Calculus

Problem 21301

Bestimmen Sie die Grenzwerte $\lim_{x \to +\infty} f(x)$ und $\lim_{x \to -\infty} f(x)$ für die Funktionen: a) $f(x)=\frac{4}{x}+2$ , b) $f(x)=3-\frac{4}{x+1}$ , c) $f(x)=\frac{1}{x-2}+2,5$ .

See Solution

Problem 21302

Bestimmen Sie die Grenzwerte von $f(x)$ für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ für die Funktionen: a) $f(x)=\frac{4}{x}+2$ , b) $f(x)=3-\frac{4}{x+1}$ , c) $f(x)=\frac{1}{x-2}+2,5$ .

See Solution

Problem 21303

Halla la constante de integración para $\int(x+1) d x=g_{(x)}$ sabiendo que $g_{(2)}=6$ .

See Solution

Problem 21304

Calculate the time for $\mathrm{NH}_{3}$ at 0.580 M to decrease by 95% using the rate law: rate = (3.3 M $^{-1}$ s $^{-1}$ )[ $\mathrm{NH}_{3}$ ] $^{2}$ . Round to 2 significant digits.

See Solution

Problem 21305

Find the intervals where the function $f(x)=2 x^{3}-3 x^{2}-12 x+20$ is increasing.

See Solution

Problem 21306

Find the series representation for the function $f(x)=\csc x$ .

See Solution

Problem 21307

Find $\lim _{x \rightarrow 0} f(x)$ given $\frac{x^{2}-x^{4}}{x^{2}} \leq f(x) \leq \frac{1}{4} x^{3}+1$ for $-2 \leq x \leq 2$ .

See Solution

Problem 21308

Bestimme die Koordinaten und die Steigung der interessanten Punkte des Graphen $f(x)=\frac{1}{3} x^{3}-2 x^{2}+9$ .

See Solution

Problem 21309

Find horizontal asymptotes of $f(x)=\frac{3(3 x+1)(x-7)}{x(3 x+1)}$ .

See Solution

Problem 21310

Determine the horizontal asymptotes of the function $f(x)=\frac{3(3 x+1)(x-7)}{x(3 x+1)}$ .

See Solution

Problem 21311

Find upper and lower bounds for $\int_{0}^{1} \frac{1}{1+x^{2}} d x$ using the Max-Min Inequality.

See Solution

Problem 21312

Show that $\int_{0}^{1} \sin \left(x^{2}\right) d x$ cannot equal 2. Which statement is true? A, B, C, or D?

See Solution

Problem 21313

Bestimme die Hoch- und Tiefpunkte der Funktion $f(x)=\frac{1}{16} x^{2}(x^{2}-24)$ .

See Solution

Problem 21314

Calculate the integral $\int_{-4}^{-2}-0.5 x \, dx$ .

See Solution

Problem 21315

Find $\int_{4}^{4} f(x) d x$ and $\int_{7}^{2} g(x) d x$ given $\int_{2}^{4} f(x) dx=6$ , $\int_{2}^{7} f(x) dx=-7$ , and $\int_{2}^{7} g(x) dx=7$ .

See Solution

Problem 21316

Find $\int_{4}^{4} f(x) d x$ given $\int_{2}^{4} f(x) dx=6$ and $\int_{2}^{7} f(x) dx=-7$ .

See Solution

Problem 21317

What is the relationship between an antiderivative $F$ of $f$ and the area function $A$ of $f$ ? Choose the correct answer.

See Solution

Problem 21318

Explain how the Fundamental Theorem of Calculus helps evaluate $\int_{a}^{b} f(x) dx$ using antiderivatives.

See Solution

Problem 21319

What is the relationship between the antiderivative $F$ of $f$ and the area function $A$ of $f$ ? Choose the correct option.

See Solution

Problem 21320

Find the number of units for max profit from $R(x)=211.2 x-0.042 x^{2}$ and $C(x)=10,000+105.6 x-0.084 x^{2}+0.00002 x^{3}$ . What is the max profit?

See Solution

Problem 21321

Avery swings a sling of length $0.698 \mathrm{~m}$ at 8.34 rev/s. Find acceleration in $\mathrm{m/s^2}$ . Then, for $0.979 \mathrm{~m}$ at 5.23 rev/s.

See Solution

Problem 21322

A wooden artifact has 35% of carbon-14 of living trees. How many years ago was it made? (Half-life: 5730 years)

See Solution

Problem 21323

A satellite orbits Earth at $289 \mathrm{~km}$ . Find its orbital radius, speed, and period in seconds and hours. Use: $\mathrm{G}=6.674 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}$ , $M_{\text{Earth}}=5.974 \times 10^{24} \mathrm{~kg}$ , $R_{\text{Earth}}=6.378 \times 10^{6} \mathrm{~m}$ .

See Solution

Problem 21324

Use $u$ -substitution to solve $\int_{3}^{4}(4x-5)^{3} dx$ and find $u$ , $du$ , $a$ , $b$ , and $f(u)$ .

See Solution

Problem 21325

Calculate the sum: $\sum_{k=0}^{5} \frac{k-1}{k !}$ .

See Solution

Problem 21326

Find the minimum marginal cost of the cost function $C(x)=5 x^{3}-9 x^{2}+7 x+2$ . Minimum marginal cost is \$ \square.

See Solution

Problem 21327

Estimate the integral $\int_{3}^{9} f(x) dx$ using 3 subintervals with right, left, and midpoint methods. Results: $R_{3}=1.4$ , $L_{3}=-1.9$ , $M_{3}=-0.3$ .

See Solution

Problem 21328

Evaluate the integrals using given values: a) $\int_{0}^{2}(f(x)+g(x)) d x$ b) $\int_{0}^{3}(f(x)-g(x)) d x$ c) $\int_{2}^{3}(3 f(x)+2 g(x)) d x$ d) Find $a$ such that $\int_{0}^{3}(a f(x)+g(x)) d x=0$ . $a=-\frac{1}{2}$

See Solution

Problem 21329

Differentiate $6 x^{\frac{3}{4}}+\frac{5}{x^{3}}-9$ with respect to $x$ .

See Solution

Problem 21330

Find the value of the series $\sum_{k=1}^{\infty} \frac{2}{(k+2) k}$ .

See Solution

Problem 21331

A wooden artifact has 40% of its original carbon-14. How many years ago was it made? (Half-life of carbon-14 is 5730 years.)

See Solution

Problem 21332

Bestimmen Sie $f^{\prime}(x)$ für die Funktionen: a) $f(x)=\frac{2}{3} x^{3}-\frac{3}{4} x^{2}+4$ , b) $f(x)=\frac{2}{x^{2}}+x^{2}$ , c) $f(x)=\sqrt[3]{x}$ , d) $f(x)=2 x+2 \sin (x)$ , e) $f(x)=\frac{2}{\sqrt{x}}+\frac{\sqrt{x}}{2}$ , f) $f(x)=x^{\frac{3}{4}}$ .

See Solution

Problem 21333

Sketch the graph of the function using its first and second derivatives: $f(x)=\frac{1}{3} x^{3}-9 x+3$ .

See Solution

Problem 21334

Find $\frac{d x}{d t}$ , $\frac{d y}{d t}$ , and $\frac{d y}{d x}$ for $x=5 t^{3}+3 t$ , $y=4 t-6 t^{2}$ .

See Solution

Problem 21335

Find $\frac{d x}{d t}$ , $\frac{d y}{d t}$ , and $\frac{d y}{d x}$ for $x=4 t e^{t}$ , $y=6 t+\sin(t)$ .

See Solution

Problem 21336

Evaluate the integral from 0 to 1: $\int_{0}^{1}\left(9 x^{e}+2 e^{x}\right) d x$

See Solution

Problem 21337

Find the derivative of the function $g(s)=\int_{9}^{s}(t-t^{9})^{7} dt$ . What is $g'(s)$ ?

See Solution

Problem 21338

Find $\frac{d x}{d t}$ , $\frac{d y}{d t}$ , and $\frac{d y}{d x}$ for $x=9 t-9 \ln (t)$ , $y=4 t^{2}-4 t^{-2}$ .

See Solution

Problem 21339

Determine if the integral $\int_{0}^{\infty} \frac{x}{(x^{2}+14)^{2}} dx$ converges or diverges, and evaluate if convergent.

See Solution

Problem 21340

Find the slope of the tangent to the curve given by $x=t^{2}+2t$ and $y=2^{t}-2t$ at the specified point.

See Solution

Problem 21341

Evaluate the integral $\int_{-5}^{5} f(x) \, dx$ where $f(x)=5$ for $-5 \leq x \leq 0$ and $f(x)=25-x^{2}$ for $0<x \leq 5$ .

See Solution

Problem 21342

Find $\frac{d y}{d x}$ for $y=4 \sqrt{x}+\frac{45}{x}$ at $x=9$ .

See Solution

Problem 21343

Find the tangent line equation to the curve at $t=-1$ : $x=t^{7}+1$ , $y=t^{8}+t$ .

See Solution

Problem 21344

Trouver $H^{\prime}(-1)$ pour $H(t)=\frac{f(t)+1}{g(t)}$ sachant que $f(-1)=5$ , $g(-1)=6$ , $f'(-1)=1$ , $g'(-1)=5$ .

See Solution

Problem 21345

Find the tangent line equation to the curve at $t=16$ for $x=\sqrt{t}$ and $y=t^{2}-2t$ .

See Solution

Problem 21346

Is the integral $\int_{-\infty}^{9} r e^{r / 3} d r$ convergent or divergent? If convergent, evaluate it.

See Solution

Problem 21347

Find the sum of the series: $\sum_{n=2}^{\infty} \frac{1}{3^{n}}$ .

See Solution

Problem 21348

Find the sum of the series $\sum_{k=0}^{\infty} \cos \left(\frac{\pi}{3}\right)^{k}$ .

See Solution

Problem 21349

Find the tangent line equation at the point on the curve for $t=\pi$ , where $x=\sin(8t)+\cos(t)$ and $y=\cos(8t)-\sin(t)$ .

See Solution

Problem 21350

Calculate the integral from 3 to 5 of $e^{-4x}$ with respect to $x$ .

See Solution

Problem 21351

Given the parametric equations $x=\sin(3t)+\cos(t)$ and $y=\cos(3t)-\sin(t)$ , find $\frac{dy}{dx}$ and its value at $t=\pi$ . Also, determine the tangent line equation at $t=\pi$ .

See Solution

Problem 21352

Find the tangent line equation to the curve $x=\sin(t)$ , $y=\cos^2(t)$ at $\left(\frac{\sqrt{2}}{2}, \frac{1}{2}\right)$ using two methods.

See Solution

Problem 21353

Calculate the sum of the series: $\sum_{k=4}^{\infty}\left(\frac{8}{7}\right)^{k+3}$ .

See Solution

Problem 21354

Calculate the average rate of change of $g(x)=4 x^{3}-7$ over the interval $[-3,3]$ .

See Solution

Problem 21355

Berechnen Sie die Ableitung $f^{\prime}(x)$ der Funktion $f(x)=x^{3}-6 x^{2}+10$ .

See Solution

Problem 21356

Calculate the sum: $\sum_{n=0}^{\infty} \frac{3^{n+2}}{2^{2 n+1}}$ .

See Solution

Problem 21357

Differentiate $9 x^{5/4} - 8 x^{-1/3}$ with respect to $x$ .

See Solution

Problem 21358

Gegeben ist die Funktion $f(x)=x^{3}-6 x^{2}+10$ . Berechnen Sie $f^{\prime}(x)$ und analysieren Sie die Nullstellen mit dem Vorzeichenwechselkriterium.

See Solution

Problem 21359

A ball is dropped from 450 m high (CN Tower). Find its velocity after 5 seconds.

See Solution

Problem 21360

A ball is dropped from 450 m. Find its velocity after 5 seconds using average velocity over 0.1 seconds.

See Solution

Problem 21361

Given the curve defined by $x=\sin(t)$ and $y=\cos^2(t)$ at point $\left(\frac{\sqrt{3}}{2}, \frac{1}{4}\right)$ , find the tangent line using two methods:
(a) Without eliminating the parameter, find $\frac{dy}{dx}$ and the tangent line equation.
(b) By eliminating the parameter, find $\frac{dy}{dx}$ and the tangent line equation.

See Solution

Problem 21362

Find the limit: $\lim _{k \rightarrow \infty}\left(\frac{\sin (\pi k)}{17}-4\right)^{2}$ .

See Solution

Problem 21363

Bestimme die Tangentengleichung von $f(x)=-x^{2}+6 x-5$ am Punkt (2|f(2)).

See Solution

Problem 21364

Find the average velocity of the object with height $h(t)=\sin t$ over the interval $\left[\frac{\pi}{6}, \frac{2 \pi}{3}\right]$ .

See Solution

Problem 21365

Find the tangent equation to the curve at the point $\left(\frac{\sqrt{2}}{2}, \frac{1}{2}\right)$ using two methods: with and without parameter elimination. $x=\sin(t), y=\cos^2(t)$

See Solution

Problem 21366

Gegeben ist die Funktion $f(x)=\frac{1}{4} x^{3}-2 x$ . Finde Nullstellen, Extrempunkte, Steigung bei $x=2$ , Wendepunkt und Krümmungsverhalten bei $x=2$ .

See Solution

Problem 21367

Find the derivative of $\ln \left(\frac{x}{k}\right)$ .

See Solution

Problem 21368

Find the instantaneous velocity at $t=5$ by evaluating the limit of average velocities as time intervals decrease. Round to one decimal place: $v=\square \mathrm{m/s}$ .

See Solution

Problem 21369

Find the fourth derivative $f^{\prime \prime \prime \prime}(x)$ if $f^{\prime \prime \prime}(x)=2 \sqrt{x}$ .

See Solution

Problem 21370

Find the tangent line equation at the point (0,9) for the curve defined by $x=t^{2}-2t$ , $y=t^{2}+2t+1$ .

See Solution

Problem 21371

Evaluate the series: $\sum_{k=1}^{\infty} \frac{(-2)^{k+2}}{17^{k}}$

See Solution

Problem 21372

Find the tangent line equation to the curve $x=\sin (\pi t)$ , $y=t^{2}+t$ at the point $(0,12)$ . $y=$

See Solution

Problem 21373

Bestimme die Kantenlängen eines Quaders mit quadratischer Grundfläche aus einem $36 \mathrm{~cm}$ Draht, um das Volumen zu maximieren.

See Solution

Problem 21374

Find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ for $x=t^{2}+8$ , $y=t^{2}+5t$ . When is the curve concave upward?

See Solution

Problem 21375

Find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ for $x=e^t$ , $y=te^{-t}$ . When is the curve concave upward?

See Solution

Problem 21376

Find the limit as $t$ approaches 10 for the expression $\frac{t^{2}+3t-130}{t^{2}-100}$ .

See Solution

Problem 21377

Find the limit as $t$ approaches 6 for the expression $\frac{t^{2}+3t-54}{t^{2}-36}$ .

See Solution

Problem 21378

Find the second derivative $f^{\prime \prime}(t)$ of the function $f(t)=e^{-b t^{2}}$ .

See Solution

Problem 21379

Explain why the function is discontinuous at $a = -1$ :
1. $f(-1)$ and $\lim_{x \to -1} f(x)$ are finite but not equal.
2. $f(-1)$ is undefined.
3. $\lim_{x \to -1^{+}} f(x)$ and $\lim_{x \to -1^{-}} f(x)$ are finite but not equal.
4. $\lim_{x \to -1} f(x)$ does not exist.
5. none of the above
Sketch the graph of $f(x)$ where $f(x) = x + 5$ for $x \leq -1$ and $f(x) = 2^{x}$ for $x > -1$ .

See Solution

Problem 21380

Differentiate to find the optimum of $5 x^{2}+4 x-16$ . Find $x=$ and state if it's a max or min with 1.d.p.

See Solution

Problem 21381

Bestimmen Sie den limes inferior und limes superior der Folge $a_{n}=\sqrt[n]{\left|\left(\frac{11}{2}\right)^{n}+(-4)^{n}\right|}$ .

See Solution

Problem 21382

Differentiate the function: $y=\frac{e^{x}}{5-e^{x}}$

See Solution

Problem 21383

Given the function $f(x)=x^{2}+7$ , find the derivative at $x=2$ , $f(2)$ , the tangent line, and graph them.

See Solution

Problem 21384

Find the limit as $x$ approaches 3 for the expression $\frac{\sqrt{x^{2}+7}-4}{x-3}$ .

See Solution

Problem 21385

Use 400 yards of fencing to enclose a lot by a river. Find $l$ in terms of $w$ , then express area $A(w) = l w$ . What is the max area?

See Solution

Problem 21386

Find the point(s) on the curve $x=3 t^{2}+3, y=t^{3}-8$ where the tangent line slope is $\frac{1}{2}$ .

See Solution

Problem 21387

Bestimme die maximale und minimale Höhe der Funktion $h(t)=\frac{1}{1000}(t^{3}-180 t^{2}+6000 t)+400$ und den Zeitpunkt des größten Höhenverlusts.

See Solution

Problem 21388

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

See Solution

Problem 21389

Find the derivative of the function $f(x) = \sqrt{x^{2} - 2x}$ . What is $f'(x)$ ?

See Solution

Problem 21390

Find the derivative $f'(2)$ for the function $f(x) = x^2 + 7$ using the limit definition of the derivative.

See Solution

Problem 21391

Approximate $\sqrt[5]{31}$ using differentials and find a linear function for $f(x)=e^{x}$ .

See Solution

Problem 21392

Berechne die Steigung von $f(x)=3 \cdot(0,5 x^{2}-1)^{5}$ bei $a=-2$ .

See Solution

Problem 21393

A rancher has 600 ft of fencing for a rectangular field divided into 2 plots. Find dimensions that maximize area.

See Solution

Problem 21394

Find a linear function that approximates $f(x)=e^{x^{2}-1}$ at the point $x=1$ .

See Solution

Problem 21395

Find the growth rate $\frac{d L}{d t}$ at $t=15, 25, 35$ weeks for $L=-37.62+3.74t-6.34 \times 10^{-4}t^{3}$ . Is it increasing or decreasing?

See Solution

Problem 21396

Find critical points of $f(x)=x-\cos x$ for $0 \leq x \leq 2\pi$ . Options: a) $0$ b) $1$ c) $\frac{3\pi}{2}$ d) $0$

See Solution

Problem 21397

Bestimmen Sie die Ableitung von $f$ für: a) $f(x)=\frac{1}{x^{5}}$ , b) $f(x)=\frac{3}{x^{4}}$ , c) $f(x)=-\frac{1}{3 x^{6}}$ , d) $f(x)=\frac{x^{4}-x+1}{x^{3}}$ .

See Solution

Problem 21398

Evaluate the integral: $\int_{0}^{\infty} \frac{2}{x^{2}+4 x+3} d x$

See Solution

Problem 21399

Calculate the integral: $\int_{0}^{1} \frac{x+1}{\sqrt{x^{2}+2 x}} d x$

See Solution

Problem 21400

Evaluate the integral $\int_{0}^{1} \frac{x+1}{\sqrt{x^{2}+2 x}} d x$ .

See Solution

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Calculus

Problem 21301

Problem 21302

Problem 21303

Halla la constante de integración para ∫(x+1)dx=g(x)\int(x+1) d x=g_{(x)}∫(x+1)dx=g(x)​ sabiendo que g(2)=6g_{(2)}=6g(2)​=6.

Problem 21304

Calculate the time for NH3\mathrm{NH}_{3}NH3​ at 0.580 M to decrease by 95% using the rate law: rate = (3.3 M−1^{-1}−1 s−1^{-1}−1)[NH3\mathrm{NH}_{3}NH3​]2^{2}2. Round to 2 significant digits.

Problem 21305

Find the intervals where the function f(x)=2x3−3x2−12x+20f(x)=2 x^{3}-3 x^{2}-12 x+20f(x)=2x3−3x2−12x+20 is increasing.

Problem 21306

Find the series representation for the function f(x)=csc⁡xf(x)=\csc xf(x)=cscx.

Problem 21307

Find lim⁡x→0f(x)\lim _{x \rightarrow 0} f(x)limx→0​f(x) given x2−x4x2≤f(x)≤14x3+1 \frac{x^{2}-x^{4}}{x^{2}} \leq f(x) \leq \frac{1}{4} x^{3}+1x2x2−x4​≤f(x)≤41​x3+1 for −2≤x≤2-2 \leq x \leq 2−2≤x≤2.

Problem 21308

Bestimme die Koordinaten und die Steigung der interessanten Punkte des Graphen f(x)=13x3−2x2+9f(x)=\frac{1}{3} x^{3}-2 x^{2}+9f(x)=31​x3−2x2+9.

Problem 21309

Find horizontal asymptotes of f(x)=3(3x+1)(x−7)x(3x+1)f(x)=\frac{3(3 x+1)(x-7)}{x(3 x+1)}f(x)=x(3x+1)3(3x+1)(x−7)​.

Problem 21310

Determine the horizontal asymptotes of the function f(x)=3(3x+1)(x−7)x(3x+1)f(x)=\frac{3(3 x+1)(x-7)}{x(3 x+1)}f(x)=x(3x+1)3(3x+1)(x−7)​.

Problem 21311

Find upper and lower bounds for ∫0111+x2dx\int_{0}^{1} \frac{1}{1+x^{2}} d x∫01​1+x21​dx using the Max-Min Inequality.

Problem 21312

Show that ∫01sin⁡(x2)dx\int_{0}^{1} \sin \left(x^{2}\right) d x∫01​sin(x2)dx cannot equal 2. Which statement is true? A, B, C, or D?

Problem 21313

Bestimme die Hoch- und Tiefpunkte der Funktion f(x)=116x2(x2−24)f(x)=\frac{1}{16} x^{2}(x^{2}-24)f(x)=161​x2(x2−24).

Problem 21314

Calculate the integral ∫−4−2−0.5x dx\int_{-4}^{-2}-0.5 x \, dx∫−4−2​−0.5xdx.

Problem 21315

Find ∫44f(x)dx\int_{4}^{4} f(x) d x∫44​f(x)dx and ∫72g(x)dx\int_{7}^{2} g(x) d x∫72​g(x)dx given ∫24f(x)dx=6\int_{2}^{4} f(x) dx=6∫24​f(x)dx=6, ∫27f(x)dx=−7\int_{2}^{7} f(x) dx=-7∫27​f(x)dx=−7, and ∫27g(x)dx=7\int_{2}^{7} g(x) dx=7∫27​g(x)dx=7.

Problem 21316

Find ∫44f(x)dx\int_{4}^{4} f(x) d x∫44​f(x)dx given ∫24f(x)dx=6\int_{2}^{4} f(x) dx=6∫24​f(x)dx=6 and ∫27f(x)dx=−7\int_{2}^{7} f(x) dx=-7∫27​f(x)dx=−7.

Problem 21317

What is the relationship between an antiderivative FFF of fff and the area function AAA of fff? Choose the correct answer.

Problem 21318

Explain how the Fundamental Theorem of Calculus helps evaluate ∫abf(x)dx\int_{a}^{b} f(x) dx∫ab​f(x)dx using antiderivatives.

Problem 21319

What is the relationship between the antiderivative FFF of fff and the area function AAA of fff? Choose the correct option.

Problem 21320

Find the number of units for max profit from R(x)=211.2x−0.042x2R(x)=211.2 x-0.042 x^{2}R(x)=211.2x−0.042x2 and C(x)=10,000+105.6x−0.084x2+0.00002x3C(x)=10,000+105.6 x-0.084 x^{2}+0.00002 x^{3}C(x)=10,000+105.6x−0.084x2+0.00002x3. What is the max profit?

Problem 21321

Avery swings a sling of length 0.698 m0.698 \mathrm{~m}0.698 m at 8.34 rev/s. Find acceleration in m/s2\mathrm{m/s^2}m/s2. Then, for 0.979 m0.979 \mathrm{~m}0.979 m at 5.23 rev/s.

Problem 21322

A wooden artifact has 35% of carbon-14 of living trees. How many years ago was it made? (Half-life: 5730 years)

Problem 21323

Problem 21324

Use uuu-substitution to solve ∫34(4x−5)3dx\int_{3}^{4}(4x-5)^{3} dx∫34​(4x−5)3dx and find uuu, dududu, aaa, bbb, and f(u)f(u)f(u).

Problem 21325

Calculate the sum: ∑k=05k−1k!\sum_{k=0}^{5} \frac{k-1}{k !}∑k=05​k!k−1​.

Problem 21326

Find the minimum marginal cost of the cost function C(x)=5x3−9x2+7x+2C(x)=5 x^{3}-9 x^{2}+7 x+2C(x)=5x3−9x2+7x+2. Minimum marginal cost is \$ \square.

Problem 21327

Estimate the integral ∫39f(x)dx\int_{3}^{9} f(x) dx∫39​f(x)dx using 3 subintervals with right, left, and midpoint methods. Results: R3=1.4R_{3}=1.4R3​=1.4, L3=−1.9L_{3}=-1.9L3​=−1.9, M3=−0.3M_{3}=-0.3M3​=−0.3.

Problem 21328

Problem 21329

Differentiate 6x34+5x3−96 x^{\frac{3}{4}}+\frac{5}{x^{3}}-96x43​+x35​−9 with respect to xxx.

Problem 21330

Find the value of the series ∑k=1∞2(k+2)k\sum_{k=1}^{\infty} \frac{2}{(k+2) k}∑k=1∞​(k+2)k2​.

Problem 21331

A wooden artifact has 40% of its original carbon-14. How many years ago was it made? (Half-life of carbon-14 is 5730 years.)

Problem 21332

Problem 21333

Sketch the graph of the function using its first and second derivatives: f(x)=13x3−9x+3f(x)=\frac{1}{3} x^{3}-9 x+3f(x)=31​x3−9x+3.

Problem 21334

Find dxdt\frac{d x}{d t}dtdx​, dydt\frac{d y}{d t}dtdy​, and dydx\frac{d y}{d x}dxdy​ for x=5t3+3tx=5 t^{3}+3 tx=5t3+3t, y=4t−6t2y=4 t-6 t^{2}y=4t−6t2.

Problem 21335

Find dxdt\frac{d x}{d t}dtdx​, dydt\frac{d y}{d t}dtdy​, and dydx\frac{d y}{d x}dxdy​ for x=4tetx=4 t e^{t}x=4tet, y=6t+sin⁡(t)y=6 t+\sin(t)y=6t+sin(t).

Problem 21336

Evaluate the integral from 0 to 1: ∫01(9xe+2ex)dx\int_{0}^{1}\left(9 x^{e}+2 e^{x}\right) d x∫01​(9xe+2ex)dx

Problem 21337

Find the derivative of the function g(s)=∫9s(t−t9)7dtg(s)=\int_{9}^{s}(t-t^{9})^{7} dtg(s)=∫9s​(t−t9)7dt. What is g′(s)g'(s)g′(s)?

Problem 21338

Find dxdt\frac{d x}{d t}dtdx​, dydt\frac{d y}{d t}dtdy​, and dydx\frac{d y}{d x}dxdy​ for x=9t−9ln⁡(t)x=9 t-9 \ln (t)x=9t−9ln(t), y=4t2−4t−2y=4 t^{2}-4 t^{-2}y=4t2−4t−2.

Problem 21339

Determine if the integral ∫0∞x(x2+14)2dx\int_{0}^{\infty} \frac{x}{(x^{2}+14)^{2}} dx∫0∞​(x2+14)2x​dx converges or diverges, and evaluate if convergent.

Problem 21340

Find the slope of the tangent to the curve given by x=t2+2tx=t^{2}+2tx=t2+2t and y=2t−2ty=2^{t}-2ty=2t−2t at the specified point.

Problem 21341

Evaluate the integral ∫−55f(x) dx\int_{-5}^{5} f(x) \, dx∫−55​f(x)dx where f(x)=5f(x)=5f(x)=5 for −5≤x≤0-5 \leq x \leq 0−5≤x≤0 and f(x)=25−x2f(x)=25-x^{2}f(x)=25−x2 for 0<x≤50<x \leq 50<x≤5.

Problem 21342

Find dydx\frac{d y}{d x}dxdy​ for y=4x+45xy=4 \sqrt{x}+\frac{45}{x}y=4x​+x45​ at x=9x=9x=9.

Halla la constante de integración para $\int(x+1) d x=g_{(x)}$ sabiendo que $g_{(2)}=6$ .

Calculate the time for $\mathrm{NH}_{3}$ at 0.580 M to decrease by 95% using the rate law: rate = (3.3 M $^{-1}$ s $^{-1}$ )[ $\mathrm{NH}_{3}$ ] $^{2}$ . Round to 2 significant digits.

Find the intervals where the function $f(x)=2 x^{3}-3 x^{2}-12 x+20$ is increasing.

Find the series representation for the function $f(x)=\csc x$ .

Find $\lim _{x \rightarrow 0} f(x)$ given $\frac{x^{2}-x^{4}}{x^{2}} \leq f(x) \leq \frac{1}{4} x^{3}+1$ for $-2 \leq x \leq 2$ .

Bestimme die Koordinaten und die Steigung der interessanten Punkte des Graphen $f(x)=\frac{1}{3} x^{3}-2 x^{2}+9$ .

Find horizontal asymptotes of $f(x)=\frac{3(3 x+1)(x-7)}{x(3 x+1)}$ .

Determine the horizontal asymptotes of the function $f(x)=\frac{3(3 x+1)(x-7)}{x(3 x+1)}$ .

Find upper and lower bounds for $\int_{0}^{1} \frac{1}{1+x^{2}} d x$ using the Max-Min Inequality.

Show that $\int_{0}^{1} \sin \left(x^{2}\right) d x$ cannot equal 2. Which statement is true? A, B, C, or D?

Bestimme die Hoch- und Tiefpunkte der Funktion $f(x)=\frac{1}{16} x^{2}(x^{2}-24)$ .

Calculate the integral $\int_{-4}^{-2}-0.5 x \, dx$ .

Find $\int_{4}^{4} f(x) d x$ and $\int_{7}^{2} g(x) d x$ given $\int_{2}^{4} f(x) dx=6$ , $\int_{2}^{7} f(x) dx=-7$ , and $\int_{2}^{7} g(x) dx=7$ .

Find $\int_{4}^{4} f(x) d x$ given $\int_{2}^{4} f(x) dx=6$ and $\int_{2}^{7} f(x) dx=-7$ .

What is the relationship between an antiderivative $F$ of $f$ and the area function $A$ of $f$ ? Choose the correct answer.

Explain how the Fundamental Theorem of Calculus helps evaluate $\int_{a}^{b} f(x) dx$ using antiderivatives.

What is the relationship between the antiderivative $F$ of $f$ and the area function $A$ of $f$ ? Choose the correct option.

Find the number of units for max profit from $R(x)=211.2 x-0.042 x^{2}$ and $C(x)=10,000+105.6 x-0.084 x^{2}+0.00002 x^{3}$ . What is the max profit?

Avery swings a sling of length $0.698 \mathrm{~m}$ at 8.34 rev/s. Find acceleration in $\mathrm{m/s^2}$ . Then, for $0.979 \mathrm{~m}$ at 5.23 rev/s.

Use $u$ -substitution to solve $\int_{3}^{4}(4x-5)^{3} dx$ and find $u$ , $du$ , $a$ , $b$ , and $f(u)$ .

Calculate the sum: $\sum_{k=0}^{5} \frac{k-1}{k !}$ .

Find the minimum marginal cost of the cost function $C(x)=5 x^{3}-9 x^{2}+7 x+2$ . Minimum marginal cost is \$ \square.

Estimate the integral $\int_{3}^{9} f(x) dx$ using 3 subintervals with right, left, and midpoint methods. Results: $R_{3}=1.4$ , $L_{3}=-1.9$ , $M_{3}=-0.3$ .

Differentiate $6 x^{\frac{3}{4}}+\frac{5}{x^{3}}-9$ with respect to $x$ .

Find the value of the series $\sum_{k=1}^{\infty} \frac{2}{(k+2) k}$ .

Sketch the graph of the function using its first and second derivatives: $f(x)=\frac{1}{3} x^{3}-9 x+3$ .

Find $\frac{d x}{d t}$ , $\frac{d y}{d t}$ , and $\frac{d y}{d x}$ for $x=5 t^{3}+3 t$ , $y=4 t-6 t^{2}$ .

Find $\frac{d x}{d t}$ , $\frac{d y}{d t}$ , and $\frac{d y}{d x}$ for $x=4 t e^{t}$ , $y=6 t+\sin(t)$ .

Evaluate the integral from 0 to 1: $\int_{0}^{1}\left(9 x^{e}+2 e^{x}\right) d x$

Find the derivative of the function $g(s)=\int_{9}^{s}(t-t^{9})^{7} dt$ . What is $g'(s)$ ?

Find $\frac{d x}{d t}$ , $\frac{d y}{d t}$ , and $\frac{d y}{d x}$ for $x=9 t-9 \ln (t)$ , $y=4 t^{2}-4 t^{-2}$ .

Determine if the integral $\int_{0}^{\infty} \frac{x}{(x^{2}+14)^{2}} dx$ converges or diverges, and evaluate if convergent.

Find the slope of the tangent to the curve given by $x=t^{2}+2t$ and $y=2^{t}-2t$ at the specified point.

Evaluate the integral $\int_{-5}^{5} f(x) \, dx$ where $f(x)=5$ for $-5 \leq x \leq 0$ and $f(x)=25-x^{2}$ for $0<x \leq 5$ .

Find $\frac{d y}{d x}$ for $y=4 \sqrt{x}+\frac{45}{x}$ at $x=9$ .

Find the tangent line equation to the curve at $t=-1$ : $x=t^{7}+1$ , $y=t^{8}+t$ .

Trouver $H^{\prime}(-1)$ pour $H(t)=\frac{f(t)+1}{g(t)}$ sachant que $f(-1)=5$ , $g(-1)=6$ , $f'(-1)=1$ , $g'(-1)=5$ .

Find the tangent line equation to the curve at $t=16$ for $x=\sqrt{t}$ and $y=t^{2}-2t$ .

Is the integral $\int_{-\infty}^{9} r e^{r / 3} d r$ convergent or divergent? If convergent, evaluate it.

Find the sum of the series: $\sum_{n=2}^{\infty} \frac{1}{3^{n}}$ .

Find the sum of the series $\sum_{k=0}^{\infty} \cos \left(\frac{\pi}{3}\right)^{k}$ .

Find the tangent line equation at the point on the curve for $t=\pi$ , where $x=\sin(8t)+\cos(t)$ and $y=\cos(8t)-\sin(t)$ .

Calculate the integral from 3 to 5 of $e^{-4x}$ with respect to $x$ .

Given the parametric equations $x=\sin(3t)+\cos(t)$ and $y=\cos(3t)-\sin(t)$ , find $\frac{dy}{dx}$ and its value at $t=\pi$ . Also, determine the tangent line equation at $t=\pi$ .

Find the tangent line equation to the curve $x=\sin(t)$ , $y=\cos^2(t)$ at $\left(\frac{\sqrt{2}}{2}, \frac{1}{2}\right)$ using two methods.

Calculate the sum of the series: $\sum_{k=4}^{\infty}\left(\frac{8}{7}\right)^{k+3}$ .

Calculate the average rate of change of $g(x)=4 x^{3}-7$ over the interval $[-3,3]$ .

Berechnen Sie die Ableitung $f^{\prime}(x)$ der Funktion $f(x)=x^{3}-6 x^{2}+10$ .

Calculate the sum: $\sum_{n=0}^{\infty} \frac{3^{n+2}}{2^{2 n+1}}$ .

Differentiate $9 x^{5/4} - 8 x^{-1/3}$ with respect to $x$ .

Gegeben ist die Funktion $f(x)=x^{3}-6 x^{2}+10$ . Berechnen Sie $f^{\prime}(x)$ und analysieren Sie die Nullstellen mit dem Vorzeichenwechselkriterium.

Find the limit: $\lim _{k \rightarrow \infty}\left(\frac{\sin (\pi k)}{17}-4\right)^{2}$ .

Bestimme die Tangentengleichung von $f(x)=-x^{2}+6 x-5$ am Punkt (2|f(2)).

Find the average velocity of the object with height $h(t)=\sin t$ over the interval $\left[\frac{\pi}{6}, \frac{2 \pi}{3}\right]$ .

Find the tangent equation to the curve at the point $\left(\frac{\sqrt{2}}{2}, \frac{1}{2}\right)$ using two methods: with and without parameter elimination. $x=\sin(t), y=\cos^2(t)$

Gegeben ist die Funktion $f(x)=\frac{1}{4} x^{3}-2 x$ . Finde Nullstellen, Extrempunkte, Steigung bei $x=2$ , Wendepunkt und Krümmungsverhalten bei $x=2$ .

Find the derivative of $\ln \left(\frac{x}{k}\right)$ .

Find the instantaneous velocity at $t=5$ by evaluating the limit of average velocities as time intervals decrease. Round to one decimal place: $v=\square \mathrm{m/s}$ .

Find the fourth derivative $f^{\prime \prime \prime \prime}(x)$ if $f^{\prime \prime \prime}(x)=2 \sqrt{x}$ .

Find the tangent line equation at the point (0,9) for the curve defined by $x=t^{2}-2t$ , $y=t^{2}+2t+1$ .

Evaluate the series: $\sum_{k=1}^{\infty} \frac{(-2)^{k+2}}{17^{k}}$

Find the tangent line equation to the curve $x=\sin (\pi t)$ , $y=t^{2}+t$ at the point $(0,12)$ . $y=$

Bestimme die Kantenlängen eines Quaders mit quadratischer Grundfläche aus einem $36 \mathrm{~cm}$ Draht, um das Volumen zu maximieren.

Find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ for $x=t^{2}+8$ , $y=t^{2}+5t$ . When is the curve concave upward?