Calculus

Problem 11301

Approximate $17^{1/4}$ using the linear approximation $L(x)$ of $f(x)=x^{1/4}$ at $a=16$ .

See Solution

Problem 11302

Estimate the temperature rise in a snake cage with $\frac{d T}{d t}=0.008^{\circ} \mathrm{C} / \mathrm{s}$ over $12 \mathrm{~s}$ . $\Delta T \approx$

See Solution

Problem 11303

Estimate the change in stopping distance $F(s)=1.1 s+0.05 \cdot 4 s^{2}$ for $s=50$ and $s=75$ mph using Linear Approximation. Give answers to two decimal places.

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Problem 11304

Estimate the change in stopping distance $\Delta F$ (in ft) per mph increase at speeds $s=50$ and $s=75$ using $F(s)=1.1s+0.054s^2$ .

See Solution

Problem 11305

Estimate $\ln 1.09$ using linearization $L(x)$ of $f(x)=\ln (x)$ at $a=1$ . Find $\ln(1.09)$ and percentage error.

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Problem 11306

Simplify the limit: $\lim _{x \rightarrow 0} \frac{\sin 5 x}{\sin 7 x}$ . Show all algebra steps.

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Problem 11307

Find the rate of change of the area $A=\pi r^{2}$ of a circular oil spill when $r=4 \mathrm{ft}$ .

See Solution

Problem 11308

Find the tangent line equation to the curve $h(t)=t^{3}-36t+4$ at the point (6,4).

See Solution

Problem 11309

Find the average speed of a watermelon falling $y=16 t^{2}$ ft in the first 5 sec and its speed at $t=5$ sec.

See Solution

Problem 11310

Simplify $\frac{f(2+h)-f(2)}{h}$ for the function $f(x) = \sqrt{16 - 3x}$ where $f(2)=3.16$ .

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Problem 11311

Find $\frac{d}{d x}(u v)$ at $x=1$ given $u(1)=3$ , $u'(1)=-5$ , $v(1)=6$ , $v'(1)=-4$ .

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Problem 11312

In a 75m building, elevator A moves down at 3 m/s and B up at 1 m/s. What's the distance change after 6 seconds?

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Problem 11313

In an 80m building, elevators A (down at 3 m/s) and B (up at 1 m/s) start moving. Find distance change after 6s.

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Problem 11314

Zeigen Sie, dass die Funktion $f(x)=e^{2x+1}$ umkehrbar ist.

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Problem 11315

A 20 ft ladder leans against a wall. If the base moves away at 8 ft/s, find the area change rate when it's 12 ft from the wall.

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Problem 11316

In a 40 m building, elevators A (down at $2 \frac{\mathrm{m}}{\mathrm{s}}$ ) and B (up at $1 \frac{\mathrm{m}}{\mathrm{s}}$ ) start moving. Find the distance change rate after 6 seconds. Answer to three decimal places.

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Problem 11317

Johanna's velocity data is given. Estimate $v^{\prime}(16)$ using the table. Also, find Bob's acceleration at $t=5$ for $B(t)=t^{3}-6t^{2}+300$ .

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Problem 11318

Solve the equation $\int_{5}^{k} \frac{x-2}{x-4} \mathrm{~d} x=k$ for the value of $k$ .

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Problem 11319

Berechne den Flächeninhalt unter $f$ von $a$ bis $b$ mit $\int_{a}^{b} f(x) d x=F(b)-F(a)$ . Bestimme: (1) $f(x)=2 x$ , $a=0.5$ , $b=3$ ; (2) $f(x)=3 x^{2}$ , $a=1$ , $b=2$ .

See Solution

Problem 11320

Berechnen Sie die folgenden Integrale und skizzieren Sie die Flächen: a) $\int_{1}^{2}(4 x^{3}) d x$ b) $\int_{-3}^{5}(2 x-4) d x$ c) $\int_{-1}^{1}(9 x^{2}-1) d x$ d) $\int_{0}^{\frac{\pi}{2}}(\sin (x)) d x$ e) $\int_{1}^{3}(\frac{1}{x^{2}}) d x$ f) $\int_{0}^{4}(x \cdot(x-1)) d x$ g) $\int_{-1}^{1}(x^{3}-x) d x$ h) $\int_{-2}^{0}(5 x^{4}-2) d x$ .

See Solution

Problem 11321

Calculate the integral: $\int_{0}^{\pi} \frac{1}{2}(\cos x + |\cos x|) \, dx$ .

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Problem 11322

Find the absolute extremum of $f(x)=1.07 x^{3}-11.17 x^{2}+23.71 x+2.13$ on $-1 \leq x \leq 8$ . Choose from (A) -27.495, (B) 5.653, (C) 16.427, (D) 24.770.

See Solution

Problem 11323

Find the number of inflection points of the polynomial $k$ if its rate of change is given by $R(x)=-3.261 x^{3}+5.362 x-1.584$ .

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Problem 11324

Ordnen Sie jeder Funktion f eine Stammfunktion F zu: f II $2x - 4$ , I $8x^{3} - 3$ , III $(x + 2)^{2}$ , IV $3(x^{2} - 2x^{3})$ , VI $\frac{x^{2} - 9}{x - 3}$ , V $2x - \frac{1}{x^{2}}$ , A $x^{3} - \frac{3}{2}x^{4} - 2$ , C $2x^{4} - 3x + 2$ , D $(x - 2)^{2}$ , E $2x^{2} + 4x + \frac{1}{3}x^{3} + C$ , B $\frac{1}{2}x^{2} + 3x$ .

See Solution

Problem 11325

1- Find vertical/horizontal asymptotes for: a. $f(x)=\frac{x^{2}+2}{x^{2}+1}$ b. $f(x)=\frac{5 x^{2}}{x^{2}-3 x-4}$ c. $f(t)=\frac{t^{2}+3 t-5}{t^{2}-5 t+6}$
2- Calculate limits: a. $\lim _{x \rightarrow 5}\left(\frac{x^{2}-3 x-10}{x-5}\right)$ b. $\lim _{x \rightarrow \infty}\left(\frac{5-x}{x^{4}-x^{3}}\right)$ c. $\lim _{x \rightarrow-\infty}\left(7 x^{2}+3 x^{3}+18000 x+6\right)$
3- Check continuity for: a. $f(x)=\frac{x^{2}-1}{x+1}$ at $x=-1$ b. $f(x)=x^{5}-x^{3}$ for all $x$
4- Determine continuity conditions for: $f(x)=\begin{cases}1-3 x & \text{if } x<4 \\ A x^{2}+2 x-3 & \text{if } x \geq 4\end{cases}$
5- Find limits: a. $\lim _{x \rightarrow \infty} \frac{2 x^{5}-4 x+10}{10 x^{5}+x-1}$ b. $\lim _{x \rightarrow 1} \frac{x^{2}+3 x+1}{4 x^{2}-9}$ c. $f(x)=\begin{cases}x^{2}+1 & \text{if } x \geq 3 \\ 2 x+4 & \text{if } x<3\end{cases}$ (limit at 3)

See Solution

Problem 11326

Find the derivative of the function $f(x) = x^{2} + 1$ .

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Problem 11327

Finde die Extremstellen der Gleichung $A_{R}^{\prime}(u)=0$ , umgeformt zu $-u^{-2}=0$ .

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Problem 11328

A 1 kg spring with $k=100$ is released at $0.1 \mathrm{~m}$ with no velocity. Graph position for $c: 10,15,20,25,30$ and identify damping types. What graphing tool are you using?

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Problem 11329

Calculate the left and right Riemann sums for $f(x)=\frac{2}{x}+2$ on $[1,5]$ with $n=4$ .

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Problem 11330

Déterminez où la fonction $f(x)=\frac{1}{6 x^{2}+5}$ est concave vers le haut ou vers le bas et trouvez les points d'inflexion.

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Problem 11331

Find the antiderivative of $x^{p}$ . For which $p$ does it hold? Choose the correct option from A, B, C, or D.

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Problem 11332

Calculate the integral $\int_{-5}^{5} x^{2} d x$ .

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Problem 11333

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

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Problem 11334

Calculate the integral of $(3x + 5)^2$ with respect to $x$ : $\int(3 x+5)^{2} d x=\square$

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Problem 11335

Find all antiderivatives of $f(x)=6 \cos x+9$ and verify by differentiating your result.

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Problem 11336

A car accelerates at $3 \mathrm{~m} / \mathrm{s}^{2}$ from $20 \mathrm{~m} / \mathrm{s}$ for 5 seconds. How far does it travel?

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Problem 11337

Evaluate the integral: $\int \frac{1}{\sqrt{x}(1+x)} dx$

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Problem 11338

Find the antiderivative $F$ of $f(t)=\csc^{2} t$ such that $F\left(\frac{\pi}{4}\right)=5$ . What is $F(t)=\square$ ?

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Problem 11339

Déterminez le domaine de $f(x)=\frac{6+6 x}{5-6 x}$ , les asymptotes, valeurs critiques et intervalles de concavité.

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Problem 11340

Solve the initial value problem: $g^{\prime}(x)=8 x\left(x^{7}-\frac{1}{8}\right)$ , with $g(1)=1$ .

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Problem 11341

Find the indefinite integral and verify by differentiating: $\int 6 \sqrt[10]{x} \, dx$

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Problem 11342

Find the max and min of $f(x)=\ln(x^{2}+x+1)$ on the interval $[-1,1]$ . Provide exact values.

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Problem 11343

Analyze the limits of $f(x)=x^{4}$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ . What do they approach?

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Problem 11344

Find the indefinite integral and verify by differentiation: $\int \frac{7+4 \cos y}{\sin ^{2} y} d y = \square$

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Problem 11345

Trouvez $x$ tel que la seconde dérivée de $f$ est nulle : $0=\frac{12 \cdot(6 x^{2}+5) \cdot(5-18 x^{2})}{(6 x^{2}+5)^{4}}$ .

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Problem 11346

Differentiate $y=\ln(\tan^{-1}(s))$ .

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Problem 11347

Differentiate $y=8 \sin^{-1}(3 \alpha)$ .

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Problem 11348

Differentiate $y=\csc^{-1}(e^{2x})$ .

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Problem 11349

Solve the initial value problem: $f'(u) = 12(\cos u - \sin u)$ with $f\left(\frac{\pi}{2}\right) = 7$ .

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Problem 11350

Find the average speed of a watermelon falling $y=16 t^{2}$ ft in the first $5$ sec and its speed at $t=5$ sec.

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Problem 11351

Find the rate of change of area $A=\pi r^{2}$ with respect to radius $r$ when $r=4 \mathrm{ft}$ .

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Problem 11352

Solve the initial value problem: $f'(u) = 8(\cos u - \sin u)$ with $f\left(\frac{\pi}{2}\right) = 2$ .

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Problem 11353

Find the position function for an object with acceleration $a(t)=-40$ , initial velocity $v(0)=22$ , and initial position $s(0)=0$ .

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Problem 11354

Determine the x-values $D$ , $E$ , and $F$ for inflection points of the function $f(x)=12 x^{5}+45 x^{4}-200 x^{3}+1$ .

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Problem 11355

Differentiate $y=\ln e^{\arctan (\alpha x^{2})} \arccos (e x^{2})$ .

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Problem 11356

Trouvez la dérivée de $f(x)=\frac{6+6 x}{5-6 x}$ .

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Problem 11357

Find the tangent line to $y=f(x)=\frac{5 \sin x}{2 \sin x+4 \cos x}$ at $a=\frac{\pi}{3}$ , $y=mx+b$ .

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Problem 11358

Find the position function from the acceleration $a(t)=0.4 t$ with initial velocity $v(0)=0$ and position $s(0)=7$ .

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Problem 11359

Differentiate $y=\ln e^{\arctan(\alpha x^{2})} \arccos(e x^{2})$ .

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Problem 11360

Find the object's velocity over time given the acceleration $a(t)=v^{\prime}(t)=g$ with $g=-9.8 \mathrm{~m} / \mathrm{s}^{2}$ .

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Problem 11361

Given $f\left(\frac{\pi}{6}\right)=-8$ and $f^{\prime}\left(\frac{\pi}{6}\right)=7$ , find $g^{\prime}(\pi / 6)$ and $h^{\prime}(\pi / 6)$ for $g(x)=f(x) \sin x$ and $h(x)=\frac{\cos x}{f(x)}$ .

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Problem 11362

Analyze the motion of a softball shot up with $34 \mathrm{~m/s}$ . Find its velocity, position, max height time, and ground strike time.

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Problem 11363

Find the derivative of $y=\sqrt{\sec \left(\frac{x}{2}\right)}$ .

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Problem 11364

Trouvez la dérivée de $f(x)=\frac{1-x e^{x}}{2+e^{x}}$ .

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Problem 11365

Find the tangent line equation for $y=x-x^{2}$ at $x=-2$ . Options: A) $y=5 x+4$ , B) $y=-3 x+4$ , C) $v=-3 x-4$ .

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Problem 11366

Untersuchen Sie die Funktionen $f(x)=x^{3}+3 x^{2}$ und $g(x)=-\frac{1}{2} x^{4}+2 x^{2}$ auf Symmetrie, Nullstellen, Extrema und Wendepunkte. Zeichnen Sie die Graphen und bestimmen Sie Schnittwinkel und Parameter.

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Problem 11367

Soit $f(x)=\frac{1}{6 x^{2}+5}$ . Trouvez les intervalles de concavité et les points d'inflexion de $f$ .

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Problem 11368

Find the tangent line for $f(x)=(2 x-2)^{1 / 5}$ at $x=2$ . Also, find where the tangent line is horizontal.

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Problem 11369

Find $g^{\prime}(x)$ if $g(x)=f\left(x^{2}\right)$ and $f^{\prime}(x)=\frac{x+4}{x-1}$ .

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Problem 11370

In einem Intervall I ist $f^{\prime \prime}(x) > 0$ . Was bedeutet das für die Graphen von $f'$ und $f$ ?

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Problem 11371

Bestimmen Sie die Kurvenrichtung und Sattelpunkte für die Funktionen: a) $f(x)=\frac{1}{3} x^{3}+x^{2}+x-2$ , b) $f(x)=-\frac{1}{6} x^{3}+x^{2}+3$ , c) $f(x)=x^{4}-2 x$ , d) $f(x)=\frac{1}{4} x^{4}-x^{3}-12 x^{2}+3 x+4$ .

See Solution

Problem 11372

Check if the integral $\int_{0}^{2} \frac{1}{(x-1)^{2}} d x$ converges or diverges.

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Problem 11373

Analyze the motion of an object in free fall with gravity $9.8 \mathrm{~m/s}^2$ :
a. Find velocity $v(t)$ , b. position $s(t)$ , c. time and height at peak, d. time to hit ground from $500 \mathrm{~m}$ at $13 \mathrm{~m/s}$ .

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Problem 11374

Find the derivative of the function $f(x)=\frac{6}{(4x^{2}+3x-5)^{2}}$ . What is $f^{\prime}(x)$ ?

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Problem 11375

Soit $f(x)=\frac{1}{6 x^{2}+5}$ . Trouvez les intervalles où $f$ est concave vers le haut/bas et les points d'inflexion.

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Problem 11376

Find the rate of change of the bacteria mass $P(t)=2+5 \tan^{-1}\left(\frac{t}{2}\right)$ when $P(t)=6$ grams.

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Problem 11377

Untersuchen Sie die Funktion $h(x)=\frac{2\left(x^{4}-16\right)}{x^{2}-4}$ bei $x=2$ und bestimmen Sie den Grenzwert.

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Problem 11378

Soit la fonction $f(x)=\frac{6+6 x}{5-6 x}$ .
a) Trouvez le domaine de $f$ . b) Déterminez les asymptotes horizontales et verticales de $f$ . c) Identifiez les valeurs critiques, les intervalles de croissance/décroissance, et les extrémums relatifs. d) Indiquez les intervalles de concavité et les points d'inflexion.

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Problem 11379

Sand falls into a conical pile with radius $r = 3h$ . If it falls at $120 \frac{ft^{3}}{min}$ , how fast is $h$ changing when $h=2$ ft?

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Problem 11380

Analyze the motion of a payload released from $400 \mathrm{~m}$ at $11 \mathrm{~m/s}$ . Find velocity, position, max height/time, and ground strike time.

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Problem 11381

Sand forms a cone with radius $r = 3h$ . If sand falls at $120 \frac{\mathrm{ft}^{3}}{\mathrm{min}}$ , find height change when $h = 2 \mathrm{ft}$ .

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Problem 11382

Bestimmen Sie die Tangenten- und Normalengleichung am Punkt $P(2 \mid-2)$ für $f(x)=-x^{2}+x$ .

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Problem 11383

Calculate the integrals: $\int_{-3}^{-2} (f(x)-g(x)) \, dx$ , $f(x)=\frac{1}{2} x^{2}$ , $g(x)=\frac{1}{2}(x^{3}+x^{2}-4x)$ .

See Solution

Problem 11384

If $F^{\prime}(x)=f(x)$ , is $\int f(x) d x=F(x)+C$ true or false? Explain your reasoning.

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Problem 11385

Find the linear approximation of $f(x)=\sqrt{x}$ at $x=0.16$ to estimate $\sqrt{0.18}$ .
Estimate the change in volume $V$ of a sphere when radius $r$ changes from $5 \mathrm{ft}$ to $5.1 \mathrm{ft}$ .

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Problem 11386

Are $f(x)=x^{3}+3$ and $g(x)=x^{3}-4$ derivatives of the same function? Explain your reasoning.

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Problem 11387

Bestimmen Sie die allgemeine Stammfunktion für: d) $f(x)=x^{3}-4 x+3$ , b) $g(x)=3 \sqrt{x}$ , ब) $h(x)=(x-1)(x-4)$ .

See Solution

Problem 11388

Find the derivative of $f(x)=(3x^{2}-6)(7x+6)$ and evaluate $f^{\prime}(4)$ .

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Problem 11389

Are $F(x)=x^{3}-4x+100$ and $G(x)=x^{3}-4x-100$ antiderivatives of the same function? Choose true/false options.

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Problem 11390

Find $y^{\prime}$ for $y=(x^{3}+3x+5)^{3}$ and the slope at $x=0.4$ , rounded to 1 decimal place.

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Problem 11391

Untersuchen Sie die Stetigkeit der Funktion $f$ an der Stelle $x_{0}=2$ , wo $f(x)=\frac{4 x^{2}-16}{2 x-4}$ für $x \neq 2$ und $f(2)=8$ .

See Solution

Problem 11392

If $F^{\prime}(x)=G^{\prime}(x)$ , does it imply $F(x)=G(x)$ ? Discuss true/false with examples.

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Problem 11393

Find the equation of the tangent line to $f(x)=\sqrt{23-x}$ at $(7,4)$ with slope $-\frac{1}{8}$ . What is $b$ ?

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Problem 11394

Find the slope of the tangent line to $p(x)=5.6 x^{1.8}$ at $x=1.2$ . Round your answer to 3 decimal places.

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Problem 11395

Find the area between $f(x)=2 \sin x+\sin (2 x)$ and the $x$ -axis from $x=0$ to $x=\pi$ .

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Problem 11396

Find the slope of the tangent line to $p(x)=5.6 x^{1.8}$ at the point $(2.3, p(2.3))$ to 3 decimal places.

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Problem 11397

Berechnen Sie die mittlere Temperatur für $f(x)=8,33+2,22 x-0,19 x^{2}$ über $[0, 12]$ und beschreiben Sie mögliche Bedingungen.

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Problem 11398

Schreibgeschwindigkeit eines Typisten: $v(t)=324-29t-4t^{2}$ ; a) Vergleiche $v(0)$ und $v(4)$ , b) mittlere Geschwindigkeit, c) Gesamtanschläge.

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Problem 11399

Berechne die durchschnittlichen täglichen Lagerkosten für $L(x)=\frac{4}{3}(x-30)^{2}$ und $K_{L}=0,35€$ .

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Problem 11400

Invest \$7000 at 5% interest compounded continuously. Find doubling time and sketch the growth graph.

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Calculus

Problem 11301

Approximate 171/417^{1/4}171/4 using the linear approximation L(x)L(x)L(x) of f(x)=x1/4f(x)=x^{1/4}f(x)=x1/4 at a=16a=16a=16.

Problem 11302

Estimate the temperature rise in a snake cage with dTdt=0.008∘C/s\frac{d T}{d t}=0.008^{\circ} \mathrm{C} / \mathrm{s}dtdT​=0.008∘C/s over 12 s12 \mathrm{~s}12 s. ΔT≈\Delta T \approxΔT≈

Problem 11303

Estimate the change in stopping distance F(s)=1.1s+0.05⋅4s2F(s)=1.1 s+0.05 \cdot 4 s^{2}F(s)=1.1s+0.05⋅4s2 for s=50s=50s=50 and s=75s=75s=75 mph using Linear Approximation. Give answers to two decimal places.

Problem 11304

Estimate the change in stopping distance ΔF\Delta FΔF (in ft) per mph increase at speeds s=50s=50s=50 and s=75s=75s=75 using F(s)=1.1s+0.054s2F(s)=1.1s+0.054s^2F(s)=1.1s+0.054s2.

Problem 11305

Estimate ln⁡1.09\ln 1.09ln1.09 using linearization L(x)L(x)L(x) of f(x)=ln⁡(x)f(x)=\ln (x)f(x)=ln(x) at a=1a=1a=1. Find ln⁡(1.09)\ln(1.09)ln(1.09) and percentage error.

Problem 11306

Simplify the limit: lim⁡x→0sin⁡5xsin⁡7x\lim _{x \rightarrow 0} \frac{\sin 5 x}{\sin 7 x}limx→0​sin7xsin5x​. Show all algebra steps.

Problem 11307

Find the rate of change of the area A=πr2A=\pi r^{2}A=πr2 of a circular oil spill when r=4ftr=4 \mathrm{ft}r=4ft.

Problem 11308

Find the tangent line equation to the curve h(t)=t3−36t+4h(t)=t^{3}-36t+4h(t)=t3−36t+4 at the point (6,4).

Problem 11309

Find the average speed of a watermelon falling y=16t2y=16 t^{2}y=16t2 ft in the first 5 sec and its speed at t=5t=5t=5 sec.

Problem 11310

Simplify f(2+h)−f(2)h\frac{f(2+h)-f(2)}{h}hf(2+h)−f(2)​ for the function f(x)=16−3xf(x) = \sqrt{16 - 3x}f(x)=16−3x​ where f(2)=3.16f(2)=3.16f(2)=3.16.

Problem 11311

Find ddx(uv)\frac{d}{d x}(u v)dxd​(uv) at x=1x=1x=1 given u(1)=3u(1)=3u(1)=3, u′(1)=−5u'(1)=-5u′(1)=−5, v(1)=6v(1)=6v(1)=6, v′(1)=−4v'(1)=-4v′(1)=−4.

Problem 11312

In a 75m building, elevator A moves down at 3 m/s and B up at 1 m/s. What's the distance change after 6 seconds?

Problem 11313

In an 80m building, elevators A (down at 3 m/s) and B (up at 1 m/s) start moving. Find distance change after 6s.

Problem 11314

Zeigen Sie, dass die Funktion f(x)=e2x+1f(x)=e^{2x+1}f(x)=e2x+1 umkehrbar ist.

Problem 11315

A 20 ft ladder leans against a wall. If the base moves away at 8 ft/s, find the area change rate when it's 12 ft from the wall.

Problem 11316

In a 40 m building, elevators A (down at 2ms2 \frac{\mathrm{m}}{\mathrm{s}}2sm​) and B (up at 1ms1 \frac{\mathrm{m}}{\mathrm{s}}1sm​) start moving. Find the distance change rate after 6 seconds. Answer to three decimal places.

Problem 11317

Johanna's velocity data is given. Estimate v′(16)v^{\prime}(16)v′(16) using the table. Also, find Bob's acceleration at t=5t=5t=5 for B(t)=t3−6t2+300B(t)=t^{3}-6t^{2}+300B(t)=t3−6t2+300.

Problem 11318

Solve the equation ∫5kx−2x−4 dx=k\int_{5}^{k} \frac{x-2}{x-4} \mathrm{~d} x=k∫5k​x−4x−2​ dx=k for the value of kkk.

Problem 11319

Berechne den Flächeninhalt unter fff von aaa bis bbb mit ∫abf(x)dx=F(b)−F(a)\int_{a}^{b} f(x) d x=F(b)-F(a)∫ab​f(x)dx=F(b)−F(a). Bestimme: (1) f(x)=2xf(x)=2 xf(x)=2x, a=0.5a=0.5a=0.5, b=3b=3b=3; (2) f(x)=3x2f(x)=3 x^{2}f(x)=3x2, a=1a=1a=1, b=2b=2b=2.

Problem 11320

Problem 11321

Calculate the integral: ∫0π12(cos⁡x+∣cos⁡x∣) dx\int_{0}^{\pi} \frac{1}{2}(\cos x + |\cos x|) \, dx∫0π​21​(cosx+∣cosx∣)dx.

Problem 11322

Find the absolute extremum of f(x)=1.07x3−11.17x2+23.71x+2.13f(x)=1.07 x^{3}-11.17 x^{2}+23.71 x+2.13f(x)=1.07x3−11.17x2+23.71x+2.13 on −1≤x≤8-1 \leq x \leq 8−1≤x≤8. Choose from (A) -27.495, (B) 5.653, (C) 16.427, (D) 24.770.

Problem 11323

Find the number of inflection points of the polynomial kkk if its rate of change is given by R(x)=−3.261x3+5.362x−1.584R(x)=-3.261 x^{3}+5.362 x-1.584R(x)=−3.261x3+5.362x−1.584.

Problem 11324

Problem 11325

Problem 11326

Find the derivative of the function f(x)=x2+1f(x) = x^{2} + 1f(x)=x2+1.

Problem 11327

Finde die Extremstellen der Gleichung AR′(u)=0A_{R}^{\prime}(u)=0AR′​(u)=0, umgeformt zu −u−2=0-u^{-2}=0−u−2=0.

Problem 11328

A 1 kg spring with k=100k=100k=100 is released at 0.1 m0.1 \mathrm{~m}0.1 m with no velocity. Graph position for c:10,15,20,25,30c: 10,15,20,25,30c:10,15,20,25,30 and identify damping types. What graphing tool are you using?

Problem 11329

Calculate the left and right Riemann sums for f(x)=2x+2f(x)=\frac{2}{x}+2f(x)=x2​+2 on [1,5][1,5][1,5] with n=4n=4n=4.

Problem 11330

Déterminez où la fonction f(x)=16x2+5f(x)=\frac{1}{6 x^{2}+5}f(x)=6x2+51​ est concave vers le haut ou vers le bas et trouvez les points d'inflexion.

Problem 11331

Find the antiderivative of xpx^{p}xp. For which ppp does it hold? Choose the correct option from A, B, C, or D.

Problem 11332

Calculate the integral ∫−55x2dx\int_{-5}^{5} x^{2} d x∫−55​x2dx.

Problem 11333

Find the derivative of f(x)=(x2−4)7f(x)=(x^{2}-4)^{7}f(x)=(x2−4)7.

Problem 11334

Calculate the integral of (3x+5)2(3x + 5)^2(3x+5)2 with respect to xxx: ∫(3x+5)2dx=□\int(3 x+5)^{2} d x=\square∫(3x+5)2dx=□

Problem 11335

Find all antiderivatives of f(x)=6cos⁡x+9f(x)=6 \cos x+9f(x)=6cosx+9 and verify by differentiating your result.

Problem 11336

A car accelerates at 3 m/s23 \mathrm{~m} / \mathrm{s}^{2}3 m/s2 from 20 m/s20 \mathrm{~m} / \mathrm{s}20 m/s for 5 seconds. How far does it travel?

Problem 11337

Evaluate the integral: ∫1x(1+x)dx\int \frac{1}{\sqrt{x}(1+x)} dx∫x​(1+x)1​dx

Problem 11338

Find the antiderivative FFF of f(t)=csc⁡2tf(t)=\csc^{2} tf(t)=csc2t such that F(π4)=5F\left(\frac{\pi}{4}\right)=5F(4π​)=5. What is F(t)=□F(t)=\squareF(t)=□?

Problem 11339

Déterminez le domaine de f(x)=6+6x5−6xf(x)=\frac{6+6 x}{5-6 x}f(x)=5−6x6+6x​, les asymptotes, valeurs critiques et intervalles de concavité.

Problem 11340

Solve the initial value problem: g′(x)=8x(x7−18)g^{\prime}(x)=8 x\left(x^{7}-\frac{1}{8}\right)g′(x)=8x(x7−81​), with g(1)=1g(1)=1g(1)=1.

Problem 11341

Find the indefinite integral and verify by differentiating: ∫6x10 dx\int 6 \sqrt[10]{x} \, dx∫610x​dx

Approximate $17^{1/4}$ using the linear approximation $L(x)$ of $f(x)=x^{1/4}$ at $a=16$ .

Estimate the temperature rise in a snake cage with $\frac{d T}{d t}=0.008^{\circ} \mathrm{C} / \mathrm{s}$ over $12 \mathrm{~s}$ . $\Delta T \approx$

Estimate the change in stopping distance $F(s)=1.1 s+0.05 \cdot 4 s^{2}$ for $s=50$ and $s=75$ mph using Linear Approximation. Give answers to two decimal places.

Estimate the change in stopping distance $\Delta F$ (in ft) per mph increase at speeds $s=50$ and $s=75$ using $F(s)=1.1s+0.054s^2$ .

Estimate $\ln 1.09$ using linearization $L(x)$ of $f(x)=\ln (x)$ at $a=1$ . Find $\ln(1.09)$ and percentage error.

Simplify the limit: $\lim _{x \rightarrow 0} \frac{\sin 5 x}{\sin 7 x}$ . Show all algebra steps.

Find the rate of change of the area $A=\pi r^{2}$ of a circular oil spill when $r=4 \mathrm{ft}$ .

Find the tangent line equation to the curve $h(t)=t^{3}-36t+4$ at the point (6,4).

Find the average speed of a watermelon falling $y=16 t^{2}$ ft in the first 5 sec and its speed at $t=5$ sec.

Simplify $\frac{f(2+h)-f(2)}{h}$ for the function $f(x) = \sqrt{16 - 3x}$ where $f(2)=3.16$ .

Find $\frac{d}{d x}(u v)$ at $x=1$ given $u(1)=3$ , $u'(1)=-5$ , $v(1)=6$ , $v'(1)=-4$ .

Zeigen Sie, dass die Funktion $f(x)=e^{2x+1}$ umkehrbar ist.

In a 40 m building, elevators A (down at $2 \frac{\mathrm{m}}{\mathrm{s}}$ ) and B (up at $1 \frac{\mathrm{m}}{\mathrm{s}}$ ) start moving. Find the distance change rate after 6 seconds. Answer to three decimal places.

Johanna's velocity data is given. Estimate $v^{\prime}(16)$ using the table. Also, find Bob's acceleration at $t=5$ for $B(t)=t^{3}-6t^{2}+300$ .

Solve the equation $\int_{5}^{k} \frac{x-2}{x-4} \mathrm{~d} x=k$ for the value of $k$ .

Berechne den Flächeninhalt unter $f$ von $a$ bis $b$ mit $\int_{a}^{b} f(x) d x=F(b)-F(a)$ . Bestimme: (1) $f(x)=2 x$ , $a=0.5$ , $b=3$ ; (2) $f(x)=3 x^{2}$ , $a=1$ , $b=2$ .

Calculate the integral: $\int_{0}^{\pi} \frac{1}{2}(\cos x + |\cos x|) \, dx$ .

Find the absolute extremum of $f(x)=1.07 x^{3}-11.17 x^{2}+23.71 x+2.13$ on $-1 \leq x \leq 8$ . Choose from (A) -27.495, (B) 5.653, (C) 16.427, (D) 24.770.

Find the number of inflection points of the polynomial $k$ if its rate of change is given by $R(x)=-3.261 x^{3}+5.362 x-1.584$ .

Find the derivative of the function $f(x) = x^{2} + 1$ .

Finde die Extremstellen der Gleichung $A_{R}^{\prime}(u)=0$ , umgeformt zu $-u^{-2}=0$ .

A 1 kg spring with $k=100$ is released at $0.1 \mathrm{~m}$ with no velocity. Graph position for $c: 10,15,20,25,30$ and identify damping types. What graphing tool are you using?

Calculate the left and right Riemann sums for $f(x)=\frac{2}{x}+2$ on $[1,5]$ with $n=4$ .

Déterminez où la fonction $f(x)=\frac{1}{6 x^{2}+5}$ est concave vers le haut ou vers le bas et trouvez les points d'inflexion.

Find the antiderivative of $x^{p}$ . For which $p$ does it hold? Choose the correct option from A, B, C, or D.

Calculate the integral $\int_{-5}^{5} x^{2} d x$ .

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

Calculate the integral of $(3x + 5)^2$ with respect to $x$ : $\int(3 x+5)^{2} d x=\square$

Find all antiderivatives of $f(x)=6 \cos x+9$ and verify by differentiating your result.

A car accelerates at $3 \mathrm{~m} / \mathrm{s}^{2}$ from $20 \mathrm{~m} / \mathrm{s}$ for 5 seconds. How far does it travel?

Evaluate the integral: $\int \frac{1}{\sqrt{x}(1+x)} dx$

Find the antiderivative $F$ of $f(t)=\csc^{2} t$ such that $F\left(\frac{\pi}{4}\right)=5$ . What is $F(t)=\square$ ?

Déterminez le domaine de $f(x)=\frac{6+6 x}{5-6 x}$ , les asymptotes, valeurs critiques et intervalles de concavité.

Solve the initial value problem: $g^{\prime}(x)=8 x\left(x^{7}-\frac{1}{8}\right)$ , with $g(1)=1$ .

Find the indefinite integral and verify by differentiating: $\int 6 \sqrt[10]{x} \, dx$

Find the max and min of $f(x)=\ln(x^{2}+x+1)$ on the interval $[-1,1]$ . Provide exact values.

Analyze the limits of $f(x)=x^{4}$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ . What do they approach?

Find the indefinite integral and verify by differentiation: $\int \frac{7+4 \cos y}{\sin ^{2} y} d y = \square$

Trouvez $x$ tel que la seconde dérivée de $f$ est nulle : $0=\frac{12 \cdot(6 x^{2}+5) \cdot(5-18 x^{2})}{(6 x^{2}+5)^{4}}$ .

Differentiate $y=\ln(\tan^{-1}(s))$ .

Differentiate $y=8 \sin^{-1}(3 \alpha)$ .

Differentiate $y=\csc^{-1}(e^{2x})$ .

Solve the initial value problem: $f'(u) = 12(\cos u - \sin u)$ with $f\left(\frac{\pi}{2}\right) = 7$ .

Find the average speed of a watermelon falling $y=16 t^{2}$ ft in the first $5$ sec and its speed at $t=5$ sec.

Find the rate of change of area $A=\pi r^{2}$ with respect to radius $r$ when $r=4 \mathrm{ft}$ .

Solve the initial value problem: $f'(u) = 8(\cos u - \sin u)$ with $f\left(\frac{\pi}{2}\right) = 2$ .

Find the position function for an object with acceleration $a(t)=-40$ , initial velocity $v(0)=22$ , and initial position $s(0)=0$ .

Determine the x-values $D$ , $E$ , and $F$ for inflection points of the function $f(x)=12 x^{5}+45 x^{4}-200 x^{3}+1$ .

Differentiate $y=\ln e^{\arctan (\alpha x^{2})} \arccos (e x^{2})$ .

Trouvez la dérivée de $f(x)=\frac{6+6 x}{5-6 x}$ .

Find the tangent line to $y=f(x)=\frac{5 \sin x}{2 \sin x+4 \cos x}$ at $a=\frac{\pi}{3}$ , $y=mx+b$ .

Find the position function from the acceleration $a(t)=0.4 t$ with initial velocity $v(0)=0$ and position $s(0)=7$ .

Differentiate $y=\ln e^{\arctan(\alpha x^{2})} \arccos(e x^{2})$ .

Find the object's velocity over time given the acceleration $a(t)=v^{\prime}(t)=g$ with $g=-9.8 \mathrm{~m} / \mathrm{s}^{2}$ .

Analyze the motion of a softball shot up with $34 \mathrm{~m/s}$ . Find its velocity, position, max height time, and ground strike time.

Find the derivative of $y=\sqrt{\sec \left(\frac{x}{2}\right)}$ .

Trouvez la dérivée de $f(x)=\frac{1-x e^{x}}{2+e^{x}}$ .

Find the tangent line equation for $y=x-x^{2}$ at $x=-2$ . Options: A) $y=5 x+4$ , B) $y=-3 x+4$ , C) $v=-3 x-4$ .

Untersuchen Sie die Funktionen $f(x)=x^{3}+3 x^{2}$ und $g(x)=-\frac{1}{2} x^{4}+2 x^{2}$ auf Symmetrie, Nullstellen, Extrema und Wendepunkte. Zeichnen Sie die Graphen und bestimmen Sie Schnittwinkel und Parameter.

Soit $f(x)=\frac{1}{6 x^{2}+5}$ . Trouvez les intervalles de concavité et les points d'inflexion de $f$ .

Find the tangent line for $f(x)=(2 x-2)^{1 / 5}$ at $x=2$ . Also, find where the tangent line is horizontal.

Find $g^{\prime}(x)$ if $g(x)=f\left(x^{2}\right)$ and $f^{\prime}(x)=\frac{x+4}{x-1}$ .

In einem Intervall I ist $f^{\prime \prime}(x) > 0$ . Was bedeutet das für die Graphen von $f'$ und $f$ ?

Check if the integral $\int_{0}^{2} \frac{1}{(x-1)^{2}} d x$ converges or diverges.

Analyze the motion of an object in free fall with gravity $9.8 \mathrm{~m/s}^2$ :
a. Find velocity $v(t)$ , b. position $s(t)$ , c. time and height at peak, d. time to hit ground from $500 \mathrm{~m}$ at $13 \mathrm{~m/s}$ .

Find the derivative of the function $f(x)=\frac{6}{(4x^{2}+3x-5)^{2}}$ . What is $f^{\prime}(x)$ ?