Calculus

Problem 25301

A cylindrical coffeepot with radius 5 in has volume change rate $-5 \pi \sqrt{h}$ . Find the height change rate equation in terms of $h$ .

See Solution

Problem 25302

Evaluate the integral from 1 to $e^{8}$ of $\frac{(\ln x)^{4}}{x}$ dx.

See Solution

Problem 25303

Find the area between $f(x)=e^{5 x}$ and $g(x)=\frac{1}{2 x}$ from $x=1$ to $x=5$ .

See Solution

Problem 25304

Find the indefinite integral of $\sec^{2} \theta + 9$ with respect to $\theta$ .

See Solution

Problem 25305

Evaluate the integrals given: a. $\int_{2}^{5} 8 f(x) d x=48$ b. $\int_{2}^{7}(f(x)-g(x)) d x=\square$

See Solution

Problem 25306

Evaluate the integrals given:
a. $\int_{2}^{5} 8 f(x) d x=48$
b. $\int_{2}^{7}(f(x)-g(x)) d x=5$
c. $\int_{2}^{5}(f(x)-g(x)) d x=\square$

See Solution

Problem 25307

Find the integral of $\tan^{5} \theta \sec^{7} \theta$ with respect to $\theta$ : $\int \tan^{5} \theta \sec^{7} \theta d \theta$ .

See Solution

Problem 25308

Find $h^{\prime}(2)$ for $h(x)=f(x) \cdot g(x)$ given $f(2)=3$ , $f^{\prime}(2)=4$ , $g(2)=-2$ , $g^{\prime}(2)=-4$ .

See Solution

Problem 25309

Evaluate the integrals based on given functions $f$ and $g$ over $[2,7]$ with specific integral values.
a. $\int_{2}^{5} 8 f(x) d x=48$ b. $\int_{2}^{7}(f(x)-g(x)) d x=5$ c. $\int_{2}^{5}(f(x)-g(x)) d x=4$ d. $\int_{5}^{7}(g(x)-f(x)) d x=\square$

See Solution

Problem 25310

Evaluate the integrals based on given conditions:
a. $\int_{2}^{5} 8 f(x) d x$ b. $\int_{2}^{7}(f(x)-g(x)) d x$ c. $\int_{2}^{5}(f(x)-g(x)) d x$ d. $\int_{5}^{7}(g(x)-f(x)) d x$ e. $\int_{5}^{7} 5 g(x) d x$

See Solution

Problem 25311

Evaluate the following integrals given $\int_{2}^{7} f(x) d x=11$ , $\int_{2}^{7} g(x) d x=6$ :
b. $\int_{2}^{7}(f(x)-g(x)) d x=5$ c. $\int_{2}^{5}(f(x)-g(x)) d x=4$ d. $\int_{5}^{7}(g(x)-f(x)) d x=-1$ e. $\int_{5}^{7} 5 g(x) d x=20$ f. $\int_{5}^{2} 3 f(x) d x=\square$

See Solution

Problem 25312

Find the water velocities at points 2 and 3 in a hydroelectric plant with a 100 cm tube and 50 cm nozzle.

See Solution

Problem 25313

Evaluate $\int_{7}^{2} g(x) d x$ given $\int_{2}^{4} f(x) d x=4$ , $\int_{2}^{7} f(x) d x=8$ , $\int_{2}^{7} g(x) d x=5$ .

See Solution

Problem 25314

Calculate the total income over 15 years from a continuous income stream with a rate of flow $f(t) = 3500$ . What is the total earned? \\square$

See Solution

Problem 25315

At age 35, you deposit \$2000 annually into an IRA earning 5\% interest, compounded continuously. What's the total at age 65?

See Solution

Problem 25316

Given $\int_{2}^{4} f(x) d x=4$ , $\int_{2}^{7} f(x) d x=8$ , and $\int_{2}^{7} g(x) d x=5$ , find:
1. $\int_{7}^{2} g(x) d x$
2. $\int_{2}^{7} 6 g(x) d x=\square$

See Solution

Problem 25317

Approximate $\int_{3}^{11} e^{x} d x$ using 4 equal subintervals and right endpoints for the Riemann Sum.

See Solution

Problem 25318

Calculate the total income from the function $f(t)=500 e^{0.05 t}$ over the first 2 years. Total income: \$ \square (round to nearest dollar).

See Solution

Problem 25319

Given $\int_{2}^{4} f(x) d x=4$ , $\int_{2}^{7} f(x) d x=8$ , and $\int_{2}^{7} g(x) d x=5$ , find:
1. $\int_{7}^{2} g(x) d x$
2. $\int_{2}^{7} 6 g(x) d x$
3. $\int_{2}^{7}[g(x)-f(x)] d x=\square$

See Solution

Problem 25320

A student makes quarterly payments of \ $3,000 into a retirement account with$ 6\%$ interest compounded continuously. Find the total after 40 years.

See Solution

Problem 25321

Calculate the future value of a continuous income stream $f(t)=1550 e^{-0.02 t}$ at $5.75\%$ interest over 7 years. What is the amount? \$ \square

See Solution

Problem 25322

You deposit \ $2000 yearly into an IRA from age 30 to 65 at a 5% continuous compound interest. What's the total value at 65? \$\square$ (Round to the nearest dollar.)

See Solution

Problem 25323

Use the value of the first integral to evaluate the two integrals:
1. $\int_{0}^{1}(10 x-2 x^{3}) dx=\square$
2. $\int_{1}^{0}(5 x-x^{3}) dx$

See Solution

Problem 25324

An object is launched upward at $39.2 \mathrm{~m/s}$ . How long is it at or above 34.3 meters?

See Solution

Problem 25325

Evaluate the integrals using $I$ : a. $\int_{0}^{1}(10x-2x^{3})dx$ b. $\int_{1}^{8}(5x-x^{3})dx$ a. $\int_{0}^{1}(10x-2x^{3})dx=\frac{9}{2}$ b. $\int_{1}^{0}(5x-x^{3})dx=\square$

See Solution

Problem 25326

Given $\int_{2}^{4} f(x) d x=4$ , $\int_{2}^{7} f(x) d x=8$ , and $\int_{2}^{7} g(x) d x=5$ , find:
1. $\int_{7}^{2} g(x) d x$
2. $\int_{2}^{7} 6 g(x) d x$
3. $\int_{2}^{7}[g(x)-f(x)] d x$
4. $\int_{2}^{7}[7 g(x)-f(x)] d x=\square$

See Solution

Problem 25327

Mickey jumps from a 1000 ft cliff at 3 ft/s. Without a parachute, when will he hit the ground? Calculate the time.

See Solution

Problem 25328

Find the integral of the function: $\int\left(2 \sqrt{x}-\frac{7}{x^{8}}\right) dx$ .

See Solution

Problem 25329

What is the total value of \$30,000/year for 25 years at a 6% continuous compounding rate?

See Solution

Problem 25330

A satellite of mass $m = 2,140 \mathrm{~kg}$ orbits Earth ( $M = 5.97 \times 10^{24} \mathrm{~kg}$ ) at $r = 9,325 \mathrm{~km}$ . Find the orbital period.

See Solution

Problem 25331

Calculate the future value of a continuous income stream $f(t)=1500 e^{-0.02 t}$ at $5.5\%$ interest for 4 years. What is the value? \$ \square

See Solution

Problem 25332

Find the derivative $\frac{d y}{d x}$ for the function $y=5 x^{3} \cos x \sin x$ .

See Solution

Problem 25333

An investor has two options: a furniture store with income $f(t)=12,000$ and a book store with $g(t)=11,000 e^{0.06 t}$ . Find their future values over 4 years.

See Solution

Problem 25334

Calculate the integral from $a$ to $b$ of $\frac{1}{3} x^{4}$ : $\int_{a}^{b} \frac{1}{3} x^{4} d x=\square$

See Solution

Problem 25335

Find the total cost of producing 220 dresses using the marginal cost function $C^{\prime}(x)=-\frac{4}{25} x+56$ . Total cost is \$\square.

See Solution

Problem 25336

Evaluate the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1} n^{2}}{3 n^{6}+2 n^{3}}$ .

See Solution

Problem 25337

Evaluate the integral: $\int \frac{e^{x}}{e^{x}-5+4 e^{x}} d x$

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Problem 25338

Find the derivative of the function $f(x)=\frac{7 x}{x^{2}+49}$ .

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Problem 25339

Find the tangent line equation to $y=x \sqrt{3 x^{2}-8}$ at the point $(2,4)$ .

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Problem 25340

Evaluate the integral from 1 to 4: $\int_{1}^{4} \frac{t^{5}-6 t^{2}}{t^{4}} d t$ .

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Problem 25341

Find the area between $f(x)=x$ and $g(x)=x^{1/n}$ for $x \geq 0$ , in terms of positive integer $n \geq 2$ . Area: $\square$ .

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Problem 25342

Find the critical points of the function $f(x)=\frac{7 x}{x^{2}+49}$ .

See Solution

Problem 25343

Find the derivative $\frac{d y}{d x}$ for the function $y=5 x^{3} \cos x \sin x$ .

See Solution

Problem 25344

Evaluate the integral from 5 to 6: $\int_{5}^{6} \frac{4 x^{2}-49}{2 x-7} d x$

See Solution

Problem 25345

Find the relative rate of change of $f(x)=110 x-0.4 x^{2}$ . The answer is $\square$ .

See Solution

Problem 25346

Find the limit: $\lim _{x \rightarrow-\infty} \frac{8 x^{3}-x^{2}}{1+x-x^{2}}$

See Solution

Problem 25347

Find the cost of installing $55 \mathrm{ft}^{2}$ of countertop using $C^{\prime}(x)=x^{\frac{2}{3}}$ and then for an extra $14 \mathrm{ft}^{2}$ .

See Solution

Problem 25348

Find the relative rate of change of the function $f(x)=120 x-0.3 x^{2}$ . What is it?

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Problem 25349

Find the absolute extrema of $f(x)=(x+1)^{\frac{2}{3}}$ on the interval $[0,3]$ .

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Problem 25350

Find the center, radius, and interval of convergence for the series: $\sum_{n=0}^{\infty} \frac{\left(\frac{1}{2}\right)^{n}\left(x-\frac{4}{5}\right)^{n}}{n !}$

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Problem 25351

Verify the Mean Value Theorem for $f(x)=\sqrt{4x-3}$ on $[1,3]$ and find a suitable $c$ .

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Problem 25352

Find values of $a$ for which $f(x, y, z)=x^{2}+y^{2}+z^{2}-x z+a y z$ is convex for all $(x, y, z)$ .

See Solution

Problem 25353

Differentiate $4 x^{2} y^{3}$ and $9 x + 9 y$ with respect to $x$ or $y$ .

See Solution

Problem 25354

Find the relative rate of change of $f(x)=234-3x$ at $x=22$ . Answer: $\square$ (round to three decimal places).

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Problem 25355

What direction does $a$ approach -4 from in the limit: $\lim _{a \rightarrow-4}\left(\int_{a}^{-3} \frac{1}{(x+4)^{2 / 3}} d x\right)$ ?

See Solution

Problem 25356

Calculate the relative rate of change of the function $f(x)=14+9 \ln x$ . The answer is $\square$ .

See Solution

Problem 25357

Evaluate the integral $\int_{0}^{2}(4 x+5) \mathrm{dx}$ using right Riemann sums. Simplify your answer to $\square$ .

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Problem 25358

What direction does $a$ approach $-4$ from? $\lim _{a \rightarrow \infty}\left(\int_{-2}^{a} \frac{1}{(x+5)^{2 / 3}} d x\right)$

See Solution

Problem 25359

Evaluate the integral using geometry: $\int_{-8}^{4} \sqrt{32-4 x-x^{2}} d x=\square$ (Use $\pi$ if needed.)

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Problem 25360

Quelles sont les dimensions du rectangle d'aire maximale inscrit sous la courbe $f(x)=\sqrt{16-x^{2}}$ ?

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Problem 25361

Find the direction of $a$ as it approaches $\infty$ in the limit: $\lim _{a \rightarrow \infty}\left(\int_{-5}^{a} \frac{1}{(x+9)^{3 / 2}} d x\right)$

See Solution

Problem 25362

Find the direction of $a$ as it approaches $\infty$ for the limit: $\lim _{a \rightarrow \infty}\left(\int_{-2}^{a} \frac{1}{(x+5)^{2 / 3}} d x\right).$

See Solution

Problem 25363

Find $b$ in terms of $c$ and $d$ such that $\int_{c}(x+b) dx=0$ for $0<c<d$ . $b=\square \text { (Type an exact answer.) }$

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Problem 25364

Evaluate the integral $\int_{1}^{7} f(x) d x$ where $f(x)=2x$ for $1 \leq x \leq 5$ and $f(x)=12-2x$ for $5<x \leq 7$ .

See Solution

Problem 25365

Find the rate of change of the base area of a cone when its radius is 6 ft and increasing at $\frac{1}{4} \mathrm{ft/hr}$ .

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Problem 25366

What direction does $a$ approach 7 from in the limit: $\lim _{a \rightarrow 7}\left(\int_{a}^{10} \frac{1}{(x-7)^{3 / 2}} d x\right)$ ?

See Solution

Problem 25367

The function $f$ is continuous and strictly increasing on $(0,1)$ with $f(0)=f\left(\frac{1}{2}\right)=f(1)$ . Which statement is true? (A) Minimum and maximum on $[0,1]$ . (B) Minimum but no maximum on $[0,1]$ . (C) Maximum but no minimum on $[0,1]$ . (D) Neither minimum nor maximum on $[0,1]$ .

See Solution

Problem 25368

Find $\frac{d}{d x} \int_{a}^{x} f(t) d t$ and $\frac{d}{d x} \int_{a}^{b} f(t) d t$ , with constants $a$ and $b$ .

See Solution

Problem 25369

Show that the sequence defined by $a_{1}=2$ and $a_{n+1}=\frac{1}{2}(a_{n}+6)$ is nondecreasing and bounded by 6. Find $\lim_{n \to \infty} a_n$ .

See Solution

Problem 25370

Déterminez l'équation de la tangente à la courbe $4 x^{2} y^{3}=9 x+9 y$ au point $(3,1)$ .

See Solution

Problem 25371

How high must a rocket go above Earth for its weight to be half? Use $F = G \frac{m_1 m_2}{r^2}$ to find $h$ .

See Solution

Problem 25372

Calculate the area between the curve $f(x) = \frac{1}{x^{3}}$ and the $x$ -axis over the interval $[-3,-2]$ . Area $=\square$ .

See Solution

Problem 25373

Determine if the series $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}}{n^{3}+1}$ converges or diverges. Show your work.

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Problem 25374

Calculate the area between the curve $f(x)=\sin x$ and the $x$ -axis from $[-2 \pi / 3,3 \pi / 4]$ . Area = $\square$ .

See Solution

Problem 25375

Simplify the expression: $\frac{d}{d x} \int_{x}^{5} \sqrt{t^{2}+3} d t = \square$

See Solution

Problem 25376

Simplify the expression: $\frac{d}{d x} \int_{-x}^{x} \sqrt{4+t^{2}} dt = \square$

See Solution

Problem 25377

Calculate the relative rate of change of $f(x) = 3x^2 - \ln x$ at $x = 5$ . Answer: $\square$ .

See Solution

Problem 25378

Find all values of $c$ that satisfy the Mean Value Theorem for $f(x)=x^{3}-3x^{2}$ on the interval $[0,3]$ .

See Solution

Problem 25379

Is the function $f(x)=\sum_{n=0}^{\infty} x e^{-n x^{2}}$ continuous at $x=0$ ? Justify your response.

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Problem 25380

Derive the position equation $s = \frac{a}{2} t^{2} + v_{0} t + s_{0}$ from the differential equation $\frac{d^{2} s}{d t^{2}} = a$ with initial conditions $v_{0}$ and $s_{0}$ .

See Solution

Problem 25381

Find the elasticity of demand using $p + 0.001x = 42$ at $p = \$22$ . What is the percentage change in demand if $p$ decreases by 10%?

See Solution

Problem 25382

Find the area function $A(x)=\int_{1}^{x} \frac{1}{t^{4}} dt$ using the Fundamental Theorem. What is $A(x)$ ?

See Solution

Problem 25383

Find the 102nd derivative of $f(x) = \ln(4x)$ .

See Solution

Problem 25384

Is the function $f(x)=\sum_{n=0}^{\infty} x e^{-n x^{2}}$ continuous at $x=0$ ? Justify your response.

See Solution

Problem 25385

Find the area function $A(x)=\int_{1}^{x} \frac{1}{t^{3}} dt$ using the Fundamental Theorem. What is $A(x)$ ?

See Solution

Problem 25386

Find the 45th derivative of $f(x)=\ln(6x)$ .

See Solution

Problem 25387

Find the derivative $f^{\prime}(x)$ of $f(x)=2 x^{2}+3 x-2$ and compute $f^{\prime}(1)$ , $f^{\prime}(4)$ , $f^{\prime}(5)$ .

See Solution

Problem 25388

Free fall on a planet: Derive $s=-\frac{1}{2} g t^{2}+v_{0} t+s_{0}$ from an initial value problem. Explain your steps.

See Solution

Problem 25389

Given $f(x)=\frac{1}{x^{2}}$ with $a=1, b=5, c=6$ , find $A(x)=\int_{1}^{x} f(t) dt$ . What is $A(x)$ ?

See Solution

Problem 25390

Is the integral $\int_{0}^{\infty} e^{6 x} d x$ convergent or divergent? If convergent, find its value.

See Solution

Problem 25391

Calculate the present value of a continuous income stream at rate $R(t)=\$ 132,000$ , over $T=20$ years with $k=3\%$ .

See Solution

Problem 25392

Differentiate $f(x)=a x^{21}$ with respect to $x$ , where $a$ is a constant.

See Solution

Problem 25393

Find the area function $A(x)=\int_{1}^{x} \frac{1}{t^{3}} dt$ using the Fundamental Theorem. What is $A(x)$ ?

See Solution

Problem 25394

Find the sales function $S(t)=9 \sqrt{t+1}$ . Calculate $S^{\prime}(t)$ , $S(15)$ , $S^{\prime}(15)$ , and estimate sales for 16 and 17 months.

See Solution

Problem 25395

Differentiate $f(x)=a x^{21}$ with respect to $x$ , where $a$ is a constant.

See Solution

Problem 25396

Check if the integral $\int_{2}^{\infty} \frac{4}{x^{4}} d x$ converges or diverges, and find its value if it converges.

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Problem 25397

Is it true or false that if $\sum_{n \in \mathbb{N}} a_{n}$ is conditionally convergent, then $\sum_{n \in \mathbb{N}} n^{k} a_{n}$ diverges for each $k \in(1,+\infty)$ ? Prove or provide a counterexample.

See Solution

Problem 25398

Find the ozone level function $P(t)=170+15t-t^{2}$ , then calculate $P^{\prime}(t)$ and evaluate $P(5)$ and $P^{\prime}(5)$ .

See Solution

Problem 25399

Find the derivative $p^{\prime}(t)$ of the tungsten consumption function $p(t)=140 t^{2}+1,075 t+14,911$ . Then, calculate $p(12)$ for 2022 and $p^{\prime}(12)$ for the rate of change. Interpret these results.

See Solution

Problem 25400

Find the derivative $P^{\prime}(t)$ of $P(t)=40+19t-4t^{2}$ and calculate $P(2)$ and $P^{\prime}(2)$ .

See Solution

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Calculus

Problem 25301

A cylindrical coffeepot with radius 5 in has volume change rate −5πh-5 \pi \sqrt{h}−5πh​. Find the height change rate equation in terms of hhh.

Problem 25302

Evaluate the integral from 1 to e8e^{8}e8 of (ln⁡x)4x\frac{(\ln x)^{4}}{x}x(lnx)4​ dx.

Problem 25303

Find the area between f(x)=e5xf(x)=e^{5 x}f(x)=e5x and g(x)=12xg(x)=\frac{1}{2 x}g(x)=2x1​ from x=1x=1x=1 to x=5x=5x=5.

Problem 25304

Find the indefinite integral of sec⁡2θ+9\sec^{2} \theta + 9sec2θ+9 with respect to θ\thetaθ.

Problem 25305

Evaluate the integrals given: a. ∫258f(x)dx=48\int_{2}^{5} 8 f(x) d x=48∫25​8f(x)dx=48 b. ∫27(f(x)−g(x))dx=□\int_{2}^{7}(f(x)-g(x)) d x=\square∫27​(f(x)−g(x))dx=□

Problem 25306

Evaluate the integrals given: a. ∫258f(x)dx=48\int_{2}^{5} 8 f(x) d x=48∫25​8f(x)dx=48 b. ∫27(f(x)−g(x))dx=5\int_{2}^{7}(f(x)-g(x)) d x=5∫27​(f(x)−g(x))dx=5 c. ∫25(f(x)−g(x))dx=□\int_{2}^{5}(f(x)-g(x)) d x=\square∫25​(f(x)−g(x))dx=□

Problem 25307

Find the integral of tan⁡5θsec⁡7θ\tan^{5} \theta \sec^{7} \thetatan5θsec7θ with respect to θ\thetaθ: ∫tan⁡5θsec⁡7θdθ\int \tan^{5} \theta \sec^{7} \theta d \theta∫tan5θsec7θdθ.

Problem 25308

Find h′(2)h^{\prime}(2)h′(2) for h(x)=f(x)⋅g(x)h(x)=f(x) \cdot g(x)h(x)=f(x)⋅g(x) given f(2)=3f(2)=3f(2)=3, f′(2)=4f^{\prime}(2)=4f′(2)=4, g(2)=−2g(2)=-2g(2)=−2, g′(2)=−4g^{\prime}(2)=-4g′(2)=−4.

Problem 25309

Problem 25310

Problem 25311

Problem 25312

Find the water velocities at points 2 and 3 in a hydroelectric plant with a 100 cm tube and 50 cm nozzle.

Problem 25313

Evaluate ∫72g(x)dx\int_{7}^{2} g(x) d x∫72​g(x)dx given ∫24f(x)dx=4\int_{2}^{4} f(x) d x=4∫24​f(x)dx=4, ∫27f(x)dx=8\int_{2}^{7} f(x) d x=8∫27​f(x)dx=8, ∫27g(x)dx=5\int_{2}^{7} g(x) d x=5∫27​g(x)dx=5.

Problem 25314

Calculate the total income over 15 years from a continuous income stream with a rate of flow f(t)=3500f(t) = 3500f(t)=3500. What is the total earned? \ \square$

Problem 25315

At age 35, you deposit \$2000 annually into an IRA earning 5\% interest, compounded continuously. What's the total at age 65?

Problem 25316

Given ∫24f(x)dx=4\int_{2}^{4} f(x) d x=4∫24​f(x)dx=4, ∫27f(x)dx=8\int_{2}^{7} f(x) d x=8∫27​f(x)dx=8, and ∫27g(x)dx=5\int_{2}^{7} g(x) d x=5∫27​g(x)dx=5, find:1. ∫72g(x)dx\int_{7}^{2} g(x) d x∫72​g(x)dx2. ∫276g(x)dx=□\int_{2}^{7} 6 g(x) d x=\square∫27​6g(x)dx=□

Problem 25317

Approximate ∫311exdx\int_{3}^{11} e^{x} d x∫311​exdx using 4 equal subintervals and right endpoints for the Riemann Sum.

Problem 25318

Calculate the total income from the function f(t)=500e0.05tf(t)=500 e^{0.05 t}f(t)=500e0.05t over the first 2 years. Total income: \$ \square (round to nearest dollar).

Problem 25319

Problem 25320

A student makes quarterly payments of \3,000intoaretirementaccountwith3,000 into a retirement account with 3,000intoaretirementaccountwith6\%$ interest compounded continuously. Find the total after 40 years.

Problem 25321

Calculate the future value of a continuous income stream f(t)=1550e−0.02tf(t)=1550 e^{-0.02 t}f(t)=1550e−0.02t at 5.75%5.75\%5.75% interest over 7 years. What is the amount? \$ \square

Problem 25322

You deposit \2000yearlyintoanIRAfromage30to65ata52000 yearly into an IRA from age 30 to 65 at a 5% continuous compound interest. What's the total value at 65? \$\square2000yearlyintoanIRAfromage30to65ata5 (Round to the nearest dollar.)

Problem 25323

Use the value of the first integral to evaluate the two integrals: 1. ∫01(10x−2x3)dx=□\int_{0}^{1}(10 x-2 x^{3}) dx=\square∫01​(10x−2x3)dx=□ 2. ∫10(5x−x3)dx\int_{1}^{0}(5 x-x^{3}) dx∫10​(5x−x3)dx

Problem 25324

An object is launched upward at 39.2 m/s39.2 \mathrm{~m/s}39.2 m/s. How long is it at or above 34.3 meters?

Problem 25325

Problem 25326

Problem 25327

Mickey jumps from a 1000 ft cliff at 3 ft/s. Without a parachute, when will he hit the ground? Calculate the time.

Problem 25328

Find the integral of the function: ∫(2x−7x8)dx\int\left(2 \sqrt{x}-\frac{7}{x^{8}}\right) dx∫(2x​−x87​)dx.

Problem 25329

What is the total value of \$30,000/year for 25 years at a 6% continuous compounding rate?

Problem 25330

A satellite of mass m=2,140 kgm = 2,140 \mathrm{~kg}m=2,140 kg orbits Earth (M=5.97×1024 kgM = 5.97 \times 10^{24} \mathrm{~kg}M=5.97×1024 kg) at r=9,325 kmr = 9,325 \mathrm{~km}r=9,325 km. Find the orbital period.

Problem 25331

Calculate the future value of a continuous income stream f(t)=1500e−0.02tf(t)=1500 e^{-0.02 t}f(t)=1500e−0.02t at 5.5%5.5\%5.5% interest for 4 years. What is the value? \$ \square

Problem 25332

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=5x3cos⁡xsin⁡xy=5 x^{3} \cos x \sin xy=5x3cosxsinx.

Problem 25333

An investor has two options: a furniture store with income f(t)=12,000f(t)=12,000f(t)=12,000 and a book store with g(t)=11,000e0.06tg(t)=11,000 e^{0.06 t}g(t)=11,000e0.06t. Find their future values over 4 years.

Problem 25334

Calculate the integral from aaa to bbb of 13x4\frac{1}{3} x^{4}31​x4: ∫ab13x4dx=□\int_{a}^{b} \frac{1}{3} x^{4} d x=\square∫ab​31​x4dx=□

Problem 25335

Find the total cost of producing 220 dresses using the marginal cost function C′(x)=−425x+56C^{\prime}(x)=-\frac{4}{25} x+56C′(x)=−254​x+56. Total cost is \$\square.

Problem 25336

Evaluate the series ∑n=1∞(−1)n+1n23n6+2n3\sum_{n=1}^{\infty} \frac{(-1)^{n+1} n^{2}}{3 n^{6}+2 n^{3}}∑n=1∞​3n6+2n3(−1)n+1n2​.

Problem 25337

Evaluate the integral: ∫exex−5+4exdx\int \frac{e^{x}}{e^{x}-5+4 e^{x}} d x∫ex−5+4exex​dx

Problem 25338

Find the derivative of the function f(x)=7xx2+49f(x)=\frac{7 x}{x^{2}+49}f(x)=x2+497x​.

Problem 25339

Find the tangent line equation to y=x3x2−8y=x \sqrt{3 x^{2}-8}y=x3x2−8​ at the point (2,4)(2,4)(2,4).

Problem 25340

Evaluate the integral from 1 to 4: ∫14t5−6t2t4dt\int_{1}^{4} \frac{t^{5}-6 t^{2}}{t^{4}} d t∫14​t4t5−6t2​dt.

Problem 25341

Find the area between f(x)=xf(x)=xf(x)=x and g(x)=x1/ng(x)=x^{1/n}g(x)=x1/n for x≥0x \geq 0x≥0, in terms of positive integer n≥2n \geq 2n≥2. Area: □\square□.

Problem 25342

Find the critical points of the function f(x)=7xx2+49f(x)=\frac{7 x}{x^{2}+49}f(x)=x2+497x​.

Problem 25343

A cylindrical coffeepot with radius 5 in has volume change rate $-5 \pi \sqrt{h}$ . Find the height change rate equation in terms of $h$ .

Evaluate the integral from 1 to $e^{8}$ of $\frac{(\ln x)^{4}}{x}$ dx.

Find the area between $f(x)=e^{5 x}$ and $g(x)=\frac{1}{2 x}$ from $x=1$ to $x=5$ .

Find the indefinite integral of $\sec^{2} \theta + 9$ with respect to $\theta$ .

Evaluate the integrals given: a. $\int_{2}^{5} 8 f(x) d x=48$ b. $\int_{2}^{7}(f(x)-g(x)) d x=\square$

Evaluate the integrals given:
a. $\int_{2}^{5} 8 f(x) d x=48$
b. $\int_{2}^{7}(f(x)-g(x)) d x=5$
c. $\int_{2}^{5}(f(x)-g(x)) d x=\square$

Find the integral of $\tan^{5} \theta \sec^{7} \theta$ with respect to $\theta$ : $\int \tan^{5} \theta \sec^{7} \theta d \theta$ .

Find $h^{\prime}(2)$ for $h(x)=f(x) \cdot g(x)$ given $f(2)=3$ , $f^{\prime}(2)=4$ , $g(2)=-2$ , $g^{\prime}(2)=-4$ .

Evaluate $\int_{7}^{2} g(x) d x$ given $\int_{2}^{4} f(x) d x=4$ , $\int_{2}^{7} f(x) d x=8$ , $\int_{2}^{7} g(x) d x=5$ .

Calculate the total income over 15 years from a continuous income stream with a rate of flow $f(t) = 3500$ . What is the total earned? \\square$

Given $\int_{2}^{4} f(x) d x=4$ , $\int_{2}^{7} f(x) d x=8$ , and $\int_{2}^{7} g(x) d x=5$ , find:
1. $\int_{7}^{2} g(x) d x$
2. $\int_{2}^{7} 6 g(x) d x=\square$

Approximate $\int_{3}^{11} e^{x} d x$ using 4 equal subintervals and right endpoints for the Riemann Sum.

Calculate the total income from the function $f(t)=500 e^{0.05 t}$ over the first 2 years. Total income: \$ \square (round to nearest dollar).

A student makes quarterly payments of \ $3,000 into a retirement account with$ 6\%$ interest compounded continuously. Find the total after 40 years.

Calculate the future value of a continuous income stream $f(t)=1550 e^{-0.02 t}$ at $5.75\%$ interest over 7 years. What is the amount? \$ \square

You deposit \ $2000 yearly into an IRA from age 30 to 65 at a 5% continuous compound interest. What's the total value at 65? \$\square$ (Round to the nearest dollar.)

Use the value of the first integral to evaluate the two integrals:
1. $\int_{0}^{1}(10 x-2 x^{3}) dx=\square$
2. $\int_{1}^{0}(5 x-x^{3}) dx$

An object is launched upward at $39.2 \mathrm{~m/s}$ . How long is it at or above 34.3 meters?

Find the integral of the function: $\int\left(2 \sqrt{x}-\frac{7}{x^{8}}\right) dx$ .

A satellite of mass $m = 2,140 \mathrm{~kg}$ orbits Earth ( $M = 5.97 \times 10^{24} \mathrm{~kg}$ ) at $r = 9,325 \mathrm{~km}$ . Find the orbital period.

Calculate the future value of a continuous income stream $f(t)=1500 e^{-0.02 t}$ at $5.5\%$ interest for 4 years. What is the value? \$ \square

Find the derivative $\frac{d y}{d x}$ for the function $y=5 x^{3} \cos x \sin x$ .

An investor has two options: a furniture store with income $f(t)=12,000$ and a book store with $g(t)=11,000 e^{0.06 t}$ . Find their future values over 4 years.

Calculate the integral from $a$ to $b$ of $\frac{1}{3} x^{4}$ : $\int_{a}^{b} \frac{1}{3} x^{4} d x=\square$

Find the total cost of producing 220 dresses using the marginal cost function $C^{\prime}(x)=-\frac{4}{25} x+56$ . Total cost is \$\square.

Evaluate the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1} n^{2}}{3 n^{6}+2 n^{3}}$ .

Evaluate the integral: $\int \frac{e^{x}}{e^{x}-5+4 e^{x}} d x$

Find the derivative of the function $f(x)=\frac{7 x}{x^{2}+49}$ .

Find the tangent line equation to $y=x \sqrt{3 x^{2}-8}$ at the point $(2,4)$ .

Evaluate the integral from 1 to 4: $\int_{1}^{4} \frac{t^{5}-6 t^{2}}{t^{4}} d t$ .

Find the area between $f(x)=x$ and $g(x)=x^{1/n}$ for $x \geq 0$ , in terms of positive integer $n \geq 2$ . Area: $\square$ .

Find the critical points of the function $f(x)=\frac{7 x}{x^{2}+49}$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=5 x^{3} \cos x \sin x$ .

Evaluate the integral from 5 to 6: $\int_{5}^{6} \frac{4 x^{2}-49}{2 x-7} d x$

Find the relative rate of change of $f(x)=110 x-0.4 x^{2}$ . The answer is $\square$ .

Find the limit: $\lim _{x \rightarrow-\infty} \frac{8 x^{3}-x^{2}}{1+x-x^{2}}$

Find the cost of installing $55 \mathrm{ft}^{2}$ of countertop using $C^{\prime}(x)=x^{\frac{2}{3}}$ and then for an extra $14 \mathrm{ft}^{2}$ .

Find the relative rate of change of the function $f(x)=120 x-0.3 x^{2}$ . What is it?

Find the absolute extrema of $f(x)=(x+1)^{\frac{2}{3}}$ on the interval $[0,3]$ .

Find the center, radius, and interval of convergence for the series: $\sum_{n=0}^{\infty} \frac{\left(\frac{1}{2}\right)^{n}\left(x-\frac{4}{5}\right)^{n}}{n !}$

Verify the Mean Value Theorem for $f(x)=\sqrt{4x-3}$ on $[1,3]$ and find a suitable $c$ .

Find values of $a$ for which $f(x, y, z)=x^{2}+y^{2}+z^{2}-x z+a y z$ is convex for all $(x, y, z)$ .

Differentiate $4 x^{2} y^{3}$ and $9 x + 9 y$ with respect to $x$ or $y$ .

Find the relative rate of change of $f(x)=234-3x$ at $x=22$ . Answer: $\square$ (round to three decimal places).

What direction does $a$ approach -4 from in the limit: $\lim _{a \rightarrow-4}\left(\int_{a}^{-3} \frac{1}{(x+4)^{2 / 3}} d x\right)$ ?

Calculate the relative rate of change of the function $f(x)=14+9 \ln x$ . The answer is $\square$ .

Evaluate the integral $\int_{0}^{2}(4 x+5) \mathrm{dx}$ using right Riemann sums. Simplify your answer to $\square$ .

What direction does $a$ approach $-4$ from? $\lim _{a \rightarrow \infty}\left(\int_{-2}^{a} \frac{1}{(x+5)^{2 / 3}} d x\right)$

Evaluate the integral using geometry: $\int_{-8}^{4} \sqrt{32-4 x-x^{2}} d x=\square$ (Use $\pi$ if needed.)

Quelles sont les dimensions du rectangle d'aire maximale inscrit sous la courbe $f(x)=\sqrt{16-x^{2}}$ ?

Find the direction of $a$ as it approaches $\infty$ in the limit: $\lim _{a \rightarrow \infty}\left(\int_{-5}^{a} \frac{1}{(x+9)^{3 / 2}} d x\right)$

Find the direction of $a$ as it approaches $\infty$ for the limit: $\lim _{a \rightarrow \infty}\left(\int_{-2}^{a} \frac{1}{(x+5)^{2 / 3}} d x\right).$

Find $b$ in terms of $c$ and $d$ such that $\int_{c}(x+b) dx=0$ for $0<c<d$ . $b=\square \text { (Type an exact answer.) }$

Evaluate the integral $\int_{1}^{7} f(x) d x$ where $f(x)=2x$ for $1 \leq x \leq 5$ and $f(x)=12-2x$ for $5<x \leq 7$ .

Find the rate of change of the base area of a cone when its radius is 6 ft and increasing at $\frac{1}{4} \mathrm{ft/hr}$ .

What direction does $a$ approach 7 from in the limit: $\lim _{a \rightarrow 7}\left(\int_{a}^{10} \frac{1}{(x-7)^{3 / 2}} d x\right)$ ?

Find $\frac{d}{d x} \int_{a}^{x} f(t) d t$ and $\frac{d}{d x} \int_{a}^{b} f(t) d t$ , with constants $a$ and $b$ .

Show that the sequence defined by $a_{1}=2$ and $a_{n+1}=\frac{1}{2}(a_{n}+6)$ is nondecreasing and bounded by 6. Find $\lim_{n \to \infty} a_n$ .

Déterminez l'équation de la tangente à la courbe $4 x^{2} y^{3}=9 x+9 y$ au point $(3,1)$ .

How high must a rocket go above Earth for its weight to be half? Use $F = G \frac{m_1 m_2}{r^2}$ to find $h$ .

Calculate the area between the curve $f(x) = \frac{1}{x^{3}}$ and the $x$ -axis over the interval $[-3,-2]$ . Area $=\square$ .

Determine if the series $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}}{n^{3}+1}$ converges or diverges. Show your work.

Calculate the area between the curve $f(x)=\sin x$ and the $x$ -axis from $[-2 \pi / 3,3 \pi / 4]$ . Area = $\square$ .