Calculus

Problem 9301

Find the tangent line equation for the function $f(x)=12 x^{2}+2 x+5$ at the point where $x=5$ .

See Solution

Problem 9302

Find the slope of the tangent line for the function $f(x)=x^{2}+12$ at the point where $x=2$ .

See Solution

Problem 9303

Find the tangent line equation for $f(x)=12 e^{x}+10 x$ at $x=0$ .

See Solution

Problem 9304

Soit $f(x)=\frac{x+3}{x-6}$ . Trouvez les intervalles de croissance et de décroissance, ainsi que les valeurs de $x$ pour les max et min relatifs.

See Solution

Problem 9305

Find the equivalent of $\int_{u=0}^{4} \int_{v=\sqrt{u}}^{2} u^{2}+v^{2} d v d u$ after reversing integration order.

See Solution

Problem 9306

Given the equation $x y=1$ , find $\frac{d y}{d x}$ using implicit differentiation and then solving for $y$ before differentiating. Are $\frac{d y}{d x}$ and $y^{\prime}$ equal?

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Problem 9307

Find the equivalent of $\int_{u=0}^{4} \int_{v=\sqrt{u}}^{2} u^{2}+v^{2} d v d u$ after reversing integration order.

See Solution

Problem 9308

Evaluate the triple integral $\int_{0}^{2} \int_{0}^{x} \int_{x}^{y} 3 x y z \, dz \, dy \, dx$ .

See Solution

Problem 9309

Déterminez les intervalles de croissance et de décroissance de $f(x)=\frac{x+3}{x-6}$ et les points de maximums et minimums relatifs.

See Solution

Problem 9310

Determine the relative extrema and inflection points of the function $y=x^{3}-6x^{2}+9x+3$ .

See Solution

Problem 9311

Analyze the function $f(x)=\frac{x^{2}+1}{x}$ for $x \neq 0$ :
1) Find zero points.
2) Compute first and second derivatives.
3) Identify extreme points.
4) Determine asymptotes.
5) Sketch the graph with asymptotes.

See Solution

Problem 9312

Approximate $(6.001)^{5}$ using $f(x)$ and $a$ , then find $L(x)=f(a)+f^{\prime}(a)(x-a)$ rounded to 2 decimals.

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Problem 9313

Find the differential $d y$ for $y=\tan(x)$ at $x=\frac{7\pi}{4}$ and $d x=\frac{\pi}{9}$ .

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Problem 9314

Find the linear approximation of $f(x)=7 x^{\frac{2}{3}}$ at $a=27$ and use it to estimate $f(29)$ .

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Problem 9315

Find the differential $d y$ for $y=\frac{\sin(2x)}{x}$ at $x=\pi$ and $d x=0.25$ .

See Solution

Problem 9316

Is the function $f(x)=\frac{x^{3}}{\sqrt{x+3}}$ continuous or discontinuous at $x=2$ ? a) continuous b) discontinuous

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Problem 9317

Find the derivative of $f(x)=4 e^{9 x}+e^{-x^{6}}$ .

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Problem 9318

Find the average rate of change of $M(t)$ over $[1,4]$ given rates over $[1,7]$ and $[4,7]$ . Choices are:
1. $\frac{6.4}{4} \mathrm{mg} / \mathrm{day}$
2. $\frac{8.4}{4} \mathrm{mg} / \mathrm{day}$
3. $\frac{7.2}{3} \mathrm{mg} / \mathrm{day}$
4. $\frac{7.2}{4} \mathrm{mg} / \mathrm{day}$
5. $3 \mathrm{mg} / \mathrm{day}$

See Solution

Problem 9319

Find the line equation for the linear approximation of $f(x)=6-2x^2$ at $a=3$ , estimate $f(3.1)$ , and compute percent error.

See Solution

Problem 9320

Approximate the change in volume $d V$ of a cube when its side increases from 12 to 12.1. Find $d V$ .

See Solution

Problem 9321

Estimate $\Delta y$ for $y=4x^2+8x+3$ using linear approximation with $\Delta x=0.4$ at $x=5$ .

See Solution

Problem 9322

A drug is given every 3 hours at $10 \mathrm{mg}$ . It decays exponentially with rate $k=0.4$ . Find $A(t)$ using $A(t)=A_{0} e^{-kt}$ .

See Solution

Problem 9323

Find the rate of change of $f(x)=\frac{x}{x^{2}-1}$ at $x=4$ . Choose from the given limit options.

See Solution

Problem 9324

Find the relative maxima, minima, and inflection points of the function $y=-x^{3}+12x-4$ .

See Solution

Problem 9325

Find the rate of change of the tumor volume $V(r)=\frac{1}{5} \pi r^{3}$ at $r=1.3 \mathrm{~cm}$ . Round up the answer. Options: a) $3 \mathrm{~cm}^{2}$ b) $4 \mathrm{~cm}^{2}$ c) $5 \mathrm{~cm}^{2}$ d) $6 \mathrm{~cm}^{2}$ .

See Solution

Problem 9326

Estimate $\Delta y$ for $y=\sin(3x)$ at $x=0$ with $\Delta x=0.2$ using linear approximation. Find error %.

See Solution

Problem 9327

Find the derivative $\frac{d y}{d t}$ of $y=f(t)=e^{t}(x^{4}+2 x^{3})$ . Choose the correct option from a) to e).

See Solution

Problem 9328

Find the derivative $\frac{d y}{d x}$ of $y=f(x)$ where $f(x)=\frac{\sin (x)}{2 x^{3}}$ . Choose correct option.

See Solution

Problem 9329

Find the line equation for the linear approximation of $f(x)=6-2x^{2}$ at $a=3$ , estimate $f(3.1)$ , and compute percent error.

See Solution

Problem 9330

Find the first and second derivatives of $y=-2 x^{8}+7$ .

See Solution

Problem 9331

A plant pot falls for 2.24 seconds. Find the height of the balcony using $h = \frac{1}{2}gt^2$ .

See Solution

Problem 9332

An object is dropped from 187 ft. Find its velocity, speed, acceleration at time $t$ , time to hit ground, and impact velocity.

See Solution

Problem 9333

Find the acceleration of a pendulum ride at 3.28 seconds, given $y=\cos \left(\frac{t}{5}\right)$ . Choose from options a) to e).

See Solution

Problem 9334

Find the derivative $\frac{d y}{d x}$ of $f(x)=(4 x^{4}+10 x^{2}+4)^{-5}$ . Choose the correct option.

See Solution

Problem 9335

An object falls from a tower 187 ft high.
a. Find velocity, speed, and acceleration at time $t$ . b. How long until it hits the ground? c. What is the velocity at impact?
Velocity at time $t$ is: $v(t) = -32t$ .

See Solution

Problem 9336

Find the second derivative of $f(x)=3 x^{-5}+2 x^{2}+8$ . Which option is correct? a) $15 x^{-3}+4 x$ b) $90 x^{-7}+4$ c) $20 x+8$ d) $90 x^{-3}+4$ e) None.

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Problem 9337

Find the rate of area increase of a rash when $r = 4 \mathrm{~cm}$ and $dr/dt = 0.1 \mathrm{~cm/day}$ . Options: a) $\pi$ , b) $0.6 \pi$ , c) $0.7 \pi$ , d) $0.8 \pi$ , e) None.

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Problem 9338

Find the derivative $\frac{d y}{d x}$ of $y=f(x)$ where $f(x)=4^{2 x} \cot \left(x^{4}\right)$ .

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Problem 9339

Find the derivative of $f(x)=\frac{e^{x^{3}+2 x}}{x^{2}}$ . Choose the correct expression from the options given.

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Problem 9340

Find the marginal utility $\mu_{x}$ from the utility function $u(x, y) = x + 2y$ .

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Problem 9341

Estimate the root of $x^{3}+3 x-15=0$ using Newton's method with initial guess $x_{0}=2$ for one iteration.

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Problem 9342

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{4 x^{3}-3}{2 x}$ .

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Problem 9343

A particle is projected at $12 \mathrm{~ms}^{-1}$ up a slope with $5 \mathrm{~ms}^{-2}$ acceleration. Find time for displacement > 8 m.

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Problem 9344

Find the derivative $f^{\prime}(4)$ for the function $f(x)=\sqrt{x}+\frac{2}{x^{2}}$ .

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Problem 9345

Find the next approximation $x_{1}$ using Newton's method for $f(x)=0$ with initial guess $x_{0}=5$ for $y=4x-5$ .

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Problem 9346

Calculate the average annual income from a 10000 PLN deposit at a continuous $8\%$ interest rate over 15 years.

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Problem 9347

Find the approximate value of $x_{0}$ using the tangent line at $(7,2)$ for the function $f$ with given properties.

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Problem 9348

Find the tangent line equation for the function $f(x)=\frac{x-13}{5x+6}$ at the point where $x=3$ .

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Problem 9349

Find the average rate of change of $f(x)=4 x^{2}+6$ from $x=0$ to $x=1$ .

See Solution

Problem 9350

Find the derivative of $P=(4+\ln x)^{0.5}$ .

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Problem 9351

Estimate the solution to $e^{-2 x}+x-\frac{4}{5}=0$ using Newton's method with initial guess $x_{0}=0$ for one iteration.

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Problem 9352

Find the average rate of change of $f(x)=4 x^{2}+6$ from $x=0$ to $x=12$ .

See Solution

Problem 9353

Find the average rate of change of $f(x)=4 x^{2}+6$ from $x=3$ to $x=6$ .

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Problem 9354

Find the approximate value of $x_{0}$ using the tangent line at $(3,3)$ for the function $f$ with given properties.

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Problem 9355

Find the average rate of change of $f(x)=4x^{2}+6$ from $x=1$ to $x=3$ .

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Problem 9356

Find the next approximation $x_{1}$ using Newton's method for $f(x)=0$ with $x_{0}=2$ and tangent line $y=5x-2$ .

See Solution

Problem 9357

Find the limit as $x$ approaches 0 for $\frac{e^{3x}-1}{x^4+6x}$ .

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Problem 9358

Find the line equation for the linear approximation of $f(x)=\sin x$ at $a=\frac{\pi}{2}$ , estimate $f(1.62)$ , and compute percent error: $100 \cdot \frac{\mid \text { approximation - exact} \mid}{\mid \text { exact } \mid}$ .

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Problem 9359

Find the linear approximation of $f(x)=x^{\frac{3}{4}}$ at $a=1296$ and estimate $f(1389)$ . Is it an overestimate or underestimate?

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Problem 9360

Find the linear approximation to $f(x)=x^{\frac{3}{4}}$ at $a=1296$ and estimate $f(1389)$ . Is it an underestimate or overestimate?

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Problem 9361

Find the next approximation $x_{1}$ using Newton's method for $f(x)=0$ with $x_{0}=4$ and tangent line $y=5x-6$ .

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Problem 9362

Find the linear approximation for $f(x)=\sec x$ at $a=0$ and estimate $f(-0.01)$ . Compute percent error: $100 \cdot \frac{\text { |approximation - exact| }}{\text { |exact |}}$ .

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Problem 9363

Find the Laurent series for $(b-a) / ((z-a)(z-b))$ where $0 < |a| < |b|$ in the region $|z| > |b|$ .

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Problem 9364

Find all points where the function $x^{4}-8 x^{3}-30 x^{2}+7$ is concave up: $x=-4, -2, 0, 2, 4, 6$ .

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Problem 9365

Find $x$ in $[0,2 \pi]$ where $y=\frac{\cos x}{2+\sin x}$ has a horizontal tangent. List smaller $x$ first. $x=$ $11 h=$

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Problem 9366

Find the fourth derivative of $f(x) = \frac{-3x}{1-x}$ .

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Problem 9367

Find the derivative of the inverse function at $-76$ for $f(x)=-3x^3+5$ . What is $\left(f^{-1}\right)^{\prime}(-76)$ ?

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Problem 9368

Find the derivative $f'(x)$ for $f(x)=\frac{4 \sin x}{4 \sin x+6 \cos x}$ and the tangent line at $a=\frac{\pi}{2}$ , $y=mx+b$ . What are $m$ and $b$ ?

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Problem 9369

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{x^{3}-2 x+4}{2300 x^{2}+65 x}$

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Problem 9370

Find the horizontal asymptote of $f(x)=2 \frac{(x+5)(3x-1)}{(3-x)(5x+2)}$ . What is $y$ ?

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Problem 9371

A chemist has a 20 mg sample of polonium-218. How long (in seconds) to decay to 16.3 mg? Half-life is 3.1 min. Describe the function type and behavior.

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Problem 9372

A chemist has a 20 mg sample of polonium-218. How long (in seconds) for it to decay to 16.3 mg? Half-life is 3.1 min. Also, describe the function's type, name, and behavior.

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Problem 9373

Find the general antiderivative of $f(x)=\frac{7 x^{2}-6 x+3}{x^{2}}$ for $x>0$ and verify by differentiation. Use $C$ for the constant.

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Problem 9374

Find the antiderivative $F$ of $f(x)=2 e^{x}-8 x$ with $F(0)=6$ and verify by graphing $f$ and $F$ .

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Problem 9375

Find the general antiderivative of $f(\theta)=5 \sin (\theta)-3 \sec (\theta) \tan (\theta)$ and check by differentiation. Use $C$ for the constant.

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Problem 9376

Calculate the limit as $x$ approaches infinity for $\frac{2300 x^{2}+65 x}{x^{3}-2 x+4}$ .

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Problem 9377

Find the tangent line equation $y=mx+b$ for $f(x)=\frac{4 \sin x}{4 \sin x+6 \cos x}$ at $x=\frac{\pi}{2}$ .

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Problem 9378

Given the function $f^{\prime \prime}(x)=42 x$ , find $f^{\prime}(x)$ and $f(x)$ using constants $C$ and $D$ .

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Problem 9379

Find the function $f$ given that $f''(x) = -2 + 12x - 12x^2$ , $f(0) = 8$ , and $f'(0) = 18$ .

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Problem 9380

Find the tangent line $y=m x+b$ to $f(x)=\frac{4 \sin x}{4 \sin x+6 \cos x}$ at $a=\frac{\pi}{2}$ . Determine $m$ and $b$ .

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Problem 9381

Gegeben ist $f(x)=x^{3}-2 x^{2}-8 x$ .
1. Finde die Nullstellen von $f$ .
2. Ist $\int_{-2}^{4} f(x) d x$ positiv, negativ oder null? Begründe.
3. Markiere den Flächeninhalt von $\int_{-2}^{0} f(x) d x$ und gib einen Ansatz zur Berechnung.

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Problem 9382

Given a fish population with a carrying capacity of 1300 and a growth rate of $230\%$ , find $p_{1}$ and $p_{2}$ starting from $p_{0}=100$ .

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Problem 9383

Given $f^{\prime \prime}(\theta)=\sin (\theta)+\cos (\theta)$ , find $f^{\prime}$ and $f$ with conditions $f^{\prime}(0)=4$ and $f(0)=1$ .

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Problem 9384

Find the limit: $\lim _{x \rightarrow 0^{+}}(|x|+7)$ .

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Problem 9385

Find the derivative of $f(a) = -a x^{3} + 4 a x$ .

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Problem 9386

Find $f^{\prime}$ given $f^{\prime \prime}(\theta)=\sin(\theta)+\cos(\theta)$ and $f^{\prime}(0)=4$ .

See Solution

Problem 9387

Differentiate the equation $2xy - y^2 = 4$ using the product rule. Find $\frac{dy}{dx}$ .

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Problem 9388

Berechne die Fläche zwischen dem Graphen von $f$ und der $x$ -Achse für die Funktionen a) bis f).

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Problem 9389

A stone is dropped from a 100 m tower. Given $g=9.8 \mathrm{~m/s}^2$ , find its velocity and height above ground at time $t$ .

See Solution

Problem 9390

Find the particle's position $s(t)$ given $a(t)=2t+3$ , $s(0)=6$ , and $v(0)=-9$ .

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Problem 9391

Find the limit: $\lim _{x \rightarrow-3} \frac{-2 x-6}{|-x-3|}$ .

See Solution

Problem 9392

Find local max and min points of $f(x)=-ax^{3}+4ax$ .

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Problem 9393

Find the limits: 1) $\lim _{x \rightarrow 0^{+}}(|x|+7)$ , 2) $\lim _{x \rightarrow-3} \frac{-2 x-6}{|-x-3|}$ , 3) $\lim _{x \rightarrow 2^{-}} \frac{3|x-2|}{x-2}$ .

See Solution

Problem 9394

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

See Solution

Problem 9395

Find the derivative of $g(x)=(x^{2}-\frac{1}{x})^{4}$ .

See Solution

Problem 9396

A stone is dropped from 100 m. Find the time to hit the ground and impact velocity. Also, time if thrown down at $7 \, \mathrm{m/s}$ .

See Solution

Problem 9397

Find $x$ and price $p$ that maximize revenue for the demand equation $p=\frac{1}{12} x^{2}-10 x+300$ , where $0 \leq x \leq 60$ .

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Problem 9398

Finde die Funktion $f(x)$ aus den Ableitungen $f^{\prime}(x)$ : a) $f^{\prime}(x)=\sin (x)+(x+2) \cos (x)$ b) $f^{\prime}(x)=2 \sin (0,5 x)+x \cdot \cos (0,5 x)$

See Solution

Problem 9399

Find the derivative of $h(x)=x^{2}(x^{2}+3x)^{5}$ .

See Solution

Problem 9400

Find the tangent line equation of $y=x+\sqrt{x}$ at the point $(1,2)$ .

See Solution

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Calculus

Problem 9301

Find the tangent line equation for the function f(x)=12x2+2x+5f(x)=12 x^{2}+2 x+5f(x)=12x2+2x+5 at the point where x=5x=5x=5.

Problem 9302

Find the slope of the tangent line for the function f(x)=x2+12f(x)=x^{2}+12f(x)=x2+12 at the point where x=2x=2x=2.

Problem 9303

Find the tangent line equation for f(x)=12ex+10xf(x)=12 e^{x}+10 xf(x)=12ex+10x at x=0x=0x=0.

Problem 9304

Soit f(x)=x+3x−6f(x)=\frac{x+3}{x-6}f(x)=x−6x+3​. Trouvez les intervalles de croissance et de décroissance, ainsi que les valeurs de xxx pour les max et min relatifs.

Problem 9305

Find the equivalent of ∫u=04∫v=u2u2+v2dvdu\int_{u=0}^{4} \int_{v=\sqrt{u}}^{2} u^{2}+v^{2} d v d u∫u=04​∫v=u​2​u2+v2dvdu after reversing integration order.

Problem 9306

Given the equation xy=1x y=1xy=1, find dydx\frac{d y}{d x}dxdy​ using implicit differentiation and then solving for yyy before differentiating. Are dydx\frac{d y}{d x}dxdy​ and y′y^{\prime}y′ equal?

Problem 9307

Find the equivalent of ∫u=04∫v=u2u2+v2dvdu\int_{u=0}^{4} \int_{v=\sqrt{u}}^{2} u^{2}+v^{2} d v d u∫u=04​∫v=u​2​u2+v2dvdu after reversing integration order.

Problem 9308

Evaluate the triple integral ∫02∫0x∫xy3xyz dz dy dx\int_{0}^{2} \int_{0}^{x} \int_{x}^{y} 3 x y z \, dz \, dy \, dx∫02​∫0x​∫xy​3xyzdzdydx.

Problem 9309

Déterminez les intervalles de croissance et de décroissance de f(x)=x+3x−6f(x)=\frac{x+3}{x-6}f(x)=x−6x+3​ et les points de maximums et minimums relatifs.

Problem 9310

Determine the relative extrema and inflection points of the function y=x3−6x2+9x+3y=x^{3}-6x^{2}+9x+3y=x3−6x2+9x+3.

Problem 9311

Analyze the function f(x)=x2+1xf(x)=\frac{x^{2}+1}{x}f(x)=xx2+1​ for x≠0x \neq 0x=0: 1) Find zero points. 2) Compute first and second derivatives. 3) Identify extreme points. 4) Determine asymptotes. 5) Sketch the graph with asymptotes.

Problem 9312

Approximate (6.001)5(6.001)^{5}(6.001)5 using f(x)f(x)f(x) and aaa, then find L(x)=f(a)+f′(a)(x−a)L(x)=f(a)+f^{\prime}(a)(x-a)L(x)=f(a)+f′(a)(x−a) rounded to 2 decimals.

Problem 9313

Find the differential dyd ydy for y=tan⁡(x)y=\tan(x)y=tan(x) at x=7π4x=\frac{7\pi}{4}x=47π​ and dx=π9d x=\frac{\pi}{9}dx=9π​.

Problem 9314

Find the linear approximation of f(x)=7x23f(x)=7 x^{\frac{2}{3}}f(x)=7x32​ at a=27a=27a=27 and use it to estimate f(29)f(29)f(29).

Problem 9315

Find the differential dyd ydy for y=sin⁡(2x)xy=\frac{\sin(2x)}{x}y=xsin(2x)​ at x=πx=\pix=π and dx=0.25d x=0.25dx=0.25.

Problem 9316

Is the function f(x)=x3x+3f(x)=\frac{x^{3}}{\sqrt{x+3}}f(x)=x+3​x3​ continuous or discontinuous at x=2x=2x=2? a) continuous b) discontinuous

Problem 9317

Find the derivative of f(x)=4e9x+e−x6f(x)=4 e^{9 x}+e^{-x^{6}}f(x)=4e9x+e−x6.

Problem 9318

Problem 9319

Find the line equation for the linear approximation of f(x)=6−2x2f(x)=6-2x^2f(x)=6−2x2 at a=3a=3a=3, estimate f(3.1)f(3.1)f(3.1), and compute percent error.

Problem 9320

Approximate the change in volume dVd VdV of a cube when its side increases from 12 to 12.1. Find dVd VdV.

Problem 9321

Estimate Δy\Delta yΔy for y=4x2+8x+3y=4x^2+8x+3y=4x2+8x+3 using linear approximation with Δx=0.4\Delta x=0.4Δx=0.4 at x=5x=5x=5.

Problem 9322

A drug is given every 3 hours at 10mg10 \mathrm{mg}10mg. It decays exponentially with rate k=0.4k=0.4k=0.4. Find A(t)A(t)A(t) using A(t)=A0e−ktA(t)=A_{0} e^{-kt}A(t)=A0​e−kt.

Problem 9323

Find the rate of change of f(x)=xx2−1f(x)=\frac{x}{x^{2}-1}f(x)=x2−1x​ at x=4x=4x=4. Choose from the given limit options.

Problem 9324

Find the relative maxima, minima, and inflection points of the function y=−x3+12x−4y=-x^{3}+12x-4y=−x3+12x−4.

Problem 9325

Find the rate of change of the tumor volume V(r)=15πr3V(r)=\frac{1}{5} \pi r^{3}V(r)=51​πr3 at r=1.3 cmr=1.3 \mathrm{~cm}r=1.3 cm. Round up the answer. Options: a) 3 cm23 \mathrm{~cm}^{2}3 cm2 b) 4 cm24 \mathrm{~cm}^{2}4 cm2 c) 5 cm25 \mathrm{~cm}^{2}5 cm2 d) 6 cm26 \mathrm{~cm}^{2}6 cm2.

Problem 9326

Estimate Δy\Delta yΔy for y=sin⁡(3x)y=\sin(3x)y=sin(3x) at x=0x=0x=0 with Δx=0.2\Delta x=0.2Δx=0.2 using linear approximation. Find error %.

Problem 9327

Find the derivative dydt\frac{d y}{d t}dtdy​ of y=f(t)=et(x4+2x3)y=f(t)=e^{t}(x^{4}+2 x^{3})y=f(t)=et(x4+2x3). Choose the correct option from a) to e).

Problem 9328

Find the derivative dydx\frac{d y}{d x}dxdy​ of y=f(x)y=f(x)y=f(x) where f(x)=sin⁡(x)2x3f(x)=\frac{\sin (x)}{2 x^{3}}f(x)=2x3sin(x)​. Choose correct option.

Problem 9329

Find the line equation for the linear approximation of f(x)=6−2x2f(x)=6-2x^{2}f(x)=6−2x2 at a=3a=3a=3, estimate f(3.1)f(3.1)f(3.1), and compute percent error.

Problem 9330

Find the first and second derivatives of y=−2x8+7y=-2 x^{8}+7y=−2x8+7.

Problem 9331

A plant pot falls for 2.24 seconds. Find the height of the balcony using h=12gt2h = \frac{1}{2}gt^2h=21​gt2.

Problem 9332

An object is dropped from 187 ft. Find its velocity, speed, acceleration at time ttt, time to hit ground, and impact velocity.

Problem 9333

Find the acceleration of a pendulum ride at 3.28 seconds, given y=cos⁡(t5)y=\cos \left(\frac{t}{5}\right)y=cos(5t​). Choose from options a) to e).

Problem 9334

Find the derivative dydx\frac{d y}{d x}dxdy​ of f(x)=(4x4+10x2+4)−5f(x)=(4 x^{4}+10 x^{2}+4)^{-5}f(x)=(4x4+10x2+4)−5. Choose the correct option.

Problem 9335

An object falls from a tower 187 ft high. a. Find velocity, speed, and acceleration at time ttt. b. How long until it hits the ground? c. What is the velocity at impact? Velocity at time ttt is: v(t)=−32tv(t) = -32tv(t)=−32t.

Problem 9336

Find the second derivative of f(x)=3x−5+2x2+8f(x)=3 x^{-5}+2 x^{2}+8f(x)=3x−5+2x2+8. Which option is correct? a) 15x−3+4x15 x^{-3}+4 x15x−3+4x b) 90x−7+490 x^{-7}+490x−7+4 c) 20x+820 x+820x+8 d) 90x−3+490 x^{-3}+490x−3+4 e) None.

Problem 9337

Find the rate of area increase of a rash when r=4 cmr = 4 \mathrm{~cm}r=4 cm and dr/dt=0.1 cm/daydr/dt = 0.1 \mathrm{~cm/day}dr/dt=0.1 cm/day. Options: a) π\piπ, b) 0.6π0.6 \pi0.6π, c) 0.7π0.7 \pi0.7π, d) 0.8π0.8 \pi0.8π, e) None.

Problem 9338

Find the derivative dydx\frac{d y}{d x}dxdy​ of y=f(x)y=f(x)y=f(x) where f(x)=42xcot⁡(x4)f(x)=4^{2 x} \cot \left(x^{4}\right)f(x)=42xcot(x4).

Problem 9339

Find the derivative of f(x)=ex3+2xx2f(x)=\frac{e^{x^{3}+2 x}}{x^{2}}f(x)=x2ex3+2x​. Choose the correct expression from the options given.

Problem 9340

Find the marginal utility μx\mu_{x}μx​ from the utility function u(x,y)=x+2yu(x, y) = x + 2yu(x,y)=x+2y.

Find the tangent line equation for the function $f(x)=12 x^{2}+2 x+5$ at the point where $x=5$ .

Find the slope of the tangent line for the function $f(x)=x^{2}+12$ at the point where $x=2$ .

Find the tangent line equation for $f(x)=12 e^{x}+10 x$ at $x=0$ .

Soit $f(x)=\frac{x+3}{x-6}$ . Trouvez les intervalles de croissance et de décroissance, ainsi que les valeurs de $x$ pour les max et min relatifs.

Find the equivalent of $\int_{u=0}^{4} \int_{v=\sqrt{u}}^{2} u^{2}+v^{2} d v d u$ after reversing integration order.

Given the equation $x y=1$ , find $\frac{d y}{d x}$ using implicit differentiation and then solving for $y$ before differentiating. Are $\frac{d y}{d x}$ and $y^{\prime}$ equal?

Find the equivalent of $\int_{u=0}^{4} \int_{v=\sqrt{u}}^{2} u^{2}+v^{2} d v d u$ after reversing integration order.

Evaluate the triple integral $\int_{0}^{2} \int_{0}^{x} \int_{x}^{y} 3 x y z \, dz \, dy \, dx$ .

Déterminez les intervalles de croissance et de décroissance de $f(x)=\frac{x+3}{x-6}$ et les points de maximums et minimums relatifs.

Determine the relative extrema and inflection points of the function $y=x^{3}-6x^{2}+9x+3$ .

Analyze the function $f(x)=\frac{x^{2}+1}{x}$ for $x \neq 0$ :
1) Find zero points.
2) Compute first and second derivatives.
3) Identify extreme points.
4) Determine asymptotes.
5) Sketch the graph with asymptotes.

Approximate $(6.001)^{5}$ using $f(x)$ and $a$ , then find $L(x)=f(a)+f^{\prime}(a)(x-a)$ rounded to 2 decimals.

Find the differential $d y$ for $y=\tan(x)$ at $x=\frac{7\pi}{4}$ and $d x=\frac{\pi}{9}$ .

Find the linear approximation of $f(x)=7 x^{\frac{2}{3}}$ at $a=27$ and use it to estimate $f(29)$ .

Find the differential $d y$ for $y=\frac{\sin(2x)}{x}$ at $x=\pi$ and $d x=0.25$ .

Is the function $f(x)=\frac{x^{3}}{\sqrt{x+3}}$ continuous or discontinuous at $x=2$ ? a) continuous b) discontinuous

Find the derivative of $f(x)=4 e^{9 x}+e^{-x^{6}}$ .

Find the line equation for the linear approximation of $f(x)=6-2x^2$ at $a=3$ , estimate $f(3.1)$ , and compute percent error.

Approximate the change in volume $d V$ of a cube when its side increases from 12 to 12.1. Find $d V$ .

Estimate $\Delta y$ for $y=4x^2+8x+3$ using linear approximation with $\Delta x=0.4$ at $x=5$ .

A drug is given every 3 hours at $10 \mathrm{mg}$ . It decays exponentially with rate $k=0.4$ . Find $A(t)$ using $A(t)=A_{0} e^{-kt}$ .

Find the rate of change of $f(x)=\frac{x}{x^{2}-1}$ at $x=4$ . Choose from the given limit options.

Find the relative maxima, minima, and inflection points of the function $y=-x^{3}+12x-4$ .

Find the rate of change of the tumor volume $V(r)=\frac{1}{5} \pi r^{3}$ at $r=1.3 \mathrm{~cm}$ . Round up the answer. Options: a) $3 \mathrm{~cm}^{2}$ b) $4 \mathrm{~cm}^{2}$ c) $5 \mathrm{~cm}^{2}$ d) $6 \mathrm{~cm}^{2}$ .

Estimate $\Delta y$ for $y=\sin(3x)$ at $x=0$ with $\Delta x=0.2$ using linear approximation. Find error %.

Find the derivative $\frac{d y}{d t}$ of $y=f(t)=e^{t}(x^{4}+2 x^{3})$ . Choose the correct option from a) to e).

Find the derivative $\frac{d y}{d x}$ of $y=f(x)$ where $f(x)=\frac{\sin (x)}{2 x^{3}}$ . Choose correct option.

Find the line equation for the linear approximation of $f(x)=6-2x^{2}$ at $a=3$ , estimate $f(3.1)$ , and compute percent error.

Find the first and second derivatives of $y=-2 x^{8}+7$ .

A plant pot falls for 2.24 seconds. Find the height of the balcony using $h = \frac{1}{2}gt^2$ .

An object is dropped from 187 ft. Find its velocity, speed, acceleration at time $t$ , time to hit ground, and impact velocity.

Find the acceleration of a pendulum ride at 3.28 seconds, given $y=\cos \left(\frac{t}{5}\right)$ . Choose from options a) to e).

Find the derivative $\frac{d y}{d x}$ of $f(x)=(4 x^{4}+10 x^{2}+4)^{-5}$ . Choose the correct option.

An object falls from a tower 187 ft high.
a. Find velocity, speed, and acceleration at time $t$ . b. How long until it hits the ground? c. What is the velocity at impact?
Velocity at time $t$ is: $v(t) = -32t$ .

Find the second derivative of $f(x)=3 x^{-5}+2 x^{2}+8$ . Which option is correct? a) $15 x^{-3}+4 x$ b) $90 x^{-7}+4$ c) $20 x+8$ d) $90 x^{-3}+4$ e) None.

Find the rate of area increase of a rash when $r = 4 \mathrm{~cm}$ and $dr/dt = 0.1 \mathrm{~cm/day}$ . Options: a) $\pi$ , b) $0.6 \pi$ , c) $0.7 \pi$ , d) $0.8 \pi$ , e) None.

Find the derivative $\frac{d y}{d x}$ of $y=f(x)$ where $f(x)=4^{2 x} \cot \left(x^{4}\right)$ .

Find the derivative of $f(x)=\frac{e^{x^{3}+2 x}}{x^{2}}$ . Choose the correct expression from the options given.

Find the marginal utility $\mu_{x}$ from the utility function $u(x, y) = x + 2y$ .

Estimate the root of $x^{3}+3 x-15=0$ using Newton's method with initial guess $x_{0}=2$ for one iteration.

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{4 x^{3}-3}{2 x}$ .

A particle is projected at $12 \mathrm{~ms}^{-1}$ up a slope with $5 \mathrm{~ms}^{-2}$ acceleration. Find time for displacement > 8 m.

Find the derivative $f^{\prime}(4)$ for the function $f(x)=\sqrt{x}+\frac{2}{x^{2}}$ .

Find the next approximation $x_{1}$ using Newton's method for $f(x)=0$ with initial guess $x_{0}=5$ for $y=4x-5$ .

Calculate the average annual income from a 10000 PLN deposit at a continuous $8\%$ interest rate over 15 years.

Find the approximate value of $x_{0}$ using the tangent line at $(7,2)$ for the function $f$ with given properties.

Find the tangent line equation for the function $f(x)=\frac{x-13}{5x+6}$ at the point where $x=3$ .

Find the average rate of change of $f(x)=4 x^{2}+6$ from $x=0$ to $x=1$ .

Find the derivative of $P=(4+\ln x)^{0.5}$ .

Estimate the solution to $e^{-2 x}+x-\frac{4}{5}=0$ using Newton's method with initial guess $x_{0}=0$ for one iteration.

Find the average rate of change of $f(x)=4 x^{2}+6$ from $x=0$ to $x=12$ .

Find the average rate of change of $f(x)=4 x^{2}+6$ from $x=3$ to $x=6$ .

Find the approximate value of $x_{0}$ using the tangent line at $(3,3)$ for the function $f$ with given properties.

Find the average rate of change of $f(x)=4x^{2}+6$ from $x=1$ to $x=3$ .

Find the next approximation $x_{1}$ using Newton's method for $f(x)=0$ with $x_{0}=2$ and tangent line $y=5x-2$ .

Find the limit as $x$ approaches 0 for $\frac{e^{3x}-1}{x^4+6x}$ .

Find the linear approximation of $f(x)=x^{\frac{3}{4}}$ at $a=1296$ and estimate $f(1389)$ . Is it an overestimate or underestimate?

Find the linear approximation to $f(x)=x^{\frac{3}{4}}$ at $a=1296$ and estimate $f(1389)$ . Is it an underestimate or overestimate?

Find the next approximation $x_{1}$ using Newton's method for $f(x)=0$ with $x_{0}=4$ and tangent line $y=5x-6$ .

Find the linear approximation for $f(x)=\sec x$ at $a=0$ and estimate $f(-0.01)$ . Compute percent error: $100 \cdot \frac{\text { |approximation - exact| }}{\text { |exact |}}$ .

Find the Laurent series for $(b-a) / ((z-a)(z-b))$ where $0 < |a| < |b|$ in the region $|z| > |b|$ .

Find all points where the function $x^{4}-8 x^{3}-30 x^{2}+7$ is concave up: $x=-4, -2, 0, 2, 4, 6$ .

Find $x$ in $[0,2 \pi]$ where $y=\frac{\cos x}{2+\sin x}$ has a horizontal tangent. List smaller $x$ first. $x=$ $11 h=$

Find the fourth derivative of $f(x) = \frac{-3x}{1-x}$ .

Find the derivative of the inverse function at $-76$ for $f(x)=-3x^3+5$ . What is $\left(f^{-1}\right)^{\prime}(-76)$ ?

Find the derivative $f'(x)$ for $f(x)=\frac{4 \sin x}{4 \sin x+6 \cos x}$ and the tangent line at $a=\frac{\pi}{2}$ , $y=mx+b$ . What are $m$ and $b$ ?

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{x^{3}-2 x+4}{2300 x^{2}+65 x}$

Find the horizontal asymptote of $f(x)=2 \frac{(x+5)(3x-1)}{(3-x)(5x+2)}$ . What is $y$ ?

Find the general antiderivative of $f(x)=\frac{7 x^{2}-6 x+3}{x^{2}}$ for $x>0$ and verify by differentiation. Use $C$ for the constant.

Find the antiderivative $F$ of $f(x)=2 e^{x}-8 x$ with $F(0)=6$ and verify by graphing $f$ and $F$ .