Calculus

Problem 20401

Given the marginal cost function $C^{\prime}(x)=-0.04 x+30$ and cost at 200 items is \$10,135.
a. Find $C^{\prime}(580)$ , round to the nearest cent. b. Integrate to find $C(x)$ . c. Compute $C(580)$ , round to the nearest dollar.

See Solution

Problem 20402

Given $C^{\prime}(x)=-0.06 x+28$ , find $C^{\prime}(590)$ , the cost function $C(x)$ , and $C(590)$ .

See Solution

Problem 20403

Is the integral $\int_{0}^{\infty} \frac{1}{\sqrt[8]{1+x}} d x$ convergent or divergent? Evaluate if convergent.

See Solution

Problem 20404

Find the average value of $f(x)=2 x^{3}-5 x+2$ on $[1,4]$ . Also, calculate $\int_{1}^{8}\left(\sqrt[3]{x^{2}}-\frac{5}{x}\right) d x$ .

See Solution

Problem 20405

A hemispherical tank with radius 6 m has water depth $h$ . When $h=3$ m and decreasing at 0.5 m/min, find the rate of area decrease.

See Solution

Problem 20406

Find the rate of change of $w$ when $x=6$ and $y=20$ , given $w=x^{2}y$ , $dx/dt=-1$ , $dy/dt=4$ .

See Solution

Problem 20407

Evaluate the series: $\sum_{n=1}^{\infty} \frac{\sqrt{n}+4}{n^{2}}$ .

See Solution

Problem 20408

A sphere's volume increases at $6 \pi$ cm³/hr. Find the diameter's increase rate when the radius is 3 cm. Use $V=\frac{4}{3} \pi r^{3}$ .

See Solution

Problem 20409

Find the exact value of the integral: $\int_{1}^{8}\left(\sqrt[3]{x^{2}}-\frac{5}{x}\right) d x$ . Show your work.

See Solution

Problem 20410

Express the limit as a definite integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[x_{i}^{} \sqrt{1+x_{i}^{3}}\right] \Delta x$ over $[2,5]$ .

See Solution

Problem 20411

Use a substitution to simplify $\int(\cos x) \ln (\sin x+15) d x$ to $\int \ln u \, du$ and evaluate it.

See Solution

Problem 20412

Evaluate the integral using limits: $\int_{-1}^{2} (4 x^{2}+x+2) \, dx$ .

See Solution

Problem 20413

Calculate the limit: $\lim _{x \rightarrow 0} \frac{6 x \tan (x)}{1-\cos (x)}$ .

See Solution

Problem 20414

Evaluate the integral using limits: $\int_{-1}^{2} (4 x^{2} + x + 2) \, dx$ .

See Solution

Problem 20415

Evaluate the integral using limits: $\int_{-1}^{2} (4 x^{2}+x+2) \, dx$ [8 points]

See Solution

Problem 20416

Graph the function $|x-2|-2$ and find the area to evaluate the integral $\int_{-1}^{4}(|x-2|-2) d x$ .

See Solution

Problem 20417

Find the rate at which the water level rises when the water is 4 cm deep in a cone with height 8 cm and radius 4 cm.

See Solution

Problem 20418

Find the surface area when the curve $x=\sqrt{a^{2}-y^{2}}$ (for $0 \leq y \leq a / 5$ ) is rotated about the $y$ -axis.

See Solution

Problem 20419

Find the value of $\int_{3}^{7} f(x) \, dx$ given the area under the curve decreases from $x=3$ to $x=7$ .

See Solution

Problem 20420

Set up an integral for the curve length of $y = x - 6 \ln(x)$ from $x = 1$ to $x = 4$ : $\int_{1}^{4}(\square) \, dx$

See Solution

Problem 20421

Evaluate the integral: $\int \frac{6 x^{2}+5 x+11}{(x+1)(x^{2}+1)} d x = \square$

See Solution

Problem 20422

Find the centripetal acceleration of Earth orbiting the Sun with a radius of $149.6$ million km.

See Solution

Problem 20423

Find relative extrema for $f(x)=x^{4}-4x^{3}+2$ using the Second Derivative Test.

See Solution

Problem 20424

Estimate the distance traveled by a car with velocity $v$ from $t=0$ to $t=8$ seconds, given $v(0)=50$ , $v(3)=40$ , $v(8)=0$ . Choices: (A) 21 ft, (B) 26 ft, (C) 150 ft, (D) 210 ft, (E) 260 ft.

See Solution

Problem 20425

Find the limit as $h$ approaches 0 of $\frac{f(x, y+h)-f(x, y)}{h}$ for $f(x, y)=y^{2}+5xy$ .

See Solution

Problem 20426

How much will $\$ 3000$ grow in 15 years at a continuous $3\%$ interest rate?

See Solution

Problem 20427

Calculate the Left Riemann Sum for $y=x+5$ on $[0,3]$ with $n=3$ .

See Solution

Problem 20428

Use the Limit Comparison Test for the series $\sum_{n=1}^{\infty} \frac{2n^{2}+n+1}{n^{4}+8n^{2}-9}$ . Find suitable $b_{n}$ and limit $L$ . Does it converge or diverge?

See Solution

Problem 20429

Determine if the series $\sum_{k=1}^{\infty} \frac{1}{9^{k}+3}$ converges using the Comparison or Limit Comparison Test.

See Solution

Problem 20430

Evaluate the integral $\int \frac{\ln x}{x^{12}} dx = \square$ using integration by parts.

See Solution

Problem 20431

A bacteria culture starts with 180 and grows to 540 in 3 hours. Find $P(t)$ , population after 9 hours, and time to reach 2630.

See Solution

Problem 20432

Determine if the series $\sum_{k=1}^{\infty} \frac{6(-1)^{k+1}}{k^{9}}$ converges. Define $a_{k}$ . Choose A, B, or C and find $\lim_{k \to \infty} a_{k}$ . Does it converge? Yes/No.

See Solution

Problem 20433

Calculate the Left Riemann Sum for $y=\frac{1}{x-1}$ on $[10,20]$ with $n=5$ .

See Solution

Problem 20434

Evaluate the integral using limits: $\int_{-1}^{2} (4 x^{2}+x+2) \, dx$ .

See Solution

Problem 20435

Find the partial derivatives of the function $f(x, y)=-5 x^{6}+3 x y^{4}-2 y^{3}$ : $f_{x}(x, y)$ and $f_{y}(x, y)$ .

See Solution

Problem 20436

Find the partial derivative of $2 e^{2 x y}$ with respect to $x$ .

See Solution

Problem 20437

Find the partial derivative $f_{x}(-2,3)$ for the function $f(x, y)=\sqrt{3 x^{2}+3 y^{2}}$ .

See Solution

Problem 20438

Calculate the Left Riemann Sum for $\int_{0}^{1} x^{3} d x$ with $n=3$ .

See Solution

Problem 20439

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

See Solution

Problem 20440

Calculate the Right Riemann Sum for $\int_{2}^{8}(x^{2}+x) dx$ with $n=2$ .

See Solution

Problem 20441

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{3}{(-4)^{n}}$ : absolute, conditional, or divergent?

See Solution

Problem 20442

Express the limits of $L_{n}$ and $R_{n}$ as $n \rightarrow \infty$ using definite integrals for given sums.

See Solution

Problem 20443

Calculate the integral of $x^{4}$ with respect to $x$ .

See Solution

Problem 20444

Evaluate the integral from -1 to 2 of the function $4 x^{2}+x+2$ .

See Solution

Problem 20445

Find the integral of the function: $\int 2 x^{-6} d x$ .

See Solution

Problem 20446

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

See Solution

Problem 20447

Find the partial derivatives of the function $f(x, y, z)=\sqrt{-x-5y-6z}$ : $f_{x}(x, y, z)$ , $f_{y}(x, y, z)$ , $f_{z}(x, y, z)$ .

See Solution

Problem 20448

Calculate the Right Riemann Sum for $\int_{0}^{2 \pi} \sin (x) dx$ with $n=4$ (in radians).

See Solution

Problem 20449

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

See Solution

Problem 20450

Find $f_x(4,2)$ and $f_y(4,2)$ for the function $f(x, y)=4 x^{5}-6 x y^{2}-3 y^{4}$ .

See Solution

Problem 20451

Is the integral $\int_{6}^{\infty} \frac{2}{(x+4)^{3 / 2}} d x$ convergent or divergent? If convergent, evaluate it.

See Solution

Problem 20452

Consider the series $a_{n}=\frac{3}{(-4)^{n}}$ . Which is true about the partial sums $S_{N}$ : $S_{111}<S_{113}<S_{112}$ or others?

See Solution

Problem 20453

Find the partial derivatives $f_{x}(x, y)$ , $f_{y}(x, y)$ , $f_{x x}(x, y)$ , and $f_{x y}(x, y)$ for $f(x, y)=-x^{3}-5 x^{2} y^{5}-4 y^{2}$ .

See Solution

Problem 20454

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{3}{(-4)^{n}}$ : absolute, conditional, or divergent?

See Solution

Problem 20455

Calculate the integral: $\int\left(3 z^{2}-2 z+6\right) d z$

See Solution

Problem 20456

Calculate the integral of the function $\sqrt{t} + 16$ with respect to $t$ .

See Solution

Problem 20457

Find the best estimate for $R'(5)$ using the given values of $R(t)$ at $t = 0, 1, 2, 3, 4, 5$ . Round to two decimal places.

See Solution

Problem 20458

Given $a_{n}=\frac{3}{(-4)^{n}}$ , which statement about partial sums $S_{N}$ is true: $S_{113}<S_{112}<S_{111}$ , $S_{111}<S_{113}<S_{112}$ , $S_{111}<S_{112}<S_{113}$ , or $S_{113}<S_{111}<S_{112}$ ?

See Solution

Problem 20459

Find the partial derivatives $f_{x}(x, y)$ , $f_{y}(x, y)$ , and $f_{x x}(x, y)$ for $f(x, y)=-x^{3}-5x^{2}y^{5}-4y^{2}$ .

See Solution

Problem 20460

Find the position function $s(t)$ for a particle with velocity $v(t)=3 t^{2}-1$ and $s(1)=2 \mathrm{~m}$ .

See Solution

Problem 20461

What is the correct integral notation for $\int (x+3) \, dx$ ?

See Solution

Problem 20462

Find the second partial derivatives of the function $f(x, y)=4 x^{3}+3 x y^{6}-3 y^{5}$ : $f_{x x}(x, y)$ and $f_{x y}(x, y)$ .

See Solution

Problem 20463

Find the second partial derivatives of the function $f(x, y)=-6 x^{5}-5 x y^{3}+y^{4}$ : $f_{x x}(x, y)$ and $f_{x y}(x, y)$ .

See Solution

Problem 20464

Find the tangent line equation for $f(x)=\frac{3 x^{2}}{(2-5 x)^{5}}$ at $x=1$ .

See Solution

Problem 20465

A firm has a cost function $C(x, y)=5x^2+9xy+6y^2+1500$ . Find total cost for $x=140$ , $y=110$ . Then, find rates of change for $x$ and $y$ .

See Solution

Problem 20466

Calculate the integral: $\int 3 \, dz =$

See Solution

Problem 20467

Evaluate the series or state if it diverges: $\sum_{k=1}^{\infty}\left[\frac{1}{7}\left(\frac{1}{4}\right)^{k}+\frac{2}{3}\left(\frac{1}{4}\right)^{k}\right]$

See Solution

Problem 20468

How much caffeine remains in Jack's bloodstream after 13 hours if he starts with $160 \mathrm{mg}$ and has a half-life of 4 hours?

See Solution

Problem 20469

Evaluate the integral: $\int \frac{1}{\sqrt{x}(6+\sqrt{x})^{2}} \, dx$

See Solution

Problem 20470

Calculate the integral: $\int_{0}^{2}\left(5-x^{2}+2 x^{3}\right) d x$

See Solution

Problem 20471

Evaluate the series or determine if it diverges: $\sum_{k=1}^{\infty}\left[\frac{2}{5}\left(\frac{1}{5}\right)^{k}+\frac{3}{5}\left(\frac{1}{6}\right)^{k}\right]$

See Solution

Problem 20472

Evaluate the integral using substitution: $\int_{\ln 2}^{\ln 4} \frac{5 e^{x}}{(e^{x}+2)^{2}} d x$ .

See Solution

Problem 20473

Evaluate the integral using substitution: $\int_{\ln 2}^{\ln 2} \frac{3 e^{x}}{(e^{x}+4)^{2}} dx$ .

See Solution

Problem 20474

Find the marginal productivity of labor $P_L$ and capital $P_K$ for the function $P(L, K)=22 L^{0.4} K^{0.6}$ . Use positive powers.

See Solution

Problem 20475

Find the marginal productivity of labor and capital for the function $P(L, K)=10 L^{0.4} K^{0.6}$ at $L=16$ , $K=15$ .

See Solution

Problem 20476

Evaluate the integral using substitution: $\int_{\ln 5}^{\ln 7} \frac{2 e^{x}}{(e^{x}+1)^{2}} d x$ .

See Solution

Problem 20477

Evaluate the series or state if it diverges:
$\sum_{k=1}^{\infty}\left[\frac{2}{7}\left(\frac{1}{5}\right)^{k}+\frac{2}{3}\left(\frac{1}{4}\right)^{k}\right]$
Choose A for a value or B for divergence.

See Solution

Problem 20478

Find the indefinite integral of $\frac{x}{\sqrt{x+11}}$ with respect to $x$ .

See Solution

Problem 20479

Evaluate the series or state if it diverges:
$\sum_{k=1}^{\infty}\left[\frac{2}{5}\left(\frac{1}{5}\right)^{k}+\frac{3}{5}\left(\frac{1}{6}\right)^{k}\right]$
Choose A for the sum or B for divergence.

See Solution

Problem 20480

Find the marginal productivity of labor and capital for the function $P(L, K)=10 L^{0.4} K^{0.6}$ with $L=16$ , $K=15$ .

See Solution

Problem 20481

Evaluate the 2nd partial sum of the series $\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(5 k+2)^{3}}$ and find the error bound.

See Solution

Problem 20482

Evaluate the 2nd partial sum of the series $\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(6 k+3)^{3}}$ and find the error bound.

See Solution

Problem 20483

Find the function $f(x)$ if $f^{(4)}(x)=48$ , $f^{\prime \prime \prime}(-1)=-96$ , $f^{\prime \prime}(1)=-48$ , $f^{\prime}(-1)=-20$ , $f(0)=24$ .

See Solution

Problem 20484

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

See Solution

Problem 20485

Evaluate the 2nd partial sum of the series $\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(5 k+2)^{3}}$ and find the error bound $\left|S-S_{2}\right|$ .

See Solution

Problem 20486

Find the partial derivative $f_{x}(2,-3)$ for the function $f(x, y)=\sqrt{4 x^{2}+3 y^{2}}$ .

See Solution

Problem 20487

Given the marginal cost function $C^{\prime}(x)=-0.04 x+27$ and cost at 150 items is \$8,075.
a. Find $C^{\prime}(520)$ , rounded to the nearest cent.
b. Integrate to find $C(x)$ .
c. Compute $C(520)$ , rounded to the nearest dollar.

See Solution

Problem 20488

Calculate the area under the curves: 1) Set up the integral. 2) For $f(x)=\sqrt{x}$ from $x=4$ to $x=9$ . 3) For $f(x)=16x^3-9x^2+12x+7$ from $[0,4]$ .

See Solution

Problem 20489

Find a function $f(x)$ where $f'(x) = -4 e^{x} - 2x$ and $f(0) = -4$ . What is $f(x)$ ?

See Solution

Problem 20490

Find the number of cars passing an intersection from 6 am to 9 am given $r(t)=300+700 t-180 t^{2}$ .

See Solution

Problem 20491

Find the total cost for producing the first 49 units given the marginal cost function $\frac{19}{\sqrt{x}}$ . Total cost: \$

See Solution

Problem 20492

Calculate the integral $\int x^{3} e^{x} d x$ using integration by parts: $\int u dv = uv - \int v du$ . Set $u=\square$ , $dv=\square dx$ .

See Solution

Problem 20493

Calculate the area under the curve $y=\frac{7}{x}$ from $x=1$ to $x=2$ .

See Solution

Problem 20494

Calculate the area under the curve $y=3 x^{5}$ from $x=0$ to $x=5$ . What is the exact value?

See Solution

Problem 20495

Calculate the area of region $R$ in the first quadrant between the $x$ -axis, $y=\ln(x)$ , and $y=5-x$ .

See Solution

Problem 20496

Find the average weekly sales for the first 5 weeks given $S(t)=9 e^{t}$ , where $S(t)$ is in hundreds of dollars.

See Solution

Problem 20497

Find the average value of $f(x)=5 \cdot x^{2}$ over the interval $0 \leq x \leq 4$ .

See Solution

Problem 20498

Find the volume of a solid with semicircular cross sections above $R$ , where $R$ is bounded by $y=\ln(x)$ and $y=5-x$ .

See Solution

Problem 20499

Determine where the function $p(x)=-(x+5)^{4}-3$ is increasing, decreasing, or constant in interval notation.

See Solution

Problem 20500

Calculate the average value of $f(x)=4 x^{5}$ over the interval $3 \leq x \leq 5$ . Round to two decimal places.

See Solution

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Calculus

Problem 20401

Given the marginal cost function C′(x)=−0.04x+30C^{\prime}(x)=-0.04 x+30C′(x)=−0.04x+30 and cost at 200 items is \$10,135. a. Find C′(580)C^{\prime}(580)C′(580), round to the nearest cent. b. Integrate to find C(x)C(x)C(x). c. Compute C(580)C(580)C(580), round to the nearest dollar.

Problem 20402

Given C′(x)=−0.06x+28C^{\prime}(x)=-0.06 x+28C′(x)=−0.06x+28, find C′(590)C^{\prime}(590)C′(590), the cost function C(x)C(x)C(x), and C(590)C(590)C(590).

Problem 20403

Is the integral ∫0∞11+x8dx\int_{0}^{\infty} \frac{1}{\sqrt[8]{1+x}} d x∫0∞​81+x​1​dx convergent or divergent? Evaluate if convergent.

Problem 20404

Find the average value of f(x)=2x3−5x+2f(x)=2 x^{3}-5 x+2f(x)=2x3−5x+2 on [1,4][1,4][1,4]. Also, calculate ∫18(x23−5x)dx\int_{1}^{8}\left(\sqrt[3]{x^{2}}-\frac{5}{x}\right) d x∫18​(3x2​−x5​)dx.

Problem 20405

A hemispherical tank with radius 6 m has water depth hhh. When h=3h=3h=3 m and decreasing at 0.5 m/min, find the rate of area decrease.

Problem 20406

Find the rate of change of www when x=6x=6x=6 and y=20y=20y=20, given w=x2yw=x^{2}yw=x2y, dx/dt=−1dx/dt=-1dx/dt=−1, dy/dt=4dy/dt=4dy/dt=4.

Problem 20407

Evaluate the series: ∑n=1∞n+4n2\sum_{n=1}^{\infty} \frac{\sqrt{n}+4}{n^{2}}∑n=1∞​n2n​+4​.

Problem 20408

A sphere's volume increases at 6π6 \pi6π cm³/hr. Find the diameter's increase rate when the radius is 3 cm. Use V=43πr3V=\frac{4}{3} \pi r^{3}V=34​πr3.

Problem 20409

Find the exact value of the integral: ∫18(x23−5x)dx\int_{1}^{8}\left(\sqrt[3]{x^{2}}-\frac{5}{x}\right) d x∫18​(3x2​−x5​)dx. Show your work.

Problem 20410

Express the limit as a definite integral: lim⁡n→∞∑i=1n[xi∗1+xi∗3]Δx\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[x_{i}^{*} \sqrt{1+x_{i}^{*3}}\right] \Delta xlimn→∞​∑i=1n​[xi∗​1+xi∗3​​]Δx over [2,5][2,5][2,5].

Problem 20411

Use a substitution to simplify ∫(cos⁡x)ln⁡(sin⁡x+15)dx\int(\cos x) \ln (\sin x+15) d x∫(cosx)ln(sinx+15)dx to ∫ln⁡u du\int \ln u \, du∫lnudu and evaluate it.

Problem 20412

Evaluate the integral using limits: ∫−12(4x2+x+2) dx\int_{-1}^{2} (4 x^{2}+x+2) \, dx∫−12​(4x2+x+2)dx.

Problem 20413

Calculate the limit: lim⁡x→06xtan⁡(x)1−cos⁡(x)\lim _{x \rightarrow 0} \frac{6 x \tan (x)}{1-\cos (x)}limx→0​1−cos(x)6xtan(x)​.

Problem 20414

Evaluate the integral using limits: ∫−12(4x2+x+2) dx\int_{-1}^{2} (4 x^{2} + x + 2) \, dx∫−12​(4x2+x+2)dx.

Problem 20415

Evaluate the integral using limits: ∫−12(4x2+x+2) dx\int_{-1}^{2} (4 x^{2}+x+2) \, dx∫−12​(4x2+x+2)dx [8 points]

Problem 20416

Graph the function ∣x−2∣−2|x-2|-2∣x−2∣−2 and find the area to evaluate the integral ∫−14(∣x−2∣−2)dx\int_{-1}^{4}(|x-2|-2) d x∫−14​(∣x−2∣−2)dx.

Problem 20417

Find the rate at which the water level rises when the water is 4 cm deep in a cone with height 8 cm and radius 4 cm.

Problem 20418

Find the surface area when the curve x=a2−y2x=\sqrt{a^{2}-y^{2}}x=a2−y2​ (for 0≤y≤a/50 \leq y \leq a / 50≤y≤a/5) is rotated about the yyy-axis.

Problem 20419

Find the value of ∫37f(x) dx\int_{3}^{7} f(x) \, dx∫37​f(x)dx given the area under the curve decreases from x=3x=3x=3 to x=7x=7x=7.

Problem 20420

Set up an integral for the curve length of y=x−6ln⁡(x)y = x - 6 \ln(x)y=x−6ln(x) from x=1x = 1x=1 to x=4x = 4x=4: ∫14(□) dx\int_{1}^{4}(\square) \, dx∫14​(□)dx

Problem 20421

Evaluate the integral: ∫6x2+5x+11(x+1)(x2+1)dx=□\int \frac{6 x^{2}+5 x+11}{(x+1)(x^{2}+1)} d x = \square∫(x+1)(x2+1)6x2+5x+11​dx=□

Problem 20422

Find the centripetal acceleration of Earth orbiting the Sun with a radius of 149.6149.6149.6 million km.

Problem 20423

Find relative extrema for f(x)=x4−4x3+2f(x)=x^{4}-4x^{3}+2f(x)=x4−4x3+2 using the Second Derivative Test.

Problem 20424

Estimate the distance traveled by a car with velocity vvv from t=0t=0t=0 to t=8t=8t=8 seconds, given v(0)=50v(0)=50v(0)=50, v(3)=40v(3)=40v(3)=40, v(8)=0v(8)=0v(8)=0. Choices: (A) 21 ft, (B) 26 ft, (C) 150 ft, (D) 210 ft, (E) 260 ft.

Problem 20425

Find the limit as hhh approaches 0 of f(x,y+h)−f(x,y)h\frac{f(x, y+h)-f(x, y)}{h}hf(x,y+h)−f(x,y)​ for f(x,y)=y2+5xyf(x, y)=y^{2}+5xyf(x,y)=y2+5xy.

Problem 20426

How much will $3000\$ 3000$3000 grow in 15 years at a continuous 3%3\%3% interest rate?

Problem 20427

Calculate the Left Riemann Sum for y=x+5y=x+5y=x+5 on [0,3][0,3][0,3] with n=3n=3n=3.

Problem 20428

Use the Limit Comparison Test for the series ∑n=1∞2n2+n+1n4+8n2−9\sum_{n=1}^{\infty} \frac{2n^{2}+n+1}{n^{4}+8n^{2}-9}n=1∑∞​n4+8n2−92n2+n+1​. Find suitable bnb_{n}bn​ and limit LLL. Does it converge or diverge?

Problem 20429

Determine if the series ∑k=1∞19k+3\sum_{k=1}^{\infty} \frac{1}{9^{k}+3}k=1∑∞​9k+31​ converges using the Comparison or Limit Comparison Test.

Problem 20430

Evaluate the integral ∫ln⁡xx12dx=□\int \frac{\ln x}{x^{12}} dx = \square∫x12lnx​dx=□ using integration by parts.

Problem 20431

A bacteria culture starts with 180 and grows to 540 in 3 hours. Find P(t)P(t)P(t), population after 9 hours, and time to reach 2630.

Problem 20432

Determine if the series ∑k=1∞6(−1)k+1k9\sum_{k=1}^{\infty} \frac{6(-1)^{k+1}}{k^{9}}k=1∑∞​k96(−1)k+1​ converges. Define aka_{k}ak​. Choose A, B, or C and find lim⁡k→∞ak\lim_{k \to \infty} a_{k}limk→∞​ak​. Does it converge? Yes/No.

Problem 20433

Calculate the Left Riemann Sum for y=1x−1y=\frac{1}{x-1}y=x−11​ on [10,20][10,20][10,20] with n=5n=5n=5.

Problem 20434

Evaluate the integral using limits: ∫−12(4x2+x+2) dx\int_{-1}^{2} (4 x^{2}+x+2) \, dx∫−12​(4x2+x+2)dx.

Problem 20435

Find the partial derivatives of the function f(x,y)=−5x6+3xy4−2y3f(x, y)=-5 x^{6}+3 x y^{4}-2 y^{3}f(x,y)=−5x6+3xy4−2y3: fx(x,y)f_{x}(x, y)fx​(x,y) and fy(x,y)f_{y}(x, y)fy​(x,y).

Problem 20436

Find the partial derivative of 2e2xy2 e^{2 x y}2e2xy with respect to xxx.

Problem 20437

Find the partial derivative fx(−2,3)f_{x}(-2,3)fx​(−2,3) for the function f(x,y)=3x2+3y2f(x, y)=\sqrt{3 x^{2}+3 y^{2}}f(x,y)=3x2+3y2​.

Problem 20438

Calculate the Left Riemann Sum for ∫01x3dx\int_{0}^{1} x^{3} d x∫01​x3dx with n=3n=3n=3.

Problem 20439

Find fx(2,4)f_{x}(2,4)fx​(2,4) and fy(2,4)f_{y}(2,4)fy​(2,4) for the function f(x,y)=−4x3−5xy5+2y2f(x, y)=-4 x^{3}-5 x y^{5}+2 y^{2}f(x,y)=−4x3−5xy5+2y2.

Problem 20440

Given the marginal cost function $C^{\prime}(x)=-0.04 x+30$ and cost at 200 items is \$10,135.
a. Find $C^{\prime}(580)$ , round to the nearest cent. b. Integrate to find $C(x)$ . c. Compute $C(580)$ , round to the nearest dollar.

Given $C^{\prime}(x)=-0.06 x+28$ , find $C^{\prime}(590)$ , the cost function $C(x)$ , and $C(590)$ .

Is the integral $\int_{0}^{\infty} \frac{1}{\sqrt[8]{1+x}} d x$ convergent or divergent? Evaluate if convergent.

Find the average value of $f(x)=2 x^{3}-5 x+2$ on $[1,4]$ . Also, calculate $\int_{1}^{8}\left(\sqrt[3]{x^{2}}-\frac{5}{x}\right) d x$ .

A hemispherical tank with radius 6 m has water depth $h$ . When $h=3$ m and decreasing at 0.5 m/min, find the rate of area decrease.

Find the rate of change of $w$ when $x=6$ and $y=20$ , given $w=x^{2}y$ , $dx/dt=-1$ , $dy/dt=4$ .

Evaluate the series: $\sum_{n=1}^{\infty} \frac{\sqrt{n}+4}{n^{2}}$ .

A sphere's volume increases at $6 \pi$ cm³/hr. Find the diameter's increase rate when the radius is 3 cm. Use $V=\frac{4}{3} \pi r^{3}$ .

Find the exact value of the integral: $\int_{1}^{8}\left(\sqrt[3]{x^{2}}-\frac{5}{x}\right) d x$ . Show your work.

Express the limit as a definite integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[x_{i}^{} \sqrt{1+x_{i}^{3}}\right] \Delta x$ over $[2,5]$ .

Use a substitution to simplify $\int(\cos x) \ln (\sin x+15) d x$ to $\int \ln u \, du$ and evaluate it.

Evaluate the integral using limits: $\int_{-1}^{2} (4 x^{2}+x+2) \, dx$ .

Calculate the limit: $\lim _{x \rightarrow 0} \frac{6 x \tan (x)}{1-\cos (x)}$ .

Evaluate the integral using limits: $\int_{-1}^{2} (4 x^{2} + x + 2) \, dx$ .

Evaluate the integral using limits: $\int_{-1}^{2} (4 x^{2}+x+2) \, dx$ [8 points]

Graph the function $|x-2|-2$ and find the area to evaluate the integral $\int_{-1}^{4}(|x-2|-2) d x$ .

Find the surface area when the curve $x=\sqrt{a^{2}-y^{2}}$ (for $0 \leq y \leq a / 5$ ) is rotated about the $y$ -axis.

Find the value of $\int_{3}^{7} f(x) \, dx$ given the area under the curve decreases from $x=3$ to $x=7$ .

Set up an integral for the curve length of $y = x - 6 \ln(x)$ from $x = 1$ to $x = 4$ : $\int_{1}^{4}(\square) \, dx$

Evaluate the integral: $\int \frac{6 x^{2}+5 x+11}{(x+1)(x^{2}+1)} d x = \square$

Find the centripetal acceleration of Earth orbiting the Sun with a radius of $149.6$ million km.

Find relative extrema for $f(x)=x^{4}-4x^{3}+2$ using the Second Derivative Test.

Estimate the distance traveled by a car with velocity $v$ from $t=0$ to $t=8$ seconds, given $v(0)=50$ , $v(3)=40$ , $v(8)=0$ . Choices: (A) 21 ft, (B) 26 ft, (C) 150 ft, (D) 210 ft, (E) 260 ft.

Find the limit as $h$ approaches 0 of $\frac{f(x, y+h)-f(x, y)}{h}$ for $f(x, y)=y^{2}+5xy$ .

How much will $\$ 3000$ grow in 15 years at a continuous $3\%$ interest rate?

Calculate the Left Riemann Sum for $y=x+5$ on $[0,3]$ with $n=3$ .

Use the Limit Comparison Test for the series $\sum_{n=1}^{\infty} \frac{2n^{2}+n+1}{n^{4}+8n^{2}-9}$ . Find suitable $b_{n}$ and limit $L$ . Does it converge or diverge?

Determine if the series $\sum_{k=1}^{\infty} \frac{1}{9^{k}+3}$ converges using the Comparison or Limit Comparison Test.

Evaluate the integral $\int \frac{\ln x}{x^{12}} dx = \square$ using integration by parts.

A bacteria culture starts with 180 and grows to 540 in 3 hours. Find $P(t)$ , population after 9 hours, and time to reach 2630.

Determine if the series $\sum_{k=1}^{\infty} \frac{6(-1)^{k+1}}{k^{9}}$ converges. Define $a_{k}$ . Choose A, B, or C and find $\lim_{k \to \infty} a_{k}$ . Does it converge? Yes/No.

Calculate the Left Riemann Sum for $y=\frac{1}{x-1}$ on $[10,20]$ with $n=5$ .

Evaluate the integral using limits: $\int_{-1}^{2} (4 x^{2}+x+2) \, dx$ .

Find the partial derivatives of the function $f(x, y)=-5 x^{6}+3 x y^{4}-2 y^{3}$ : $f_{x}(x, y)$ and $f_{y}(x, y)$ .

Find the partial derivative of $2 e^{2 x y}$ with respect to $x$ .

Find the partial derivative $f_{x}(-2,3)$ for the function $f(x, y)=\sqrt{3 x^{2}+3 y^{2}}$ .

Calculate the Left Riemann Sum for $\int_{0}^{1} x^{3} d x$ with $n=3$ .

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

Calculate the Right Riemann Sum for $\int_{2}^{8}(x^{2}+x) dx$ with $n=2$ .

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{3}{(-4)^{n}}$ : absolute, conditional, or divergent?

Express the limits of $L_{n}$ and $R_{n}$ as $n \rightarrow \infty$ using definite integrals for given sums.

Calculate the integral of $x^{4}$ with respect to $x$ .

Evaluate the integral from -1 to 2 of the function $4 x^{2}+x+2$ .

Find the integral of the function: $\int 2 x^{-6} d x$ .

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

Find the partial derivatives of the function $f(x, y, z)=\sqrt{-x-5y-6z}$ : $f_{x}(x, y, z)$ , $f_{y}(x, y, z)$ , $f_{z}(x, y, z)$ .

Calculate the Right Riemann Sum for $\int_{0}^{2 \pi} \sin (x) dx$ with $n=4$ (in radians).

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

Find $f_x(4,2)$ and $f_y(4,2)$ for the function $f(x, y)=4 x^{5}-6 x y^{2}-3 y^{4}$ .

Is the integral $\int_{6}^{\infty} \frac{2}{(x+4)^{3 / 2}} d x$ convergent or divergent? If convergent, evaluate it.

Consider the series $a_{n}=\frac{3}{(-4)^{n}}$ . Which is true about the partial sums $S_{N}$ : $S_{111}<S_{113}<S_{112}$ or others?

Find the partial derivatives $f_{x}(x, y)$ , $f_{y}(x, y)$ , $f_{x x}(x, y)$ , and $f_{x y}(x, y)$ for $f(x, y)=-x^{3}-5 x^{2} y^{5}-4 y^{2}$ .

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{3}{(-4)^{n}}$ : absolute, conditional, or divergent?

Calculate the integral: $\int\left(3 z^{2}-2 z+6\right) d z$

Calculate the integral of the function $\sqrt{t} + 16$ with respect to $t$ .

Find the best estimate for $R'(5)$ using the given values of $R(t)$ at $t = 0, 1, 2, 3, 4, 5$ . Round to two decimal places.

Find the partial derivatives $f_{x}(x, y)$ , $f_{y}(x, y)$ , and $f_{x x}(x, y)$ for $f(x, y)=-x^{3}-5x^{2}y^{5}-4y^{2}$ .

Find the position function $s(t)$ for a particle with velocity $v(t)=3 t^{2}-1$ and $s(1)=2 \mathrm{~m}$ .

What is the correct integral notation for $\int (x+3) \, dx$ ?

Find the second partial derivatives of the function $f(x, y)=4 x^{3}+3 x y^{6}-3 y^{5}$ : $f_{x x}(x, y)$ and $f_{x y}(x, y)$ .

Find the second partial derivatives of the function $f(x, y)=-6 x^{5}-5 x y^{3}+y^{4}$ : $f_{x x}(x, y)$ and $f_{x y}(x, y)$ .

Find the tangent line equation for $f(x)=\frac{3 x^{2}}{(2-5 x)^{5}}$ at $x=1$ .

A firm has a cost function $C(x, y)=5x^2+9xy+6y^2+1500$ . Find total cost for $x=140$ , $y=110$ . Then, find rates of change for $x$ and $y$ .

Calculate the integral: $\int 3 \, dz =$

Evaluate the series or state if it diverges: $\sum_{k=1}^{\infty}\left[\frac{1}{7}\left(\frac{1}{4}\right)^{k}+\frac{2}{3}\left(\frac{1}{4}\right)^{k}\right]$

How much caffeine remains in Jack's bloodstream after 13 hours if he starts with $160 \mathrm{mg}$ and has a half-life of 4 hours?

Evaluate the integral: $\int \frac{1}{\sqrt{x}(6+\sqrt{x})^{2}} \, dx$

Calculate the integral: $\int_{0}^{2}\left(5-x^{2}+2 x^{3}\right) d x$

Evaluate the series or determine if it diverges: $\sum_{k=1}^{\infty}\left[\frac{2}{5}\left(\frac{1}{5}\right)^{k}+\frac{3}{5}\left(\frac{1}{6}\right)^{k}\right]$

Evaluate the integral using substitution: $\int_{\ln 2}^{\ln 4} \frac{5 e^{x}}{(e^{x}+2)^{2}} d x$ .

Evaluate the integral using substitution: $\int_{\ln 2}^{\ln 2} \frac{3 e^{x}}{(e^{x}+4)^{2}} dx$ .

Find the marginal productivity of labor $P_L$ and capital $P_K$ for the function $P(L, K)=22 L^{0.4} K^{0.6}$ . Use positive powers.