Calculus

Problem 28001

Find the tangent line to $f(x)=x^{2}$ at $x=3$ . Choose the correct equation: a. $y=6 x-9$ , b. $y=3 x-6$ , c. $y=2 x-6$ , d. $y=2 x-3$ .

See Solution

Problem 28002

Find the integral of the function: $\int \frac{2 x}{(x-1)^{2}} d x$

See Solution

Problem 28003

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

See Solution

Problem 28004

Finde die Tangentengleichung an die Funktion $f(x) = x^3 - 3x$ durch den Wendepunkt.

See Solution

Problem 28005

A \$4,000 deposit earns 5.5% interest, compounded continuously, for 30 years. What is the final amount received?

See Solution

Problem 28006

Find the volume of revolution for the region between $x=y^{2}$ , $x=6-y$ , and the $x$ -axis using different methods.

See Solution

Problem 28007

Find the limit as $x$ approaches 0 for the expression $\frac{2 x}{\tan ^{2} \sqrt{x}}$ .

See Solution

Problem 28008

Find the area between the curve $y=x$ , the $x$ -axis, and the lines $x=0$ and $x=a$ where $a>0$ .

See Solution

Problem 28009

Find $\frac{d y}{d x}$ for the function $y = \sin(2x) + 2x$ .

See Solution

Problem 28010

Calculate the integral $\int_{0}^{2} x \, dx$ .

See Solution

Problem 28011

Find the general solution to the equation $\ddot{x}+\omega_{n}^{2} x=0$ .

See Solution

Problem 28012

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

See Solution

Problem 28013

Find the limit as $x$ approaches 0 for $\frac{2x}{\tan^{2}(\sqrt{x})}$ using L'Hôpital's rule.

See Solution

Problem 28014

Find turning points for these functions: a) $y=12 x-2 x^{2}$ (first derivative) b) $y=7-2 x^{4}$ (second derivative).

See Solution

Problem 28015

Calculate the integral $\int 2 x \cos \left(x^{2}-5\right)$ .

See Solution

Problem 28016

Differentiate $x \sin x + \cos x$ and evaluate $\int_{0}^{\frac{\pi}{2}} x \cos x \, dx$ . Also, find $y$ if $\frac{d y}{d x} = \frac{1}{\sqrt{2-x}}$ and $y=-2$ when $x=1$ .

See Solution

Problem 28017

Calculate the integrals: 1) $\int 7 \sin (x) dx$ , 2) $\int 9 \sin (3x) dx$ , 3) $\int 7 \cos (5x) dx$ .

See Solution

Problem 28018

Verify that $x_{p}=a \frac{\cos (\omega t)}{\omega_{n}^{2}-\omega^{2}}$ solves $\hat{x}+\omega_{1 t}^{2} x=a \cos (\omega t)$ and $y_{p}=b \frac{\sin (\omega t)}{\omega_{n}^{2}-\omega^{2}}$ solves $y+\omega_{n}^{2} y=b \sin (\omega t)$ for $\omega \neq \pm \omega_{n}$ .

See Solution

Problem 28019

Find the oblique asymptotes, domain, range, vertical asymptotes, behavior near them, and end behavior for $g(x) = \frac{x^{3} - 8}{x^{2} - 3x - 4}$ .

See Solution

Problem 28020

Evaluate the integral from 0 to 2: $\int_{0}^{2} x \, dx$ .

See Solution

Problem 28021

Calculate the integral: $\int(\cos x + \sin x) \, dx = \ldots$

See Solution

Problem 28022

Find $\frac{d y}{d x}$ for the equation $\sin 2 x + 2 x = 0$ .

See Solution

Problem 28023

Calculate the integral from 0 to 1 of $\frac{1}{1+x^{2}}$ with respect to $x$ .

See Solution

Problem 28024

Find the area between the curve $y=x$ , the $x$ -axis, and the lines $x=0$ and $x=a$ , where $a>0$ .

See Solution

Problem 28025

Calculate the integral of $\int \frac{e^{\tan^{-1} x}}{1+x^{2}} \, dx$ .

See Solution

Problem 28026

A particle's acceleration is $a=(16-6t) \mathrm{ms}^{-2}$ . Given $x=-5$ , $v=1$ at $t=1$ , find speed $U$ at $t=T$ .

See Solution

Problem 28027

Show that the function $f(x)=\sin \left(\frac{1}{x}\right)$ is bounded for $x \neq 0$ .

See Solution

Problem 28028

Show that the sequence $\frac{\sqrt{n}}{\sqrt{n}+2}$ is a Cauchy sequence using its definition.

See Solution

Problem 28029

Find the limit as $x$ approaches 3 for $f(g(x))$ where $f(x)=\frac{1}{2x-6}$ and $g(x)=\sqrt{x-2}$ .

See Solution

Problem 28030

Evaluate the integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin ^{2}(x) \, dx$ .

See Solution

Problem 28031

Calculate the value of the integral $\int_{0}^{\frac{\pi}{6}}(\sec x+\tan x)^{2} d x$ .

See Solution

Problem 28032

Calculate the integral $\int_{0}^{2} x \, dx$ . Choose: a) 1 b) 0 c) 2 d) None

See Solution

Problem 28033

Determine if the following statements are true or false and justify:
1. Is $\mathbb{R}$ closed?
2. Is $\{1,2\}$ not closed in $\mathbb{R}$ ?
3. Is $\bigcap_{n=1}^{\infty}\left[2,2+\frac{1}{n}\right]=\phi$ ?
4. Is $f(x)=\sin \left(\frac{1}{x}\right)$ bounded in $\mathbb{R}$ ?
5. If $f(x)$ is continuous at $x=c$ , is it differentiable at $x=c$ ?
6. Is $-\frac{d}{d x} f(x)$ at $x=c$ unique if it exists?

See Solution

Problem 28034

Berechnen Sie die Ableitung $f^{\prime}(x)$ der Funktion $f(x)=4 x^{2}-2$ an der Stelle $x=3$ .

See Solution

Problem 28035

Find relative extrema of $g(x) = x + 2 \sin x$ on $(0, 2\pi)$ using the second derivative test.

See Solution

Problem 28036

A helicopter drops a bag from $669 \mathrm{~m}$ at $100 \mathrm{~m/s}$ . What is the bag's impact velocity? Options: a. $153 \mathrm{~m/s}$ b. $122 \mathrm{~m/s}$ c. $199 \mathrm{~m/s}$ d. $229 \mathrm{~m/s}$ e. $76 \mathrm{~m/s}$

See Solution

Problem 28037

Bestimme $f^{\prime}(1)$ und $f^{\prime}(2)$ für $f(x)=3 x^{2}$ ; sowie $f^{\prime}(3)$ und $f^{\prime}(-2)$ für $f(x)=4 x^{2}$ .

See Solution

Problem 28038

Find the derivative of $3 \sin (5 x) + 7$ . What is $\frac{d}{d x}(3 \sin (5 x)+7)$ ?

See Solution

Problem 28039

Find the derivative of $y=(x^{3}-9)^{4}$ with respect to $x$ : $\frac{d y}{d x}$ .

See Solution

Problem 28040

Find the derivative of $\sqrt{a-2x}$ with respect to $x$ , where $a$ is a constant.

See Solution

Problem 28041

Find the sum of the series: $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots$ What does it converge to?

See Solution

Problem 28042

Find the derivative of $y=(x+1) \sqrt{2 x+5}$ with respect to $x$ : $\frac{d y}{d x}$ .

See Solution

Problem 28043

Find the derivative of $y=\frac{x}{x-2}$ , which is $\frac{d y}{d x}$ . Options: $-\frac{2}{(x-2)^{2}}$ , $-\frac{2}{x-2}$ , $\frac{2}{(x-2)^{2}}$ , $\frac{2}{x-2}$ .

See Solution

Problem 28044

Find the derivative of $(x-1)(x+3)$ . Choices: $2x-2$ , $-1$ , $1$ , $2x+2$ .

See Solution

Problem 28045

Find the derivative of $y=\frac{\sqrt{2 x-1}}{x-1}$ , i.e., compute $\frac{d y}{d x}$ .

See Solution

Problem 28046

Find the derivative $\frac{d y}{d x}$ for $y=\frac{x}{\sqrt{2 x+3}}$ .

See Solution

Problem 28047

Find $\frac{d s}{d t}$ for $s=4-4 e^{-t}-\frac{2}{5} t$ . Choose the correct option from the given choices.

See Solution

Problem 28048

Find the value of $m$ for which the line $x+y=m$ is normal to the curve $y^{2}=12 x$ .

See Solution

Problem 28049

Find the value of $\frac{d}{d x}\left(e^{x} \tan x\right)$ when $x=0$ . Options: -1, 0, 2, 1.

See Solution

Problem 28050

Find $a$ , $b$ , and $c$ in the derivative equation: $\frac{d(12 e^{3 x}+2 x^{2}-7)}{d x}=a e^{3 x}+b x+c$ .

See Solution

Problem 28051

Find the derivative of $e^{x} \sin x + e^{x} \cos x$ . What is it?

See Solution

Problem 28052

Find the derivative of $y=6 x^{2}+\cos (3 x)-x^{2} \sin (x)$ . What is $\frac{d y}{d x}=$ ?

See Solution

Problem 28053

Find $a$ in the equation: $\frac{d\left(e^{\cos (x)}\right)}{d x}=a e^{\cos (x)} \sin (x)$ .

See Solution

Problem 28054

Find the derivative of $15 x^{2}+\sec (2 x)-x^{4} \cos (2 x)$ and match it with the options provided.

See Solution

Problem 28055

Bestimme den Wert von $x$ , um die Querschnittsfläche einer U-förmigen Rinne aus einer 40 cm breiten Blechtafel zu maximieren.

See Solution

Problem 28056

Find the degree of the differential equation $\frac{d y}{d x}=\frac{3 x-7}{\frac{d y}{d x}}$ . Options: 1, 2, 3, Cannot be determined.

See Solution

Problem 28057

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{2 x} \sin x}{3 x}$ . What is the value? Options: $1/2$ , $1/5$ , $1/3$ , $1/4$ .

See Solution

Problem 28058

Solve the equation $(2+\sin x) \frac{d y}{d x}+(y+1) \cos x=0$ with $y(0)=1$ to find $y(\pi / 2)$ . Options: $\frac{-1}{3}$ , $\frac{1}{3}$ , $\frac{4}{3}$ , $\frac{-2}{3}$ .

See Solution

Problem 28059

Calculate the integral from 0 to 1 of $\frac{1}{1+x^{2}}$ dx. What is the value? Options: 1, 0, $\frac{\pi}{4}$ , $\frac{\pi}{2}$ .

See Solution

Problem 28060

Find the limit: $\lim _{x \rightarrow 1} \frac{\tan (x-1)}{x-1} = ?$ Options: O2, O1, O1, O0.

See Solution

Problem 28061

Find the order of the differential equation with the general solution $y=(c_{1}+c_{2}) \cos (x+c_{3})-c_{4} e^{x}+c_{5}$ .

See Solution

Problem 28062

Find $n$ in the equation: $\frac{d\left(e^{\sin (x)}\right)}{d x}=e^{\sin (x)} \cos ^{n}(x)$ .

See Solution

Problem 28063

Leite die Funktion $f(x)=\frac{1}{2}\left(e^{-x}+e^{x}\right)$ ab.

See Solution

Problem 28064

Find the limits of the piecewise function $f(x)=\left\{\begin{array}{ll}2 x-1 & x>1 \\ 3 x-1 & x \leq 1\end{array}\right.$ as $x$ approaches 1.

See Solution

Problem 28065

Find the derivative of $\sqrt{1+e^{x}}$ and select the correct answer from the options: 1) $\frac{e^{x}}{\sqrt{1+e^{x}}}$ 2) $\frac{e^{x}}{2 \sqrt{1+e^{x}}}$ 3) $\frac{2 e^{x}}{\sqrt{1+e^{x}}}$ 4) $\frac{1}{2 \sqrt{1+e^{x}}}$

See Solution

Problem 28066

Find $a$ and $b$ in $y' = e^x(a \cos(2x) + b \sin(2x))$ for $y = e^x(12 \cos(2x) - 9 \sin(2x))$ .

See Solution

Problem 28067

Find $a$ and $b$ in the derivative $y' = e^x(a \cos(2x) + b \sin(2x))$ for $y = e^x(3 \sin(2x) + 6 \cos(2x))$ .

See Solution

Problem 28068

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{3}-8}{x-2}=$ ? (Options: 1, 12, 20, 10)

See Solution

Problem 28069

Find the derivative of $y=e^{5 x^{4}}$ in the form $\frac{d y}{d x}=a x^{n} e^{b x^{4}}$ . Identify $a$ , $n$ , and $b$ .

See Solution

Problem 28070

Bestimme die Monotonieintervalle und die Extrempunkte der Funktionen: a) $f(x)=4 x^{2}+3$ , b) $f(x)=18 x^{5}$ , c) $f(x)=\frac{1}{3} x^{3}-4 x+2$ .

See Solution

Problem 28071

Berechne die Verkaufszahlen für $f(t)=4-\frac{400}{t}$ in den ersten 800 und 807 Tagen und bestimme $f^{\prime}(800)$ .

See Solution

Problem 28072

Berechne den Unterschied zwischen maximaler und minimaler Temperatur an einem Sommertag mit $T(t)=\frac{1}{1000}(-14 t^{3}+411 t^{2}-2218 t+19089)$ .

See Solution

Problem 28073

Find the derivative of $y=e^{\frac{1}{\sqrt{x}}}$ with respect to $x$ .

See Solution

Problem 28074

Find $\alpha$ in the equation $\frac{d\left(e^{x^{2}}-e^{2 x}\right)}{d x}=\alpha\left(x e^{x^{2}}-e^{2 x}\right)$ .

See Solution

Problem 28075

Calculate the integral: $\int \cos ^{2} x \, dx =$

See Solution

Problem 28076

Is $f(x)=x^{4}+8 x^{3}+19 x^{2}+12 x$ at $x=-2$ a max or min? Justify your answer.

See Solution

Problem 28077

Berechne für den Weg $s(t)=0,4 t^{2}$ nach 10 s: a) zurückgelegter Weg, b) Durchschnittsgeschwindigkeit, c) Geschwindigkeit.

See Solution

Problem 28078

$\sin x - x + \frac{1}{2} = 0$ denkleminin $[0, \pi]$ aralığındaki kökünü basit iterasyonla iki adımda bulun.

See Solution

Problem 28079

Find the limit as $x$ approaches $\frac{7}{6}$ from the left for $\frac{16x}{7-6x}$ .

See Solution

Problem 28080

Find the limit as $x$ approaches -1 for $f(x)$ and determine if $f(-1)$ is defined.

See Solution

Problem 28081

Find the tangent line equation for the curve $y=2 x^{2}-4 x+1$ at $x=3$ . Choose the correct option.

See Solution

Problem 28082

Berechne den Differenzenquotienten für die Funktionen $f$ in den angegebenen Intervallen $I$ .

See Solution

Problem 28083

Find the limit as $z$ approaches $\frac{7}{6}$ from the right for $\frac{16 z}{7-6 z}$ .

See Solution

Problem 28084

Identify which statements about limits are true:
1. $\lim _{x \rightarrow a^{+}} f(x)=-6$ implies $\lim _{x \rightarrow a} f(x)=-6$ .
2. $\lim _{x \rightarrow a} f(x)=-6$ and $\lim _{x \rightarrow a^{+}} f(x)=-6$ imply $\lim _{x \rightarrow a^{-}} f(x)=-6$ .
3. $\lim _{x \rightarrow a} f(x)=-6$ implies $\lim _{x \rightarrow a^{+}} f(x)=-6$ .
4. $\lim _{x \rightarrow a} f(x)=-6$ implies $\lim _{x \rightarrow a^{-}} f(x)=-6$ .
5. $\lim _{x \rightarrow a^{-}} f(x)=-6$ implies $\lim _{x \rightarrow a^{+}} f(x)=-6$ .
6. $\lim _{x \rightarrow a^{-}} f(x)=-6$ implies $\lim _{x \rightarrow a} f(x)=-6$ .
7. $\lim _{x \rightarrow a^{+}} f(x)=-6$ implies $\lim _{x \rightarrow a^{-}} f(x)=-6$ .
8. $\lim _{x \rightarrow a^{+}} f(x)=-6$ and $\lim _{x \rightarrow a^{-}} f(x)=-6$ imply $\lim _{x \rightarrow a} f(x)=-6$ .
9. $\lim _{x \rightarrow a} f(x)=-6$ and $\lim _{x \rightarrow a^{-}} f(x)=-6$ imply $\lim _{x \rightarrow a^{+}} f(x)=-6$ .

See Solution

Problem 28085

Finde die 1. und 2. Ableitung von $f(x) = 5e^{7x^2 + 4x}$ .

See Solution

Problem 28086

Evaluate the limit: $\lim _{x \rightarrow 3} \frac{3 x-1}{x-3}$ . Choices: a. does not exist b. 2 c. -2 d. 1

See Solution

Problem 28087

Find where the graph of $f(x)=x^{4}+2 x^{3}-100 x^{2}+30 x-4$ is concave up and down.

See Solution

Problem 28088

Given the function $h(t)=-4.9 t^{2}+78.4 t+11$ , find the velocity at $t=6$ , minimum height, and impact velocity.

See Solution

Problem 28089

Determine the convergence of the series $\sum_{n=1}^{\infty} e^{\frac{1}{n}}$ . Options: a) 1, b) $\frac{1}{e-1}$ , c) e, d) diverges.

See Solution

Problem 28090

Is the function $f(t)=\frac{2}{e^{t}-e^{-t}}$ continuous everywhere? True or False? Justify and find its interval(s) of continuity.

See Solution

Problem 28091

Find the limit: $\lim _{x \rightarrow 0} x^{8}$ . Show that for $\varepsilon>0$ , choose $\delta=\sqrt[8]{\varepsilon}$ .

See Solution

Problem 28092

Find the volume of the solid formed by revolving the triangle bounded by $3x + 4y = 12$ and the $x$ -axis around the $x$ -axis.

See Solution

Problem 28093

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

See Solution

Problem 28094

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

See Solution

Problem 28095

Calculate the surface area from revolving $y=\sqrt{x+1}$ , for $1 \leq x \leq 19$ , around the specified axis.

See Solution

Problem 28096

Find the arc length of the curve $y=3 x^{3/2}-1$ from $x=0$ to $x=1$ .

See Solution

Problem 28097

Evaluate the integral $\int_{-2}^{2}\left(x^{3} \cos \frac{x}{2}+1\right) \sqrt{4-x^{2}} d x$ .

See Solution

Problem 28098

Determine if $\lim _{x \rightarrow 5} f(x)$ exists for $f(x)=\left\{\begin{array}{cc}\frac{x^{2}-25}{x-5} & x \neq 5 \\ 0 & x=5\end{array}\right.$ . Explain your reasoning.

See Solution

Problem 28099

Calculate the work done by a force moving an object $12 \mathrm{~m}$ , given the force varies from $5$ N to $2$ N.

See Solution

Problem 28100

Find the volume of the solid formed by revolving the area between $y=x$ and $y=x^{2}$ around the $y$ -axis using the shell method.

See Solution

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Calculus

Problem 28001

Find the tangent line to f(x)=x2f(x)=x^{2}f(x)=x2 at x=3x=3x=3. Choose the correct equation: a. y=6x−9y=6 x-9y=6x−9, b. y=3x−6y=3 x-6y=3x−6, c. y=2x−6y=2 x-6y=2x−6, d. y=2x−3y=2 x-3y=2x−3.

Problem 28002

Find the integral of the function: ∫2x(x−1)2dx\int \frac{2 x}{(x-1)^{2}} d x∫(x−1)22x​dx

Problem 28003

Find the limit: lim⁡x→2x2−4x2−4x+4\lim _{x \rightarrow 2} \frac{x^{2}-4}{x^{2}-4 x+4}limx→2​x2−4x+4x2−4​

Problem 28004

Finde die Tangentengleichung an die Funktion f(x)=x3−3xf(x) = x^3 - 3xf(x)=x3−3x durch den Wendepunkt.

Problem 28005

A \$4,000 deposit earns 5.5% interest, compounded continuously, for 30 years. What is the final amount received?

Problem 28006

Find the volume of revolution for the region between x=y2x=y^{2}x=y2, x=6−yx=6-yx=6−y, and the xxx-axis using different methods.

Problem 28007

Find the limit as xxx approaches 0 for the expression 2xtan⁡2x\frac{2 x}{\tan ^{2} \sqrt{x}}tan2x​2x​.

Problem 28008

Find the area between the curve y=xy=xy=x, the xxx-axis, and the lines x=0x=0x=0 and x=ax=ax=a where a>0a>0a>0.

Problem 28009

Find dydx\frac{d y}{d x}dxdy​ for the function y=sin⁡(2x)+2xy = \sin(2x) + 2xy=sin(2x)+2x.

Problem 28010

Calculate the integral ∫02x dx\int_{0}^{2} x \, dx∫02​xdx.

Problem 28011

Find the general solution to the equation x¨+ωn2x=0\ddot{x}+\omega_{n}^{2} x=0x¨+ωn2​x=0.

Problem 28012

Calculate the integral ∫0111+x2dx\int_{0}^{1} \frac{1}{1+x^{2}} d x∫01​1+x21​dx.

Problem 28013

Find the limit as xxx approaches 0 for 2xtan⁡2(x)\frac{2x}{\tan^{2}(\sqrt{x})}tan2(x​)2x​ using L'Hôpital's rule.

Problem 28014

Find turning points for these functions: a) y=12x−2x2y=12 x-2 x^{2}y=12x−2x2 (first derivative) b) y=7−2x4y=7-2 x^{4}y=7−2x4 (second derivative).

Problem 28015

Calculate the integral ∫2xcos⁡(x2−5)\int 2 x \cos \left(x^{2}-5\right)∫2xcos(x2−5).

Problem 28016

Differentiate xsin⁡x+cos⁡xx \sin x + \cos xxsinx+cosx and evaluate ∫0π2xcos⁡x dx\int_{0}^{\frac{\pi}{2}} x \cos x \, dx∫02π​​xcosxdx. Also, find yyy if dydx=12−x\frac{d y}{d x} = \frac{1}{\sqrt{2-x}}dxdy​=2−x​1​ and y=−2y=-2y=−2 when x=1x=1x=1.

Problem 28017

Calculate the integrals: 1) ∫7sin⁡(x)dx\int 7 \sin (x) dx∫7sin(x)dx, 2) ∫9sin⁡(3x)dx\int 9 \sin (3x) dx∫9sin(3x)dx, 3) ∫7cos⁡(5x)dx\int 7 \cos (5x) dx∫7cos(5x)dx.

Problem 28018

Problem 28019

Find the oblique asymptotes, domain, range, vertical asymptotes, behavior near them, and end behavior for g(x)=x3−8x2−3x−4g(x) = \frac{x^{3} - 8}{x^{2} - 3x - 4}g(x)=x2−3x−4x3−8​.

Problem 28020

Evaluate the integral from 0 to 2: ∫02x dx\int_{0}^{2} x \, dx∫02​xdx.

Problem 28021

Calculate the integral: ∫(cos⁡x+sin⁡x) dx=…\int(\cos x + \sin x) \, dx = \ldots∫(cosx+sinx)dx=…

Problem 28022

Find dydx\frac{d y}{d x}dxdy​ for the equation sin⁡2x+2x=0\sin 2 x + 2 x = 0sin2x+2x=0.

Problem 28023

Calculate the integral from 0 to 1 of 11+x2\frac{1}{1+x^{2}}1+x21​ with respect to xxx.

Problem 28024

Find the area between the curve y=xy=xy=x, the xxx-axis, and the lines x=0x=0x=0 and x=ax=ax=a, where a>0a>0a>0.

Problem 28025

Calculate the integral of ∫etan⁡−1x1+x2 dx\int \frac{e^{\tan^{-1} x}}{1+x^{2}} \, dx∫1+x2etan−1x​dx.

Problem 28026

A particle's acceleration is a=(16−6t)ms−2a=(16-6t) \mathrm{ms}^{-2}a=(16−6t)ms−2. Given x=−5x=-5x=−5, v=1v=1v=1 at t=1t=1t=1, find speed UUU at t=Tt=Tt=T.

Problem 28027

Show that the function f(x)=sin⁡(1x)f(x)=\sin \left(\frac{1}{x}\right)f(x)=sin(x1​) is bounded for x≠0x \neq 0x=0.

Problem 28028

Show that the sequence nn+2\frac{\sqrt{n}}{\sqrt{n}+2}n​+2n​​ is a Cauchy sequence using its definition.

Problem 28029

Find the limit as xxx approaches 3 for f(g(x))f(g(x))f(g(x)) where f(x)=12x−6f(x)=\frac{1}{2x-6}f(x)=2x−61​ and g(x)=x−2g(x)=\sqrt{x-2}g(x)=x−2​.

Problem 28030

Evaluate the integral ∫−π4π4sin⁡2(x) dx\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin ^{2}(x) \, dx∫−4π​4π​​sin2(x)dx.

Problem 28031

Calculate the value of the integral ∫0π6(sec⁡x+tan⁡x)2dx\int_{0}^{\frac{\pi}{6}}(\sec x+\tan x)^{2} d x∫06π​​(secx+tanx)2dx.

Problem 28032

Calculate the integral ∫02x dx\int_{0}^{2} x \, dx∫02​xdx. Choose: a) 1 b) 0 c) 2 d) None

Problem 28033

Problem 28034

Berechnen Sie die Ableitung f′(x)f^{\prime}(x)f′(x) der Funktion f(x)=4x2−2f(x)=4 x^{2}-2f(x)=4x2−2 an der Stelle x=3x=3x=3.

Problem 28035

Find relative extrema of g(x)=x+2sin⁡xg(x) = x + 2 \sin xg(x)=x+2sinx on (0,2π)(0, 2\pi)(0,2π) using the second derivative test.

Problem 28036

Problem 28037

Bestimme f′(1)f^{\prime}(1)f′(1) und f′(2)f^{\prime}(2)f′(2) für f(x)=3x2f(x)=3 x^{2}f(x)=3x2; sowie f′(3)f^{\prime}(3)f′(3) und f′(−2)f^{\prime}(-2)f′(−2) für f(x)=4x2f(x)=4 x^{2}f(x)=4x2.

Problem 28038

Find the derivative of 3sin⁡(5x)+73 \sin (5 x) + 73sin(5x)+7. What is ddx(3sin⁡(5x)+7)\frac{d}{d x}(3 \sin (5 x)+7)dxd​(3sin(5x)+7)?

Problem 28039

Find the derivative of y=(x3−9)4y=(x^{3}-9)^{4}y=(x3−9)4 with respect to xxx: dydx\frac{d y}{d x}dxdy​.

Problem 28040

Find the derivative of a−2x\sqrt{a-2x}a−2x​ with respect to xxx, where aaa is a constant.

Problem 28041

Find the sum of the series: 1+12+14+18+…1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots1+21​+41​+81​+… What does it converge to?

Find the tangent line to $f(x)=x^{2}$ at $x=3$ . Choose the correct equation: a. $y=6 x-9$ , b. $y=3 x-6$ , c. $y=2 x-6$ , d. $y=2 x-3$ .

Find the integral of the function: $\int \frac{2 x}{(x-1)^{2}} d x$

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

Finde die Tangentengleichung an die Funktion $f(x) = x^3 - 3x$ durch den Wendepunkt.

Find the volume of revolution for the region between $x=y^{2}$ , $x=6-y$ , and the $x$ -axis using different methods.

Find the limit as $x$ approaches 0 for the expression $\frac{2 x}{\tan ^{2} \sqrt{x}}$ .

Find the area between the curve $y=x$ , the $x$ -axis, and the lines $x=0$ and $x=a$ where $a>0$ .

Find $\frac{d y}{d x}$ for the function $y = \sin(2x) + 2x$ .

Calculate the integral $\int_{0}^{2} x \, dx$ .

Find the general solution to the equation $\ddot{x}+\omega_{n}^{2} x=0$ .

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

Find the limit as $x$ approaches 0 for $\frac{2x}{\tan^{2}(\sqrt{x})}$ using L'Hôpital's rule.

Find turning points for these functions: a) $y=12 x-2 x^{2}$ (first derivative) b) $y=7-2 x^{4}$ (second derivative).

Calculate the integral $\int 2 x \cos \left(x^{2}-5\right)$ .

Differentiate $x \sin x + \cos x$ and evaluate $\int_{0}^{\frac{\pi}{2}} x \cos x \, dx$ . Also, find $y$ if $\frac{d y}{d x} = \frac{1}{\sqrt{2-x}}$ and $y=-2$ when $x=1$ .

Calculate the integrals: 1) $\int 7 \sin (x) dx$ , 2) $\int 9 \sin (3x) dx$ , 3) $\int 7 \cos (5x) dx$ .

Find the oblique asymptotes, domain, range, vertical asymptotes, behavior near them, and end behavior for $g(x) = \frac{x^{3} - 8}{x^{2} - 3x - 4}$ .

Evaluate the integral from 0 to 2: $\int_{0}^{2} x \, dx$ .

Calculate the integral: $\int(\cos x + \sin x) \, dx = \ldots$

Find $\frac{d y}{d x}$ for the equation $\sin 2 x + 2 x = 0$ .

Calculate the integral from 0 to 1 of $\frac{1}{1+x^{2}}$ with respect to $x$ .

Find the area between the curve $y=x$ , the $x$ -axis, and the lines $x=0$ and $x=a$ , where $a>0$ .

Calculate the integral of $\int \frac{e^{\tan^{-1} x}}{1+x^{2}} \, dx$ .

A particle's acceleration is $a=(16-6t) \mathrm{ms}^{-2}$ . Given $x=-5$ , $v=1$ at $t=1$ , find speed $U$ at $t=T$ .

Show that the function $f(x)=\sin \left(\frac{1}{x}\right)$ is bounded for $x \neq 0$ .

Show that the sequence $\frac{\sqrt{n}}{\sqrt{n}+2}$ is a Cauchy sequence using its definition.

Find the limit as $x$ approaches 3 for $f(g(x))$ where $f(x)=\frac{1}{2x-6}$ and $g(x)=\sqrt{x-2}$ .

Evaluate the integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin ^{2}(x) \, dx$ .

Calculate the value of the integral $\int_{0}^{\frac{\pi}{6}}(\sec x+\tan x)^{2} d x$ .

Calculate the integral $\int_{0}^{2} x \, dx$ . Choose: a) 1 b) 0 c) 2 d) None

Berechnen Sie die Ableitung $f^{\prime}(x)$ der Funktion $f(x)=4 x^{2}-2$ an der Stelle $x=3$ .

Find relative extrema of $g(x) = x + 2 \sin x$ on $(0, 2\pi)$ using the second derivative test.

Bestimme $f^{\prime}(1)$ und $f^{\prime}(2)$ für $f(x)=3 x^{2}$ ; sowie $f^{\prime}(3)$ und $f^{\prime}(-2)$ für $f(x)=4 x^{2}$ .

Find the derivative of $3 \sin (5 x) + 7$ . What is $\frac{d}{d x}(3 \sin (5 x)+7)$ ?

Find the derivative of $y=(x^{3}-9)^{4}$ with respect to $x$ : $\frac{d y}{d x}$ .

Find the derivative of $\sqrt{a-2x}$ with respect to $x$ , where $a$ is a constant.

Find the sum of the series: $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots$ What does it converge to?

Find the derivative of $y=(x+1) \sqrt{2 x+5}$ with respect to $x$ : $\frac{d y}{d x}$ .

Find the derivative of $y=\frac{x}{x-2}$ , which is $\frac{d y}{d x}$ . Options: $-\frac{2}{(x-2)^{2}}$ , $-\frac{2}{x-2}$ , $\frac{2}{(x-2)^{2}}$ , $\frac{2}{x-2}$ .

Find the derivative of $(x-1)(x+3)$ . Choices: $2x-2$ , $-1$ , $1$ , $2x+2$ .

Find the derivative of $y=\frac{\sqrt{2 x-1}}{x-1}$ , i.e., compute $\frac{d y}{d x}$ .

Find the derivative $\frac{d y}{d x}$ for $y=\frac{x}{\sqrt{2 x+3}}$ .

Find $\frac{d s}{d t}$ for $s=4-4 e^{-t}-\frac{2}{5} t$ . Choose the correct option from the given choices.

Find the value of $m$ for which the line $x+y=m$ is normal to the curve $y^{2}=12 x$ .

Find the value of $\frac{d}{d x}\left(e^{x} \tan x\right)$ when $x=0$ . Options: -1, 0, 2, 1.

Find $a$ , $b$ , and $c$ in the derivative equation: $\frac{d(12 e^{3 x}+2 x^{2}-7)}{d x}=a e^{3 x}+b x+c$ .

Find the derivative of $e^{x} \sin x + e^{x} \cos x$ . What is it?

Find the derivative of $y=6 x^{2}+\cos (3 x)-x^{2} \sin (x)$ . What is $\frac{d y}{d x}=$ ?

Find $a$ in the equation: $\frac{d\left(e^{\cos (x)}\right)}{d x}=a e^{\cos (x)} \sin (x)$ .

Find the derivative of $15 x^{2}+\sec (2 x)-x^{4} \cos (2 x)$ and match it with the options provided.

Bestimme den Wert von $x$ , um die Querschnittsfläche einer U-förmigen Rinne aus einer 40 cm breiten Blechtafel zu maximieren.

Find the degree of the differential equation $\frac{d y}{d x}=\frac{3 x-7}{\frac{d y}{d x}}$ . Options: 1, 2, 3, Cannot be determined.

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{2 x} \sin x}{3 x}$ . What is the value? Options: $1/2$ , $1/5$ , $1/3$ , $1/4$ .

Solve the equation $(2+\sin x) \frac{d y}{d x}+(y+1) \cos x=0$ with $y(0)=1$ to find $y(\pi / 2)$ . Options: $\frac{-1}{3}$ , $\frac{1}{3}$ , $\frac{4}{3}$ , $\frac{-2}{3}$ .

Calculate the integral from 0 to 1 of $\frac{1}{1+x^{2}}$ dx. What is the value? Options: 1, 0, $\frac{\pi}{4}$ , $\frac{\pi}{2}$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{\tan (x-1)}{x-1} = ?$ Options: O2, O1, O1, O0.

Find the order of the differential equation with the general solution $y=(c_{1}+c_{2}) \cos (x+c_{3})-c_{4} e^{x}+c_{5}$ .

Find $n$ in the equation: $\frac{d\left(e^{\sin (x)}\right)}{d x}=e^{\sin (x)} \cos ^{n}(x)$ .

Leite die Funktion $f(x)=\frac{1}{2}\left(e^{-x}+e^{x}\right)$ ab.

Find the limits of the piecewise function $f(x)=\left\{\begin{array}{ll}2 x-1 & x>1 \\ 3 x-1 & x \leq 1\end{array}\right.$ as $x$ approaches 1.

Find $a$ and $b$ in $y' = e^x(a \cos(2x) + b \sin(2x))$ for $y = e^x(12 \cos(2x) - 9 \sin(2x))$ .

Find $a$ and $b$ in the derivative $y' = e^x(a \cos(2x) + b \sin(2x))$ for $y = e^x(3 \sin(2x) + 6 \cos(2x))$ .

Find the limit: $\lim _{x \rightarrow 2} \frac{x^{3}-8}{x-2}=$ ? (Options: 1, 12, 20, 10)

Find the derivative of $y=e^{5 x^{4}}$ in the form $\frac{d y}{d x}=a x^{n} e^{b x^{4}}$ . Identify $a$ , $n$ , and $b$ .

Bestimme die Monotonieintervalle und die Extrempunkte der Funktionen: a) $f(x)=4 x^{2}+3$ , b) $f(x)=18 x^{5}$ , c) $f(x)=\frac{1}{3} x^{3}-4 x+2$ .

Berechne die Verkaufszahlen für $f(t)=4-\frac{400}{t}$ in den ersten 800 und 807 Tagen und bestimme $f^{\prime}(800)$ .

Berechne den Unterschied zwischen maximaler und minimaler Temperatur an einem Sommertag mit $T(t)=\frac{1}{1000}(-14 t^{3}+411 t^{2}-2218 t+19089)$ .

Find the derivative of $y=e^{\frac{1}{\sqrt{x}}}$ with respect to $x$ .

Find $\alpha$ in the equation $\frac{d\left(e^{x^{2}}-e^{2 x}\right)}{d x}=\alpha\left(x e^{x^{2}}-e^{2 x}\right)$ .