Calculus

Problem 18301

Determine the series and interval of convergence for the function $f(x) = \frac{x + a}{x^{2} + a^{2}}, a > 0$

See Solution

Problem 18302

Calculate the average rate of change of $f(x)=-2 x^{3}+5 x^{2}$ between $x=-3$ and $x=2$ .

See Solution

Problem 18303

Determine if the series $\sum_{n=1}^{\infty} \frac{9}{n(n+3)}$ converges or diverges, and find the sum if it converges.

See Solution

Problem 18304

A rocket reaches 507 m. What is its velocity when it returns to the start, with gravity at -9.8 m/s²?

See Solution

Problem 18305

Which statements about the function $f(x)=\frac{1}{(x-5)^{2}}$ are true? I, II, III, or combinations?

See Solution

Problem 18306

Find the Taylor series for the function $\frac{1}{\sqrt{1+x^{4}}}$ centered at 0.

See Solution

Problem 18307

Find the Taylor series expansion at 0 for $\frac{1}{\sqrt{1+x^{4}}}$ using a known series.

See Solution

Problem 18308

Evaluate the integral: $\int_{-\pi / 4}^{3 \pi / 4} 4 \sec \theta \tan \theta \, d\theta$

See Solution

Problem 18309

Evaluate the integral from 1 to 3: $\int_{1}^{3} 4 e^{x-1} d x$ using the Fundamental Theorem of Calculus.

See Solution

Problem 18310

Calculate the definite integral using Riemann sums: $\int_{-4}^{4} x^{3} \, dx$ .

See Solution

Problem 18311

Evaluate the limit using I'Hopital's Rule: $\lim _{x \rightarrow \infty} x \sin \left(\frac{16}{x}\right)$ .

See Solution

Problem 18312

Find the derivative of $y$ where $y=-\sin^{-1}(7x^2+4)$ .

See Solution

Problem 18313

Evaluate the integral from 0 to 15 of $4 \sqrt{x} \, dx$ .

See Solution

Problem 18314

Find the derivative of $y=\tan^{-1}\left(\frac{6x}{7}\right)$ .

See Solution

Problem 18315

Evaluate the integral from 0 to $\frac{\pi}{2}$ of $18 \sin x$ dx.

See Solution

Problem 18316

Check if the Mean Value Theorem applies to $f(x)=3 \ln (x)+9$ on $[1,17]$ . If yes, find $c$ values in $[1,17]$ . If no, enter DNE. $c=$

See Solution

Problem 18317

Express the limit as a definite integral: $\lim _{\Delta x_{k} \rightarrow 0} \sum_{k=1}^{n}(3 \bar{x}_{k}^{2}-6 \bar{x}_{k}+13) \Delta x_{k}$ over $[-5,4]$ .

See Solution

Problem 18318

Calculate the integral $\int_{0}^{\pi / 2} 18 \sin x \, dx$ .

See Solution

Problem 18319

Calculate the integral $\int_{0}^{16} 4 \sqrt{x} \, dx$ .

See Solution

Problem 18320

Find $\frac{dy}{dx}$ for $y=\tan^{-1}\left(\frac{6x}{7}\right)$ .

See Solution

Problem 18321

Evaluate the integral $\int_{1}^{2}\left(t+\frac{1}{t}\right)^{2} dt$ .

See Solution

Problem 18322

Find the derivative of $y=-\cos^{-1}\left(\frac{1}{x^{5}}\right)$ with respect to $x$ .

See Solution

Problem 18323

Find the limit: $\lim _{n \rightarrow \infty} \frac{n^{2} \cos (n)}{8+n^{2}}$ .

See Solution

Problem 18324

Estimate the area under $f(x)=x^{2}$ from $x=0$ to $x=4$ using a right sum with 2 equal-width rectangles.

See Solution

Problem 18325

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(1+\frac{2}{x^{5}}\right)^{x}$ .

See Solution

Problem 18326

Check if the Mean Value Theorem applies to $f(x)=4 \sin^{-1} x$ on $[-1,1]$ . If yes, find $c$ values; otherwise, enter DNE.

See Solution

Problem 18327

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(1+\frac{2}{x^{5}}\right)^{x}$ .

See Solution

Problem 18328

A function $f(x)$ is increasing at a decreasing rate. What are the signs of $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ ?

See Solution

Problem 18329

Calculate the integral $\int_{0}^{4} \frac{t^{2}+1}{\sqrt{t}} d t$ .

See Solution

Problem 18330

Evaluate the integral from 0 to 4 of $\frac{t^{2}+1}{\sqrt{t}}$ dt.

See Solution

Problem 18331

Find the derivative of $y=\int_{0}^{x} \sqrt{6 t+3} \, dt$ .

See Solution

Problem 18332

Find the first eight terms of the partial sums for the series $\sum_{n=1}^{\infty} \frac{7}{\sqrt[3]{n}}$ and check convergence.

See Solution

Problem 18333

Estimate the change in atmospheric pressure as altitude rises from $z=9 \mathrm{~km}$ to $z=9.03 \mathrm{~km}$ using $P(z)=1000 e^{-\frac{z}{10}}$ and linear approximation.

See Solution

Problem 18334

Find the rates of change of revenue $R(x)$ and cost $C(x)$ at $x=35$ with $dx/dt=30$ units/day.

See Solution

Problem 18335

Find $\frac{d y}{d t}$ given $y^{3}=2 x^{3}+11$ , $\frac{d x}{d t}=3$ , $x=2$ , $y=3$ . Round to two decimal places.

See Solution

Problem 18336

Given the function $f(x)=\frac{x^{4}}{4}-x^{3}-4$ , find:
a) Intervals of concavity. b) Inflection points as ordered pairs. c) Critical numbers and relative extrema. d) $x$ -values where $f^{\prime}(x)$ has relative extrema.

See Solution

Problem 18337

Two people walk from a right angle of a triangle. If the area changes at $2 \mathrm{~m}^{2}/\mathrm{s}$ , find their speed when $5 \mathrm{~m}$ away.

See Solution

Problem 18338

Two planes approach an airport: one from the north at $218 \mathrm{~km/hr}$ and another from the east at $278 \mathrm{~km/hr}$ . Find the rate of distance change when the northbound plane is $28 \mathrm{~km}$ and the eastbound plane is $24 \mathrm{~km}$ from the airport.

See Solution

Problem 18339

Find the partial derivative $f_{x}(x, y)$ for $f(x, y)=4 x^{2}+5 x^{2} y+12 x y^{2}$ and evaluate at $x=5$ , $y=7$ .

See Solution

Problem 18340

Find $x$ where the tangent line of $f(x)=2 x^{3}+9 x^{2}-60 x+4$ is horizontal. What is the smaller value?

See Solution

Problem 18341

Find the partial derivative $f_{x}(x, y)$ for $f(x, y)=4 x^{2}+5 x^{2} y+12 x y^{2}$ at $x=5$ , $y=1$ .

See Solution

Problem 18342

Find inflection points of $f(x)=x^{2} e^{14 x}$ at $C$ and $D$ . Determine concavity in intervals: $(-\infty, C]$ , $[C, D]$ , $[D, \infty)$ .

See Solution

Problem 18343

Solve $f^{\prime \prime}(t)=e^{4 t}+3 t^{1/6}$ with $f^{\prime}(0)=3$ and $f(0)=2$ . Find $f^{\prime}(t)$ and $f(t)$ .

See Solution

Problem 18344

Consider $a_{n}=\frac{3 n}{7 n+1}$ . (a) Is $\{a_{n}\}$ convergent or divergent? (b) Is $\sum_{n=1}^{\infty} a_{n}$ convergent or divergent?

See Solution

Problem 18345

Find the integral of $(1-3x)^9$ with respect to $x$ .

See Solution

Problem 18346

Find inflection points of $f(x)=x^{2} e^{14 x}$ at $C$ and $D$ . Determine concavity for intervals: $(-\infty, C]$ , $[C, D]$ , $[D, \infty)$ .

See Solution

Problem 18347

Evaluate the series $\sum_{n=4}^{\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)$ .

See Solution

Problem 18348

Evaluate the integral: $\int \frac{\ln (x+1)}{(x+1)} d x$

See Solution

Problem 18349

Find the first eight terms of the partial sums for the series $\sum_{n=1}^{\infty} \frac{8}{\sqrt[3]{n}}$ to four decimal places.

See Solution

Problem 18350

Determine if the series $\sum_{n=1}^{\infty} \frac{12}{n(n+3)}$ converges or diverges, and find its sum if convergent.

See Solution

Problem 18351

Calculate the integral of $(\ln x)^{2}$ with respect to $x$ : $\int(\ln x)^{2} d x$ .

See Solution

Problem 18352

Given the function $f(x)=4 \sin x+5 \cos x$ on $(-\pi, \pi)$ , find concavity, inflection points, critical numbers, and extrema.

See Solution

Problem 18353

Calculate the integral: $\int x^{3}(1+x^{4}) \, dx$

See Solution

Problem 18354

Given the function $f(x)=x^{2} \ln x$ , find intervals of concavity, inflection points, and critical numbers for relative extrema.

See Solution

Problem 18355

Find the absolute extrema of $f(x)=3 x^{2/3}-2 x$ on $[-1,1]$ . Max at $x=\square$ , min at $x=\square$ .

See Solution

Problem 18356

Find the intervals where the function $f(x)=x^{4}-10 x^{2}$ is increasing.

See Solution

Problem 18357

Find the intervals where the function $f(x)=x^{4}-30 x^{2}$ is increasing.

See Solution

Problem 18358

Find the derivative of the function $f(x)=3 \tan ^{-1}\left(6 e^{x}\right)$ , i.e., compute $f^{\prime}(x)$ .

See Solution

Problem 18359

Find the intervals where the function $f(x)=x^{4}-6 x^{2}$ is increasing.

See Solution

Problem 18360

Find the absolute min and max of the function $f(x)=2 x^{3}+6 x^{2}-144 x+2$ on the interval $-6 \leq x \leq 5$ .

See Solution

Problem 18361

Calculate the integral: $\int \sec 2 x \tan 2 x \, dx$

See Solution

Problem 18362

Find the absolute minimum of the function $f(x)=x^{3}-3 x^{2}+3 x+7$ on the interval $[-1,2]$ .

See Solution

Problem 18363

Evaluate the integral: $\int \frac{3 x}{\sqrt{1+3 x^{2}}} d x$

See Solution

Problem 18364

Evaluate the integral: $\int(7 \cos 5 x - 5 \sin 7 x) \, dx$

See Solution

Problem 18365

Calculate the integral: $\int 5 \sin ^{3} 4 x \cos 4 x \, dx$

See Solution

Problem 18366

Find intervals where the function $f(x)=\frac{1}{6} x^{6}+x^{5}-5 x^{4}$ is concave up.

See Solution

Problem 18367

A company's expenditure rate is $E(x)=6x+3$ (in hundreds). Find total expenditure for 12 days. \ $\square$ (Simplify.)

See Solution

Problem 18368

Find the total cost to drill a 300-meter oil well with fixed costs of 1,000,000 riyals and marginal cost $C'(x)=5000+12x$ .

See Solution

Problem 18369

Calculate the integral $\int x^{2} \cos 4 x^{3} \, dx$ .

See Solution

Problem 18370

A company's expenditure rate is $E(x)=6x+3$ . Find total spending for 12 days and spending from days 12 to 27.

See Solution

Problem 18371

Find $f(6)$ , $f'(6)$ , and $f''(6)$ for a twice-differentiable function $f$ with inflection points at $x=-1.4, 2.8, 8$ .

See Solution

Problem 18372

A 2g culture grows at $W'(t)= 0.2e^{0.1t}$ . Find its weight increase in the first 6 hours and from 6 to 12 hours.

See Solution

Problem 18373

Me-Tube's demand function is $x=f(p)=850-2p$ . Find the rate of change and relative rate of change at $p=\$200$ .

See Solution

Problem 18374

Find the total depreciation after 3 years for the function $f(t)=\frac{6000(0.3+0.28 t)}{(1+0.3 t+0.14 t^{2})^{2}}$ , rounded to the nearest cent.

See Solution

Problem 18375

Use logarithmic differentiation to find the derivative of: 37. $y=(x+5)(x+9)$ , 38. $y=(3x+5)(4x+9)$ .

See Solution

Problem 18376

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

See Solution

Problem 18377

Find the antiderivative of $f(x)=\sec (x)(6 \tan (x)-6 \sec (x))$ as $F(x)=$ .

See Solution

Problem 18378

Find the antiderivative $F(x)$ of $f(x)=\frac{5 x^{7}+9 x^{-2}}{x^{5}}$ for $x \neq 0$ .
a) Simplify $f(x)$ as $\beta x^{\alpha}$ terms.
b) Find $F(x)=$ .

See Solution

Problem 18379

Find $f^{\prime}(t)$ for $f^{\prime \prime}(t)=e^{2 t}+2 t^{1/5}$ with $f^{\prime}(0)=4$ , $f(0)=4$ . Then find $f(t)$ .

See Solution

Problem 18380

Calculate the integral: $\int x \sqrt{4-x} \, dx$

See Solution

Problem 18381

Find the total depreciation after 3 years for the function $f(t)=\frac{6000(0.3+0.28 t)}{(1+0.3 t+0.14 t^{2})^{2}}$ . Round to the nearest cent.

See Solution

Problem 18382

Calculate the integral: $\int \frac{x^{2}+2 x^{3}}{x^{4}+x^{2}} \, dx$

See Solution

Problem 18383

A wrench is dropped from a 1,500 m cliff on Mars. Find the velocity $v(t)$ , displacement $s(t)$ , and height $h$ .

See Solution

Problem 18384

Find the function $f(x)$ given that $f^{\prime \prime}(x)=3 x^{3}+4 \cos (x)$ , $f^{\prime}(0)=7$ , and $f(0)=4$ .

See Solution

Problem 18385

For the population model given by $N_{t+1}=\frac{1}{2+N_{t}^{3}}$ , find: a) the updating function $f(x)=$ b) the equation $g(x)=0$ for fixed points. c) a closed interval $[a, b]$ with an equilibrium point. d) use Newton's method to approximate an equilibrium point: provide $\left[x_{0}, x_{4}\right]$ .

See Solution

Problem 18386

Calculate the integral $\int_{0}^{2} x^{2} \sqrt{9-x^{3}} \, dx$ .

See Solution

Problem 18387

Demand for toy trucks at price $p$ : a. $f(45)=10,000$ : At $\$45$ , the demand is toy trucks. b. $f^{\prime}(45)=-300$ : At $\$45$ , the change in demand is toy trucks per dollar. c. $\frac{f^{\prime}(45)}{f(45)}=-0.03$ : At $\$45$ , the percentage change in demand is $\%$ toy trucks per dollar.

See Solution

Problem 18388

Evaluate the integral: $\int_{1}^{2} \frac{2 x+1}{\sqrt{x^{2}+x}} d x$

See Solution

Problem 18389

Calculate the integral $\int \frac{x+e^{x}}{x^{2}+e^{2 x}} d x$ .

See Solution

Problem 18390

Find the integral of $\frac{e^{\sqrt[3]{x}}}{x^{\frac{2}{3}}}$ with respect to $x$ .

See Solution

Problem 18391

Find the integral of $\frac{7}{(2 x+3)^{3}}$ with respect to $x$ .

See Solution

Problem 18392

Finde eine Stammfunktion für die Funktionen: a) $f(x)=2 x^{3}+x^{2}$ , b) $f(x)=\frac{5}{6} x^{4}+3 x-1$ , c) $f(x)=2(0,5 x^{3}+0,2 x-3)$ .

See Solution

Problem 18393

Find the limit as $x$ approaches 1 for the expression $\frac{\frac{x^{2}-1}{x^{2}(x+1)-(x+1)}}{\lim_{x \to 1}(x+1)-x-1} \frac{x^{2}-1}{(x+1)(x^{2}-1)}$ .

See Solution

Problem 18394

Calculate the integral from -1 to 1 of the function $2 - x^{2}$ .

See Solution

Problem 18395

Sequoia Publishing's cost for $x$ books is $C(x)=2600+8x$ . Revenue is $R(x)=50x-0.15x^2$ . Find $C'(x)$ and $R'(x)$ . Approximate revenue for the 201st book.

See Solution

Problem 18396

Evaluate the integral $\int_{0}^{6} \int_{\frac{x}{3}}^{2} x \sqrt{y^{3}+1} \, dy \, dx$ by reversing the order of integration.

See Solution

Problem 18397

Berechnen Sie die Integrale $\int_{0}^{3} x^{2} dx$ und $\int_{-1}^{3}(3x^{2}-2x) dx$ . Erklären Sie die Flächenberechnung von $f(x)=x^{3}-x$ und bestimmen Sie die Fläche unter dem Graphen.

See Solution

Problem 18398

Find the max and min of $f(x)=2 x^{3}-15 x^{2}+1$ for $-1 \leq x \leq 10$ . What are their values and where do they occur?

See Solution

Problem 18399

Estimate the area from 0 to 5 under $f(x)=64-x^{2}$ using 5 rectangles and right endpoints.

See Solution

Problem 18400

Approximate the area under $y=\frac{5}{x^{2}}$ from 1 to 2 using 5 rectangles with left endpoints. Round to nearest hundredth.

See Solution

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Calculus

Problem 18301

Determine the series and interval of convergence for the function f(x)=x+ax2+a2,a>0 f(x) = \frac{x + a}{x^{2} + a^{2}}, a > 0 f(x)=x2+a2x+a​,a>0

Problem 18302

Calculate the average rate of change of f(x)=−2x3+5x2f(x)=-2 x^{3}+5 x^{2}f(x)=−2x3+5x2 between x=−3x=-3x=−3 and x=2x=2x=2.

Problem 18303

Determine if the series ∑n=1∞9n(n+3)\sum_{n=1}^{\infty} \frac{9}{n(n+3)}n=1∑∞​n(n+3)9​ converges or diverges, and find the sum if it converges.

Problem 18304

A rocket reaches 507 m. What is its velocity when it returns to the start, with gravity at -9.8 m/s²?

Problem 18305

Which statements about the function f(x)=1(x−5)2f(x)=\frac{1}{(x-5)^{2}}f(x)=(x−5)21​ are true? I, II, III, or combinations?

Problem 18306

Find the Taylor series for the function 11+x4\frac{1}{\sqrt{1+x^{4}}}1+x4​1​ centered at 0.

Problem 18307

Find the Taylor series expansion at 0 for 11+x4\frac{1}{\sqrt{1+x^{4}}}1+x4​1​ using a known series.

Problem 18308

Evaluate the integral: ∫−π/43π/44sec⁡θtan⁡θ dθ\int_{-\pi / 4}^{3 \pi / 4} 4 \sec \theta \tan \theta \, d\theta∫−π/43π/4​4secθtanθdθ

Problem 18309

Evaluate the integral from 1 to 3: ∫134ex−1dx\int_{1}^{3} 4 e^{x-1} d x∫13​4ex−1dx using the Fundamental Theorem of Calculus.

Problem 18310

Calculate the definite integral using Riemann sums: ∫−44x3 dx\int_{-4}^{4} x^{3} \, dx∫−44​x3dx.

Problem 18311

Evaluate the limit using I'Hopital's Rule: lim⁡x→∞xsin⁡(16x)\lim _{x \rightarrow \infty} x \sin \left(\frac{16}{x}\right)x→∞lim​xsin(x16​).

Problem 18312

Find the derivative of yyy where y=−sin⁡−1(7x2+4)y=-\sin^{-1}(7x^2+4)y=−sin−1(7x2+4).

Problem 18313

Evaluate the integral from 0 to 15 of 4x dx4 \sqrt{x} \, dx4x​dx.

Problem 18314

Find the derivative of y=tan⁡−1(6x7)y=\tan^{-1}\left(\frac{6x}{7}\right)y=tan−1(76x​).

Problem 18315

Evaluate the integral from 0 to π2\frac{\pi}{2}2π​ of 18sin⁡x18 \sin x18sinx dx.

Problem 18316

Check if the Mean Value Theorem applies to f(x)=3ln⁡(x)+9f(x)=3 \ln (x)+9f(x)=3ln(x)+9 on [1,17][1,17][1,17]. If yes, find ccc values in [1,17][1,17][1,17]. If no, enter DNE. c= c= c=

Problem 18317

Express the limit as a definite integral: lim⁡Δxk→0∑k=1n(3xˉk2−6xˉk+13)Δxk\lim _{\Delta x_{k} \rightarrow 0} \sum_{k=1}^{n}(3 \bar{x}_{k}^{2}-6 \bar{x}_{k}+13) \Delta x_{k}limΔxk​→0​∑k=1n​(3xˉk2​−6xˉk​+13)Δxk​ over [−5,4][-5,4][−5,4].

Problem 18318

Calculate the integral ∫0π/218sin⁡x dx\int_{0}^{\pi / 2} 18 \sin x \, dx∫0π/2​18sinxdx.

Problem 18319

Calculate the integral ∫0164x dx\int_{0}^{16} 4 \sqrt{x} \, dx∫016​4x​dx.

Problem 18320

Find dydx\frac{dy}{dx}dxdy​ for y=tan⁡−1(6x7)y=\tan^{-1}\left(\frac{6x}{7}\right)y=tan−1(76x​).

Problem 18321

Evaluate the integral ∫12(t+1t)2dt\int_{1}^{2}\left(t+\frac{1}{t}\right)^{2} dt∫12​(t+t1​)2dt.

Problem 18322

Find the derivative of y=−cos⁡−1(1x5)y=-\cos^{-1}\left(\frac{1}{x^{5}}\right)y=−cos−1(x51​) with respect to xxx.

Problem 18323

Find the limit: lim⁡n→∞n2cos⁡(n)8+n2\lim _{n \rightarrow \infty} \frac{n^{2} \cos (n)}{8+n^{2}}limn→∞​8+n2n2cos(n)​.

Problem 18324

Estimate the area under f(x)=x2f(x)=x^{2}f(x)=x2 from x=0x=0x=0 to x=4x=4x=4 using a right sum with 2 equal-width rectangles.

Problem 18325

Find the limit as xxx approaches infinity: lim⁡x→∞(1+2x5)x\lim _{x \rightarrow \infty}\left(1+\frac{2}{x^{5}}\right)^{x}limx→∞​(1+x52​)x.

Problem 18326

Check if the Mean Value Theorem applies to f(x)=4sin⁡−1xf(x)=4 \sin^{-1} xf(x)=4sin−1x on [−1,1][-1,1][−1,1]. If yes, find ccc values; otherwise, enter DNE.

Problem 18327

Find the limit as xxx approaches infinity: lim⁡x→∞(1+2x5)x\lim _{x \rightarrow \infty}\left(1+\frac{2}{x^{5}}\right)^{x}limx→∞​(1+x52​)x.

Problem 18328

A function f(x)f(x)f(x) is increasing at a decreasing rate. What are the signs of f′(x)f^{\prime}(x)f′(x) and f′′(x)f^{\prime \prime}(x)f′′(x)?

Problem 18329

Calculate the integral ∫04t2+1tdt\int_{0}^{4} \frac{t^{2}+1}{\sqrt{t}} d t∫04​t​t2+1​dt.

Problem 18330

Evaluate the integral from 0 to 4 of t2+1t\frac{t^{2}+1}{\sqrt{t}}t​t2+1​ dt.

Problem 18331

Find the derivative of y=∫0x6t+3 dty=\int_{0}^{x} \sqrt{6 t+3} \, dty=∫0x​6t+3​dt.

Problem 18332

Find the first eight terms of the partial sums for the series ∑n=1∞7n3\sum_{n=1}^{\infty} \frac{7}{\sqrt[3]{n}}∑n=1∞​3n​7​ and check convergence.

Problem 18333

Estimate the change in atmospheric pressure as altitude rises from z=9 kmz=9 \mathrm{~km}z=9 km to z=9.03 kmz=9.03 \mathrm{~km}z=9.03 km using P(z)=1000e−z10P(z)=1000 e^{-\frac{z}{10}}P(z)=1000e−10z​ and linear approximation.

Problem 18334

Find the rates of change of revenue R(x)R(x)R(x) and cost C(x)C(x)C(x) at x=35x=35x=35 with dx/dt=30dx/dt=30dx/dt=30 units/day.

Problem 18335

Find dydt\frac{d y}{d t}dtdy​ given y3=2x3+11y^{3}=2 x^{3}+11y3=2x3+11, dxdt=3\frac{d x}{d t}=3dtdx​=3, x=2x=2x=2, y=3y=3y=3. Round to two decimal places.

Problem 18336

Given the function f(x)=x44−x3−4f(x)=\frac{x^{4}}{4}-x^{3}-4f(x)=4x4​−x3−4, find:a) Intervals of concavity. b) Inflection points as ordered pairs. c) Critical numbers and relative extrema. d) xxx-values where f′(x)f^{\prime}(x)f′(x) has relative extrema.

Problem 18337

Two people walk from a right angle of a triangle. If the area changes at 2 m2/s2 \mathrm{~m}^{2}/\mathrm{s}2 m2/s, find their speed when 5 m5 \mathrm{~m}5 m away.

Problem 18338

Problem 18339

Find the partial derivative fx(x,y)f_{x}(x, y)fx​(x,y) for f(x,y)=4x2+5x2y+12xy2f(x, y)=4 x^{2}+5 x^{2} y+12 x y^{2}f(x,y)=4x2+5x2y+12xy2 and evaluate at x=5x=5x=5, y=7y=7y=7.

Problem 18340

Find xxx where the tangent line of f(x)=2x3+9x2−60x+4f(x)=2 x^{3}+9 x^{2}-60 x+4f(x)=2x3+9x2−60x+4 is horizontal. What is the smaller value?

Determine the series and interval of convergence for the function $f(x) = \frac{x + a}{x^{2} + a^{2}}, a > 0$

Calculate the average rate of change of $f(x)=-2 x^{3}+5 x^{2}$ between $x=-3$ and $x=2$ .

Determine if the series $\sum_{n=1}^{\infty} \frac{9}{n(n+3)}$ converges or diverges, and find the sum if it converges.

Which statements about the function $f(x)=\frac{1}{(x-5)^{2}}$ are true? I, II, III, or combinations?

Find the Taylor series for the function $\frac{1}{\sqrt{1+x^{4}}}$ centered at 0.

Find the Taylor series expansion at 0 for $\frac{1}{\sqrt{1+x^{4}}}$ using a known series.

Evaluate the integral: $\int_{-\pi / 4}^{3 \pi / 4} 4 \sec \theta \tan \theta \, d\theta$

Evaluate the integral from 1 to 3: $\int_{1}^{3} 4 e^{x-1} d x$ using the Fundamental Theorem of Calculus.

Calculate the definite integral using Riemann sums: $\int_{-4}^{4} x^{3} \, dx$ .

Evaluate the limit using I'Hopital's Rule: $\lim _{x \rightarrow \infty} x \sin \left(\frac{16}{x}\right)$ .

Find the derivative of $y$ where $y=-\sin^{-1}(7x^2+4)$ .

Evaluate the integral from 0 to 15 of $4 \sqrt{x} \, dx$ .

Find the derivative of $y=\tan^{-1}\left(\frac{6x}{7}\right)$ .

Evaluate the integral from 0 to $\frac{\pi}{2}$ of $18 \sin x$ dx.

Check if the Mean Value Theorem applies to $f(x)=3 \ln (x)+9$ on $[1,17]$ . If yes, find $c$ values in $[1,17]$ . If no, enter DNE. $c=$

Express the limit as a definite integral: $\lim _{\Delta x_{k} \rightarrow 0} \sum_{k=1}^{n}(3 \bar{x}_{k}^{2}-6 \bar{x}_{k}+13) \Delta x_{k}$ over $[-5,4]$ .

Calculate the integral $\int_{0}^{\pi / 2} 18 \sin x \, dx$ .

Calculate the integral $\int_{0}^{16} 4 \sqrt{x} \, dx$ .

Find $\frac{dy}{dx}$ for $y=\tan^{-1}\left(\frac{6x}{7}\right)$ .

Evaluate the integral $\int_{1}^{2}\left(t+\frac{1}{t}\right)^{2} dt$ .

Find the derivative of $y=-\cos^{-1}\left(\frac{1}{x^{5}}\right)$ with respect to $x$ .

Find the limit: $\lim _{n \rightarrow \infty} \frac{n^{2} \cos (n)}{8+n^{2}}$ .

Estimate the area under $f(x)=x^{2}$ from $x=0$ to $x=4$ using a right sum with 2 equal-width rectangles.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(1+\frac{2}{x^{5}}\right)^{x}$ .

Check if the Mean Value Theorem applies to $f(x)=4 \sin^{-1} x$ on $[-1,1]$ . If yes, find $c$ values; otherwise, enter DNE.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(1+\frac{2}{x^{5}}\right)^{x}$ .

A function $f(x)$ is increasing at a decreasing rate. What are the signs of $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ ?

Calculate the integral $\int_{0}^{4} \frac{t^{2}+1}{\sqrt{t}} d t$ .

Evaluate the integral from 0 to 4 of $\frac{t^{2}+1}{\sqrt{t}}$ dt.

Find the derivative of $y=\int_{0}^{x} \sqrt{6 t+3} \, dt$ .

Find the first eight terms of the partial sums for the series $\sum_{n=1}^{\infty} \frac{7}{\sqrt[3]{n}}$ and check convergence.

Estimate the change in atmospheric pressure as altitude rises from $z=9 \mathrm{~km}$ to $z=9.03 \mathrm{~km}$ using $P(z)=1000 e^{-\frac{z}{10}}$ and linear approximation.

Find the rates of change of revenue $R(x)$ and cost $C(x)$ at $x=35$ with $dx/dt=30$ units/day.

Find $\frac{d y}{d t}$ given $y^{3}=2 x^{3}+11$ , $\frac{d x}{d t}=3$ , $x=2$ , $y=3$ . Round to two decimal places.

Given the function $f(x)=\frac{x^{4}}{4}-x^{3}-4$ , find:
a) Intervals of concavity. b) Inflection points as ordered pairs. c) Critical numbers and relative extrema. d) $x$ -values where $f^{\prime}(x)$ has relative extrema.

Two people walk from a right angle of a triangle. If the area changes at $2 \mathrm{~m}^{2}/\mathrm{s}$ , find their speed when $5 \mathrm{~m}$ away.

Find the partial derivative $f_{x}(x, y)$ for $f(x, y)=4 x^{2}+5 x^{2} y+12 x y^{2}$ and evaluate at $x=5$ , $y=7$ .

Find $x$ where the tangent line of $f(x)=2 x^{3}+9 x^{2}-60 x+4$ is horizontal. What is the smaller value?

Find the partial derivative $f_{x}(x, y)$ for $f(x, y)=4 x^{2}+5 x^{2} y+12 x y^{2}$ at $x=5$ , $y=1$ .

Find inflection points of $f(x)=x^{2} e^{14 x}$ at $C$ and $D$ . Determine concavity in intervals: $(-\infty, C]$ , $[C, D]$ , $[D, \infty)$ .

Solve $f^{\prime \prime}(t)=e^{4 t}+3 t^{1/6}$ with $f^{\prime}(0)=3$ and $f(0)=2$ . Find $f^{\prime}(t)$ and $f(t)$ .

Consider $a_{n}=\frac{3 n}{7 n+1}$ . (a) Is $\{a_{n}\}$ convergent or divergent? (b) Is $\sum_{n=1}^{\infty} a_{n}$ convergent or divergent?

Find the integral of $(1-3x)^9$ with respect to $x$ .

Find inflection points of $f(x)=x^{2} e^{14 x}$ at $C$ and $D$ . Determine concavity for intervals: $(-\infty, C]$ , $[C, D]$ , $[D, \infty)$ .

Evaluate the series $\sum_{n=4}^{\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)$ .

Evaluate the integral: $\int \frac{\ln (x+1)}{(x+1)} d x$

Find the first eight terms of the partial sums for the series $\sum_{n=1}^{\infty} \frac{8}{\sqrt[3]{n}}$ to four decimal places.

Determine if the series $\sum_{n=1}^{\infty} \frac{12}{n(n+3)}$ converges or diverges, and find its sum if convergent.

Calculate the integral of $(\ln x)^{2}$ with respect to $x$ : $\int(\ln x)^{2} d x$ .

Given the function $f(x)=4 \sin x+5 \cos x$ on $(-\pi, \pi)$ , find concavity, inflection points, critical numbers, and extrema.

Calculate the integral: $\int x^{3}(1+x^{4}) \, dx$

Given the function $f(x)=x^{2} \ln x$ , find intervals of concavity, inflection points, and critical numbers for relative extrema.

Find the absolute extrema of $f(x)=3 x^{2/3}-2 x$ on $[-1,1]$ . Max at $x=\square$ , min at $x=\square$ .

Find the intervals where the function $f(x)=x^{4}-10 x^{2}$ is increasing.

Find the intervals where the function $f(x)=x^{4}-30 x^{2}$ is increasing.

Find the derivative of the function $f(x)=3 \tan ^{-1}\left(6 e^{x}\right)$ , i.e., compute $f^{\prime}(x)$ .

Find the intervals where the function $f(x)=x^{4}-6 x^{2}$ is increasing.

Find the absolute min and max of the function $f(x)=2 x^{3}+6 x^{2}-144 x+2$ on the interval $-6 \leq x \leq 5$ .

Calculate the integral: $\int \sec 2 x \tan 2 x \, dx$

Find the absolute minimum of the function $f(x)=x^{3}-3 x^{2}+3 x+7$ on the interval $[-1,2]$ .

Evaluate the integral: $\int \frac{3 x}{\sqrt{1+3 x^{2}}} d x$

Evaluate the integral: $\int(7 \cos 5 x - 5 \sin 7 x) \, dx$

Calculate the integral: $\int 5 \sin ^{3} 4 x \cos 4 x \, dx$

Find intervals where the function $f(x)=\frac{1}{6} x^{6}+x^{5}-5 x^{4}$ is concave up.

A company's expenditure rate is $E(x)=6x+3$ (in hundreds). Find total expenditure for 12 days. \ $\square$ (Simplify.)

Find the total cost to drill a 300-meter oil well with fixed costs of 1,000,000 riyals and marginal cost $C'(x)=5000+12x$ .

Calculate the integral $\int x^{2} \cos 4 x^{3} \, dx$ .

A company's expenditure rate is $E(x)=6x+3$ . Find total spending for 12 days and spending from days 12 to 27.

Find $f(6)$ , $f'(6)$ , and $f''(6)$ for a twice-differentiable function $f$ with inflection points at $x=-1.4, 2.8, 8$ .

A 2g culture grows at $W'(t)= 0.2e^{0.1t}$ . Find its weight increase in the first 6 hours and from 6 to 12 hours.

Me-Tube's demand function is $x=f(p)=850-2p$ . Find the rate of change and relative rate of change at $p=\$200$ .

Find the total depreciation after 3 years for the function $f(t)=\frac{6000(0.3+0.28 t)}{(1+0.3 t+0.14 t^{2})^{2}}$ , rounded to the nearest cent.