Calculus

Problem 20301

Find the tangent line equations at the specified points for each curve. a) $y=3x^2$ at $x=1$ b) $y=\sqrt{x-5}$ at $x=9$ c) $y=\frac{8}{\sqrt{x+11}}$ at $x=5$

See Solution

Problem 20302

Find tangent line equations for $y=\cos x$ at $x=-\frac{\pi}{2}, 0, \frac{\pi}{2}$ . Graph over $[-\frac{3\pi}{2}, 2\pi]$ .

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Problem 20303

Find the integration norm of the polynomial $f(x) = -9x - 2$ on $C[0,1]$ . What is $\|f\|$ ?

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Problem 20304

Find the largest $\theta$ in $0<\theta<2\pi$ where $N(\theta)=\frac{30}{5+2(\sin 2\theta-2\cos 2\theta)^{2}}$ is maximized.

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Problem 20305

Find tangent line equations for $y=\cos x$ at $x=-\frac{\pi}{2}, 0, \frac{\pi}{2}$ . Graph the curve and tangents.

See Solution

Problem 20306

Find the integration distance $\langle f, g\rangle=\int_{0}^{1} f(x) g(x) d x$ for $f(x)=7x+2$ , $g(x)=-2x-4$ .

See Solution

Problem 20307

Calculate the present value of a \$7,000 annual investment over 7 years at a continuous interest rate of 0.9%.

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Problem 20308

State the Fundamental Theorem of Calculus and define $F(x)=\int_{2}^{5x} \cos(t^3) dt$ .

See Solution

Problem 20309

Find a function $f(x)$ where $f'(x) = 3x^2 + 6$ and $f(0) = 5$ . What is $f(x)$ ?

See Solution

Problem 20310

Find tangent line equations for $y=\cos x$ at $x=-\frac{\pi}{2}, 0, \frac{\pi}{2}$ . Graph the curve and tangents.

See Solution

Problem 20311

Find the volume of the solid formed by revolving the region $R$ (bounded by $f(x)=6x-2x^2$ and the $x$ -axis from $0$ to $1$ ) around the $y$ -axis. Express your answer in terms of $\pi$ .

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Problem 20312

Find the solution for $y^{\prime}=x^{2}+16x-2$ with $y=12$ at $x=0$ . What is $y=?$

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Problem 20313

Approximate the root of $x^{3}+x+3=0$ using Newton's method. Given $x_{1} = -1$ and $x_{2} = -\frac{5}{4}$ , find $x_{3}$ .

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Problem 20314

The function $s(t)=8 t(t+2)$ gives distance in km. Find average velocity for: i) $t=3$ to $t=4$ , ii) $t=3$ to $t=3.1$ , iii) $t=3$ to $t=3.01$ . Then estimate instantaneous velocity at $t=3$ .

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Problem 20315

A foreign language student learns $N(t)=20t-t^{2}$ terms after $t$ hours.
a) How many terms are learned in the $3^{\text{rd}}$ hour? b) What is the learning rate at $t=2$ hours?

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Problem 20316

Zeigen Sie, dass das Volumen zwischen $f(x) = \sqrt{25-(x-3)^2}$ und $g(x) = \frac{1}{144}x^3 - \frac{1}{8}x^2 + 7$ mindestens 500 ml beträgt:
$\int_{a}^{b}[f(x)-g(x)] dx \geq 500$

See Solution

Problem 20317

Find the derivative $f^{\prime}(x)$ and critical numbers of $f(x)=x^{3}-27x-6$ .

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Problem 20318

Find the population size at time $t=5$ for the model $\frac{d y}{d t}=0.1 y$ , given $y(0)=350$ .

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Problem 20319

If $h(x)$ is a continuous increasing function, what can we conclude about its maxima and minima?

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Problem 20320

Solve $\frac{d y}{d x}=-0.4 y$ with initial condition $y(0)=6$ . Find $y(x)=$ .

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Problem 20321

Find $f(x)$ given $\frac{d y}{d x}=36 y x^{5}$ and the $y$ -intercept is 3. What is $f(x)$ ?

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Problem 20322

Find the demand function $D(x)$ given that $D^{\prime}(x)=-\frac{3000}{x^{2}}$ and $D(6)=505$ .

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Problem 20323

Leiten Sie die Funktion $f$ einmal ab: a) $f(x)=(3 x-1)^{2}$ , b) $f(t)=2 \cos (\pi \cdot t+2)$ , c) $f(x)=\sqrt{x} \cdot(3 x-2)$ , d) $f(x)=x \cdot \sqrt{x+1}$ , e) $f(t)=(t+1) \cdot \sin (t)$ , f) $f(t)=\sqrt{2 t+1}$ .

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Problem 20324

Find the tangent line equation to the curve $3 x^{3}+2 x^{2} y^{2}+3 y^{3}=8$ at the point $(1,1)$ .

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Problem 20325

A foreign language student learns $N(t)=20t-t^{2}$ terms. a) How many terms are learned in the 3rd hour? b) What is the rate of change at $t=2$ hr?

See Solution

Problem 20326

Calculate the value of the integral $\int_{0}^{4} t^{-1 / 2} d t$ .

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Problem 20327

Find $F^{\prime}(\pi)$ if $F(x)=-\int_{0}^{x} \cos (t) dt$ .

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Problem 20328

Berechne das Volumen des Drehkörpers, der durch die Rotation der Fläche zwischen $f(x)=3 \sqrt{x}$ und $g(x)=\sqrt{4-x}$ um die $x$ -Achse entsteht.

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Problem 20329

Calculate the value of the integral from 1 to 2: $\int_{1}^{2} 4 x^{3} d x$ .

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Problem 20330

Find the limit: $\lim _{t \rightarrow \infty} t^{5}+t^{4}-t^{3}+t+5$ . What is the answer?

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Problem 20331

Find the velocity after $2.3 \mathrm{~s}$ and the distance fallen during this time on a free-fall ride.

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Problem 20332

Find the demand function $D(x)$ given $D'(x)=-\frac{3000}{x^2}$ and $D(1.50)=2003$ .

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Problem 20333

Evaluate the integral: $\int_{0}^{1} x^{8}(3+2 x^{9})^{5} \, dx$

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Problem 20334

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

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Problem 20335

Find the demand function $D(x)$ given that $D'(x)=-\frac{7000}{x^{2}}$ and demand is \$3.50 per unit.

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Problem 20336

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

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Problem 20337

Find the demand function $D(x)$ given $D^{\prime}(x)=-\frac{10000}{x^{2}}$ and $D(20)=501$ .

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Problem 20338

Find where $f(x)=x^{3}+8x+7$ is increasing, decreasing, and its local extrema.

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Problem 20339

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

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Problem 20340

Find $f(x)$ if $\frac{d y}{d x}=68 y x^{16}$ and the $y$ -intercept is 6. What is $f(x)$ ?

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Problem 20341

Evaluate the integral $\int_{-\pi}^{\pi} \sin x \, dx$ . Options: 0, 2, -2, $\pi$ .

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Problem 20342

Bestimmen Sie $f^{\prime}(x)$ für: a) $f(x)=\frac{1}{x}(2x+1)^{4}$ , b) $f(x)=x\sqrt{3x-2}$ , c) $f(x)=x\cos(2x)$ .

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Problem 20343

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

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Problem 20344

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x}$ .

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Problem 20345

Find the area under the curve $y=x^{1/2}+\frac{1}{x}$ from $x=4$ to $x=9$ .

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Problem 20346

Solve the equation $\frac{d y}{d x}=-4 y$ with initial condition $y(0)=3$ . Find $y(x)=$ .

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Problem 20347

Find the limit: $\lim _{x \rightarrow 0} \frac{7 x-x^{2}}{x}$ .

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Problem 20348

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{5-x}-\sqrt{3+x}}{x-1}$ .

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Problem 20349

Find the population size at time $t=5$ given $\frac{d y}{d t}=0.1 y$ and $y(0)=600$ . Round to the nearest whole number.

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Problem 20350

Calculate the future value of an 8-year continuous income stream of \$160,000 at a continuous compounding rate of 4.1%.

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Problem 20351

Evaluate the integral from 0 to a: $\int_{0}^{a} x^{2} d x$ and find the result.

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Problem 20352

Evaluate the integral of $\cos (x+1)$ with respect to $x$ . What is the result?

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Problem 20353

Find the limit: $\lim _{x \rightarrow 1} \frac{(x-1)}{\sqrt{x}-1}$ .

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Problem 20354

Calculate the future value of a 14-year continuous income stream of \$220,000 at a continuous compounding rate of 4\%.

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Problem 20355

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{3}{(-4)^{n}}$ : converges conditionally, absolutely, or diverges?

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Problem 20356

Find $\frac{d y}{d x}$ for $y=\sin u$ and $u=4 x+3$ . Use $\frac{d y}{d x}=f^{\prime}(g(x)) g^{\prime}(x)$ .

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Problem 20357

Find the limit: $\lim _{x \rightarrow 0} \frac{(2 x+1)^{\frac{1}{3}}-1}{x}$ .

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Problem 20358

Evaluate the integral: $\int \frac{e^{x+2}}{2} dx$ . What is the result?

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Problem 20359

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{5-x}-\sqrt{3+x}}{x-1}$ .

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Problem 20360

Find the limit: $\lim _{t \rightarrow 0} \tan \left(\frac{11}{4} \pi-\frac{\pi \sin t}{t}\right)$ . Simplify your answer.

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Problem 20361

Find the partial derivatives of the function $f(x, y)=x^{6}-3xy^{2}-6y^{4}$ : $f_{x}(x, y)$ and $f_{y}(x, y)$ .

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Problem 20362

Find the population size at $t=5$ for the model $\frac{d y}{d t}=0.2 y$ with $y(0)=350$ . Round to nearest whole number.

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Problem 20363

Find the equilibrium quantity where demand $d(x)=500-0.5 x^{2}$ and supply $s(x)=0.3 x^{2}$ . What is the consumer surplus?

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Problem 20364

Find the volume of the solid formed by revolving the region $R$ (bounded by $f(x)=2 \sin(2x^2)+2$ and the $x$ -axis) around the $y$ -axis.

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Problem 20365

Calculate $\lim _{h \rightarrow 0} \frac{f(x, y+h)-f(x, y)}{h}$ for $f(x, y)=y^{2}+7 x y$ .

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Problem 20366

Find the limits: a) $\lim _{x \rightarrow 1} \frac{(x-1)}{\sqrt{x}-1}$ b) $\lim _{x \rightarrow 0} \frac{(2x+1)^{\frac{1}{3}}-1}{x}$ c) $\lim _{x \rightarrow 0} \frac{(343+x)^{\frac{1}{3}}-7}{x}$

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Problem 20367

Given $a_{n}=\frac{3}{(-4)^{n}}$ , determine the order of partial sums: $S_{N}=a_{1}+a_{2}+\ldots+a_{N}$ . Which is true?

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Problem 20368

Calculate the partial derivative of $5 e^{4 x y}$ with respect to $x$ .

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Problem 20369

Find the limit: $\lim _{x \rightarrow 0} \frac{(343+x)^{\frac{1}{3}}-7}{x}$

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Problem 20370

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin x}{x}$ . What is the value?

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Problem 20371

Find $f_{x}(4,3)$ and $f_{y}(4,3)$ for the function $f(x, y)=-2 x^{4}-6 x y^{3}+3 y^{2}$ .

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Problem 20372

Find the derivative of $y=\frac{5x-8}{x^2+6x}$ . What is $y'=\square$ ?

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Problem 20373

Find the present value of a 12-year continuous income stream of \$170,000 at a continuous rate of 4.9\%. Round to the nearest dollar.

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Problem 20374

Lucky Larry wins \$8,000,000 paid as \$320,000/year for 25 years. Find the future and present value at 7% interest, compounded continuously.

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Problem 20375

Evaluate the limit: $\lim _{x \rightarrow 4} \frac{|x-4|}{x-4}$ . Does it exist?

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Problem 20376

Find the partial derivatives of the function $f(x, y, z)=\sqrt{-6 x+6 y+2 z}$ : $f_{x}(x, y, z)$ , $f_{y}(x, y, z)$ , $f_{z}(x, y, z)$ .

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Problem 20377

Find the Elasticity of Demand for $D(p)=\frac{200}{p}$ at price \$79. Is it Unitary, Elastic, or Inelastic? What should we do to increase revenue?

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Problem 20378

Find $f_{x}(2,3)$ and $f_{y}(2,3)$ for the function $f(x, y)=x^{2}+4xy^{3}+5y^{4}$ .

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Problem 20379

Find the Elasticity of Demand for $D(p)=\frac{200}{p}$ at price \$79. Is it Unitary, Elastic, or Inelastic? To increase revenue, should we keep, raise, or lower prices?

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Problem 20380

Given the demand function $D(p)=\sqrt{150-2p}$ , find:
1. Sales at $p=73$ .
2. Elasticity function $E(p)=-\frac{p}{D(p)}(D'(p))$ .
3. Elasticity at $p=73$ .
4. Type of demand at this price.
5. Price strategy for revenue.
6. Price to maximize revenue.

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Problem 20381

Find the second partial derivatives of the function $f(x, y)=x^{3}-2xy^{5}-6y^{2}$ : $f_{xx}(x, y)$ and $f_{xy}(x, y)$ .

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Problem 20382

Calculate the integral $A=\int_{1}^{5} \frac{1}{\sqrt{2 x-1}} d x$ .

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Problem 20383

Find the marginal productivity of labor $P_L$ and capital $P_K$ for the function $P(L, K)=27 L^{0.5} K^{0.5}$ .

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Problem 20384

Approximate the area under $f(x)=\frac{4}{x}$ from $x=1$ to $x=5$ using 8 right rectangles. Provide a fraction.

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Problem 20385

Approximate the area under $f(x) = x^{2}$ on $[0,2]$ using a right-endpoint method with $n=4$ subintervals.

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Problem 20386

A company's weekly sales $q$ relate to price $p$ as $q=\frac{13950}{\sqrt[3]{p^{10}}}$ . Currently, $p=5$ .
a. Find the point elasticity of demand $\varepsilon=\square$ .
b. If price decreases by $1\%$ , revenue changes by \$.
c. If price decreases by $5\%$ , revenue changes by \$

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Problem 20387

Find a function $f(x)$ where $f'(x) = e^x + 4x$ and $f(0) = 5$ . What is $f(x)$ ?

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Problem 20388

A box starts from rest and slides down a ramp with a height of 4.0m. What is its speed at the bottom?

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Problem 20389

Find the marginal cost $C^{\prime}(x)=-0.06 x+21$ .
a. Compute $C^{\prime}(580)$ . Round to the nearest cent.
b. Integrate to find cost function $C(x)$ .
c. Compute $C(580)$ . Round to the nearest dollar.

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Problem 20390

Bestimme die Ableitungen von $f(x)$ für: a) $x^{3}$ , b) $x^{5}$ , c) $x^{2 n}$ , d) $x$ , e) $x^{n+4}$ , f) $x^{2016}$ .

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Problem 20391

Berechne die Steigung von $f$ bei $x_0$ mit der h-Methode für: a) $f(x)=x^2$ , $x_0=1$ ; b) $f(x)=2x^2$ , $x_0=-1$ ; c) $f(x)=x^3$ , $x_0=2$ ; d) $f(x)=2x$ , $x_0=1$ .

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Problem 20392

Determine the max and min of the cubic function $f(x) = 6x^{3} - 37x^{2} + 54x - 8$ .

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Problem 20393

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x+9}=$

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Problem 20394

Encuentra la derivada de $f(x)=(x^{5}-2x^{2})^{3}(4x^{3}-5)^{4}$ usando la regla del producto.

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Problem 20395

What does the function $f(x)=a(b)^{x}$ approach as $x$ increases, given that $0<b<1$ ?

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Problem 20396

A biologist has a 366g radioactive sample. Find its mass after 2 hours with a decay rate of $5\%$ per hour. Round to the nearest tenth.

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Problem 20397

Evaluate the integral: $\int \frac{x^{2}-7}{x^{3}-2 x^{2}+x} d x$ using partial fractions.

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Problem 20398

Find $k$ so that $f(x)=\frac{k}{(x+1)^{2}}$ (for $2 \leq x \leq 3$ ) is a probability density function.

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Problem 20399

Determine which two of the following sequences converge:
1. $\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{n !}$
2. $\sum_{n=1}^{\infty}\left(1+\frac{1}{n^{2}}\right)^{n^{3}}$
3. $\sum_{n=1}^{\infty}\left(\frac{n-1}{n+1}\right)^{n^{2}+n}$
4. $\sum_{n=1}^{\infty} \frac{3^{3}+1}{\sqrt[3]{n^{10}+n}}$
5. $\sum_{n=1}^{\infty} \frac{n !}{(2 n) !}$
6. $\sum_{n=1}^{\infty}\left(\frac{n^{2}}{2 n+1}\right)^{n}$

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Problem 20400

Find the marginal cost at $x=570$ from $C'(x)=-0.02x+27$ , then integrate to get $C(x)$ and compute $C(570)$ .

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Calculus

Problem 20301

Find the tangent line equations at the specified points for each curve. a) y=3x2y=3x^2y=3x2 at x=1x=1x=1 b) y=x−5y=\sqrt{x-5}y=x−5​ at x=9x=9x=9 c) y=8x+11y=\frac{8}{\sqrt{x+11}}y=x+11​8​ at x=5x=5x=5

Problem 20302

Find tangent line equations for y=cos⁡xy=\cos xy=cosx at x=−π2,0,π2x=-\frac{\pi}{2}, 0, \frac{\pi}{2}x=−2π​,0,2π​. Graph over [−3π2,2π][-\frac{3\pi}{2}, 2\pi][−23π​,2π].

Problem 20303

Find the integration norm of the polynomial f(x)=−9x−2f(x) = -9x - 2f(x)=−9x−2 on C[0,1]C[0,1]C[0,1]. What is ∥f∥\|f\|∥f∥?

Problem 20304

Find the largest θ\thetaθ in 0<θ<2π0<\theta<2\pi0<θ<2π where N(θ)=305+2(sin⁡2θ−2cos⁡2θ)2N(\theta)=\frac{30}{5+2(\sin 2\theta-2\cos 2\theta)^{2}}N(θ)=5+2(sin2θ−2cos2θ)230​ is maximized.

Problem 20305

Find tangent line equations for y=cos⁡xy=\cos xy=cosx at x=−π2,0,π2x=-\frac{\pi}{2}, 0, \frac{\pi}{2}x=−2π​,0,2π​. Graph the curve and tangents.

Problem 20306

Find the integration distance ⟨f,g⟩=∫01f(x)g(x)dx\langle f, g\rangle=\int_{0}^{1} f(x) g(x) d x⟨f,g⟩=∫01​f(x)g(x)dx for f(x)=7x+2f(x)=7x+2f(x)=7x+2, g(x)=−2x−4g(x)=-2x-4g(x)=−2x−4.

Problem 20307

Calculate the present value of a \$7,000 annual investment over 7 years at a continuous interest rate of 0.9%.

Problem 20308

State the Fundamental Theorem of Calculus and define F(x)=∫25xcos⁡(t3)dtF(x)=\int_{2}^{5x} \cos(t^3) dtF(x)=∫25x​cos(t3)dt.

Problem 20309

Find a function f(x)f(x)f(x) where f′(x)=3x2+6f'(x) = 3x^2 + 6f′(x)=3x2+6 and f(0)=5f(0) = 5f(0)=5. What is f(x)f(x)f(x)?

Problem 20310

Find tangent line equations for y=cos⁡xy=\cos xy=cosx at x=−π2,0,π2x=-\frac{\pi}{2}, 0, \frac{\pi}{2}x=−2π​,0,2π​. Graph the curve and tangents.

Problem 20311

Find the volume of the solid formed by revolving the region RRR (bounded by f(x)=6x−2x2f(x)=6x-2x^2f(x)=6x−2x2 and the xxx-axis from 000 to 111) around the yyy-axis. Express your answer in terms of π\piπ.

Problem 20312

Find the solution for y′=x2+16x−2y^{\prime}=x^{2}+16x-2y′=x2+16x−2 with y=12y=12y=12 at x=0x=0x=0. What is y=?y=?y=?

Problem 20313

Approximate the root of x3+x+3=0x^{3}+x+3=0x3+x+3=0 using Newton's method. Given x1=−1x_{1} = -1x1​=−1 and x2=−54x_{2} = -\frac{5}{4}x2​=−45​, find x3x_{3}x3​.

Problem 20314

The function s(t)=8t(t+2)s(t)=8 t(t+2)s(t)=8t(t+2) gives distance in km. Find average velocity for: i) t=3t=3t=3 to t=4t=4t=4, ii) t=3t=3t=3 to t=3.1t=3.1t=3.1, iii) t=3t=3t=3 to t=3.01t=3.01t=3.01. Then estimate instantaneous velocity at t=3t=3t=3.

Problem 20315

A foreign language student learns N(t)=20t−t2N(t)=20t-t^{2}N(t)=20t−t2 terms after ttt hours. a) How many terms are learned in the 3rd3^{\text{rd}}3rd hour? b) What is the learning rate at t=2t=2t=2 hours?

Problem 20316

Problem 20317

Find the derivative f′(x)f^{\prime}(x)f′(x) and critical numbers of f(x)=x3−27x−6f(x)=x^{3}-27x-6f(x)=x3−27x−6.

Problem 20318

Find the population size at time t=5t=5t=5 for the model dydt=0.1y\frac{d y}{d t}=0.1 ydtdy​=0.1y, given y(0)=350y(0)=350y(0)=350.

Problem 20319

If h(x)h(x)h(x) is a continuous increasing function, what can we conclude about its maxima and minima?

Problem 20320

Solve dydx=−0.4y\frac{d y}{d x}=-0.4 ydxdy​=−0.4y with initial condition y(0)=6y(0)=6y(0)=6. Find y(x)=y(x)=y(x)=.

Problem 20321

Find f(x)f(x)f(x) given dydx=36yx5\frac{d y}{d x}=36 y x^{5}dxdy​=36yx5 and the yyy-intercept is 3. What is f(x)f(x)f(x)?

Problem 20322

Find the demand function D(x)D(x)D(x) given that D′(x)=−3000x2D^{\prime}(x)=-\frac{3000}{x^{2}}D′(x)=−x23000​ and D(6)=505D(6)=505D(6)=505.

Problem 20323

Problem 20324

Find the tangent line equation to the curve 3x3+2x2y2+3y3=83 x^{3}+2 x^{2} y^{2}+3 y^{3}=83x3+2x2y2+3y3=8 at the point (1,1)(1,1)(1,1).

Problem 20325

A foreign language student learns N(t)=20t−t2N(t)=20t-t^{2}N(t)=20t−t2 terms. a) How many terms are learned in the 3rd hour? b) What is the rate of change at t=2t=2t=2 hr?

Problem 20326

Calculate the value of the integral ∫04t−1/2dt\int_{0}^{4} t^{-1 / 2} d t∫04​t−1/2dt.

Problem 20327

Find F′(π)F^{\prime}(\pi)F′(π) if F(x)=−∫0xcos⁡(t)dtF(x)=-\int_{0}^{x} \cos (t) dtF(x)=−∫0x​cos(t)dt.

Problem 20328

Berechne das Volumen des Drehkörpers, der durch die Rotation der Fläche zwischen f(x)=3xf(x)=3 \sqrt{x}f(x)=3x​ und g(x)=4−xg(x)=\sqrt{4-x}g(x)=4−x​ um die xxx-Achse entsteht.

Problem 20329

Calculate the value of the integral from 1 to 2: ∫124x3dx\int_{1}^{2} 4 x^{3} d x∫12​4x3dx.

Problem 20330

Find the limit: lim⁡t→∞t5+t4−t3+t+5\lim _{t \rightarrow \infty} t^{5}+t^{4}-t^{3}+t+5limt→∞​t5+t4−t3+t+5. What is the answer?

Problem 20331

Find the velocity after 2.3 s2.3 \mathrm{~s}2.3 s and the distance fallen during this time on a free-fall ride.

Problem 20332

Find the demand function D(x)D(x)D(x) given D′(x)=−3000x2D'(x)=-\frac{3000}{x^2}D′(x)=−x23000​ and D(1.50)=2003D(1.50)=2003D(1.50)=2003.

Problem 20333

Evaluate the integral: ∫01x8(3+2x9)5 dx\int_{0}^{1} x^{8}(3+2 x^{9})^{5} \, dx∫01​x8(3+2x9)5dx

Problem 20334

Find the limit: lim⁡x→−24−x22+x\lim _{x \rightarrow-2} \frac{4-x^{2}}{2+x}limx→−2​2+x4−x2​.

Problem 20335

Find the demand function D(x)D(x)D(x) given that D′(x)=−7000x2D'(x)=-\frac{7000}{x^{2}}D′(x)=−x27000​ and demand is \$3.50 per unit.

Problem 20336

Find the limit: lim⁡x→−433x2+x−43x+4\lim _{x \rightarrow-\frac{4}{3}} \frac{3 x^{2}+x-4}{3 x+4}limx→−34​​3x+43x2+x−4​.

Problem 20337

Find the demand function D(x)D(x)D(x) given D′(x)=−10000x2D^{\prime}(x)=-\frac{10000}{x^{2}}D′(x)=−x210000​ and D(20)=501D(20)=501D(20)=501.

Problem 20338

Find where f(x)=x3+8x+7f(x)=x^{3}+8x+7f(x)=x3+8x+7 is increasing, decreasing, and its local extrema.

Problem 20339

Find the limit as xxx approaches -2 for the expression 4−x22+x\frac{4-x^{2}}{2+x}2+x4−x2​.

Problem 20340

Find f(x)f(x)f(x) if dydx=68yx16\frac{d y}{d x}=68 y x^{16}dxdy​=68yx16 and the yyy-intercept is 6. What is f(x)f(x)f(x)?

Problem 20341

Find the tangent line equations at the specified points for each curve. a) $y=3x^2$ at $x=1$ b) $y=\sqrt{x-5}$ at $x=9$ c) $y=\frac{8}{\sqrt{x+11}}$ at $x=5$

Find tangent line equations for $y=\cos x$ at $x=-\frac{\pi}{2}, 0, \frac{\pi}{2}$ . Graph over $[-\frac{3\pi}{2}, 2\pi]$ .

Find the integration norm of the polynomial $f(x) = -9x - 2$ on $C[0,1]$ . What is $\|f\|$ ?

Find the largest $\theta$ in $0<\theta<2\pi$ where $N(\theta)=\frac{30}{5+2(\sin 2\theta-2\cos 2\theta)^{2}}$ is maximized.

Find tangent line equations for $y=\cos x$ at $x=-\frac{\pi}{2}, 0, \frac{\pi}{2}$ . Graph the curve and tangents.

Find the integration distance $\langle f, g\rangle=\int_{0}^{1} f(x) g(x) d x$ for $f(x)=7x+2$ , $g(x)=-2x-4$ .

State the Fundamental Theorem of Calculus and define $F(x)=\int_{2}^{5x} \cos(t^3) dt$ .

Find a function $f(x)$ where $f'(x) = 3x^2 + 6$ and $f(0) = 5$ . What is $f(x)$ ?

Find tangent line equations for $y=\cos x$ at $x=-\frac{\pi}{2}, 0, \frac{\pi}{2}$ . Graph the curve and tangents.

Find the volume of the solid formed by revolving the region $R$ (bounded by $f(x)=6x-2x^2$ and the $x$ -axis from $0$ to $1$ ) around the $y$ -axis. Express your answer in terms of $\pi$ .

Find the solution for $y^{\prime}=x^{2}+16x-2$ with $y=12$ at $x=0$ . What is $y=?$

Approximate the root of $x^{3}+x+3=0$ using Newton's method. Given $x_{1} = -1$ and $x_{2} = -\frac{5}{4}$ , find $x_{3}$ .

The function $s(t)=8 t(t+2)$ gives distance in km. Find average velocity for: i) $t=3$ to $t=4$ , ii) $t=3$ to $t=3.1$ , iii) $t=3$ to $t=3.01$ . Then estimate instantaneous velocity at $t=3$ .

A foreign language student learns $N(t)=20t-t^{2}$ terms after $t$ hours.
a) How many terms are learned in the $3^{\text{rd}}$ hour? b) What is the learning rate at $t=2$ hours?

Find the derivative $f^{\prime}(x)$ and critical numbers of $f(x)=x^{3}-27x-6$ .

Find the population size at time $t=5$ for the model $\frac{d y}{d t}=0.1 y$ , given $y(0)=350$ .

If $h(x)$ is a continuous increasing function, what can we conclude about its maxima and minima?

Solve $\frac{d y}{d x}=-0.4 y$ with initial condition $y(0)=6$ . Find $y(x)=$ .

Find $f(x)$ given $\frac{d y}{d x}=36 y x^{5}$ and the $y$ -intercept is 3. What is $f(x)$ ?

Find the demand function $D(x)$ given that $D^{\prime}(x)=-\frac{3000}{x^{2}}$ and $D(6)=505$ .

Find the tangent line equation to the curve $3 x^{3}+2 x^{2} y^{2}+3 y^{3}=8$ at the point $(1,1)$ .

A foreign language student learns $N(t)=20t-t^{2}$ terms. a) How many terms are learned in the 3rd hour? b) What is the rate of change at $t=2$ hr?

Calculate the value of the integral $\int_{0}^{4} t^{-1 / 2} d t$ .

Find $F^{\prime}(\pi)$ if $F(x)=-\int_{0}^{x} \cos (t) dt$ .

Berechne das Volumen des Drehkörpers, der durch die Rotation der Fläche zwischen $f(x)=3 \sqrt{x}$ und $g(x)=\sqrt{4-x}$ um die $x$ -Achse entsteht.

Calculate the value of the integral from 1 to 2: $\int_{1}^{2} 4 x^{3} d x$ .

Find the limit: $\lim _{t \rightarrow \infty} t^{5}+t^{4}-t^{3}+t+5$ . What is the answer?

Find the velocity after $2.3 \mathrm{~s}$ and the distance fallen during this time on a free-fall ride.

Find the demand function $D(x)$ given $D'(x)=-\frac{3000}{x^2}$ and $D(1.50)=2003$ .

Evaluate the integral: $\int_{0}^{1} x^{8}(3+2 x^{9})^{5} \, dx$

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

Find the demand function $D(x)$ given that $D'(x)=-\frac{7000}{x^{2}}$ and demand is \$3.50 per unit.

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

Find the demand function $D(x)$ given $D^{\prime}(x)=-\frac{10000}{x^{2}}$ and $D(20)=501$ .

Find where $f(x)=x^{3}+8x+7$ is increasing, decreasing, and its local extrema.

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

Find $f(x)$ if $\frac{d y}{d x}=68 y x^{16}$ and the $y$ -intercept is 6. What is $f(x)$ ?

Evaluate the integral $\int_{-\pi}^{\pi} \sin x \, dx$ . Options: 0, 2, -2, $\pi$ .

Bestimmen Sie $f^{\prime}(x)$ für: a) $f(x)=\frac{1}{x}(2x+1)^{4}$ , b) $f(x)=x\sqrt{3x-2}$ , c) $f(x)=x\cos(2x)$ .

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

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x}$ .

Find the area under the curve $y=x^{1/2}+\frac{1}{x}$ from $x=4$ to $x=9$ .

Solve the equation $\frac{d y}{d x}=-4 y$ with initial condition $y(0)=3$ . Find $y(x)=$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{7 x-x^{2}}{x}$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{5-x}-\sqrt{3+x}}{x-1}$ .

Find the population size at time $t=5$ given $\frac{d y}{d t}=0.1 y$ and $y(0)=600$ . Round to the nearest whole number.

Evaluate the integral from 0 to a: $\int_{0}^{a} x^{2} d x$ and find the result.

Evaluate the integral of $\cos (x+1)$ with respect to $x$ . What is the result?

Find the limit: $\lim _{x \rightarrow 1} \frac{(x-1)}{\sqrt{x}-1}$ .

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{3}{(-4)^{n}}$ : converges conditionally, absolutely, or diverges?

Find $\frac{d y}{d x}$ for $y=\sin u$ and $u=4 x+3$ . Use $\frac{d y}{d x}=f^{\prime}(g(x)) g^{\prime}(x)$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{(2 x+1)^{\frac{1}{3}}-1}{x}$ .

Evaluate the integral: $\int \frac{e^{x+2}}{2} dx$ . What is the result?

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{5-x}-\sqrt{3+x}}{x-1}$ .

Find the limit: $\lim _{t \rightarrow 0} \tan \left(\frac{11}{4} \pi-\frac{\pi \sin t}{t}\right)$ . Simplify your answer.

Find the partial derivatives of the function $f(x, y)=x^{6}-3xy^{2}-6y^{4}$ : $f_{x}(x, y)$ and $f_{y}(x, y)$ .

Find the population size at $t=5$ for the model $\frac{d y}{d t}=0.2 y$ with $y(0)=350$ . Round to nearest whole number.

Find the equilibrium quantity where demand $d(x)=500-0.5 x^{2}$ and supply $s(x)=0.3 x^{2}$ . What is the consumer surplus?

Find the volume of the solid formed by revolving the region $R$ (bounded by $f(x)=2 \sin(2x^2)+2$ and the $x$ -axis) around the $y$ -axis.

Calculate $\lim _{h \rightarrow 0} \frac{f(x, y+h)-f(x, y)}{h}$ for $f(x, y)=y^{2}+7 x y$ .

Given $a_{n}=\frac{3}{(-4)^{n}}$ , determine the order of partial sums: $S_{N}=a_{1}+a_{2}+\ldots+a_{N}$ . Which is true?

Calculate the partial derivative of $5 e^{4 x y}$ with respect to $x$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{(343+x)^{\frac{1}{3}}-7}{x}$

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin x}{x}$ . What is the value?

Find $f_{x}(4,3)$ and $f_{y}(4,3)$ for the function $f(x, y)=-2 x^{4}-6 x y^{3}+3 y^{2}$ .

Find the derivative of $y=\frac{5x-8}{x^2+6x}$ . What is $y'=\square$ ?