Calculus

Problem 30301

Leiten Sie die Funktionen ab und bestimmen Sie die ersten Ableitungen für die angegebenen $f(x)$ :
1. a) $f(x)=e^{x}+x^{3}$ b) $f(x)=4 x^{2}-e^{x}$ c) $f(x)=2 e^{x}-\frac{1}{4} x^{4}$ d) $f(x)=-e^{x}+4$
2. Leiten Sie zweimal ab: a) $f(x)=-e^{x}+\frac{1}{2} x^{2}$ b) $f(x)=3 e^{x}+2 x^{3}-x+5$ c) $f(x)=\frac{2}{x}-3 e^{x}$ d) $f(x)=0,3 e^{x}-\sqrt{x}$
3. Bestimmen Sie die erste Ableitung für $a$ und $b$ reelle Zahlen: a) $f(x)=e^{x}-a x^{2}$ b) $f(x)=a e^{x}+x^{b}$ c) $f(x)=\frac{e^{x}}{a}+b^{2} x^{2}$ d) $f(x)=-a e^{x}+a$

See Solution

Problem 30302

Find $\lim \sqrt{f(x)}$ as $x$ approaches $c$ if $\lim f(x) = 4$ .

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Problem 30303

Find the limit: $\lim _{x \rightarrow -\infty} \arctan x$ .

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Problem 30304

Find the limit as $x$ approaches $-\infty$ for the function $e^{x}$ .

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Problem 30305

Find the limit as $x$ approaches $-\infty$ for $\frac{1}{x^{p}}$ where $p$ is a positive integer constant.

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Problem 30306

Find the limit: $\lim _{n \rightarrow \infty} \frac{\ln \left(\frac{4 n^{2}+a}{4 n^{2}+1}\right)}{\frac{1}{2 n^{2}-1}}$ .

See Solution

Problem 30307

Find the limit: $\lim _{x \rightarrow \infty} \arctan x$ .

See Solution

Problem 30308

Calculate the integral $\int \frac{x^{5}-3 x^{-3}+x}{2} dx$ .

See Solution

Problem 30309

Find the limit: $\lim _{x \rightarrow \infty} \frac{2-x^{2}}{1+2 x}$ .

See Solution

Problem 30310

Find the function $y$ if $\frac{d y}{d x}=\frac{1}{6 x^{2}}$ .

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Problem 30311

Bestimmen Sie die Wendestellen und das Krümmungsverhalten für die Funktionen: a) $f(x)=x^{3}-3 x^{2}$ , b) $f(x)=x^{4}-24 x^{2}+8 x$ , c) $f(x)=x^{4}+24 x^{2}+8 x$ , d) $f(x)=\frac{1}{12} x^{4}+\frac{1}{6} x^{3}-x^{2}+5 x$ , e) $f(x)=x^{4}-12 x^{2}+4 x$ , f) $f(x)=6 x^{5}-15 x^{4}+10 x^{3}$ .

See Solution

Problem 30312

Calculate the limit: $\lim _{n \rightarrow \infty} \frac{4 \cdot 6+9 \cdot 4+14 \cdot 5+\cdots+(5 n-1)(n+2)}{n^{3}}.$

See Solution

Problem 30313

Find the limit: $\lim _{n \rightarrow \infty}\left(\frac{4 n^{2}+a}{4 n^{2}+1}\right)^{2 n^{2}-1}=e^{5}$ .

See Solution

Problem 30314

Find the limit as $x$ approaches infinity for $\frac{1}{x^{r}}$ where $p$ is a positive integer and $r$ is a positive real number.

See Solution

Problem 30315

Find the limit as $x$ approaches 0 from the right of $\frac{1}{x^{p}}$ .

See Solution

Problem 30316

Find the partial derivatives of $f(x, y)=x^{3}-4xy^{2}+3y$ : $f_{xx}, f_{yy}, f_{yx}$ .

See Solution

Problem 30317

Find possible values of $k$ given that $\int_{1}^{2} k x^{2}+a \, dx=11$ and $\int_{1}^{k} \frac{6}{x^{2}} \, dx=a$ .

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Problem 30318

Express $\mathrm{f}(x)=\frac{6+x}{\sqrt[3]{x}}$ as $a x^{n}+b x^{m}$ and find $\int \frac{6+x}{\sqrt[3]{x}} \mathrm{~d} x$ .

See Solution

Problem 30319

Find the partial derivative $f_{y}$ of the function $f(x, y) = 2 x^{4} y^{2} + 8 x$ .

See Solution

Problem 30320

Find the mixed partial derivative $f_{xy}$ for the function $f(x, y)=2 x^{4} y^{2}+8 x$ .

See Solution

Problem 30321

Find the partial derivative $f_x$ of the function $f(x, y)=2 x^{4} y^{2}+8 x$ .

See Solution

Problem 30322

Find the second partial derivative $f_{xx}$ of the function $f(x, y) = 2x^4y^2 + 8x$ .

See Solution

Problem 30323

Find the slope of the tangent line for $y=2^{\cos x}$ at $x=\frac{\pi}{2}$ . Choices: $\ln 2$ , $\ln \frac{1}{2}$ , $1 / \ln 3$ , $1 / \ln 2$ .

See Solution

Problem 30324

Calculate the sum: $\sum_{k=1}^{10^{7}-1}\left(\frac{1}{k^{5}}-\frac{1}{(k+1)^{5}}\right)$

See Solution

Problem 30325

Find the slope of the tangent line for $y=3^{\cos x}$ at $x=\frac{\pi}{2}$ .

See Solution

Problem 30326

Find $\frac{d f^{-1}}{d x}$ at $x=5$ for $f(x)=x^{2}-3x+5$ , with $x \geq 2$ . Options: $1/5$ , $1/3$ , $1/4$ , $1/6$ .

See Solution

Problem 30327

Calculate $\int_{1}^{2} \frac{\ln x^{3}}{x} d x$ . What is the result?

See Solution

Problem 30328

Find the slope of the tangent line of $y=3^{\sin x}$ at $x=0$ . Choose from: $\ln 2$ , $\ln 3$ , $1 / \ln 3$ , $1 / \ln 2$ .

See Solution

Problem 30329

Evaluate the integral $\int_{1}^{2} \frac{2^{1 / x}}{x^{2}} d x$ . What is the result?

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Problem 30330

Evaluate the integral $\int_{1}^{4} \frac{2^{\sqrt{x}}}{\sqrt{x}} d x$ . What is the result?

See Solution

Problem 30331

Find the slope of the tangent line for $y=3^{\cos x}$ at $x=\frac{\pi}{2}$ .

See Solution

Problem 30332

Find the tangent line equation to $f^{-1}(x)$ at $x=12.282$ for $f(x)=3^{\frac{x-1}{4}}+5$ .

See Solution

Problem 30333

Calculate the average growth rate of $h(t)=\frac{260}{1+24(0.9)^{t}}$ over 30 days and the instantaneous rate at $t=90$ days.

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Problem 30334

Find the limit as $x$ approaches 2 from the left for the expression $\frac{|x|}{x}$ .

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Problem 30335

Evaluate the integral from $\frac{\pi}{3}$ to $\frac{\pi}{6}$ of $\frac{\cos x - x \sin x}{x \cos x} \, dx$ and choose the answer.

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Problem 30336

Find $\frac{d f^{-1}}{d x}$ at $x=f(2)$ for $f(x)=2 x^{3}$ . Options: a. 16 b. $\frac{1}{24}$ c. none d. 4 e. $\frac{1}{8}$

See Solution

Problem 30337

Evaluate the integral $\int_{\pi / 3}^{\pi / 6} \frac{\cos x-x \sin x}{x \cos x} d x=$ a. $-\frac{\ln 3}{2}-\ln 2$ b. $-\frac{\ln 3}{2}+\ln 2$ c. $\frac{\ln 3}{2}+\ln 2$ d. none e. $\frac{\ln 3}{2}-\ln 2$

See Solution

Problem 30338

If $f'(x) = 3x - 1$ , find $\frac{d f^{-1}}{d x}$ at $x = f(2)$ . Choices: a) 3, b) $\frac{1}{5}$ , c) 5, d) none, e) $\frac{1}{3}$ .

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Problem 30339

Evaluate the integral $\int_{2}^{e^{2}} \frac{1+\ln x}{x \ln x} d x=$ a. $1-\ln 2$ b. $1+\ln 2$ c. $3+\ln 2$ d. $2-\ln (\ln 2)$

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Problem 30340

Which statement is true for the graph of $y=|\ln x|$ ? a. Asymptotic to negative $x$ -axis b. Range $(-\infty, \infty)$ c. Crosses $y=c$ once d. Crosses y-axis once e. Domain $(0, \infty)$

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Problem 30341

Find the derivative $y'$ , where $y=\ln \left(\frac{(x^{2}+1)^{5}}{\sqrt{1-x}}\right)$ .

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Problem 30342

Find the limit as $h$ approaches 0 for $\frac{\sqrt{x+h+9} - \sqrt{x+9}}{h}$ .

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Problem 30343

Find the derivative $y'$ , where $y=\frac{\ln (x)}{1+\ln (x)}$ . Choose from the options given.

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Problem 30344

Simplify the expression: $\frac{\frac{-9}{(x+h)+5} - \frac{-9}{x+5}}{h}$ .

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Problem 30345

Calculate the integral $\int_{\pi / 3}^{\pi / 6} \frac{\cos x - x \sin x}{x \cos x} dx$ . Choose the correct answer from options a-e.

See Solution

Problem 30346

Find $y^{\prime}$ if $y=\frac{\ln (x)}{1+\ln (x)}$ . Choose from: a. $\frac{1}{(1+\ln (x))^{2}}$ , b. $\frac{x}{(1+\ln (x))^{2}}$ , c. $\frac{1}{x(1+\ln (x))^{2}}$ , d. $-\frac{1}{x(1+\ln (x))^{2}}$ , e. $-\frac{x}{(1+\ln (x))^{2}}$ .

See Solution

Problem 30347

Find the limit as $x$ approaches infinity for $\frac{x+7 \sin x}{-2 x+13}$ .

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Problem 30348

Set up $R_{n}$ and find $\lim _{n \rightarrow \infty} R_{n}$ for $f(x)=1-x^{2}$ on the interval $[0,2]$ .

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Problem 30349

Find the limit: $\lim _{x \rightarrow 0} \frac{|\sin x|}{x}$ .

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Problem 30350

Find the limit: $\lim _{x \rightarrow \infty} \frac{4}{e^{-x}+e^{-2 x}+2}=$ a. $\infty$ b. 0 c. 2 d. none e. 4

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Problem 30351

Find the limit: $\lim _{x \rightarrow \infty} \frac{(5-3 x)^{2}}{2 x^{2}+5 x+1}$ . What is the value?

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Problem 30352

Set up $R_{N}$ and find $\lim_{N \rightarrow \infty} R_{N}$ for a.) $f(x)=1-x^{2}$ over $[0,1]$ .

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Problem 30353

Given a continuous function $g(x)$ on $[0,1]$ with $g(0)=1$ and $g(1)=0$ , which statement is NOT necessarily true? a. If $a=b$ , then $g(a)=g(b)$ b. There exists $h$ in $[0,1]$ such that $g(h)=\frac{1}{2}$ c. There exists $h$ in $[0,1]$ such that $g(h)=\frac{3}{2}$ d. none e. For $h$ in $(0,1)$ , $\lim_{x \to h} g(x)=g(h)$

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Problem 30354

Let $g(x)$ be continuous on $[0,1]$ with $g(0)=1$ and $g(1)=0$ . Which is NOT necessarily true? a. $g(h)=\frac{1}{2}$ for some $h$ in $[0,1]$ . b. $\lim_{x \to h} g(x)=g(h)$ for all $h$ in $(0,1)$ . c. If $a=b$ , then $g(a)=g(b)$ for all $a,b$ in $[0,1]$ . d. none. e. There exists $h$ in $[0,1]$ such that...

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Problem 30355

Find the limit: $\lim _{x \rightarrow 0} \frac{x^{2} \sec x}{\sin x}=$ a. 2 b. 0 c. none d. 1 e. 3

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Problem 30356

A continuous function $f$ on $[-3,6]$ with $f(-3)=-1$ and $f(6)=3$ must satisfy which statement? a, b, c, d, or e?

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Problem 30357

Find the limit: $\lim _{x \rightarrow 0} x^{2} e^{\sin \left(\frac{1}{x}\right)}$ .

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Problem 30358

Given a continuous function $g(x)$ on $[0,1]$ with $g(0)=1$ and $g(1)=0$ , which statement is NOT necessarily true? a. $\lim _{x \rightarrow h} g(x)=g(h)$ for all $h$ in $(0,1)$ b. none c. $\exists h \in [0,1]$ such that $g(h)=\frac{3}{2}$ d. $\exists h \in [0,1]$ such that $g(h)=\frac{1}{2}$ e. $a=b \implies g(a)=g(b)$ for all $a,b \in [0,1]$

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Problem 30359

Find the derivative of the function $y=3 x^{3}-x^{2}+4 x-1$ . What is $\frac{d y}{d x}$ ?

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Problem 30360

Find the derivative $y^{\prime}$ of $y=\sqrt{\ln \sqrt{t}}$ . Choose from the options given.

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Problem 30361

Find the derivative of $y=\sin^{-1}(x^2)$ . What is $\frac{d y}{d x}$ ?

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Problem 30362

Find the derivative of $y=\tan^{-1}(\sqrt{5-x^{2}})$ . What is $\frac{d y}{d x}$ ?

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Problem 30363

Find where the graph of $y=x^{3}+x^{2}-27 x$ is concave down: a. $x \geq \frac{-1}{3}$ b. $x \leq 0$ c. $x \geq 0$ d. $-6 \geq x \geq 2$ e. None.

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Problem 30364

Find the derivative of $y=x^{3} \cdot x^{7}$ using the Product Rule and select the correct answer.

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Problem 30365

Find the derivative of $y=\frac{x^{5}}{x^{3}}$ using the Quotient Rule. Choose the correct answer from the options.

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Problem 30366

Find the derivative of $y=\frac{x^{5}}{x^{3}}$ using the Quotient Rule or by dividing first.

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Problem 30367

Divide $x^{5}$ by $x^{3}$ , simplify to get $x^{2}$ , then find $\frac{d y}{d x}$ and verify both methods give the same result.

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Problem 30368

A dog slows from $16 \mathrm{~m/s}$ [S] to $4.0 \mathrm{~m/s}$ [S] in $4.0 \mathrm{~s}$ . Find its displacement.

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Problem 30369

Find the derivative of $y=\frac{x^{5}}{x^{3}}$ using the Quotient Rule or by dividing first.

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Problem 30370

Divide $x^{8}$ by $x^{4}$ and simplify. Then find the derivative $\frac{d y}{d x}$ and check if both results match.

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Problem 30371

Identify $u$ and $d v$ for the integral $\int \ln (2 x) d x$ using integration by parts.

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Problem 30372

Find the rate of change of average cost for producing $x$ belts, given $C(x)=780+31x-0.062x^2$ , at $x=174$ .

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Problem 30373

Find the derivative of $y=\frac{x^{8}}{x^{4}}$ using the Quotient Rule and select the correct expression.

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Problem 30374

Divide $x^{8}$ by $x^{4}$ , simplify to get $x^{4}$ . Find its derivative and confirm $\frac{d y}{d x}=\square$ .

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Problem 30375

Find the integral using the tabular method. Include constant C. $\int x^{2}(x-2)^{3 / 2} d x$

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Problem 30376

Find the derivative of $y=\frac{x^{8}}{x^{4}}$ using the Quotient Rule and select the correct derivative form.

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Problem 30377

Wasserbecken: Gegeben ist $z(t)=t^{3}-12 t^{2}+35 t$ für $t \in [0 ; 8]$ .
a) Finde Zeitpunkte, wo Wasser nicht fließt und Intervalle für Zu- bzw. Abfluss. b) Bestimme den Zeitpunkt der maximalen Zulaufrate. c) Finde den Zeitpunkt der stärksten Änderung der Zulaufrate. d) Skizziere den Graphen von $z(t)$ . e) Berechne die Wassermenge nach 4 Stunden. f) Gibt es einen Zeitpunkt mit der Anfangswassermenge? g) Bestimme die maximale Wassermenge und wann sie erreicht wird.

See Solution

Problem 30378

Find the rate of change of average cost when producing 174 belts, given $C(x)=780+31x-0.062x^{2}$ .

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Problem 30379

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{(8(4+h)^{2}+1)-(8(4)^{2}+1)}{h}.$

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Problem 30380

Given the temperature function $T(t)=\frac{4 t}{t^{2}+3}+986$ , find: (a) $T'(t)$ , (b) $T(1)$ , (c) $T'(1)$ .

See Solution

Problem 30381

Compute the integral: $\int 17 \arctan x \, dx$ using the simplest method and include the constant $C$ .

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Problem 30382

Find the temperature function $T(t)=\frac{6t}{t^2+1}+986$ .
(a) Compute $T'(t)$ . (b) Determine $T(1)$ . (c) Find $T'(1)$ .

See Solution

Problem 30383

Differentiate the function $y=(5x+8)^{5}$ . What is $\frac{dy}{dx}=\square$ ?

See Solution

Problem 30384

Find the instantaneous rate of change of water volume $V(t)=\frac{(120-t)^{2}}{9}$ after 60 minutes. Include units.

See Solution

Problem 30385

Differentiate the function $y=(8-x)^{73}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 30386

Differentiate $y=\sqrt{2-3 x}$ using the Chain Rule. Find $\frac{d y}{d x}=$

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Problem 30387

Differentiate the function $y=\frac{1}{(7 x+9)^{4}}$ . Find $\frac{d y}{d x}$ .

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Problem 30388

Differentiate the function $y=(2x^{2}-6)^{-12}$ . Find $\frac{dy}{dx}$ .

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Problem 30389

Differentiate the function $G(x)=\sqrt[4]{x^{7}+2 x}$ . Find $G^{\prime}(x)=\square$ .

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Problem 30390

Evaluate the integral from $e$ to $e^{2}$ of $\frac{1+\ln x}{x \ln x} dx$ and choose the correct option.

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Problem 30391

Găsiți valorile lui $x$ , $y$ , $z$ pentru profit maxim din $f(x, y, z)=x y^{2} z^{3}(14-x-2 y-3 z)$ .

See Solution

Problem 30392

Differentiate the function $y=\sqrt{x^{2}+3}$ . Find $\frac{d y}{d x}$ .

See Solution

Problem 30393

Find $y^{\prime}$ for $y=\frac{x \ln (x)}{1+\ln (x)}$ . Choose from the options: a, b, c, d, e.

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Problem 30394

Differentiate the function $f(x)=\left(\frac{x+3}{x+1}\right)^{4}$ . Find $f^{\prime}(x)$ .

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Problem 30395

Differentiate the function $y=(19-x)^{52}$ . Find $\frac{d y}{d x}=\square$ .

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Problem 30396

Găsiți valorile $x, y, z$ pentru profit maxim dat de $f(x, y, z)=x y^{2} z^{3}(14-x-2 y-3 z)$ . Răspuns: $f_{\max }=128$ pentru $x=y=z=2$ .

See Solution

Problem 30397

Differentiate the function $G(x)=\sqrt[5]{x^{7}+3x}$ . Find $G^{\prime}(x)$ .

See Solution

Problem 30398

Differentiate the function $y=\frac{1}{(3 x-16)^{4}}$ . Find $\frac{d y}{d x}=\square$ .

See Solution

Problem 30399

Differentiate the function $f(x)=\left(\frac{x-5}{x+6}\right)^{9}$ . Find $f^{\prime}(x)$ .

See Solution

Problem 30400

Find the curve equation with $\frac{d y}{d x}=72 y x^{17}$ and $y$ -intercept 2. What is $y(x)=$ ?

See Solution

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Calculus

Problem 30301

Problem 30302

Find lim⁡f(x)\lim \sqrt{f(x)}limf(x)​ as xxx approaches ccc if lim⁡f(x)=4\lim f(x) = 4limf(x)=4.

Problem 30303

Find the limit: lim⁡x→−∞arctan⁡x\lim _{x \rightarrow -\infty} \arctan xlimx→−∞​arctanx.

Problem 30304

Find the limit as xxx approaches −∞-\infty−∞ for the function exe^{x}ex.

Problem 30305

Find the limit as xxx approaches −∞-\infty−∞ for 1xp\frac{1}{x^{p}}xp1​ where ppp is a positive integer constant.

Problem 30306

Find the limit: lim⁡n→∞ln⁡(4n2+a4n2+1)12n2−1\lim _{n \rightarrow \infty} \frac{\ln \left(\frac{4 n^{2}+a}{4 n^{2}+1}\right)}{\frac{1}{2 n^{2}-1}}limn→∞​2n2−11​ln(4n2+14n2+a​)​.

Problem 30307

Find the limit: lim⁡x→∞arctan⁡x\lim _{x \rightarrow \infty} \arctan xlimx→∞​arctanx.

Problem 30308

Calculate the integral ∫x5−3x−3+x2dx\int \frac{x^{5}-3 x^{-3}+x}{2} dx∫2x5−3x−3+x​dx.

Problem 30309

Find the limit: lim⁡x→∞2−x21+2x\lim _{x \rightarrow \infty} \frac{2-x^{2}}{1+2 x}limx→∞​1+2x2−x2​.

Problem 30310

Find the function yyy if dydx=16x2\frac{d y}{d x}=\frac{1}{6 x^{2}}dxdy​=6x21​.

Problem 30311

Problem 30312

Calculate the limit: lim⁡n→∞4⋅6+9⋅4+14⋅5+⋯+(5n−1)(n+2)n3.\lim _{n \rightarrow \infty} \frac{4 \cdot 6+9 \cdot 4+14 \cdot 5+\cdots+(5 n-1)(n+2)}{n^{3}}.n→∞lim​n34⋅6+9⋅4+14⋅5+⋯+(5n−1)(n+2)​.

Problem 30313

Find the limit: lim⁡n→∞(4n2+a4n2+1)2n2−1=e5\lim _{n \rightarrow \infty}\left(\frac{4 n^{2}+a}{4 n^{2}+1}\right)^{2 n^{2}-1}=e^{5}limn→∞​(4n2+14n2+a​)2n2−1=e5.

Problem 30314

Find the limit as xxx approaches infinity for 1xr\frac{1}{x^{r}}xr1​ where ppp is a positive integer and rrr is a positive real number.

Problem 30315

Find the limit as xxx approaches 0 from the right of 1xp\frac{1}{x^{p}}xp1​.

Problem 30316

Find the partial derivatives of f(x,y)=x3−4xy2+3yf(x, y)=x^{3}-4xy^{2}+3yf(x,y)=x3−4xy2+3y: fxx,fyy,fyxf_{xx}, f_{yy}, f_{yx}fxx​,fyy​,fyx​.

Problem 30317

Find possible values of kkk given that ∫12kx2+a dx=11\int_{1}^{2} k x^{2}+a \, dx=11∫12​kx2+adx=11 and ∫1k6x2 dx=a\int_{1}^{k} \frac{6}{x^{2}} \, dx=a∫1k​x26​dx=a.

Problem 30318

Express f(x)=6+xx3\mathrm{f}(x)=\frac{6+x}{\sqrt[3]{x}}f(x)=3x​6+x​ as axn+bxma x^{n}+b x^{m}axn+bxm and find ∫6+xx3 dx\int \frac{6+x}{\sqrt[3]{x}} \mathrm{~d} x∫3x​6+x​ dx.

Problem 30319

Find the partial derivative fyf_{y}fy​ of the function f(x,y)=2x4y2+8xf(x, y) = 2 x^{4} y^{2} + 8 xf(x,y)=2x4y2+8x.

Problem 30320

Find the mixed partial derivative fxyf_{xy}fxy​ for the function f(x,y)=2x4y2+8xf(x, y)=2 x^{4} y^{2}+8 xf(x,y)=2x4y2+8x.

Problem 30321

Find the partial derivative fxf_xfx​ of the function f(x,y)=2x4y2+8xf(x, y)=2 x^{4} y^{2}+8 xf(x,y)=2x4y2+8x.

Problem 30322

Find the second partial derivative fxxf_{xx}fxx​ of the function f(x,y)=2x4y2+8xf(x, y) = 2x^4y^2 + 8xf(x,y)=2x4y2+8x.

Problem 30323

Find the slope of the tangent line for y=2cos⁡xy=2^{\cos x}y=2cosx at x=π2x=\frac{\pi}{2}x=2π​. Choices: ln⁡2\ln 2ln2, ln⁡12\ln \frac{1}{2}ln21​, 1/ln⁡31 / \ln 31/ln3, 1/ln⁡21 / \ln 21/ln2.

Problem 30324

Calculate the sum: ∑k=1107−1(1k5−1(k+1)5)\sum_{k=1}^{10^{7}-1}\left(\frac{1}{k^{5}}-\frac{1}{(k+1)^{5}}\right)∑k=1107−1​(k51​−(k+1)51​)

Problem 30325

Find the slope of the tangent line for y=3cos⁡xy=3^{\cos x}y=3cosx at x=π2x=\frac{\pi}{2}x=2π​.

Problem 30326

Find df−1dx\frac{d f^{-1}}{d x}dxdf−1​ at x=5x=5x=5 for f(x)=x2−3x+5f(x)=x^{2}-3x+5f(x)=x2−3x+5, with x≥2x \geq 2x≥2. Options: 1/51/51/5, 1/31/31/3, 1/41/41/4, 1/61/61/6.

Problem 30327

Calculate ∫12ln⁡x3xdx\int_{1}^{2} \frac{\ln x^{3}}{x} d x∫12​xlnx3​dx. What is the result?

Problem 30328

Find the slope of the tangent line of y=3sin⁡xy=3^{\sin x}y=3sinx at x=0x=0x=0. Choose from: ln⁡2\ln 2ln2, ln⁡3\ln 3ln3, 1/ln⁡31 / \ln 31/ln3, 1/ln⁡21 / \ln 21/ln2.

Problem 30329

Evaluate the integral ∫1221/xx2dx\int_{1}^{2} \frac{2^{1 / x}}{x^{2}} d x∫12​x221/x​dx. What is the result?

Problem 30330

Evaluate the integral ∫142xxdx\int_{1}^{4} \frac{2^{\sqrt{x}}}{\sqrt{x}} d x∫14​x​2x​​dx. What is the result?

Problem 30331

Find the slope of the tangent line for y=3cos⁡xy=3^{\cos x}y=3cosx at x=π2x=\frac{\pi}{2}x=2π​.

Problem 30332

Find the tangent line equation to f−1(x)f^{-1}(x)f−1(x) at x=12.282x=12.282x=12.282 for f(x)=3x−14+5f(x)=3^{\frac{x-1}{4}}+5f(x)=34x−1​+5.

Problem 30333

Calculate the average growth rate of h(t)=2601+24(0.9)th(t)=\frac{260}{1+24(0.9)^{t}}h(t)=1+24(0.9)t260​ over 30 days and the instantaneous rate at t=90t=90t=90 days.

Problem 30334

Find the limit as xxx approaches 2 from the left for the expression ∣x∣x\frac{|x|}{x}x∣x∣​.

Problem 30335

Evaluate the integral from π3\frac{\pi}{3}3π​ to π6\frac{\pi}{6}6π​ of cos⁡x−xsin⁡xxcos⁡x dx\frac{\cos x - x \sin x}{x \cos x} \, dxxcosxcosx−xsinx​dx and choose the answer.

Problem 30336

Find df−1dx\frac{d f^{-1}}{d x}dxdf−1​ at x=f(2)x=f(2)x=f(2) for f(x)=2x3f(x)=2 x^{3}f(x)=2x3. Options: a. 16 b. 124\frac{1}{24}241​ c. none d. 4 e. 18\frac{1}{8}81​

Problem 30337

Problem 30338

If f′(x)=3x−1f'(x) = 3x - 1f′(x)=3x−1, find df−1dx\frac{d f^{-1}}{d x}dxdf−1​ at x=f(2)x = f(2)x=f(2). Choices: a) 3, b) 15\frac{1}{5}51​, c) 5, d) none, e) 13\frac{1}{3}31​.

Problem 30339

Evaluate the integral ∫2e21+ln⁡xxln⁡xdx=\int_{2}^{e^{2}} \frac{1+\ln x}{x \ln x} d x=∫2e2​xlnx1+lnx​dx= a. 1−ln⁡21-\ln 21−ln2 b. 1+ln⁡21+\ln 21+ln2 c. 3+ln⁡23+\ln 23+ln2 d. 2−ln⁡(ln⁡2)2-\ln (\ln 2)2−ln(ln2)

Problem 30340

Which statement is true for the graph of y=∣ln⁡x∣y=|\ln x|y=∣lnx∣? a. Asymptotic to negative xxx-axis b. Range (−∞,∞)(-\infty, \infty)(−∞,∞) c. Crosses y=cy=cy=c once d. Crosses y-axis once e. Domain (0,∞)(0, \infty)(0,∞)

Problem 30341

Find the derivative y′y'y′, where y=ln⁡((x2+1)51−x)y=\ln \left(\frac{(x^{2}+1)^{5}}{\sqrt{1-x}}\right)y=ln(1−x​(x2+1)5​).

Find $\lim \sqrt{f(x)}$ as $x$ approaches $c$ if $\lim f(x) = 4$ .

Find the limit: $\lim _{x \rightarrow -\infty} \arctan x$ .

Find the limit as $x$ approaches $-\infty$ for the function $e^{x}$ .

Find the limit as $x$ approaches $-\infty$ for $\frac{1}{x^{p}}$ where $p$ is a positive integer constant.

Find the limit: $\lim _{n \rightarrow \infty} \frac{\ln \left(\frac{4 n^{2}+a}{4 n^{2}+1}\right)}{\frac{1}{2 n^{2}-1}}$ .

Find the limit: $\lim _{x \rightarrow \infty} \arctan x$ .

Calculate the integral $\int \frac{x^{5}-3 x^{-3}+x}{2} dx$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{2-x^{2}}{1+2 x}$ .

Find the function $y$ if $\frac{d y}{d x}=\frac{1}{6 x^{2}}$ .

Calculate the limit: $\lim _{n \rightarrow \infty} \frac{4 \cdot 6+9 \cdot 4+14 \cdot 5+\cdots+(5 n-1)(n+2)}{n^{3}}.$

Find the limit: $\lim _{n \rightarrow \infty}\left(\frac{4 n^{2}+a}{4 n^{2}+1}\right)^{2 n^{2}-1}=e^{5}$ .

Find the limit as $x$ approaches infinity for $\frac{1}{x^{r}}$ where $p$ is a positive integer and $r$ is a positive real number.

Find the limit as $x$ approaches 0 from the right of $\frac{1}{x^{p}}$ .

Find the partial derivatives of $f(x, y)=x^{3}-4xy^{2}+3y$ : $f_{xx}, f_{yy}, f_{yx}$ .

Find possible values of $k$ given that $\int_{1}^{2} k x^{2}+a \, dx=11$ and $\int_{1}^{k} \frac{6}{x^{2}} \, dx=a$ .

Express $\mathrm{f}(x)=\frac{6+x}{\sqrt[3]{x}}$ as $a x^{n}+b x^{m}$ and find $\int \frac{6+x}{\sqrt[3]{x}} \mathrm{~d} x$ .

Find the partial derivative $f_{y}$ of the function $f(x, y) = 2 x^{4} y^{2} + 8 x$ .

Find the mixed partial derivative $f_{xy}$ for the function $f(x, y)=2 x^{4} y^{2}+8 x$ .

Find the partial derivative $f_x$ of the function $f(x, y)=2 x^{4} y^{2}+8 x$ .

Find the second partial derivative $f_{xx}$ of the function $f(x, y) = 2x^4y^2 + 8x$ .

Find the slope of the tangent line for $y=2^{\cos x}$ at $x=\frac{\pi}{2}$ . Choices: $\ln 2$ , $\ln \frac{1}{2}$ , $1 / \ln 3$ , $1 / \ln 2$ .

Calculate the sum: $\sum_{k=1}^{10^{7}-1}\left(\frac{1}{k^{5}}-\frac{1}{(k+1)^{5}}\right)$

Find the slope of the tangent line for $y=3^{\cos x}$ at $x=\frac{\pi}{2}$ .

Find $\frac{d f^{-1}}{d x}$ at $x=5$ for $f(x)=x^{2}-3x+5$ , with $x \geq 2$ . Options: $1/5$ , $1/3$ , $1/4$ , $1/6$ .

Calculate $\int_{1}^{2} \frac{\ln x^{3}}{x} d x$ . What is the result?

Find the slope of the tangent line of $y=3^{\sin x}$ at $x=0$ . Choose from: $\ln 2$ , $\ln 3$ , $1 / \ln 3$ , $1 / \ln 2$ .

Evaluate the integral $\int_{1}^{2} \frac{2^{1 / x}}{x^{2}} d x$ . What is the result?

Evaluate the integral $\int_{1}^{4} \frac{2^{\sqrt{x}}}{\sqrt{x}} d x$ . What is the result?

Find the slope of the tangent line for $y=3^{\cos x}$ at $x=\frac{\pi}{2}$ .

Find the tangent line equation to $f^{-1}(x)$ at $x=12.282$ for $f(x)=3^{\frac{x-1}{4}}+5$ .

Calculate the average growth rate of $h(t)=\frac{260}{1+24(0.9)^{t}}$ over 30 days and the instantaneous rate at $t=90$ days.

Find the limit as $x$ approaches 2 from the left for the expression $\frac{|x|}{x}$ .

Evaluate the integral from $\frac{\pi}{3}$ to $\frac{\pi}{6}$ of $\frac{\cos x - x \sin x}{x \cos x} \, dx$ and choose the answer.

Find $\frac{d f^{-1}}{d x}$ at $x=f(2)$ for $f(x)=2 x^{3}$ . Options: a. 16 b. $\frac{1}{24}$ c. none d. 4 e. $\frac{1}{8}$

If $f'(x) = 3x - 1$ , find $\frac{d f^{-1}}{d x}$ at $x = f(2)$ . Choices: a) 3, b) $\frac{1}{5}$ , c) 5, d) none, e) $\frac{1}{3}$ .

Evaluate the integral $\int_{2}^{e^{2}} \frac{1+\ln x}{x \ln x} d x=$ a. $1-\ln 2$ b. $1+\ln 2$ c. $3+\ln 2$ d. $2-\ln (\ln 2)$

Which statement is true for the graph of $y=|\ln x|$ ? a. Asymptotic to negative $x$ -axis b. Range $(-\infty, \infty)$ c. Crosses $y=c$ once d. Crosses y-axis once e. Domain $(0, \infty)$

Find the derivative $y'$ , where $y=\ln \left(\frac{(x^{2}+1)^{5}}{\sqrt{1-x}}\right)$ .

Find the limit as $h$ approaches 0 for $\frac{\sqrt{x+h+9} - \sqrt{x+9}}{h}$ .

Find the derivative $y'$ , where $y=\frac{\ln (x)}{1+\ln (x)}$ . Choose from the options given.

Simplify the expression: $\frac{\frac{-9}{(x+h)+5} - \frac{-9}{x+5}}{h}$ .

Calculate the integral $\int_{\pi / 3}^{\pi / 6} \frac{\cos x - x \sin x}{x \cos x} dx$ . Choose the correct answer from options a-e.

Find the limit as $x$ approaches infinity for $\frac{x+7 \sin x}{-2 x+13}$ .

Set up $R_{n}$ and find $\lim _{n \rightarrow \infty} R_{n}$ for $f(x)=1-x^{2}$ on the interval $[0,2]$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{|\sin x|}{x}$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{4}{e^{-x}+e^{-2 x}+2}=$ a. $\infty$ b. 0 c. 2 d. none e. 4

Find the limit: $\lim _{x \rightarrow \infty} \frac{(5-3 x)^{2}}{2 x^{2}+5 x+1}$ . What is the value?

Set up $R_{N}$ and find $\lim_{N \rightarrow \infty} R_{N}$ for a.) $f(x)=1-x^{2}$ over $[0,1]$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{x^{2} \sec x}{\sin x}=$ a. 2 b. 0 c. none d. 1 e. 3

A continuous function $f$ on $[-3,6]$ with $f(-3)=-1$ and $f(6)=3$ must satisfy which statement? a, b, c, d, or e?

Find the limit: $\lim _{x \rightarrow 0} x^{2} e^{\sin \left(\frac{1}{x}\right)}$ .

Find the derivative of the function $y=3 x^{3}-x^{2}+4 x-1$ . What is $\frac{d y}{d x}$ ?

Find the derivative $y^{\prime}$ of $y=\sqrt{\ln \sqrt{t}}$ . Choose from the options given.

Find the derivative of $y=\sin^{-1}(x^2)$ . What is $\frac{d y}{d x}$ ?

Find the derivative of $y=\tan^{-1}(\sqrt{5-x^{2}})$ . What is $\frac{d y}{d x}$ ?

Find where the graph of $y=x^{3}+x^{2}-27 x$ is concave down: a. $x \geq \frac{-1}{3}$ b. $x \leq 0$ c. $x \geq 0$ d. $-6 \geq x \geq 2$ e. None.

Find the derivative of $y=x^{3} \cdot x^{7}$ using the Product Rule and select the correct answer.

Find the derivative of $y=\frac{x^{5}}{x^{3}}$ using the Quotient Rule. Choose the correct answer from the options.

Find the derivative of $y=\frac{x^{5}}{x^{3}}$ using the Quotient Rule or by dividing first.

Divide $x^{5}$ by $x^{3}$ , simplify to get $x^{2}$ , then find $\frac{d y}{d x}$ and verify both methods give the same result.

A dog slows from $16 \mathrm{~m/s}$ [S] to $4.0 \mathrm{~m/s}$ [S] in $4.0 \mathrm{~s}$ . Find its displacement.

Find the derivative of $y=\frac{x^{5}}{x^{3}}$ using the Quotient Rule or by dividing first.

Divide $x^{8}$ by $x^{4}$ and simplify. Then find the derivative $\frac{d y}{d x}$ and check if both results match.

Identify $u$ and $d v$ for the integral $\int \ln (2 x) d x$ using integration by parts.

Find the rate of change of average cost for producing $x$ belts, given $C(x)=780+31x-0.062x^2$ , at $x=174$ .

Find the derivative of $y=\frac{x^{8}}{x^{4}}$ using the Quotient Rule and select the correct expression.

Divide $x^{8}$ by $x^{4}$ , simplify to get $x^{4}$ . Find its derivative and confirm $\frac{d y}{d x}=\square$ .