Calculus

Problem 23301

Find the average rate of change of $g(x)=-2 x^{3}+5 x^{2}$ between $x=1$ and $x=2$ . Simplify your answer.

See Solution

Problem 23302

Find $x$ where the derivative $\mathrm{f}^{\prime}(x)=2$ for the function $\mathrm{f}(x)=2 x^{2}-3 x+1$ .

See Solution

Problem 23303

Untersuchen Sie die Funktionen auf Monotonie: a) $f(x)=-0,25 x^{2}+4$ , b) $f(x)=x^{3}-3 x$ , c) $f(x)=2 e^{x}-1$ , d) $f(x)=2 \cos (x)$ für $0 \leq x \leq 2 \pi$ .

See Solution

Problem 23304

Find the derivative of the function $\frac{20}{3 \sqrt[5]{t}}$ with respect to $t$ .

See Solution

Problem 23305

Find the limits of these sequences: $a_n=\sqrt{n^2+n}-n$ , $b_n=\frac{n^2+n+1}{n^2+n \sin n+1}$ , $c_n=\frac{n!(n+5)+2^n}{(n+1)!+3^n}$ , $d_n=\sqrt[n]{3^n+4^n}$ , $e_n=\frac{n^3-3n+7}{7n+1}$ , $t_n=\sqrt{n^3-n^2+1}-n$ for $n \in \mathbb{N}$ .

See Solution

Problem 23306

Find the derivative of $a(t)=\int_{t}^{-1} e^{w+w^{2}} d w$ using the Fundamental Theorem of Calculus.

See Solution

Problem 23307

Bestimmen Sie die Grenzwerte: a) $\lim _{x \rightarrow \infty} \frac{\ln (x)}{x^{2}}$ b) $\lim _{x \rightarrow 0}\left((x^{2}+2 x) \cdot \ln (x)\right)$ c) $\lim _{x \rightarrow \infty} \frac{\ln (x)}{3 x^{2}-2 x-4}$

See Solution

Problem 23308

Evaluate the integral: $\int_{0}^{3 \pi / 4}|\cos x| d x$ [8 points]

See Solution

Problem 23309

Find the area of one petal of the rose $r = \cos 2\theta$ using $A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta$ . Determine $\alpha$ and $\beta$ for one petal.

See Solution

Problem 23310

Find the area of one loop of the lemniscate given by $r^{2}=4 \cos 2 \theta$ . Use $A = \frac{1}{2}\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (r(\theta))^2 d\theta$ .

See Solution

Problem 23311

Find the $x$ -coordinate of point $\mathrm{P}$ on the curve $y=2-x^{3}$ where the normal has a gradient of 3.

See Solution

Problem 23312

A cow's eye level is 4 ft. A billboard is 9 ft high, 12 ft above ground. Find $\theta(x)$ and $x_{0}$ to maximize $\theta$ .

See Solution

Problem 23313

Berechne den Gesamtinhalt der Fläche zwischen $f(x)=\frac{1}{2} x^{2}-\frac{5}{2} x+2$ und der $x$ -Achse im Intervall $[0 ; 3]$ .

See Solution

Problem 23314

Find the area of the inner loop of the limaçon with the polar equation $r = 2 \sin \theta - 1$ .

See Solution

Problem 23315

f. Bestimme, wann das U-Boot mit maximaler Geschwindigkeit auftaucht, gegeben $h(x)=-x^{3}+7,9 x^{2}-15,47 x-0,043$ für $0 \leq x \leq 5$ . g. a) Finde die Strecke des Flugzeugs nach 60 Sekunden mit $y=2x$ . b) Finde die Strecke nach $x$ Sekunden.

See Solution

Problem 23316

e. Finde die Tangentengleichung an $x=2$ für die Funktion $x^{3}-2 x^{2}-3 x+5$ . f. Bestimme, wann das U-Boot mit der Funktion $h(x)=-x^{3}+7,9 x^{2}-15,47 x-0,043$ maximale Geschwindigkeit hat (in Minuten). g. a) Wie weit fliegt ein Flugzeug nach einer Minute mit $y=2 x$ ? b) Wie weit nach $x$ Sekunden? h. Zeige, dass $(x-a)^{3}$ bei $x=a$ eine Wendestelle hat. i. Bestimme die Intervallgrenzen für die Blumenwiese und das Gemüsebeet zwischen $f(x)=x+1$ und $g(x)=x^{2}-4 x+5$ , und berechne die Flächeninhalte.

See Solution

Problem 23317

Find the half-life of a radioactive substance with a decay rate of $9.7\%$ per day using the exponential decay model.

See Solution

Problem 23318

Find the interest rate for \$2000 invested continuously, growing to \$2449 in 5 years. Round to the nearest hundredth.

See Solution

Problem 23319

Find the hourly growth rate for a bacteria population that grows from 1500 to 1836 in 6 hours. Express as a percentage.

See Solution

Problem 23320

Calculate $\Delta G$ using $\Delta G^\circ$ and $Q = \frac{[\text{CO}_2][\text{H}_2]^2}{[\text{CH}_4][\text{O}_2]}$ .

See Solution

Problem 23321

Find $\lim _{x \rightarrow 0} x \cos \left(\frac{1}{x}\right)$ given $-x \leq x \cos \left(\frac{1}{x}\right) \leq x$ for $x>0$ .

See Solution

Problem 23322

Find the local minimum and maximum points of the curve $y=2x^{3}+3x^{2}-12x-10$ .

See Solution

Problem 23323

Use the Intermediate Value Theorem to show $f(x)=3x^{3}+3x^{2}-4x-5$ has a zero between $x=1$ and $x=2$ .

See Solution

Problem 23324

A differentiable function $f$ on $[-2, 7]$ has $f'(c)=0$ for $-2<c<7$ and $f(c)<f(7)$ . Which statement is true?

See Solution

Problem 23325

Find $c$ such that the instantaneous rate of change of $f(x)=2\sqrt{x}$ equals the average rate of change from $1$ to $4$ .

See Solution

Problem 23326

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\cos 4 x \tan 4 x}{6 x}$ .

See Solution

Problem 23327

Find the equation of the tangent line to $f(x)=4 x^{3}-5 x+3$ at $x=-1$ .

See Solution

Problem 23328

Find the average rate of change of $f(x)=1-x^{2}$ from $x=-2$ to $x=-1$ . Choices: A. 3 B. 0 C. -3 D. undefined

See Solution

Problem 23329

Find the limit: $\lim _{x \rightarrow 0} \frac{2 x^{2}+1 - 1}{x^{2}}$ for the function $f(x)=2 x^{2}+1$ .

See Solution

Problem 23330

Find the radius that maximizes the volume of a cylinder with height + circumference = 30 cm, where $V=\pi r^{2} h$ .

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Problem 23331

Find $\lim _{x \rightarrow a} \frac{x^{2}-a^{2}}{x^{4}-a^{4}}$ for $x \neq a$ .

See Solution

Problem 23332

Find the coordinates of the point on the curve $y = \sqrt{x}$ where the gradient equals $2$ .

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Problem 23333

Find the limit as $x$ approaches infinity for the expression: $\frac{x^{2}-5 x+10}{x^{3}-3 x^{2}+13}$ . Show your work.

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Problem 23334

Find the limit: $\lim _{x \rightarrow-\infty} \frac{-7 x^{2}-8 x+9}{-4 x^{2}+7 x+15}$ . Show your work.

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Problem 23335

A 0.270 kg block compresses a spring (k = 4.13 x 10^3 N/m) by 0.0900 m. How high does it rise after release? Answer in m.

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Problem 23336

Evaluate the function $g(x)=\frac{x+\cos x}{x+3}$ and determine which statements about limits and asymptotes are true.

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Problem 23337

Find the slope of the tangent line for $f(x)=3 \sqrt[3]{x^{2}}+2 x$ at $x=-8$ .

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Problem 23338

Find the slope of the normal line to $h(x)=\frac{3 x}{\sqrt[3]{x}}$ at $x=-8$ .

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Problem 23339

Find the limit: $\lim _{x \rightarrow-2} \frac{1}{x+2}$ . Show your work.

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Problem 23340

Find the limit: $\lim _{x \rightarrow(\pi / 2)^{+}} \tan x$ . Show your work.

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Problem 23341

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \sqrt{\frac{4 x^{2}}{3+25 x^{2}}}$ . Show your work.

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Problem 23342

Determine if the function $f(x)=x^{3}+3x^{2}$ is increasing or decreasing at $x=-3$ and $x=0$ .

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Problem 23343

A 0.20 kg ball is thrown up from a 30.0 m building at 22.0 m/s. Find its total energy, max height, and impact velocity.

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Problem 23344

Calculate the sum of the series: $\sum_{n=0}^{\infty}(-1)^{n} \frac{4}{9^{n}}$ .

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Problem 23345

Medikamenten-Konzentration im Blut
Gegeben ist die Funktion $c(t)=t^{3}-17 t^{2}+63 t+81$ für $0 \leq t \leq 9$ .
a) Skizzieren Sie $c$ , finden Sie die Anfangskonzentration, den Abbauzeitpunkt, und Zeitintervalle für Anstieg/Abfall. b) Bestimmen Sie Zeit für Maximalkonzentration. c) Erklären Sie $c'$ im Kontext. d) Finden Sie den Zeitraum, in dem $c > 100 \mathrm{mg} / \mathrm{l}$ .

See Solution

Problem 23346

Modellieren Sie das Wachstum einer Rattenpopulation, die bei 3500 Tieren beginnt und mit 200 Tieren/Tag wächst. Wie lange bis >50000?

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Problem 23347

Check if the series $\sum_{n=1}^{\infty} \frac{7}{n(n+3)}$ converges or diverges, and find the sum if it converges.

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Problem 23348

How much water is left in a 1000-liter tank after 1 hour if pumped out at $3 - 3 e^{-0.15 t}$ L/min?

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Problem 23349

What is the value of the definite integral $\int_{1}^{3} 2 x \, dx$ ? (a) 1 (b) 2 (c) 4 (d) 8 (e) 10

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Problem 23350

Calculate the area under the curve of $f(x)=x^{2}+7$ from $x=0$ to $x=6$ .

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Problem 23351

Determine if the series $\sum_{n=1}^{\infty} \frac{5 n}{n^{2}+2}$ converges or diverges.

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Problem 23352

Determine if the series $\sum_{n=1}^{\infty} \frac{3+7 \cos n}{n^{3}}$ converges or diverges.

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Problem 23353

A radioactive substance decays from 9.30 mg to 7.40 mg in 11.7 years. Find its half-life rounded to 2 significant digits.

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Problem 23354

Determine if the series converges or diverges: $\sum_{n=1}^{\infty} \frac{4 \sqrt{n}}{8 n^{3 / 2}+7 n-8}$

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Problem 23355

Determine if the series converges or diverges: $\sum_{n=1}^{\infty} \frac{4(n l)^{2}}{(2 n) !}$

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Problem 23356

Estimate the area between $f(x)$ and the $x$ -axis from $0$ to $20$ using left and right sums. Round to one decimal place. Area $\approx$

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Problem 23357

A coffee cup starts at $135^{\circ} \mathrm{F}$ in a freezer at $0^{\circ} \mathrm{F}$ . After 10 min it's $60^{\circ} \mathrm{F}$ . Find temp after 15 min.

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Problem 23358

Determine if the series converges or diverges: $\sum_{n=1}^{\infty}\left(\frac{10 n}{\ln n+9 n+9}\right)^{n}$

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Problem 23359

Find the function $f$ given $f^{\prime \prime}(x)=x^{2}+1$ , $f^{\prime}(0)=4$ , and $f(0)=2$ . Which is $f(x)$ ?

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Problem 23360

After substituting $u=2 e^{x}-3$ , what does the integral $\int 6 e^{x} \sqrt{2 e^{x}-3} d x$ become?

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Problem 23361

Find the indefinite integral $\int \sin (x) e^{\cos (x)} d x$ . Which option is correct? (a) (b) (c) (d) (e)

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Problem 23362

Calculate the series sum: $S = \sum_{n=4}^{\infty} 64\left(\frac{1}{5}\right)^{n-1}$ . Choose the correct answer.

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Problem 23363

Find the average rate of change of $p(x) = \sqrt{15x}$ over the interval $[0.01, 1.01]$ .

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Problem 23364

Determine if the series $\sum_{n=1}^{\infty}(-6)^{-n}$ converges absolutely, conditionally, or diverges.

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Problem 23365

Find the difference quotient, $\frac{f(x+h)-f(x)}{h}$ , for the function $f(x)=6x-4$ .

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Problem 23366

En jeger skyter horisontalt mot en blink 98.1 m unna, $180 \mathrm{~cm}$ over bakken. Hvor langt under blinken treffer kulen? $28.4$ $\mathrm{cm}$

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Problem 23367

Determine if the series converges absolutely, conditionally, or diverges: $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{3 n^{1 / 2}+8}$

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Problem 23368

Sketch the graph of $f$ with given properties: asymptotes, local extrema, and inflection points. Key points include:
- Continuous on $(-\infty, 3)$ and $(3, \infty)$ - $f(-1)=5$ - $\lim_{x \rightarrow 3^{-}} f(x)=-\infty$ - $\lim_{x \rightarrow 3^{+}} f(x)=-\infty$ - $\lim_{x \rightarrow -\infty} f(x)=-2$ - $\lim_{x \rightarrow +\infty} f(x)=4$ - $f^{\prime}(x)<0$ on $(-1,3)$ - $f^{\prime}(x)>0$ on $(-\infty,-1) \cup (3, \infty)$ - $f^{\prime \prime}(x)<0$ on $(-3,3) \cup (3, \infty)$ - $f^{\prime \prime}(x)>0$ on $(-\infty,-3)$

See Solution

Problem 23369

Find the value of the integral $\int_{1}^{3} f^{\prime}(x) dx$ given $f(0)=-2$ , $f(1)=2$ , $f(2)=-3$ , $f(3)=4$ . Choices: (a) 1 (b) 2 (c) 3 (d) 6 (e) 9.

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Problem 23370

Sketch the graph of $f$ with these properties: continuous on $(-\infty, 3)$ and $(3, \infty)$ , $f(-1)=5$ , $\lim_{x \to 3^{-}} f(x)=-\infty$ , $\lim_{x \to 3^{+}} f(x)=-\infty$ , $\lim_{x \to -\infty} f(x)=-2$ , $\lim_{x \to +\infty} f(x)=4$ , $f'(x)<0$ on $(-1,3)$ , $f'(x)>0$ on $(-\infty,-1) \cup (3, \infty)$ , $f''(x)<0$ on $(-3,3) \cup (3, \infty)$ , $f''(x)>0$ on $(-\infty,-3)$ .

See Solution

Problem 23371

Ein Radfahrer legt die Strecke $s(t)$ in km zurück.
a) Mittlere Änderungsrate von $s$ in $[2; 5]$ ist 15. Was bedeutet das und wie weit ist der Radfahrer gefahren? b) Momentane Änderungsrate $s'(2)$ ist 18. Was bedeutet das und wie lange für die nächsten $1,5 \mathrm{~km}$ bei konstanter Rate?

See Solution

Problem 23372

Determine the truth of these statements about function $g$ : I. $a < 0$ for normal lines on $(-\infty, -2)$ ; II. $g'(x) > 0$ on $(-2, 2)$ and $(3, \infty)$ ; III. $g'$ is undefined at $x = -2$ and $x = 2$ .

See Solution

Problem 23373

Ein Taucher ist nach 10 Minuten in 24 m Tiefe.
a) Die mittlere Änderungsrate von $f$ im Intervall $[10 ; 12]$ ist 3. Berechne $f(12)$ . b) Die momentane Änderungsrate nach 10 Minuten ist 2,8. Welche Aussagen sind richtig oder falsch? $f^{\prime}(10)=2,8$ $f(11)=26,8$

See Solution

Problem 23374

Find total revenue for 100 units using $R^{\prime}(q)=150-12 \sqrt{q}$ and marginal revenue at 100 units.

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Problem 23375

Find the total cost for producing 20 units given $C^{\prime}(q)=q^{2}-17q+70$ , with $C(0)=550$ . What are fixed and variable costs?

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Problem 23376

The marginal cost $C^{\prime}(q)$ for $q$ units is given. If fixed cost is \$17,000, find total cost for 400 units and the increase for 401 units.

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Problem 23377

Evaluate the integral $\int_{-2}^{-1} x \sqrt{2+x} d x$ using $u=2+x$ . Which option is correct?

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Problem 23378

Find the Maclaurin series for the function $e^{8x}$ .

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Problem 23379

Find the value of the definite integral $\int_{0}^{3} f(x) dx$ for the piecewise function $f(x)$ . Options: (a) 4, (b) 6, (c) 8, (d) 10, (e) 11.

See Solution

Problem 23380

Find the Taylor series at $x=0$ for $f(x)=x^{10} \sin x$ using power series operations.

See Solution

Problem 23381

Determine the horizontal asymptotes of the function $f(x)=\frac{10 x^{2}-76 x+96}{15 x^{2}-24 x}$ .

See Solution

Problem 23382

Given $g(x)<0$ for all $x$ and $f'(x)=(x^{2}-4)g(x)$ , which statement about $f$ is true? Options: a, b, c, d, e.

See Solution

Problem 23383

How much to deposit now to have \$800,000 in 30 years at 6% interest compounded continuously?

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Problem 23384

Find the indefinite integral $\int 10 x\left(x^{2}+5\right)^{4} d x$ . Which option is correct?

See Solution

Problem 23385

Estimate the area between $f(x)$ and the $x$ -axis from $0$ to $20$ using left and right sums. Round to one decimal place.

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Problem 23386

Find the Laplace transform of the function $t^{5}$ .

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Problem 23387

Find the inverse Laplace transform of $\frac{s-1}{s^{2}-s-2}$ .

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Problem 23388

A satellite orbits Earth at $252 \mathrm{~km}$ . Find its orbital radius, speed, and period (in seconds and hours).

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Problem 23389

Identify the error in the equation: $\int_{0}^{\pi} \sec ^{2} x \, dx = \tan (\pi) - \tan (0)$ .

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Problem 23390

Calculate the average rate of change of $g(x)=-2 x^{3}+3 x^{2}$ between $x=1$ and $x=3$ . Simplify your result.

See Solution

Problem 23391

Find the tangent line equation for $y=x^{2}+1$ at point $(3,10)$ using the limit definition of slope. Show work.

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Problem 23392

Find two positive numbers $x$ and $y$ such that $x^2 + y = 75$ and maximize the product $P = xy$ .

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Problem 23393

A ski jumper leaps with a horizontal speed of $25.9 \mathrm{~m/s}$ and lands after 5.7 seconds. Find her takeoff height.

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Problem 23394

Fungus size is $L(t)=3.3t+1.2\cos\left(\frac{2\pi t}{24}\right)$ . Find $\frac{\mathrm{dL}}{\mathrm{dt}}$ and its max/min values.

See Solution

Problem 23395

Calculate the Left Riemann Sum $L_8$ for $f(x) = x^{2}-2x+1$ on the interval $[0,2]$ .

See Solution

Problem 23396

Find the limit: $\lim _{x \rightarrow \pi / 4} \frac{\tan x-1}{x-\pi / 4}$ .

See Solution

Problem 23397

Gegeben ist die Funktion $f_{a}(x)=-x^{3}+a^{2} \cdot x, a>0$ . Finde Nullstellen, Extremstellen und Wendestellen.

See Solution

Problem 23398

Estimate the distance a car traveled in 30 seconds using a trapezium with speeds of 3 and 25 units. Is it an underestimate or overestimate?

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Problem 23399

Show that $f^{\prime \prime}(0)>0$ for $m=2$ in Hill's equation $f(P)=\frac{P^{m}}{k^{m}+P^{m}}$ . Find $f(P)$ for $m=2$ .

See Solution

Problem 23400

Find velocity and acceleration at $t=1 \mathrm{~s}$ for $s(t)=t^{2}-3t$ and $s(t)=\sqrt{t^{2}+3}$ .

See Solution

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Calculus

Problem 23301

Find the average rate of change of g(x)=−2x3+5x2g(x)=-2 x^{3}+5 x^{2}g(x)=−2x3+5x2 between x=1x=1x=1 and x=2x=2x=2. Simplify your answer.

Problem 23302

Find xxx where the derivative f′(x)=2\mathrm{f}^{\prime}(x)=2f′(x)=2 for the function f(x)=2x2−3x+1\mathrm{f}(x)=2 x^{2}-3 x+1f(x)=2x2−3x+1.

Problem 23303

Untersuchen Sie die Funktionen auf Monotonie: a) f(x)=−0,25x2+4f(x)=-0,25 x^{2}+4f(x)=−0,25x2+4, b) f(x)=x3−3xf(x)=x^{3}-3 xf(x)=x3−3x, c) f(x)=2ex−1f(x)=2 e^{x}-1f(x)=2ex−1, d) f(x)=2cos⁡(x)f(x)=2 \cos (x)f(x)=2cos(x) für 0≤x≤2π0 \leq x \leq 2 \pi0≤x≤2π.

Problem 23304

Find the derivative of the function 203t5\frac{20}{3 \sqrt[5]{t}}35t​20​ with respect to ttt.

Problem 23305

Problem 23306

Find the derivative of a(t)=∫t−1ew+w2dwa(t)=\int_{t}^{-1} e^{w+w^{2}} d wa(t)=∫t−1​ew+w2dw using the Fundamental Theorem of Calculus.

Problem 23307

Problem 23308

Evaluate the integral: ∫03π/4∣cos⁡x∣dx\int_{0}^{3 \pi / 4}|\cos x| d x∫03π/4​∣cosx∣dx [8 points]

Problem 23309

Find the area of one petal of the rose r=cos⁡2θ r = \cos 2\theta r=cos2θ using A=12∫αβr2 dθ A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta A=21​∫αβ​r2dθ. Determine α \alpha α and β \beta β for one petal.

Problem 23310

Find the area of one loop of the lemniscate given by r2=4cos⁡2θr^{2}=4 \cos 2 \thetar2=4cos2θ. Use A=12∫−π4π4(r(θ))2dθA = \frac{1}{2}\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (r(\theta))^2 d\thetaA=21​∫−4π​4π​​(r(θ))2dθ.

Problem 23311

Find the xxx-coordinate of point P\mathrm{P}P on the curve y=2−x3y=2-x^{3}y=2−x3 where the normal has a gradient of 3.

Problem 23312

A cow's eye level is 4 ft. A billboard is 9 ft high, 12 ft above ground. Find θ(x)\theta(x)θ(x) and x0x_{0}x0​ to maximize θ\thetaθ.

Problem 23313

Berechne den Gesamtinhalt der Fläche zwischen f(x)=12x2−52x+2f(x)=\frac{1}{2} x^{2}-\frac{5}{2} x+2f(x)=21​x2−25​x+2 und der xxx-Achse im Intervall [0;3][0 ; 3][0;3].

Problem 23314

Find the area of the inner loop of the limaçon with the polar equation r=2sin⁡θ−1r = 2 \sin \theta - 1r=2sinθ−1.

Problem 23315

Problem 23316

Problem 23317

Find the half-life of a radioactive substance with a decay rate of 9.7%9.7\%9.7% per day using the exponential decay model.

Problem 23318

Find the interest rate for \$2000 invested continuously, growing to \$2449 in 5 years. Round to the nearest hundredth.

Problem 23319

Find the hourly growth rate for a bacteria population that grows from 1500 to 1836 in 6 hours. Express as a percentage.

Problem 23320

Calculate ΔG\Delta GΔG using ΔG∘\Delta G^\circΔG∘ and Q=[CO2][H2]2[CH4][O2]Q = \frac{[\text{CO}_2][\text{H}_2]^2}{[\text{CH}_4][\text{O}_2]}Q=[CH4​][O2​][CO2​][H2​]2​.

Problem 23321

Find lim⁡x→0xcos⁡(1x)\lim _{x \rightarrow 0} x \cos \left(\frac{1}{x}\right)limx→0​xcos(x1​) given −x≤xcos⁡(1x)≤x-x \leq x \cos \left(\frac{1}{x}\right) \leq x−x≤xcos(x1​)≤x for x>0x>0x>0.

Problem 23322

Find the local minimum and maximum points of the curve y=2x3+3x2−12x−10y=2x^{3}+3x^{2}-12x-10y=2x3+3x2−12x−10.

Problem 23323

Use the Intermediate Value Theorem to show f(x)=3x3+3x2−4x−5f(x)=3x^{3}+3x^{2}-4x-5f(x)=3x3+3x2−4x−5 has a zero between x=1x=1x=1 and x=2x=2x=2.

Problem 23324

A differentiable function fff on [−2,7][-2, 7][−2,7] has f′(c)=0f'(c)=0f′(c)=0 for −2<c<7-2<c<7−2<c<7 and f(c)<f(7)f(c)<f(7)f(c)<f(7). Which statement is true?

Problem 23325

Find ccc such that the instantaneous rate of change of f(x)=2xf(x)=2\sqrt{x}f(x)=2x​ equals the average rate of change from 111 to 444.

Problem 23326

Calculate the limit: lim⁡x→0cos⁡4xtan⁡4x6x\lim _{x \rightarrow 0} \frac{\cos 4 x \tan 4 x}{6 x}limx→0​6xcos4xtan4x​.

Problem 23327

Find the equation of the tangent line to f(x)=4x3−5x+3f(x)=4 x^{3}-5 x+3f(x)=4x3−5x+3 at x=−1x=-1x=−1.

Problem 23328

Find the average rate of change of f(x)=1−x2f(x)=1-x^{2}f(x)=1−x2 from x=−2x=-2x=−2 to x=−1x=-1x=−1. Choices: A. 3 B. 0 C. -3 D. undefined

Problem 23329

Find the limit: lim⁡x→02x2+1−1x2\lim _{x \rightarrow 0} \frac{2 x^{2}+1 - 1}{x^{2}}limx→0​x22x2+1−1​ for the function f(x)=2x2+1f(x)=2 x^{2}+1f(x)=2x2+1.

Problem 23330

Find the radius that maximizes the volume of a cylinder with height + circumference = 30 cm, where V=πr2hV=\pi r^{2} hV=πr2h.

Problem 23331

Find lim⁡x→ax2−a2x4−a4\lim _{x \rightarrow a} \frac{x^{2}-a^{2}}{x^{4}-a^{4}}limx→a​x4−a4x2−a2​ for x≠ax \neq ax=a.

Problem 23332

Find the coordinates of the point on the curve y=xy = \sqrt{x}y=x​ where the gradient equals 222.

Problem 23333

Find the limit as xxx approaches infinity for the expression: x2−5x+10x3−3x2+13\frac{x^{2}-5 x+10}{x^{3}-3 x^{2}+13}x3−3x2+13x2−5x+10​. Show your work.

Problem 23334

Find the limit: lim⁡x→−∞−7x2−8x+9−4x2+7x+15\lim _{x \rightarrow-\infty} \frac{-7 x^{2}-8 x+9}{-4 x^{2}+7 x+15}limx→−∞​−4x2+7x+15−7x2−8x+9​. Show your work.

Problem 23335

A 0.270 kg block compresses a spring (k = 4.13 x 10^3 N/m) by 0.0900 m. How high does it rise after release? Answer in m.

Problem 23336

Evaluate the function g(x)=x+cos⁡xx+3g(x)=\frac{x+\cos x}{x+3}g(x)=x+3x+cosx​ and determine which statements about limits and asymptotes are true.

Problem 23337

Find the slope of the tangent line for f(x)=3x23+2xf(x)=3 \sqrt[3]{x^{2}}+2 xf(x)=33x2​+2x at x=−8x=-8x=−8.

Problem 23338

Find the slope of the normal line to h(x)=3xx3h(x)=\frac{3 x}{\sqrt[3]{x}}h(x)=3x​3x​ at x=−8x=-8x=−8.

Problem 23339

Find the limit: lim⁡x→−21x+2\lim _{x \rightarrow-2} \frac{1}{x+2}limx→−2​x+21​. Show your work.

Problem 23340

Find the limit: lim⁡x→(π/2)+tan⁡x\lim _{x \rightarrow(\pi / 2)^{+}} \tan xlimx→(π/2)+​tanx. Show your work.

Problem 23341

Find the limit as xxx approaches infinity: lim⁡x→∞4x23+25x2\lim _{x \rightarrow \infty} \sqrt{\frac{4 x^{2}}{3+25 x^{2}}}limx→∞​3+25x24x2​​. Show your work.

Problem 23342

Find the average rate of change of $g(x)=-2 x^{3}+5 x^{2}$ between $x=1$ and $x=2$ . Simplify your answer.

Find $x$ where the derivative $\mathrm{f}^{\prime}(x)=2$ for the function $\mathrm{f}(x)=2 x^{2}-3 x+1$ .

Untersuchen Sie die Funktionen auf Monotonie: a) $f(x)=-0,25 x^{2}+4$ , b) $f(x)=x^{3}-3 x$ , c) $f(x)=2 e^{x}-1$ , d) $f(x)=2 \cos (x)$ für $0 \leq x \leq 2 \pi$ .

Find the derivative of the function $\frac{20}{3 \sqrt[5]{t}}$ with respect to $t$ .

Find the derivative of $a(t)=\int_{t}^{-1} e^{w+w^{2}} d w$ using the Fundamental Theorem of Calculus.

Evaluate the integral: $\int_{0}^{3 \pi / 4}|\cos x| d x$ [8 points]

Find the area of one petal of the rose $r = \cos 2\theta$ using $A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta$ . Determine $\alpha$ and $\beta$ for one petal.

Find the area of one loop of the lemniscate given by $r^{2}=4 \cos 2 \theta$ . Use $A = \frac{1}{2}\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (r(\theta))^2 d\theta$ .

Find the $x$ -coordinate of point $\mathrm{P}$ on the curve $y=2-x^{3}$ where the normal has a gradient of 3.

A cow's eye level is 4 ft. A billboard is 9 ft high, 12 ft above ground. Find $\theta(x)$ and $x_{0}$ to maximize $\theta$ .

Berechne den Gesamtinhalt der Fläche zwischen $f(x)=\frac{1}{2} x^{2}-\frac{5}{2} x+2$ und der $x$ -Achse im Intervall $[0 ; 3]$ .

Find the area of the inner loop of the limaçon with the polar equation $r = 2 \sin \theta - 1$ .

Find the half-life of a radioactive substance with a decay rate of $9.7\%$ per day using the exponential decay model.

Calculate $\Delta G$ using $\Delta G^\circ$ and $Q = \frac{[\text{CO}_2][\text{H}_2]^2}{[\text{CH}_4][\text{O}_2]}$ .

Find $\lim _{x \rightarrow 0} x \cos \left(\frac{1}{x}\right)$ given $-x \leq x \cos \left(\frac{1}{x}\right) \leq x$ for $x>0$ .

Find the local minimum and maximum points of the curve $y=2x^{3}+3x^{2}-12x-10$ .

Use the Intermediate Value Theorem to show $f(x)=3x^{3}+3x^{2}-4x-5$ has a zero between $x=1$ and $x=2$ .

A differentiable function $f$ on $[-2, 7]$ has $f'(c)=0$ for $-2<c<7$ and $f(c)<f(7)$ . Which statement is true?

Find $c$ such that the instantaneous rate of change of $f(x)=2\sqrt{x}$ equals the average rate of change from $1$ to $4$ .

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\cos 4 x \tan 4 x}{6 x}$ .

Find the equation of the tangent line to $f(x)=4 x^{3}-5 x+3$ at $x=-1$ .

Find the average rate of change of $f(x)=1-x^{2}$ from $x=-2$ to $x=-1$ . Choices: A. 3 B. 0 C. -3 D. undefined

Find the limit: $\lim _{x \rightarrow 0} \frac{2 x^{2}+1 - 1}{x^{2}}$ for the function $f(x)=2 x^{2}+1$ .

Find the radius that maximizes the volume of a cylinder with height + circumference = 30 cm, where $V=\pi r^{2} h$ .

Find $\lim _{x \rightarrow a} \frac{x^{2}-a^{2}}{x^{4}-a^{4}}$ for $x \neq a$ .

Find the coordinates of the point on the curve $y = \sqrt{x}$ where the gradient equals $2$ .

Find the limit as $x$ approaches infinity for the expression: $\frac{x^{2}-5 x+10}{x^{3}-3 x^{2}+13}$ . Show your work.

Find the limit: $\lim _{x \rightarrow-\infty} \frac{-7 x^{2}-8 x+9}{-4 x^{2}+7 x+15}$ . Show your work.

Evaluate the function $g(x)=\frac{x+\cos x}{x+3}$ and determine which statements about limits and asymptotes are true.

Find the slope of the tangent line for $f(x)=3 \sqrt[3]{x^{2}}+2 x$ at $x=-8$ .

Find the slope of the normal line to $h(x)=\frac{3 x}{\sqrt[3]{x}}$ at $x=-8$ .

Find the limit: $\lim _{x \rightarrow-2} \frac{1}{x+2}$ . Show your work.

Find the limit: $\lim _{x \rightarrow(\pi / 2)^{+}} \tan x$ . Show your work.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \sqrt{\frac{4 x^{2}}{3+25 x^{2}}}$ . Show your work.

Determine if the function $f(x)=x^{3}+3x^{2}$ is increasing or decreasing at $x=-3$ and $x=0$ .

Calculate the sum of the series: $\sum_{n=0}^{\infty}(-1)^{n} \frac{4}{9^{n}}$ .

Check if the series $\sum_{n=1}^{\infty} \frac{7}{n(n+3)}$ converges or diverges, and find the sum if it converges.

How much water is left in a 1000-liter tank after 1 hour if pumped out at $3 - 3 e^{-0.15 t}$ L/min?

What is the value of the definite integral $\int_{1}^{3} 2 x \, dx$ ? (a) 1 (b) 2 (c) 4 (d) 8 (e) 10

Calculate the area under the curve of $f(x)=x^{2}+7$ from $x=0$ to $x=6$ .

Determine if the series $\sum_{n=1}^{\infty} \frac{5 n}{n^{2}+2}$ converges or diverges.

Determine if the series $\sum_{n=1}^{\infty} \frac{3+7 \cos n}{n^{3}}$ converges or diverges.

Determine if the series converges or diverges: $\sum_{n=1}^{\infty} \frac{4 \sqrt{n}}{8 n^{3 / 2}+7 n-8}$

Determine if the series converges or diverges: $\sum_{n=1}^{\infty} \frac{4(n l)^{2}}{(2 n) !}$

Estimate the area between $f(x)$ and the $x$ -axis from $0$ to $20$ using left and right sums. Round to one decimal place. Area $\approx$

A coffee cup starts at $135^{\circ} \mathrm{F}$ in a freezer at $0^{\circ} \mathrm{F}$ . After 10 min it's $60^{\circ} \mathrm{F}$ . Find temp after 15 min.

Determine if the series converges or diverges: $\sum_{n=1}^{\infty}\left(\frac{10 n}{\ln n+9 n+9}\right)^{n}$

Find the function $f$ given $f^{\prime \prime}(x)=x^{2}+1$ , $f^{\prime}(0)=4$ , and $f(0)=2$ . Which is $f(x)$ ?

After substituting $u=2 e^{x}-3$ , what does the integral $\int 6 e^{x} \sqrt{2 e^{x}-3} d x$ become?

Find the indefinite integral $\int \sin (x) e^{\cos (x)} d x$ . Which option is correct? (a) (b) (c) (d) (e)

Calculate the series sum: $S = \sum_{n=4}^{\infty} 64\left(\frac{1}{5}\right)^{n-1}$ . Choose the correct answer.

Find the average rate of change of $p(x) = \sqrt{15x}$ over the interval $[0.01, 1.01]$ .

Determine if the series $\sum_{n=1}^{\infty}(-6)^{-n}$ converges absolutely, conditionally, or diverges.

Find the difference quotient, $\frac{f(x+h)-f(x)}{h}$ , for the function $f(x)=6x-4$ .

En jeger skyter horisontalt mot en blink 98.1 m unna, $180 \mathrm{~cm}$ over bakken. Hvor langt under blinken treffer kulen? $28.4$ $\mathrm{cm}$

Determine if the series converges absolutely, conditionally, or diverges: $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{3 n^{1 / 2}+8}$

Find the value of the integral $\int_{1}^{3} f^{\prime}(x) dx$ given $f(0)=-2$ , $f(1)=2$ , $f(2)=-3$ , $f(3)=4$ . Choices: (a) 1 (b) 2 (c) 3 (d) 6 (e) 9.

Determine the truth of these statements about function $g$ : I. $a < 0$ for normal lines on $(-\infty, -2)$ ; II. $g'(x) > 0$ on $(-2, 2)$ and $(3, \infty)$ ; III. $g'$ is undefined at $x = -2$ and $x = 2$ .

Find total revenue for 100 units using $R^{\prime}(q)=150-12 \sqrt{q}$ and marginal revenue at 100 units.