Calculus

Problem 19301

Find the area between $y=3-x^{2}$ and $y=\sqrt{x}$ from $x=0$ to $x=1$ .

See Solution

Problem 19302

Évaluez les intégrales suivantes : a) $\int x \sqrt{1+4 x} d x$ , b) $\int \frac{1+2 x}{1+16 x^{2}} d x$ , c) $\int(4 t+2) e^{t^{2}+t} d x$ , d) $\int \frac{e^{x}}{e^{x}+e^{-x}} d x$ , e) $\int x^{2} 2^{x} d x$ , f) $\int x^{2} e^{4 x} d x$ , g) $\int 3 t^{3} \sin \left(t^{2}\right) d t$ , h) $\int x^{2}(\ln x)^{2} d x$ . Trouvez le profit maximal d'un appareil acheté à \ $.2500 avec$ R $et$ C $donnés par$ \frac{d R}{d t}=100(18-3 \sqrt{t}) $et$ \frac{d C}{d t}=100(2+\sqrt{t})$.

See Solution

Problem 19303

Find the area between the curves $y=2 x^{3}-3 x^{2}-9 x$ and $y=x^{3}-2 x^{2}-3 x$ . Provide the exact value.

See Solution

Problem 19304

Find the tangent line to $f(x)=x^{2}-5x+4$ at $x=5$ , the normal at $x=3$ , and their intersection point $P$ .

See Solution

Problem 19305

An object moves with velocity $v(t)=t(8-t)$ . Its initial position is -3. Find its position after 16 sec and total distance traveled.

See Solution

Problem 19306

Find the general antiderivative of $f(x)=4 e^{x}+\sec^{2} x+x^{7/3}+x^{3/7}$ .

See Solution

Problem 19307

Find the sky diver's velocity $v(t)=59(1-e^{-0.23 t})$ after 2 seconds and 5 seconds, rounding to the nearest whole number.

See Solution

Problem 19308

Choose the integral for the area bounded by $y=x^{2}$ , $x=2$ , and the $x$ -axis: $\int_{0}^{2} x^{2} d x$ .

See Solution

Problem 19309

Find the area above the $x$ -axis and below $y=4x^2+4x+3$ from $x=0$ to $x=2$ . Enter an exact value.

See Solution

Problem 19310

Find the tangent and normal lines at point $P(4,2.5)$ on the curve $y=\sqrt{x}+\frac{1}{\sqrt{x}}$ , then calculate the area of triangle $ABP$ .

See Solution

Problem 19311

Find the initial temperature and the temperature after 18 minutes for $T(x)=-5+27 e^{-0.041 x}$ . Round to the nearest degree.

See Solution

Problem 19312

Find the area above the $x$ -axis under the curve $y=1-x^{4}$ . Enter the exact value, e.g., "3+1/2".

See Solution

Problem 19313

Find the area between $y=2x+1$ and $y=-x^{2}$ from $x=0$ to $x=1$ .

See Solution

Problem 19314

Evaluate the integral from -3 to 0: $\int_{-3}^{0}\left(8 x-e^{x}\right) d x$

See Solution

Problem 19315

Find the area between $f(x)=1-x^{2}$ and $g(x)=4-4x^{2}$ from $x=0$ to $x=1$ .

See Solution

Problem 19316

An object moves with velocity $v(t)=t(6-t)$ , starting 5 units left of the origin. Find its position and distance after 18 seconds.

See Solution

Problem 19317

Find the total area between the curve $y=f(x)=x^{2}-x-2$ and the $x$ -axis from $x=-7$ to $x=6$ . Calculate:
$\int_{-7}^{6} f(x) \, dx$
Interpret this integral in terms of areas.

See Solution

Problem 19318

Find the tangent line equation at point $P(4,2.5)$ on the curve $y=\sqrt{x}+\frac{1}{\sqrt{x}}$ in the form $a x+b y+c=0$ .

See Solution

Problem 19319

Find the tangent and normal lines at point $P(4,2.5)$ on the curve $y=\sqrt{x}+\frac{1}{\sqrt{x}}$ in the form $a x+b y+c=0$ .

See Solution

Problem 19320

Find the tangent line equation at point $P(4,2.5)$ on the curve $y=\sqrt{x}+\frac{1}{\sqrt{x}}$ in the form $a x+b y+c=0$ .

See Solution

Problem 19321

Rewrite the integral $\int x(x+1)^{11} dx$ using $u = x + 1$ and $du$ .

See Solution

Problem 19322

Eine Rennrodlerin fährt im Eiskanal. Gegeben sind: Durchschnittsgeschwindigkeit $35 \frac{\mathrm{m}}{\mathrm{s}}$ (20s-40s) und $w(20)=600$ .
a) Bestimme $w^{\prime}(30)$ in $\frac{m}{s}$ . b) Finde $w(40)$ . c) Zeige, dass es ein $t$ mit $w^{\prime}(t)>35$ gibt. d) Was sagt $w(30)$ aus, wenn $w(20)=600$ ?

See Solution

Problem 19323

Rewrite the integral using $u$ and $du$ : $\int 18 \sec^2(x) \tan^2(x) dx, \quad u=\tan(x)$ .

See Solution

Problem 19324

Find the general function $f$ such that $f^{\prime \prime}(x)=8 x+\cos x+3$ .

See Solution

Problem 19325

Differentiate the equation $V=-3 x-1.1 \ln (x)+1.7 y-1.6 \ln (y)$ implicitly to find $\frac{d y}{d x}$ . Evaluate at $x=5$ , $y=5$ . Round to three decimal places.

See Solution

Problem 19326

Find the integral: $\int \frac{3 x+1}{(3 x^{2}+2 x)^{3}} dx$

See Solution

Problem 19327

What are the units of $\int_{a}^{b} f(t) dt$ if $f(t)$ is in m/s² and $t$ is in seconds?

See Solution

Problem 19328

Estimate $f(6.4)$ using linear approximation with $f(6)=3.4$ and $f'(6)=2.3$ . Round to three decimal places.

See Solution

Problem 19329

Find the box dimensions with a length of 12 cm and volume of 1500 cm³ that minimize surface area.

See Solution

Problem 19330

Compute the flux $\iint_{S} \mathbf{F} \cdot d \mathbf{S}$ for $\mathbf{F}(x,y,z)=(x^{2}, y, z)$ using the divergence theorem on the unit sphere $S$ .

See Solution

Problem 19331

A substance grows at $17\%$ per day. If it starts at 719 grams, find its mass after 6 days. Round to the nearest tenth.

See Solution

Problem 19332

Consider the system:
$\dot{\theta}_{1}=E-\sin \theta_{1}+K \sin \left(\theta_{2}-\theta_{1}\right), \quad \dot{\theta}_{2}=E+\sin \theta_{2}+K \sin \left(\theta_{1}-\theta_{2}\right)$
where $E, K \geq 0$ .
a) Find and classify fixed points. b) Show periodic solutions exist for large $E$ and identify the bifurcation type. c) Determine the bifurcation curve in $(E, K)$ space for these solutions.

See Solution

Problem 19333

Find the integral: $\int 25 z^{2}\left(125 z^{3}+1\right)^{12} d z$

See Solution

Problem 19334

A radioactive substance starts at $259 \mathrm{~kg}$ and decays at $6\%$ per day. Find its mass after 3 days.

See Solution

Problem 19335

A biologist has a 5483-gram sample of a radioactive substance. Find the mass after 5 hours with a decay rate of $12\%$ per hour.

See Solution

Problem 19336

Find the horizontal asymptotes for $f(x)=\frac{x^{2}+4}{x^{2}-4}$ . Options: $y=1$ , $x=4$ , $x=-2$ , $y=-1$ .

See Solution

Problem 19337

Find the stationary points of the curve defined by $f(x)=2 x^{3}-\frac{1}{2} x^{2}-x+2$ .

See Solution

Problem 19338

Bestimme die ersten beiden Ableitungen von $f(x)=-\frac{1}{6} x^{3}+\frac{3}{4} x^{2}$ und zeichne den Graphen.

See Solution

Problem 19339

Find the derivative of $f(x)=-4 x^{2}-x$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 19340

Find the derivative of $g(x)=\int_{1-2 x}^{1+2 x} t \sin t \, dt$ .

See Solution

Problem 19341

Find all lines with slope -1 that are tangent to the curve $y=\frac{1}{x-1}$ .

See Solution

Problem 19342

An object dropped from a 100-m tower has height $100 - 4.9t^2$ m. Find its falling speed at $t = 2$ sec.

See Solution

Problem 19343

Compute the outward flux of $\mathbf{F}(x, y, z)=1 x \mathbf{i}+5 y \mathbf{j}+3 z \mathbf{k}$ across the prism $-1 \leq x \leq 6, -1 \leq y \leq 7, -1 \leq z \leq 7$ .

See Solution

Problem 19344

Find the derivative of $f(x)=x^{2}-5x-4$ . Which option is correct: (a) $-4$ , (b) $7$ , (c) $x^{2}-5x-4$ , or (d) $2x-5$ ?

See Solution

Problem 19345

Find the derivatives of $f(x)=4-x^{2}$ at $x=-3$ , $0$ , and $1$ : $f^{\prime}(-3)$ , $f^{\prime}(0)$ , $f^{\prime}(1)$ .

See Solution

Problem 19346

A substance grows at a rate of $16\%$ daily. If it starts at 48 grams, find its mass after 2 days. Round to the nearest tenth.

See Solution

Problem 19347

Compute the indefinite integral: $\int \frac{\sin 2 x}{\sin x} \, dx$ .

See Solution

Problem 19348

Find $\frac{d w}{d q}$ if $w=\frac{5^{q}}{\ln 5}$ .

See Solution

Problem 19349

Find the derivative of the function $f(t)=\frac{e^{8 t}}{t^{3}+10}$ , denoted as $f^{\prime}(t)=L_{t}$ .

See Solution

Problem 19350

Find the derivative $\frac{d w}{d q}$ if $w=\frac{5^{q}}{\ln 5}$ .

See Solution

Problem 19351

Find the derivative of the function $f(t)=\frac{e^{8 t}}{t^{3}+10}$ . What is $f^{\prime}(t)$ ?

See Solution

Problem 19352

Evaluate the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-x-12}{4 x^{2}+11 x-3}$ . If it doesn't exist, write DNE.

See Solution

Problem 19353

Find the derivative of the function $j(x)=\sqrt[5]{x} \ln x$ . What is $j^{\prime}(x)$ ?

See Solution

Problem 19354

A soda can's temperature $T(x) = -3 + 21 e^{-0.025 x}$ ; find $T(0)$ and $T(18)$ . Round to the nearest degree.

See Solution

Problem 19355

Calculate the half-life of a radioactive element with decay function $A(t)=A_{0} e^{-0.0355 t}$ . Answer: $\square$ years.

See Solution

Problem 19356

Find the half-life of a radioactive element with decay function $A(t)=A_{0} e^{-0.021 t}$ . Round to the nearest tenth.

See Solution

Problem 19357

Find the total increase in a fruit fly population from day 1 to day 12 given $g(t)=4 e^{0.1 t}$ and initial population of 380.

See Solution

Problem 19358

Find $g^{\prime}(-1)$ , $g^{\prime}(2)$ , and $g^{\prime}(\sqrt{3})$ for $g(t)=\frac{1}{t^{2}}$ .

See Solution

Problem 19359

Find the derivative of the function $f(t)=\frac{e^{8 t}}{t^{3}+10}$ . What is $f^{\prime}(t)$ ?

See Solution

Problem 19360

Find the number of fruit flies added from day 4 to day 14, given $g(t)=5 e^{0.02 t}$ and initial population 340.

See Solution

Problem 19361

Evaluate the integral from -3 to 1 of the constant -7: $\int_{-3}^{1}(-7) d p$ .

See Solution

Problem 19362

How long to double \$3000 at a continuous interest rate of 1.75\%? Round to the nearest hundredth.

See Solution

Problem 19363

Evaluate the integral from -2 to 4 of the function $5t - 6$ .

See Solution

Problem 19364

Evaluate the integral from 0 to 1 of $(5 x^{2}-4 x+9)$ . What is the result? $\int_{0}^{1}(5 x^{2}-4 x+9) d x=\square$

See Solution

Problem 19365

Find the indefinite integral using substitution: $\int 6(6 x-5)^{3} d x$

See Solution

Problem 19366

Evaluate the integral: $\int \frac{9 e^{9 t}}{9+e^{9 t}} d t$

See Solution

Problem 19367

Find the indefinite integral using substitution: $\int \frac{x^{3}+2 x}{x^{4}+4 x^{2}+2} d x$

See Solution

Problem 19368

Find the derivative $S^{\prime}(t)$ , compute $S(3)$ and $S^{\prime}(3)$ , and interpret $S(11)$ and $S^{\prime}(11)$ .

See Solution

Problem 19369

Find the indefinite integral using substitution: $\int \frac{5 x+9}{(5 x^{2}+18 x)^{3}} d x$

See Solution

Problem 19370

Find the indefinite integral using substitution: $\int \frac{x^{3}+2 x}{x^{4}+4 x^{2}+3} d x$

See Solution

Problem 19371

Find the indefinite integral using substitution: $\int \frac{x^{3}}{2 x^{4}+1} d x$

See Solution

Problem 19372

Find the indefinite integral using substitution: $\int\left(\sqrt{t^{5}+2 t}\right)\left(5 t^{4}+2\right) d t$ .

See Solution

Problem 19373

Find the indefinite integral using substitution: $\int z \sqrt{7 z^{2}-5} \, dz$

See Solution

Problem 19374

Find the indefinite integral using substitution: $\int 6 x^{5} e^{8 x^{6}} d x$

See Solution

Problem 19375

Find the function $f$ given that $f'(x)=\frac{8}{1+x^{2}}+2 \sinh x+9 \operatorname{sech}^{2} x+9^{x} \ln 9$ and $f(0)=14$ .

See Solution

Problem 19376

Find the function $f$ given $f'(x)=\frac{8}{1+x^{2}}+2 \sinh x+9 \operatorname{sech}^{2} x+9^{x} \ln 9$ and $f(0)=14$ .

See Solution

Problem 19377

Calculate the integral: $\int(15 \sqrt{x}+\sqrt{2}) d x$ .

See Solution

Problem 19378

Evaluate the integral: $\int 3 x\left(x^{2}+2\right) d x$

See Solution

Problem 19379

Evaluate the integral: $\int \frac{4}{x^{7}} d x$

See Solution

Problem 19380

Evaluate the integral: $\int\left(-22.5 t^{-3.5}-8 t^{-1}\right) dt$

See Solution

Problem 19381

Evaluate the integral $\int \frac{1}{9 x^{7}} d x$ .

See Solution

Problem 19382

Find the integral of the function: $\int 3 e^{-0.5 x} \mathrm{dx}$ .

See Solution

Problem 19383

Approximate the integral $\int_{3}^{8} \sqrt{x} \, dx$ using Riemann sums with 5 subintervals: a) right endpoints, b) left endpoints, c) midpoints. Round answers to two decimal places.

See Solution

Problem 19384

Evaluate the integral: $\int \frac{1+7 t^{6}}{2 t} d t$

See Solution

Problem 19385

Evaluate the integral: $\int\left(e^{10 u}+4 u\right) d u$

See Solution

Problem 19386

Calculate the integral of $(9x + 5)^{2}$ with respect to $x$ .

See Solution

Problem 19387

Evaluate the integral of the function: $\int 13^{x} d x$ .

See Solution

Problem 19388

Find the function $f(x)$ where $f^{\prime \prime}(x)=x^{2}+\sin (x)$ , given $f(0)=2$ and $f^{\prime}(0)=0$ .

See Solution

Problem 19389

Find the derivative of $f(x)=3x-5$ . Is it (a) $-3$ , (b) $3$ , (c) $3x-5$ , or (d) $3$ ?

See Solution

Problem 19390

Find the equation of the curve where the slope is $f^{\prime}(x)=4 x^{2}+3 x-6$ and it passes through $(0,4)$ .

See Solution

Problem 19391

Identify the divergent series from the following:
I: 24-12+6-3+...
II: -0.2+1-5+25-...
III: 64+32+16+8+...
IV: $\frac{1}{9}-\frac{1}{3}+1-3+...$
Options: I only, I and III, III only, II and IV, I, III and IV.

See Solution

Problem 19392

Find $p^{\prime}(1)$ , $p^{\prime}(3)$ , and $p^{\prime}(2/3)$ for $p(\theta)=\sqrt{3\theta}$ .

See Solution

Problem 19393

Find the limit of the sum: $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n}\left[\left(\frac{i}{n}\right)^{2}+2\left(\frac{i}{n}\right)+5\right]$

See Solution

Problem 19394

Compute the limit of the sum: $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n}\left[\left(\frac{i}{n}\right)^{2}+2\left(\frac{i}{n}\right)+5\right]$

See Solution

Problem 19395

Evaluate the integral: $\int \frac{\sec ^{2}(1 / x)}{x^{2}} d x$

See Solution

Problem 19396

Differentiate the function $g(t)=\frac{9-5 t}{7 t+6}$ .

See Solution

Problem 19397

Determine if the series $\sum_{k=1}^{\infty} \frac{(-1)^{k}}{\frac{5}{7}}$ converges absolutely, conditionally, or diverges.

See Solution

Problem 19398

Determine if the series $\sum_{k=1}^{\infty} \frac{5 \cos k}{6 k^{4}}$ converges absolutely, conditionally, or diverges.

See Solution

Problem 19399

Determine if the series $\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^{\frac{5}{7}}}$ converges absolutely, conditionally, or diverges.

See Solution

Problem 19400

Determine if the series $\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k^{4}}{k^{7}+1}$ converges absolutely, conditionally, or diverges.

See Solution

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Calculus

Problem 19301

Find the area between y=3−x2y=3-x^{2}y=3−x2 and y=xy=\sqrt{x}y=x​ from x=0x=0x=0 to x=1x=1x=1.

Problem 19302

Problem 19303

Find the area between the curves y=2x3−3x2−9xy=2 x^{3}-3 x^{2}-9 xy=2x3−3x2−9x and y=x3−2x2−3xy=x^{3}-2 x^{2}-3 xy=x3−2x2−3x. Provide the exact value.

Problem 19304

Find the tangent line to f(x)=x2−5x+4f(x)=x^{2}-5x+4f(x)=x2−5x+4 at x=5x=5x=5, the normal at x=3x=3x=3, and their intersection point PPP.

Problem 19305

An object moves with velocity v(t)=t(8−t)v(t)=t(8-t)v(t)=t(8−t). Its initial position is -3. Find its position after 16 sec and total distance traveled.

Problem 19306

Find the general antiderivative of f(x)=4ex+sec⁡2x+x7/3+x3/7f(x)=4 e^{x}+\sec^{2} x+x^{7/3}+x^{3/7}f(x)=4ex+sec2x+x7/3+x3/7.

Problem 19307

Find the sky diver's velocity v(t)=59(1−e−0.23t)v(t)=59(1-e^{-0.23 t})v(t)=59(1−e−0.23t) after 2 seconds and 5 seconds, rounding to the nearest whole number.

Problem 19308

Choose the integral for the area bounded by y=x2y=x^{2}y=x2, x=2x=2x=2, and the xxx-axis: ∫02x2dx\int_{0}^{2} x^{2} d x∫02​x2dx.

Problem 19309

Find the area above the xxx-axis and below y=4x2+4x+3y=4x^2+4x+3y=4x2+4x+3 from x=0x=0x=0 to x=2x=2x=2. Enter an exact value.

Problem 19310

Find the tangent and normal lines at point P(4,2.5)P(4,2.5)P(4,2.5) on the curve y=x+1xy=\sqrt{x}+\frac{1}{\sqrt{x}}y=x​+x​1​, then calculate the area of triangle ABPABPABP.

Problem 19311

Find the initial temperature and the temperature after 18 minutes for T(x)=−5+27e−0.041xT(x)=-5+27 e^{-0.041 x}T(x)=−5+27e−0.041x. Round to the nearest degree.

Problem 19312

Find the area above the xxx-axis under the curve y=1−x4y=1-x^{4}y=1−x4. Enter the exact value, e.g., "3+1/2".

Problem 19313

Find the area between y=2x+1y=2x+1y=2x+1 and y=−x2y=-x^{2}y=−x2 from x=0x=0x=0 to x=1x=1x=1.

Problem 19314

Evaluate the integral from -3 to 0: ∫−30(8x−ex)dx\int_{-3}^{0}\left(8 x-e^{x}\right) d x∫−30​(8x−ex)dx

Problem 19315

Find the area between f(x)=1−x2f(x)=1-x^{2}f(x)=1−x2 and g(x)=4−4x2g(x)=4-4x^{2}g(x)=4−4x2 from x=0x=0x=0 to x=1x=1x=1.

Problem 19316

An object moves with velocity v(t)=t(6−t)v(t)=t(6-t)v(t)=t(6−t), starting 5 units left of the origin. Find its position and distance after 18 seconds.

Problem 19317

Find the total area between the curve y=f(x)=x2−x−2y=f(x)=x^{2}-x-2y=f(x)=x2−x−2 and the xxx-axis from x=−7x=-7x=−7 to x=6x=6x=6. Calculate: ∫−76f(x) dx \int_{-7}^{6} f(x) \, dx ∫−76​f(x)dx Interpret this integral in terms of areas.

Problem 19318

Find the tangent line equation at point P(4,2.5)P(4,2.5)P(4,2.5) on the curve y=x+1xy=\sqrt{x}+\frac{1}{\sqrt{x}}y=x​+x​1​ in the form ax+by+c=0a x+b y+c=0ax+by+c=0.

Problem 19319

Find the tangent and normal lines at point P(4,2.5)P(4,2.5)P(4,2.5) on the curve y=x+1xy=\sqrt{x}+\frac{1}{\sqrt{x}}y=x​+x​1​ in the form ax+by+c=0a x+b y+c=0ax+by+c=0.

Problem 19320

Find the tangent line equation at point P(4,2.5)P(4,2.5)P(4,2.5) on the curve y=x+1xy=\sqrt{x}+\frac{1}{\sqrt{x}}y=x​+x​1​ in the form ax+by+c=0a x+b y+c=0ax+by+c=0.

Problem 19321

Rewrite the integral ∫x(x+1)11dx\int x(x+1)^{11} dx∫x(x+1)11dx using u=x+1u = x + 1u=x+1 and dududu.

Problem 19322

Problem 19323

Rewrite the integral using uuu and dududu: ∫18sec⁡2(x)tan⁡2(x)dx,u=tan⁡(x)\int 18 \sec^2(x) \tan^2(x) dx, \quad u=\tan(x)∫18sec2(x)tan2(x)dx,u=tan(x).

Problem 19324

Find the general function fff such that f′′(x)=8x+cos⁡x+3f^{\prime \prime}(x)=8 x+\cos x+3f′′(x)=8x+cosx+3.

Problem 19325

Differentiate the equation V=−3x−1.1ln⁡(x)+1.7y−1.6ln⁡(y)V=-3 x-1.1 \ln (x)+1.7 y-1.6 \ln (y)V=−3x−1.1ln(x)+1.7y−1.6ln(y) implicitly to find dydx\frac{d y}{d x}dxdy​. Evaluate at x=5x=5x=5, y=5y=5y=5. Round to three decimal places.

Problem 19326

Find the integral: ∫3x+1(3x2+2x)3dx\int \frac{3 x+1}{(3 x^{2}+2 x)^{3}} dx∫(3x2+2x)33x+1​dx

Problem 19327

What are the units of ∫abf(t)dt\int_{a}^{b} f(t) dt∫ab​f(t)dt if f(t)f(t)f(t) is in m/s² and ttt is in seconds?

Problem 19328

Estimate f(6.4)f(6.4)f(6.4) using linear approximation with f(6)=3.4f(6)=3.4f(6)=3.4 and f′(6)=2.3f'(6)=2.3f′(6)=2.3. Round to three decimal places.

Problem 19329

Find the box dimensions with a length of 12 cm and volume of 1500 cm³ that minimize surface area.

Problem 19330

Compute the flux ∬SF⋅dS\iint_{S} \mathbf{F} \cdot d \mathbf{S}∬S​F⋅dS for F(x,y,z)=(x2,y,z)\mathbf{F}(x,y,z)=(x^{2}, y, z)F(x,y,z)=(x2,y,z) using the divergence theorem on the unit sphere SSS.

Problem 19331

A substance grows at 17%17\%17% per day. If it starts at 719 grams, find its mass after 6 days. Round to the nearest tenth.

Problem 19332

Problem 19333

Find the integral: ∫25z2(125z3+1)12dz\int 25 z^{2}\left(125 z^{3}+1\right)^{12} d z∫25z2(125z3+1)12dz

Problem 19334

A radioactive substance starts at 259 kg259 \mathrm{~kg}259 kg and decays at 6%6\%6% per day. Find its mass after 3 days.

Problem 19335

A biologist has a 5483-gram sample of a radioactive substance. Find the mass after 5 hours with a decay rate of 12%12\%12% per hour.

Problem 19336

Find the horizontal asymptotes for f(x)=x2+4x2−4f(x)=\frac{x^{2}+4}{x^{2}-4}f(x)=x2−4x2+4​. Options: y=1y=1y=1, x=4x=4x=4, x=−2x=-2x=−2, y=−1y=-1y=−1.

Problem 19337

Find the stationary points of the curve defined by f(x)=2x3−12x2−x+2f(x)=2 x^{3}-\frac{1}{2} x^{2}-x+2f(x)=2x3−21​x2−x+2.

Problem 19338

Bestimme die ersten beiden Ableitungen von f(x)=−16x3+34x2f(x)=-\frac{1}{6} x^{3}+\frac{3}{4} x^{2}f(x)=−61​x3+43​x2 und zeichne den Graphen.

Problem 19339

Find the derivative of f(x)=−4x2−xf(x)=-4 x^{2}-xf(x)=−4x2−x. What is f′(x)f^{\prime}(x)f′(x)?

Problem 19340

Find the derivative of g(x)=∫1−2x1+2xtsin⁡t dtg(x)=\int_{1-2 x}^{1+2 x} t \sin t \, dtg(x)=∫1−2x1+2x​tsintdt.

Problem 19341

Find all lines with slope -1 that are tangent to the curve y=1x−1y=\frac{1}{x-1}y=x−11​.

Find the area between $y=3-x^{2}$ and $y=\sqrt{x}$ from $x=0$ to $x=1$ .

Find the area between the curves $y=2 x^{3}-3 x^{2}-9 x$ and $y=x^{3}-2 x^{2}-3 x$ . Provide the exact value.

Find the tangent line to $f(x)=x^{2}-5x+4$ at $x=5$ , the normal at $x=3$ , and their intersection point $P$ .

An object moves with velocity $v(t)=t(8-t)$ . Its initial position is -3. Find its position after 16 sec and total distance traveled.

Find the general antiderivative of $f(x)=4 e^{x}+\sec^{2} x+x^{7/3}+x^{3/7}$ .

Find the sky diver's velocity $v(t)=59(1-e^{-0.23 t})$ after 2 seconds and 5 seconds, rounding to the nearest whole number.

Choose the integral for the area bounded by $y=x^{2}$ , $x=2$ , and the $x$ -axis: $\int_{0}^{2} x^{2} d x$ .

Find the area above the $x$ -axis and below $y=4x^2+4x+3$ from $x=0$ to $x=2$ . Enter an exact value.

Find the tangent and normal lines at point $P(4,2.5)$ on the curve $y=\sqrt{x}+\frac{1}{\sqrt{x}}$ , then calculate the area of triangle $ABP$ .

Find the initial temperature and the temperature after 18 minutes for $T(x)=-5+27 e^{-0.041 x}$ . Round to the nearest degree.

Find the area above the $x$ -axis under the curve $y=1-x^{4}$ . Enter the exact value, e.g., "3+1/2".

Find the area between $y=2x+1$ and $y=-x^{2}$ from $x=0$ to $x=1$ .

Evaluate the integral from -3 to 0: $\int_{-3}^{0}\left(8 x-e^{x}\right) d x$

Find the area between $f(x)=1-x^{2}$ and $g(x)=4-4x^{2}$ from $x=0$ to $x=1$ .

An object moves with velocity $v(t)=t(6-t)$ , starting 5 units left of the origin. Find its position and distance after 18 seconds.

Find the total area between the curve $y=f(x)=x^{2}-x-2$ and the $x$ -axis from $x=-7$ to $x=6$ . Calculate:
$\int_{-7}^{6} f(x) \, dx$
Interpret this integral in terms of areas.

Find the tangent line equation at point $P(4,2.5)$ on the curve $y=\sqrt{x}+\frac{1}{\sqrt{x}}$ in the form $a x+b y+c=0$ .

Find the tangent and normal lines at point $P(4,2.5)$ on the curve $y=\sqrt{x}+\frac{1}{\sqrt{x}}$ in the form $a x+b y+c=0$ .

Find the tangent line equation at point $P(4,2.5)$ on the curve $y=\sqrt{x}+\frac{1}{\sqrt{x}}$ in the form $a x+b y+c=0$ .

Rewrite the integral $\int x(x+1)^{11} dx$ using $u = x + 1$ and $du$ .

Rewrite the integral using $u$ and $du$ : $\int 18 \sec^2(x) \tan^2(x) dx, \quad u=\tan(x)$ .

Find the general function $f$ such that $f^{\prime \prime}(x)=8 x+\cos x+3$ .

Differentiate the equation $V=-3 x-1.1 \ln (x)+1.7 y-1.6 \ln (y)$ implicitly to find $\frac{d y}{d x}$ . Evaluate at $x=5$ , $y=5$ . Round to three decimal places.

Find the integral: $\int \frac{3 x+1}{(3 x^{2}+2 x)^{3}} dx$

What are the units of $\int_{a}^{b} f(t) dt$ if $f(t)$ is in m/s² and $t$ is in seconds?

Estimate $f(6.4)$ using linear approximation with $f(6)=3.4$ and $f'(6)=2.3$ . Round to three decimal places.

Compute the flux $\iint_{S} \mathbf{F} \cdot d \mathbf{S}$ for $\mathbf{F}(x,y,z)=(x^{2}, y, z)$ using the divergence theorem on the unit sphere $S$ .

A substance grows at $17\%$ per day. If it starts at 719 grams, find its mass after 6 days. Round to the nearest tenth.

Find the integral: $\int 25 z^{2}\left(125 z^{3}+1\right)^{12} d z$

A radioactive substance starts at $259 \mathrm{~kg}$ and decays at $6\%$ per day. Find its mass after 3 days.

A biologist has a 5483-gram sample of a radioactive substance. Find the mass after 5 hours with a decay rate of $12\%$ per hour.

Find the horizontal asymptotes for $f(x)=\frac{x^{2}+4}{x^{2}-4}$ . Options: $y=1$ , $x=4$ , $x=-2$ , $y=-1$ .

Find the stationary points of the curve defined by $f(x)=2 x^{3}-\frac{1}{2} x^{2}-x+2$ .

Bestimme die ersten beiden Ableitungen von $f(x)=-\frac{1}{6} x^{3}+\frac{3}{4} x^{2}$ und zeichne den Graphen.

Find the derivative of $f(x)=-4 x^{2}-x$ . What is $f^{\prime}(x)$ ?

Find the derivative of $g(x)=\int_{1-2 x}^{1+2 x} t \sin t \, dt$ .

Find all lines with slope -1 that are tangent to the curve $y=\frac{1}{x-1}$ .

An object dropped from a 100-m tower has height $100 - 4.9t^2$ m. Find its falling speed at $t = 2$ sec.

Compute the outward flux of $\mathbf{F}(x, y, z)=1 x \mathbf{i}+5 y \mathbf{j}+3 z \mathbf{k}$ across the prism $-1 \leq x \leq 6, -1 \leq y \leq 7, -1 \leq z \leq 7$ .

Find the derivative of $f(x)=x^{2}-5x-4$ . Which option is correct: (a) $-4$ , (b) $7$ , (c) $x^{2}-5x-4$ , or (d) $2x-5$ ?

Find the derivatives of $f(x)=4-x^{2}$ at $x=-3$ , $0$ , and $1$ : $f^{\prime}(-3)$ , $f^{\prime}(0)$ , $f^{\prime}(1)$ .

A substance grows at a rate of $16\%$ daily. If it starts at 48 grams, find its mass after 2 days. Round to the nearest tenth.

Compute the indefinite integral: $\int \frac{\sin 2 x}{\sin x} \, dx$ .

Find $\frac{d w}{d q}$ if $w=\frac{5^{q}}{\ln 5}$ .

Find the derivative of the function $f(t)=\frac{e^{8 t}}{t^{3}+10}$ , denoted as $f^{\prime}(t)=L_{t}$ .

Find the derivative $\frac{d w}{d q}$ if $w=\frac{5^{q}}{\ln 5}$ .

Find the derivative of the function $f(t)=\frac{e^{8 t}}{t^{3}+10}$ . What is $f^{\prime}(t)$ ?

Evaluate the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-x-12}{4 x^{2}+11 x-3}$ . If it doesn't exist, write DNE.

Find the derivative of the function $j(x)=\sqrt[5]{x} \ln x$ . What is $j^{\prime}(x)$ ?

A soda can's temperature $T(x) = -3 + 21 e^{-0.025 x}$ ; find $T(0)$ and $T(18)$ . Round to the nearest degree.

Calculate the half-life of a radioactive element with decay function $A(t)=A_{0} e^{-0.0355 t}$ . Answer: $\square$ years.

Find the half-life of a radioactive element with decay function $A(t)=A_{0} e^{-0.021 t}$ . Round to the nearest tenth.

Find the total increase in a fruit fly population from day 1 to day 12 given $g(t)=4 e^{0.1 t}$ and initial population of 380.

Find $g^{\prime}(-1)$ , $g^{\prime}(2)$ , and $g^{\prime}(\sqrt{3})$ for $g(t)=\frac{1}{t^{2}}$ .

Find the derivative of the function $f(t)=\frac{e^{8 t}}{t^{3}+10}$ . What is $f^{\prime}(t)$ ?

Find the number of fruit flies added from day 4 to day 14, given $g(t)=5 e^{0.02 t}$ and initial population 340.

Evaluate the integral from -3 to 1 of the constant -7: $\int_{-3}^{1}(-7) d p$ .

Evaluate the integral from -2 to 4 of the function $5t - 6$ .

Evaluate the integral from 0 to 1 of $(5 x^{2}-4 x+9)$ . What is the result? $\int_{0}^{1}(5 x^{2}-4 x+9) d x=\square$

Find the indefinite integral using substitution: $\int 6(6 x-5)^{3} d x$

Evaluate the integral: $\int \frac{9 e^{9 t}}{9+e^{9 t}} d t$

Find the indefinite integral using substitution: $\int \frac{x^{3}+2 x}{x^{4}+4 x^{2}+2} d x$

Find the derivative $S^{\prime}(t)$ , compute $S(3)$ and $S^{\prime}(3)$ , and interpret $S(11)$ and $S^{\prime}(11)$ .

Find the indefinite integral using substitution: $\int \frac{5 x+9}{(5 x^{2}+18 x)^{3}} d x$

Find the indefinite integral using substitution: $\int \frac{x^{3}+2 x}{x^{4}+4 x^{2}+3} d x$

Find the indefinite integral using substitution: $\int \frac{x^{3}}{2 x^{4}+1} d x$

Find the indefinite integral using substitution: $\int\left(\sqrt{t^{5}+2 t}\right)\left(5 t^{4}+2\right) d t$ .

Find the indefinite integral using substitution: $\int z \sqrt{7 z^{2}-5} \, dz$

Find the indefinite integral using substitution: $\int 6 x^{5} e^{8 x^{6}} d x$