Calculus

Problem 10301

Evaluate the triple integral: $\int_{0}^{2 \pi} \int_{0}^{\arctan \left(\frac{1}{2}\right)} \int_{0}^{\csc (\phi)} \rho^{3} \sin ^{2} \phi \cos \theta d \rho d \phi d \theta$ .

See Solution

Problem 10302

Determine the radius and interval of convergence for the series $\sum_{n=0}^{\infty}(-1)^{n} \frac{n}{2^{n}}(x-1)^{n}$ .

See Solution

Problem 10303

Find the derivative of the function $f(x) = 0 + x$ .

See Solution

Problem 10304

Berechnen Sie die Ableitung von $f(x)=0,7x$ .

See Solution

Problem 10305

Find $\frac{d y}{d x}$ for the equation $-13 x^{7}+5 x^{65} y+y^{8}=-7$ . Then, find the tangent line at $(1,1)$ in $y=m x+b$ format.

See Solution

Problem 10306

Find $\alpha \in \mathbb{R}$ for $F(x) = \alpha \sin(x^2 + 3) + c$ where $f(x) = x \cos(x^2 + 3)$ . Answer: $\alpha = \frac{1}{2}$ .

See Solution

Problem 10307

Bestimmen Sie die Ableitung von $f(x) = e^{x} + x$ .

See Solution

Problem 10308

Evaluate the integral from 1 to 2 of the function $x^{3} - \frac{1}{x^{2}}$ .

See Solution

Problem 10309

Differentiate $y=\csc x(x+\cot x)$ and find $y'$ .

See Solution

Problem 10310

Find the derivative of $\tan \left(\frac{\pi}{6}\right)$ .

See Solution

Problem 10311

Écrivez l'intégrale pour l'aire entre l'axe des $x$ , $f(x)=-e^{0,1 x}-3$ , $x=1$ et $x=3$ . Ne pas évaluer.

See Solution

Problem 10312

Écrivez l'intégrale pour l'aire entre $f(x)=2 x-3$ et $g(x)=-x^{2}+7 x+3$ sans l'évaluer.

See Solution

Problem 10313

Find the tangent line equation for the curve $x e^{2}+3 x+4 y=4$ .

See Solution

Problem 10314

Gravel is dumped at $50 \mathrm{ft}^{3}/\mathrm{m}$ . Find the height increase rate when the pile is 10 ft tall.

See Solution

Problem 10315

Bestimme $f^{\prime}(1)$ und $f^{\prime}(2)$ für die Funktion $f(x)=3 x^{2}$ .

See Solution

Problem 10316

Estimate the population of country $X$ in 22 years, given $N(t)=500 e^{0.02 t}$ for an initial population of 500 million.

See Solution

Problem 10317

Bestimme die Ableitung von $f(x)=4 e^{3 x}+x^{-6}$ .

See Solution

Problem 10318

Calculate the integral $\int_{0}^{a}(a x-x^{2}) dx$ .

See Solution

Problem 10319

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{x}-1}{\sin (15 x)}$ .

See Solution

Problem 10320

Find the limit as $x$ approaches -4 for $\frac{f(x)}{g(x)}$ given $f(-4)=0$ , $g(-4)=0$ , $f'(-4)=-5$ , $g'(-4)=3$ .

See Solution

Problem 10321

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{2}-x-2}{-5 x+10}=$

See Solution

Problem 10322

Evaluate the limit using L'Hôpital's rule: $\lim _{x \rightarrow 0} \frac{3^{x}-11^{x}}{x}$ .

See Solution

Problem 10323

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{15 x^{2}}{e^{10 x}}=$

See Solution

Problem 10324

A cone has a radius of 12 units and height of 8 units. Water flows in at 24 cubic units/sec. Find the radius change rate at 6 units depth.

See Solution

Problem 10325

Evaluate the limit using L'Hôpital's rule: $\lim _{s \rightarrow 25} \frac{25-s}{5-\sqrt{s}}=$

See Solution

Problem 10326

Evaluate the limit using L'Hôpital's rule: $\lim _{x \rightarrow 0} \frac{e^{x}+2 x-1}{3 x}=$

See Solution

Problem 10327

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{(-2+h)^{2}-4}{h}=$

See Solution

Problem 10328

Berechnen Sie die Ableitung von $f(x) = 3x^2 + 2 \ln(x)$ bei $x=2$ .

See Solution

Problem 10329

Find the limit as $x$ approaches 1 for the piecewise function $f(x)=\left\{\begin{array}{l}3 x-1, x \leq 1 \\ \frac{3}{2} x^{2}, x>1\end{array}\right.$ .

See Solution

Problem 10330

Find the slope of the secant line between points $x=a$ and $x=1$ for the function $f(x)=2(x-3)^{2}+1$ .

See Solution

Problem 10331

Evaluate the limit using L'Hôpital's rule:
$\lim _{y \rightarrow 4} \frac{6(y^{2}-1)}{6y^{2}(y-1)^{3}}=$

See Solution

Problem 10332

Evaluate the limit using L'Hôpital's rule: $\lim _{x \rightarrow-5} \frac{x^{2}+6 x+5}{x+5}.$

See Solution

Problem 10333

Find the slope of the curve $4(x^{2}+y^{2})^{2}=25xy^{2}$ at the point $(1,2)$ .

See Solution

Problem 10334

Aubrey invested \$ 61,000 at a continuous interest rate of 1.9\%. How long to reach \$ 73,600? Round to the nearest tenth.

See Solution

Problem 10335

Evaluate the limit using L'Hôpital's rule: $\lim _{a \rightarrow 1} \frac{a^{3}-a}{a^{2}-1}=$

See Solution

Problem 10336

Bestimme den Wert von $x$ , der das Volumen einer Schachtel maximiert, wenn $x$ von Ecken eines $16 \mathrm{~cm} \times 10 \mathrm{~cm}$ Rechtecks ausgeschnitten wird.

See Solution

Problem 10337

Identify where L'Hospital's Rule applies for these limits:
$\lim _{x \rightarrow 0} \frac{x+\tan x}{\sin x}$ , $\lim _{x \rightarrow 0} \frac{\sin (7 x)}{7 x}$ , $\lim _{x \rightarrow \infty} \frac{e^{-x}}{x}$ , $\lim _{x \rightarrow \infty} \frac{\ln x}{\sqrt{x}}$ .

See Solution

Problem 10338

Grayson wants to know how much to invest at $4\%$ interest to reach \$15,000 in 5 years.

See Solution

Problem 10339

Given $f(7)=9$ , $f^{\prime}(7)=-8$ , and $f^{\prime}(x)$ continuous, find:
1. $\lim _{x \rightarrow 7} \frac{f(x)-9}{x^{2}-49}$
2. $\lim _{x \rightarrow 7} \frac{\ln (f(x)-8)}{x-7}$

See Solution

Problem 10340

Bestimmen Sie die Koordinaten von Punkt $P$ mit maximalem Flächeninhalt für die Funktionen a) $f(x)=4-\frac{1}{2} x^{3}$ , b) $f(x)=\frac{1}{x^{2}}$ , c) $f(x)=\sqrt{4-x}$ . Finden Sie die Extrema durch Ableitungen und setzen Sie diese gleich Null.

See Solution

Problem 10341

Bestimme den Wert von $\mathrm{x}$ für maximales Volumen einer Schachtel und die Koordinaten von $P$ für maximalen Flächeninhalt. a) $f(x)=4-\frac{1}{2} x^{3} ; x \in[0 ; 2]$ b) $f(x)=\frac{1}{x^{2}} ; x \in[1 ; 3]$ c) $f(x)=\sqrt{4-x} ; x \in[0 ; 4]$

See Solution

Problem 10342

Bestimmen Sie die Steigung von $f_{t}(x)$ an $x_{0}$ für die folgenden Funktionen: a) $f_{t}(x)=(x+t)(x-t)^{2}$ bei $P(0,8)$ , b) $f_{t}(x)=t \cdot 2^{x}$ bei $P(-2,1)$ , c) $f_{t}(x)=e^{x-t}$ bei $P(-1,e^{2})$ .

See Solution

Problem 10343

Bestimmen Sie die Steigung von $f_{t}(x)$ für $x_{0}$ in Abhängigkeit von $t$ für die folgenden Funktionen: a) $f_{t}(x)=t-\frac{t}{3} x$ , $x_{0}=-3$ b) $f_{t}(x)=\frac{t}{4} x^{3}-t x^{2}+t x+1$ , $x_{0}=1$ c) $f_{t}(x)=t \cdot e^{t x}$ , $x_{0}=0$

See Solution

Problem 10344

Find the derivative of $y=\frac{\sqrt{1-x}}{x^{4}}+4(x+3)^{3}$ and show $y'=\frac{7 x-8}{2 x^{5}(1-x)^{\frac{1}{2}}}+12(x+3)^{2}$ .

See Solution

Problem 10345

Find the absolute extrema of $f(x)=0.002 x^{2}+3.6 x-30$ over the interval $(-\infty, \infty)$ .

See Solution

Problem 10346

A quantity starts at 7800 and grows continuously at $0.8\%$ per decade. Find its value after 59 years, rounded to two decimal places.

See Solution

Problem 10347

Berechnen Sie die Steigung von $f_{t}(x)=x^{3}-12 t^{2} x$ im Ursprung in Abhängigkeit von $t$ .

See Solution

Problem 10348

Find the derivative of the composite function $h(x)=f(g(x))$ at $x=2$ using the graphs of $y=f(x)$ and $y=g(x)$ .

See Solution

Problem 10349

Find the limit: $\lim _{x \rightarrow 1}\left(e^{x^{5}}-e^{x^{2}}\right)$ .

See Solution

Problem 10350

Find the limit as $x$ approaches 5 from the left of $\frac{x^{2}-4x-5}{x^{2}-25}$ .

See Solution

Problem 10351

Determine the horizontal asymptotes of the function $f(x)=\frac{3x^{2}-2}{x^{2}+4}$ .

See Solution

Problem 10352

Find the derivative of $d(z)=\left(\frac{e^{z^{7}-5 z^{3}}{(2 z^{7}+7 z^{2})^{5}}\right)^{26}$.

See Solution

Problem 10353

Trouver la dérivée de $f(x)=3 x^{2}-1$ à $x=1$ .

See Solution

Problem 10354

Find $\frac{d y}{d x}$ for $y=u^{3}-2 u+1$ with $u=2 \sqrt{x}$ at $x=4$ .

See Solution

Problem 10355

Evaluate the integral: $\int_{2}^{4} \int_{0}^{y}\left(\frac{1}{2}+3 y\right) d x d y$

See Solution

Problem 10356

Calculate the volume of a box with dimensions 2 m, 5 m, and 6 m using a double integral. Use calculus for full credit.

See Solution

Problem 10357

Evaluate the integral $\int_{R}(x^{2}-2 y^{2}) d A$ over the first quadrant between circles of radius 4 and 5.

See Solution

Problem 10358

Find the average rate of change of $f(x)=2x^{2}+1$ over these intervals: (a) 0 to 2, (b) 3 to 5, (c) -3 to 0.

See Solution

Problem 10359

Find the average rate of change of $f(x)=2x^{2}+1$ over these intervals: (a) 0 to 2, (b) 3 to 5, (c) -3 to 0.

See Solution

Problem 10360

Find the derivative of $y=\left(-5 x^{3}-3\right)^{3}$ with respect to $x$ using the chain rule.

See Solution

Problem 10361

Find the derivative of $y=\left(-5 x^{3}-3\right)^{3}$ with respect to $x$ .

See Solution

Problem 10362

Find the coffee temperature after 30 minutes using $f(t)=57(0.9)^{t}+21$ . What is the asymptote and its meaning?

See Solution

Problem 10363

Calculate the integral: $\int\left(6 t \sqrt{t}-9 \sqrt[7]{t^{6}}\right) d t$

See Solution

Problem 10364

Evaluate the integral: $\int 10 \, dx$

See Solution

Problem 10365

Calculate the indefinite integral: $\int\left(6 t \sqrt{t}-9 \sqrt[7]{t^{6}}\right) d t$

See Solution

Problem 10366

Find the derivative of $\ln(x^2 + 3)$ .

See Solution

Problem 10367

Find the indefinite integral: $\int\left(10 t \sqrt{t}-7 \sqrt[7]{t^{6}}\right) dt$

See Solution

Problem 10368

Evaluate the integral: $\int\left(7 x^{4 / 5}-6 x^{5 / 2}\right) d x$

See Solution

Problem 10369

Find the average rate of change of $f(x) = \sqrt{-7x - 1}$ from $x_1 = -8$ to $x_2 = -5$ , rounded to the nearest hundredth.

See Solution

Problem 10370

Evaluate the integral: $\int 12 \, dx$

See Solution

Problem 10371

What must be true if $F(x)$ and $G(x)$ are both antiderivatives of $f(x)$ ?
A. Not possible for two antiderivatives. B. They are the same function. C. They differ only by a constant. D. They differ by a factor of $x^{2}$ .

See Solution

Problem 10372

Evaluate the integral: $\int\left(-9 t^{-2.5}-8 t^{-1}\right) d t$ .

See Solution

Problem 10373

Find the derivative of $y=((x+5)^{5}-1)^{4}$ using the chain rule.

See Solution

Problem 10374

Water drains from a cone (height 12 ft, diameter 8 ft) into a cylinder (area $400 \pi$ ft²).
(a) Find volume $V$ of cone as a function of $h$ . (b) Rate of volume change when $h=3$ ? Include units. (c) Rate of depth change $y$ in cylinder when $h=3$ ? Include units.

See Solution

Problem 10375

A quantity starts at 2300 and grows at $45\%$ per minute. Find its value after 0.2 hours, rounded to the nearest hundredth.

See Solution

Problem 10376

A quantity starts at 2300 and grows at $45\%$ per minute. Find its value after 0.2 hours, rounded to the nearest hundredth.

See Solution

Problem 10377

Evaluate the triple integral domain and find the average value $\bar{f}$ if the integral equals $24\pi$ .

See Solution

Problem 10378

A quantity starts at 6200 and decays at $5.5\%$ monthly. Find its value after 4 years, rounded to the nearest hundredth.

See Solution

Problem 10379

Determine the long run behavior of these functions:
1. $f(x)=\frac{x^{3}+1}{x^{2}+2}$
2. $f(x)=\frac{x^{2}+1}{x^{2}+2}$
3. $f(x)=\frac{x^{2}+1}{x^{3}+2}$

See Solution

Problem 10380

Approximate $f(10)$ using the tangent line at $(8,2)$ for the function $f(x)=\sqrt[3]{x}$ .

See Solution

Problem 10381

Find the velocity of the particle at the first time it is at the origin, given $x(t)=\cos \sqrt{t}$ .

See Solution

Problem 10382

Calculate the compound amount and interest for \$130,000 at 5.4% for 16 years with continuous compounding.

See Solution

Problem 10383

A sky diver jumps from $1,251 \mathrm{~m}$ . How long (in s) until he hits the ground if $\mathrm{a}=9.715 \mathrm{~m/s}^{2}$ ?

See Solution

Problem 10384

Given the drug concentration formula $C=\frac{5 \cdot t}{21+t^{2}}$ :
a. Find $C$ after 3 hours (round to 3 decimals). b. How long until $C$ drops to $0.7\%$ ? (round to 2 decimals) c. Determine the end behavior of $C$ as $t \rightarrow \infty$ .

See Solution

Problem 10385

Determine the horizontal asymptote for the function $f(x)=\frac{2}{x-3}$ .

See Solution

Problem 10386

Determine where the function $f(x)=\sin x+\sin ^{3} x$ increases/decreases and find local minima/maxima for $-\pi<x<\pi$ .

See Solution

Problem 10387

Find Sarah's acceleration $x \mathrm{mph} / \mathrm{h}$ between 7:00 pm (15 mph) and 7:20 pm (50 mph).

See Solution

Problem 10388

Find where the prize wheel stops, starting at 22, spinning at $116.30 \mathrm{rpm}$ , slowing at $1.900 \mathrm{rad/s^2}$ .

See Solution

Problem 10389

Find the acceleration $x$ in mph/h for Sarah driving from 20 mph to 40 mph between 7:00 pm and 7:10 pm.

See Solution

Problem 10390

Find the derivative of $y=-6(7x^{2}+3)^{-6}$ . What is $\frac{dy}{dx}$ ?

See Solution

Problem 10391

Find the average rate of change of $f(x)$ from $x_{1}=1$ to $x_{2}=4$ , rounding to the nearest hundredth. $f(x)=\sqrt{9x+1}$

See Solution

Problem 10392

Find the derivative of $y=(6x^4-4x^2+3)^4$ . What is $u=g(x)$ if $y=f(u)$ ?

See Solution

Problem 10393

Find the derivative of $f(t)=8 \sqrt{4 t^{2}+9}$ . What is $u=g(t)$ if you write $f$ as $f=h(u)$ ?

See Solution

Problem 10394

Find Sarah's acceleration $x$ in mph/h between 10:00 pm (30 mph) and 10:15 pm (50 mph).

See Solution

Problem 10395

Find the rate of change of $A=1700\left(1+\frac{r}{36,500}\right)^{1095}$ with respect to $r$ . Choices: (a) $6 \%$ , (b) $2 \%$ , (c) $8 \%$ .

See Solution

Problem 10396

Find values of $x$ where the tangent line of $f(x)=\sqrt{x^{3}-9 x^{2}+24 x+1}$ is horizontal.

See Solution

Problem 10397

Find the derivative $f^{\prime}\left(\frac{\pi}{2}\right)$ for the function $f(x)=\frac{x^{3}+2 \cos (x)}{3 \sin (x)}$ .

See Solution

Problem 10398

Compute the derivative $h^{\prime}(\pi / 3)$ for the function $h(t)=e^{-5 t} \cos (5 t)$ .

See Solution

Problem 10399

A company has a fixed cost of \ $400,000 and \$100 per bike. What does$ \overline{C}(4000)=200$ mean? What's the horizontal asymptote?

See Solution

Problem 10400

Find the derivative of $y = \tan \left(\cot \frac{x^{2}}{2^{x}}\right)$ with respect to $x$ .

See Solution

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Calculus

Problem 10301

Problem 10302

Determine the radius and interval of convergence for the series ∑n=0∞(−1)nn2n(x−1)n\sum_{n=0}^{\infty}(-1)^{n} \frac{n}{2^{n}}(x-1)^{n}∑n=0∞​(−1)n2nn​(x−1)n.

Problem 10303

Find the derivative of the function f(x)=0+xf(x) = 0 + xf(x)=0+x.

Problem 10304

Berechnen Sie die Ableitung von f(x)=0,7xf(x)=0,7xf(x)=0,7x.

Problem 10305

Find dydx\frac{d y}{d x}dxdy​ for the equation −13x7+5x65y+y8=−7-13 x^{7}+5 x^{65} y+y^{8}=-7−13x7+5x65y+y8=−7. Then, find the tangent line at (1,1)(1,1)(1,1) in y=mx+by=m x+by=mx+b format.

Problem 10306

Find α∈R\alpha \in \mathbb{R}α∈R for F(x)=αsin⁡(x2+3)+cF(x) = \alpha \sin(x^2 + 3) + cF(x)=αsin(x2+3)+c where f(x)=xcos⁡(x2+3)f(x) = x \cos(x^2 + 3)f(x)=xcos(x2+3). Answer: α=12\alpha = \frac{1}{2}α=21​.

Problem 10307

Bestimmen Sie die Ableitung von f(x)=ex+xf(x) = e^{x} + xf(x)=ex+x.

Problem 10308

Evaluate the integral from 1 to 2 of the function x3−1x2x^{3} - \frac{1}{x^{2}}x3−x21​.

Problem 10309

Differentiate y=csc⁡x(x+cot⁡x)y=\csc x(x+\cot x)y=cscx(x+cotx) and find y′y'y′.

Problem 10310

Find the derivative of tan⁡(π6)\tan \left(\frac{\pi}{6}\right)tan(6π​).

Problem 10311

Écrivez l'intégrale pour l'aire entre l'axe des xxx, f(x)=−e0,1x−3f(x)=-e^{0,1 x}-3f(x)=−e0,1x−3, x=1x=1x=1 et x=3x=3x=3. Ne pas évaluer.

Problem 10312

Écrivez l'intégrale pour l'aire entre f(x)=2x−3f(x)=2 x-3f(x)=2x−3 et g(x)=−x2+7x+3g(x)=-x^{2}+7 x+3g(x)=−x2+7x+3 sans l'évaluer.

Problem 10313

Find the tangent line equation for the curve xe2+3x+4y=4x e^{2}+3 x+4 y=4xe2+3x+4y=4.

Problem 10314

Gravel is dumped at 50ft3/m50 \mathrm{ft}^{3}/\mathrm{m}50ft3/m. Find the height increase rate when the pile is 10 ft tall.

Problem 10315

Bestimme f′(1)f^{\prime}(1)f′(1) und f′(2)f^{\prime}(2)f′(2) für die Funktion f(x)=3x2f(x)=3 x^{2}f(x)=3x2.

Problem 10316

Estimate the population of country XXX in 22 years, given N(t)=500e0.02tN(t)=500 e^{0.02 t}N(t)=500e0.02t for an initial population of 500 million.

Problem 10317

Bestimme die Ableitung von f(x)=4e3x+x−6f(x)=4 e^{3 x}+x^{-6}f(x)=4e3x+x−6.

Problem 10318

Calculate the integral ∫0a(ax−x2)dx\int_{0}^{a}(a x-x^{2}) dx∫0a​(ax−x2)dx.

Problem 10319

Find the limit: lim⁡x→0ex−1sin⁡(15x)\lim _{x \rightarrow 0} \frac{e^{x}-1}{\sin (15 x)}limx→0​sin(15x)ex−1​.

Problem 10320

Find the limit as xxx approaches -4 for f(x)g(x)\frac{f(x)}{g(x)}g(x)f(x)​ given f(−4)=0f(-4)=0f(−4)=0, g(−4)=0g(-4)=0g(−4)=0, f′(−4)=−5f'(-4)=-5f′(−4)=−5, g′(−4)=3g'(-4)=3g′(−4)=3.

Problem 10321

Find the limit: lim⁡x→2x2−x−2−5x+10=\lim _{x \rightarrow 2} \frac{x^{2}-x-2}{-5 x+10}=x→2lim​−5x+10x2−x−2​=

Problem 10322

Evaluate the limit using L'Hôpital's rule: lim⁡x→03x−11xx\lim _{x \rightarrow 0} \frac{3^{x}-11^{x}}{x}x→0lim​x3x−11x​.

Problem 10323

Evaluate the limit: lim⁡x→∞15x2e10x=\lim _{x \rightarrow \infty} \frac{15 x^{2}}{e^{10 x}}=limx→∞​e10x15x2​=

Problem 10324

A cone has a radius of 12 units and height of 8 units. Water flows in at 24 cubic units/sec. Find the radius change rate at 6 units depth.

Problem 10325

Evaluate the limit using L'Hôpital's rule: lim⁡s→2525−s5−s=\lim _{s \rightarrow 25} \frac{25-s}{5-\sqrt{s}}=s→25lim​5−s​25−s​=

Problem 10326

Evaluate the limit using L'Hôpital's rule: lim⁡x→0ex+2x−13x=\lim _{x \rightarrow 0} \frac{e^{x}+2 x-1}{3 x}=x→0lim​3xex+2x−1​=

Problem 10327

Evaluate the limit: lim⁡h→0(−2+h)2−4h=\lim _{h \rightarrow 0} \frac{(-2+h)^{2}-4}{h}=limh→0​h(−2+h)2−4​=

Problem 10328

Berechnen Sie die Ableitung von f(x)=3x2+2ln⁡(x)f(x) = 3x^2 + 2 \ln(x)f(x)=3x2+2ln(x) bei x=2x=2x=2.

Problem 10329

Find the limit as xxx approaches 1 for the piecewise function f(x)={3x−1,x≤132x2,x>1f(x)=\left\{\begin{array}{l}3 x-1, x \leq 1 \\ \frac{3}{2} x^{2}, x>1\end{array}\right.f(x)={3x−1,x≤123​x2,x>1​.

Problem 10330

Find the slope of the secant line between points x=ax=ax=a and x=1x=1x=1 for the function f(x)=2(x−3)2+1f(x)=2(x-3)^{2}+1f(x)=2(x−3)2+1.

Problem 10331

Evaluate the limit using L'Hôpital's rule: lim⁡y→46(y2−1)6y2(y−1)3= \lim _{y \rightarrow 4} \frac{6(y^{2}-1)}{6y^{2}(y-1)^{3}}= y→4lim​6y2(y−1)36(y2−1)​=

Problem 10332

Evaluate the limit using L'Hôpital's rule: lim⁡x→−5x2+6x+5x+5.\lim _{x \rightarrow-5} \frac{x^{2}+6 x+5}{x+5}.x→−5lim​x+5x2+6x+5​.

Problem 10333

Find the slope of the curve 4(x2+y2)2=25xy24(x^{2}+y^{2})^{2}=25xy^{2}4(x2+y2)2=25xy2 at the point (1,2)(1,2)(1,2).

Problem 10334

Aubrey invested \$ 61,000 at a continuous interest rate of 1.9\%. How long to reach \$ 73,600? Round to the nearest tenth.

Problem 10335

Evaluate the limit using L'Hôpital's rule: lim⁡a→1a3−aa2−1=\lim _{a \rightarrow 1} \frac{a^{3}-a}{a^{2}-1}=a→1lim​a2−1a3−a​=

Problem 10336

Bestimme den Wert von xxx, der das Volumen einer Schachtel maximiert, wenn xxx von Ecken eines 16 cm×10 cm16 \mathrm{~cm} \times 10 \mathrm{~cm}16 cm×10 cm Rechtecks ausgeschnitten wird.

Problem 10337

Problem 10338

Grayson wants to know how much to invest at 4%4\%4% interest to reach \$15,000 in 5 years.

Problem 10339

Problem 10340

Problem 10341

Problem 10342

Problem 10343

Determine the radius and interval of convergence for the series $\sum_{n=0}^{\infty}(-1)^{n} \frac{n}{2^{n}}(x-1)^{n}$ .

Find the derivative of the function $f(x) = 0 + x$ .

Berechnen Sie die Ableitung von $f(x)=0,7x$ .

Find $\frac{d y}{d x}$ for the equation $-13 x^{7}+5 x^{65} y+y^{8}=-7$ . Then, find the tangent line at $(1,1)$ in $y=m x+b$ format.

Find $\alpha \in \mathbb{R}$ for $F(x) = \alpha \sin(x^2 + 3) + c$ where $f(x) = x \cos(x^2 + 3)$ . Answer: $\alpha = \frac{1}{2}$ .

Bestimmen Sie die Ableitung von $f(x) = e^{x} + x$ .

Evaluate the integral from 1 to 2 of the function $x^{3} - \frac{1}{x^{2}}$ .

Differentiate $y=\csc x(x+\cot x)$ and find $y'$ .

Find the derivative of $\tan \left(\frac{\pi}{6}\right)$ .

Écrivez l'intégrale pour l'aire entre l'axe des $x$ , $f(x)=-e^{0,1 x}-3$ , $x=1$ et $x=3$ . Ne pas évaluer.

Écrivez l'intégrale pour l'aire entre $f(x)=2 x-3$ et $g(x)=-x^{2}+7 x+3$ sans l'évaluer.

Find the tangent line equation for the curve $x e^{2}+3 x+4 y=4$ .

Gravel is dumped at $50 \mathrm{ft}^{3}/\mathrm{m}$ . Find the height increase rate when the pile is 10 ft tall.

Bestimme $f^{\prime}(1)$ und $f^{\prime}(2)$ für die Funktion $f(x)=3 x^{2}$ .

Estimate the population of country $X$ in 22 years, given $N(t)=500 e^{0.02 t}$ for an initial population of 500 million.

Bestimme die Ableitung von $f(x)=4 e^{3 x}+x^{-6}$ .

Calculate the integral $\int_{0}^{a}(a x-x^{2}) dx$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{x}-1}{\sin (15 x)}$ .

Find the limit as $x$ approaches -4 for $\frac{f(x)}{g(x)}$ given $f(-4)=0$ , $g(-4)=0$ , $f'(-4)=-5$ , $g'(-4)=3$ .

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{2}-x-2}{-5 x+10}=$

Evaluate the limit using L'Hôpital's rule: $\lim _{x \rightarrow 0} \frac{3^{x}-11^{x}}{x}$ .

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{15 x^{2}}{e^{10 x}}=$

Evaluate the limit using L'Hôpital's rule: $\lim _{s \rightarrow 25} \frac{25-s}{5-\sqrt{s}}=$

Evaluate the limit using L'Hôpital's rule: $\lim _{x \rightarrow 0} \frac{e^{x}+2 x-1}{3 x}=$

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{(-2+h)^{2}-4}{h}=$

Berechnen Sie die Ableitung von $f(x) = 3x^2 + 2 \ln(x)$ bei $x=2$ .

Find the limit as $x$ approaches 1 for the piecewise function $f(x)=\left\{\begin{array}{l}3 x-1, x \leq 1 \\ \frac{3}{2} x^{2}, x>1\end{array}\right.$ .

Find the slope of the secant line between points $x=a$ and $x=1$ for the function $f(x)=2(x-3)^{2}+1$ .

Evaluate the limit using L'Hôpital's rule:
$\lim _{y \rightarrow 4} \frac{6(y^{2}-1)}{6y^{2}(y-1)^{3}}=$

Evaluate the limit using L'Hôpital's rule: $\lim _{x \rightarrow-5} \frac{x^{2}+6 x+5}{x+5}.$

Find the slope of the curve $4(x^{2}+y^{2})^{2}=25xy^{2}$ at the point $(1,2)$ .

Evaluate the limit using L'Hôpital's rule: $\lim _{a \rightarrow 1} \frac{a^{3}-a}{a^{2}-1}=$

Bestimme den Wert von $x$ , der das Volumen einer Schachtel maximiert, wenn $x$ von Ecken eines $16 \mathrm{~cm} \times 10 \mathrm{~cm}$ Rechtecks ausgeschnitten wird.

Grayson wants to know how much to invest at $4\%$ interest to reach \$15,000 in 5 years.

Find the derivative of $y=\frac{\sqrt{1-x}}{x^{4}}+4(x+3)^{3}$ and show $y'=\frac{7 x-8}{2 x^{5}(1-x)^{\frac{1}{2}}}+12(x+3)^{2}$ .

Find the absolute extrema of $f(x)=0.002 x^{2}+3.6 x-30$ over the interval $(-\infty, \infty)$ .

A quantity starts at 7800 and grows continuously at $0.8\%$ per decade. Find its value after 59 years, rounded to two decimal places.

Berechnen Sie die Steigung von $f_{t}(x)=x^{3}-12 t^{2} x$ im Ursprung in Abhängigkeit von $t$ .

Find the derivative of the composite function $h(x)=f(g(x))$ at $x=2$ using the graphs of $y=f(x)$ and $y=g(x)$ .

Find the limit: $\lim _{x \rightarrow 1}\left(e^{x^{5}}-e^{x^{2}}\right)$ .

Find the limit as $x$ approaches 5 from the left of $\frac{x^{2}-4x-5}{x^{2}-25}$ .

Determine the horizontal asymptotes of the function $f(x)=\frac{3x^{2}-2}{x^{2}+4}$ .

Trouver la dérivée de $f(x)=3 x^{2}-1$ à $x=1$ .

Find $\frac{d y}{d x}$ for $y=u^{3}-2 u+1$ with $u=2 \sqrt{x}$ at $x=4$ .

Evaluate the integral: $\int_{2}^{4} \int_{0}^{y}\left(\frac{1}{2}+3 y\right) d x d y$

Evaluate the integral $\int_{R}(x^{2}-2 y^{2}) d A$ over the first quadrant between circles of radius 4 and 5.

Find the average rate of change of $f(x)=2x^{2}+1$ over these intervals: (a) 0 to 2, (b) 3 to 5, (c) -3 to 0.

Find the average rate of change of $f(x)=2x^{2}+1$ over these intervals: (a) 0 to 2, (b) 3 to 5, (c) -3 to 0.

Find the derivative of $y=\left(-5 x^{3}-3\right)^{3}$ with respect to $x$ using the chain rule.

Find the derivative of $y=\left(-5 x^{3}-3\right)^{3}$ with respect to $x$ .

Find the coffee temperature after 30 minutes using $f(t)=57(0.9)^{t}+21$ . What is the asymptote and its meaning?

Calculate the integral: $\int\left(6 t \sqrt{t}-9 \sqrt[7]{t^{6}}\right) d t$

Evaluate the integral: $\int 10 \, dx$

Calculate the indefinite integral: $\int\left(6 t \sqrt{t}-9 \sqrt[7]{t^{6}}\right) d t$

Find the derivative of $\ln(x^2 + 3)$ .

Find the indefinite integral: $\int\left(10 t \sqrt{t}-7 \sqrt[7]{t^{6}}\right) dt$

Evaluate the integral: $\int\left(7 x^{4 / 5}-6 x^{5 / 2}\right) d x$

Find the average rate of change of $f(x) = \sqrt{-7x - 1}$ from $x_1 = -8$ to $x_2 = -5$ , rounded to the nearest hundredth.

Evaluate the integral: $\int 12 \, dx$

What must be true if $F(x)$ and $G(x)$ are both antiderivatives of $f(x)$ ?
A. Not possible for two antiderivatives. B. They are the same function. C. They differ only by a constant. D. They differ by a factor of $x^{2}$ .

Evaluate the integral: $\int\left(-9 t^{-2.5}-8 t^{-1}\right) d t$ .

Find the derivative of $y=((x+5)^{5}-1)^{4}$ using the chain rule.

Water drains from a cone (height 12 ft, diameter 8 ft) into a cylinder (area $400 \pi$ ft²).
(a) Find volume $V$ of cone as a function of $h$ . (b) Rate of volume change when $h=3$ ? Include units. (c) Rate of depth change $y$ in cylinder when $h=3$ ? Include units.