Calculus

Problem 19401

Show that the series $\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k}{2 k+1}$ diverges. Which Alternating Series Test condition fails? Identify $a_{k}$ .

See Solution

Problem 19402

Evaluate the integral: $\int \frac{d x}{x(\ln (x))^{7}}$ .

See Solution

Problem 19403

Find the integral: $\int x^{8} \sin \left(x^{9}+5\right) d x$

See Solution

Problem 19404

Find the volume of the solid formed by revolving $f(x)=(x-1)^{1/2}$ around the $x$ -axis from $x=1$ to $x=3$ . Round to hundredths.

See Solution

Problem 19405

Find the limit: $\varliminf_{x_{0} \rightarrow 3} \frac{\sqrt{x+1}-2}{x-3} \times \frac{\sqrt{x+1}+2}{\sqrt{x+1}+2}$ .

See Solution

Problem 19406

Compute the integral $\int_{-2}^{-1} \left[\frac{1}{4x} - \frac{1}{x}\right] dx$ .

See Solution

Problem 19407

Find the derivative $f^{\prime}(x)$ of $f(x)=(-x-4)(-3 x^{2}-4 x-4)$ . Choose from the options: (a) $9 x^{2}+32 x+20$ (b) $-9 x^{2}+32 x+12$ (c) $9 x^{2}-16 x+20$ (d) $-9 x^{2}-16 x+12$

See Solution

Problem 19408

Evaluate the integral: $\int \frac{e^{x}}{\left(e^{x}+1\right)^{18}} d x$

See Solution

Problem 19409

Find the critical values of the function $f(x)=5 x^{2} \ln (x)$ for $x>0$ . If none, write 'NONE'.

See Solution

Problem 19410

Calculate the integral $\int_{-1}^{0}[\frac{1}{4}x - x] dx$ .

See Solution

Problem 19411

Evaluate the integral $\int_{1}^{2} \frac{x}{(x^{2}+8)^{3}} \, dx$ .

See Solution

Problem 19412

Calculate the integral $\int_{0}^{1}(x-\frac{1}{4}x) dx$ .

See Solution

Problem 19413

Calculate the integral of $f(x) - h(x)$ from 0 to 1, where $f(x) = x$ and $h(x) = \frac{1}{4}x$ :
$\int_{0}^{1}[f(x)-h(x)] dx$

See Solution

Problem 19414

Calculate the integral $\int_{1}^{2}[g(x)-h(x)] dx$ for $g(x)= \frac{1}{x}$ and $h(x)= \frac{1}{4x}$ .

See Solution

Problem 19415

Given the function $f(x)=5 x^{2} \ln (x)$ for $x>0$ , find critical values, intervals of increase/decrease, local maxima/minima, concavity, and inflection points.

See Solution

Problem 19416

Find the area between the line $y=x$ and the parabola $y=x^2-2$ .

See Solution

Problem 19417

Find the derivative of the function $y=e^{x} \tan x$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 19418

Evaluate the integral $\int_{-1/2}^{0} 2x^{2}(-x) \, dx$ .

See Solution

Problem 19419

A plane flies south at $800 \mathrm{~km/h}$ and another flies east at $900 \mathrm{~km/h}$ . At 5 pm, the east plane is $1600 \mathrm{~km}$ from the airport. Find the rate of change of distance between the planes. Round to one decimal place.

See Solution

Problem 19420

Find the volume change rate $V$ of a sphere with radius increasing at $6 \mathrm{~cm/s}$ when radius is $5 \mathrm{~cm}$ .

See Solution

Problem 19421

Find the rate of radius increase of a balloon inflating at $4.3 \mathrm{ft}^3/\mathrm{min}$ when the radius is $1.5 \mathrm{ft}$ .

See Solution

Problem 19422

Calculate the integral $\int_{0}^{1/2} (1 - 2x^{2} - x) \, dx$ .

See Solution

Problem 19423

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{\frac{4}{2-(-1+\Delta x)^{2}}-4}{\Delta x}$ .

See Solution

Problem 19424

Calculate the integral $\int_{-2}^{-1}[h(x)-g(x)] dx$ with $h(x)= \frac{1}{4x}$ and $g(x)= \frac{1}{x}$ .

See Solution

Problem 19425

How fast is a rock falling after 4.1 seconds if it accelerates at $9.81 \mathrm{~m} / \mathrm{s}^{2}$ ?

See Solution

Problem 19426

Find the function $f(x)$ and the number $c$ such that $\lim _{\Delta x \rightarrow 0} \frac{\frac{4}{2-(-1+\Delta x)^{2}}-4}{\Delta x} = f^{\prime}(c)$ .

See Solution

Problem 19427

Estimate the area under $f(x)=17-x^{2}$ on $[0,4]$ using 4 equal subintervals and right endpoints.

See Solution

Problem 19428

Estimate the area under $f(x)=21-x^{2}$ on $[0,4]$ using 4 subintervals and right endpoints.

See Solution

Problem 19429

Find the area using the limit: area = $\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{2 n} \sin\left(\frac{i \pi}{2 n}\right)$ without evaluating it.

See Solution

Problem 19430

Plot $f(x)=x^{4}-6 x^{3}$ . How many inflection points does it have? Choices: 0, 1, 2, 3.

See Solution

Problem 19431

Find how many values of $x$ in $[-10,10]$ give a horizontal tangent for $f(x)=\frac{x^{3}+3 x^{2}+1}{x^{2}+2}$ .

See Solution

Problem 19432

Determine the extrema of the function $f(x)=x^{5}-4 x^{3}+x-1$ . Choose from: A. I and II B. III and IV C. II and IV D. I, II, III, and IV.

See Solution

Problem 19433

Find the height $h$ and radius $r$ of a can with volume 550 cm³ that minimizes cost. Minimum cost in cents?

See Solution

Problem 19434

Find the slope of the tangent line for $f(x)=2 x \sin x$ at $x=\frac{\pi}{2}$ .

See Solution

Problem 19435

Find the derivative $y^{\prime}$ of $y=x^{\ln (x)}$ at $x=4$ and round to four decimal places.

See Solution

Problem 19436

Determine if the series $\sum_{k=4}^{\infty} \frac{1}{(k-3)^{4}}$ converges or diverges and justify your answer.

See Solution

Problem 19437

How fast is a rock falling after 4.1 seconds if it accelerates at $9.81 \mathrm{~m} / \mathrm{s}^{2}$ ?

See Solution

Problem 19438

Evaluate the integral I = ∫(1/cos²(θ) + 2sin(θ)) dθ from 0 to π/3.

See Solution

Problem 19439

Identify the region with area given by $\text{area}=\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{2 n} \tan\left(\frac{i \pi}{2 n}\right)$ without calculating the limit.

See Solution

Problem 19440

Evaluate the integral $I=\int_{0}^{1} f(x) d x$ for a linear function $f$ with slope $m$ and intercept $b$ .

See Solution

Problem 19441

Find $f^{\prime}(-1)$ given $36 + 3f(x) + x^2(f(x))^3 = 0$ and $f(-1) = -3$ .

See Solution

Problem 19442

Find the radius and height of a right circular cone with slant height 33 that maximizes its volume.

See Solution

Problem 19443

Find the limit as $h$ approaches 0 for $\frac{\sec (x+h)-\sec x}{h}$ .

See Solution

Problem 19444

a. With 100 m of fencing, find dimensions to maximize area of a pen against a barn. b. For 4 identical pens of area 100 m² each, find dimensions to minimize fencing used.

See Solution

Problem 19445

Evaluate the limit: $\lim _{n \rightarrow \infty}\left|\frac{\frac{x^{n}}{n^{4} 8^{n}}+1}{\frac{x^{n}}{n^{4} 8^{n}}}\right|$

See Solution

Problem 19446

Identify the extrema of the function $f(x)=x^{2}+6 x+4$ : I. absolute max II. absolute min III. local max IV. local min. Options: A. III B. IV C. II D. II and IV.

See Solution

Problem 19447

Find the volume of the solid formed by rotating region $R$ (bounded by $f(x)=3$ and $g(x)=x+1$ from $x=0$ to $x=1$ ) around the $x$ -axis.

See Solution

Problem 19448

Gravel is dumped at 40 ft³/min into a cone with base diameter = height. Find the height's rate of increase when it's 23 ft.

See Solution

Problem 19449

Find the tangent line equation to $y=8 \ln \left(x^{3}-26\right)$ at (3,0). $y=$

See Solution

Problem 19450

Identify the region with area given by the limit $\text { area }=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{4 n} \sin \left(\frac{i \pi}{4 n}\right)$ without calculating it.

See Solution

Problem 19451

Find the correct expression for the derivative of $f(x)=\frac{2 x}{x^{2}+1}$ . Options are given.

See Solution

Problem 19452

Calculate $\ln 3$ to four decimal places. What is $\ln 3$ ?

See Solution

Problem 19453

A pork roast cools from $180^{\circ} \mathrm{F}$ to $170^{\circ} \mathrm{F}$ in 20 min. Find $T(t)$ and time to reach $140^{\circ} \mathrm{F}$ .

See Solution

Problem 19454

Trouver la dérivée seconde de $f(x)=\frac{2x^2}{x^2-1}$ .

See Solution

Problem 19455

Calculate the integral $I=\int_{0}^{\pi / 3} 2 \sin (3 t) d t$ .

See Solution

Problem 19456

Find the tangent line equation at $x=2$ given $f(2)=-1$ and $f'(2)=3$ . Choose from the options.

See Solution

Problem 19457

Find the volume of the solid formed by revolving the area between $y=3x$ , $y=0$ , and $x=2$ around the $x$ -axis.

See Solution

Problem 19458

Evaluate the integral $\int \frac{(\sqrt{x}+1)^{8}}{2 \sqrt{x}} d x=\int(\square) d u$ .

See Solution

Problem 19459

How long will it take for a material decaying at $0.099\%$ per year to reduce to $30\%$ of its original amount?

See Solution

Problem 19460

Calculate the volume of the solid formed by revolving the area between $y=2x$ , $y=0$ , and $x=4$ around the $x$ -axis. The volume is $\square$ cubic units.

See Solution

Problem 19461

A photographer is 5 m from a stunt. The stunt moves at $2.5 \, \text{m/s}$ when 1 m away. Can he take a clear picture if speed must be $\leq 1 \, \text{m/s}$ ?

See Solution

Problem 19462

How long will it take for a town's population to double if it grows at a constant rate of $12\%$ ?

See Solution

Problem 19463

Find the derivative $f'(\theta)$ if $f(\theta)=2 \sqrt{\theta}+3 \sin \theta$ . Choose from the options given.

See Solution

Problem 19464

Find the volume of a solid with base $y=26 \sqrt{\cos x}$ on $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and isosceles right triangle cross sections.

See Solution

Problem 19465

Identify the region with area given by $\text{area}=\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{5 n} \sin\left(\frac{i \pi}{5 n}\right)$ without solving the limit.

See Solution

Problem 19466

Determine the horizontal asymptote for the function $h(x)=\frac{3}{x^{2}-4x}$ .

See Solution

Problem 19467

Find the volume of the solid formed by revolving the region bounded by $y=3-4x$ , $y=0$ , and $x=0$ around the $x$ -axis.

See Solution

Problem 19468

Determine the end behavior asymptote of the function $h(x) = \frac{3}{x^2 - 4x}$ with the denominator factored as $(x - 0)(x - 4)$ .

See Solution

Problem 19469

Find the volume of the solid formed by revolving the area between $y=2x$ and $y=8\sqrt{x}$ around the $x$ -axis.

See Solution

Problem 19470

Find the volume of the solid formed by revolving the region bounded by $y=3-3x$ , $y=0$ , and $x=0$ around the $x$ -axis.

See Solution

Problem 19471

Calculate the volume of the solid formed by revolving the area between $y=4x$ and $y=12\sqrt{x}$ around the $x$ -axis.

See Solution

Problem 19472

Calculate the integral: $\int \frac{e^{u}}{\left(1-e^{u}\right)^{2}} dx$

See Solution

Problem 19473

Find the volume of the solid formed by revolving the region $R$ (bounded by $y=x$ , $y=2x$ , and $y=8$ ) around the $y$ -axis.

See Solution

Problem 19474

Find the value(s) of $a$ for which the slope of $g(x)=a \sqrt{x}+4 x$ is 3 at $x=4$ .

See Solution

Problem 19475

Calculer $\frac{d y}{d x}$ pour la courbe $y^{4}-y^{2}+\left(x+\frac{1}{2}\right)^{2}=x+\frac{1}{2}$ .

See Solution

Problem 19476

Calculate the integral: $\int x^{3}(x^{2}+1)^{2023} \, dx$

See Solution

Problem 19477

Find the differential $d y$ for $y=\tan(5x+5)$ at $x=4$ for $d x=0.1$ and $d x=0.2$ .

See Solution

Problem 19478

Encuentra la derivada de $y=\left(\frac{3}{4}\right)^{x}$ . ¿Cuál es $y^{\prime}$ ?

See Solution

Problem 19479

Encuentra la derivada de la función $y=\frac{\ln x}{\sqrt{x}}$ .

See Solution

Problem 19480

Encuentra la derivada de $y=\sqrt{x} \cdot e^{x}$ .

See Solution

Problem 19481

Find the area enclosed by the cardioid $r=1+\cos \theta$ using a double integral. (20 points)

See Solution

Problem 19482

Halla la derivada de la función $y=\left(\frac{3}{4}\right)^{x}$ .

See Solution

Problem 19483

Find $d y$ for $y=\tan(5x+5)$ at $x=4$ for $d x=0.1$ and $d x=0.2$ .

See Solution

Problem 19484

Calculate $e^{2.72}$ and round to four decimal places.

See Solution

Problem 19485

Find the indefinite integral $\int \frac{x+2}{(4 x-5)^{2}} d x$ using the substitution $u=4 x-5$ .

See Solution

Problem 19486

Find $d y$ for $y=\tan(5x+5)$ at $x=4$ for $d x=0.1$ and $d x=0.2$ .

See Solution

Problem 19487

Determina la mínima utilidad usando $I=8q$ y $C=5q^{2}-\frac{2}{3}q^{3}-20$ para $q$ en miles.

See Solution

Problem 19488

Evaluate the integral $\int \sqrt{x}(\sqrt[5]{x}-x^{2}) \, dx$ .

See Solution

Problem 19489

Encuentra el número de unidades $x$ que maximiza la utilidad, dado que el costo total es $C(x)=x^{2}+20x+700$ y el precio es \$100 por unidad.

See Solution

Problem 19490

Given $s(x) = f(x) - g(x)$ , find $s^{\prime}(2)$ using $f^{\prime}(2)$ and $g^{\prime}(2)$ .

See Solution

Problem 19491

Find $h^{\prime}(1)$ for $h(x) = f(x) g(x)$ , where $f(x) = -4$ and $g(x) = 2$ .

See Solution

Problem 19492

Evaluate the integral using the substitution $u=9+x^{4}$ : $\int x^{3}(9+x^{4})^{5} dx.$

See Solution

Problem 19493

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

See Solution

Problem 19494

Find the three critical numbers of the function $F(x)=x^{4/5}(5x-20)^{2}$ in increasing order.

See Solution

Problem 19495

Evaluate the integral $\int_{0}^{\sqrt{\ln (5)}} x e^{x^{2}} d x$ and find the result.

See Solution

Problem 19496

Find $g^{\prime}(2)$ for $g(x)=\int_{x^{2}}^{4} 3 t \, dt$ . Your answer should be a number.

See Solution

Problem 19497

Calculate the integral $\int_{5}^{2} f^{\prime}(x) \, dx$ using the provided values of $f$ and $f^{\prime}$ .

See Solution

Problem 19498

A package dropped from a hot-air balloon at 80 ft, ascending at 12 ft/sec, takes how long to hit the ground? Use $\frac{d v}{d t}=-32 \mathrm{ft/s^2}$ .

See Solution

Problem 19499

Evaluate the integral $\int_{-\pi}^{\pi} f(x) \, dx$ where $f(x) = 2x^4$ for $-\pi \leq x < 0$ and $f(x) = 5\sin(x)$ for $0 \leq x \leq \pi$ .

See Solution

Problem 19500

Find $g^{\prime}(x)$ for $g(x)=\int_{2}^{x} \sin (t) e^{t} dt$ . Which option is correct?

See Solution

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Calculus

Problem 19401

Show that the series ∑k=1∞(−1)k+1k2k+1\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k}{2 k+1}∑k=1∞​(−1)k+12k+1k​ diverges. Which Alternating Series Test condition fails? Identify aka_{k}ak​.

Problem 19402

Evaluate the integral: ∫dxx(ln⁡(x))7\int \frac{d x}{x(\ln (x))^{7}}∫x(ln(x))7dx​.

Problem 19403

Find the integral: ∫x8sin⁡(x9+5)dx\int x^{8} \sin \left(x^{9}+5\right) d x∫x8sin(x9+5)dx

Problem 19404

Find the volume of the solid formed by revolving f(x)=(x−1)1/2f(x)=(x-1)^{1/2}f(x)=(x−1)1/2 around the xxx-axis from x=1x=1x=1 to x=3x=3x=3. Round to hundredths.

Problem 19405

Find the limit: lim‾⁡x0→3x+1−2x−3×x+1+2x+1+2\varliminf_{x_{0} \rightarrow 3} \frac{\sqrt{x+1}-2}{x-3} \times \frac{\sqrt{x+1}+2}{\sqrt{x+1}+2}lim​x0​→3​x−3x+1​−2​×x+1​+2x+1​+2​.

Problem 19406

Compute the integral ∫−2−1[14x−1x]dx\int_{-2}^{-1} \left[\frac{1}{4x} - \frac{1}{x}\right] dx∫−2−1​[4x1​−x1​]dx.

Problem 19407

Problem 19408

Evaluate the integral: ∫ex(ex+1)18dx\int \frac{e^{x}}{\left(e^{x}+1\right)^{18}} d x∫(ex+1)18ex​dx

Problem 19409

Find the critical values of the function f(x)=5x2ln⁡(x)f(x)=5 x^{2} \ln (x)f(x)=5x2ln(x) for x>0x>0x>0. If none, write 'NONE'.

Problem 19410

Calculate the integral ∫−10[14x−x]dx\int_{-1}^{0}[\frac{1}{4}x - x] dx∫−10​[41​x−x]dx.

Problem 19411

Evaluate the integral ∫12x(x2+8)3 dx\int_{1}^{2} \frac{x}{(x^{2}+8)^{3}} \, dx∫12​(x2+8)3x​dx.

Problem 19412

Calculate the integral ∫01(x−14x)dx\int_{0}^{1}(x-\frac{1}{4}x) dx∫01​(x−41​x)dx.

Problem 19413

Calculate the integral of f(x)−h(x)f(x) - h(x)f(x)−h(x) from 0 to 1, where f(x)=xf(x) = xf(x)=x and h(x)=14xh(x) = \frac{1}{4}xh(x)=41​x: ∫01[f(x)−h(x)]dx\int_{0}^{1}[f(x)-h(x)] dx∫01​[f(x)−h(x)]dx

Problem 19414

Calculate the integral ∫12[g(x)−h(x)]dx\int_{1}^{2}[g(x)-h(x)] dx∫12​[g(x)−h(x)]dx for g(x)=1xg(x)= \frac{1}{x}g(x)=x1​ and h(x)=14xh(x)= \frac{1}{4x}h(x)=4x1​.

Problem 19415

Given the function f(x)=5x2ln⁡(x)f(x)=5 x^{2} \ln (x)f(x)=5x2ln(x) for x>0x>0x>0, find critical values, intervals of increase/decrease, local maxima/minima, concavity, and inflection points.

Problem 19416

Find the area between the line y=xy=xy=x and the parabola y=x2−2y=x^2-2y=x2−2.

Problem 19417

Find the derivative of the function y=extan⁡xy=e^{x} \tan xy=extanx. What is dydx\frac{d y}{d x}dxdy​?

Problem 19418

Evaluate the integral ∫−1/202x2(−x) dx\int_{-1/2}^{0} 2x^{2}(-x) \, dx∫−1/20​2x2(−x)dx.

Problem 19419

A plane flies south at 800 km/h800 \mathrm{~km/h}800 km/h and another flies east at 900 km/h900 \mathrm{~km/h}900 km/h. At 5 pm, the east plane is 1600 km1600 \mathrm{~km}1600 km from the airport. Find the rate of change of distance between the planes. Round to one decimal place.

Problem 19420

Find the volume change rate VVV of a sphere with radius increasing at 6 cm/s6 \mathrm{~cm/s}6 cm/s when radius is 5 cm5 \mathrm{~cm}5 cm.

Problem 19421

Find the rate of radius increase of a balloon inflating at 4.3ft3/min4.3 \mathrm{ft}^3/\mathrm{min}4.3ft3/min when the radius is 1.5ft1.5 \mathrm{ft}1.5ft.

Problem 19422

Calculate the integral ∫01/2(1−2x2−x) dx\int_{0}^{1/2} (1 - 2x^{2} - x) \, dx∫01/2​(1−2x2−x)dx.

Problem 19423

Find the limit: lim⁡Δx→042−(−1+Δx)2−4Δx\lim _{\Delta x \rightarrow 0} \frac{\frac{4}{2-(-1+\Delta x)^{2}}-4}{\Delta x}limΔx→0​Δx2−(−1+Δx)24​−4​.

Problem 19424

Calculate the integral ∫−2−1[h(x)−g(x)]dx\int_{-2}^{-1}[h(x)-g(x)] dx∫−2−1​[h(x)−g(x)]dx with h(x)=14xh(x)= \frac{1}{4x}h(x)=4x1​ and g(x)=1xg(x)= \frac{1}{x}g(x)=x1​.

Problem 19425

How fast is a rock falling after 4.1 seconds if it accelerates at 9.81 m/s29.81 \mathrm{~m} / \mathrm{s}^{2}9.81 m/s2?

Problem 19426

Find the function f(x)f(x)f(x) and the number ccc such that lim⁡Δx→042−(−1+Δx)2−4Δx=f′(c)\lim _{\Delta x \rightarrow 0} \frac{\frac{4}{2-(-1+\Delta x)^{2}}-4}{\Delta x} = f^{\prime}(c)limΔx→0​Δx2−(−1+Δx)24​−4​=f′(c).

Problem 19427

Estimate the area under f(x)=17−x2f(x)=17-x^{2}f(x)=17−x2 on [0,4][0,4][0,4] using 4 equal subintervals and right endpoints.

Problem 19428

Estimate the area under f(x)=21−x2f(x)=21-x^{2}f(x)=21−x2 on [0,4][0,4][0,4] using 4 subintervals and right endpoints.

Problem 19429

Find the area using the limit: area = lim⁡n→∞∑i=1nπ2nsin⁡(iπ2n)\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{2 n} \sin\left(\frac{i \pi}{2 n}\right)limn→∞​∑i=1n​2nπ​sin(2niπ​) without evaluating it.

Problem 19430

Plot f(x)=x4−6x3f(x)=x^{4}-6 x^{3}f(x)=x4−6x3. How many inflection points does it have? Choices: 0, 1, 2, 3.

Problem 19431

Find how many values of xxx in [−10,10][-10,10][−10,10] give a horizontal tangent for f(x)=x3+3x2+1x2+2f(x)=\frac{x^{3}+3 x^{2}+1}{x^{2}+2}f(x)=x2+2x3+3x2+1​.

Problem 19432

Determine the extrema of the function f(x)=x5−4x3+x−1f(x)=x^{5}-4 x^{3}+x-1f(x)=x5−4x3+x−1. Choose from: A. I and II B. III and IV C. II and IV D. I, II, III, and IV.

Problem 19433

Find the height hhh and radius rrr of a can with volume 550 cm³ that minimizes cost. Minimum cost in cents?

Problem 19434

Find the slope of the tangent line for f(x)=2xsin⁡xf(x)=2 x \sin xf(x)=2xsinx at x=π2x=\frac{\pi}{2}x=2π​.

Problem 19435

Find the derivative y′y^{\prime}y′ of y=xln⁡(x)y=x^{\ln (x)}y=xln(x) at x=4x=4x=4 and round to four decimal places.

Problem 19436

Determine if the series ∑k=4∞1(k−3)4\sum_{k=4}^{\infty} \frac{1}{(k-3)^{4}}∑k=4∞​(k−3)41​ converges or diverges and justify your answer.

Problem 19437

How fast is a rock falling after 4.1 seconds if it accelerates at 9.81 m/s29.81 \mathrm{~m} / \mathrm{s}^{2}9.81 m/s2?

Problem 19438

Evaluate the integral I = ∫(1/cos²(θ) + 2sin(θ)) dθ from 0 to π/3.

Problem 19439

Identify the region with area given by area=lim⁡n→∞∑i=1nπ2ntan⁡(iπ2n)\text{area}=\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{2 n} \tan\left(\frac{i \pi}{2 n}\right)area=n→∞lim​i=1∑n​2nπ​tan(2niπ​) without calculating the limit.

Problem 19440

Evaluate the integral I=∫01f(x)dxI=\int_{0}^{1} f(x) d xI=∫01​f(x)dx for a linear function fff with slope mmm and intercept bbb.

Show that the series $\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k}{2 k+1}$ diverges. Which Alternating Series Test condition fails? Identify $a_{k}$ .

Evaluate the integral: $\int \frac{d x}{x(\ln (x))^{7}}$ .

Find the integral: $\int x^{8} \sin \left(x^{9}+5\right) d x$

Find the volume of the solid formed by revolving $f(x)=(x-1)^{1/2}$ around the $x$ -axis from $x=1$ to $x=3$ . Round to hundredths.

Find the limit: $\varliminf_{x_{0} \rightarrow 3} \frac{\sqrt{x+1}-2}{x-3} \times \frac{\sqrt{x+1}+2}{\sqrt{x+1}+2}$ .

Compute the integral $\int_{-2}^{-1} \left[\frac{1}{4x} - \frac{1}{x}\right] dx$ .

Evaluate the integral: $\int \frac{e^{x}}{\left(e^{x}+1\right)^{18}} d x$

Find the critical values of the function $f(x)=5 x^{2} \ln (x)$ for $x>0$ . If none, write 'NONE'.

Calculate the integral $\int_{-1}^{0}[\frac{1}{4}x - x] dx$ .

Evaluate the integral $\int_{1}^{2} \frac{x}{(x^{2}+8)^{3}} \, dx$ .

Calculate the integral $\int_{0}^{1}(x-\frac{1}{4}x) dx$ .

Calculate the integral of $f(x) - h(x)$ from 0 to 1, where $f(x) = x$ and $h(x) = \frac{1}{4}x$ :
$\int_{0}^{1}[f(x)-h(x)] dx$

Calculate the integral $\int_{1}^{2}[g(x)-h(x)] dx$ for $g(x)= \frac{1}{x}$ and $h(x)= \frac{1}{4x}$ .

Given the function $f(x)=5 x^{2} \ln (x)$ for $x>0$ , find critical values, intervals of increase/decrease, local maxima/minima, concavity, and inflection points.

Find the area between the line $y=x$ and the parabola $y=x^2-2$ .

Find the derivative of the function $y=e^{x} \tan x$ . What is $\frac{d y}{d x}$ ?

Evaluate the integral $\int_{-1/2}^{0} 2x^{2}(-x) \, dx$ .

A plane flies south at $800 \mathrm{~km/h}$ and another flies east at $900 \mathrm{~km/h}$ . At 5 pm, the east plane is $1600 \mathrm{~km}$ from the airport. Find the rate of change of distance between the planes. Round to one decimal place.

Find the volume change rate $V$ of a sphere with radius increasing at $6 \mathrm{~cm/s}$ when radius is $5 \mathrm{~cm}$ .

Find the rate of radius increase of a balloon inflating at $4.3 \mathrm{ft}^3/\mathrm{min}$ when the radius is $1.5 \mathrm{ft}$ .

Calculate the integral $\int_{0}^{1/2} (1 - 2x^{2} - x) \, dx$ .

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{\frac{4}{2-(-1+\Delta x)^{2}}-4}{\Delta x}$ .

Calculate the integral $\int_{-2}^{-1}[h(x)-g(x)] dx$ with $h(x)= \frac{1}{4x}$ and $g(x)= \frac{1}{x}$ .

How fast is a rock falling after 4.1 seconds if it accelerates at $9.81 \mathrm{~m} / \mathrm{s}^{2}$ ?

Find the function $f(x)$ and the number $c$ such that $\lim _{\Delta x \rightarrow 0} \frac{\frac{4}{2-(-1+\Delta x)^{2}}-4}{\Delta x} = f^{\prime}(c)$ .

Estimate the area under $f(x)=17-x^{2}$ on $[0,4]$ using 4 equal subintervals and right endpoints.

Estimate the area under $f(x)=21-x^{2}$ on $[0,4]$ using 4 subintervals and right endpoints.

Find the area using the limit: area = $\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{2 n} \sin\left(\frac{i \pi}{2 n}\right)$ without evaluating it.

Plot $f(x)=x^{4}-6 x^{3}$ . How many inflection points does it have? Choices: 0, 1, 2, 3.

Find how many values of $x$ in $[-10,10]$ give a horizontal tangent for $f(x)=\frac{x^{3}+3 x^{2}+1}{x^{2}+2}$ .

Determine the extrema of the function $f(x)=x^{5}-4 x^{3}+x-1$ . Choose from: A. I and II B. III and IV C. II and IV D. I, II, III, and IV.

Find the height $h$ and radius $r$ of a can with volume 550 cm³ that minimizes cost. Minimum cost in cents?

Find the slope of the tangent line for $f(x)=2 x \sin x$ at $x=\frac{\pi}{2}$ .

Find the derivative $y^{\prime}$ of $y=x^{\ln (x)}$ at $x=4$ and round to four decimal places.

Determine if the series $\sum_{k=4}^{\infty} \frac{1}{(k-3)^{4}}$ converges or diverges and justify your answer.

How fast is a rock falling after 4.1 seconds if it accelerates at $9.81 \mathrm{~m} / \mathrm{s}^{2}$ ?

Identify the region with area given by $\text{area}=\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{2 n} \tan\left(\frac{i \pi}{2 n}\right)$ without calculating the limit.

Evaluate the integral $I=\int_{0}^{1} f(x) d x$ for a linear function $f$ with slope $m$ and intercept $b$ .

Find $f^{\prime}(-1)$ given $36 + 3f(x) + x^2(f(x))^3 = 0$ and $f(-1) = -3$ .

Find the limit as $h$ approaches 0 for $\frac{\sec (x+h)-\sec x}{h}$ .

Evaluate the limit: $\lim _{n \rightarrow \infty}\left|\frac{\frac{x^{n}}{n^{4} 8^{n}}+1}{\frac{x^{n}}{n^{4} 8^{n}}}\right|$

Identify the extrema of the function $f(x)=x^{2}+6 x+4$ : I. absolute max II. absolute min III. local max IV. local min. Options: A. III B. IV C. II D. II and IV.

Find the volume of the solid formed by rotating region $R$ (bounded by $f(x)=3$ and $g(x)=x+1$ from $x=0$ to $x=1$ ) around the $x$ -axis.

Find the tangent line equation to $y=8 \ln \left(x^{3}-26\right)$ at (3,0). $y=$

Identify the region with area given by the limit $\text { area }=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{4 n} \sin \left(\frac{i \pi}{4 n}\right)$ without calculating it.

Find the correct expression for the derivative of $f(x)=\frac{2 x}{x^{2}+1}$ . Options are given.

Calculate $\ln 3$ to four decimal places. What is $\ln 3$ ?

A pork roast cools from $180^{\circ} \mathrm{F}$ to $170^{\circ} \mathrm{F}$ in 20 min. Find $T(t)$ and time to reach $140^{\circ} \mathrm{F}$ .

Trouver la dérivée seconde de $f(x)=\frac{2x^2}{x^2-1}$ .

Calculate the integral $I=\int_{0}^{\pi / 3} 2 \sin (3 t) d t$ .

Find the tangent line equation at $x=2$ given $f(2)=-1$ and $f'(2)=3$ . Choose from the options.

Find the volume of the solid formed by revolving the area between $y=3x$ , $y=0$ , and $x=2$ around the $x$ -axis.

Evaluate the integral $\int \frac{(\sqrt{x}+1)^{8}}{2 \sqrt{x}} d x=\int(\square) d u$ .

How long will it take for a material decaying at $0.099\%$ per year to reduce to $30\%$ of its original amount?

Calculate the volume of the solid formed by revolving the area between $y=2x$ , $y=0$ , and $x=4$ around the $x$ -axis. The volume is $\square$ cubic units.

A photographer is 5 m from a stunt. The stunt moves at $2.5 \, \text{m/s}$ when 1 m away. Can he take a clear picture if speed must be $\leq 1 \, \text{m/s}$ ?

How long will it take for a town's population to double if it grows at a constant rate of $12\%$ ?

Find the derivative $f'(\theta)$ if $f(\theta)=2 \sqrt{\theta}+3 \sin \theta$ . Choose from the options given.

Find the volume of a solid with base $y=26 \sqrt{\cos x}$ on $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and isosceles right triangle cross sections.

Identify the region with area given by $\text{area}=\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{5 n} \sin\left(\frac{i \pi}{5 n}\right)$ without solving the limit.

Determine the horizontal asymptote for the function $h(x)=\frac{3}{x^{2}-4x}$ .

Find the volume of the solid formed by revolving the region bounded by $y=3-4x$ , $y=0$ , and $x=0$ around the $x$ -axis.

Determine the end behavior asymptote of the function $h(x) = \frac{3}{x^2 - 4x}$ with the denominator factored as $(x - 0)(x - 4)$ .

Find the volume of the solid formed by revolving the area between $y=2x$ and $y=8\sqrt{x}$ around the $x$ -axis.

Find the volume of the solid formed by revolving the region bounded by $y=3-3x$ , $y=0$ , and $x=0$ around the $x$ -axis.

Calculate the volume of the solid formed by revolving the area between $y=4x$ and $y=12\sqrt{x}$ around the $x$ -axis.

Calculate the integral: $\int \frac{e^{u}}{\left(1-e^{u}\right)^{2}} dx$

Find the volume of the solid formed by revolving the region $R$ (bounded by $y=x$ , $y=2x$ , and $y=8$ ) around the $y$ -axis.

Find the value(s) of $a$ for which the slope of $g(x)=a \sqrt{x}+4 x$ is 3 at $x=4$ .