Calculus

Problem 28401

Find $f_{y y}(x, y)$ for the function $f(x, y)=\ln(1+4x^{3}y^{2})$ .

See Solution

Problem 28402

Find the partial derivative $\mathbf{P}_{x}(7,8)$ and $\mathbf{P}_{x}(2,8)$ for the profit function $P(x, y)=130 x+300 y-16 x^{2}+4 x y-14 y^{2}-600$ .

See Solution

Problem 28403

Find the integral of the function: $\int (e^{3} x + e x^{3} + 3 e x) \, dx$ .

See Solution

Problem 28404

Differentiate the function: $\frac{1}{2} e^{3} x^{2} + \frac{1}{4} e x^{4} + \frac{3}{2} e^{2}$ .

See Solution

Problem 28405

Finde die Funktion $f$ für die Ableitungen: a) $f^{\prime}(x)=e^{x}+x+2$ , b) $f^{\prime}(x)=e^{2} x+e x^{2}+2 e x$ .

See Solution

Problem 28406

Find why the acceleration is undefined at $t=0$ for the position function $s^3 + t^2s = 2t$ and velocity $\frac{ds}{dt} = \frac{2 - 2st}{3s^2 + t^2}$ .

See Solution

Problem 28407

Find the intervals where the function $y = -x^5$ is increasing and decreasing. Choose A, B, or C for your answer.

See Solution

Problem 28408

Find the curve length of $y=\frac{4 \sqrt{2}}{3} x^{\frac{3}{2}}-1$ for $0 \leq x \leq 1$ .

See Solution

Problem 28409

Check if the series $\sum_{k=1}^{\infty} \frac{2^{k+2}}{3^{k+1}}$ converges or diverges.

See Solution

Problem 28410

Find intervals where the function $y = \frac{1}{9x}$ is increasing or decreasing. Choose A, B, or C with your answers.

See Solution

Problem 28411

Determine if $x^{3}+y^{2}=3 x y$ has a horizontal tangent at $(2,4)$ .

See Solution

Problem 28412

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

See Solution

Problem 28413

Determine the truth of these statements for the curve $\sin y = x + y$ : I. Vertical tangent if $\cos y - 1 = 0$ . II. No horizontal tangents if $\cos y - 1 \neq 0$ . III. No horizontal tangents if $\frac{dy}{dx} \neq 0$ .

See Solution

Problem 28414

Determine if the series converges and find its sum: $\sum_{k=1}^{\infty} \frac{2^{k+2}}{3^{k+1}}$ (Enter DNE if no sum exists.)

See Solution

Problem 28415

Find values of $x$ for which the series $\sum_{n=0}^{\infty} \frac{(x+1)^{n}}{2^{n}}$ converges and its sum.

See Solution

Problem 28416

Find the derivative of $y=x^{3}-12 x+4$ and determine the coordinates of the turning points $A$ and $B$ .

See Solution

Problem 28417

Find the average speed of a flare modeled by $h(t)=-16 t^{2}+92 t+3.5$ from 1 to 3 seconds.

See Solution

Problem 28418

Find the sum of the series: $\sum_{n=1}^{\infty} (3 x^{2n} - 22)$ .

See Solution

Problem 28419

Determine if the function $f(x)=\frac{1}{x^{2}-1}$ has a max, min, or is increasing (YES/NO for each).

See Solution

Problem 28420

Finde die Extremstellen von $f(x) = \frac{x}{1+x^2}$ . Bestimme die erste und zweite Ableitung und analysiere die kritischen Punkte.

See Solution

Problem 28421

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+9}-\sqrt{x^{2}-9}}{6 x}$ .

See Solution

Problem 28422

Find a curve through (1,1) with length integral $L=\int_{1}^{4} \sqrt{1+\frac{1}{4x}} \, dx$ .

See Solution

Problem 28423

Find the curve length for $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} d t$ over $-\pi / 4 \leq y \leq \pi / 4$ .

See Solution

Problem 28424

Determine limits of sequences: $a_n=\left(1+\frac{1}{5n+1}\right)^{6n+1}$ , $b_n$ , $c_n$ , $d_n$ and $a_n=\left(\sqrt{n^2+4}-n\right) \prod_{k=1}^{n}\left(1+\frac{1}{k+1}\right)$ .

See Solution

Problem 28425

Find the curve length for $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} d t$ , where $-\pi / 4 \leq y \leq \pi / 4$ .

See Solution

Problem 28426

Find $\frac{d y}{d x}$ for $y=\sin (u)$ and $u=2 x-4$ using the chain rule: $\frac{d y}{d x}=\frac{d y}{d u} \frac{d u}{d x}$ .

See Solution

Problem 28427

Calculate the limit: $\lim _{x \rightarrow 0} \frac{x^{3}}{\sin \left(x^{2}\right)}$ or state if it doesn't exist.

See Solution

Problem 28428

Calculate the curve length for $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} dt$ in the range $-\pi / 4 \leq y \leq \pi / 4$ . Options: a-2, b-4, c-8, d-None.

See Solution

Problem 28429

Find the velocity of the object at $t=\frac{\pi}{4}$ , given $S(t)=3+7 \cos t$ .

See Solution

Problem 28430

Determine if the function $f(x)=\frac{1}{1-x^{2}}$ has a max, min, and if it's increasing (YES/NO for each).

See Solution

Problem 28431

Given $y=\sin(4x^2-4x-1)$ , find $\frac{dy}{dx}$ using the chain rule: $\frac{dy}{du}=\square$ , $\frac{du}{dx}=\square$ .

See Solution

Problem 28432

Find the radius of convergence for the series $\sum_{n=1}^{\infty} 6^{n}\left(n+\frac{1}{n}\right) x^{n}$ .

See Solution

Problem 28433

Find the limit: $\lim _{x \rightarrow 3^{+}} \frac{x^{2}+2 x-3}{|x+3|}$ . Does it exist?

See Solution

Problem 28434

Find the limit: $\lim _{x \rightarrow 1} \arctan \left(\frac{x^{2}}{1-x^{2}}\right)$ . Does it exist?

See Solution

Problem 28435

Find the limit: $\lim _{x \rightarrow 0} \frac{x^{3}}{\sin(3 x^{2})}$ ; state if it does not exist.

See Solution

Problem 28436

Find the average rate of change of $f(x)=4x^{2}+1$ on the interval $[2,2+h]$ .

See Solution

Problem 28437

Find the derivative of $y=x^{-7}$ for $x \neq 0$ . What is $y'$ ?

See Solution

Problem 28438

Find the slope of the inverse function $\mathrm{g}^{-1}$ at the origin, given $g(0)=0$ and $g'(0)=2$ . Options: a-2, b-0, c-1/2, d-undefined, e-None.

See Solution

Problem 28439

Is the function $f(x)=\left\{\begin{array}{cc}(x-1) \sin \left(\frac{1}{x-1}\right) & x \neq 1 \\ -1 & x=1\end{array}\right.$ continuous on $\mathbb{R}$ ?

See Solution

Problem 28440

Find where the tangent line to $f(x) = x^{3} \ln(x)$ is horizontal and if it intersects at $(1,0)$ for $x > 1$ .

See Solution

Problem 28441

Check if the Mean Value Theorem applies to $f(x) = \sin x$ on $[0, \pi]$ and find $c$ where $f'(c) = \frac{f(\pi) - f(0)}{\pi}$ .

See Solution

Problem 28442

Find the limit as $x$ approaches $-\infty$ for $\left(\frac{x^{3}-8}{x^{4}+16}\right)^{10}$ .

See Solution

Problem 28443

Check if the Mean Value Theorem applies to $f(x)=-x^{3}+4x^{2}-3$ on $[0,4]$ and find $c$ where $f'(c) = \frac{f(4) - f(0)}{4}$ .

See Solution

Problem 28444

Find the surface area $S= \int 2 \pi x d s$ for $y=(1/3)(x^{2}+2)^{3/2}$ , $0 \leq x \leq \sqrt{2}$ .

See Solution

Problem 28445

Find the limit as $x$ approaches $-\infty$ for: $\sqrt{x^{2}+x}-x$ .

See Solution

Problem 28446

Find the derivative of $y=\left(x^{6}+4\right)^{7}$ using the Chain Rule. What is $y^{\prime}=$ ?

See Solution

Problem 28447

Find the derivative of $y=2\left(x^{2}+2 x\right)^{-3}$ with respect to $x$ : $\frac{d y}{d x}=$

See Solution

Problem 28448

Find the derivative $f'(x)$ of the function $f(x)=(x^{3}+5x+7)^{2}$ and calculate $f'(3)$ .

See Solution

Problem 28449

Find the derivative $\frac{d y}{d x}$ for $y=(4 x-3)^{-2}$ using the chain rule.

See Solution

Problem 28450

Find the derivative of $y=(16x+25)^{\frac{1}{2}}$ . What is $y'$ ?

See Solution

Problem 28451

Find the derivative of $f(x)=\tan(5x)$ and calculate $f^{\prime}(4)$ , rounding to the nearest hundredth.

See Solution

Problem 28452

Find the derivative of the function $f(x)=\sqrt{5x+10}$ , denoted as $f^{\prime}(x)$ .

See Solution

Problem 28453

Find the derivatives: (a) $f(x)=\sin^5(x)$ , (b) $f(x)=\sin(x^5)$ . What are $f^{\prime}(x)$ for each?

See Solution

Problem 28454

Find the derivative $d y / d x$ for the function $y=(\tan (x))^{-1}$ .

See Solution

Problem 28455

Find the derivative of $y=(9+\cos^2 x)^{13}$ with respect to $x$ : $\frac{dy}{dx}=$

See Solution

Problem 28456

Find where $f^{\prime}(x)=0$ for: a. $f(x)=x^{3}+6x^{2}+1$ , c. $f(x)=(2x-1)^{2}(x^{2}-9)$ .

See Solution

Problem 28457

Evaluate these integrals using table 8.1 and appropriate methods or identities: a. $\int \frac{2^{\sqrt{x}}}{2 \sqrt{x}} d x$ b. $\int \frac{d x}{\sqrt{2 x-x^{2}}}$

See Solution

Problem 28458

Find the derivatives $f'(x)$ for $f(x)=\sin \frac{1}{x}$ and $g'(x)$ for $g(x)=\frac{1}{\sin x}$ .

See Solution

Problem 28459

Find the derivative of the function $g(t)=6(-6 \sin t-2 \cos t)^{6}$ . What is $g^{\prime}(t)=\square$ ?

See Solution

Problem 28460

Evaluate the following integrals using integration by parts: a) $\int x^{3} e^{-2 x} d x$ b) $\int e^{x} \cos x d x$

See Solution

Problem 28461

Evaluate these integrals: a) $\int 8 \cos^{3} 2\theta \sin 2\theta \, d\theta$ b) $\int \sec^{2} \theta \tan \theta \, d\theta$ .

See Solution

Problem 28462

Exercice 2 : Limites et continuité (5 points) A. Pour $f(x)=\frac{\sqrt{3 x^{2}+1}-2}{x-1}$ , justifiez la discontinuité en 1 et montrez le prolongement. B. Pour $g(x)=x^{3}-3 x-4$ , calculez les limites aux extrêmes, dérivée, variations, unique solution de $g(x)=0$ , et son signe.

See Solution

Problem 28463

Find the derivative of the function $f(x)=\sin(x^{2}+81)$ . What is $f^{\prime}(x)=\square$ ?

See Solution

Problem 28464

Find the length of the curve $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} dt$ for $-\pi / 4 \leq y \leq \pi / 4$ . Options: a) 2, c) 8.

See Solution

Problem 28465

Find the tangent line equation for $f(x)=2 x(3-4 x)^{3}$ at $x=2$ . Tangent line: $y=$

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Problem 28466

Find the derivative of the function $f(x)=4 \cos (4 x-9)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 28467

Find the velocity of a particle with displacement $s(t)=12+\frac{1}{4} \sin (8 \pi t)$ after $t$ seconds. $v(t)=$

See Solution

Problem 28468

Find the derivative $d y / d t$ for the function $y=\cot ^{3}(\pi-t)$ .

See Solution

Problem 28469

Calculate the length of the curve $y=(x / 2)^{2 / 3}$ between $x=0$ and $x=2$ .

See Solution

Problem 28470

Rewrite $y=\cot(5x-6)$ as $y=f(u)$ and $u=g(x)$ , then find $\frac{dy}{dx}$ in terms of $x$ .

See Solution

Problem 28471

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the function $y=6 x \sin x$ .

See Solution

Problem 28472

Find the limit: $\lim _{h \rightarrow 0} \frac{(5+h)^{2}-25}{h}$

See Solution

Problem 28473

Find the limit as $x$ approaches -4 for the expression $x^{2}+2x+6$ .

See Solution

Problem 28474

Find the derivative $D_{x} y$ for the function $y=\frac{x+8}{x-8}$ .

See Solution

Problem 28475

A force on an object moving along the x-axis is given by $F_{x}=(3 x^{2}+2 x) \, N$ . Find the work done from $x=0$ to $x=5 \, m$ .

See Solution

Problem 28476

Find the limit as x approaches 0: $\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x}$

See Solution

Problem 28477

Berechne die Ableitung von $f(x)=(x-2) \cdot \sqrt{2x+3}$ .

See Solution

Problem 28478

Find the slope of the curve $y=3 \sqrt{x}$ at $x=36$ .

See Solution

Problem 28479

Let $a_{n}=\frac{n}{n+2}$ . Find the smallest $M$ such that: (a) $|a_{n}-1| \leq 0.001$ for $n \geq M$ , (b) $|a_{n}-1| \leq 0.00001$ for $n \geq M$ , (c) $|a_{n}-1|<t$ for all $n>M$ , find $M$ as a function of $t$ .

See Solution

Problem 28480

Find the limit: If $f(x)=\frac{\sin (4 x)}{x}$ , what is $\lim _{x \rightarrow 0} f(x)$ ?

See Solution

Problem 28481

Express $y$ as $y=f(u)$ and $u=g(x)$ , then calculate $\frac{dy}{dx}$ for $y=(2x+9)^{3}$ .

See Solution

Problem 28482

Find the derivative $D_{x} y$ for the function $y=(4-8 x^{2})(3 x^{2}-96)$ .

See Solution

Problem 28483

Find the rate of change of current $A(t)=5 \sin(120\pi t)$ at $t=2$ seconds.

See Solution

Problem 28484

Find the derivative $\mathrm{dy} / \mathrm{dt}$ for the function $y=\cos ^{4}(\pi t-19)$ .

See Solution

Problem 28485

Find the limit as $x$ approaches 1 from the right: $\lim _{x \rightarrow 1^{+}} \frac{1}{(x-1)^{2}}$ .

See Solution

Problem 28486

Find the derivative $D_{x} y$ for the function $y=3 x(2 x+4)^{4}$ .

See Solution

Problem 28487

Find the derivative of the function: $s = t^{3} - \csc t + 20$ .

See Solution

Problem 28488

Find the tangent line equation to the curve $y=x^{2}-x$ at the point (-4, 20).

See Solution

Problem 28489

A car moving at $42 \mathrm{ft/sec}$ has position $s=42t-3t^2$ . How long until it stops after braking?

See Solution

Problem 28490

Calculate the limit: $\lim _{x \rightarrow 4} \frac{x^{2}-16}{x^{2}-9 x+20}$

See Solution

Problem 28491

Find the derivative $D_{x} y$ for the function $y=\frac{1}{2} x^{6}-\frac{1}{5} x^{5}$ .

See Solution

Problem 28492

Find the speed and acceleration of the body at $t=2$ for the function $s=6 t^{2}+2 t+4$ where $0 \leq t \leq 2$ .

See Solution

Problem 28493

Zbadaj zbieżność szeregów dla $\alpha \in \mathbb{R}$ : a) $\sum_{n=1}^{+\infty}\left(\frac{15^{n}+n}{100 \cdot 14^{n}-7}\right)^{\alpha}$ b) $\sum_{n=1}^{+\infty}\left(\frac{\sqrt[n]{2023}-1}{n}\right)^{\alpha}$ .

See Solution

Problem 28494

Find the instantaneous rate of change of $f(x)=3x^{2}-5x+1$ at $x=4$ using the difference quotient. Simplify first!

See Solution

Problem 28495

Find the derivative of $f(x) = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$ using the quotient rule.

See Solution

Problem 28496

Find the derivative $\frac{dy}{dx}$ of $y=\frac{3-\sqrt{x}}{(2+x)^{2}}$ using the quotient rule.

See Solution

Problem 28497

Solve $4^{2x}=8^{x-1}$ and find the instantaneous mass change of a $2000 \mathrm{~g}$ substance after 1.5 days.

See Solution

Problem 28498

Determine if $(-1,-1)$ is a max, min, or neither for $f(x)=4x^4+4x^3-2x^2+1$ using rates of change with $h=\pm 0.001$ .

See Solution

Problem 28499

A rectangle's width grows at $2 \mathrm{~cm/s}$ and length at $4 \mathrm{~cm/s}$ . Find area change rate with width $9 \mathrm{~cm}$ and length $15 \mathrm{~cm}$ .

See Solution

Problem 28500

A square's area grows at $23 \mathrm{~cm}^{2} / \mathrm{s}$ . Find the side length growth rate when it is $10 \mathrm{~cm}$ .

See Solution

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Calculus

Problem 28401

Find fyy(x,y)f_{y y}(x, y)fyy​(x,y) for the function f(x,y)=ln⁡(1+4x3y2)f(x, y)=\ln(1+4x^{3}y^{2})f(x,y)=ln(1+4x3y2).

Problem 28402

Find the partial derivative Px(7,8)\mathbf{P}_{x}(7,8)Px​(7,8) and Px(2,8)\mathbf{P}_{x}(2,8)Px​(2,8) for the profit function P(x,y)=130x+300y−16x2+4xy−14y2−600P(x, y)=130 x+300 y-16 x^{2}+4 x y-14 y^{2}-600P(x,y)=130x+300y−16x2+4xy−14y2−600.

Problem 28403

Find the integral of the function: ∫(e3x+ex3+3ex) dx\int (e^{3} x + e x^{3} + 3 e x) \, dx∫(e3x+ex3+3ex)dx.

Problem 28404

Differentiate the function: 12e3x2+14ex4+32e2\frac{1}{2} e^{3} x^{2} + \frac{1}{4} e x^{4} + \frac{3}{2} e^{2}21​e3x2+41​ex4+23​e2.

Problem 28405

Finde die Funktion fff für die Ableitungen: a) f′(x)=ex+x+2f^{\prime}(x)=e^{x}+x+2f′(x)=ex+x+2, b) f′(x)=e2x+ex2+2exf^{\prime}(x)=e^{2} x+e x^{2}+2 e xf′(x)=e2x+ex2+2ex.

Problem 28406

Find why the acceleration is undefined at t=0t=0t=0 for the position function s3+t2s=2ts^3 + t^2s = 2ts3+t2s=2t and velocity dsdt=2−2st3s2+t2\frac{ds}{dt} = \frac{2 - 2st}{3s^2 + t^2}dtds​=3s2+t22−2st​.

Problem 28407

Find the intervals where the function y=−x5y = -x^5y=−x5 is increasing and decreasing. Choose A, B, or C for your answer.

Problem 28408

Find the curve length of y=423x32−1y=\frac{4 \sqrt{2}}{3} x^{\frac{3}{2}}-1y=342​​x23​−1 for 0≤x≤10 \leq x \leq 10≤x≤1.

Problem 28409

Check if the series ∑k=1∞2k+23k+1\sum_{k=1}^{\infty} \frac{2^{k+2}}{3^{k+1}}∑k=1∞​3k+12k+2​ converges or diverges.

Problem 28410

Find intervals where the function y=19xy = \frac{1}{9x}y=9x1​ is increasing or decreasing. Choose A, B, or C with your answers.

Problem 28411

Determine if x3+y2=3xyx^{3}+y^{2}=3 x yx3+y2=3xy has a horizontal tangent at (2,4)(2,4)(2,4).

Problem 28412

Find the derivative of the function f(x)=ex+e−xf(x) = e^{x} + e^{-x}f(x)=ex+e−x.

Problem 28413

Problem 28414

Determine if the series converges and find its sum: ∑k=1∞2k+23k+1\sum_{k=1}^{\infty} \frac{2^{k+2}}{3^{k+1}}k=1∑∞​3k+12k+2​ (Enter DNE if no sum exists.)

Problem 28415

Find values of xxx for which the series ∑n=0∞(x+1)n2n\sum_{n=0}^{\infty} \frac{(x+1)^{n}}{2^{n}}∑n=0∞​2n(x+1)n​ converges and its sum.

Problem 28416

Find the derivative of y=x3−12x+4y=x^{3}-12 x+4y=x3−12x+4 and determine the coordinates of the turning points AAA and BBB.

Problem 28417

Find the average speed of a flare modeled by h(t)=−16t2+92t+3.5h(t)=-16 t^{2}+92 t+3.5h(t)=−16t2+92t+3.5 from 1 to 3 seconds.

Problem 28418

Find the sum of the series: ∑n=1∞(3x2n−22)\sum_{n=1}^{\infty} (3 x^{2n} - 22)n=1∑∞​(3x2n−22).

Problem 28419

Determine if the function f(x)=1x2−1f(x)=\frac{1}{x^{2}-1}f(x)=x2−11​ has a max, min, or is increasing (YES/NO for each).

Problem 28420

Finde die Extremstellen von f(x)=x1+x2 f(x) = \frac{x}{1+x^2} f(x)=1+x2x​. Bestimme die erste und zweite Ableitung und analysiere die kritischen Punkte.

Problem 28421

Find the limit: lim⁡x→∞x2+9−x2−96x\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+9}-\sqrt{x^{2}-9}}{6 x}limx→∞​6xx2+9​−x2−9​​.

Problem 28422

Find a curve through (1,1) with length integral L=∫141+14x dxL=\int_{1}^{4} \sqrt{1+\frac{1}{4x}} \, dxL=∫14​1+4x1​​dx.

Problem 28423

Find the curve length for x=∫0ysec⁡4t−1dtx=\int_{0}^{y} \sqrt{\sec ^{4} t-1} d tx=∫0y​sec4t−1​dt over −π/4≤y≤π/4-\pi / 4 \leq y \leq \pi / 4−π/4≤y≤π/4.

Problem 28424

Determine limits of sequences: an=(1+15n+1)6n+1a_n=\left(1+\frac{1}{5n+1}\right)^{6n+1}an​=(1+5n+11​)6n+1, bnb_nbn​, cnc_ncn​, dnd_ndn​ and an=(n2+4−n)∏k=1n(1+1k+1)a_n=\left(\sqrt{n^2+4}-n\right) \prod_{k=1}^{n}\left(1+\frac{1}{k+1}\right)an​=(n2+4​−n)∏k=1n​(1+k+11​).

Problem 28425

Find the curve length for x=∫0ysec⁡4t−1dtx=\int_{0}^{y} \sqrt{\sec ^{4} t-1} d tx=∫0y​sec4t−1​dt, where −π/4≤y≤π/4-\pi / 4 \leq y \leq \pi / 4−π/4≤y≤π/4.

Problem 28426

Find dydx\frac{d y}{d x}dxdy​ for y=sin⁡(u)y=\sin (u)y=sin(u) and u=2x−4u=2 x-4u=2x−4 using the chain rule: dydx=dydududx\frac{d y}{d x}=\frac{d y}{d u} \frac{d u}{d x}dxdy​=dudy​dxdu​.

Problem 28427

Calculate the limit: lim⁡x→0x3sin⁡(x2)\lim _{x \rightarrow 0} \frac{x^{3}}{\sin \left(x^{2}\right)}x→0lim​sin(x2)x3​ or state if it doesn't exist.

Problem 28428

Calculate the curve length for x=∫0ysec⁡4t−1dtx=\int_{0}^{y} \sqrt{\sec ^{4} t-1} dtx=∫0y​sec4t−1​dt in the range −π/4≤y≤π/4-\pi / 4 \leq y \leq \pi / 4−π/4≤y≤π/4. Options: a-2, b-4, c-8, d-None.

Problem 28429

Find the velocity of the object at t=π4t=\frac{\pi}{4}t=4π​, given S(t)=3+7cos⁡tS(t)=3+7 \cos tS(t)=3+7cost.

Problem 28430

Determine if the function f(x)=11−x2f(x)=\frac{1}{1-x^{2}}f(x)=1−x21​ has a max, min, and if it's increasing (YES/NO for each).

Problem 28431

Given y=sin⁡(4x2−4x−1)y=\sin(4x^2-4x-1)y=sin(4x2−4x−1), find dydx\frac{dy}{dx}dxdy​ using the chain rule: dydu=□\frac{dy}{du}=\squaredudy​=□, dudx=□\frac{du}{dx}=\squaredxdu​=□.

Problem 28432

Find the radius of convergence for the series ∑n=1∞6n(n+1n)xn\sum_{n=1}^{\infty} 6^{n}\left(n+\frac{1}{n}\right) x^{n}∑n=1∞​6n(n+n1​)xn.

Problem 28433

Find the limit: lim⁡x→3+x2+2x−3∣x+3∣\lim _{x \rightarrow 3^{+}} \frac{x^{2}+2 x-3}{|x+3|}limx→3+​∣x+3∣x2+2x−3​. Does it exist?

Problem 28434

Find the limit: lim⁡x→1arctan⁡(x21−x2)\lim _{x \rightarrow 1} \arctan \left(\frac{x^{2}}{1-x^{2}}\right)x→1lim​arctan(1−x2x2​). Does it exist?

Problem 28435

Find the limit: lim⁡x→0x3sin⁡(3x2)\lim _{x \rightarrow 0} \frac{x^{3}}{\sin(3 x^{2})}limx→0​sin(3x2)x3​; state if it does not exist.

Problem 28436

Find the average rate of change of f(x)=4x2+1f(x)=4x^{2}+1f(x)=4x2+1 on the interval [2,2+h][2,2+h][2,2+h].

Problem 28437

Find the derivative of y=x−7y=x^{-7}y=x−7 for x≠0x \neq 0x=0. What is y′y'y′?

Problem 28438

Find the slope of the inverse function g−1\mathrm{g}^{-1}g−1 at the origin, given g(0)=0g(0)=0g(0)=0 and g′(0)=2g'(0)=2g′(0)=2. Options: a-2, b-0, c-1/2, d-undefined, e-None.

Problem 28439

Is the function f(x)={(x−1)sin⁡(1x−1)x≠1−1x=1f(x)=\left\{\begin{array}{cc}(x-1) \sin \left(\frac{1}{x-1}\right) & x \neq 1 \\ -1 & x=1\end{array}\right.f(x)={(x−1)sin(x−11​)−1​x=1x=1​ continuous on R\mathbb{R}R?

Problem 28440

Find where the tangent line to f(x)=x3ln⁡(x)f(x) = x^{3} \ln(x)f(x)=x3ln(x) is horizontal and if it intersects at (1,0)(1,0)(1,0) for x>1x > 1x>1.

Find $f_{y y}(x, y)$ for the function $f(x, y)=\ln(1+4x^{3}y^{2})$ .

Find the partial derivative $\mathbf{P}_{x}(7,8)$ and $\mathbf{P}_{x}(2,8)$ for the profit function $P(x, y)=130 x+300 y-16 x^{2}+4 x y-14 y^{2}-600$ .

Find the integral of the function: $\int (e^{3} x + e x^{3} + 3 e x) \, dx$ .

Differentiate the function: $\frac{1}{2} e^{3} x^{2} + \frac{1}{4} e x^{4} + \frac{3}{2} e^{2}$ .

Finde die Funktion $f$ für die Ableitungen: a) $f^{\prime}(x)=e^{x}+x+2$ , b) $f^{\prime}(x)=e^{2} x+e x^{2}+2 e x$ .

Find why the acceleration is undefined at $t=0$ for the position function $s^3 + t^2s = 2t$ and velocity $\frac{ds}{dt} = \frac{2 - 2st}{3s^2 + t^2}$ .

Find the intervals where the function $y = -x^5$ is increasing and decreasing. Choose A, B, or C for your answer.

Find the curve length of $y=\frac{4 \sqrt{2}}{3} x^{\frac{3}{2}}-1$ for $0 \leq x \leq 1$ .

Check if the series $\sum_{k=1}^{\infty} \frac{2^{k+2}}{3^{k+1}}$ converges or diverges.

Find intervals where the function $y = \frac{1}{9x}$ is increasing or decreasing. Choose A, B, or C with your answers.

Determine if $x^{3}+y^{2}=3 x y$ has a horizontal tangent at $(2,4)$ .

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

Determine if the series converges and find its sum: $\sum_{k=1}^{\infty} \frac{2^{k+2}}{3^{k+1}}$ (Enter DNE if no sum exists.)

Find values of $x$ for which the series $\sum_{n=0}^{\infty} \frac{(x+1)^{n}}{2^{n}}$ converges and its sum.

Find the derivative of $y=x^{3}-12 x+4$ and determine the coordinates of the turning points $A$ and $B$ .

Find the average speed of a flare modeled by $h(t)=-16 t^{2}+92 t+3.5$ from 1 to 3 seconds.

Find the sum of the series: $\sum_{n=1}^{\infty} (3 x^{2n} - 22)$ .

Determine if the function $f(x)=\frac{1}{x^{2}-1}$ has a max, min, or is increasing (YES/NO for each).

Finde die Extremstellen von $f(x) = \frac{x}{1+x^2}$ . Bestimme die erste und zweite Ableitung und analysiere die kritischen Punkte.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+9}-\sqrt{x^{2}-9}}{6 x}$ .

Find a curve through (1,1) with length integral $L=\int_{1}^{4} \sqrt{1+\frac{1}{4x}} \, dx$ .

Find the curve length for $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} d t$ over $-\pi / 4 \leq y \leq \pi / 4$ .

Determine limits of sequences: $a_n=\left(1+\frac{1}{5n+1}\right)^{6n+1}$ , $b_n$ , $c_n$ , $d_n$ and $a_n=\left(\sqrt{n^2+4}-n\right) \prod_{k=1}^{n}\left(1+\frac{1}{k+1}\right)$ .

Find the curve length for $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} d t$ , where $-\pi / 4 \leq y \leq \pi / 4$ .

Find $\frac{d y}{d x}$ for $y=\sin (u)$ and $u=2 x-4$ using the chain rule: $\frac{d y}{d x}=\frac{d y}{d u} \frac{d u}{d x}$ .

Calculate the limit: $\lim _{x \rightarrow 0} \frac{x^{3}}{\sin \left(x^{2}\right)}$ or state if it doesn't exist.

Calculate the curve length for $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} dt$ in the range $-\pi / 4 \leq y \leq \pi / 4$ . Options: a-2, b-4, c-8, d-None.

Find the velocity of the object at $t=\frac{\pi}{4}$ , given $S(t)=3+7 \cos t$ .

Determine if the function $f(x)=\frac{1}{1-x^{2}}$ has a max, min, and if it's increasing (YES/NO for each).

Given $y=\sin(4x^2-4x-1)$ , find $\frac{dy}{dx}$ using the chain rule: $\frac{dy}{du}=\square$ , $\frac{du}{dx}=\square$ .

Find the radius of convergence for the series $\sum_{n=1}^{\infty} 6^{n}\left(n+\frac{1}{n}\right) x^{n}$ .

Find the limit: $\lim _{x \rightarrow 3^{+}} \frac{x^{2}+2 x-3}{|x+3|}$ . Does it exist?

Find the limit: $\lim _{x \rightarrow 1} \arctan \left(\frac{x^{2}}{1-x^{2}}\right)$ . Does it exist?

Find the limit: $\lim _{x \rightarrow 0} \frac{x^{3}}{\sin(3 x^{2})}$ ; state if it does not exist.

Find the average rate of change of $f(x)=4x^{2}+1$ on the interval $[2,2+h]$ .

Find the derivative of $y=x^{-7}$ for $x \neq 0$ . What is $y'$ ?

Find the slope of the inverse function $\mathrm{g}^{-1}$ at the origin, given $g(0)=0$ and $g'(0)=2$ . Options: a-2, b-0, c-1/2, d-undefined, e-None.

Is the function $f(x)=\left\{\begin{array}{cc}(x-1) \sin \left(\frac{1}{x-1}\right) & x \neq 1 \\ -1 & x=1\end{array}\right.$ continuous on $\mathbb{R}$ ?

Find where the tangent line to $f(x) = x^{3} \ln(x)$ is horizontal and if it intersects at $(1,0)$ for $x > 1$ .

Check if the Mean Value Theorem applies to $f(x) = \sin x$ on $[0, \pi]$ and find $c$ where $f'(c) = \frac{f(\pi) - f(0)}{\pi}$ .

Find the limit as $x$ approaches $-\infty$ for $\left(\frac{x^{3}-8}{x^{4}+16}\right)^{10}$ .

Check if the Mean Value Theorem applies to $f(x)=-x^{3}+4x^{2}-3$ on $[0,4]$ and find $c$ where $f'(c) = \frac{f(4) - f(0)}{4}$ .

Find the surface area $S= \int 2 \pi x d s$ for $y=(1/3)(x^{2}+2)^{3/2}$ , $0 \leq x \leq \sqrt{2}$ .

Find the limit as $x$ approaches $-\infty$ for: $\sqrt{x^{2}+x}-x$ .

Find the derivative of $y=\left(x^{6}+4\right)^{7}$ using the Chain Rule. What is $y^{\prime}=$ ?

Find the derivative of $y=2\left(x^{2}+2 x\right)^{-3}$ with respect to $x$ : $\frac{d y}{d x}=$

Find the derivative $f'(x)$ of the function $f(x)=(x^{3}+5x+7)^{2}$ and calculate $f'(3)$ .

Find the derivative $\frac{d y}{d x}$ for $y=(4 x-3)^{-2}$ using the chain rule.

Find the derivative of $y=(16x+25)^{\frac{1}{2}}$ . What is $y'$ ?

Find the derivative of $f(x)=\tan(5x)$ and calculate $f^{\prime}(4)$ , rounding to the nearest hundredth.

Find the derivative of the function $f(x)=\sqrt{5x+10}$ , denoted as $f^{\prime}(x)$ .

Find the derivatives: (a) $f(x)=\sin^5(x)$ , (b) $f(x)=\sin(x^5)$ . What are $f^{\prime}(x)$ for each?

Find the derivative $d y / d x$ for the function $y=(\tan (x))^{-1}$ .

Find the derivative of $y=(9+\cos^2 x)^{13}$ with respect to $x$ : $\frac{dy}{dx}=$

Find where $f^{\prime}(x)=0$ for: a. $f(x)=x^{3}+6x^{2}+1$ , c. $f(x)=(2x-1)^{2}(x^{2}-9)$ .

Evaluate these integrals using table 8.1 and appropriate methods or identities: a. $\int \frac{2^{\sqrt{x}}}{2 \sqrt{x}} d x$ b. $\int \frac{d x}{\sqrt{2 x-x^{2}}}$

Find the derivatives $f'(x)$ for $f(x)=\sin \frac{1}{x}$ and $g'(x)$ for $g(x)=\frac{1}{\sin x}$ .

Find the derivative of the function $g(t)=6(-6 \sin t-2 \cos t)^{6}$ . What is $g^{\prime}(t)=\square$ ?

Evaluate the following integrals using integration by parts: a) $\int x^{3} e^{-2 x} d x$ b) $\int e^{x} \cos x d x$

Evaluate these integrals: a) $\int 8 \cos^{3} 2\theta \sin 2\theta \, d\theta$ b) $\int \sec^{2} \theta \tan \theta \, d\theta$ .

Find the derivative of the function $f(x)=\sin(x^{2}+81)$ . What is $f^{\prime}(x)=\square$ ?

Find the length of the curve $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} dt$ for $-\pi / 4 \leq y \leq \pi / 4$ . Options: a) 2, c) 8.

Find the tangent line equation for $f(x)=2 x(3-4 x)^{3}$ at $x=2$ . Tangent line: $y=$

Find the derivative of the function $f(x)=4 \cos (4 x-9)$ . What is $f^{\prime}(x)$ ?

Find the velocity of a particle with displacement $s(t)=12+\frac{1}{4} \sin (8 \pi t)$ after $t$ seconds. $v(t)=$

Find the derivative $d y / d t$ for the function $y=\cot ^{3}(\pi-t)$ .

Calculate the length of the curve $y=(x / 2)^{2 / 3}$ between $x=0$ and $x=2$ .

Rewrite $y=\cot(5x-6)$ as $y=f(u)$ and $u=g(x)$ , then find $\frac{dy}{dx}$ in terms of $x$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the function $y=6 x \sin x$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{(5+h)^{2}-25}{h}$

Find the limit as $x$ approaches -4 for the expression $x^{2}+2x+6$ .

Find the derivative $D_{x} y$ for the function $y=\frac{x+8}{x-8}$ .