Calculus

Problem 28601

Find the velocity equation in $\mathrm{ms}^{-1}$ from the displacement model $x=A \cos (\omega t)+B \sin (\omega t)$ at time $t$ .

See Solution

Problem 28602

Determine the arc length function for $f(x)=\frac{x^{3}}{12}+\frac{1}{x}$ from point $A=(1,\frac{13}{12})$ .

See Solution

Problem 28603

Bestimme die Extrempunkte der Funktionen: a) $f(x)=2 x-e^{x}$ , b) $f(x)=x^{2} \cdot e^{2 x}$ .

See Solution

Problem 28604

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

See Solution

Problem 28605

Calculate the curve length of $y=(x / 2)^{2 / 3}$ from $x=0$ to $x=2$ . Options: a) $\frac{2}{27}(\sqrt{10}-1)$ b) $\frac{27}{2}(1+\sqrt{10})$ c) $\frac{27}{4}(10 \sqrt{10}-1)$ d) $\frac{2}{27}(10 \sqrt{10}-1)$

See Solution

Problem 28606

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

See Solution

Problem 28607

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

See Solution

Problem 28608

Sia $f$ derivabile in zero con $f(0)=0$ e $f'(0)=4$ . Qual è il limite $\ell = \lim_{x \to 0^{+}} \frac{f(x)}{-1+\sqrt{1+6x}}$ ? (a) C non esiste. (b) $\ell=0$ . (c) $\ell=4/3$ . (d) $\ell=4$ . (e) Nessuna delle precedenti.

See Solution

Problem 28609

Find a curve through (1,1) with length $L=\frac{1}{2} \int_{1}^{4} \sqrt{\frac{4x+1}{x}} dx$ . Options: a. $-\sqrt{x}$ , b. $\frac{1}{2x}$ , c. $-\frac{1}{2\sqrt{x}}$ , d. None.

See Solution

Problem 28610

Find the time $t$ when the particle at $x(t)=16t-3t^{3}$ is momentarily at rest. Options: 51.30 s, 55.30 s, 57.3 s.

See Solution

Problem 28611

Oblicz granicę ciągu: $\lim _{n \rightarrow \infty}\left(\frac{2 n^{2}+3}{n^{2}-6}\right)^{n}$

See Solution

Problem 28612

Use the Ratio or Root Test on the series $\sum_{n=1}^{\infty} \frac{n^{10}}{(-10)^{n+1}}$ .

See Solution

Problem 28613

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

See Solution

Problem 28614

Find the surface area from revolving $y=\frac{1}{\pi} \sin (3 x)$ , $0 \leq x \leq \frac{\pi}{6}$ , about the $x$ -axis.

See Solution

Problem 28615

Find where the function $y=\frac{1}{3}(x+1)^{3}-2$ is increasing, decreasing, or constant.

See Solution

Problem 28616

Evaluate the integral $I=\int_{0}^{1}(6x-5x^{2/3})dx$ . What is the value of $I$ ?

See Solution

Problem 28617

Find the limit: $\lim _{x \rightarrow+\infty} \frac{\sin \left(\frac{1}{x}\right)}{x}$ . Choose one: does not exist, $+\infty$ , 1, 0.

See Solution

Problem 28618

Find the surface area of the curve $y=\frac{1}{\pi} \sin (3 x)$ from $0$ to $\frac{\pi}{6}$ rotated about the $x$ -axis.

See Solution

Problem 28619

Oblicz granicę funkcji: $\lim _{x \rightarrow 0} \frac{\sin \left(3 x^{2}\right)}{x^{2}-\sin ^{3}(x)}$ przy użyciu reguły de hospitala.

See Solution

Problem 28620

Find the surface area of the curve $x=3 \sqrt{y}$ , for $\frac{1}{9} \leq y \leq \frac{2}{9}$ , revolved about the $y$ -axis.

See Solution

Problem 28621

Estimate the following using linear approximation: 8. $\ln (1.1)$ 9. $\sqrt{8.9}$ 10. $\sec (0.1)$

See Solution

Problem 28622

Find the length of the curve $y=\frac{2}{5}(x+3)^{5/2}$ for $-2 \leq x \leq -1$ . Choose the correct integral.

See Solution

Problem 28623

Find the length of the curve $y=\frac{2}{3}(x+1)^{3 / 2}$ for $4 \leq x \leq 5$ . Select the correct answer.

See Solution

Problem 28624

A Toyota Corolla's distance from a stop sign is $x(t)=1.50t^2-0.0500t^3$ . Find average velocity for: (a) $t=0$ to $t=2.00$ s, (b) $t=0$ to $t=4.00$ s, (c) $t=2.00$ s to $t=4.00$ s.

See Solution

Problem 28625

Is the function $f(x)=\left\{\begin{array}{c}\frac{x^{4}-625}{x^{2}-25} \text { if } x \neq 5 \\ 25 \text { if } x=5\end{array}\right.$ continuous at $x=5$ ? If not, why?

See Solution

Problem 28626

Find the surface area of the curve $y=\frac{1}{\pi} \sin (3 x)$ from $0$ to $\frac{\pi}{6}$ revolved around the $x$ -axis.

See Solution

Problem 28627

Find the linear approximation of $g(z)=\sqrt[4]{z}$ at $z=2$ and use it to estimate $\sqrt[4]{3}$ and $\sqrt[4]{10}$ .

See Solution

Problem 28628

Calculate the limit: $\lim _{x \rightarrow 1^{-}} \frac{\left|x^{2}-1\right|}{x-1}$ . Options: (1) -2, 0, 2, $+\infty$ .

See Solution

Problem 28629

Find the limit as $x$ approaches 2 from the left for $\frac{[x]}{x}$ . What is the value?

See Solution

Problem 28630

Find the limit as $x$ approaches 2 from the left of $\frac{[x]}{x}$ . What is the result? Options: 2, 1, $\frac{1}{2}$ .

See Solution

Problem 28631

Find the surface area of the curve $y=\frac{1}{\pi} \sin (3 x)$ from $0$ to $\frac{\pi}{6}$ revolved around the $x$ -axis.

See Solution

Problem 28632

Find the limit as $x$ approaches infinity for $\frac{3x^2 + x + 1}{2x^3 + x + 3}$ .

See Solution

Problem 28633

Find the limit as $x$ approaches 2 from the left of the expression $\frac{[x]}{x}$ . What is the value?

See Solution

Problem 28634

A car's distance from a light is $x(t)=b t^{2}-c^{3}$ with $b=2.40 \mathrm{~m/s}^{2}$ and $c=0.120 \mathrm{~m/s}^{3}$ . Find: (a) average velocity from $t=0$ to $t=10.0 \mathrm{~s}$ , (b) instantaneous velocity at $t=0, t=5.0 \mathrm{~s}$ , and $t=10.0 \mathrm{~s}$ , (c) when is the car at rest again?

See Solution

Problem 28635

Calculate the average velocity of a car with $x(t)=bt^2-ct^3$ for $b=2.40 \mathrm{~m/s}^2$ , $c=0.120 \mathrm{~m/s}^3$ over given intervals.

See Solution

Problem 28636

A $0.18 \mathrm{~kg}$ pine cone falls from $14 \mathrm{~m}$ . Find its speed without air resistance and work done by friction if it hits at $10.2 \mathrm{~m/s}$ .

See Solution

Problem 28637

A car's distance from a light is $x(t)=b t^{2}-c t^{3}$ with $b=2.40 \mathrm{~m/s}^{2}$ and $c=0.120 \mathrm{~m/s}^{3}$ . Find: (a) average velocity from $t=0$ to $t=10.0 \mathrm{~s}$ , (b) instantaneous velocity at $t=0, 5.0, 10.0 \mathrm{~s}$ , (c) time when car is again at rest.

See Solution

Problem 28638

Find the limit: $\lim _{x \rightarrow \infty} \frac{3 x^{2}+x+1}{2 x^{3}+x+3}$ . What is it?

See Solution

Problem 28639

Solve the equation $y^{\prime \prime}-4 y^{\prime}+4=0$ with conditions $y(0)=1$ and $y^{\prime}(0)=0$ .

See Solution

Problem 28640

Find the integral of the function $e^{\sqrt{x}}$ .

See Solution

Problem 28641

Bestimme die Ableitungsfunktion von $f(x)=\frac{1}{x}$ für $x \neq 0$ . Was ist $f'(x)$ ?

See Solution

Problem 28642

Find the surface area generated by revolving the curve $x=\sqrt{4y-y^{2}}$ from $y=1$ to $y=2$ around the y-axis.

See Solution

Problem 28643

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{4}$ .

See Solution

Problem 28644

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

See Solution

Problem 28645

Find the limit as $x$ approaches 2 from the left: $\lim _{x \rightarrow 2^{-}} \frac{[x]}{x}=$ . Choose from 2, 1, $\frac{1}{2}$ , $\frac{3}{2}$ .

See Solution

Problem 28646

Find $\frac{d f^{-1}}{d x}$ for $f(x)=\frac{1}{(1-x)^{2}}$ at $x=\frac{1}{4}$ , where $x>1$ . Choices: -4, 4, $\frac{1}{4}$ , none, $-\frac{1}{4}$ .

See Solution

Problem 28647

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{4}$ .

See Solution

Problem 28648

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

See Solution

Problem 28649

Find the surface area generated by revolving $x=\sqrt{y}$ , for $2 \leq y \leq 6$ , around the $y$ -axis.

See Solution

Problem 28650

Find $\left(f^{-1}\right)^{\prime}(4)$ given $f(2)=4, f(4)=5, f^{\prime}(2)=\frac{1}{3}, f^{\prime}(5)=\frac{1}{5}$ .

See Solution

Problem 28651

Find the surface area generated by revolving the curve $x=2 \sqrt{4-y}$ from $y=0$ to $y=\frac{15}{4}$ about the $y$ -axis. Choose the correct answer from the options provided.

See Solution

Problem 28652

Find the area of the surface formed by revolving the curve $x=2 \sqrt{4-y}$ , for $0 \leq y \leq \frac{15}{4}$ , around the y-axis.

See Solution

Problem 28653

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{3}$ . Choices: $\sqrt{2}$ , $\frac{\sqrt{3}}{2}$ , $\sqrt{\frac{3}{2}}$ , $\frac{1}{\sqrt{2}}$ , none.

See Solution

Problem 28654

Find the surface area generated by revolving $y=\frac{x^{3}}{9}$ , for $0 \leq x \leq 2$ , around the x-axis.

See Solution

Problem 28655

Find the curve with positive slope passing through (1,2) and length $L=\int_{1}^{4} \sqrt{1+4 x^{2}} d x$ .

See Solution

Problem 28656

Find $\left(f^{-1}\right)^{\prime}(4)$ given $f(2)=4$ , $f(4)=5$ , $f^{\prime}(2)=\frac{1}{3}$ , $f^{\prime}(5)=\frac{1}{5}$ .

See Solution

Problem 28657

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

See Solution

Problem 28658

Find the length of the curve $x = \int_{0}^{y} \sqrt{\sec^{4} t - 1} \, dt$ from $y = -\frac{\pi}{3}$ to $y = \frac{\pi}{6}$ .

See Solution

Problem 28659

Find the length of the curve $x = \int_{0}^{y} \sqrt{\sec^{4}t - 1} \, dt$ from $y = -\frac{\pi}{4}$ to $y = \frac{\pi}{4}$ .

See Solution

Problem 28660

Find the area of the surface formed by revolving $y=\sqrt{x+1}$ from $x=1$ to $x=5$ around the x-axis.

See Solution

Problem 28661

Find the surface area from revolving the curve $x=\frac{y^{3}}{3}$ , $0 \leq y \leq 1$ about the y-axis.

See Solution

Problem 28662

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{3}$ . Options: $\frac{1}{\sqrt{2}}$ , $\frac{\sqrt{3}}{2}$ , $\sqrt{\frac{3}{2}}$ , $\sqrt{2}$ .

See Solution

Problem 28663

Find the length of the curve defined by $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} d t$ from $y=-\frac{\pi}{6}$ to $y=\frac{\pi}{6}$ .

See Solution

Problem 28664

Find the surface area of the curve $x=\sqrt{y}$ from $y=2$ to $y=6$ when revolved around the $y$ -axis.

See Solution

Problem 28665

Find the surface area generated by revolving $y=\sqrt{2x+1}$ from $0 \leq x \leq 3$ about the x-axis.

See Solution

Problem 28666

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{2}$ .

See Solution

Problem 28667

Find the surface area generated by revolving $x=\sqrt{4y-y^{2}}$ from $y=1$ to $y=2$ around the y-axis.

See Solution

Problem 28668

Find the surface area from revolving $x=\sqrt{2y-1}$ , $\frac{5}{8} \leq y \leq 1$ around the y-axis.

See Solution

Problem 28669

Find the length of the curve $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} \, dt$ from $y=-\frac{\pi}{6}$ to $y=\frac{\pi}{3}$ . Choices: 0, 10, $\frac{4}{\sqrt{3}}$ , $2\sqrt{3}$ , none.

See Solution

Problem 28670

Find the curve with positive derivative and length $L=\int_{1}^{4} \sqrt{1+4 x^{2}} d x$ passing through $(1,0)$ .

See Solution

Problem 28671

Find the length of the curve $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} dt$ from $y=-\frac{\pi}{6}$ to $y=\frac{\pi}{3}$ .

See Solution

Problem 28672

Find $\left(f^{-1}\right)^{\prime}(4)$ given $f(2)=4$ , $f(4)=5$ , $f^{\prime}(2)=\frac{1}{3}$ , $f^{\prime}(5)=\frac{1}{5}$ .

See Solution

Problem 28673

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{4}$ .

See Solution

Problem 28674

Find $\left(f^{-1}\right)^{\prime}(4)$ given $f(2)=4$ , $f(4)=5$ , $f^{\prime}(2)=\frac{1}{3}$ , $f^{\prime}(5)=\frac{1}{5}$ .

See Solution

Problem 28675

Find the length of the curve defined by $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{6}$ .

See Solution

Problem 28676

Find the surface area from revolving $x=\frac{y^{3}}{3}, 0 \leq y \leq 1$ around the y-axis.

See Solution

Problem 28677

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{6}$ .

See Solution

Problem 28678

Find $\left(f^{-1}\right)^{\prime}(4)$ given $f(2)=4$ , $f(4)=5$ , $f^{\prime}(2)=\frac{1}{3}$ , $f^{\prime}(5)=\frac{1}{5}$ .

See Solution

Problem 28679

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{6}$ .

See Solution

Problem 28680

Find the area of the surface generated by revolving $y=\sqrt{x+1}$ , for $1 \leq x \leq 5$ , around the x-axis.

See Solution

Problem 28681

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{3}$ .

See Solution

Problem 28682

Find $\left(f^{-1}\right)^{\prime}(4)$ given $f(2)=4$ , $f(4)=5$ , $f^{\prime}(2)=\frac{1}{3}$ , $f^{\prime}(5)=\frac{1}{5}$ .

See Solution

Problem 28683

Find the surface area generated by revolving $x=\sqrt{y}$ from $y=2$ to $y=6$ around the $y$ -axis.

See Solution

Problem 28684

Find the surface area of revolution for $y=\sqrt{2x+1}$ from $0$ to $3$ about the $x$ -axis. Options: $\frac{14 \pi \sqrt{2}}{3}$ , $\frac{28 \pi \sqrt{2}}{3}$ , none, $\frac{28 \pi}{3}$ , $28 \pi \sqrt{2}$ .

See Solution

Problem 28685

Find the surface area generated by revolving the curve $x=\sqrt{y}$ from $y=2$ to $y=6$ about the y-axis.

See Solution

Problem 28686

Find the surface area of the curve $x=\sqrt{y}$ , for $2 \leq y \leq 6$ , revolved around the y-axis.

See Solution

Problem 28687

Find the surface area from revolving $x=\sqrt{y}, 2 \leq y \leq 6$ around the $y$ -axis. Choices include $\frac{49 \pi}{6}$ , $\frac{98 \pi}{18}$ , $\frac{98 \pi}{6}$ , $\frac{49 \pi}{81}$ , none.

See Solution

Problem 28688

Find the length of the curve defined by $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{6}$ .

See Solution

Problem 28689

Find the surface area from revolving $x=\frac{y^{3}}{3}, 0 \leq y \leq 1$ around the y-axis. Options include: $\frac{49 \pi}{81}(\sqrt{8}-1)$ , $\frac{98 \pi}{18}(2 \sqrt{2}-1)$ , none, $\frac{\pi}{9}(2 \sqrt{2}-1)$ , $\frac{\pi}{9}(\sqrt{2}-1)$ .

See Solution

Problem 28690

Find the curve with positive derivative and length $L=\int_{1}^{4} \sqrt{1+4 x^{2}} d x$ , passing through $(1,0)$ .

See Solution

Problem 28691

Find the surface area from revolving $y=\sqrt{2x+1}$ , $0 \leq x \leq 3$ , about the $x$ -axis. Choose the correct answer:
1. $\frac{14 \pi \sqrt{2}}{3}$
2. $\frac{28 \pi \sqrt{2}}{3}$
3. None
4. $\frac{28 \pi}{3}$
5. $28 \pi \sqrt{2}$

See Solution

Problem 28692

Find the length of the curve defined by $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} d t$ from $y=-\frac{\pi}{4}$ to $y=\frac{\pi}{4}$ .

See Solution

Problem 28693

Find the curve with positive derivative and length $L = \int_{1}^{4} \sqrt{1 + 4x^2} \, dx$ passing through $(0,1)$ . Choose one:
1. $y = x^2 + 1$
2. $y = x^2 - 1$
3. $y = x^2$
4. $y = 2x$
5. none

See Solution

Problem 28694

Find the surface area of the curve $x=\sqrt{2y-1}$ , for $\frac{5}{8} \leq y \leq 1$ , revolved around the y-axis.

See Solution

Problem 28695

Find the surface area of the curve $y=\sqrt{2x+1}$ , $0 \leq x \leq 3$ , rotated around the x-axis. Options: $\frac{14 \pi \sqrt{2}}{3}, \frac{28 \pi \sqrt{2}}{3}, \text{none}, \frac{28 \pi}{3}, 28 \pi \sqrt{2}$

See Solution

Problem 28696

Bestimmen Sie die Ableitungsfunktion $f^{\prime}$ und berechnen Sie $f^{\prime}(2)$ für: a) $f(x)=2 x^{2}$ , b) $f(x)=4 x^{2}$ , c) $f(x)=-3 x^{2}$ , d) $f(x)=\frac{1}{2} x^{2}$ .

See Solution

Problem 28697

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{3}$ .

See Solution

Problem 28698

Find the position $s(t)$ , displacement, and distance for the following velocity functions over $[0,1]$ :
1. $v(t)=4 t^{2}-5 t$ , $s(0)=5$
2. $v(t)=\sin(t)$ , $s(\pi)=-2$

See Solution

Problem 28699

Find the surface area of the curve $x=\sqrt{y}$ from $y=2$ to $y=6$ revolved around the y-axis.

See Solution

Problem 28700

Find the surface area generated by revolving $x=\sqrt{y}$ from $y=2$ to $y=6$ around the $y$ -axis.

See Solution

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Calculus

Problem 28601

Find the velocity equation in ms−1\mathrm{ms}^{-1}ms−1 from the displacement model x=Acos⁡(ωt)+Bsin⁡(ωt)x=A \cos (\omega t)+B \sin (\omega t)x=Acos(ωt)+Bsin(ωt) at time ttt.

Problem 28602

Determine the arc length function for f(x)=x312+1xf(x)=\frac{x^{3}}{12}+\frac{1}{x}f(x)=12x3​+x1​ from point A=(1,1312)A=(1,\frac{13}{12})A=(1,1213​).

Problem 28603

Bestimme die Extrempunkte der Funktionen: a) f(x)=2x−exf(x)=2 x-e^{x}f(x)=2x−ex, b) f(x)=x2⋅e2xf(x)=x^{2} \cdot e^{2 x}f(x)=x2⋅e2x.

Problem 28604

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

Problem 28605

Problem 28606

Find the curve length for x=∫0ysec⁡4t−1dtx=\int_{0}^{y} \sqrt{\sec ^{4} t-1} dtx=∫0y​sec4t−1​dt, with −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 28607

Find the slope of the inverse function g−1g^{-1}g−1 at the origin if 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 28608

Problem 28609

Find a curve through (1,1) with length L=12∫144x+1xdxL=\frac{1}{2} \int_{1}^{4} \sqrt{\frac{4x+1}{x}} dxL=21​∫14​x4x+1​​dx. Options: a. −x-\sqrt{x}−x​, b. 12x\frac{1}{2x}2x1​, c. −12x-\frac{1}{2\sqrt{x}}−2x​1​, d. None.

Problem 28610

Find the time ttt when the particle at x(t)=16t−3t3x(t)=16t-3t^{3}x(t)=16t−3t3 is momentarily at rest. Options: 51.30 s, 55.30 s, 57.3 s.

Problem 28611

Oblicz granicę ciągu: lim⁡n→∞(2n2+3n2−6)n\lim _{n \rightarrow \infty}\left(\frac{2 n^{2}+3}{n^{2}-6}\right)^{n}n→∞lim​(n2−62n2+3​)n

Problem 28612

Use the Ratio or Root Test on the series ∑n=1∞n10(−10)n+1\sum_{n=1}^{\infty} \frac{n^{10}}{(-10)^{n+1}}∑n=1∞​(−10)n+1n10​.

Problem 28613

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

Problem 28614

Find the surface area from revolving y=1πsin⁡(3x)y=\frac{1}{\pi} \sin (3 x)y=π1​sin(3x), 0≤x≤π60 \leq x \leq \frac{\pi}{6}0≤x≤6π​, about the xxx-axis.

Problem 28615

Find where the function y=13(x+1)3−2y=\frac{1}{3}(x+1)^{3}-2y=31​(x+1)3−2 is increasing, decreasing, or constant.

Problem 28616

Evaluate the integral I=∫01(6x−5x2/3)dxI=\int_{0}^{1}(6x-5x^{2/3})dxI=∫01​(6x−5x2/3)dx. What is the value of III?

Problem 28617

Find the limit: lim⁡x→+∞sin⁡(1x)x\lim _{x \rightarrow+\infty} \frac{\sin \left(\frac{1}{x}\right)}{x}limx→+∞​xsin(x1​)​. Choose one: does not exist, +∞+\infty+∞, 1, 0.

Problem 28618

Find the surface area of the curve y=1πsin⁡(3x)y=\frac{1}{\pi} \sin (3 x)y=π1​sin(3x) from 000 to π6\frac{\pi}{6}6π​ rotated about the xxx-axis.

Problem 28619

Oblicz granicę funkcji: lim⁡x→0sin⁡(3x2)x2−sin⁡3(x)\lim _{x \rightarrow 0} \frac{\sin \left(3 x^{2}\right)}{x^{2}-\sin ^{3}(x)}x→0lim​x2−sin3(x)sin(3x2)​ przy użyciu reguły de hospitala.

Problem 28620

Find the surface area of the curve x=3yx=3 \sqrt{y}x=3y​, for 19≤y≤29\frac{1}{9} \leq y \leq \frac{2}{9}91​≤y≤92​, revolved about the yyy-axis.

Problem 28621

Estimate the following using linear approximation: 8. ln⁡(1.1)\ln (1.1)ln(1.1) 9. 8.9\sqrt{8.9}8.9​ 10. sec⁡(0.1)\sec (0.1)sec(0.1)

Problem 28622

Find the length of the curve y=25(x+3)5/2y=\frac{2}{5}(x+3)^{5/2}y=52​(x+3)5/2 for −2≤x≤−1-2 \leq x \leq -1−2≤x≤−1. Choose the correct integral.

Problem 28623

Find the length of the curve y=23(x+1)3/2y=\frac{2}{3}(x+1)^{3 / 2}y=32​(x+1)3/2 for 4≤x≤54 \leq x \leq 54≤x≤5. Select the correct answer.

Problem 28624

A Toyota Corolla's distance from a stop sign is x(t)=1.50t2−0.0500t3x(t)=1.50t^2-0.0500t^3x(t)=1.50t2−0.0500t3. Find average velocity for: (a) t=0t=0t=0 to t=2.00t=2.00t=2.00 s, (b) t=0t=0t=0 to t=4.00t=4.00t=4.00 s, (c) t=2.00t=2.00t=2.00 s to t=4.00t=4.00t=4.00 s.

Problem 28625

Is the function f(x)={x4−625x2−25 if x≠525 if x=5f(x)=\left\{\begin{array}{c}\frac{x^{4}-625}{x^{2}-25} \text { if } x \neq 5 \\ 25 \text { if } x=5\end{array}\right.f(x)={x2−25x4−625​ if x=525 if x=5​ continuous at x=5x=5x=5? If not, why?

Problem 28626

Find the surface area of the curve y=1πsin⁡(3x)y=\frac{1}{\pi} \sin (3 x)y=π1​sin(3x) from 000 to π6\frac{\pi}{6}6π​ revolved around the xxx-axis.

Problem 28627

Find the linear approximation of g(z)=z4g(z)=\sqrt[4]{z}g(z)=4z​ at z=2z=2z=2 and use it to estimate 34\sqrt[4]{3}43​ and 104\sqrt[4]{10}410​.

Problem 28628

Calculate the limit: lim⁡x→1−∣x2−1∣x−1\lim _{x \rightarrow 1^{-}} \frac{\left|x^{2}-1\right|}{x-1}x→1−lim​x−1∣∣​x2−1∣∣​​. Options: (1) -2, 0, 2, +∞+\infty+∞.

Problem 28629

Find the limit as xxx approaches 2 from the left for [x]x\frac{[x]}{x}x[x]​. What is the value?

Problem 28630

Find the limit as xxx approaches 2 from the left of [x]x\frac{[x]}{x}x[x]​. What is the result? Options: 2, 1, 12\frac{1}{2}21​.

Problem 28631

Find the surface area of the curve y=1πsin⁡(3x)y=\frac{1}{\pi} \sin (3 x)y=π1​sin(3x) from 000 to π6\frac{\pi}{6}6π​ revolved around the xxx-axis.

Problem 28632

Find the limit as xxx approaches infinity for 3x2+x+12x3+x+3\frac{3x^2 + x + 1}{2x^3 + x + 3}2x3+x+33x2+x+1​.

Problem 28633

Find the limit as xxx approaches 2 from the left of the expression [x]x\frac{[x]}{x}x[x]​. What is the value?

Problem 28634

Problem 28635

Calculate the average velocity of a car with x(t)=bt2−ct3x(t)=bt^2-ct^3x(t)=bt2−ct3 for b=2.40 m/s2b=2.40 \mathrm{~m/s}^2b=2.40 m/s2, c=0.120 m/s3c=0.120 \mathrm{~m/s}^3c=0.120 m/s3 over given intervals.

Problem 28636

A 0.18 kg0.18 \mathrm{~kg}0.18 kg pine cone falls from 14 m14 \mathrm{~m}14 m. Find its speed without air resistance and work done by friction if it hits at 10.2 m/s10.2 \mathrm{~m/s}10.2 m/s.

Problem 28637

Problem 28638

Find the limit: lim⁡x→∞3x2+x+12x3+x+3\lim _{x \rightarrow \infty} \frac{3 x^{2}+x+1}{2 x^{3}+x+3}limx→∞​2x3+x+33x2+x+1​. What is it?

Problem 28639

Solve the equation y′′−4y′+4=0y^{\prime \prime}-4 y^{\prime}+4=0y′′−4y′+4=0 with conditions y(0)=1y(0)=1y(0)=1 and y′(0)=0y^{\prime}(0)=0y′(0)=0.

Problem 28640

Find the integral of the function exe^{\sqrt{x}}ex​.

Problem 28641

Bestimme die Ableitungsfunktion von f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ für x≠0x \neq 0x=0. Was ist f′(x)f'(x)f′(x)?

Problem 28642

Find the velocity equation in $\mathrm{ms}^{-1}$ from the displacement model $x=A \cos (\omega t)+B \sin (\omega t)$ at time $t$ .

Determine the arc length function for $f(x)=\frac{x^{3}}{12}+\frac{1}{x}$ from point $A=(1,\frac{13}{12})$ .

Bestimme die Extrempunkte der Funktionen: a) $f(x)=2 x-e^{x}$ , b) $f(x)=x^{2} \cdot e^{2 x}$ .

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

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

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

Find a curve through (1,1) with length $L=\frac{1}{2} \int_{1}^{4} \sqrt{\frac{4x+1}{x}} dx$ . Options: a. $-\sqrt{x}$ , b. $\frac{1}{2x}$ , c. $-\frac{1}{2\sqrt{x}}$ , d. None.

Find the time $t$ when the particle at $x(t)=16t-3t^{3}$ is momentarily at rest. Options: 51.30 s, 55.30 s, 57.3 s.

Oblicz granicę ciągu: $\lim _{n \rightarrow \infty}\left(\frac{2 n^{2}+3}{n^{2}-6}\right)^{n}$

Use the Ratio or Root Test on the series $\sum_{n=1}^{\infty} \frac{n^{10}}{(-10)^{n+1}}$ .

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

Find the surface area from revolving $y=\frac{1}{\pi} \sin (3 x)$ , $0 \leq x \leq \frac{\pi}{6}$ , about the $x$ -axis.

Find where the function $y=\frac{1}{3}(x+1)^{3}-2$ is increasing, decreasing, or constant.

Evaluate the integral $I=\int_{0}^{1}(6x-5x^{2/3})dx$ . What is the value of $I$ ?

Find the limit: $\lim _{x \rightarrow+\infty} \frac{\sin \left(\frac{1}{x}\right)}{x}$ . Choose one: does not exist, $+\infty$ , 1, 0.

Find the surface area of the curve $y=\frac{1}{\pi} \sin (3 x)$ from $0$ to $\frac{\pi}{6}$ rotated about the $x$ -axis.

Oblicz granicę funkcji: $\lim _{x \rightarrow 0} \frac{\sin \left(3 x^{2}\right)}{x^{2}-\sin ^{3}(x)}$ przy użyciu reguły de hospitala.

Find the surface area of the curve $x=3 \sqrt{y}$ , for $\frac{1}{9} \leq y \leq \frac{2}{9}$ , revolved about the $y$ -axis.

Estimate the following using linear approximation: 8. $\ln (1.1)$ 9. $\sqrt{8.9}$ 10. $\sec (0.1)$

Find the length of the curve $y=\frac{2}{5}(x+3)^{5/2}$ for $-2 \leq x \leq -1$ . Choose the correct integral.

Find the length of the curve $y=\frac{2}{3}(x+1)^{3 / 2}$ for $4 \leq x \leq 5$ . Select the correct answer.

A Toyota Corolla's distance from a stop sign is $x(t)=1.50t^2-0.0500t^3$ . Find average velocity for: (a) $t=0$ to $t=2.00$ s, (b) $t=0$ to $t=4.00$ s, (c) $t=2.00$ s to $t=4.00$ s.

Is the function $f(x)=\left\{\begin{array}{c}\frac{x^{4}-625}{x^{2}-25} \text { if } x \neq 5 \\ 25 \text { if } x=5\end{array}\right.$ continuous at $x=5$ ? If not, why?

Find the surface area of the curve $y=\frac{1}{\pi} \sin (3 x)$ from $0$ to $\frac{\pi}{6}$ revolved around the $x$ -axis.

Find the linear approximation of $g(z)=\sqrt[4]{z}$ at $z=2$ and use it to estimate $\sqrt[4]{3}$ and $\sqrt[4]{10}$ .

Calculate the limit: $\lim _{x \rightarrow 1^{-}} \frac{\left|x^{2}-1\right|}{x-1}$ . Options: (1) -2, 0, 2, $+\infty$ .

Find the limit as $x$ approaches 2 from the left for $\frac{[x]}{x}$ . What is the value?

Find the limit as $x$ approaches 2 from the left of $\frac{[x]}{x}$ . What is the result? Options: 2, 1, $\frac{1}{2}$ .

Find the surface area of the curve $y=\frac{1}{\pi} \sin (3 x)$ from $0$ to $\frac{\pi}{6}$ revolved around the $x$ -axis.

Find the limit as $x$ approaches infinity for $\frac{3x^2 + x + 1}{2x^3 + x + 3}$ .

Find the limit as $x$ approaches 2 from the left of the expression $\frac{[x]}{x}$ . What is the value?

Calculate the average velocity of a car with $x(t)=bt^2-ct^3$ for $b=2.40 \mathrm{~m/s}^2$ , $c=0.120 \mathrm{~m/s}^3$ over given intervals.

A $0.18 \mathrm{~kg}$ pine cone falls from $14 \mathrm{~m}$ . Find its speed without air resistance and work done by friction if it hits at $10.2 \mathrm{~m/s}$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{3 x^{2}+x+1}{2 x^{3}+x+3}$ . What is it?

Solve the equation $y^{\prime \prime}-4 y^{\prime}+4=0$ with conditions $y(0)=1$ and $y^{\prime}(0)=0$ .

Find the integral of the function $e^{\sqrt{x}}$ .

Bestimme die Ableitungsfunktion von $f(x)=\frac{1}{x}$ für $x \neq 0$ . Was ist $f'(x)$ ?

Find the surface area generated by revolving the curve $x=\sqrt{4y-y^{2}}$ from $y=1$ to $y=2$ around the y-axis.

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{4}$ .

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

Find the limit as $x$ approaches 2 from the left: $\lim _{x \rightarrow 2^{-}} \frac{[x]}{x}=$ . Choose from 2, 1, $\frac{1}{2}$ , $\frac{3}{2}$ .

Find $\frac{d f^{-1}}{d x}$ for $f(x)=\frac{1}{(1-x)^{2}}$ at $x=\frac{1}{4}$ , where $x>1$ . Choices: -4, 4, $\frac{1}{4}$ , none, $-\frac{1}{4}$ .

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{4}$ .

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

Find the surface area generated by revolving $x=\sqrt{y}$ , for $2 \leq y \leq 6$ , around the $y$ -axis.

Find $\left(f^{-1}\right)^{\prime}(4)$ given $f(2)=4, f(4)=5, f^{\prime}(2)=\frac{1}{3}, f^{\prime}(5)=\frac{1}{5}$ .

Find the surface area generated by revolving the curve $x=2 \sqrt{4-y}$ from $y=0$ to $y=\frac{15}{4}$ about the $y$ -axis. Choose the correct answer from the options provided.

Find the area of the surface formed by revolving the curve $x=2 \sqrt{4-y}$ , for $0 \leq y \leq \frac{15}{4}$ , around the y-axis.

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{3}$ . Choices: $\sqrt{2}$ , $\frac{\sqrt{3}}{2}$ , $\sqrt{\frac{3}{2}}$ , $\frac{1}{\sqrt{2}}$ , none.

Find the surface area generated by revolving $y=\frac{x^{3}}{9}$ , for $0 \leq x \leq 2$ , around the x-axis.

Find the curve with positive slope passing through (1,2) and length $L=\int_{1}^{4} \sqrt{1+4 x^{2}} d x$ .

Find $\left(f^{-1}\right)^{\prime}(4)$ given $f(2)=4$ , $f(4)=5$ , $f^{\prime}(2)=\frac{1}{3}$ , $f^{\prime}(5)=\frac{1}{5}$ .

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

Find the length of the curve $x = \int_{0}^{y} \sqrt{\sec^{4} t - 1} \, dt$ from $y = -\frac{\pi}{3}$ to $y = \frac{\pi}{6}$ .

Find the length of the curve $x = \int_{0}^{y} \sqrt{\sec^{4}t - 1} \, dt$ from $y = -\frac{\pi}{4}$ to $y = \frac{\pi}{4}$ .

Find the area of the surface formed by revolving $y=\sqrt{x+1}$ from $x=1$ to $x=5$ around the x-axis.

Find the surface area from revolving the curve $x=\frac{y^{3}}{3}$ , $0 \leq y \leq 1$ about the y-axis.

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{3}$ . Options: $\frac{1}{\sqrt{2}}$ , $\frac{\sqrt{3}}{2}$ , $\sqrt{\frac{3}{2}}$ , $\sqrt{2}$ .

Find the length of the curve defined by $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} d t$ from $y=-\frac{\pi}{6}$ to $y=\frac{\pi}{6}$ .

Find the surface area of the curve $x=\sqrt{y}$ from $y=2$ to $y=6$ when revolved around the $y$ -axis.

Find the surface area generated by revolving $y=\sqrt{2x+1}$ from $0 \leq x \leq 3$ about the x-axis.

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{2}$ .

Find the surface area generated by revolving $x=\sqrt{4y-y^{2}}$ from $y=1$ to $y=2$ around the y-axis.

Find the surface area from revolving $x=\sqrt{2y-1}$ , $\frac{5}{8} \leq y \leq 1$ around the y-axis.

Find the length of the curve $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} \, dt$ from $y=-\frac{\pi}{6}$ to $y=\frac{\pi}{3}$ . Choices: 0, 10, $\frac{4}{\sqrt{3}}$ , $2\sqrt{3}$ , none.

Find the curve with positive derivative and length $L=\int_{1}^{4} \sqrt{1+4 x^{2}} d x$ passing through $(1,0)$ .

Find the length of the curve $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} dt$ from $y=-\frac{\pi}{6}$ to $y=\frac{\pi}{3}$ .

Find $\left(f^{-1}\right)^{\prime}(4)$ given $f(2)=4$ , $f(4)=5$ , $f^{\prime}(2)=\frac{1}{3}$ , $f^{\prime}(5)=\frac{1}{5}$ .

Find the length of the curve $y=\int_{0}^{x} \sqrt{\cos 2 t} d t$ from $x=0$ to $x=\frac{\pi}{4}$ .

Find $\left(f^{-1}\right)^{\prime}(4)$ given $f(2)=4$ , $f(4)=5$ , $f^{\prime}(2)=\frac{1}{3}$ , $f^{\prime}(5)=\frac{1}{5}$ .