Calculus

Problem 32201

Differentiate the function $f(x) = \ln x^{2}$ .

See Solution

Problem 32202

Find the second derivative $F''(2)$ , where $F(x)=\int_{1}^{x} f(t) dt$ and $f(t)=\int_{1}^{t^{3}} \frac{\sqrt{6+u^{6}}}{u} du$ .

See Solution

Problem 32203

Does the series $\sum_{n=1}^{\infty}\left(\frac{1}{n^{2}}+1\right)^{2}$ converge or diverge? a. converge b. diverge

See Solution

Problem 32204

Find $\delta > 0$ so that $0 < |x - 5| < \delta \Rightarrow |2x - 4 - 6| < 0.2$ . Use the graph.

See Solution

Problem 32205

Determine the convergence of $\sum_{n=1}^{\infty} \frac{4}{3^{n}+\sqrt{n}}$ using comparison tests.

See Solution

Problem 32206

Determine if the series $\sum_{n=1}^{\infty} \frac{\cos ^{2}(n)}{n^{3 / 2}}$ converges or diverges using comparison tests.

See Solution

Problem 32207

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}$ using comparison tests.

See Solution

Problem 32208

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}$ using comparison tests.

See Solution

Problem 32209

Differentiate the function $f(x) = \log_{2}(3x)$ .

See Solution

Problem 32210

Calculate the sum: $\sum_{n=1}^{\infty} \frac{1+(-2)^{n}}{3^{n}}$ .

See Solution

Problem 32211

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{4 n^{3}+n^{2}+1}{n^{4}+n+3}$ .

See Solution

Problem 32212

Determine whether the series $\sum_{n=1}^{\infty} \frac{4 n^{3}+n^{2}+1}{n^{4}+n+3}$ converges or diverges.

See Solution

Problem 32213

Determine if the series $\sum_{n=3}^{\infty} \frac{e^{-n}}{n^{2}+2 n}$ converges or diverges.

See Solution

Problem 32214

Check if the series $\sum_{n=3}^{\infty} \frac{e^{-n}}{n^{2}+2 n}$ converges or diverges. Options: a, b, c, d, e.

See Solution

Problem 32215

Determine the convergence of $\sum_{n=3}^{\infty} \frac{e^{-n}}{n^{2}+2n}$ using comparison tests.

See Solution

Problem 32216

Determine whether the series $\sum_{n=1}^{\infty} \frac{4}{3^{n}+\sqrt{n}}$ converges or diverges using comparison tests.

See Solution

Problem 32217

Find the area function $A(x)=\int_{0}^{x} f(t) dt$ for the line with slope 1 and y-intercept 4 on $[0, x]$ .

See Solution

Problem 32218

Find all local maximum values of the sine integral function $\operatorname{Si}(x)$ for $x$ in $[-10, 10]$ . List them in order.

See Solution

Problem 32219

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}$ using comparison tests.

See Solution

Problem 32220

Solve the equation $\frac{d y}{d x}=\frac{4 x y+6 y^{2}}{2 x^{2}+4 x y}$ .

See Solution

Problem 32221

Find the slope of the tangent line of $f(x)=e^{\tan x}$ at $x=0$ . Choose from: $\frac{-1}{e}$ , 1, $-1$ , $e$ .

See Solution

Problem 32222

Find the slope of the tangent line for $f(x)=e^{\tan x}$ at $x=0$ . Options: 1, $e$ , $-1$ , $\frac{-1}{e}$ .

See Solution

Problem 32223

Find the slope of the tangent line of $f(x)=e^{\tan x}$ at $x=0$ . Options: e, -1, 1, $\frac{-1}{e}$ .

See Solution

Problem 32224

1.

See Solution

Problem 32225

Find the derivative $f^{\prime}(0)$ for the function $f(x)=\ln(x^{2}+3x+\cos x)$ .

See Solution

Problem 32226

Find $f^{\prime}(2)$ if $f(x)=\ln(x^{2}-3)$ . Choices: e, 4, 2, 1.

See Solution

Problem 32227

Calculate the integral from 0 to $\frac{\pi}{6}$ of $\sec(2x)$ dx and choose the correct answer.

See Solution

Problem 32228

Find $y^{\prime}(e)$ if $y=x^{\ln x}$ .

See Solution

Problem 32229

Find the slope of the tangent line of $f(x)=e^{\tan x}$ at $x=0$ . Choices: $-1$ , e, $\frac{-1}{e}$ , 1.

See Solution

Problem 32230

Find the integral $\int \frac{\sec ^{2}(2 x)}{\tan (2 x)} d x$ . What is the correct answer?

See Solution

Problem 32231

Find $f^{\prime}(0)$ if $f(x)=\int_{1}^{e^{2 x}} (t^{2}+\ln t) dt$ .

See Solution

Problem 32232

Find $f^{\prime}(2)$ for the function $f(x)=\ln \left(x^{2}-3\right)$ .

See Solution

Problem 32233

Find the integral of $\sec (5 x)$ : $\int \sec (5 x) d x$ . Choose the correct answer from the options given.

See Solution

Problem 32234

Find $f^{\prime}(2)$ for $f(x)=\ln(x^{2}-3)$ . Choices: 1, 2, $e$ , 4.

See Solution

Problem 32235

Find the first derivative $\frac{d y}{d x}$ at the point $(1,1)$ for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ .

See Solution

Problem 32236

Find the derivative $f^{\prime}(2)$ for the function $f(x)=\ln \left(x^{2}-3\right)$ .

See Solution

Problem 32237

Find the tangent line to $f(x)=e^{2 x}$ at the point $(0,1)$ . Choose the correct equation from the options given.

See Solution

Problem 32238

Fjellområde fikk snø fra midnatt til 11:00. Funksjonen $s(x)=-0.017 x^{3}+2.0 x$ beskriver snødybden.
a) Tegn $s$ for $0 \leq x \leq 11$ b) Snødybden etter 2 timer? c) Største snødybde? d) Når var snødybden $3 \mathrm{~cm}$ ? e) Gjennomsnittlig minskning fra 06:00 til 10:00? f) Momentan vekstfart kl. 03:00 og 09:00?
Bruk GeoGebra og lever inn et Word-dokument med forklaring. Frist: 24.1 kl. 22:00.

See Solution

Problem 32239

Find $\left(f^{-1}\right)^{\prime}(1)$ given $f(-1)=1$ , $f(1)=0$ , $f'(-1)=\frac{1}{5}$ , $f'(1)=\frac{-1}{5}$ .

See Solution

Problem 32240

Find the integral: $\int \frac{\sec ^{2}(2 x)}{\tan (2 x)} d x$ . Choose the correct answer from the options.

See Solution

Problem 32241

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ given $x^{2}+y^{2}=1$ . Choose the correct option.

See Solution

Problem 32242

Find the integral $\int \frac{\sec ^{2}(2 x)}{\tan (2 x)} d x$ .

See Solution

Problem 32243

Find $f^{\prime}(0)$ for the function $f(x)=\int_{1}^{e^{2 x}} t^{2}+\ln t \, dt$ .

See Solution

Problem 32244

Find the derivative $f^{\prime}(0)$ for the function $f(x)=\ln \left(x^{2}+3 x+\cos x\right)$ .

See Solution

Problem 32245

Find the first derivative $\frac{d y}{d x}$ at the point (1,1) for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ .

See Solution

Problem 32246

Find the first derivative $\frac{d y}{d x}$ of $x^{2} y^{4}-2 y=3$ at the point $(2,1)$ . Options: 2, 20, $-2$ , $\frac{1}{2}$ .

See Solution

Problem 32247

Find the derivative of $y=\sin \left(e^{x^{2}}\right)$ : $\frac{d y}{d x}$ .

See Solution

Problem 32248

Find the derivative of $y=e^{x}(\sin x+\cos x)$ with respect to $x$ .

See Solution

Problem 32249

Find the derivative $\frac{d y}{d x}$ if $y=\sin \left(e^{x^{2}}\right)$ .

See Solution

Problem 32250

Find the derivative of $y=e^{x}(\sin x+\cos x)$ with respect to $x$ .

See Solution

Problem 32251

Find the first derivative $\frac{d y}{d x}$ for $x=y \cos (2 x)$ at the point $\left(\frac{\pi}{2}, \frac{-\pi}{2}\right)$ .

See Solution

Problem 32252

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ given $x^{2}+y^{2}=1$ . Choices include: $\frac{-x}{y^{2}}$ , $\frac{x^{2}+y^{2}}{y^{3}}$ , $\frac{x^{2}-y^{2}}{y^{3}}$ , $\frac{-1}{y^{3}}$ .

See Solution

Problem 32253

Find the tangent line to the curve $f(x)=e^{2x}$ at the point $(0,1)$ . Choose the correct equation.

See Solution

Problem 32254

Find $f^{\prime}(2)$ if $f(x)=\ln \left(x^{2}-3\right)$ .

See Solution

Problem 32255

Evaluate the integral $\int_{0}^{1} x^{2} 2^{x^{3}} d x$ and choose the correct answer: a) $\frac{1}{\ln (4)}$ , b) $\frac{1}{\ln (8)}$ , c) $\frac{1}{\ln (9)}$ , d) $\frac{1}{\ln (2)}$ , e) none.

See Solution

Problem 32256

Find $\frac{d y}{d x}$ if $e^{y} = 6 x + 3 y$ . Choices: A) $\frac{6 x}{e^{y}+3}$ B) $\frac{6}{e^{v}+3}$ C) $\frac{6}{e^{y}-3}$ D) $\frac{6}{e^{y}+3 y}$ .

See Solution

Problem 32257

Evaluate the integral from $e$ to $e^{2}$ of $\frac{1+\ln x}{x \ln x} \, dx = ?$ Options: a. $\ln (\ln 2)$ b. $1-\ln 2$ c. $1+\ln 2$ d. $3+\ln 2$

See Solution

Problem 32258

Find the slope of the tangent line of $f(x)=e^{\tan x}$ at $x=0$ . Options: $\frac{-1}{e}$ , $-1$ , 1, e.

See Solution

Problem 32259

Find $f^{\prime}(0)$ if $f(x)=\int_{1}^{e^{2 x}} t^{2}+\ln t \, dt$ .

See Solution

Problem 32260

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

See Solution

Problem 32261

Find the derivative of $y=e^{x}(\sin x+\cos x)$ with respect to $x$ .

See Solution

Problem 32262

Find the derivative of the function $f(x)=\ln \left(e^{-x}+1\right)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 32263

1

See Solution

Problem 32264

Find the tangent line to the curve $f(x)=e^{2 x}$ at the point $(0,1)$ . Which equation is correct?

See Solution

Problem 32265

Find the derivative of $y=e^{x}(\sin x+\cos x)$ with respect to $x$ .

See Solution

Problem 32266

Fjellområde snødybde: $s(x)=-0.017 x^{3}+2.0 x$ . a) Tegn graf for $0 \leq x \leq 11$ . b) Snødybde etter 2 timer? c) Maks snødybde? d) Når $s(x)=3$ cm? e) Gjennomsnittlig nedgang fra 06:00 til 10:00? f) Momentan vekstfart kl. 03:00 og 09:00? Bruk GeoGebra. Innlevering innen 24.1. kl. 22:00.

See Solution

Problem 32267

Find the tangent line to $f(x)=e^{2 x}$ at the point $(0,1)$ . Which equation is correct? $y=-2x-1$ , $y=2x-2$ , $y=-2x-2$ , $y=2x+1$ .

See Solution

Problem 32268

Find $\frac{d y}{d x}$ for $y = \cos(2x)$ at $\left(\frac{\pi}{2}, -\frac{\pi}{2}\right)$ . Choices: 1, $\frac{1}{2}$ , -1, 0.

See Solution

Problem 32269

Evaluate the integral $\int_{0}^{1} x^{2} 2^{x^{3}} d x$ and choose the correct answer from the options given.

See Solution

Problem 32270

Find the derivative of $f(x)=\ln(2 e^{-x} \sin x)$ , i.e., $f^{\prime}(x)$ .

See Solution

Problem 32271

Find $f^{\prime}(2)$ for the function $f(x)=\ln \left(x^{2}-3\right)$ . Choices: $e$ , 1, 2, 4.

See Solution

Problem 32272

Evaluate the integral $\int_{0}^{1} x^{2} 2^{x^{3}} d x$ .

See Solution

Problem 32273

Find $f^{\prime}(e)$ if $f(x)=x^{\ln x}$ . Options: 1, 0, $e$ , 2.

See Solution

Problem 32274

0.2

See Solution

Problem 32275

Find $f^{\prime}(2)$ if $f(x)=\ln \left(x^{2}-3\right)$ . Choices: 2, 4, 1, $e$ .

See Solution

Problem 32276

Calculate the integral from $e$ to $e^{2}$ of $\frac{4(\ln x)^{3}}{x} \, dx$ . Provide a number only.

See Solution

Problem 32277

Calculate the integral $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \tan x \, dx$ . Choose from: a. $\ln (4)$ , b. $\ln (\sqrt{2})$ , c. $\ln (4)$ , d. $\ln (2)$ , e. none.

See Solution

Problem 32278

Find the slope of the tangent line for $f(x)=e^{\tan x}$ at $x=0$ . Choices: $-1$ , 1, $e$ , $\frac{-1}{e}$ .

See Solution

Problem 32279

Find the derivative $f'(0)$ for $f(x)=\ln(x^2 + 3x + \cos x)$ .

See Solution

Problem 32280

Find $f^{\prime}(0)$ for $f(x)=\ln (\sec (x)+\tan (x))$ . Options: 1, $\frac{\pi}{2}$ , $-1$ , 0.

See Solution

Problem 32281

Find the value of $\left(f^{-1}\right)^{\prime}(1)$ given $f(-1)=1$ , $f(1)=0$ , $f'(-1)=\frac{1}{5}$ , $f'(1)=\frac{-1}{5}$ . Write a number only.

See Solution

Problem 32282

Find the tangent line equation to $f(x)=e^{2 x}$ at the point $(0,1)$ . Choose from the options provided.

See Solution

Problem 32283

-5

See Solution

Problem 32284

Find the first derivative $\frac{d y}{d x}$ at the point $(2,1)$ for the equation $x^{2} y^{4}-2 y=3$ .

See Solution

Problem 32285

Find $\frac{d y}{d x}$ for $x^{2} y^{4}-2 y=3$ at the point $(2,1)$ . Choose from: 2, 20, $-2$ , $\frac{1}{2}$ .

See Solution

Problem 32286

Find the derivative $f'(0)$ for the function $f(x)=\ln(x^{2}+3x+\cos x)$ . Output a number only.

See Solution

Problem 32287

Find the derivative of $f(x)=\ln \left(2 e^{-x} \sin x\right)$ , i.e., find $f^{\prime}(x)$ .

See Solution

Problem 32288

Find the derivative $f'(0)$ if $f(x)=\ln (\sec (x)+\tan (x))$ . Choices: 0, 1, $\frac{\pi}{2}$ , $-1$ .

See Solution

Problem 32289

Find the derivative of the function $f(x)=\ln \left(e^{-x}+1\right)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 32290

Find $f'(x)$ if $f(x)=\ln(e^{-x}+1)$ . Options: $\frac{1}{e^{-x}+1}$ , $\frac{-1}{e^{-x}+1}$ , $\frac{-1}{1+e^{x}}$ , $\frac{e^{-x}}{e^{-x}+1}$ .

See Solution

Problem 32291

Find the tangent line to the curve $f(x)=e^{2 x}$ at the point $(0,1)$ . Choose the correct equation.

See Solution

Problem 32292

Find the tangent line to the curve $f(x)=e^{2x}$ at the point $(0,1)$ . Choose the correct equation.

See Solution

Problem 32293

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

See Solution

Problem 32294

Find the slope of the tangent line of $f(x)=e^{\tan x}$ at $x=0$ . Choices: $-1$ , 1, $\frac{-1}{e}$ , e.

See Solution

Problem 32295

Find the derivative of $y=e^{x}(\sin x+\cos x)$ with respect to $x$ .

See Solution

Problem 32296

Find the derivative $f^{\prime}(0)$ for the function $f(x)=\ln(x^{2}+3x+\cos x)$ . Write a number only.

See Solution

Problem 32297

Evaluate the integral $\int_{0}^{\frac{\pi}{6}} \sec (2 x) d x$ and choose the correct answer from the options.

See Solution

Problem 32298

Find the derivative of the function $f(x)=\ln \left(2 e^{-x} \sin x\right)$ .

See Solution

Problem 32299

Find $\left(f^{-1}\right)^{\prime}(1)$ given $f(-1)=1$ , $f(1)=0$ , $f^{\prime}(-1)=\frac{1}{5}$ , $f^{\prime}(1)=\frac{-1}{5}$ . Write number only.

See Solution

Problem 32300

Find the derivative $f^{\prime}(0)$ for $f(x)=\ln (\sec (x)+\tan (x))$ . Choices: 1, $\frac{\pi}{2}$ , $-1$ , 0.

See Solution

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Calculus

Problem 32201

Differentiate the function f(x)=ln⁡x2f(x) = \ln x^{2}f(x)=lnx2.

Problem 32202

Find the second derivative F′′(2)F''(2)F′′(2), where F(x)=∫1xf(t)dtF(x)=\int_{1}^{x} f(t) dtF(x)=∫1x​f(t)dt and f(t)=∫1t36+u6uduf(t)=\int_{1}^{t^{3}} \frac{\sqrt{6+u^{6}}}{u} duf(t)=∫1t3​u6+u6​​du.

Problem 32203

Does the series ∑n=1∞(1n2+1)2\sum_{n=1}^{\infty}\left(\frac{1}{n^{2}}+1\right)^{2}n=1∑∞​(n21​+1)2 converge or diverge? a. converge b. diverge

Problem 32204

Find δ>0\delta > 0δ>0 so that 0<∣x−5∣<δ⇒∣2x−4−6∣<0.20 < |x - 5| < \delta \Rightarrow |2x - 4 - 6| < 0.20<∣x−5∣<δ⇒∣2x−4−6∣<0.2. Use the graph.

Problem 32205

Determine the convergence of ∑n=1∞43n+n\sum_{n=1}^{\infty} \frac{4}{3^{n}+\sqrt{n}}n=1∑∞​3n+n​4​ using comparison tests.

Problem 32206

Determine if the series ∑n=1∞cos⁡2(n)n3/2\sum_{n=1}^{\infty} \frac{\cos ^{2}(n)}{n^{3 / 2}}∑n=1∞​n3/2cos2(n)​ converges or diverges using comparison tests.

Problem 32207

Determine the convergence of the series ∑n=1∞n+1n2+1\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}∑n=1∞​n2+1n+1​​ using comparison tests.

Problem 32208

Determine the convergence of the series ∑n=1∞n+1n2+1\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}∑n=1∞​n2+1n+1​​ using comparison tests.

Problem 32209

Differentiate the function f(x)=log⁡2(3x)f(x) = \log_{2}(3x)f(x)=log2​(3x).

Problem 32210

Calculate the sum: ∑n=1∞1+(−2)n3n\sum_{n=1}^{\infty} \frac{1+(-2)^{n}}{3^{n}}∑n=1∞​3n1+(−2)n​.

Problem 32211

Determine the convergence of the series ∑n=1∞4n3+n2+1n4+n+3\sum_{n=1}^{\infty} \frac{4 n^{3}+n^{2}+1}{n^{4}+n+3}∑n=1∞​n4+n+34n3+n2+1​.

Problem 32212

Determine whether the series ∑n=1∞4n3+n2+1n4+n+3\sum_{n=1}^{\infty} \frac{4 n^{3}+n^{2}+1}{n^{4}+n+3}∑n=1∞​n4+n+34n3+n2+1​ converges or diverges.

Problem 32213

Determine if the series ∑n=3∞e−nn2+2n\sum_{n=3}^{\infty} \frac{e^{-n}}{n^{2}+2 n}∑n=3∞​n2+2ne−n​ converges or diverges.

Problem 32214

Check if the series ∑n=3∞e−nn2+2n\sum_{n=3}^{\infty} \frac{e^{-n}}{n^{2}+2 n}n=3∑∞​n2+2ne−n​ converges or diverges. Options: a, b, c, d, e.

Problem 32215

Determine the convergence of ∑n=3∞e−nn2+2n\sum_{n=3}^{\infty} \frac{e^{-n}}{n^{2}+2n}n=3∑∞​n2+2ne−n​ using comparison tests.

Problem 32216

Determine whether the series ∑n=1∞43n+n\sum_{n=1}^{\infty} \frac{4}{3^{n}+\sqrt{n}}∑n=1∞​3n+n​4​ converges or diverges using comparison tests.

Problem 32217

Find the area function A(x)=∫0xf(t)dtA(x)=\int_{0}^{x} f(t) dtA(x)=∫0x​f(t)dt for the line with slope 1 and y-intercept 4 on [0,x][0, x][0,x].

Problem 32218

Find all local maximum values of the sine integral function Si⁡(x)\operatorname{Si}(x)Si(x) for xxx in [−10,10][-10, 10][−10,10]. List them in order.

Problem 32219

Determine the convergence of the series ∑n=1∞n+1n2+1\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}∑n=1∞​n2+1n+1​​ using comparison tests.

Problem 32220

Solve the equation dydx=4xy+6y22x2+4xy\frac{d y}{d x}=\frac{4 x y+6 y^{2}}{2 x^{2}+4 x y}dxdy​=2x2+4xy4xy+6y2​.

Problem 32221

Find the slope of the tangent line of f(x)=etan⁡xf(x)=e^{\tan x}f(x)=etanx at x=0x=0x=0. Choose from: −1e\frac{-1}{e}e−1​, 1, −1-1−1, eee.

Problem 32222

Find the slope of the tangent line for f(x)=etan⁡xf(x)=e^{\tan x}f(x)=etanx at x=0x=0x=0. Options: 1, eee, −1-1−1, −1e\frac{-1}{e}e−1​.

Problem 32223

Find the slope of the tangent line of f(x)=etan⁡xf(x)=e^{\tan x}f(x)=etanx at x=0x=0x=0. Options: e, -1, 1, −1e\frac{-1}{e}e−1​.

Problem 32224

1.

Problem 32225

Find the derivative f′(0)f^{\prime}(0)f′(0) for the function f(x)=ln⁡(x2+3x+cos⁡x)f(x)=\ln(x^{2}+3x+\cos x)f(x)=ln(x2+3x+cosx).

Problem 32226

Find f′(2)f^{\prime}(2)f′(2) if f(x)=ln⁡(x2−3)f(x)=\ln(x^{2}-3)f(x)=ln(x2−3). Choices: e, 4, 2, 1.

Problem 32227

Calculate the integral from 0 to π6\frac{\pi}{6}6π​ of sec⁡(2x)\sec(2x)sec(2x) dx and choose the correct answer.

Problem 32228

Find y′(e)y^{\prime}(e)y′(e) if y=xln⁡xy=x^{\ln x}y=xlnx.

Problem 32229

Find the slope of the tangent line of f(x)=etan⁡xf(x)=e^{\tan x}f(x)=etanx at x=0x=0x=0. Choices: −1-1−1, e, −1e\frac{-1}{e}e−1​, 1.

Problem 32230

Find the integral ∫sec⁡2(2x)tan⁡(2x)dx\int \frac{\sec ^{2}(2 x)}{\tan (2 x)} d x∫tan(2x)sec2(2x)​dx. What is the correct answer?

Problem 32231

Find f′(0)f^{\prime}(0)f′(0) if f(x)=∫1e2x(t2+ln⁡t)dtf(x)=\int_{1}^{e^{2 x}} (t^{2}+\ln t) dtf(x)=∫1e2x​(t2+lnt)dt.

Problem 32232

Find f′(2)f^{\prime}(2)f′(2) for the function f(x)=ln⁡(x2−3)f(x)=\ln \left(x^{2}-3\right)f(x)=ln(x2−3).

Problem 32233

Find the integral of sec⁡(5x)\sec (5 x)sec(5x): ∫sec⁡(5x)dx\int \sec (5 x) d x∫sec(5x)dx. Choose the correct answer from the options given.

Problem 32234

Find f′(2)f^{\prime}(2)f′(2) for f(x)=ln⁡(x2−3)f(x)=\ln(x^{2}-3)f(x)=ln(x2−3). Choices: 1, 2, eee, 4.

Problem 32235

Find the first derivative dydx\frac{d y}{d x}dxdy​ at the point (1,1)(1,1)(1,1) for the equation x23+y23=2yx^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}x32​+y32​=2y​.

Problem 32236

Find the derivative f′(2)f^{\prime}(2)f′(2) for the function f(x)=ln⁡(x2−3)f(x)=\ln \left(x^{2}-3\right)f(x)=ln(x2−3).

Problem 32237

Find the tangent line to f(x)=e2xf(x)=e^{2 x}f(x)=e2x at the point (0,1)(0,1)(0,1). Choose the correct equation from the options given.

Problem 32238

Problem 32239

Find (f−1)′(1)\left(f^{-1}\right)^{\prime}(1)(f−1)′(1) given f(−1)=1f(-1)=1f(−1)=1, f(1)=0f(1)=0f(1)=0, f′(−1)=15f'(-1)=\frac{1}{5}f′(−1)=51​, f′(1)=−15f'(1)=\frac{-1}{5}f′(1)=5−1​.

Problem 32240

Find the integral: ∫sec⁡2(2x)tan⁡(2x)dx\int \frac{\sec ^{2}(2 x)}{\tan (2 x)} d x∫tan(2x)sec2(2x)​dx. Choose the correct answer from the options.

Differentiate the function $f(x) = \ln x^{2}$ .

Find the second derivative $F''(2)$ , where $F(x)=\int_{1}^{x} f(t) dt$ and $f(t)=\int_{1}^{t^{3}} \frac{\sqrt{6+u^{6}}}{u} du$ .

Does the series $\sum_{n=1}^{\infty}\left(\frac{1}{n^{2}}+1\right)^{2}$ converge or diverge? a. converge b. diverge

Find $\delta > 0$ so that $0 < |x - 5| < \delta \Rightarrow |2x - 4 - 6| < 0.2$ . Use the graph.

Determine the convergence of $\sum_{n=1}^{\infty} \frac{4}{3^{n}+\sqrt{n}}$ using comparison tests.

Determine if the series $\sum_{n=1}^{\infty} \frac{\cos ^{2}(n)}{n^{3 / 2}}$ converges or diverges using comparison tests.

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}$ using comparison tests.

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}$ using comparison tests.

Differentiate the function $f(x) = \log_{2}(3x)$ .

Calculate the sum: $\sum_{n=1}^{\infty} \frac{1+(-2)^{n}}{3^{n}}$ .

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{4 n^{3}+n^{2}+1}{n^{4}+n+3}$ .

Determine whether the series $\sum_{n=1}^{\infty} \frac{4 n^{3}+n^{2}+1}{n^{4}+n+3}$ converges or diverges.

Determine if the series $\sum_{n=3}^{\infty} \frac{e^{-n}}{n^{2}+2 n}$ converges or diverges.

Check if the series $\sum_{n=3}^{\infty} \frac{e^{-n}}{n^{2}+2 n}$ converges or diverges. Options: a, b, c, d, e.

Determine the convergence of $\sum_{n=3}^{\infty} \frac{e^{-n}}{n^{2}+2n}$ using comparison tests.

Determine whether the series $\sum_{n=1}^{\infty} \frac{4}{3^{n}+\sqrt{n}}$ converges or diverges using comparison tests.

Find the area function $A(x)=\int_{0}^{x} f(t) dt$ for the line with slope 1 and y-intercept 4 on $[0, x]$ .

Find all local maximum values of the sine integral function $\operatorname{Si}(x)$ for $x$ in $[-10, 10]$ . List them in order.

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}$ using comparison tests.

Solve the equation $\frac{d y}{d x}=\frac{4 x y+6 y^{2}}{2 x^{2}+4 x y}$ .

Find the slope of the tangent line of $f(x)=e^{\tan x}$ at $x=0$ . Choose from: $\frac{-1}{e}$ , 1, $-1$ , $e$ .

Find the slope of the tangent line for $f(x)=e^{\tan x}$ at $x=0$ . Options: 1, $e$ , $-1$ , $\frac{-1}{e}$ .

Find the slope of the tangent line of $f(x)=e^{\tan x}$ at $x=0$ . Options: e, -1, 1, $\frac{-1}{e}$ .

Find the derivative $f^{\prime}(0)$ for the function $f(x)=\ln(x^{2}+3x+\cos x)$ .

Find $f^{\prime}(2)$ if $f(x)=\ln(x^{2}-3)$ . Choices: e, 4, 2, 1.

Calculate the integral from 0 to $\frac{\pi}{6}$ of $\sec(2x)$ dx and choose the correct answer.

Find $y^{\prime}(e)$ if $y=x^{\ln x}$ .

Find the slope of the tangent line of $f(x)=e^{\tan x}$ at $x=0$ . Choices: $-1$ , e, $\frac{-1}{e}$ , 1.

Find the integral $\int \frac{\sec ^{2}(2 x)}{\tan (2 x)} d x$ . What is the correct answer?

Find $f^{\prime}(0)$ if $f(x)=\int_{1}^{e^{2 x}} (t^{2}+\ln t) dt$ .

Find $f^{\prime}(2)$ for the function $f(x)=\ln \left(x^{2}-3\right)$ .

Find the integral of $\sec (5 x)$ : $\int \sec (5 x) d x$ . Choose the correct answer from the options given.

Find $f^{\prime}(2)$ for $f(x)=\ln(x^{2}-3)$ . Choices: 1, 2, $e$ , 4.

Find the first derivative $\frac{d y}{d x}$ at the point $(1,1)$ for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ .

Find the derivative $f^{\prime}(2)$ for the function $f(x)=\ln \left(x^{2}-3\right)$ .

Find the tangent line to $f(x)=e^{2 x}$ at the point $(0,1)$ . Choose the correct equation from the options given.

Find $\left(f^{-1}\right)^{\prime}(1)$ given $f(-1)=1$ , $f(1)=0$ , $f'(-1)=\frac{1}{5}$ , $f'(1)=\frac{-1}{5}$ .

Find the integral: $\int \frac{\sec ^{2}(2 x)}{\tan (2 x)} d x$ . Choose the correct answer from the options.

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ given $x^{2}+y^{2}=1$ . Choose the correct option.

Find the integral $\int \frac{\sec ^{2}(2 x)}{\tan (2 x)} d x$ .

Find $f^{\prime}(0)$ for the function $f(x)=\int_{1}^{e^{2 x}} t^{2}+\ln t \, dt$ .

Find the derivative $f^{\prime}(0)$ for the function $f(x)=\ln \left(x^{2}+3 x+\cos x\right)$ .

Find the first derivative $\frac{d y}{d x}$ at the point (1,1) for the equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2 \sqrt{y}$ .

Find the first derivative $\frac{d y}{d x}$ of $x^{2} y^{4}-2 y=3$ at the point $(2,1)$ . Options: 2, 20, $-2$ , $\frac{1}{2}$ .

Find the derivative of $y=\sin \left(e^{x^{2}}\right)$ : $\frac{d y}{d x}$ .

Find the derivative of $y=e^{x}(\sin x+\cos x)$ with respect to $x$ .

Find the derivative $\frac{d y}{d x}$ if $y=\sin \left(e^{x^{2}}\right)$ .

Find the derivative of $y=e^{x}(\sin x+\cos x)$ with respect to $x$ .

Find the first derivative $\frac{d y}{d x}$ for $x=y \cos (2 x)$ at the point $\left(\frac{\pi}{2}, \frac{-\pi}{2}\right)$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ given $x^{2}+y^{2}=1$ . Choices include: $\frac{-x}{y^{2}}$ , $\frac{x^{2}+y^{2}}{y^{3}}$ , $\frac{x^{2}-y^{2}}{y^{3}}$ , $\frac{-1}{y^{3}}$ .

Find the tangent line to the curve $f(x)=e^{2x}$ at the point $(0,1)$ . Choose the correct equation.

Find $f^{\prime}(2)$ if $f(x)=\ln \left(x^{2}-3\right)$ .

Evaluate the integral $\int_{0}^{1} x^{2} 2^{x^{3}} d x$ and choose the correct answer: a) $\frac{1}{\ln (4)}$ , b) $\frac{1}{\ln (8)}$ , c) $\frac{1}{\ln (9)}$ , d) $\frac{1}{\ln (2)}$ , e) none.

Find $\frac{d y}{d x}$ if $e^{y} = 6 x + 3 y$ . Choices: A) $\frac{6 x}{e^{y}+3}$ B) $\frac{6}{e^{v}+3}$ C) $\frac{6}{e^{y}-3}$ D) $\frac{6}{e^{y}+3 y}$ .

Evaluate the integral from $e$ to $e^{2}$ of $\frac{1+\ln x}{x \ln x} \, dx = ?$ Options: a. $\ln (\ln 2)$ b. $1-\ln 2$ c. $1+\ln 2$ d. $3+\ln 2$

Find the slope of the tangent line of $f(x)=e^{\tan x}$ at $x=0$ . Options: $\frac{-1}{e}$ , $-1$ , 1, e.

Find $f^{\prime}(0)$ if $f(x)=\int_{1}^{e^{2 x}} t^{2}+\ln t \, dt$ .

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

Find the derivative of $y=e^{x}(\sin x+\cos x)$ with respect to $x$ .

Find the derivative of the function $f(x)=\ln \left(e^{-x}+1\right)$ . What is $f^{\prime}(x)$ ?

Find the tangent line to the curve $f(x)=e^{2 x}$ at the point $(0,1)$ . Which equation is correct?

Find the derivative of $y=e^{x}(\sin x+\cos x)$ with respect to $x$ .

Find the tangent line to $f(x)=e^{2 x}$ at the point $(0,1)$ . Which equation is correct? $y=-2x-1$ , $y=2x-2$ , $y=-2x-2$ , $y=2x+1$ .

Find $\frac{d y}{d x}$ for $y = \cos(2x)$ at $\left(\frac{\pi}{2}, -\frac{\pi}{2}\right)$ . Choices: 1, $\frac{1}{2}$ , -1, 0.

Evaluate the integral $\int_{0}^{1} x^{2} 2^{x^{3}} d x$ and choose the correct answer from the options given.

Find the derivative of $f(x)=\ln(2 e^{-x} \sin x)$ , i.e., $f^{\prime}(x)$ .

Find $f^{\prime}(2)$ for the function $f(x)=\ln \left(x^{2}-3\right)$ . Choices: $e$ , 1, 2, 4.

Evaluate the integral $\int_{0}^{1} x^{2} 2^{x^{3}} d x$ .

Find $f^{\prime}(e)$ if $f(x)=x^{\ln x}$ . Options: 1, 0, $e$ , 2.