \, dx$ .

See Solution

Problem 5270

Find all $x$ values for horizontal tangents of the function $f(x)=3 x^{3}-18 x^{2}+36 x-25$ . Select "None" if none exist.

See Solution

Problem 5271

Find the derivative of $h(t)=(t^{4}-1)^{9}(t^{3}+1)^{8}$ .

See Solution

Problem 5272

Find the intervals where the function $f(x)=3 x^{3}-18 x^{2}+36 x-25$ is increasing or decreasing.

See Solution

Problem 5273

Compute the limit as $x$ approaches 0 for $\frac{3 x^{3}-4 x^{2}}{11 x^{2}-5 x^{3}}$ . Options: $-\frac{4}{11}$ or $\frac{3}{5}$ .

See Solution

Problem 5274

Find the tangent slope function for $j(x) = \frac{4 x^{5}}{5} - x^{4} + x^{2}$ . Choose from the given options.

See Solution

Problem 5275

Determine the intervals where revenue $R(x)=12.396 x-0.001 x^{2}$ is increasing or decreasing for $0 \leq x \leq 9300$ .

See Solution

Problem 5276

Find values of $x$ for horizontal tangents of $f(x)=\frac{x+7}{x+4}$ . Select "None" if none exist.

See Solution

Problem 5277

Find the intervals where the function $f(x)=\frac{x+7}{x+4}$ is increasing or decreasing. Use $\varnothing$ for empty intervals.

See Solution

Problem 5278

Find $x$ where $f'(x) = -2$ for $f(x) = 4x^3 - 6x + 1$ . Choose from: 42, 43, 19, Undefined.

See Solution

Problem 5279

Find $x$ where $f^{\prime}(x)=-2$ for $f(x)=4x^{3}-6x+1$ . Choose from: 42, 43, 19, Undefined.

See Solution

Problem 5280

Bestimme die Flächeninhaltsfunktion von $f(x)=(2x+3)^5$ mit unterer Grenze 0.

See Solution

Problem 5281

Find local extrema of $f(x)=8x^{2}+160x-4$ using the First Derivative Test. Provide results as ordered pairs.

See Solution

Problem 5282

Find local extrema of the function $f(x)=4 x^{\frac{5}{2}}-80 x-8$ using the First Derivative Test.

See Solution

Problem 5283

Find local extrema of the function $f(x)=5 x^{2}-60 x-1$ using the First Derivative Test. Provide as ordered pairs.

See Solution

Problem 5284

Find local extrema for the function $f(x)=9x^{2}+180x+1$ using the First Derivative Test.

See Solution

Problem 5285

Bestimmen Sie die Flächeninhaltsfunktion von $f(x)=(7 x+11)^{-4}$ .

See Solution

Problem 5286

Find the critical values of the function $f(x)=4 x^{\frac{5}{2}}-80 x-8$ .

See Solution

Problem 5287

Bestimmen Sie die Nullstellen und den Gesamtinhalt der Fläche zwischen dem Graphen von $f$ und der X-Achse über den Intervallen: a) $f(x)=x^{3}+2 x^{2}-3 x$ , $I=[-2 ; 2,5]$ b) $f(x)=(x+2)(x-1)^{2}$ , $I=[-2 ; 2]$ c) $f(x)=(x-1)(x+2)(x-3)$ , $I=[-1 ; 2]$ d) $f(x)=x^{4}+x^{2}-2$ , $I=[-2 ; 3]$

See Solution

Problem 5288

Find the critical values of the function $f(x)=36x+\frac{49}{x}$ .

See Solution

Problem 5289

Find the world population increase rates in 1920, 1951, and 2000 using the model $P(t)=(1436.53)(1.01395)^{t}$ .

See Solution

Problem 5290

Find local extrema of the function $f(x)=36x+\frac{49}{x}$ using the First Derivative Test. Provide ordered pairs.

See Solution

Problem 5291

Find the derivative of $y=3 \cdot \sin(5 \cdot x)$ . What is $\frac{dy}{dx}$ ?

See Solution

Problem 5292

Bestimme die Flächeninhaltsfunktion für $f(x)=\sin(2x+3)$ .

See Solution

Problem 5293

Find the derivative of $y=3 \cdot \sin (5 \cdot x)$ .

See Solution

Problem 5294

Find local extrema for the function $f(x)=16x+\frac{25}{x}$ using the First Derivative Test.

See Solution

Problem 5295

Find local extrema of the function $f(x)=-5 x^{2}+70 x-4$ using the First Derivative Test. Provide as ordered pairs.

See Solution

Problem 5296

Bestimmen Sie die Flächeninhaltsfunktion für $f(x)=e^{-(2x+3)}$ .

See Solution

Problem 5297

Find local extrema for the function $f(x)=16x+\frac{49}{x}$ using the First Derivative Test. Provide ordered pairs.

See Solution

Problem 5298

Find local extrema of the function $f(x)=4x+\frac{9}{x}$ using the First Derivative Test. Provide results as ordered pairs.

See Solution

Problem 5299

Find the derivative of $y=\frac{e^{7 u}-e^{-7 u}}{e^{7 u}+e^{-7 u}}$ .

See Solution

Problem 5300

Find the critical values of the function $f(x)=-2 x^{\frac{3}{2}}+12 x+10$ .

See Solution

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Calculus

Problem 5201

Find the limit: lim⁡x→01x−1+1x+1x\lim _{x \rightarrow 0} \frac{\frac{1}{x-1}+\frac{1}{x+1}}{x}limx→0​xx−11​+x+11​​.

Problem 5202

Find the limit: lim⁡y→2y2−y−2y2−4y+4\lim _{y \rightarrow 2} \frac{y^{2}-y-2}{y^{2}-4 y+4}limy→2​y2−4y+4y2−y−2​.

Problem 5203

Find the limit of g(x)=∣x∣xg(x)=\frac{|x|}{x}g(x)=x∣x∣​ as xxx approaches 0 from the left, right, and directly.

Problem 5204

Problem 5205

Find the limit: lim⁡x→25x−5x−25\lim _{x \rightarrow 25} \frac{\sqrt{x}-5}{x-25}limx→25​x−25x​−5​.

Problem 5206

Find the limit of h(x)=xh(x)=\sqrt{x}h(x)=x​ for x≥0x \geq 0x≥0 and h(x)=−xh(x)=\sqrt{-x}h(x)=−x​ for x<0x<0x<0 as xxx approaches 000.

Problem 5207

Find lim⁡x→42f(x)g(x)\lim _{x \rightarrow 4} 2 f(x) g(x)limx→4​2f(x)g(x) if lim⁡x→4f(x)=5\lim _{x \rightarrow 4} f(x) = 5limx→4​f(x)=5 and lim⁡x→4g(x)=−3\lim _{x \rightarrow 4} g(x) = -3limx→4​g(x)=−3.

Problem 5208

Find the limit: lim⁡x→1x−1x+3−2\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{x+3}-2}limx→1​x+3​−2x−1​

Problem 5209

Find the limit: lim⁡x→44−x5−x2+9\lim _{x \rightarrow 4} \frac{4-x}{5-\sqrt{x^{2}+9}}limx→4​5−x2+9​4−x​

Problem 5210

Find the interval of convergence for the series 1+23x+49x2+…1+\frac{2}{3} x+\frac{4}{9} x^{2}+\ldots1+32​x+94​x2+…. Choices: (a) −32<x<32-\frac{3}{2}<x<\frac{3}{2}−23​<x<23​, (b) −3<x<3-3<x<3−3<x<3, (c) −9<x<9-9<x<9−9<x<9, (d) None.

Problem 5211

Rewrite the function f(x)=x(x−3)f(x)=x(x-3)f(x)=x(x−3) and find its derivative f′(x)f'(x)f′(x).

Problem 5212

Find the derivative of f(x)=5x3−x35f(x)=\frac{5}{x^{3}}-\frac{x^{3}}{5}f(x)=x35​−5x3​.

Problem 5213

Rewrite f(x)=(x+4)2f(x)=(x+4)^{2}f(x)=(x+4)2 and find its derivative f′(x)f'(x)f′(x).

Problem 5214

Forme den Funktionsterm f(x)=x4+1x2f(x)=\sqrt[4]{x}+\frac{1}{x^{2}}f(x)=4x​+x21​ um und leite ihn anschließend ab.

Problem 5215

Formen Sie die Funktion f(x)=x⋅(x+1x)f(x)=x \cdot\left(x+\frac{1}{x}\right)f(x)=x⋅(x+x1​) um und bestimmen Sie die Ableitung.

Problem 5216

Rewrite f(x)=2x−3x5f(x) = \frac{2}{\sqrt{x}} - \frac{3}{x^{5}}f(x)=x​2​−x53​ and find f′(x)f'(x)f′(x).

Problem 5217

Find the intervals where the function f(x)=3x3+9x2−72xf(x)=3 x^{3}+9 x^{2}-72 xf(x)=3x3+9x2−72x is increasing or decreasing. Use ∅\varnothing∅ for empty intervals.

Problem 5218

Bestimme die erste und zweite Ableitung der Funktionen: 1. f(x)=3xcos⁡(x)f(x)=3x \cos(x)f(x)=3xcos(x), 2. g(x)=(x2−3)sin⁡(x)g(x)=(x^{2}-3) \sin(x)g(x)=(x2−3)sin(x), 3. h(x)=xcos⁡(x)h(x)=\sqrt{x} \cos(x)h(x)=x​cos(x), 4. j(x)=(x−1)x4j(x)=(\sqrt{x}-1)x^{4}j(x)=(x​−1)x4.

Problem 5219

Find the limit lim⁡x→0−f(x)\lim _{x \rightarrow 0^{-}} f(x)limx→0−​f(x) where f(x)=⌊x+4⌋f(x)=\lfloor x+4\rfloorf(x)=⌊x+4⌋ for x<0x<0x<0, x2x^{2}x2 for 0≤x<70 \leq x<70≤x<7, and 2−x2-x2−x for x≥7x \geq 7x≥7. Options: a) 3 b) 4 c) 5 d) None.

Problem 5220

Find if the statement is true or false: lim⁡x→0.79−f(x)=0\lim _{x \rightarrow 0.79^{-}} f(x)=0limx→0.79−​f(x)=0. a) True b) False

Problem 5221

Überprüfen Sie die Integrationsregeln für k=2k=2k=2, f(x)=x2f(x)=x^{2}f(x)=x2 und g(x)=−x+1g(x)=-x+1g(x)=−x+1. Prüfen Sie (3) und (4).

Problem 5222

Determine if the statement is true or false: lim⁡x→3−f(x)=4 \lim _{x \rightarrow 3^{-}} f(x)=4 x→3−lim​f(x)=4 a) True b) False

Problem 5223

Find the intervals where revenue R(x)=214.956x−0.009x2R(x)=214.956 x-0.009 x^{2}R(x)=214.956x−0.009x2 is increasing or decreasing for 0≤x≤180000 \leq x \leq 180000≤x≤18000.

Problem 5224

Find the derivative of y=x(5−3x)y=x(5^{-3x})y=x(5−3x). What is dydx=?\frac{d y}{d x}=?dxdy​=?

Problem 5225

Find the derivative of P=t6ln⁡(t+6)P=t^{6} \ln(t+6)P=t6ln(t+6) with respect to ttt.

Problem 5226

Find the derivative of f(t)=te3+5t3f(t)=t e^{3+5 t^{3}}f(t)=te3+5t3 with respect to ttt.

Problem 5227

Find the intervals where the function f(x)=x2−6x+10f(x)=x^{2}-6 x+10f(x)=x2−6x+10 is increasing or decreasing. Use ∅\varnothing∅ for empty intervals.

Problem 5228

Find the derivative of f(z)=(6z+9)e6zf(z)=(\sqrt{6 z+9}) e^{6 z}f(z)=(6z+9​)e6z with respect to zzz.

Problem 5229

Find the total number of ants on the paraboloid surface z=x2+y2\mathrm{z}=x^{2}+y^{2}z=x2+y2 using the density f(x,y,z)=32(x2+y2)f(x, y, z)=32\left(x^{2}+y^{2}\right)f(x,y,z)=32(x2+y2). Use polar coordinates for integration.

Problem 5230

Find values of xxx for horizontal tangents of f(x)=2x2−12x+14f(x)=2x^{2}-12x+14f(x)=2x2−12x+14. Select "None" if none exist.

Problem 5231

Find where the function f(x)=−2x3−6x2+90xf(x)=-2 x^{3}-6 x^{2}+90 xf(x)=−2x3−6x2+90x is increasing or decreasing. Use ∅\varnothing∅ for empty intervals.

Problem 5232

Gegeben ist die Funktion f(x)=1x⋅sin⁡xf(x)=\frac{1}{x} \cdot \sin xf(x)=x1​⋅sinx. a) Vervollständige die Wertetabelle. b) Bestimme lim⁡x→−∞f(x)\lim _{x \rightarrow-\infty} f(x)limx→−∞​f(x) und lim⁡x→+∞f(x)\lim _{x \rightarrow+\infty} f(x)limx→+∞​f(x).

Problem 5233

Find dAdt\frac{d A}{d t}dtdA​ for a circle's area AAA with radius rrr as it expands, in terms of drdt\frac{d r}{d t}dtdr​.

Problem 5234

(a) Find dAdt\frac{d A}{d t}dtdA​ for a circle's area AAA with radius rrr expanding over time. (b) If the radius of an oil spill increases at 2 m/s2 \mathrm{~m/s}2 m/s, how fast is the area increasing when r=38 mr = 38 \mathrm{~m}r=38 m?

Problem 5235

Bestimme lim⁡x→±∞1x\lim _{x \rightarrow \pm \infty} \frac{1}{x}limx→±∞​x1​ und lim⁡x→±∞sin⁡x\lim _{x \rightarrow \pm \infty} \sin xlimx→±∞​sinx.

Problem 5236

How fast is the area of an oil spill increasing at a radius of 38 m if the radius grows at 2 m/s?

Problem 5237

(a) Find dAdt\frac{d A}{d t}dtdA​ for a circle area AAA with radius rrr expanding at rate drdt\frac{d r}{d t}dtdr​. (b) If the radius of an oil spill increases at 2 m/s2 \mathrm{~m/s}2 m/s, how fast is the area increasing when r=38 mr=38 \mathrm{~m}r=38 m? m2/s\mathrm{m}^{2}/\mathrm{s}m2/s

Problem 5238

Find the integral of 3x2\frac{3}{x^{2}}x23​ with respect to xxx.

Problem 5239

Bestimmen Sie die Funktion f(x)f(x)f(x) mit f′(0)=3f'(0)=3f′(0)=3, f(0)=0f(0)=0f(0)=0, Wendestelle bei x=2x=2x=2 mit Steigung -1.

Problem 5240

Find the derivative dydt\frac{d y}{d t}dtdy​ for y=(2.6t−3t2)(4t+1.4)y=(2.6 t-3 t^{2})(4 t+1.4)y=(2.6t−3t2)(4t+1.4).

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

Find the limit: $\lim _{y \rightarrow 2} \frac{y^{2}-y-2}{y^{2}-4 y+4}$ .

Find the limit of $g(x)=\frac{|x|}{x}$ as $x$ approaches 0 from the left, right, and directly.

Find the limit: $\lim _{x \rightarrow 25} \frac{\sqrt{x}-5}{x-25}$ .

Find the limit of $h(x)=\sqrt{x}$ for $x \geq 0$ and $h(x)=\sqrt{-x}$ for $x<0$ as $x$ approaches $0$ .

Find $\lim _{x \rightarrow 4} 2 f(x) g(x)$ if $\lim _{x \rightarrow 4} f(x) = 5$ and $\lim _{x \rightarrow 4} g(x) = -3$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{x+3}-2}$

Find the limit: $\lim _{x \rightarrow 4} \frac{4-x}{5-\sqrt{x^{2}+9}}$

Find the interval of convergence for the series $1+\frac{2}{3} x+\frac{4}{9} x^{2}+\ldots$ . Choices: (a) $-\frac{3}{2}<x<\frac{3}{2}$ , (b) $-3<x<3$ , (c) $-9<x<9$ , (d) None.

Rewrite the function $f(x)=x(x-3)$ and find its derivative $f'(x)$ .

Find the derivative of $f(x)=\frac{5}{x^{3}}-\frac{x^{3}}{5}$ .

Rewrite $f(x)=(x+4)^{2}$ and find its derivative $f'(x)$ .

Forme den Funktionsterm $f(x)=\sqrt[4]{x}+\frac{1}{x^{2}}$ um und leite ihn anschließend ab.

Formen Sie die Funktion $f(x)=x \cdot\left(x+\frac{1}{x}\right)$ um und bestimmen Sie die Ableitung.

Rewrite $f(x) = \frac{2}{\sqrt{x}} - \frac{3}{x^{5}}$ and find $f'(x)$ .

Find the intervals where the function $f(x)=3 x^{3}+9 x^{2}-72 x$ is increasing or decreasing. Use $\varnothing$ for empty intervals.

Bestimme die erste und zweite Ableitung der Funktionen: 1. $f(x)=3x \cos(x)$ , 2. $g(x)=(x^{2}-3) \sin(x)$ , 3. $h(x)=\sqrt{x} \cos(x)$ , 4. $j(x)=(\sqrt{x}-1)x^{4}$ .

Find the limit $\lim _{x \rightarrow 0^{-}} f(x)$ where $f(x)=\lfloor x+4\rfloor$ for $x<0$ , $x^{2}$ for $0 \leq x<7$ , and $2-x$ for $x \geq 7$ . Options: a) 3 b) 4 c) 5 d) None.

Find if the statement is true or false: $\lim _{x \rightarrow 0.79^{-}} f(x)=0$ . a) True b) False

Überprüfen Sie die Integrationsregeln für $k=2$ , $f(x)=x^{2}$ und $g(x)=-x+1$ . Prüfen Sie (3) und (4).

Determine if the statement is true or false:
$\lim _{x \rightarrow 3^{-}} f(x)=4$
a) True b) False

Find the intervals where revenue $R(x)=214.956 x-0.009 x^{2}$ is increasing or decreasing for $0 \leq x \leq 18000$ .

Find the derivative of $y=x(5^{-3x})$ . What is $\frac{d y}{d x}=?$

Find the derivative of $P=t^{6} \ln(t+6)$ with respect to $t$ .

Find the derivative of $f(t)=t e^{3+5 t^{3}}$ with respect to $t$ .

Find the intervals where the function $f(x)=x^{2}-6 x+10$ is increasing or decreasing. Use $\varnothing$ for empty intervals.

Find the derivative of $f(z)=(\sqrt{6 z+9}) e^{6 z}$ with respect to $z$ .

Find the total number of ants on the paraboloid surface $\mathrm{z}=x^{2}+y^{2}$ using the density $f(x, y, z)=32\left(x^{2}+y^{2}\right)$ . Use polar coordinates for integration.

Find values of $x$ for horizontal tangents of $f(x)=2x^{2}-12x+14$ . Select "None" if none exist.

Find where the function $f(x)=-2 x^{3}-6 x^{2}+90 x$ is increasing or decreasing. Use $\varnothing$ for empty intervals.

Gegeben ist die Funktion $f(x)=\frac{1}{x} \cdot \sin x$ . a) Vervollständige die Wertetabelle. b) Bestimme $\lim _{x \rightarrow-\infty} f(x)$ und $\lim _{x \rightarrow+\infty} f(x)$ .

Find $\frac{d A}{d t}$ for a circle's area $A$ with radius $r$ as it expands, in terms of $\frac{d r}{d t}$ .

(a) Find $\frac{d A}{d t}$ for a circle's area $A$ with radius $r$ expanding over time.
(b) If the radius of an oil spill increases at $2 \mathrm{~m/s}$ , how fast is the area increasing when $r = 38 \mathrm{~m}$ ?

Bestimme $\lim _{x \rightarrow \pm \infty} \frac{1}{x}$ und $\lim _{x \rightarrow \pm \infty} \sin x$ .

(a) Find $\frac{d A}{d t}$ for a circle area $A$ with radius $r$ expanding at rate $\frac{d r}{d t}$ . (b) If the radius of an oil spill increases at $2 \mathrm{~m/s}$ , how fast is the area increasing when $r=38 \mathrm{~m}$ ? $\mathrm{m}^{2}/\mathrm{s}$

Find the integral of $\frac{3}{x^{2}}$ with respect to $x$ .

Bestimmen Sie die Funktion $f(x)$ mit $f'(0)=3$ , $f(0)=0$ , Wendestelle bei $x=2$ mit Steigung -1.

Find the derivative $\frac{d y}{d t}$ for $y=(2.6 t-3 t^{2})(4 t+1.4)$ .

Find the derivative of the vector function $\mathbf{r}(t)=t^{3} \mathbf{i}-\sqrt{t} \mathbf{j}$ .

Find the square size $x$ (in inches) to cut from corners to maximize the volume $V(x)=x(14-2x)(12-2x)$ of a box.

Integrate: $\int \frac{\left(6 x^{2}-4\right) \ln \left(x^{3}-2 x\right)}{x^{3}-2 x} d x$

Given $4 x^{2}+16 y^{2}=100$ , find $\frac{d x}{d t}$ for (a) $x=3$ , $y=2$ , $\frac{d y}{d t}=\frac{1}{4}$ and (b) $\frac{d x}{d t}=4$ , $x=-3$ , $y=2$ .

Find the derivative $f^{\prime}(x)$ and the values of $x$ where $f^{\prime}(x)=0$ for $f(x)=\frac{x}{x^{2}+4}$ .

Gegeben ist die Funktion $f(x)=\frac{1}{9}(3 x+2)^{3}$ . Bestimme die Steigung in $P(2, f(2))$ , Punkte mit waagerechter Tangente und wo die Tangente die Steigung 1 hat.

Find the derivative of the vector function $\mathbf{q}(t)=\left\langle e^{t^{2}}, t \sin (2 t), \frac{t}{1-4 t^{3}}\right\rangle$ .

The AROC of $f(x)=3x+b$ over $[2,2+h]$ is positive for any $h>0$ and constant $b$ . True or False?

Find $f^{\prime}(x)$ , the tangent line at $x=0$ for $f(x)=3 e^{x^{2}-7 x+5}$ , and where the tangent is horizontal.

Find the derivative $g^{\prime}(x)$ for the function $g(x)=9 x e^{5 x}$ .

Find $\frac{d y}{d x}$ using the chain rule for $y=\sin u$ and $u=3 x-6$ .

Bestimme die Bedeutung von $f(t)=(60-2 t)^{3}$ und $f^{\prime}(2)$ für einen Würfel mit Kantenlängen, die sich verringern.

Ein Pkw beschleunigt. Skizziere den Geschwindigkeitsgraphen v, untersuche $v'$ und interpretiere $v(2)=3$ , $v'(12)=1$ .

Is it true or false that the AROC of $f(x)=P(0.2)^{x}$ on $[3,3+h]$ is positive for any $h>0$ ?

Bestimme $f(t)=(60-2t)^{3}$ und $f^{\prime}(2)$ für ein Würfelmodell mit Kantenlängen, die sich pro Minute verkürzen.

Bestimmen Sie die Stammfunktion von $f(x)=(x+3)(x+2)$ .

Find $\lim _{t \rightarrow 31} Y(t)$ where $Y(t)$ is the year at time $t$ days from Dec 1, 2022. Options: 2022, 2023, limit doesn't exist, not enough info.

Bestimmen Sie die Stammfunktion von $f(x)=\frac{x^{2}+2 x+1}{x}$ nach Vereinfachung des Ausdrucks.

Find values of $a$ and $b$ so that the function $f(x)=\begin{cases} x^{2}+4x+5 & \text{if } x \leq 3 \\ ax+b & \text{if } x>3 \end{cases}$ is differentiable everywhere. $a=$ , $b=$ .

Finde eine Stammfunktion von $f(x)=(3 x+4)^{3}$ und erkläre, warum es verschiedene Ergebnisse geben kann.

Calculate the integral of the function: $\int(3 x^{4}-6 x+8) \, dx$ .

Find the derivative of $y=6 \cot n \theta$ .

Find the slope of the tangent line to $y(b)$ at $b = 6$ , where $y(b) = b^{7} - b^{5} + 4b^{2}$ .

Calculate the integral of the function: $\int 2 x^{3} \, dx$ .

Find all $x$ values for horizontal tangents of the function $f(x)=3 x^{3}-18 x^{2}+36 x-25$ . Select "None" if none exist.

Find the derivative of $h(t)=(t^{4}-1)^{9}(t^{3}+1)^{8}$ .

Find the intervals where the function $f(x)=3 x^{3}-18 x^{2}+36 x-25$ is increasing or decreasing.

Compute the limit as $x$ approaches 0 for $\frac{3 x^{3}-4 x^{2}}{11 x^{2}-5 x^{3}}$ . Options: $-\frac{4}{11}$ or $\frac{3}{5}$ .

Find the tangent slope function for $j(x) = \frac{4 x^{5}}{5} - x^{4} + x^{2}$ . Choose from the given options.