Calculus

Problem 31301

Given $y=2x-\sin(kx)$ and $x=2kt$ , find: (a) $\frac{dy}{dx}=$ (in terms of $k$ and $x$ ) (b) $\frac{dx}{dt}=$ (in terms of $k$ and/or $t$ ) (c) If $k=1$ , at $t=\frac{\pi}{2}$ , find $\frac{dy}{dt}=$ (to 1 decimal place)

See Solution

Problem 31302

Find the current $i$ at $t=10$ s, given $q=10-10 e^{-0.1 t}$ and $i=\frac{d q}{d t}$ .

See Solution

Problem 31303

Given $y=2 x-\sin (k x)$ and $x=2 k t$ , find:
(a) $\frac{d y}{d x}=2-k \cos (k x)$ , answer in terms of $k$ and $x$ .
(b) $\frac{d x}{d t}=$ , answer in terms of $k$ and/or $t$ .

See Solution

Problem 31304

Given $y=4x+\cos(kx)$ and $x=6kt$ , find: (a) $\frac{dy}{dx}$ in terms of $k$ and $x$ ; (b) $\frac{dx}{dt}$ in terms of $k$ and/or $t$ ; (c) For $k=2$ , at $t=\pi$ , find $\frac{dy}{dt}$ .

See Solution

Problem 31305

Given $y=2x-\sin(kx)$ and $x=2kt$ , find: (a) $\frac{dy}{dx}=2-k\cos(kx)$ (Answer in terms of $k$ and $x$ ) (b) $\frac{dx}{dt}=2k$ (Answer in terms of $k$ and/or $t$ ) (c) If $k=1$ , at $t=\frac{\pi}{2}$ , find $\frac{dy}{dt}$ . (Answer in 1 decimal place or integer)

See Solution

Problem 31306

Find the slopes of secant lines between given points and the tangent line at $(-1, f(-1))$ for $f(x)=x^{2}+x$ .

See Solution

Problem 31307

Find the current $i$ (in A) at $t=10 \mathrm{~s}$ for the charge $q=10-10 e^{-0.1 t}$ , where $i=\frac{d q}{d t}$ .

See Solution

Problem 31308

Find the instantaneous velocity function $v=f^{\prime}(x)$ for $y=f(x)=9x^{2}-7x$ and evaluate it at $x=1,3,5$ seconds.

See Solution

Problem 31309

An object moves along the $y$ axis with $y=f(x)=x^{2}+x$ . Find: (A) average velocity from $x=3$ to $6$ , (B) from $3$ to $3+h$ , (C) instantaneous velocity at $x=3$ .

See Solution

Problem 31310

Find the derivative of $f(x)=9x+9\sqrt{x}$ using the definition. State the domains of $f(x)$ and $f'(x)$ .

See Solution

Problem 31311

Find the gradient and tangent line equation for $y=e^{\sin x+\cos x}$ at $x=\frac{\pi}{2}$ .

See Solution

Problem 31312

Describe the solid of revolution represented by the integral $\pi \int_{0}^{1}\left(x^{8}-x^{10}\right) d x$ .

See Solution

Problem 31313

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

See Solution

Problem 31314

Find the average revenue change from selling 1,000 to 1,050 car seats using $R(x)=64x-0.020x^2$ .

See Solution

Problem 31315

Evaluate the integral: $\int (x^{3} \ln x + x) \, dx$ .

See Solution

Problem 31316

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

See Solution

Problem 31317

Calculate the integral $\int_{0}^{1} \ln (x+1) \, dx$ .

See Solution

Problem 31318

Set up an integral for the volume of the solid formed by rotating the region bounded by $x=\sqrt{7-y}$ , $y=0$ , $x=0$ about the $y$ -axis: $\int_{0}^{7}(\sqrt{7-y})^2 d y$

See Solution

Problem 31319

Find the average change in revenue from selling 1,000 to 1,050 car seats with $R(x)=56x-0.020x^{2}$ . Also, find $R^{\prime}(x)$ and the revenue at $x=1000$ .

See Solution

Problem 31320

Find the volume $V$ of the solid formed by rotating the area between $x=2\sqrt{7y}$ , $x=0$ , $y=3$ around the $y$ -axis. $V=$

See Solution

Problem 31321

Given $f(x)=\frac{5}{x}$ , find $A$ in $\frac{f(x+h)-f(x)}{h} = \frac{A}{x(x+h)}$ . Then calculate $f^{\prime}(1)$ , $f^{\prime}(2)$ , $f^{\prime}(3)$ .

See Solution

Problem 31322

Find the derivative $P^{\prime}(t)$ of $P(t)=190+15t-3t^{2}$ and calculate $P(2)$ and $P^{\prime}(2)$ .

See Solution

Problem 31323

Find the marginal revenue function for $R(x)=x(28-0.05 x)$ . What is $R^{\prime}(x)=\square$ ?

See Solution

Problem 31324

Find the marginal cost function for $C(x)=177+1.7x$ . What is $C'(x)=\square$ ?

See Solution

Problem 31325

Find the current $i$ for the capacitor with charge $q=4000 e^{-62.5 t} \mu \mathrm{C}$ using $i=\frac{d q}{d t}$ .

See Solution

Problem 31326

Find average velocities for $s(t)=-16 t^{2}+100 t$ over intervals $[2,3]$ , $[2.9,3]$ , $[2.99,3]$ , $[2.999,3]$ , and $[2.9999,3]$ . Predict instantaneous velocity at $t=3$ .

See Solution

Problem 31327

Find the marginal profit function given $C(x)=259+0.5 x$ and $R(x)=9 x-0.03 x^{2}$ . What is $P^{\prime}(x)=\square$ ?

See Solution

Problem 31328

Find the rate of change of price $p=15-\ln \left(x^{4}\right)$ with respect to the number of jerseys sold $x$ .

See Solution

Problem 31329

Find the integral of the inverse sine function: $\int \sin^{-1} x \, dx$ .

See Solution

Problem 31330

Find the derivative of $f(x)=(2x-3)(4x-5)$ using the product or quotient rule. What is $f'(x)$ ?

See Solution

Problem 31331

Find the integral of the inverse sine function: $\int \sin^{-1} x \, dx$ .

See Solution

Problem 31332

Find the cost of producing the 31st food processor using $C(x)=2400+60x-0.3x^2$ . Exact cost: \ $\square$ .

See Solution

Problem 31333

Find the antiderivative $F(t)$ of $f(t)=10 \sec ^{2}(t)-2 t^{3}$ with $F(0)=0$ . Calculate $F(0.6)$ .

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Problem 31334

Identify eigenfunctions of the operator $d / d x$ from the list and determine their eigenvalues: a. $x^{2}$ , b. $e^{-\alpha x}$ , c. $e^{-\alpha \times 2}$ , d. $\cos (n x)+i \sin (n x)$ .

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Problem 31335

Calculate the integral of arcsin(ax) with respect to x: $\int \arcsin(ax) \, dx$

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Problem 31336

Find the cost of producing the 21st food processor using $C(x)=1600+60x-0.7x^2$ . Exact cost: \ $\square$ .

See Solution

Problem 31337

Find the cost of producing the 51st food processor using $C(x)=1500+90x-0.1x^{2}$ and marginal cost.

See Solution

Problem 31338

Find the cost of producing the 51st food processor using $C(x)=1500+90x-0.1x^2$ . Exact and approximate costs.

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Problem 31339

Find the cost of producing the 31st food processor using $C(x)=1700+30x-0.4x^{2}$ . Exact and marginal costs needed.

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Problem 31340

Find the integral of the inverse sine function: $\int \sin^{-1}(x) \, dx$ .

See Solution

Problem 31341

Find the voltage across a $2-\mathrm{H}$ inductor with current $i=e^{3t}$ at $t=1$ seconds. Use $v=L \frac{d i}{d t}$ .

See Solution

Problem 31342

Find the function $f(t)$ given that $f''(t) = 2 e^{t} + 3 \sin(t)$ , with conditions $f(0) = 6$ and $f(\pi) = 9$ .

See Solution

Problem 31343

Evaluate the limits:
1. $\lim _{x \rightarrow 3^{-}} \frac{|x-3|-0}{x-3}$
2. $\lim _{x \rightarrow 3^{+}} \frac{|x-3|-0}{x-3}$ .
Is $f(x)$ differentiable at 3?

See Solution

Problem 31344

Find how fast the current $I$ is changing when $R=6.25 \Omega$ and $R$ changes at $0.250 \Omega/\mathrm{min}$ .

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Problem 31345

Calculate the integral: $\int 4 x \sec ^{2} 2 x \, dx$

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Problem 31346

Find $f(3)$ for the function with $f^{\prime \prime}(x)=5x+6\sin(x)$ , given $f(0)=2$ and $f^{\prime}(0)=3$ .

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Problem 31347

Find the derivative $f^{\prime}(x)$ of $f(x)=x^{2}+9x-8$ and calculate $f^{\prime}(1)$ , $f^{\prime}(2)$ , $f^{\prime}(3)$ .

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Problem 31348

Normalize the wavefunction $\psi=\cos (\pi x / 2 a)$ for a particle in the region $-\mathrm{a}<\mathrm{x}<\mathrm{a}$ .

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Problem 31349

Does the limit $\lim_{x \rightarrow 3} f(x)$ exist? If yes, find it; if no, explain why using the graph details.

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Problem 31350

Find the derivative $r^{\prime}(x)$ for $r(x)=2+3x^{2}$ and calculate $r^{\prime}(1)$ , $r^{\prime}(2)$ , $r^{\prime}(3)$ .

See Solution

Problem 31351

Find $f^{\prime}(x)$ from $f^{\prime \prime}(x)=8 \cos (x)$ using constant $\mathrm{C}$ . Then find $f(x)$ using constant $\mathrm{D}$ .

See Solution

Problem 31352

Find the volume of the solid formed by revolving the area between $y=4x-x^2$ and $y=2x$ around the $y$ -axis. Volume $=\square$ (use $\pi$ ).

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Problem 31353

Find the volume of the solid formed by revolving the area between $y=4x-x^{2}$ and $y=2x$ about the $y$ -axis and $x=2$ .

See Solution

Problem 31354

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

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Problem 31355

Find the values for (A) $f(-1)$ , (B) $f(-2+h)$ , and (C) the limit as $h \to 0$ for $f(x)=8-x^{2}$ .

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Problem 31356

Find the volume of the solid formed by revolving the area between $y=12x-x^{2}$ and $y=x$ around the $y$ -axis. $\text{Volume} = \square$

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Problem 31357

Calculate the integral: $\int\left(x^{2}-5 x\right) e^{3} d x$

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Problem 31358

Calculate the integral: $\int\left(x^{2}+x+1\right) e^{x} d x$ .

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Problem 31359

Find $f^{\prime}(x)$ for $f(x)=2x^{2}+x-1$ , then calculate $f^{\prime}(1)$ , $f^{\prime}(2)$ , and $f^{\prime}(8)$ .

See Solution

Problem 31360

Evaluate the integral: $\int x^{2} e^{4 x} d x$

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Problem 31361

Find the derivative $s^{\prime}(x)$ of $s(x)=6x-5$ and calculate $s^{\prime}(1)$ , $s^{\prime}(2)$ , and $s^{\prime}(3)$ .

See Solution

Problem 31362

Find the sales function $S(t)=3 \sqrt{t+1}$ , then compute $S^{\prime}(t)$ , $S(15)$ , $S^{\prime}(15)$ , and estimate $S(16)$ .

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Problem 31363

Find the volume of the solid formed by revolving the area in the first quadrant bounded by $y=x^{2}$ , $x=1$ , and the $x$ -axis around $x=-1$ . Use the washer method to set up the integral for volume: $v=\int_{\square} \square$

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Problem 31364

Calculate the integral: $\int\left(x^{2}+x+1\right) e^{2} dx$

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Problem 31365

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

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Problem 31366

Indicate T or F for each statement below (all must be correct for credit):
1. If $\lim _{x \rightarrow 1} f(x)=0$ and $\lim _{x \rightarrow 1} g(x)=4$ , then $\lim _{x \rightarrow 1}[f(x) / g(x)]$ does not exist.
2. If $f(x)$ is continuous at $a$ , then $f(x)$ is differentiable at $a$ .
3. If $\lim _{x \rightarrow 1} f(x)=4$ and $\lim _{x \rightarrow 1} g(x)=0$ , then $\lim _{x \rightarrow 1}[f(x) / g(x)]$ does not exist.
4. $\lim _{x \rightarrow 2} \frac{x^{2}+4 x-11}{x^{2}+4 x-10}=\frac{\lim _{x \rightarrow 2} x^{2}+4 x-11}{\lim _{x \rightarrow 2} x^{2}+4 x-10}$ .
5. $\lim _{x \rightarrow 2} \frac{x^{2}+4 x-12}{x^{2}+4 x-12}=\frac{\lim _{x \rightarrow 2} x^{2}+4 x-12}{\lim _{x \rightarrow 2} x^{2}+4 x-12}$ .

See Solution

Problem 31367

Calculate the integral: $\int x^{4} \cos x \, dx$

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Problem 31368

Calculate the integral: $\int x^{2} \cos(a x) \, dx$

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Problem 31369

Find the integral: $\int x^{2} e^{6 x} d x$

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Problem 31370

Find the limit: $\lim _{x \rightarrow 2^{+}} \frac{1}{|2-x|}$ .

See Solution

Problem 31371

Find the tangent line equation for $y=2 x^{2}-3 x+5$ at $x=4$ .

See Solution

Problem 31372

What does $C^{\prime}(500)=80$ mean for the cost function $C$ related to shredding 500 pounds of documents?

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Problem 31373

Find the volume of the solid formed by revolving the area under $y=x^2$ from $x=0$ to $x=1$ around $x=-1$ using the washer method.

See Solution

Problem 31374

Find the volumes of solids formed by revolving the region between $y=2\sqrt{x}$ and $y=\frac{x^2}{4}$ around a) the $x$ -axis and b) the $y$ -axis.

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Problem 31375

Find the limit as $x$ approaches 2 from the left for the expression $\frac{x}{x^{2}-4}$ .

See Solution

Problem 31376

Find the limit: $\lim _{x \rightarrow 3} \frac{x}{x-3}$ .

See Solution

Problem 31377

Find the derivative of $f(x)=-8x-6$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 31378

Find the limit as $x$ approaches 3 from the right for the expression $\frac{x}{x-3}$ .

See Solution

Problem 31379

Evaluate the integral: $\int_{1}^{4} \sec^{-1} \sqrt{x} \, dx$

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Problem 31380

Find the derivative of $f(x)=\frac{1}{\sqrt{x^{15}}}$ using the power rule. $f^{\prime}(x)=$

See Solution

Problem 31381

Find the derivative of $f(x)=x^{\frac{12}{13}}$ using the power rule: $f^{\prime}(x)=\square$

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Problem 31382

Find the volume using the shell method for the region revolved around: a. x-axis, b. y=1, c. y=8/5, d. y=-2/5.

See Solution

Problem 31383

Find the derivative of $f(x)=x^{10}$ using the power rule. What is $f^{\prime}(x)=?$

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Problem 31384

Find the derivative of $f(x)=\frac{1}{\sqrt[9]{x}}$ using the power rule. $f^{\prime}(x)=\square$

See Solution

Problem 31385

Find volumes using the shell method for the region defined by $x = 12(y^2 - y^3)$ revolving around:
a. $x$ -axis, volume = $\frac{6 \pi}{5}$ ;
b. $y=1$ ;
c. $y=\frac{8}{5}$ ;
d. $y=-\frac{2}{5}$ .
Type exact answers in terms of $\pi$ .

See Solution

Problem 31386

Find the derivative of $f(x)=\sqrt[13]{x}$ using the power rule. What is $f^{\prime}(x)=?$

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Problem 31387

Find the derivative of the constant function $f(x)=5^{4}$ . Simplify your answer: $f^{\prime}(x)=\square$ .

See Solution

Problem 31388

Find the derivative of $f(x)=x^{8}$ at $x=\frac{1}{2}$ . What is $f^{\prime}\left(\frac{1}{2}\right)=\square$ ?

See Solution

Problem 31389

Calculate the integral: $\int x \sin ^{-1}\left(\frac{1}{x}\right) d x$

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Problem 31390

Find the derivative of $f(x)=\frac{1}{x}$ at $x=\frac{7}{10}$ . What is $f^{\prime}\left(\frac{7}{10}\right)$ ?

See Solution

Problem 31391

Find the slope of the curve $y=x^{3}$ at $x=4$ . What is $f^{\prime}(4)=\square$ ?

See Solution

Problem 31392

Find the tangent line to $f(x)=\frac{2 x-3}{x+3}$ at $x=-2$ : $y=9 x+\square$ .

See Solution

Problem 31393

Find $f(2)$ and $f^{\prime}(2)$ for the function $f(x)=\frac{1}{x^{4}}$ . What is $f(2)=\square$ ?

See Solution

Problem 31394

Find the tangent line equation to $y=f(x)$ at $x=-\frac{1}{2}$ for $f(x)=x^{5}$ . What is it?

See Solution

Problem 31395

Find the derivative of $f(x)=\sqrt{x}$ at $x=\frac{1}{49}$ . What is $f^{\prime}\left(\frac{1}{49}\right)$ ?

See Solution

Problem 31396

Differentiate $x^{9/10}$ with respect to $x$ . What is $\frac{d}{d x}\left(x^{9 / 10}\right)=\square$ ?

See Solution

Problem 31397

Find the derivative of the function $f(x)=2 x^{3}$ using the three-step method. What is $f^{\prime}(x)=\square$ ?

See Solution

Problem 31398

Find the tangent line equation for $y=f(x)$ at $x=\frac{1}{225}$ where $f(x)=\sqrt{x}$ .

See Solution

Problem 31399

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-4 x^{2}-x-4$ and simplify. $\frac{f(x+h)-f(x)}{h}=\square$

See Solution

Problem 31400

Find the point on $y=\sqrt{x}$ where the tangent is parallel to $y=\frac{x}{22}$ . Answer as an ordered pair.

See Solution

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Calculus

Problem 31301

Problem 31302

Find the current iii at t=10t=10t=10 s, given q=10−10e−0.1tq=10-10 e^{-0.1 t}q=10−10e−0.1t and i=dqdti=\frac{d q}{d t}i=dtdq​.

Problem 31303

Given y=2x−sin⁡(kx)y=2 x-\sin (k x)y=2x−sin(kx) and x=2ktx=2 k tx=2kt, find:(a) dydx=2−kcos⁡(kx)\frac{d y}{d x}=2-k \cos (k x)dxdy​=2−kcos(kx), answer in terms of kkk and xxx.(b) dxdt=\frac{d x}{d t}=dtdx​=, answer in terms of kkk and/or ttt.

Problem 31304

Given y=4x+cos⁡(kx)y=4x+\cos(kx)y=4x+cos(kx) and x=6ktx=6ktx=6kt, find: (a) dydx\frac{dy}{dx}dxdy​ in terms of kkk and xxx; (b) dxdt\frac{dx}{dt}dtdx​ in terms of kkk and/or ttt; (c) For k=2k=2k=2, at t=πt=\pit=π, find dydt\frac{dy}{dt}dtdy​.

Problem 31305

Problem 31306

Find the slopes of secant lines between given points and the tangent line at (−1,f(−1))(-1, f(-1))(−1,f(−1)) for f(x)=x2+xf(x)=x^{2}+xf(x)=x2+x.

Problem 31307

Find the current iii (in A) at t=10 st=10 \mathrm{~s}t=10 s for the charge q=10−10e−0.1tq=10-10 e^{-0.1 t}q=10−10e−0.1t, where i=dqdti=\frac{d q}{d t}i=dtdq​.

Problem 31308

Find the instantaneous velocity function v=f′(x)v=f^{\prime}(x)v=f′(x) for y=f(x)=9x2−7xy=f(x)=9x^{2}-7xy=f(x)=9x2−7x and evaluate it at x=1,3,5x=1,3,5x=1,3,5 seconds.

Problem 31309

An object moves along the yyy axis with y=f(x)=x2+xy=f(x)=x^{2}+xy=f(x)=x2+x. Find: (A) average velocity from x=3x=3x=3 to 666, (B) from 333 to 3+h3+h3+h, (C) instantaneous velocity at x=3x=3x=3.

Problem 31310

Find the derivative of f(x)=9x+9xf(x)=9x+9\sqrt{x}f(x)=9x+9x​ using the definition. State the domains of f(x)f(x)f(x) and f′(x)f'(x)f′(x).

Problem 31311

Find the gradient and tangent line equation for y=esin⁡x+cos⁡xy=e^{\sin x+\cos x}y=esinx+cosx at x=π2x=\frac{\pi}{2}x=2π​.

Problem 31312

Describe the solid of revolution represented by the integral π∫01(x8−x10)dx\pi \int_{0}^{1}\left(x^{8}-x^{10}\right) d xπ∫01​(x8−x10)dx.

Problem 31313

Calculate the integral: ∫x2ln⁡x dx\int x^{2} \ln x \, dx∫x2lnxdx

Problem 31314

Find the average revenue change from selling 1,000 to 1,050 car seats using R(x)=64x−0.020x2R(x)=64x-0.020x^2R(x)=64x−0.020x2.

Problem 31315

Evaluate the integral: ∫(x3ln⁡x+x) dx\int (x^{3} \ln x + x) \, dx∫(x3lnx+x)dx.

Problem 31316

Calculate the integral: ∫x2tan⁡x dx\int x^{2} \tan x \, dx∫x2tanxdx

Problem 31317

Calculate the integral ∫01ln⁡(x+1) dx\int_{0}^{1} \ln (x+1) \, dx∫01​ln(x+1)dx.

Problem 31318

Set up an integral for the volume of the solid formed by rotating the region bounded by x=7−yx=\sqrt{7-y}x=7−y​, y=0y=0y=0, x=0x=0x=0 about the yyy-axis: ∫07(7−y)2dy\int_{0}^{7}(\sqrt{7-y})^2 d y∫07​(7−y​)2dy

Problem 31319

Find the average change in revenue from selling 1,000 to 1,050 car seats with R(x)=56x−0.020x2R(x)=56x-0.020x^{2}R(x)=56x−0.020x2. Also, find R′(x)R^{\prime}(x)R′(x) and the revenue at x=1000x=1000x=1000.

Problem 31320

Find the volume VVV of the solid formed by rotating the area between x=27yx=2\sqrt{7y}x=27y​, x=0x=0x=0, y=3y=3y=3 around the yyy-axis. V=V=V=

Problem 31321

Given f(x)=5xf(x)=\frac{5}{x}f(x)=x5​, find AAA in f(x+h)−f(x)h=Ax(x+h)\frac{f(x+h)-f(x)}{h} = \frac{A}{x(x+h)}hf(x+h)−f(x)​=x(x+h)A​. Then calculate f′(1)f^{\prime}(1)f′(1), f′(2)f^{\prime}(2)f′(2), f′(3)f^{\prime}(3)f′(3).

Problem 31322

Find the derivative P′(t)P^{\prime}(t)P′(t) of P(t)=190+15t−3t2P(t)=190+15t-3t^{2}P(t)=190+15t−3t2 and calculate P(2)P(2)P(2) and P′(2)P^{\prime}(2)P′(2).

Problem 31323

Find the marginal revenue function for R(x)=x(28−0.05x)R(x)=x(28-0.05 x)R(x)=x(28−0.05x). What is R′(x)=□R^{\prime}(x)=\squareR′(x)=□?

Problem 31324

Find the marginal cost function for C(x)=177+1.7xC(x)=177+1.7xC(x)=177+1.7x. What is C′(x)=□C'(x)=\squareC′(x)=□?

Problem 31325

Find the current iii for the capacitor with charge q=4000e−62.5tμCq=4000 e^{-62.5 t} \mu \mathrm{C}q=4000e−62.5tμC using i=dqdti=\frac{d q}{d t}i=dtdq​.

Problem 31326

Find average velocities for s(t)=−16t2+100ts(t)=-16 t^{2}+100 ts(t)=−16t2+100t over intervals [2,3][2,3][2,3], [2.9,3][2.9,3][2.9,3], [2.99,3][2.99,3][2.99,3], [2.999,3][2.999,3][2.999,3], and [2.9999,3][2.9999,3][2.9999,3]. Predict instantaneous velocity at t=3t=3t=3.

Problem 31327

Find the marginal profit function given C(x)=259+0.5xC(x)=259+0.5 xC(x)=259+0.5x and R(x)=9x−0.03x2R(x)=9 x-0.03 x^{2}R(x)=9x−0.03x2. What is P′(x)=□P^{\prime}(x)=\squareP′(x)=□?

Problem 31328

Find the rate of change of price p=15−ln⁡(x4)p=15-\ln \left(x^{4}\right)p=15−ln(x4) with respect to the number of jerseys sold xxx.

Problem 31329

Find the integral of the inverse sine function: ∫sin⁡−1x dx\int \sin^{-1} x \, dx∫sin−1xdx.

Problem 31330

Find the derivative of f(x)=(2x−3)(4x−5)f(x)=(2x-3)(4x-5)f(x)=(2x−3)(4x−5) using the product or quotient rule. What is f′(x)f'(x)f′(x)?

Problem 31331

Find the integral of the inverse sine function: ∫sin⁡−1x dx\int \sin^{-1} x \, dx∫sin−1xdx.

Problem 31332

Find the cost of producing the 31st food processor using C(x)=2400+60x−0.3x2C(x)=2400+60x-0.3x^2C(x)=2400+60x−0.3x2. Exact cost: \□\square□.

Problem 31333

Find the antiderivative F(t)F(t)F(t) of f(t)=10sec⁡2(t)−2t3f(t)=10 \sec ^{2}(t)-2 t^{3}f(t)=10sec2(t)−2t3 with F(0)=0F(0)=0F(0)=0. Calculate F(0.6)F(0.6)F(0.6).

Problem 31334

Identify eigenfunctions of the operator d/dxd / d xd/dx from the list and determine their eigenvalues: a. x2x^{2}x2, b. e−αxe^{-\alpha x}e−αx, c. e−α×2e^{-\alpha \times 2}e−α×2, d. cos⁡(nx)+isin⁡(nx)\cos (n x)+i \sin (n x)cos(nx)+isin(nx).

Problem 31335

Calculate the integral of arcsin(ax) with respect to x: ∫arcsin⁡(ax) dx\int \arcsin(ax) \, dx∫arcsin(ax)dx

Problem 31336

Find the cost of producing the 21st food processor using C(x)=1600+60x−0.7x2C(x)=1600+60x-0.7x^2C(x)=1600+60x−0.7x2. Exact cost: \□\square□.

Problem 31337

Find the cost of producing the 51st food processor using C(x)=1500+90x−0.1x2C(x)=1500+90x-0.1x^{2}C(x)=1500+90x−0.1x2 and marginal cost.

Problem 31338

Find the cost of producing the 51st food processor using C(x)=1500+90x−0.1x2C(x)=1500+90x-0.1x^2C(x)=1500+90x−0.1x2. Exact and approximate costs.

Problem 31339

Find the cost of producing the 31st food processor using C(x)=1700+30x−0.4x2C(x)=1700+30x-0.4x^{2}C(x)=1700+30x−0.4x2. Exact and marginal costs needed.

Problem 31340

Find the integral of the inverse sine function: ∫sin⁡−1(x) dx\int \sin^{-1}(x) \, dx∫sin−1(x)dx.

Problem 31341

Find the current $i$ at $t=10$ s, given $q=10-10 e^{-0.1 t}$ and $i=\frac{d q}{d t}$ .

Given $y=2 x-\sin (k x)$ and $x=2 k t$ , find:
(a) $\frac{d y}{d x}=2-k \cos (k x)$ , answer in terms of $k$ and $x$ .
(b) $\frac{d x}{d t}=$ , answer in terms of $k$ and/or $t$ .

Given $y=4x+\cos(kx)$ and $x=6kt$ , find: (a) $\frac{dy}{dx}$ in terms of $k$ and $x$ ; (b) $\frac{dx}{dt}$ in terms of $k$ and/or $t$ ; (c) For $k=2$ , at $t=\pi$ , find $\frac{dy}{dt}$ .

Find the slopes of secant lines between given points and the tangent line at $(-1, f(-1))$ for $f(x)=x^{2}+x$ .

Find the current $i$ (in A) at $t=10 \mathrm{~s}$ for the charge $q=10-10 e^{-0.1 t}$ , where $i=\frac{d q}{d t}$ .

Find the instantaneous velocity function $v=f^{\prime}(x)$ for $y=f(x)=9x^{2}-7x$ and evaluate it at $x=1,3,5$ seconds.

An object moves along the $y$ axis with $y=f(x)=x^{2}+x$ . Find: (A) average velocity from $x=3$ to $6$ , (B) from $3$ to $3+h$ , (C) instantaneous velocity at $x=3$ .

Find the derivative of $f(x)=9x+9\sqrt{x}$ using the definition. State the domains of $f(x)$ and $f'(x)$ .

Find the gradient and tangent line equation for $y=e^{\sin x+\cos x}$ at $x=\frac{\pi}{2}$ .

Describe the solid of revolution represented by the integral $\pi \int_{0}^{1}\left(x^{8}-x^{10}\right) d x$ .

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

Find the average revenue change from selling 1,000 to 1,050 car seats using $R(x)=64x-0.020x^2$ .

Evaluate the integral: $\int (x^{3} \ln x + x) \, dx$ .

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

Calculate the integral $\int_{0}^{1} \ln (x+1) \, dx$ .

Set up an integral for the volume of the solid formed by rotating the region bounded by $x=\sqrt{7-y}$ , $y=0$ , $x=0$ about the $y$ -axis: $\int_{0}^{7}(\sqrt{7-y})^2 d y$

Find the average change in revenue from selling 1,000 to 1,050 car seats with $R(x)=56x-0.020x^{2}$ . Also, find $R^{\prime}(x)$ and the revenue at $x=1000$ .

Find the volume $V$ of the solid formed by rotating the area between $x=2\sqrt{7y}$ , $x=0$ , $y=3$ around the $y$ -axis. $V=$

Given $f(x)=\frac{5}{x}$ , find $A$ in $\frac{f(x+h)-f(x)}{h} = \frac{A}{x(x+h)}$ . Then calculate $f^{\prime}(1)$ , $f^{\prime}(2)$ , $f^{\prime}(3)$ .

Find the derivative $P^{\prime}(t)$ of $P(t)=190+15t-3t^{2}$ and calculate $P(2)$ and $P^{\prime}(2)$ .

Find the marginal revenue function for $R(x)=x(28-0.05 x)$ . What is $R^{\prime}(x)=\square$ ?

Find the marginal cost function for $C(x)=177+1.7x$ . What is $C'(x)=\square$ ?

Find the current $i$ for the capacitor with charge $q=4000 e^{-62.5 t} \mu \mathrm{C}$ using $i=\frac{d q}{d t}$ .

Find average velocities for $s(t)=-16 t^{2}+100 t$ over intervals $[2,3]$ , $[2.9,3]$ , $[2.99,3]$ , $[2.999,3]$ , and $[2.9999,3]$ . Predict instantaneous velocity at $t=3$ .

Find the marginal profit function given $C(x)=259+0.5 x$ and $R(x)=9 x-0.03 x^{2}$ . What is $P^{\prime}(x)=\square$ ?

Find the rate of change of price $p=15-\ln \left(x^{4}\right)$ with respect to the number of jerseys sold $x$ .

Find the integral of the inverse sine function: $\int \sin^{-1} x \, dx$ .

Find the derivative of $f(x)=(2x-3)(4x-5)$ using the product or quotient rule. What is $f'(x)$ ?

Find the integral of the inverse sine function: $\int \sin^{-1} x \, dx$ .

Find the cost of producing the 31st food processor using $C(x)=2400+60x-0.3x^2$ . Exact cost: \ $\square$ .

Find the antiderivative $F(t)$ of $f(t)=10 \sec ^{2}(t)-2 t^{3}$ with $F(0)=0$ . Calculate $F(0.6)$ .

Identify eigenfunctions of the operator $d / d x$ from the list and determine their eigenvalues: a. $x^{2}$ , b. $e^{-\alpha x}$ , c. $e^{-\alpha \times 2}$ , d. $\cos (n x)+i \sin (n x)$ .

Calculate the integral of arcsin(ax) with respect to x: $\int \arcsin(ax) \, dx$

Find the cost of producing the 21st food processor using $C(x)=1600+60x-0.7x^2$ . Exact cost: \ $\square$ .

Find the cost of producing the 51st food processor using $C(x)=1500+90x-0.1x^{2}$ and marginal cost.

Find the cost of producing the 51st food processor using $C(x)=1500+90x-0.1x^2$ . Exact and approximate costs.

Find the cost of producing the 31st food processor using $C(x)=1700+30x-0.4x^{2}$ . Exact and marginal costs needed.

Find the integral of the inverse sine function: $\int \sin^{-1}(x) \, dx$ .

Find the voltage across a $2-\mathrm{H}$ inductor with current $i=e^{3t}$ at $t=1$ seconds. Use $v=L \frac{d i}{d t}$ .

Find the function $f(t)$ given that $f''(t) = 2 e^{t} + 3 \sin(t)$ , with conditions $f(0) = 6$ and $f(\pi) = 9$ .

Evaluate the limits:
1. $\lim _{x \rightarrow 3^{-}} \frac{|x-3|-0}{x-3}$
2. $\lim _{x \rightarrow 3^{+}} \frac{|x-3|-0}{x-3}$ .
Is $f(x)$ differentiable at 3?

Find how fast the current $I$ is changing when $R=6.25 \Omega$ and $R$ changes at $0.250 \Omega/\mathrm{min}$ .

Calculate the integral: $\int 4 x \sec ^{2} 2 x \, dx$

Find $f(3)$ for the function with $f^{\prime \prime}(x)=5x+6\sin(x)$ , given $f(0)=2$ and $f^{\prime}(0)=3$ .

Find the derivative $f^{\prime}(x)$ of $f(x)=x^{2}+9x-8$ and calculate $f^{\prime}(1)$ , $f^{\prime}(2)$ , $f^{\prime}(3)$ .

Normalize the wavefunction $\psi=\cos (\pi x / 2 a)$ for a particle in the region $-\mathrm{a}<\mathrm{x}<\mathrm{a}$ .

Does the limit $\lim_{x \rightarrow 3} f(x)$ exist? If yes, find it; if no, explain why using the graph details.

Find the derivative $r^{\prime}(x)$ for $r(x)=2+3x^{2}$ and calculate $r^{\prime}(1)$ , $r^{\prime}(2)$ , $r^{\prime}(3)$ .

Find $f^{\prime}(x)$ from $f^{\prime \prime}(x)=8 \cos (x)$ using constant $\mathrm{C}$ . Then find $f(x)$ using constant $\mathrm{D}$ .

Find the volume of the solid formed by revolving the area between $y=4x-x^2$ and $y=2x$ around the $y$ -axis. Volume $=\square$ (use $\pi$ ).

Find the volume of the solid formed by revolving the area between $y=4x-x^{2}$ and $y=2x$ about the $y$ -axis and $x=2$ .

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

Find the values for (A) $f(-1)$ , (B) $f(-2+h)$ , and (C) the limit as $h \to 0$ for $f(x)=8-x^{2}$ .

Find the volume of the solid formed by revolving the area between $y=12x-x^{2}$ and $y=x$ around the $y$ -axis. $\text{Volume} = \square$

Calculate the integral: $\int\left(x^{2}-5 x\right) e^{3} d x$

Calculate the integral: $\int\left(x^{2}+x+1\right) e^{x} d x$ .

Find $f^{\prime}(x)$ for $f(x)=2x^{2}+x-1$ , then calculate $f^{\prime}(1)$ , $f^{\prime}(2)$ , and $f^{\prime}(8)$ .

Evaluate the integral: $\int x^{2} e^{4 x} d x$

Find the derivative $s^{\prime}(x)$ of $s(x)=6x-5$ and calculate $s^{\prime}(1)$ , $s^{\prime}(2)$ , and $s^{\prime}(3)$ .

Find the sales function $S(t)=3 \sqrt{t+1}$ , then compute $S^{\prime}(t)$ , $S(15)$ , $S^{\prime}(15)$ , and estimate $S(16)$ .

Find the volume of the solid formed by revolving the area in the first quadrant bounded by $y=x^{2}$ , $x=1$ , and the $x$ -axis around $x=-1$ . Use the washer method to set up the integral for volume: $v=\int_{\square} \square$

Calculate the integral: $\int\left(x^{2}+x+1\right) e^{2} dx$

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

Calculate the integral: $\int x^{4} \cos x \, dx$

Calculate the integral: $\int x^{2} \cos(a x) \, dx$

Find the integral: $\int x^{2} e^{6 x} d x$

Find the limit: $\lim _{x \rightarrow 2^{+}} \frac{1}{|2-x|}$ .

Find the tangent line equation for $y=2 x^{2}-3 x+5$ at $x=4$ .

What does $C^{\prime}(500)=80$ mean for the cost function $C$ related to shredding 500 pounds of documents?

Find the volume of the solid formed by revolving the area under $y=x^2$ from $x=0$ to $x=1$ around $x=-1$ using the washer method.

Find the volumes of solids formed by revolving the region between $y=2\sqrt{x}$ and $y=\frac{x^2}{4}$ around a) the $x$ -axis and b) the $y$ -axis.