Calculus

Problem 31401

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{\frac{1}{9}}$ . What is $\frac{\mathrm{dy}}{\mathrm{dx}}$ ?

See Solution

Problem 31402

Find the integral $I=\int x(3 \sec ^{2} x + 2 \tan ^{2} x) \, dx$ .

See Solution

Problem 31403

Find the local minimum and maximum of $f(x)=-2x^3+24x^2-42x+7$ . What are their $x$ values and corresponding function values?

See Solution

Problem 31404

Find the limit as $x$ approaches -1 for $(x+2)^{4}(4x^{2})$ . Identify the Limit Laws used (A-F).

See Solution

Problem 31405

Find the trapezoidal sum for $\int_{3}^{7} 2 \sqrt{x} \, dx$ using 4 equal subintervals. Round to the nearest thousandth.

See Solution

Problem 31406

Calculate the average value of $f(x)=\frac{6}{9-4 x}$ from $x=3$ to $x=9$ as a constant times $\ln 3$ .

See Solution

Problem 31407

Find the value of $\int_{-8}^{-6} f\left(\frac{1}{2} x-3\right) d x$ given that $\int_{-7}^{-6} f(u) d u=-6$ .

See Solution

Problem 31408

Find the value of $\int_{-1}^{2} f(x) d x$ given $\int_{4}^{2} f(x) d x=-4$ and $\int_{-1}^{4} f(x) d x=3$ .

See Solution

Problem 31409

Find $c(15)+\int_{15}^{24} c^{\prime}(n) d n$ using given data and interpret it regarding snow plowing costs.

See Solution

Problem 31410

If $g'(x) = f(x)$ and $g(-12) = a$ , $g(-3) = b$ , $g(1) = c$ , $g(4) = d$ , find $\int_{-6}^{2} f\left(\frac{1}{2} x\right) dx$ in terms of $a$ , $b$ , $c$ , and/or $d$ .

See Solution

Problem 31411

Find the value of $\int_{-3}^{10} \frac{1}{3} f(x) d x$ given $\int_{-2}^{-3} f(x) d x=-6$ and $\int_{-2}^{10} f(x) d x=3$ .

See Solution

Problem 31412

Find the value of $\int_{9}^{8}(f(x)+2) d x$ given $\int_{3}^{9} f(x) dx=19$ and $\int_{3}^{8} f(x) dx=10$ .

See Solution

Problem 31413

Finde die Wendepunkte für die Funktionen $f(x)=x^{3}+1$ , $f(x)=x^{4}-2x^{3}$ und $f(x)=-3x^{3}+12x+3$ .

See Solution

Problem 31414

Berechnen Sie die Fläche zwischen dem Graphen der Funktion $f: x \mapsto x^{2}-2 x$ und dem Intervall $[-1 ; 2]$ und zeigen Sie, dass die Teilflächen gleich sind.

See Solution

Problem 31415

计算从 $x=4$ 到 $x=10$ 的函数 $f(x)$ 的积分，其中 $f(x)$ 由线段 $AB$ , $BC$ , 和 $CD$ 组成。

See Solution

Problem 31416

Find $\frac{d f^{-1}}{d x}$ at $x=f(3)$ for $f(x)=x^{2}-3x+2$ , where $x>1$ . Choices: a. $\frac{1}{3}$ b. 2 c. 3 d. none e. $\frac{1}{2}$ .

See Solution

Problem 31417

Bestimme die Wendepunkte der Funktionen: $f(x)=-3x^{3}+12x+3$ , $f(x)=3x^{3}+9x^{2}-27x+3$ , $f(x)=-3x^{2}+2x+x^{3}$ , $f(x)=2+x-6x^{2}+x^{4}$ .

See Solution

Problem 31418

Oblicz granice ciągów:
1. $\frac{6 \cdot 7^{n}+3^{n}}{4 \cdot 6^{n}-2^{n}+7^{n}}$
2. $\sqrt{7 n^{2}-2 n+3}-\sqrt{7 n^{2}+6 n+1}$
3. $\sqrt[n]{5^{n}+\left(\frac{5}{7}\right)^{n}+7^{n}}$
4. $\left(\frac{3 n+3}{3 n+1}\right)^{7 n+6}$

See Solution

Problem 31419

Find the surface area for revolving $y=10 \sqrt{\sin x}$ , $\frac{\pi}{4} \leq x \leq \frac{\pi}{2}$ around the $x$ -axis.

See Solution

Problem 31420

Zbadaj zbieżność szeregów: 7. $\sum_{n=1}^{\infty} \frac{\sqrt[n]{n}}{\log 50}$ , 8. $\sum_{n=1}^{\infty} \frac{7^{n}}{n^{2} 4^{n}}$ , 9. $\sum_{n=1}^{\infty}\left(\frac{n+6}{2 n+7}\right)^{n}$ , 10. $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{4 n-1}$ . Podaj kryteria.

See Solution

Problem 31421

Berechne die Flächeninhalte, die von den Funktionen $f(x)=-0,5 x^{2}+2$ und $g(x)=(x+2)^{2}-3,5$ eingeschlossen werden.

See Solution

Problem 31422

Untersuchen Sie die Funktionen $f(x)=x^{3}-x$ und $g(x)=3 e^{x}+x$ auf Punktsymmetrie, Krümmung und Sattelpunkte.

See Solution

Problem 31423

Bestimme das globale Minimum der Funktion $d(x)=\sqrt{x^{2}+(-x^{2}+4x+12)^{2}}$ für $-2 \leq x \leq 6$ .

See Solution

Problem 31424

Find the length of the curve defined by $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} \, dt$ from $y=-\frac{\pi}{6}$ to $y=\frac{\pi}{3}$ . Select one: 0, $2 \sqrt{3}$ , none, 1, $\frac{4}{\sqrt{3}}$ .

See Solution

Problem 31425

Find the derivative of $\frac{1}{(3-x)^{4}}$ with respect to $x$ .

See Solution

Problem 31426

Find the derivative of the function $\sqrt{5x - 3}$ with respect to $x$ .

See Solution

Problem 31427

Evaluate the following limits as $n$ approaches infinity:
1. $\lim _{n \rightarrow \infty} \frac{6 \cdot 7^{n}+3^{n}}{4 \cdot 6^{n}-2^{n}+7^{n}}$
2. $\lim _{n \rightarrow \infty} \sqrt{7 n^{2}-2 n+3}-\sqrt{7 n^{2}+6 n+1}$
3. $\lim _{n \rightarrow \infty} \sqrt[n]{5^{n}+\left(\frac{5}{7}\right)^{n}+7^{n}}$
4. $\lim _{n \rightarrow \infty}\left(\frac{3 n+3}{3 n+1}\right)^{7 n+6}$

See Solution

Problem 31428

Find the derivative of $f(x) = (6x + 5)^{\frac{5}{3}}$ .

See Solution

Problem 31429

Find the derivatives of: a) $x \tan x$ , b) $x^{2} e^{-x}$ , c) $50^{-2 x} \sin 3 x$ , d) $3 x^{1/2} \cos 2 x$ .

See Solution

Problem 31430

Find $z_s$ at $(t, s)=(0,0)$ for $z=\sin 2x \cos 3y$ , where $x=t+s$ , $y=s-t$ . Options: a. 2 b. 1 c. $0^x$ d. -1

See Solution

Problem 31431

Find the inflection point of $f(x)=-0.1 x^{3}+3 x^{2}+1100$ . Choose from: a. $(10,1300)$ b. $(20,1300)$ c. $(20,1500)$ d. $(0,1100)$ e. $(10,30)$

See Solution

Problem 31432

Differentiate the function $f(x) = 2 x^{6}(1+x)^{5}$ .

See Solution

Problem 31433

Find all $x$ where relative minima occur for $y=x^{3}-3x^{2}-72x+1$ . Options: a. 6, b. 0, c. None, d. -4, e. 1.

See Solution

Problem 31434

Determine the behavior of the function $f(x)=\frac{1}{x^{2}+1}$ : increasing, decreasing, or neither for $x<0$ and $x>0$ .

See Solution

Problem 31435

Find the critical points of $y=x^{3}-3 x^{2}-45 x+6$ . Choose from the options given.

See Solution

Problem 31436

Oblicz granicę ciągu $a_{n}=\frac{\left(3 n^{4}+n^{2}+1\right)^{4}}{(n+4)^{16}}$ dla $n \to \infty$ .

See Solution

Problem 31437

Find the intervals where the function $f(x)=\frac{x^{4}}{12}-2 x^{2}$ is concave up. Options: a. $(-\infty,-2),(2, \infty)$ b. none c. $(2, \infty)$ d. $(-2,2)$ e. $(-4,4)$

See Solution

Problem 31438

Budd's plane cost per hour follows $\frac{dp}{dt} = 10 - \frac{600}{t^2}$ .
a) Find $h$ for local minimum.
b) Prove cost is \$200/hour for 10 hours if \$230/hour for 4 hours.

See Solution

Problem 31439

Oblicz granicę ciągu $a_{n}=\left(\frac{4 n+3}{4 n+1}\right)^{7 n+6}$ .

See Solution

Problem 31440

Find the derivative of the function $f(x)=(x^{3}+2x)\sqrt{x}$ . What is $f'(x)$ ?

See Solution

Problem 31441

Oblicz granicę ciągu $a_{n}=\sqrt{2 n^{2}+6 n+4}-\sqrt{2 n^{2}+3 n-2}$ .

See Solution

Problem 31442

Zbadać zbieżność szeregu $\sum_{n=1}^{\infty} \frac{n^{2} 4^{n}}{3^{n}}$ .

See Solution

Problem 31443

Zbadaj zbieżność szeregu $\sum_{n=1}^{\infty} \frac{5 n^{4}+2}{3 n^{4}+11}$ .

See Solution

Problem 31444

Find the derivative of the function $f(x)=x^{5/2}$ .

See Solution

Problem 31445

Find the derivative of $f(x)=x^{3}+8$ at $x=2$ .

See Solution

Problem 31446

Zbadać zbieżność szeregu $\sum_{n=1}^{\infty}\left(\frac{7n+6}{4n+2}\right)^{n}$ .

See Solution

Problem 31447

Find the slope of the curve $y=\sqrt{2 x^{2}+1}$ at the point $(2,3)$ .

See Solution

Problem 31448

Differentiate $y=(x^{2}+4)^{3}$ .

See Solution

Problem 31449

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=x^{2}-7x$ and simplify.

See Solution

Problem 31450

Find the second derivative of the function $f(x)=9 x^{2}+2 x-2$ .

See Solution

Problem 31451

Find the limit: $\lim _{x \rightarrow 3} \frac{\sqrt{x}-4}{x^{3}+27}$ .

See Solution

Problem 31452

Oblicz granicę ciągu $a_{n}=\sqrt[n]{2^{n}+\left(\frac{2}{3}\right)^{n}+3^{n}}$ .

See Solution

Problem 31453

Determine if the limit exists and find its value: $\lim _{x \rightarrow 2} \frac{x^{2}+2 x-8}{x-2}$

See Solution

Problem 31454

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

See Solution

Problem 31455

Differentiate the function: $y=\frac{8}{x^{3}}-\frac{8}{x}$ .

See Solution

Problem 31456

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

See Solution

Problem 31457

Find $\lim _{x \rightarrow 8}[f(x)-g(x)]$ given $\lim _{x \rightarrow 8} f(x)=4$ and $\lim _{x \rightarrow 8} g(x)=-5$ .

See Solution

Problem 31458

Find points of inflection for the function $f(x)=-x^{3}+5x+2$ .

See Solution

Problem 31459

Find the integral of $\frac{\sin x}{\cos ^{3} x}$ with respect to $x$ .

See Solution

Problem 31460

Find $\mathrm{dy} / \mathrm{dx}$ using $y=\cos(\sqrt{11} x)$ and $u=\sqrt{11} x$ with substitution and chain rule.

See Solution

Problem 31461

Find the inflection point(s) of the function $y=2 x^{3}-3 x^{2}-12 x+17$ .

See Solution

Problem 31462

Find the relative extrema of the function $f(x)=x^{3}-12x+4$ .

See Solution

Problem 31463

Determine where the function $f(x)=4 x^{4}-3 x^{3}+5 x-10$ is concave up.

See Solution

Problem 31464

Find the $x$ -coordinates of all relative extrema for $f(x)=\frac{1}{4} x^{4}+\frac{2}{3} x^{3}-\frac{3}{2} x^{2}+4$ .

See Solution

Problem 31465

Berechnen Sie die Steigung von $f$ bei $x_{0}$ für: a) $f(x)=x^{2}, x_{0}=-1$ und b) $f(x)=0,5 x^{2}, x_{0}=2$ .

See Solution

Problem 31466

1. Don won a lottery of $\$ 1000$ weekly for 25 years. How much to invest now at $4\%$ weekly compounding?
2. An annuity pays $\$ 1200$ yearly for 15 years at $5.2\%$ annual compounding. Find the total interest earned.
3. Noah has two annuity options: - A: $\$ 1000$ yearly at $6.25\%$ annually. - B: $\$ 500$ semi-annually at $6.25\%$ semi-annually. Which is better? Show calculations.

See Solution

Problem 31467

Find the length of the curve $x=\tan(3y)$ for $\frac{\pi}{18} \leq y \leq \frac{\pi}{9}$ .

See Solution

Problem 31468

Find the area of the region between $y=e^{x}$ , $y=2$ , and $x=-1$ . Then, find volumes when this region is revolved around $x=-1$ and $y=-1$ .

See Solution

Problem 31469

Find the integral of $\sec ^{2}(x) \tan ^{2}(x)$ with respect to $x$ .

See Solution

Problem 31470

Find the volume of a solid with base under $y=e^{x}$ , $x=1$ , and square cross sections perpendicular to the $x$ -axis.

See Solution

Problem 31471

Calculate the area between the $x$ -axis and the curve $y=x^{3}-7 x^{2}+10 x$ .

See Solution

Problem 31472

Find the area between $f(x)=\sin x$ and $g(x)=-\sin x$ for $0 \leq x \leq \pi$ , volume when revolved about $y=3$ , and $k$ if volume is $8\pi$ .

See Solution

Problem 31473

Find $f_{xz}$ for $f(x, y, z)=e^{xyz}$ at the point (1, -1, -1). Choices: a. $e$ , b. $0^{x}$ , c. $-e$ , d. None.

See Solution

Problem 31474

Find limits for the function $f(x)=2 x e^{2 x}$ : a. $\lim _{x \rightarrow-\infty} f(x)=$ , b. $\lim _{x \rightarrow \infty} f(x)=$ .

See Solution

Problem 31475

Check if the function $f$ is continuous at $x=2$ , where $f(x)=\frac{x^{2}-4}{x-2}$ for $x \neq 2$ and $f(2)=1$ .

See Solution

Problem 31476

Find the infinite limit: $\lim _{x \rightarrow 0}\left(\ln \left(x^{2}\right)-x^{-2}\right)$ . Is it $\infty$ or $-\infty$ ?

See Solution

Problem 31477

Find the integral $\int(4x-x^{3})dx$ and evaluate $\int_{-2}^{2}(4x+x^{3})dx$ . Explain area under $y=4x-x^{3}$ .

See Solution

Problem 31478

Find the volumes using the shell method for the region revolving around: a. $x$ -axis, b. $y=1$ , c. $y=\frac{8}{5}$ , d. $y=-\frac{2}{5}$ . Answer in terms of $\pi$ .

See Solution

Problem 31479

Evaluate the integral: $\int \frac{\sec x \, dx}{\sqrt{\ln (\sec x + \tan x)}}$ .

See Solution

Problem 31480

Find the critical points of the function $y=x^{3}-3 x^{2}-45 x+6$ . Which option is correct?

See Solution

Problem 31481

For the function described, find the limits: (a) $\lim _{x \rightarrow 1} f(x)$ and (b) $\lim _{x \rightarrow 3^{-}} f(x)$ .

See Solution

Problem 31482

Find the $\mathrm{x}$ -values for relative minima of $y=x^{3}-3 x^{2}-72 x+1$ : a. 1 b. 6 c. -4 d. None e. 0

See Solution

Problem 31483

Determine the values of $x$ for the convergence of the series $\sum_{n=0}^{\infty} \frac{\cos ^{n}(x)}{7^{n}}$ and find its sum.

See Solution

Problem 31484

Determine the values of $x$ for convergence of the series $\sum_{n=0}^{\infty} \frac{\cos ^{n}(x)}{7^{n}}$ and find its sum.

See Solution

Problem 31485

Use the shell method to find the volume by revolving the region bounded by $y=3x$ , $y=0$ , and $x=2$ around various lines.

See Solution

Problem 31486

Find the relative maximum point of the function $f(x)=\frac{x^{2}-2}{x^{2}-1}$ using $f'(x)$ and $f''(x)$ .

See Solution

Problem 31487

Find the total sales formula for a company with daily growth rate $10 e^{0.02 t}$ and initial sales of 800 items.

See Solution

Problem 31488

Find the derivative of the function $f(x)=3\left(\frac{1}{3}\right)^{x}$ .

See Solution

Problem 31489

A firework launched at $128 \mathrm{ft/s}$ with gravity $-16 \mathrm{ft/s}^2$ reaches what maximum height? Options: $1,024 \mathrm{ft}$ , $448 \mathrm{ft}$ , $256 \mathrm{ft}$ , $512 \mathrm{ft}$

See Solution

Problem 31490

Find the limit: $\lim _{x \rightarrow 4} f(x)$ given that $10 x-46 \leq f(x) \leq x^{2}+2 x-30$ .

See Solution

Problem 31491

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=0$ , $x=\frac{\pi}{4}$ around $x=-1$ .

See Solution

Problem 31492

Determine where the function $f(x)=\frac{x^{2}-2}{x^{2}-1}$ is concave down based on $f^{\prime \prime}=\frac{2\left(3 x^{2}+1\right)}{\left(x^{2}-1\right)^{3}}$ . Choose the correct interval.

See Solution

Problem 31493

Determine where the function $f(x)=\frac{x^{2}}{x-1}$ is concave up based on $f^{\prime \prime}=\frac{2}{(x-1)^{3}}$ .

See Solution

Problem 31494

Calculate the area under the curve of $f(x)=\left\{\begin{array}{ll}5, & \text { if } x<1 \\ 5 x^{2}, & \text { if } x \geq 1\end{array}\right.$ from $x=-3$ to $x=3$ .

See Solution

Problem 31495

Find the value of $c$ for $f(x)=x^{2}+x$ on $[0,2]$ using the Mean Value Theorem. Options: 1, 2, $\frac{3}{4}$ , 0, none, $\frac{3}{2}$ .

See Solution

Problem 31496

Find the $y$ -coordinate on the ellipse $4x^{2}+25y^{2}=100$ where $x=1$ , then find the tangent line equation at that point.

See Solution

Problem 31497

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{6x + 25} - 5}{x}$

See Solution

Problem 31498

Find the limit as $b$ approaches 4 for the expression $\frac{\frac{1}{b}-\frac{1}{4}}{b-4}$ .

See Solution

Problem 31499

Find the infinite limit: $\lim _{x \rightarrow 4} \frac{x^{2}+8 x}{x^{2}-8 x+16}$

See Solution

Problem 31500

Find the average rate of change of $f(x)=x^{2}+10x$ over $[3,3+h]$ as an expression in terms of $h$ .

See Solution

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Calculus

Problem 31401

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=x19y=x^{\frac{1}{9}}y=x91​. What is dydx\frac{\mathrm{dy}}{\mathrm{dx}}dxdy​?

Problem 31402

Find the integral I=∫x(3sec⁡2x+2tan⁡2x) dxI=\int x(3 \sec ^{2} x + 2 \tan ^{2} x) \, dxI=∫x(3sec2x+2tan2x)dx.

Problem 31403

Find the local minimum and maximum of f(x)=−2x3+24x2−42x+7f(x)=-2x^3+24x^2-42x+7f(x)=−2x3+24x2−42x+7. What are their xxx values and corresponding function values?

Problem 31404

Find the limit as xxx approaches -1 for (x+2)4(4x2)(x+2)^{4}(4x^{2})(x+2)4(4x2). Identify the Limit Laws used (A-F).

Problem 31405

Find the trapezoidal sum for ∫372x dx\int_{3}^{7} 2 \sqrt{x} \, dx∫37​2x​dx using 4 equal subintervals. Round to the nearest thousandth.

Problem 31406

Calculate the average value of f(x)=69−4xf(x)=\frac{6}{9-4 x}f(x)=9−4x6​ from x=3x=3x=3 to x=9x=9x=9 as a constant times ln⁡3\ln 3ln3.

Problem 31407

Find the value of ∫−8−6f(12x−3)dx\int_{-8}^{-6} f\left(\frac{1}{2} x-3\right) d x∫−8−6​f(21​x−3)dx given that ∫−7−6f(u)du=−6\int_{-7}^{-6} f(u) d u=-6∫−7−6​f(u)du=−6.

Problem 31408

Find the value of ∫−12f(x)dx\int_{-1}^{2} f(x) d x∫−12​f(x)dx given ∫42f(x)dx=−4\int_{4}^{2} f(x) d x=-4∫42​f(x)dx=−4 and ∫−14f(x)dx=3\int_{-1}^{4} f(x) d x=3∫−14​f(x)dx=3.

Problem 31409

Find c(15)+∫1524c′(n)dnc(15)+\int_{15}^{24} c^{\prime}(n) d nc(15)+∫1524​c′(n)dn using given data and interpret it regarding snow plowing costs.

Problem 31410

If g′(x)=f(x)g'(x) = f(x)g′(x)=f(x) and g(−12)=ag(-12) = ag(−12)=a, g(−3)=bg(-3) = bg(−3)=b, g(1)=cg(1) = cg(1)=c, g(4)=dg(4) = dg(4)=d, find ∫−62f(12x)dx\int_{-6}^{2} f\left(\frac{1}{2} x\right) dx∫−62​f(21​x)dx in terms of aaa, bbb, ccc, and/or ddd.

Problem 31411

Find the value of ∫−31013f(x)dx\int_{-3}^{10} \frac{1}{3} f(x) d x∫−310​31​f(x)dx given ∫−2−3f(x)dx=−6\int_{-2}^{-3} f(x) d x=-6∫−2−3​f(x)dx=−6 and ∫−210f(x)dx=3\int_{-2}^{10} f(x) d x=3∫−210​f(x)dx=3.

Problem 31412

Find the value of ∫98(f(x)+2)dx\int_{9}^{8}(f(x)+2) d x∫98​(f(x)+2)dx given ∫39f(x)dx=19\int_{3}^{9} f(x) dx=19∫39​f(x)dx=19 and ∫38f(x)dx=10\int_{3}^{8} f(x) dx=10∫38​f(x)dx=10.

Problem 31413

Finde die Wendepunkte für die Funktionen f(x)=x3+1f(x)=x^{3}+1f(x)=x3+1, f(x)=x4−2x3f(x)=x^{4}-2x^{3}f(x)=x4−2x3 und f(x)=−3x3+12x+3f(x)=-3x^{3}+12x+3f(x)=−3x3+12x+3.

Problem 31414

Berechnen Sie die Fläche zwischen dem Graphen der Funktion f:x↦x2−2xf: x \mapsto x^{2}-2 xf:x↦x2−2x und dem Intervall [−1;2][-1 ; 2][−1;2] und zeigen Sie, dass die Teilflächen gleich sind.

Problem 31415

计算从 x=4x=4x=4 到 x=10x=10x=10 的函数 f(x)f(x)f(x) 的积分，其中 f(x)f(x)f(x) 由线段 ABABAB, BCBCBC, 和 CDCDCD 组成。

Problem 31416

Find df−1dx\frac{d f^{-1}}{d x}dxdf−1​ at x=f(3)x=f(3)x=f(3) for f(x)=x2−3x+2f(x)=x^{2}-3x+2f(x)=x2−3x+2, where x>1x>1x>1. Choices: a. 13\frac{1}{3}31​ b. 2 c. 3 d. none e. 12\frac{1}{2}21​.

Problem 31417

Bestimme die Wendepunkte der Funktionen: f(x)=−3x3+12x+3f(x)=-3x^{3}+12x+3f(x)=−3x3+12x+3, f(x)=3x3+9x2−27x+3f(x)=3x^{3}+9x^{2}-27x+3f(x)=3x3+9x2−27x+3, f(x)=−3x2+2x+x3f(x)=-3x^{2}+2x+x^{3}f(x)=−3x2+2x+x3, f(x)=2+x−6x2+x4f(x)=2+x-6x^{2}+x^{4}f(x)=2+x−6x2+x4.

Problem 31418

Problem 31419

Find the surface area for revolving y=10sin⁡xy=10 \sqrt{\sin x}y=10sinx​, π4≤x≤π2\frac{\pi}{4} \leq x \leq \frac{\pi}{2}4π​≤x≤2π​ around the xxx-axis.

Problem 31420

Problem 31421

Berechne die Flächeninhalte, die von den Funktionen f(x)=−0,5x2+2f(x)=-0,5 x^{2}+2f(x)=−0,5x2+2 und g(x)=(x+2)2−3,5g(x)=(x+2)^{2}-3,5g(x)=(x+2)2−3,5 eingeschlossen werden.

Problem 31422

Untersuchen Sie die Funktionen f(x)=x3−xf(x)=x^{3}-xf(x)=x3−x und g(x)=3ex+xg(x)=3 e^{x}+xg(x)=3ex+x auf Punktsymmetrie, Krümmung und Sattelpunkte.

Problem 31423

Bestimme das globale Minimum der Funktion d(x)=x2+(−x2+4x+12)2d(x)=\sqrt{x^{2}+(-x^{2}+4x+12)^{2}}d(x)=x2+(−x2+4x+12)2​ für −2≤x≤6-2 \leq x \leq 6−2≤x≤6.

Problem 31424

Find the length of the curve defined by x=∫0ysec⁡4t−1 dtx=\int_{0}^{y} \sqrt{\sec ^{4} t-1} \, dtx=∫0y​sec4t−1​dt from y=−π6y=-\frac{\pi}{6}y=−6π​ to y=π3y=\frac{\pi}{3}y=3π​. Select one: 0, 232 \sqrt{3}23​, none, 1, 43\frac{4}{\sqrt{3}}3​4​.

Problem 31425

Find the derivative of 1(3−x)4\frac{1}{(3-x)^{4}}(3−x)41​ with respect to xxx.

Problem 31426

Find the derivative of the function 5x−3\sqrt{5x - 3}5x−3​ with respect to xxx.

Problem 31427

Problem 31428

Find the derivative of f(x)=(6x+5)53f(x) = (6x + 5)^{\frac{5}{3}}f(x)=(6x+5)35​.

Problem 31429

Find the derivatives of: a) xtan⁡xx \tan xxtanx, b) x2e−xx^{2} e^{-x}x2e−x, c) 50−2xsin⁡3x50^{-2 x} \sin 3 x50−2xsin3x, d) 3x1/2cos⁡2x3 x^{1/2} \cos 2 x3x1/2cos2x.

Problem 31430

Find zsz_szs​ at (t,s)=(0,0)(t, s)=(0,0)(t,s)=(0,0) for z=sin⁡2xcos⁡3yz=\sin 2x \cos 3yz=sin2xcos3y, where x=t+sx=t+sx=t+s, y=s−ty=s-ty=s−t. Options: a. 2 b. 1 c. 0x0^x0x d. -1

Problem 31431

Find the inflection point of f(x)=−0.1x3+3x2+1100f(x)=-0.1 x^{3}+3 x^{2}+1100f(x)=−0.1x3+3x2+1100. Choose from: a. (10,1300)(10,1300)(10,1300) b. (20,1300)(20,1300)(20,1300) c. (20,1500)(20,1500)(20,1500) d. (0,1100)(0,1100)(0,1100) e. (10,30)(10,30)(10,30)

Problem 31432

Differentiate the function f(x)=2x6(1+x)5f(x) = 2 x^{6}(1+x)^{5}f(x)=2x6(1+x)5.

Problem 31433

Find all xxx where relative minima occur for y=x3−3x2−72x+1y=x^{3}-3x^{2}-72x+1y=x3−3x2−72x+1. Options: a. 6, b. 0, c. None, d. -4, e. 1.

Problem 31434

Determine the behavior of the function f(x)=1x2+1f(x)=\frac{1}{x^{2}+1}f(x)=x2+11​: increasing, decreasing, or neither for x<0x<0x<0 and x>0x>0x>0.

Problem 31435

Find the critical points of y=x3−3x2−45x+6y=x^{3}-3 x^{2}-45 x+6y=x3−3x2−45x+6. Choose from the options given.

Problem 31436

Oblicz granicę ciągu an=(3n4+n2+1)4(n+4)16a_{n}=\frac{\left(3 n^{4}+n^{2}+1\right)^{4}}{(n+4)^{16}}an​=(n+4)16(3n4+n2+1)4​ dla n→∞n \to \inftyn→∞.

Problem 31437

Find the intervals where the function f(x)=x412−2x2f(x)=\frac{x^{4}}{12}-2 x^{2}f(x)=12x4​−2x2 is concave up. Options: a. (−∞,−2),(2,∞)(-\infty,-2),(2, \infty)(−∞,−2),(2,∞) b. none c. (2,∞)(2, \infty)(2,∞) d. (−2,2)(-2,2)(−2,2) e. (−4,4)(-4,4)(−4,4)

Problem 31438

Budd's plane cost per hour follows dpdt=10−600t2\frac{dp}{dt} = 10 - \frac{600}{t^2}dtdp​=10−t2600​. a) Find hhh for local minimum. b) Prove cost is \$200/hour for 10 hours if \$230/hour for 4 hours.

Problem 31439

Oblicz granicę ciągu an=(4n+34n+1)7n+6a_{n}=\left(\frac{4 n+3}{4 n+1}\right)^{7 n+6}an​=(4n+14n+3​)7n+6.

Problem 31440

Find the derivative of the function f(x)=(x3+2x)xf(x)=(x^{3}+2x)\sqrt{x}f(x)=(x3+2x)x​. What is f′(x)f'(x)f′(x)?

Problem 31441

Oblicz granicę ciągu an=2n2+6n+4−2n2+3n−2a_{n}=\sqrt{2 n^{2}+6 n+4}-\sqrt{2 n^{2}+3 n-2}an​=2n2+6n+4​−2n2+3n−2​.

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{\frac{1}{9}}$ . What is $\frac{\mathrm{dy}}{\mathrm{dx}}$ ?

Find the integral $I=\int x(3 \sec ^{2} x + 2 \tan ^{2} x) \, dx$ .

Find the local minimum and maximum of $f(x)=-2x^3+24x^2-42x+7$ . What are their $x$ values and corresponding function values?

Find the limit as $x$ approaches -1 for $(x+2)^{4}(4x^{2})$ . Identify the Limit Laws used (A-F).

Find the trapezoidal sum for $\int_{3}^{7} 2 \sqrt{x} \, dx$ using 4 equal subintervals. Round to the nearest thousandth.

Calculate the average value of $f(x)=\frac{6}{9-4 x}$ from $x=3$ to $x=9$ as a constant times $\ln 3$ .

Find the value of $\int_{-8}^{-6} f\left(\frac{1}{2} x-3\right) d x$ given that $\int_{-7}^{-6} f(u) d u=-6$ .

Find the value of $\int_{-1}^{2} f(x) d x$ given $\int_{4}^{2} f(x) d x=-4$ and $\int_{-1}^{4} f(x) d x=3$ .

Find $c(15)+\int_{15}^{24} c^{\prime}(n) d n$ using given data and interpret it regarding snow plowing costs.

If $g'(x) = f(x)$ and $g(-12) = a$ , $g(-3) = b$ , $g(1) = c$ , $g(4) = d$ , find $\int_{-6}^{2} f\left(\frac{1}{2} x\right) dx$ in terms of $a$ , $b$ , $c$ , and/or $d$ .

Find the value of $\int_{-3}^{10} \frac{1}{3} f(x) d x$ given $\int_{-2}^{-3} f(x) d x=-6$ and $\int_{-2}^{10} f(x) d x=3$ .

Find the value of $\int_{9}^{8}(f(x)+2) d x$ given $\int_{3}^{9} f(x) dx=19$ and $\int_{3}^{8} f(x) dx=10$ .

Finde die Wendepunkte für die Funktionen $f(x)=x^{3}+1$ , $f(x)=x^{4}-2x^{3}$ und $f(x)=-3x^{3}+12x+3$ .

Berechnen Sie die Fläche zwischen dem Graphen der Funktion $f: x \mapsto x^{2}-2 x$ und dem Intervall $[-1 ; 2]$ und zeigen Sie, dass die Teilflächen gleich sind.

计算从 $x=4$ 到 $x=10$ 的函数 $f(x)$ 的积分，其中 $f(x)$ 由线段 $AB$ , $BC$ , 和 $CD$ 组成。

Find $\frac{d f^{-1}}{d x}$ at $x=f(3)$ for $f(x)=x^{2}-3x+2$ , where $x>1$ . Choices: a. $\frac{1}{3}$ b. 2 c. 3 d. none e. $\frac{1}{2}$ .

Bestimme die Wendepunkte der Funktionen: $f(x)=-3x^{3}+12x+3$ , $f(x)=3x^{3}+9x^{2}-27x+3$ , $f(x)=-3x^{2}+2x+x^{3}$ , $f(x)=2+x-6x^{2}+x^{4}$ .

Find the surface area for revolving $y=10 \sqrt{\sin x}$ , $\frac{\pi}{4} \leq x \leq \frac{\pi}{2}$ around the $x$ -axis.

Berechne die Flächeninhalte, die von den Funktionen $f(x)=-0,5 x^{2}+2$ und $g(x)=(x+2)^{2}-3,5$ eingeschlossen werden.

Untersuchen Sie die Funktionen $f(x)=x^{3}-x$ und $g(x)=3 e^{x}+x$ auf Punktsymmetrie, Krümmung und Sattelpunkte.

Bestimme das globale Minimum der Funktion $d(x)=\sqrt{x^{2}+(-x^{2}+4x+12)^{2}}$ für $-2 \leq x \leq 6$ .

Find the length of the curve defined by $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} \, dt$ from $y=-\frac{\pi}{6}$ to $y=\frac{\pi}{3}$ . Select one: 0, $2 \sqrt{3}$ , none, 1, $\frac{4}{\sqrt{3}}$ .

Find the derivative of $\frac{1}{(3-x)^{4}}$ with respect to $x$ .

Find the derivative of the function $\sqrt{5x - 3}$ with respect to $x$ .

Find the derivative of $f(x) = (6x + 5)^{\frac{5}{3}}$ .

Find the derivatives of: a) $x \tan x$ , b) $x^{2} e^{-x}$ , c) $50^{-2 x} \sin 3 x$ , d) $3 x^{1/2} \cos 2 x$ .

Find $z_s$ at $(t, s)=(0,0)$ for $z=\sin 2x \cos 3y$ , where $x=t+s$ , $y=s-t$ . Options: a. 2 b. 1 c. $0^x$ d. -1

Find the inflection point of $f(x)=-0.1 x^{3}+3 x^{2}+1100$ . Choose from: a. $(10,1300)$ b. $(20,1300)$ c. $(20,1500)$ d. $(0,1100)$ e. $(10,30)$

Differentiate the function $f(x) = 2 x^{6}(1+x)^{5}$ .

Find all $x$ where relative minima occur for $y=x^{3}-3x^{2}-72x+1$ . Options: a. 6, b. 0, c. None, d. -4, e. 1.

Determine the behavior of the function $f(x)=\frac{1}{x^{2}+1}$ : increasing, decreasing, or neither for $x<0$ and $x>0$ .

Find the critical points of $y=x^{3}-3 x^{2}-45 x+6$ . Choose from the options given.

Oblicz granicę ciągu $a_{n}=\frac{\left(3 n^{4}+n^{2}+1\right)^{4}}{(n+4)^{16}}$ dla $n \to \infty$ .

Find the intervals where the function $f(x)=\frac{x^{4}}{12}-2 x^{2}$ is concave up. Options: a. $(-\infty,-2),(2, \infty)$ b. none c. $(2, \infty)$ d. $(-2,2)$ e. $(-4,4)$

Budd's plane cost per hour follows $\frac{dp}{dt} = 10 - \frac{600}{t^2}$ .
a) Find $h$ for local minimum.
b) Prove cost is \$200/hour for 10 hours if \$230/hour for 4 hours.

Oblicz granicę ciągu $a_{n}=\left(\frac{4 n+3}{4 n+1}\right)^{7 n+6}$ .

Find the derivative of the function $f(x)=(x^{3}+2x)\sqrt{x}$ . What is $f'(x)$ ?

Oblicz granicę ciągu $a_{n}=\sqrt{2 n^{2}+6 n+4}-\sqrt{2 n^{2}+3 n-2}$ .

Zbadać zbieżność szeregu $\sum_{n=1}^{\infty} \frac{n^{2} 4^{n}}{3^{n}}$ .

Zbadaj zbieżność szeregu $\sum_{n=1}^{\infty} \frac{5 n^{4}+2}{3 n^{4}+11}$ .

Find the derivative of the function $f(x)=x^{5/2}$ .

Find the derivative of $f(x)=x^{3}+8$ at $x=2$ .

Zbadać zbieżność szeregu $\sum_{n=1}^{\infty}\left(\frac{7n+6}{4n+2}\right)^{n}$ .

Find the slope of the curve $y=\sqrt{2 x^{2}+1}$ at the point $(2,3)$ .

Differentiate $y=(x^{2}+4)^{3}$ .

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=x^{2}-7x$ and simplify.

Find the second derivative of the function $f(x)=9 x^{2}+2 x-2$ .

Find the limit: $\lim _{x \rightarrow 3} \frac{\sqrt{x}-4}{x^{3}+27}$ .

Oblicz granicę ciągu $a_{n}=\sqrt[n]{2^{n}+\left(\frac{2}{3}\right)^{n}+3^{n}}$ .

Determine if the limit exists and find its value: $\lim _{x \rightarrow 2} \frac{x^{2}+2 x-8}{x-2}$

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

Differentiate the function: $y=\frac{8}{x^{3}}-\frac{8}{x}$ .

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

Find $\lim _{x \rightarrow 8}[f(x)-g(x)]$ given $\lim _{x \rightarrow 8} f(x)=4$ and $\lim _{x \rightarrow 8} g(x)=-5$ .

Find points of inflection for the function $f(x)=-x^{3}+5x+2$ .

Find the integral of $\frac{\sin x}{\cos ^{3} x}$ with respect to $x$ .

Find $\mathrm{dy} / \mathrm{dx}$ using $y=\cos(\sqrt{11} x)$ and $u=\sqrt{11} x$ with substitution and chain rule.

Find the inflection point(s) of the function $y=2 x^{3}-3 x^{2}-12 x+17$ .

Find the relative extrema of the function $f(x)=x^{3}-12x+4$ .

Determine where the function $f(x)=4 x^{4}-3 x^{3}+5 x-10$ is concave up.

Find the $x$ -coordinates of all relative extrema for $f(x)=\frac{1}{4} x^{4}+\frac{2}{3} x^{3}-\frac{3}{2} x^{2}+4$ .

Berechnen Sie die Steigung von $f$ bei $x_{0}$ für: a) $f(x)=x^{2}, x_{0}=-1$ und b) $f(x)=0,5 x^{2}, x_{0}=2$ .

Find the length of the curve $x=\tan(3y)$ for $\frac{\pi}{18} \leq y \leq \frac{\pi}{9}$ .

Find the area of the region between $y=e^{x}$ , $y=2$ , and $x=-1$ . Then, find volumes when this region is revolved around $x=-1$ and $y=-1$ .

Find the integral of $\sec ^{2}(x) \tan ^{2}(x)$ with respect to $x$ .

Find the volume of a solid with base under $y=e^{x}$ , $x=1$ , and square cross sections perpendicular to the $x$ -axis.

Calculate the area between the $x$ -axis and the curve $y=x^{3}-7 x^{2}+10 x$ .

Find the area between $f(x)=\sin x$ and $g(x)=-\sin x$ for $0 \leq x \leq \pi$ , volume when revolved about $y=3$ , and $k$ if volume is $8\pi$ .

Find $f_{xz}$ for $f(x, y, z)=e^{xyz}$ at the point (1, -1, -1). Choices: a. $e$ , b. $0^{x}$ , c. $-e$ , d. None.

Find limits for the function $f(x)=2 x e^{2 x}$ : a. $\lim _{x \rightarrow-\infty} f(x)=$ , b. $\lim _{x \rightarrow \infty} f(x)=$ .