Calculus

Problem 33201

Find the second derivative if $\frac{d y}{d x}=e^{x}-3 x^{2}$ . What is $\frac{d^{2} y}{d x^{2}}$ ?

See Solution

Problem 33202

Find the derivative of $y=\frac{9}{\sqrt[4]{x^{3}}}$ .

See Solution

Problem 33203

Find the average slope of the function $f(x)=2 x^{3}-5 x$ on $[-4,4]$ and the two values of $c$ where $f^{\prime}(c)$ equals it.

See Solution

Problem 33204

Find $y'$ if $y=2^{3x+\ln x}$ . Choose the correct derivative from the options provided.

See Solution

Problem 33205

Evaluate the integral $\int_{1}^{e} \frac{6 \ln x+4}{x} d x$ . Choose from: 8, None, 10, 5, 7.

See Solution

Problem 33206

Find the average slope of $f(x)=\frac{1}{x}$ on $[2,9]$ and the value of $c$ in $(2,9)$ where $f^{\prime}(c)$ equals this slope.

See Solution

Problem 33207

Evaluate the integral from 1 to e of $\frac{2 \ln x + 3}{x} \, dx$ : $3 \bigcirc$ , $4 \bigcirc$ , None $\bigcirc$ , $5 \bigcirc$ .

See Solution

Problem 33208

Find $\frac{d y}{d x}$ for the equation $\ln (2 x+4 y)=\sin y$ .

See Solution

Problem 33209

Find $\frac{d y}{d x}$ for the equation $\ln (3 x+4 y)=\sin y$ . Choose the correct option.

See Solution

Problem 33210

Evaluate the integral $\int \frac{\sec (4 x) \tan (4 x)}{2+\sec (4 x)} d x$ . What is the result?

See Solution

Problem 33211

Find the derivative of $y=3 x^{3}-2 x$ and locate the stationary points.
(a) $\frac{d y}{d x}=$
(b) The stationary point(s) is/are at

See Solution

Problem 33212

Evaluate the integral $\int \frac{\sec (3 x) \tan (3 x)}{2+\sec (3 x)} d x$ . What is the result?

See Solution

Problem 33213

Find the average slope of $f(x)=1-3x^{2}$ on $[-1,6]$ and the unique $c$ where $f'(c)$ equals this slope.

See Solution

Problem 33214

Find the derivative $y^{\prime}$ if $y=5^{3x+\ln x}$ .

See Solution

Problem 33215

Find the slope of the tangent line to the curve $y=x e^{4 x}-e^{7 x}$ at $x=0$ . Choices: 60, None, $-7$ , $7$ .

See Solution

Problem 33216

Find the marginal revenue $R$ from renting $x$ apartments, given $R=2 x(900+34 x-x^{2})$ , when $x=14$ .

See Solution

Problem 33217

Find the average slope of $f(x)=6 \sqrt{x}+6$ on $[2,9]$ and the value of $c$ in $(2,9)$ where $f'(c)$ equals this slope.

See Solution

Problem 33218

Find the derivative of $y=3^{2 x+\ln x}$ . What is $y^{\prime}$ ?

See Solution

Problem 33219

Evaluate the integral $\int \frac{\sec (3 x) \tan (3 x)}{2+\sec (3 x)} d x$ . What is the result?

See Solution

Problem 33220

Find the slope of the tangent line to the curve $y=x e^{5 x}-e^{3 x}$ at $x=0$ . Choose: $-2$ , $-3$ , None, 2, 3.

See Solution

Problem 33221

Find $\frac{d y}{d x}$ for $\ln (3 x+4 y)=\sin y$ . Choose the correct expression for $\frac{d y}{d x}$ .

See Solution

Problem 33222

Find the nature of $y$ at $x=0$ given $\frac{d y}{d x}=\sin (x)-x$ and $\frac{d y}{d x}=\frac{d^{2} y}{d x^{2}}=0$ .

See Solution

Problem 33223

Find $\frac{d y}{d x}$ for $\ln (3 x+4 y)=\sin y$ . What is the derivative?

See Solution

Problem 33224

Find the slope of the tangent line to the curve $y=x e^{4 x}-e^{7 x}$ at $x=0$ . Options: -6, 60, 0, -7, 7.

See Solution

Problem 33225

Find the second derivative of $y=f(x)$ where $\frac{d y}{d x}=e^{x}-15 x^{4}$ . Determine the nature of $y$ at $x=-0.45$ .

See Solution

Problem 33226

Find the derivative of the function $f(x) = 4x(x^2 + 1)$ .

See Solution

Problem 33227

Find the average slope of $f(x)=2 x^{3}-6 x^{2}-90 x+6$ on $[-5,8]$ and the two values of $c$ where $f'(c)$ equals this slope.

See Solution

Problem 33228

Find the slope of the tangent line to the curve $y=x e^{5 x}-e^{3 x}$ at $x=0$ .

See Solution

Problem 33229

Find $\frac{d y}{d x}$ given that $\ln(2x + 5y) = \sin y$ .

See Solution

Problem 33230

Find the average rate of change of $f(x)=x^{3}-5 x+7$ from (a) -3 to -2 and (b) 2 to 4.

See Solution

Problem 33231

Find the derivative $y^{\prime}$ if $y=2^{3 x+\ln x}$ . Choose from the options provided.

See Solution

Problem 33232

Evaluate the integral $\int_{1}^{e} \frac{6 \ln x + 4}{x} \, dx$ . Choose: 10, 7, None, 5, 8.

See Solution

Problem 33233

Find the average rate of change of $g(x)=4x^{2}-2$ from -6 to 3, and the secant line through $(-6, g(-6))$ and $(3, g(3))$ .

See Solution

Problem 33234

Find the derivative of the function $e^{5x}$ .

See Solution

Problem 33235

Evaluate the integral $\int_{1}^{e} \frac{2 \ln x+3}{x} d x$ . Choose: 4, 5, 2, None, 3.

See Solution

Problem 33236

Evaluate the integral $\int \frac{\sec (3 x) \tan (3 x)}{2+\sec (3 x)} d x$ .

See Solution

Problem 33237

Find the derivative of the function $f(x) = x^{2}\left(1-\frac{2}{x^{3}}\right)$ .

See Solution

Problem 33238

The average cost per hour for $x$ lawn mowers is $\bar{C}(x)=0.2 x^{2}+21 x-273+\frac{2800}{x}$ . Find $x$ to minimize cost and the minimum cost.

See Solution

Problem 33239

Find where the function $f$ is increasing based on its behavior from $(-2,-4)$ to $(2,4)$ .

See Solution

Problem 33240

Find the limit and check if the function is continuous at $y=1$ :
$\lim _{y \rightarrow 1} \sec \left(y \sec ^{2} y-\tan ^{2} y-1\right)$

See Solution

Problem 33241

Find the derivative of $\frac{1}{2} \ln (x)$ and match it with the correct option: $\frac{2}{x}$ , $\frac{\ln (x^{2})}{4}$ , $\ln (\sqrt{x})$ , $\frac{1}{2 x}$ .

See Solution

Problem 33242

Find the slope of the tangent line to the curve $y=x^{2} e^{5 x-3}$ at $x=1$ .

See Solution

Problem 33243

Find the derivative $f'(0)$ for the function $f(x)=\ln (\sec (x)+\tan (x))$ .

See Solution

Problem 33244

Find the derivative of $y=\frac{1}{7 x^{2}}+\frac{1}{7 x}$ .

See Solution

Problem 33245

Find the derivative $\frac{d y}{d x}$ for the equation $\tan (2 y)=\ln (x y)+2 x$ .

See Solution

Problem 33246

Evaluate these limits: (a) $\lim _{x \rightarrow \infty}(-22 x^{2}-30 x^{3})=$ ; (b) $\lim _{x \rightarrow -\infty}(-22 x^{2}-30 x^{3})=$ .

See Solution

Problem 33247

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{4 x^{3}-8 x^{2}-3 x}{11-10 x-11 x^{3}}$ .

See Solution

Problem 33248

Find the slope of the tangent line to the curve $y=x^{2} e^{2 x-3}$ at $x=1$ .

See Solution

Problem 33249

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{5 x^{2}-4 x+10}{7 x+8}$ .

See Solution

Problem 33250

Evaluate the integral $\int \frac{9}{x(3+\ln x)} d x$ .

See Solution

Problem 33251

Calculate the integral from 1 to 16 of $\frac{16}{x \sqrt{\ln x}} \, dx$ .

See Solution

Problem 33252

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{4+10 x}{3-11 x}$ .

See Solution

Problem 33253

Evaluate the integral $\int \frac{2}{x(4+\ln x)} d x$ . Choose the correct answer from the options given.

See Solution

Problem 33254

Evaluate the integral $\int_{0}^{1} x^{2} e^{4 x^{3}} d x$ . Choose the correct answer from the options provided.

See Solution

Problem 33255

Find the absolute maximum and minimum of the function $y=f(x)$ from the graph, and identify local maxima/minima.

See Solution

Problem 33256

Find the derivative $y'$ , where $y=\ln(\sec(e^{2x}))$ . Options include $e^{2x} \sec(e^{2x}) \tan(e^{2x})$ , etc.

See Solution

Problem 33257

Find $\frac{d y}{d x}$ for $\tan(2y) = \ln(xy) + 2x$ . Choose the correct derivative expression.

See Solution

Problem 33258

Find $\frac{d y}{d x}$ for the equation $\tan(3y) = \ln(xy) + 2x$ . Choose the correct derivative expression.

See Solution

Problem 33259

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{4 x+4}{2 x^{2}-3 x+3}$ .

See Solution

Problem 33260

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{10+2 x^{2}}}{8+4 x}=$ ; (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{10+2 x^{2}}}{8+4 x}=$

See Solution

Problem 33261

Find the limit and check continuity at the point:
$\lim _{x \rightarrow 0} \sec \left[\cos x+5 \pi \tan \left(\frac{\pi}{4 \sec x}\right)-1\right]$
Is it equal to $\square$ or does it not exist?

See Solution

Problem 33262

Evaluate the integral $\int_{0}^{1} x^{2} e^{2 x^{3}} d x$ . Choose the correct answer from the options provided.

See Solution

Problem 33263

Calculate the integral $\int_{2}^{16} \frac{d x}{x \sqrt{\ln x}}$ .

See Solution

Problem 33264

Find the derivative of $y=\ln \left(\sec \left(e^{2 x}\right)\right)$ . What is $y^{\prime}$ ?

See Solution

Problem 33265

Find the absolute maximum and minimum of the function $y=f(x)$ from the graph. Identify local maxima and minima.

See Solution

Problem 33266

Find the limit and check if the function is continuous at that point:
$\lim _{x \rightarrow 0} 9 \sec \left[9 \cos x+6 \pi \tan \left(\frac{\pi}{4 \sec x}\right)-9\right]$
A. $\lim _{x \rightarrow 0} 9 \sec \left[9 \cos x+6 \pi \tan \left(\frac{\pi}{4 \sec x}\right)-9\right]=\square$
B. The limit does not exist.

See Solution

Problem 33267

Graph $f(x)=x^3-2x+4$ on $(-2,2)$ , find local maxima/minima, and determine where it is increasing/decreasing.

See Solution

Problem 33268

Find the derivative of the inverse function $f^{-1}(x)$ at $x=6$ for $f(x)=x^{3}-2$ .

See Solution

Problem 33269

Evaluate the integral $\int \frac{\sec (3 x) \tan (3 x)}{2+\sec (3 x)} d x$ . What is the result?

See Solution

Problem 33270

Find the derivative of $y$ and set it equal to the derivative of $4x(x^2+1)$ .

See Solution

Problem 33271

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{11 x+11}{10 x^{2}-2 x+10}$ .

See Solution

Problem 33272

Find the limit as $x$ approaches infinity: $\lim_{x \rightarrow \infty} \frac{\sqrt{3+3x^{2}}}{9+2x}$

See Solution

Problem 33273

Find the integral: $\int \frac{4 \sin y}{1 - \cos y} \, dy$ .

See Solution

Problem 33274

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{4}{e^{x}+2}=$ (b) $\lim _{x \rightarrow-\infty} \frac{4}{e^{x}+2}=$

See Solution

Problem 33275

Find the limit of $f(x)=\frac{10}{x}-9$ as $x \to \infty$ and $x \to -\infty$ . What is $\lim _{x \rightarrow \infty} f(x)$ ?

See Solution

Problem 33276

Evaluate the limit as $x$ approaches infinity: $\lim_{x \rightarrow \infty} \frac{5 + 10x}{11 - 11x}$

See Solution

Problem 33277

Evaluate these limits: (a) $\lim _{x \rightarrow \infty}(-23 x^{2}-35 x^{3})=$ (b) $\lim _{x \rightarrow -\infty}(-23 x^{2}-35 x^{3})=$

See Solution

Problem 33278

Find the limit: $\lim _{x \rightarrow-\infty}\left(\frac{1-x^{3}}{x^{2}+9 x}\right)^{7}$ and simplify your answer to $\infty$ or $-\infty$ .

See Solution

Problem 33279

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{2+9 x}{10-7 x}=$ , (b) $\lim _{x \rightarrow-\infty} \frac{2+9 x}{10-7 x}=$ .

See Solution

Problem 33280

Find the limit as $x$ approaches infinity for $\frac{\sin 3 x}{8 x}$ . Simplify your answer.

See Solution

Problem 33281

Find the limit as $x$ approaches infinity: $\lim_{x \rightarrow \infty} \frac{\sqrt{2+5x^{2}}}{5+2x}$ .

See Solution

Problem 33282

Find the limit as $x$ approaches infinity for $\frac{(10-x)(2+5 x)}{(3-10 x)(11+11 x)}$ .

See Solution

Problem 33283

Find the limit of $f(x)=\frac{-5+\frac{5}{x}}{9-\frac{9}{x^{2}}}$ as $x \to \infty$ and $x \to -\infty$ .

See Solution

Problem 33284

Find the capacitor current at $t=3$ seconds in a series RC-circuit with $v=V_{s}(1-e^{-\frac{t}{RC}})$ , where $i=C \frac{d v}{d t}$ .

See Solution

Problem 33285

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{4 x^{3}-2 x^{2}-9 x}{9-9 x-10 x^{3}}$ .

See Solution

Problem 33286

Is the integral $\int \sin a x d x$ equal to $\frac{-1}{a} \cos a x + c$ ? True or False?

See Solution

Problem 33287

Find the derivative of the function: $3 e^{-2 x^{2}+7 x}$ .

See Solution

Problem 33288

Find $\frac{d f^{-1}}{d x}$ at $x=3$ for $f(x)=x^{2}-4x+3$ where $x \geq 2$ . Choices: $1/4$ , 4, $1/2$ , $0.1/3$ .

See Solution

Problem 33289

Find the limit $L=\lim _{x \rightarrow 3} \frac{9}{x}$ and the largest $\delta>0$ for $|f(x)-L|<0.7$ .

See Solution

Problem 33290

Given the cost function $C(x)=62500+400 x+x^{2}$ , find: a) Cost at $x=1850$ b) Average cost at $x=1850$ c) Marginal cost at $x=1850$ d) Production level that minimizes average cost e) Minimal average cost

See Solution

Problem 33291

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin ^{2} x}{2 x}$ . Choose one: a. 1 b. 0 c. 3

See Solution

Problem 33292

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{2 x^{3}-2 x^{2}-6 x}{4-10 x-4 x^{3}}$ ; (b) $\lim _{x \rightarrow-\infty} \frac{2 x^{3}-2 x^{2}-6 x}{4-10 x-4 x^{3}}$ .

See Solution

Problem 33293

Find the limit of $f(x)=\frac{7x^{3}+3}{x^{3}-x^{2}+x+5}$ as $x \rightarrow \infty$ and $x \rightarrow -\infty$ .

See Solution

Problem 33294

Find $\frac{d f^{-1}}{d x}$ at $x=3$ for $f(x)=x^{2}-4x+3$ , where $x \geq 2$ . Choices: 4, $1/4$ , $1/2$ , $1/3$ .

See Solution

Problem 33295

Find the slope of the tangent line for $y=3^{\cos x}$ at $x=\frac{\pi}{2}$ .

See Solution

Problem 33296

Find $\frac{d y}{d x}$ for $\ln (2 x + 4 y) = \sin y$ . Choose the correct expression for $\frac{d y}{d x}$ .

See Solution

Problem 33297

Find the limits: (a) $\lim _{x \rightarrow \infty} \frac{(2-x)(5+5 x)}{(3-6 x)(3+3 x)}$ and (b) $\lim _{x \rightarrow-\infty} \frac{(2-x)(5+5 x)}{(3-6 x)(3+3 x)}$ .

See Solution

Problem 33298

Find the limits of $f(x)=\frac{4 x^{7}+2 x^{6}+5}{8 x^{8}}$ as $x \rightarrow \infty$ and $x \rightarrow -\infty$ .

See Solution

Problem 33299

Cut squares of side length $x$ from a 28 in cardboard to make an open-top box. Find $V(x)$ and max volume.
$x=$
Maximum Volume $=$

See Solution

Problem 33300

Transform the PDE $k(u_{xx}+u_{yy})=u_{t}$ using $u(x,t)=X(x)Y(y)T(t)$ into ODEs. True or False?

See Solution

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Calculus

Problem 33201

Find the second derivative if dydx=ex−3x2\frac{d y}{d x}=e^{x}-3 x^{2}dxdy​=ex−3x2. What is d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​?

Problem 33202

Find the derivative of y=9x34 y=\frac{9}{\sqrt[4]{x^{3}}} y=4x3​9​.

Problem 33203

Find the average slope of the function f(x)=2x3−5xf(x)=2 x^{3}-5 xf(x)=2x3−5x on [−4,4][-4,4][−4,4] and the two values of ccc where f′(c)f^{\prime}(c)f′(c) equals it.

Problem 33204

Find y′y'y′ if y=23x+ln⁡xy=2^{3x+\ln x}y=23x+lnx. Choose the correct derivative from the options provided.

Problem 33205

Evaluate the integral ∫1e6ln⁡x+4xdx\int_{1}^{e} \frac{6 \ln x+4}{x} d x∫1e​x6lnx+4​dx. Choose from: 8, None, 10, 5, 7.

Problem 33206

Find the average slope of f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ on [2,9][2,9][2,9] and the value of ccc in (2,9)(2,9)(2,9) where f′(c)f^{\prime}(c)f′(c) equals this slope.

Problem 33207

Evaluate the integral from 1 to e of 2ln⁡x+3x dx\frac{2 \ln x + 3}{x} \, dxx2lnx+3​dx: 3◯3 \bigcirc3◯, 4◯4 \bigcirc4◯, None ◯\bigcirc◯, 5◯5 \bigcirc5◯.

Problem 33208

Find dydx\frac{d y}{d x}dxdy​ for the equation ln⁡(2x+4y)=sin⁡y\ln (2 x+4 y)=\sin yln(2x+4y)=siny.

Problem 33209

Find dydx\frac{d y}{d x}dxdy​ for the equation ln⁡(3x+4y)=sin⁡y\ln (3 x+4 y)=\sin yln(3x+4y)=siny. Choose the correct option.

Problem 33210

Evaluate the integral ∫sec⁡(4x)tan⁡(4x)2+sec⁡(4x)dx\int \frac{\sec (4 x) \tan (4 x)}{2+\sec (4 x)} d x∫2+sec(4x)sec(4x)tan(4x)​dx. What is the result?

Problem 33211

Find the derivative of y=3x3−2xy=3 x^{3}-2 xy=3x3−2x and locate the stationary points. (a) dydx=\frac{d y}{d x}=dxdy​= (b) The stationary point(s) is/are at

Problem 33212

Evaluate the integral ∫sec⁡(3x)tan⁡(3x)2+sec⁡(3x)dx\int \frac{\sec (3 x) \tan (3 x)}{2+\sec (3 x)} d x∫2+sec(3x)sec(3x)tan(3x)​dx. What is the result?

Problem 33213

Find the average slope of f(x)=1−3x2f(x)=1-3x^{2}f(x)=1−3x2 on [−1,6][-1,6][−1,6] and the unique ccc where f′(c)f'(c)f′(c) equals this slope.

Problem 33214

Find the derivative y′y^{\prime}y′ if y=53x+ln⁡xy=5^{3x+\ln x}y=53x+lnx.

Problem 33215

Find the slope of the tangent line to the curve y=xe4x−e7xy=x e^{4 x}-e^{7 x}y=xe4x−e7x at x=0x=0x=0. Choices: 60, None, −7-7−7, 777.

Problem 33216

Find the marginal revenue RRR from renting xxx apartments, given R=2x(900+34x−x2)R=2 x(900+34 x-x^{2})R=2x(900+34x−x2), when x=14x=14x=14.

Problem 33217

Find the average slope of f(x)=6x+6f(x)=6 \sqrt{x}+6f(x)=6x​+6 on [2,9][2,9][2,9] and the value of ccc in (2,9)(2,9)(2,9) where f′(c)f'(c)f′(c) equals this slope.

Problem 33218

Find the derivative of y=32x+ln⁡xy=3^{2 x+\ln x}y=32x+lnx. What is y′y^{\prime}y′?

Problem 33219

Evaluate the integral ∫sec⁡(3x)tan⁡(3x)2+sec⁡(3x)dx\int \frac{\sec (3 x) \tan (3 x)}{2+\sec (3 x)} d x∫2+sec(3x)sec(3x)tan(3x)​dx. What is the result?

Problem 33220

Find the slope of the tangent line to the curve y=xe5x−e3xy=x e^{5 x}-e^{3 x}y=xe5x−e3x at x=0x=0x=0. Choose: −2-2−2, −3-3−3, None, 2, 3.

Problem 33221

Find dydx\frac{d y}{d x}dxdy​ for ln⁡(3x+4y)=sin⁡y\ln (3 x+4 y)=\sin yln(3x+4y)=siny. Choose the correct expression for dydx\frac{d y}{d x}dxdy​.

Problem 33222

Find the nature of yyy at x=0x=0x=0 given dydx=sin⁡(x)−x\frac{d y}{d x}=\sin (x)-xdxdy​=sin(x)−x and dydx=d2ydx2=0\frac{d y}{d x}=\frac{d^{2} y}{d x^{2}}=0dxdy​=dx2d2y​=0.

Problem 33223

Find dydx\frac{d y}{d x}dxdy​ for ln⁡(3x+4y)=sin⁡y\ln (3 x+4 y)=\sin yln(3x+4y)=siny. What is the derivative?

Problem 33224

Find the slope of the tangent line to the curve y=xe4x−e7xy=x e^{4 x}-e^{7 x}y=xe4x−e7x at x=0x=0x=0. Options: -6, 60, 0, -7, 7.

Problem 33225

Find the second derivative of y=f(x)y=f(x)y=f(x) where dydx=ex−15x4\frac{d y}{d x}=e^{x}-15 x^{4}dxdy​=ex−15x4. Determine the nature of yyy at x=−0.45x=-0.45x=−0.45.

Problem 33226

Find the derivative of the function f(x)=4x(x2+1)f(x) = 4x(x^2 + 1)f(x)=4x(x2+1).

Problem 33227

Find the average slope of f(x)=2x3−6x2−90x+6f(x)=2 x^{3}-6 x^{2}-90 x+6f(x)=2x3−6x2−90x+6 on [−5,8][-5,8][−5,8] and the two values of ccc where f′(c)f'(c)f′(c) equals this slope.

Problem 33228

Find the slope of the tangent line to the curve y=xe5x−e3xy=x e^{5 x}-e^{3 x}y=xe5x−e3x at x=0x=0x=0.

Problem 33229

Find dydx\frac{d y}{d x}dxdy​ given that ln⁡(2x+5y)=sin⁡y\ln(2x + 5y) = \sin yln(2x+5y)=siny.

Problem 33230

Find the average rate of change of f(x)=x3−5x+7f(x)=x^{3}-5 x+7f(x)=x3−5x+7 from (a) -3 to -2 and (b) 2 to 4.

Problem 33231

Find the derivative y′y^{\prime}y′ if y=23x+ln⁡xy=2^{3 x+\ln x}y=23x+lnx. Choose from the options provided.

Problem 33232

Evaluate the integral ∫1e6ln⁡x+4x dx\int_{1}^{e} \frac{6 \ln x + 4}{x} \, dx∫1e​x6lnx+4​dx. Choose: 10, 7, None, 5, 8.

Problem 33233

Find the average rate of change of g(x)=4x2−2g(x)=4x^{2}-2g(x)=4x2−2 from -6 to 3, and the secant line through (−6,g(−6))(-6, g(-6))(−6,g(−6)) and (3,g(3))(3, g(3))(3,g(3)).

Problem 33234

Find the derivative of the function e5xe^{5x}e5x.

Problem 33235

Evaluate the integral ∫1e2ln⁡x+3xdx\int_{1}^{e} \frac{2 \ln x+3}{x} d x∫1e​x2lnx+3​dx. Choose: 4, 5, 2, None, 3.

Problem 33236

Evaluate the integral ∫sec⁡(3x)tan⁡(3x)2+sec⁡(3x)dx\int \frac{\sec (3 x) \tan (3 x)}{2+\sec (3 x)} d x∫2+sec(3x)sec(3x)tan(3x)​dx.

Problem 33237

Find the derivative of the function f(x)=x2(1−2x3)f(x) = x^{2}\left(1-\frac{2}{x^{3}}\right)f(x)=x2(1−x32​).

Problem 33238

The average cost per hour for xxx lawn mowers is Cˉ(x)=0.2x2+21x−273+2800x\bar{C}(x)=0.2 x^{2}+21 x-273+\frac{2800}{x}Cˉ(x)=0.2x2+21x−273+x2800​. Find xxx to minimize cost and the minimum cost.

Problem 33239

Find where the function f f f is increasing based on its behavior from (−2,−4) (-2,-4) (−2,−4) to (2,4) (2,4) (2,4).

Problem 33240

Find the second derivative if $\frac{d y}{d x}=e^{x}-3 x^{2}$ . What is $\frac{d^{2} y}{d x^{2}}$ ?

Find the derivative of $y=\frac{9}{\sqrt[4]{x^{3}}}$ .

Find the average slope of the function $f(x)=2 x^{3}-5 x$ on $[-4,4]$ and the two values of $c$ where $f^{\prime}(c)$ equals it.

Find $y'$ if $y=2^{3x+\ln x}$ . Choose the correct derivative from the options provided.

Evaluate the integral $\int_{1}^{e} \frac{6 \ln x+4}{x} d x$ . Choose from: 8, None, 10, 5, 7.

Find the average slope of $f(x)=\frac{1}{x}$ on $[2,9]$ and the value of $c$ in $(2,9)$ where $f^{\prime}(c)$ equals this slope.

Evaluate the integral from 1 to e of $\frac{2 \ln x + 3}{x} \, dx$ : $3 \bigcirc$ , $4 \bigcirc$ , None $\bigcirc$ , $5 \bigcirc$ .

Find $\frac{d y}{d x}$ for the equation $\ln (2 x+4 y)=\sin y$ .

Find $\frac{d y}{d x}$ for the equation $\ln (3 x+4 y)=\sin y$ . Choose the correct option.

Evaluate the integral $\int \frac{\sec (4 x) \tan (4 x)}{2+\sec (4 x)} d x$ . What is the result?

Find the derivative of $y=3 x^{3}-2 x$ and locate the stationary points.
(a) $\frac{d y}{d x}=$
(b) The stationary point(s) is/are at

Evaluate the integral $\int \frac{\sec (3 x) \tan (3 x)}{2+\sec (3 x)} d x$ . What is the result?

Find the average slope of $f(x)=1-3x^{2}$ on $[-1,6]$ and the unique $c$ where $f'(c)$ equals this slope.

Find the derivative $y^{\prime}$ if $y=5^{3x+\ln x}$ .

Find the slope of the tangent line to the curve $y=x e^{4 x}-e^{7 x}$ at $x=0$ . Choices: 60, None, $-7$ , $7$ .

Find the marginal revenue $R$ from renting $x$ apartments, given $R=2 x(900+34 x-x^{2})$ , when $x=14$ .

Find the average slope of $f(x)=6 \sqrt{x}+6$ on $[2,9]$ and the value of $c$ in $(2,9)$ where $f'(c)$ equals this slope.

Find the derivative of $y=3^{2 x+\ln x}$ . What is $y^{\prime}$ ?

Evaluate the integral $\int \frac{\sec (3 x) \tan (3 x)}{2+\sec (3 x)} d x$ . What is the result?

Find the slope of the tangent line to the curve $y=x e^{5 x}-e^{3 x}$ at $x=0$ . Choose: $-2$ , $-3$ , None, 2, 3.

Find $\frac{d y}{d x}$ for $\ln (3 x+4 y)=\sin y$ . Choose the correct expression for $\frac{d y}{d x}$ .

Find the nature of $y$ at $x=0$ given $\frac{d y}{d x}=\sin (x)-x$ and $\frac{d y}{d x}=\frac{d^{2} y}{d x^{2}}=0$ .

Find $\frac{d y}{d x}$ for $\ln (3 x+4 y)=\sin y$ . What is the derivative?

Find the slope of the tangent line to the curve $y=x e^{4 x}-e^{7 x}$ at $x=0$ . Options: -6, 60, 0, -7, 7.

Find the second derivative of $y=f(x)$ where $\frac{d y}{d x}=e^{x}-15 x^{4}$ . Determine the nature of $y$ at $x=-0.45$ .

Find the derivative of the function $f(x) = 4x(x^2 + 1)$ .

Find the average slope of $f(x)=2 x^{3}-6 x^{2}-90 x+6$ on $[-5,8]$ and the two values of $c$ where $f'(c)$ equals this slope.

Find the slope of the tangent line to the curve $y=x e^{5 x}-e^{3 x}$ at $x=0$ .

Find $\frac{d y}{d x}$ given that $\ln(2x + 5y) = \sin y$ .

Find the average rate of change of $f(x)=x^{3}-5 x+7$ from (a) -3 to -2 and (b) 2 to 4.

Find the derivative $y^{\prime}$ if $y=2^{3 x+\ln x}$ . Choose from the options provided.

Evaluate the integral $\int_{1}^{e} \frac{6 \ln x + 4}{x} \, dx$ . Choose: 10, 7, None, 5, 8.

Find the average rate of change of $g(x)=4x^{2}-2$ from -6 to 3, and the secant line through $(-6, g(-6))$ and $(3, g(3))$ .

Find the derivative of the function $e^{5x}$ .

Evaluate the integral $\int_{1}^{e} \frac{2 \ln x+3}{x} d x$ . Choose: 4, 5, 2, None, 3.

Evaluate the integral $\int \frac{\sec (3 x) \tan (3 x)}{2+\sec (3 x)} d x$ .

Find the derivative of the function $f(x) = x^{2}\left(1-\frac{2}{x^{3}}\right)$ .

The average cost per hour for $x$ lawn mowers is $\bar{C}(x)=0.2 x^{2}+21 x-273+\frac{2800}{x}$ . Find $x$ to minimize cost and the minimum cost.

Find where the function $f$ is increasing based on its behavior from $(-2,-4)$ to $(2,4)$ .

Find the limit and check if the function is continuous at $y=1$ :
$\lim _{y \rightarrow 1} \sec \left(y \sec ^{2} y-\tan ^{2} y-1\right)$

Find the derivative of $\frac{1}{2} \ln (x)$ and match it with the correct option: $\frac{2}{x}$ , $\frac{\ln (x^{2})}{4}$ , $\ln (\sqrt{x})$ , $\frac{1}{2 x}$ .

Find the slope of the tangent line to the curve $y=x^{2} e^{5 x-3}$ at $x=1$ .

Find the derivative $f'(0)$ for the function $f(x)=\ln (\sec (x)+\tan (x))$ .

Find the derivative of $y=\frac{1}{7 x^{2}}+\frac{1}{7 x}$ .

Find the derivative $\frac{d y}{d x}$ for the equation $\tan (2 y)=\ln (x y)+2 x$ .

Evaluate these limits: (a) $\lim _{x \rightarrow \infty}(-22 x^{2}-30 x^{3})=$ ; (b) $\lim _{x \rightarrow -\infty}(-22 x^{2}-30 x^{3})=$ .

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{4 x^{3}-8 x^{2}-3 x}{11-10 x-11 x^{3}}$ .

Find the slope of the tangent line to the curve $y=x^{2} e^{2 x-3}$ at $x=1$ .

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{5 x^{2}-4 x+10}{7 x+8}$ .

Evaluate the integral $\int \frac{9}{x(3+\ln x)} d x$ .

Calculate the integral from 1 to 16 of $\frac{16}{x \sqrt{\ln x}} \, dx$ .

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{4+10 x}{3-11 x}$ .

Evaluate the integral $\int \frac{2}{x(4+\ln x)} d x$ . Choose the correct answer from the options given.

Evaluate the integral $\int_{0}^{1} x^{2} e^{4 x^{3}} d x$ . Choose the correct answer from the options provided.

Find the absolute maximum and minimum of the function $y=f(x)$ from the graph, and identify local maxima/minima.

Find the derivative $y'$ , where $y=\ln(\sec(e^{2x}))$ . Options include $e^{2x} \sec(e^{2x}) \tan(e^{2x})$ , etc.

Find $\frac{d y}{d x}$ for $\tan(2y) = \ln(xy) + 2x$ . Choose the correct derivative expression.

Find $\frac{d y}{d x}$ for the equation $\tan(3y) = \ln(xy) + 2x$ . Choose the correct derivative expression.

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{4 x+4}{2 x^{2}-3 x+3}$ .

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{10+2 x^{2}}}{8+4 x}=$ ; (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{10+2 x^{2}}}{8+4 x}=$

Find the limit and check continuity at the point:
$\lim _{x \rightarrow 0} \sec \left[\cos x+5 \pi \tan \left(\frac{\pi}{4 \sec x}\right)-1\right]$
Is it equal to $\square$ or does it not exist?

Evaluate the integral $\int_{0}^{1} x^{2} e^{2 x^{3}} d x$ . Choose the correct answer from the options provided.

Calculate the integral $\int_{2}^{16} \frac{d x}{x \sqrt{\ln x}}$ .

Find the derivative of $y=\ln \left(\sec \left(e^{2 x}\right)\right)$ . What is $y^{\prime}$ ?

Find the absolute maximum and minimum of the function $y=f(x)$ from the graph. Identify local maxima and minima.

Graph $f(x)=x^3-2x+4$ on $(-2,2)$ , find local maxima/minima, and determine where it is increasing/decreasing.

Find the derivative of the inverse function $f^{-1}(x)$ at $x=6$ for $f(x)=x^{3}-2$ .

Evaluate the integral $\int \frac{\sec (3 x) \tan (3 x)}{2+\sec (3 x)} d x$ . What is the result?

Find the derivative of $y$ and set it equal to the derivative of $4x(x^2+1)$ .

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{11 x+11}{10 x^{2}-2 x+10}$ .

Find the limit as $x$ approaches infinity: $\lim_{x \rightarrow \infty} \frac{\sqrt{3+3x^{2}}}{9+2x}$

Find the integral: $\int \frac{4 \sin y}{1 - \cos y} \, dy$ .

Evaluate these limits: (a) $\lim _{x \rightarrow \infty} \frac{4}{e^{x}+2}=$ (b) $\lim _{x \rightarrow-\infty} \frac{4}{e^{x}+2}=$