Calculus

Problem 15301

Find the general antiderivative of $f(x)=\frac{8}{x}$ for $x \neq 0$ . Use $C$ as the constant. $F(x)=$

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Problem 15302

Bestimmen Sie die Ableitung von $f$ an $x_{0}$ für: a) $f(x)=x^{2}$ , $x_{0}=3$ ; b) $f(x)=2 x^{2}$ , $x_{0}=1$ .

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Problem 15303

Find the second order partial derivative, $\frac{\partial^{2} f}{\partial x \partial y}$ , for $f=\sqrt{9 x^{4}+9 y^{3}}$ .

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Problem 15304

Find the antiderivatives: a. $f(x)=\sqrt{x}$ , b. $f(x)=x^{n}$ for $n \neq -1$ , c. $f(x)=x^{-1}$ for $x>0$ .

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Problem 15305

Justin has 120 m of fencing for a rectangular garden.
(a) Find the area function $A(x)$ in terms of $x$ .
(b) What side length $x$ maximizes the area?
(c) What is the maximum area?

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Problem 15306

Find the critical points of $f(x)=4 x^{3}-\frac{23}{2} x^{2}+5 x$ . Are they $x=\frac{1}{4}, \frac{5}{3}$ or none?

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Problem 15307

Find the critical points of $f(x)=3x^2+5x-4$ . Derivative: $f'(x)=6x+5$ . Critical points: A. $x=-\frac{5}{6}$ , B. none.

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Problem 15308

Find the critical points of the function $f(x)=\frac{x}{x^{2}+49}$ by determining its derivative.

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Problem 15309

Ashley has 120 m of fencing for three sides of a rectangle by a river.
(a) Find area function $A(x)$ in terms of $x$ .
(b) What $x$ maximizes the area?
(c) What is the maximum area?

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Problem 15310

Berechnen Sie den Flächeninhalt $A$ zwischen den Funktionen $f(x)=6-\frac{3}{\pi} x$ und $g(x)=3 \cdot \sin \left(\frac{x}{2}\right)$ an der $y$ -Achse.

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Problem 15311

Find critical points of $f(x)=x^{2} \sqrt{x+18}$ . Choices: A. $x=0,-14 \frac{2}{5}$ ; B. No critical points.

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Problem 15312

Calculate the integral from 3 to 8 of the function $-20 x^{2}+5 x^{8 / 3}$ .

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Problem 15313

A school uses 360 m of fencing for 3 sides of a playground.
(a) Find the area function $A(x)$ .
(b) What length $x$ maximizes the area?
(c) What is the maximum area?

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Problem 15314

Find the absolute extreme values of $f(x)=x^{2}-7$ on $[-3,4]$ . What are the maximum and minimum values?

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Problem 15315

Find the absolute extreme values of $f(x)=9 x^{\frac{2}{3}}-x$ on $[0,729]$ . What are the max and min?

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Problem 15316

Oblicz granicę dla ciągu $a_{n}=\frac{\sqrt{n^{2}+\sqrt{n+1}}-\sqrt{n^{2}-\sqrt{n+1}}}{\sqrt{n+1}-\sqrt{n}}$ . Jeśli granica nie istnieje, wyjaśnij dlaczego.

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Problem 15317

Evaluate the series $\sum_{n=3}^{\infty}(-2)^{n+1}(n)\left(5^{-n}\right)$ .

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Problem 15318

Find the minimum surface area of a box with volume $20 \mathrm{ft}^{3}$ , given $S(x)=x^{2}+\frac{80}{x}$ . What are the dimensions?

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Problem 15319

Find the minimum surface area of a box with volume $20 \mathrm{ft}^{3}$ , given $S(x)=x^{2}+\frac{80}{x}$ . What are the dimensions?

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Problem 15320

Find the minimum surface area of a box with volume $20 \mathrm{ft}^{3}$ given $S(x)=x^{2}+\frac{80}{x}$ . What are the dimensions?

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Problem 15321

A company sells $x$ wigits weekly; price-demand is $p(x)=20-x$ .
(a) Find the marginal revenue function. (b) Estimate additional revenue from producing 6 wigits instead of 5.

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Problem 15322

Bestimme die Tangentengleichung $t$ an $f$ im Punkt $P$ und wo sie die $x$ -Achse schneidet für die gegebenen Funktionen.

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Problem 15323

Find the linearization $L(x)$ of $f(x)=\sqrt{x+4}$ at $x=1$ and use it to approximate $\sqrt{4.9}$ . Is it an overestimate or underestimate?

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Problem 15324

A cylindrical can has volume $V$ (in $\mathrm{cm}^{3}$ ) and radius $r$ (in $\mathrm{cm}$ ).
a) Find surface area $S$ in terms of $r$ and $V$ .
b) Calculate $\frac{d S}{d V}$ for $r=10.7 \mathrm{~cm}$ and $V=2410 \mathrm{~cm}^{3}$ , rounded to 3 significant figures.
c) Find $\frac{d S}{d r}$ for $r=10.7 \mathrm{~cm}$ and $V=2410 \mathrm{~cm}^{3}$ , rounded to 3 significant figures.
d) Determine the change in radius needed to reduce surface area by $17 \mathrm{~cm}^{2}$ , keeping $V=2410 \mathrm{~cm}^{3}$ , using linear approximation.
e) Repeat (d) using the function $S$ directly.

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Problem 15325

Estimate the error in using the first six terms to approximate the series sum: $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{(0.01)^{n}}{n}$ .

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Problem 15326

Skizzieren Sie den Graphen von $h(x)=\frac{4}{x^{\frac{3}{2}}}$ und berechnen Sie den Flächeninhalt $A_{u}$ über $[0,25; u]$ . Ist $A_{u}$ endlich für $u \rightarrow \infty$ ?

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Problem 15327

Evaluate the series: $\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{(n+1)^{3}}$ .

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Problem 15328

Calculate the integral $I=\int_{0}^{5}\left|4 x-x^{2}\right| d x$ .

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Problem 15329

Estimate the sum of the series $\sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n^{2}+3}$ with an error < 0.001 using the alternating series estimation theorem.

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Problem 15330

Find the minimum surface area of a box with volume $20 \mathrm{ft}^{3}$ , given $S(x)=x^{2}+\frac{80}{x}$ . What are the dimensions?

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Problem 15331

Estimate the distance traveled by a braking car using a midpoint sum with 6 equal intervals from the velocity graph points: (0.5,42.75), (1.5,30), (2.5,21), (3.5,13.5), (4.5,7.5), (5.5,3).

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Problem 15332

A car's velocity after $t$ hours is $v(t)=12+4t-t^{2}$ . How many miles does it cover in the first 2 hours?

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Problem 15333

Find the additional profit for selling $x$ BabCo Lounge Chairs instead of 10, and the marginal profit at $x=10$ .

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Problem 15334

Find the $y$ -intercept of the curve through $(1,2)$ with slope $\frac{d y}{d x}=8 x-3$ .

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Problem 15335

Acme Office Supplies has a cost function for $x$ file cabinets: $C(x)=1020+50x-x^{2}$ .
(a) Find the cost difference between producing 8 and 7 cabinets. (b) Determine the marginal cost function. (c) Use the marginal cost to estimate the cost difference for 8 vs. 7 cabinets.

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Problem 15336

Find the growth rate of $L(t)=2.5 t+1.5 \cos \left(\frac{2 \pi t}{24}\right)$ and its max/min values.

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Problem 15337

Find the radius of convergence and values of $x$ for absolute and conditional convergence of $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x+12)^{n}}{n(12)^{n}}$ .

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Problem 15338

Find the radius of convergence and the values of $x$ for absolute and conditional convergence of: a. $\sum_{n=0}^{\infty}(-1)^{n}(6 x+8)^{n}$

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Problem 15339

Check if the series $\sum_{n=1}^{\infty} \frac{(n !)^{2} 5^{n}}{(2 n) !}$ converges using the ratio test or another method.

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Problem 15340

Calculate the growth rate $\frac{\mathrm{dL}}{\mathrm{dt}}$ for $L(t)=2.5 t+1.5 \cos \left(\frac{2 \pi t}{24}\right)$ . What are the max and min growth rates?

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Problem 15341

Calculate the indefinite integral and use $C$ for the constant of integration: $\int\left(6+\frac{2}{5} x^{2}+\frac{3}{8} x^{3}\right) d x$

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Problem 15342

Find Medtronic's marginal profit when producing and selling 1000 insulin pumps using the profit function $P(x)=2200 x-0.1 x^{2}-350$ .

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Problem 15343

Calculate the integral $\int_{2}^{4}(y^{2}-3y+5) \, dy$ .

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Problem 15344

Find the growth rate $\frac{dL}{dt}$ for $L(t)=2.5t+1.5\cos\left(\frac{2\pi t}{24}\right)$ . What are the max and min growth rates?

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Problem 15345

Find the limit: $\lim _{n \rightarrow \infty}\left(\frac{7 n+5}{7 n-3}\right)^{\sqrt{n}}$

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Problem 15346

Prüfen Sie, ob $\mathrm{F}$ eine Stammfunktion von $\mathrm{f}$ ist für die gegebenen Funktionen.

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Problem 15347

Determine if the series $\sum_{n=1}^{\infty} \frac{1 \cdot 4 \cdot 7 \cdots (3n-2)}{3 \cdot 5 \cdots (2n+1)}$ converges or diverges using the ratio test.

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Problem 15348

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin ^{2}(x)}{\ln (4 x+1)}$ .

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Problem 15349

Find the velocity of a $540 \mathrm{~g}$ grapefruit falling $0.930 \mathrm{~m}$ . Answer in $\mathrm{m/s}$ as a three-digit number.

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Problem 15350

Estimate $\int_{3}^{9} f(x) d x$ using 3 subintervals with right, left, and midpoint Riemann sums. Use values from $f(x)$ .

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Problem 15351

Find the limit: $\lim _{n \rightarrow+\infty}\left(\frac{7 n+5}{7 n-3}\right)^{\sqrt{n}}$ .

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Problem 15352

Analyze the series $\sum_{n=1}^{\infty} \frac{1 \cdot 4 \cdot 7 \cdots (3n-2)}{3 \cdot 5 \cdots (2n+1)}$ for convergence using the ratio test.

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Problem 15353

Find the general antiderivative of $f(\theta)=7 \sin (\theta)-5 \sec (\theta) \tan (\theta)$ and check by differentiation. Use $C$ for the constant. $F(\theta)=\square$

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Problem 15354

Calculate the integral: $\int \frac{t+1+t^{2}}{\sqrt{t}} \, dt$

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Problem 15355

Find the rectangle dimensions (in m) with area $343 \mathrm{~m}^{2}$ and minimal perimeter. Enter as x,y.

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Problem 15356

A tour guide's profit for $n$ people is $P(n)=n(46-0.5 n)-92$ . a. Find $n$ to maximize profit with a bus of 92. b. For a bus of 42, how many people maximize profit?

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Problem 15357

Calculate the limit of $a_{n}=\frac{1^{2}}{n^{3}}+\frac{3^{2}}{n^{3}}+\frac{5^{2}}{n^{3}}+\cdots+\frac{(2 n-1)^{2}}{n^{3}}$ .

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Problem 15358

Estimate $\int_{3}^{9} f(x) dx$ using right, left, and midpoint Riemann sums with 3 intervals. Function is increasing.

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Problem 15359

Solve the integral using integration by parts: $\int\left[4 x^{3} \ln (2 x)\right] d x$

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Problem 15360

Estimate the distance traveled by the car (in feet) using a midpoint sum with 6 equal intervals from the points: (0.5,42.75), (1.5,30), (2.5,21), (3.5,13.5), (4.5,7.5), (5.5,3).

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Problem 15361

Differentiate $y=\frac{\ln x}{x^{5}}$ and show your work.

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Problem 15362

A tour guide's profit is $P(n)=n(46-0.5 n)-92$ . Find $n$ for max profit with 92 and 42 max capacity buses. Derivative: $P^{\prime}(n)=-n+46$ .

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Problem 15363

Differentiate $y=x^{4} \ln x-\frac{1}{3} x^{3}$ . Show your work.

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Problem 15364

Find the derivative of the function $f(x)=e^{x^{3}+7 x}$ . What is $f^{\prime}(x)$ ?

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Problem 15365

Find the function $f(x)$ with $f''(x)=0$ , $f'(2)=-1$ , and $f(2)=-3$ . What is $f(x)$ ?

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Problem 15366

Find the derivative of $f(x)=\ln(e^{6x})$ . What is $f'(x)=\square$ ?

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Problem 15367

Explain why Rolle's Theorem doesn't apply to $f(x)=|x|$ on $[-a, a]$ for any $a>0$ . Choose the correct option.

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Problem 15368

Find the derivative of the function $f(x)=e^{6+e^{x}}$ .

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Problem 15369

Evaluate the integral $\int_{e}^{b} \frac{1}{x \ln (x)} d x$ .

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Problem 15370

Find the derivative of the function $f(x)=8 e x$ . What is $f^{\prime}(x)$ ?

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Problem 15371

Find the derivative of the function $f(x)=e^{x} \ln(x^{6})$ . What is $f'(x)$ ?

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Problem 15372

Differentiate $y=\ln \left[\frac{1-x}{(x+2)^{4}}\right]$ . Show your work.

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Problem 15373

Find the derivative of $f(z)=\frac{e^{z}}{z^{6}-5}$ .

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Problem 15374

Given the function $f(x)=\ln(x^{3}+40)$ , find: (a) $f^{\prime}(x)$ and (b) $f^{\prime}(2)$ .

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Problem 15375

Find the tangent line $y=m x+b$ for $y=x \cos (3 x)$ at $x=\pi$ . Calculate $m=$ and $b=$ .

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Problem 15376

Bestimme das unbestimmte Integral von $h(x)=(x-2)^{2}+2x$ .

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Problem 15377

Find $F^{\prime}(x)$ if $F(x)=f(g(x))$ , where $f^{\prime}(x)=\sqrt{3x+7}$ and $g(x)=x^{2}-3$ .

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Problem 15378

Find the absolute extremum of the function $f(x)=7 x^{3}-2$ .
Choose: A. The absolute minimum is $\square$ at $x=\square$ . B. There is no absolute minimum.

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Problem 15379

Find the extrema of the function $f(x)=\frac{x^{2}-1}{x^{2}+1}$ on the interval $[-1,3]$ .

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Problem 15380

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

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Problem 15381

Find the derivative of $f(x)=x^{3} \ln (x)-x^{3}$ and evaluate $f^{\prime}(e)$ .

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Problem 15382

Find the max and min values of $f(x)=x^{2}-8x-1$ on the interval $[2,8]$ . Give the values and corresponding $x$ .

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Problem 15383

Find the derivative $f^{\prime}(x)$ and evaluate $f^{\prime}(e)$ for $f(x)=x^{3} \ln (x)-x^{3}$ .

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Problem 15384

Find the tangent line equation for $y=4^{x^{2}-2 x+1}$ at $x=4$ . $y=$

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Problem 15385

Find the limit $\lim _{x \rightarrow 0} \frac{f(x)}{\tan (2 x)}$ given $\lim _{x \rightarrow 0} f(x)=0$ and $\lim _{x \rightarrow 0} f^{\prime}(x)=3$ .

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Problem 15386

Find the tangent line equation for $f(x)=x \ln (x-6)$ at $x=7$ .

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Problem 15387

Find the points $c$ where the Mean Value Theorem applies for $f(x)=x^{3}$ on the interval $[-2,2]$ .

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Problem 15388

Find the derivative of the function $f(x)=2\left(-4 x^{2}-4 x-3\right)^{3}$ .

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Problem 15389

Is the function $y=f(t)$ , representing the age of the oldest living person since 1950, continuous? Justify your answer.

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Problem 15390

Find the derivative of the function $f(x)=x-e^{-x}$ . What is $f^{\prime}(x)=\square$ ?

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Problem 15391

Given $f(x)$ with $f^{\prime \prime}(-1)=f^{\prime \prime}(0)=f^{\prime \prime}(1)=0$ , find the true statement about its inflection points.

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Problem 15392

Find the interval where the function $f$ is decreasing given its derivative $f'(x)=(x-3)(x+2)^{2}(x-5)^{3}$ .

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Problem 15393

Find the derivative of the function $f(x)=e^{\left[\left(x^{3}\right) / 3\right]}$ . What is $f^{\prime}(x)$ ?

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Problem 15394

Find the derivative of the function $f(x)=(-7 x-3)^{4}$ , denoted as $f^{\prime}(x)$ .

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Problem 15395

Determine if Rolle's Theorem applies to $g(x)=x^{3}+2x^{2}-15x-36$ on $[-3,4]$ and find guaranteed point(s).

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Problem 15396

Find the derivative of $f(x)=\ln (-x)$ . What is $f^{\prime}(x)=?$

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Problem 15397

Find the derivative of the function $y=\frac{5}{-5 x^{2}-8 x-9}$ with respect to $x$ .

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Problem 15398

Differentiate $y=\log_{8}\left(\frac{x^{2}}{\sqrt[8]{x+1}}\right)$ . Show your work.

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Problem 15399

Determine if the Mean Value Theorem applies to $f(x)=-3-x^{2}$ on $[-2,1]$ and find any guaranteed points.

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Problem 15400

Find the derivative $\frac{d y}{d x}$ of the function $y=\frac{5}{-5 x^{2}-8 x-9}$ .

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Calculus

Problem 15301

Find the general antiderivative of f(x)=8xf(x)=\frac{8}{x}f(x)=x8​ for x≠0x \neq 0x=0. Use CCC as the constant. F(x)=F(x)=F(x)=

Problem 15302

Bestimmen Sie die Ableitung von fff an x0x_{0}x0​ für: a) f(x)=x2f(x)=x^{2}f(x)=x2, x0=3x_{0}=3x0​=3; b) f(x)=2x2f(x)=2 x^{2}f(x)=2x2, x0=1x_{0}=1x0​=1.

Problem 15303

Find the second order partial derivative, ∂2f∂x∂y\frac{\partial^{2} f}{\partial x \partial y}∂x∂y∂2f​, for f=9x4+9y3f=\sqrt{9 x^{4}+9 y^{3}}f=9x4+9y3​.

Problem 15304

Find the antiderivatives: a. f(x)=xf(x)=\sqrt{x}f(x)=x​, b. f(x)=xnf(x)=x^{n}f(x)=xn for n≠−1n \neq -1n=−1, c. f(x)=x−1f(x)=x^{-1}f(x)=x−1 for x>0x>0x>0.

Problem 15305

Justin has 120 m of fencing for a rectangular garden. (a) Find the area function A(x)A(x)A(x) in terms of xxx.(b) What side length xxx maximizes the area?(c) What is the maximum area?

Problem 15306

Find the critical points of f(x)=4x3−232x2+5xf(x)=4 x^{3}-\frac{23}{2} x^{2}+5 xf(x)=4x3−223​x2+5x. Are they x=14,53x=\frac{1}{4}, \frac{5}{3}x=41​,35​ or none?

Problem 15307

Find the critical points of f(x)=3x2+5x−4f(x)=3x^2+5x-4f(x)=3x2+5x−4. Derivative: f′(x)=6x+5f'(x)=6x+5f′(x)=6x+5. Critical points: A. x=−56x=-\frac{5}{6}x=−65​, B. none.

Problem 15308

Find the critical points of the function f(x)=xx2+49f(x)=\frac{x}{x^{2}+49}f(x)=x2+49x​ by determining its derivative.

Problem 15309

Ashley has 120 m of fencing for three sides of a rectangle by a river.(a) Find area function A(x)A(x)A(x) in terms of xxx.(b) What xxx maximizes the area?(c) What is the maximum area?

Problem 15310

Berechnen Sie den Flächeninhalt AAA zwischen den Funktionen f(x)=6−3πxf(x)=6-\frac{3}{\pi} xf(x)=6−π3​x und g(x)=3⋅sin⁡(x2)g(x)=3 \cdot \sin \left(\frac{x}{2}\right)g(x)=3⋅sin(2x​) an der yyy-Achse.

Problem 15311

Find critical points of f(x)=x2x+18f(x)=x^{2} \sqrt{x+18}f(x)=x2x+18​. Choices: A. x=0,−1425x=0,-14 \frac{2}{5}x=0,−1452​; B. No critical points.

Problem 15312

Calculate the integral from 3 to 8 of the function −20x2+5x8/3-20 x^{2}+5 x^{8 / 3}−20x2+5x8/3.

Problem 15313

A school uses 360 m of fencing for 3 sides of a playground. (a) Find the area function A(x)A(x)A(x). (b) What length xxx maximizes the area? (c) What is the maximum area?

Problem 15314

Find the absolute extreme values of f(x)=x2−7f(x)=x^{2}-7f(x)=x2−7 on [−3,4][-3,4][−3,4]. What are the maximum and minimum values?

Problem 15315

Find the absolute extreme values of f(x)=9x23−xf(x)=9 x^{\frac{2}{3}}-xf(x)=9x32​−x on [0,729][0,729][0,729]. What are the max and min?

Problem 15316

Oblicz granicę dla ciągu an=n2+n+1−n2−n+1n+1−na_{n}=\frac{\sqrt{n^{2}+\sqrt{n+1}}-\sqrt{n^{2}-\sqrt{n+1}}}{\sqrt{n+1}-\sqrt{n}}an​=n+1​−n​n2+n+1​​−n2−n+1​​​. Jeśli granica nie istnieje, wyjaśnij dlaczego.

Problem 15317

Evaluate the series ∑n=3∞(−2)n+1(n)(5−n)\sum_{n=3}^{\infty}(-2)^{n+1}(n)\left(5^{-n}\right)∑n=3∞​(−2)n+1(n)(5−n).

Problem 15318

Find the minimum surface area of a box with volume 20ft320 \mathrm{ft}^{3}20ft3, given S(x)=x2+80xS(x)=x^{2}+\frac{80}{x}S(x)=x2+x80​. What are the dimensions?

Problem 15319

Find the minimum surface area of a box with volume 20ft320 \mathrm{ft}^{3}20ft3, given S(x)=x2+80xS(x)=x^{2}+\frac{80}{x}S(x)=x2+x80​. What are the dimensions?

Problem 15320

Find the minimum surface area of a box with volume 20ft320 \mathrm{ft}^{3}20ft3 given S(x)=x2+80xS(x)=x^{2}+\frac{80}{x}S(x)=x2+x80​. What are the dimensions?

Problem 15321

A company sells xxx wigits weekly; price-demand is p(x)=20−xp(x)=20-xp(x)=20−x. (a) Find the marginal revenue function. (b) Estimate additional revenue from producing 6 wigits instead of 5.

Problem 15322

Bestimme die Tangentengleichung ttt an fff im Punkt PPP und wo sie die xxx-Achse schneidet für die gegebenen Funktionen.

Problem 15323

Find the linearization L(x)L(x)L(x) of f(x)=x+4f(x)=\sqrt{x+4}f(x)=x+4​ at x=1x=1x=1 and use it to approximate 4.9\sqrt{4.9}4.9​. Is it an overestimate or underestimate?

Problem 15324

Problem 15325

Estimate the error in using the first six terms to approximate the series sum: ∑n=1∞(−1)n+1(0.01)nn\sum_{n=1}^{\infty}(-1)^{n+1} \frac{(0.01)^{n}}{n}n=1∑∞​(−1)n+1n(0.01)n​.

Problem 15326

Skizzieren Sie den Graphen von h(x)=4x32h(x)=\frac{4}{x^{\frac{3}{2}}}h(x)=x23​4​ und berechnen Sie den Flächeninhalt AuA_{u}Au​ über [0,25;u][0,25; u][0,25;u]. Ist AuA_{u}Au​ endlich für u→∞u \rightarrow \inftyu→∞?

Problem 15327

Evaluate the series: ∑n=1∞(−1)nn(n+1)3\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{(n+1)^{3}}∑n=1∞​(n+1)3(−1)nn​.

Problem 15328

Calculate the integral I=∫05∣4x−x2∣dxI=\int_{0}^{5}\left|4 x-x^{2}\right| d xI=∫05​∣∣​4x−x2∣∣​dx.

Problem 15329

Estimate the sum of the series ∑n=1∞(−1)n1n2+3\sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n^{2}+3}n=1∑∞​(−1)nn2+31​ with an error < 0.001 using the alternating series estimation theorem.

Problem 15330

Find the minimum surface area of a box with volume 20ft320 \mathrm{ft}^{3}20ft3, given S(x)=x2+80xS(x)=x^{2}+\frac{80}{x}S(x)=x2+x80​. What are the dimensions?

Problem 15331

Estimate the distance traveled by a braking car using a midpoint sum with 6 equal intervals from the velocity graph points: (0.5,42.75), (1.5,30), (2.5,21), (3.5,13.5), (4.5,7.5), (5.5,3).

Problem 15332

A car's velocity after ttt hours is v(t)=12+4t−t2v(t)=12+4t-t^{2}v(t)=12+4t−t2. How many miles does it cover in the first 2 hours?

Problem 15333

Find the additional profit for selling xxx BabCo Lounge Chairs instead of 10, and the marginal profit at x=10x=10x=10.

Problem 15334

Find the yyy-intercept of the curve through (1,2)(1,2)(1,2) with slope dydx=8x−3\frac{d y}{d x}=8 x-3dxdy​=8x−3.

Problem 15335

Problem 15336

Find the growth rate of L(t)=2.5t+1.5cos⁡(2πt24)L(t)=2.5 t+1.5 \cos \left(\frac{2 \pi t}{24}\right)L(t)=2.5t+1.5cos(242πt​) and its max/min values.

Problem 15337

Find the radius of convergence and values of xxx for absolute and conditional convergence of ∑n=1∞(−1)n+1(x+12)nn(12)n\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x+12)^{n}}{n(12)^{n}}∑n=1∞​n(12)n(−1)n+1(x+12)n​.

Problem 15338

Find the radius of convergence and the values of xxx for absolute and conditional convergence of: a. ∑n=0∞(−1)n(6x+8)n\sum_{n=0}^{\infty}(-1)^{n}(6 x+8)^{n}∑n=0∞​(−1)n(6x+8)n

Problem 15339

Check if the series ∑n=1∞(n!)25n(2n)!\sum_{n=1}^{\infty} \frac{(n !)^{2} 5^{n}}{(2 n) !}∑n=1∞​(2n)!(n!)25n​ converges using the ratio test or another method.

Problem 15340

Calculate the growth rate dLdt\frac{\mathrm{dL}}{\mathrm{dt}}dtdL​ for L(t)=2.5t+1.5cos⁡(2πt24)L(t)=2.5 t+1.5 \cos \left(\frac{2 \pi t}{24}\right)L(t)=2.5t+1.5cos(242πt​). What are the max and min growth rates?

Problem 15341

Find the general antiderivative of $f(x)=\frac{8}{x}$ for $x \neq 0$ . Use $C$ as the constant. $F(x)=$

Bestimmen Sie die Ableitung von $f$ an $x_{0}$ für: a) $f(x)=x^{2}$ , $x_{0}=3$ ; b) $f(x)=2 x^{2}$ , $x_{0}=1$ .

Find the second order partial derivative, $\frac{\partial^{2} f}{\partial x \partial y}$ , for $f=\sqrt{9 x^{4}+9 y^{3}}$ .

Find the antiderivatives: a. $f(x)=\sqrt{x}$ , b. $f(x)=x^{n}$ for $n \neq -1$ , c. $f(x)=x^{-1}$ for $x>0$ .

Justin has 120 m of fencing for a rectangular garden.
(a) Find the area function $A(x)$ in terms of $x$ .
(b) What side length $x$ maximizes the area?
(c) What is the maximum area?

Find the critical points of $f(x)=4 x^{3}-\frac{23}{2} x^{2}+5 x$ . Are they $x=\frac{1}{4}, \frac{5}{3}$ or none?

Find the critical points of $f(x)=3x^2+5x-4$ . Derivative: $f'(x)=6x+5$ . Critical points: A. $x=-\frac{5}{6}$ , B. none.

Find the critical points of the function $f(x)=\frac{x}{x^{2}+49}$ by determining its derivative.

Ashley has 120 m of fencing for three sides of a rectangle by a river.
(a) Find area function $A(x)$ in terms of $x$ .
(b) What $x$ maximizes the area?
(c) What is the maximum area?

Berechnen Sie den Flächeninhalt $A$ zwischen den Funktionen $f(x)=6-\frac{3}{\pi} x$ und $g(x)=3 \cdot \sin \left(\frac{x}{2}\right)$ an der $y$ -Achse.

Find critical points of $f(x)=x^{2} \sqrt{x+18}$ . Choices: A. $x=0,-14 \frac{2}{5}$ ; B. No critical points.

Calculate the integral from 3 to 8 of the function $-20 x^{2}+5 x^{8 / 3}$ .

A school uses 360 m of fencing for 3 sides of a playground.
(a) Find the area function $A(x)$ .
(b) What length $x$ maximizes the area?
(c) What is the maximum area?

Find the absolute extreme values of $f(x)=x^{2}-7$ on $[-3,4]$ . What are the maximum and minimum values?

Find the absolute extreme values of $f(x)=9 x^{\frac{2}{3}}-x$ on $[0,729]$ . What are the max and min?

Oblicz granicę dla ciągu $a_{n}=\frac{\sqrt{n^{2}+\sqrt{n+1}}-\sqrt{n^{2}-\sqrt{n+1}}}{\sqrt{n+1}-\sqrt{n}}$ . Jeśli granica nie istnieje, wyjaśnij dlaczego.

Evaluate the series $\sum_{n=3}^{\infty}(-2)^{n+1}(n)\left(5^{-n}\right)$ .

Find the minimum surface area of a box with volume $20 \mathrm{ft}^{3}$ , given $S(x)=x^{2}+\frac{80}{x}$ . What are the dimensions?

Find the minimum surface area of a box with volume $20 \mathrm{ft}^{3}$ , given $S(x)=x^{2}+\frac{80}{x}$ . What are the dimensions?

Find the minimum surface area of a box with volume $20 \mathrm{ft}^{3}$ given $S(x)=x^{2}+\frac{80}{x}$ . What are the dimensions?

A company sells $x$ wigits weekly; price-demand is $p(x)=20-x$ .
(a) Find the marginal revenue function. (b) Estimate additional revenue from producing 6 wigits instead of 5.

Bestimme die Tangentengleichung $t$ an $f$ im Punkt $P$ und wo sie die $x$ -Achse schneidet für die gegebenen Funktionen.

Find the linearization $L(x)$ of $f(x)=\sqrt{x+4}$ at $x=1$ and use it to approximate $\sqrt{4.9}$ . Is it an overestimate or underestimate?

Estimate the error in using the first six terms to approximate the series sum: $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{(0.01)^{n}}{n}$ .

Skizzieren Sie den Graphen von $h(x)=\frac{4}{x^{\frac{3}{2}}}$ und berechnen Sie den Flächeninhalt $A_{u}$ über $[0,25; u]$ . Ist $A_{u}$ endlich für $u \rightarrow \infty$ ?

Evaluate the series: $\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{(n+1)^{3}}$ .

Calculate the integral $I=\int_{0}^{5}\left|4 x-x^{2}\right| d x$ .

Estimate the sum of the series $\sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n^{2}+3}$ with an error < 0.001 using the alternating series estimation theorem.

Find the minimum surface area of a box with volume $20 \mathrm{ft}^{3}$ , given $S(x)=x^{2}+\frac{80}{x}$ . What are the dimensions?

A car's velocity after $t$ hours is $v(t)=12+4t-t^{2}$ . How many miles does it cover in the first 2 hours?

Find the additional profit for selling $x$ BabCo Lounge Chairs instead of 10, and the marginal profit at $x=10$ .

Find the $y$ -intercept of the curve through $(1,2)$ with slope $\frac{d y}{d x}=8 x-3$ .

Find the growth rate of $L(t)=2.5 t+1.5 \cos \left(\frac{2 \pi t}{24}\right)$ and its max/min values.

Find the radius of convergence and values of $x$ for absolute and conditional convergence of $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x+12)^{n}}{n(12)^{n}}$ .

Find the radius of convergence and the values of $x$ for absolute and conditional convergence of: a. $\sum_{n=0}^{\infty}(-1)^{n}(6 x+8)^{n}$

Check if the series $\sum_{n=1}^{\infty} \frac{(n !)^{2} 5^{n}}{(2 n) !}$ converges using the ratio test or another method.

Calculate the growth rate $\frac{\mathrm{dL}}{\mathrm{dt}}$ for $L(t)=2.5 t+1.5 \cos \left(\frac{2 \pi t}{24}\right)$ . What are the max and min growth rates?

Calculate the indefinite integral and use $C$ for the constant of integration: $\int\left(6+\frac{2}{5} x^{2}+\frac{3}{8} x^{3}\right) d x$

Find Medtronic's marginal profit when producing and selling 1000 insulin pumps using the profit function $P(x)=2200 x-0.1 x^{2}-350$ .

Calculate the integral $\int_{2}^{4}(y^{2}-3y+5) \, dy$ .

Find the growth rate $\frac{dL}{dt}$ for $L(t)=2.5t+1.5\cos\left(\frac{2\pi t}{24}\right)$ . What are the max and min growth rates?

Find the limit: $\lim _{n \rightarrow \infty}\left(\frac{7 n+5}{7 n-3}\right)^{\sqrt{n}}$

Prüfen Sie, ob $\mathrm{F}$ eine Stammfunktion von $\mathrm{f}$ ist für die gegebenen Funktionen.

Determine if the series $\sum_{n=1}^{\infty} \frac{1 \cdot 4 \cdot 7 \cdots (3n-2)}{3 \cdot 5 \cdots (2n+1)}$ converges or diverges using the ratio test.

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin ^{2}(x)}{\ln (4 x+1)}$ .

Find the velocity of a $540 \mathrm{~g}$ grapefruit falling $0.930 \mathrm{~m}$ . Answer in $\mathrm{m/s}$ as a three-digit number.

Estimate $\int_{3}^{9} f(x) d x$ using 3 subintervals with right, left, and midpoint Riemann sums. Use values from $f(x)$ .

Find the limit: $\lim _{n \rightarrow+\infty}\left(\frac{7 n+5}{7 n-3}\right)^{\sqrt{n}}$ .

Analyze the series $\sum_{n=1}^{\infty} \frac{1 \cdot 4 \cdot 7 \cdots (3n-2)}{3 \cdot 5 \cdots (2n+1)}$ for convergence using the ratio test.

Find the general antiderivative of $f(\theta)=7 \sin (\theta)-5 \sec (\theta) \tan (\theta)$ and check by differentiation. Use $C$ for the constant. $F(\theta)=\square$

Calculate the integral: $\int \frac{t+1+t^{2}}{\sqrt{t}} \, dt$

Find the rectangle dimensions (in m) with area $343 \mathrm{~m}^{2}$ and minimal perimeter. Enter as x,y.

A tour guide's profit for $n$ people is $P(n)=n(46-0.5 n)-92$ . a. Find $n$ to maximize profit with a bus of 92. b. For a bus of 42, how many people maximize profit?

Calculate the limit of $a_{n}=\frac{1^{2}}{n^{3}}+\frac{3^{2}}{n^{3}}+\frac{5^{2}}{n^{3}}+\cdots+\frac{(2 n-1)^{2}}{n^{3}}$ .

Estimate $\int_{3}^{9} f(x) dx$ using right, left, and midpoint Riemann sums with 3 intervals. Function is increasing.

Solve the integral using integration by parts: $\int\left[4 x^{3} \ln (2 x)\right] d x$

Differentiate $y=\frac{\ln x}{x^{5}}$ and show your work.

A tour guide's profit is $P(n)=n(46-0.5 n)-92$ . Find $n$ for max profit with 92 and 42 max capacity buses. Derivative: $P^{\prime}(n)=-n+46$ .

Differentiate $y=x^{4} \ln x-\frac{1}{3} x^{3}$ . Show your work.

Find the derivative of the function $f(x)=e^{x^{3}+7 x}$ . What is $f^{\prime}(x)$ ?

Find the function $f(x)$ with $f''(x)=0$ , $f'(2)=-1$ , and $f(2)=-3$ . What is $f(x)$ ?

Find the derivative of $f(x)=\ln(e^{6x})$ . What is $f'(x)=\square$ ?

Explain why Rolle's Theorem doesn't apply to $f(x)=|x|$ on $[-a, a]$ for any $a>0$ . Choose the correct option.

Find the derivative of the function $f(x)=e^{6+e^{x}}$ .

Evaluate the integral $\int_{e}^{b} \frac{1}{x \ln (x)} d x$ .

Find the derivative of the function $f(x)=8 e x$ . What is $f^{\prime}(x)$ ?

Find the derivative of the function $f(x)=e^{x} \ln(x^{6})$ . What is $f'(x)$ ?

Differentiate $y=\ln \left[\frac{1-x}{(x+2)^{4}}\right]$ . Show your work.

Find the derivative of $f(z)=\frac{e^{z}}{z^{6}-5}$ .

Given the function $f(x)=\ln(x^{3}+40)$ , find: (a) $f^{\prime}(x)$ and (b) $f^{\prime}(2)$ .