Calculus

Problem 26301

Find the derivative of $f(x) = \left(\frac{x+5}{x^{2}+2}\right)^{2}$ and explain each step.

See Solution

Problem 26302

Find the integral for the volume of a triangle revolving about the $x$ -axis given $y=3-x$ . Choices include various integrals.

See Solution

Problem 26303

You have two machines with data rates $f(t)$ and $g(t)$ . For $s$ hours, express data processed by the first machine as a definite integral: $\int_0^s f(t) \, dt$ .

See Solution

Problem 26304

Let $f(x)$ be a positive differentiable function where the area under $f(x)$ from $0$ to $x$ equals its arclength. Which ODE does $f(x)$ satisfy? Choose one:
1. $(f'(x))^2 = 1 + (f(x))^2$
2. $(f'(x))^2 = 1 - (f(x))^2$
3. $(f'(x))^2 = (f(x))^2 - 1$
4. $(f'(x))^2 = (1 + f(x))^2$
5. $(f'(x))^2 = (1 - f(x))^2$

See Solution

Problem 26305

Find the derivative of the function $y=\frac{1}{8^{x}}$ .

See Solution

Problem 26306

Calculate the integral $\int_{-1}^{5}(3 x^{2}-7) \, dx$ .

See Solution

Problem 26307

Find the slope of the tangent line to $y=x e^{x}$ at $x=\ln 5$ . Options: (A) $5 \ln 5$ (B) $5 \ln 5+5$ (C) $e^{5}(\ln 5)+e^{5}$ (D) $5+\frac{2 \ln 5}{e}$

See Solution

Problem 26308

You have two machines with rates $f(t)=9-t$ and $g(t)=2t+1$ . How long to use the first machine to maximize data processed in 6 hours?

See Solution

Problem 26309

Find the values of $r$ for which $y=e^{r x}$ solves the equation $2 y^{\prime \prime}+y^{\prime}-y=0$ .

See Solution

Problem 26310

Berechnen Sie die Anzahl der Erkrankten am 10. Tag mit der Funktion $f(x)=-\frac{1}{250} x^{3}+\frac{1}{10} x^{2}$ . Bestimmen Sie das Ende der Epidemie, den Definitions- und Wertebereich, Grenzwerte und die Symmetrie.

See Solution

Problem 26311

Berechne die Ableitung von $f(x)=3 x^{2}+12 x-8$ bei $x_{0}=-6$ als Grenzwert.

See Solution

Problem 26312

Bestimme die Ableitung von $f(x)=x^{2}-8x$ bei $x_{0}=3$ als Grenzwert von $\lim_{h \to 0} \frac{f(3+h)-f(3)}{h}$ .

See Solution

Problem 26313

Find $\frac{dy}{dx}$ when $x=2$ for the equation $x^{2} y+y^{2}+4=0$ .

See Solution

Problem 26314

Determine the left and right end behavior of the function $f(x)=\frac{x^{3}-5 x^{2}+3 x-1}{x^{4}-7 x+2}$ .

See Solution

Problem 26315

Bestimme die Ableitung von $f(x)=4(x-7)^{2}+8$ an der Stelle $x_0=6$ als Grenzwert.

See Solution

Problem 26316

Find the $x$ values for local minima in a quartic function (answer as $x=a, b, c$ ). Are they $-2, -1, 0, 1, 2$ ?

See Solution

Problem 26317

Given velocity data, find the object's acceleration, distance in 4s, and distance from 4s to 7s. Choices included.

See Solution

Problem 26318

A woman 1.5 m tall runs away from a 3 m pole at 1.8 m/s. Find the speed of her shadow's tip when she's 15 m from the pole.

See Solution

Problem 26319

Find the derivative of $f(x)=x^{2}$ at $x=-4$ using first principles.

See Solution

Problem 26320

Find the derivative of the following functions with respect to $\mathrm{x}$ : i) $f(x)=x$ , ii) $f(x)=x^{3}$ , iii) $f(x)=x^{4}$ , iv) $f(x)=x^{5}$ .

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Problem 26321

Find the derivative of $f(x)=x^{2}$ using first principles. Calculate $f'(-11)$ , $f'(0)$ , and $f'(5)$ .

See Solution

Problem 26322

Find $g'(x)$ for $g(x)=(6x^{3}+4x^{2}-3)(4x^{2}+2x+7)$ .

See Solution

Problem 26323

Find the derivative of the function $f(x)=7 x^{2}-4 x+2$ , i.e., compute $f^{\prime}(x)$ .

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Problem 26324

Find the derivative of $f(x)=x^{39}$ and generalize it for $f(x)=x^{n}$ , where $n$ is a positive integer.

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Problem 26325

Find the derivatives: a) $f(x)=x^{13}$ , b) $g(x)=x^{-5}$ . Calculate $f^{\prime}(x)$ and $g^{\prime}(x)$ .

See Solution

Problem 26326

How many times larger is a cell population growing at $100\%$ per day today compared to yesterday? Options: 2, 2.25, 2.5, 2.75.

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Problem 26327

Find $\frac{d y}{d x}$ at $x=-2$ for $y=x^{4}(3 x+7)^{3}$ and at $x=-1$ for $y=(2 x+1)^{5}(3 x+2)^{4}$ .

See Solution

Problem 26328

Find the derivative $h'(x)$ for $h(x)=(x^{2}+3)(4-2x^{3})$ .

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Problem 26329

Find $\frac{d y}{d x}$ for $x=-2$ in $y=x^{4}(3 x+7)^{3}$ and for $x=-1$ in $y=(2 x+1)^{5}(3 x+2)^{4}$ .

See Solution

Problem 26330

Calculate the average rate of change of $f(x)=\tan(3x)$ from 0 to $\frac{\pi}{9}$ .

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Problem 26331

Find the second derivative of these functions: a) $y=x^{3}-2 x^{2}$ , b) $y=x^{5}$ , c) $y=x^{-8}$ , d) $y=\sqrt{x}$ .

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Problem 26332

Find $f^{\prime}(g(-1))$ where $f(x)=2 x^{2}+x+3$ and $g(x)=4 x^{2}+5 x-12$ .

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Problem 26333

i) Find the 9th derivative of $y=x^{9}$ . ii) Find the 11th derivative of $y=13 x^{11}$ . e) Use implicit differentiation on $y^{2}-x y=3$ .

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Problem 26334

A particle moves with $s(t)=t^{3}-15 t^{2}+63 t-49$ . Find velocity, acceleration, critical points, and speeding/slowing intervals.

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Problem 26335

Find the max/min of $f(x)=x^{3}-x^{2}$ for $x \in [-2,1]$ . Set $f'(x)=0$ to locate critical points.

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Problem 26336

Find the motion of a particle at time $t=1$ given $y(t)=t^{3}-4t^{2}+4t+3$ . Choose the correct statement about its movement.

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Problem 26337

Solve $\frac{d y}{d x}=2-y$ with $y=1$ at $x=1$ . Find $y=$ (A) $2-e^{x-1}$ (B) $2-e^{1-x}$ (C) $2-e^{-x}$ (D) $2+e^{-x}$ .

See Solution

Problem 26338

Find the derivative of $2(\sin \sqrt{x})^{2}$ . Choose from: (A) $4 \cos \left(\frac{1}{2 \sqrt{x}}\right)$ , (B) $4 \sin \sqrt{x} \cos \sqrt{x}$ , (C) $\frac{2 \sin \sqrt{x}}{\sqrt{x}}$ , (D) $\frac{2 \sin \sqrt{x} \cos \sqrt{x}}{\sqrt{x}}$ .

See Solution

Problem 26339

Find interval where $f'(c)=2$ using values from $f(0)=8$ , $f(4)=0$ , $f(8)=2$ , $f(12)=10$ , $f(16)=1$ . Options: (A) (0,4) (B) (4,8) (C) (8,12) (D) (12,16).

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Problem 26340

At time $t=0$ , a tank fills with water at a rate of $1 W(i)$ ft/hr. What does $W^{\prime}(2)>3$ mean?

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Problem 26341

Find the value of $\int_{0}^{12} f(x) d x$ given $\int_{0}^{17} f(x) d x=8$ , $\int_{17}^{20} f(x) d x=-3$ , $\int_{13}^{20} f(x) d x=7$ .

See Solution

Problem 26342

Find $h$ such that $\int_{-1}^{h} x^{2} dx = \lim_{n \to \infty} \sum_{k=1}^{n} \left(1+\frac{2k}{n}\right)^{2} \frac{2}{n}$ . Options: (A) $b=2$ , (B) $h=3$ , (C) $b$ could be any real number, (D) no such $b$ .

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Problem 26343

Find $\int_{-2}^{4} \frac{f(x)}{8} d x$ if the average value of $f$ on $[-2,4]$ is 12. Options: (A) $\frac{3}{2}$ (B) 3 (C) 9 (D) 72.

See Solution

Problem 26344

Find the solution to $\frac{d y}{d x}=\frac{2 y}{2 x+1}$ with $y(0)=e$ for $x>-\frac{1}{2}$ . Options: (A) (B) (C) (D)

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Problem 26345

Find a point c for the MVT conclusion for these functions and intervals: 1. $y=x^{-1}$ , [2,8]; 2. $y=\sqrt{x}$ , [9,25]; 3. $y=\cos x-\sin x$ , [0,2\pi]; 4. $y=\frac{x}{x+2}$ , [1,4]; 5. $y=x^{3}$ , [-4,5]; 6. $y=x \ln x$ , [1,2]; 7. $y=e^{-2x}$ , [0,3]; 8. $y=e^{x}-x$ , [-1,1].

See Solution

Problem 26346

Find the expression for $F(x) = \int_{1}^{x} (\tan(5t)\sec(5t) - 1) \, dt$ . Options: (A) $\frac{1}{5} \sec (5 x) - 1$ , (B) $\frac{1}{5} \sec (5 x) - x$ , (C) $\tan (5 x) \sec (5 x)$ , (D) $\tan (5 x) \sec (5 x) - 1$ .

See Solution

Problem 26347

Find $\frac{d y}{d x}$ at the point $(2,1)$ for the equation $x^{2}+x y-3 y=3$ . Choices: (A) 5, (B) 4, (C) $\frac{7}{3}$ , (D) 2.

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Problem 26348

Determine the max and min values of $f(x)=\frac{x^{2}-9}{x^{2}+9}$ on the interval $[-5,5]$ .

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Problem 26349

Find $c$ from the Mean Value Theorem for $f(x)=x \sqrt{x+5}$ on $[-5,0]$ and calculate $f^{\prime}(c)$ .

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Problem 26350

Find the max and min values of $f(x)=x \sqrt{64-x^{2}}$ on $[-8,8]$ . a) Max b) Min

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Problem 26351

Find the max and min of $f(x)=10 \cos (x)+5 \sin (2 x)$ on $[0, \frac{\pi}{2}]$ .

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Problem 26352

Calculate the limit of $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=6 x^{2}+x+3$ .

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Problem 26353

Find $c$ guaranteed by the Mean Value Theorem for $f(x)=3 \sqrt[4]{x}$ on $[1,16]$ .

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Problem 26354

Evaluate the Riemann sum for $f(x)=2 x^{2}-5$ on $[-1,2]$ using 3 rectangles with midpoints for height.

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Problem 26355

Find all numbers $c$ in $[-1,2]$ for $f(x)=2 x^{3}-2 x^{2}+3 x-3$ that satisfy the Mean Value Theorem.

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Problem 26356

Find local extrema of $f(x)=3 x^{3}-9 x^{2}-216 x+7$ for $x \in (-\infty, \infty)$ . List min, max values or DNE.

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Problem 26357

Given the function $f(x)=4 x^{3}+9 x^{2}-12 x+4$ , find: a) intervals where $f$ is increasing, b) intervals where $f$ is decreasing, c) local maxima as ordered pairs.

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Problem 26358

Find the derivative of $y=e^{kx}$ and determine the values of $k$ for which this holds true.

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Problem 26359

Find the derivative $f^{\prime}(x)$ and values of $x$ where the tangent line of $f(x)=\frac{x}{(7x-8)^{6}}$ is horizontal.

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Problem 26360

Find $h'(1)$ and $p'(3)$ for $h(x)=f(g(x))$ and $p(x)=g(f(x))$ using the given function values and derivatives.

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Problem 26361

Given $f(x)=4 x^{3}+15 x^{2}-18 x+2$ , find intervals for increasing, decreasing, local maxima, minima, and concavity.

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Problem 26362

Find the next approximation $x_{2}$ using Newton's method for $f(x)=4-2x+\sin(x)$ with $x_{1}=3.5$ , rounded to 3 decimals.

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Problem 26363

Use Newton's method to approximate solutions for $3x^{3} + 3x - 5 = 0$ with initial guess $x_{1} = 1$ . Find $x_{2}$ and $x_{3}$ . Round to 3 decimal places.

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Problem 26364

Estimate the area under $f(x) = 1 + 2^{\frac{x}{6}}$ from $a = 0$ to $b = 8$ using rectangles (4 left, 4 right, 8 left, 8 right). Round to 3 decimal places.

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Problem 26365

Estimate the area under $f(x) = 7 - \frac{x^2}{36}$ from $x = 0$ to $x = 12$ using 6 rectangles. Round to 3 decimal places.

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Problem 26366

Estimate the area under $f(x) = 4x - \ln(x)$ from $x = 1$ to $x = 5$ using 4 rectangles. Round to three decimal places.

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Problem 26367

Evaluate the Riemann sum for $f(x)=4 x^{2}-3$ on $[-1,2]$ using 3 rectangles with midpoints for height.

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Problem 26368

Determine the max and min of $f(x)=8 \cos (x)+4 \sin (2 x)$ on $[0, \frac{\pi}{2}]$ .

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Problem 26369

Find the max and min of $f(x)=10 \cos (x)+5 \sin (2 x)$ on $[0, \frac{\pi}{2}]$ .

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Problem 26370

Find function $f$ such that $f^{\prime \prime}(x)=\sin x+\cos x$ , $f^{\prime}(0)=2$ , and $f(0)=9$ .

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Problem 26371

Estimate the area under $f(x) = \sqrt{2x}$ from $x = 0$ to $x = 4$ using 4 rectangles (left, right, midpoints). Round to 3 decimal places. Is each estimate an underestimate or overestimate?

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Problem 26372

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

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Problem 26373

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\sqrt{14 x}$ , where $h \neq 0$ . Simplify fully.

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Problem 26374

Find the limit as $x$ approaches 1 from the left: $\lim _{x \rightarrow 1^{-}} \frac{x^{2}+4 x-5}{x-1}$ .

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Problem 26375

Bestimmen Sie die Extremstellen der Funktionen: a) $f(x)=e^{3 x}-2 x$ , b) $f(x)=(x^{2}-8)e^{x}$ , c) $f(x)=e^{2 x+e^{-x}}$ .

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Problem 26376

Berechnen Sie das Volumen des Rotationshyperboloids für $f(x)=\frac{1}{x}$ , $x \geq 1$ und zeigen Sie, dass es endlich ist.

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Problem 26377

Find critical points, increasing/decreasing intervals, local extremes, concavity, and inflection points for $f(x)=\frac{x^{5}-4 x^{4}+10 x}{x}$ .

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Problem 26378

Calculate $\lim _{x \rightarrow 0} \frac{\text{perimeter of } N O P}{\text{perimeter of } M O P}$ for points $M=(1,0), N=(0,1), O=(0,0), P=(x, y)$ with $y=\sqrt{x}$ .

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Problem 26379

Bestimmen Sie die Extremstellen der Funktionen: a) $f(x)=e^{3 x}-2 x$ b) $f(x)=(x^{2}-8)e^{x}$

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Problem 26380

Calculate these limits: 1) $\lim _{x \rightarrow 1^{-}} \frac{x^{2}+4 x-5}{x-1}$ , 2) $\lim _{x \rightarrow-\infty} \frac{5 x^{2}-7}{8 x^{3}+2 x^{2}+1}$ , 3) $\lim _{x \rightarrow+\infty} \sqrt{1+x^{2}}-x$ , 4) $\lim _{x \rightarrow 0^{-}} \frac{|x| \sqrt{x+1}}{x}$ .

See Solution

Problem 26381

Find the limit as $x$ approaches 1 for the expression $\frac{x^{n}-1}{x-1}$ .

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Problem 26382

Find the limit as $x$ approaches 0 for $\frac{1}{x^{2}}$ .

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Problem 26383

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+1}-1}{\sqrt{x^{2}+25}-5}$ .

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Problem 26384

Berechnen Sie die mittlere Steigung von $f$ über den Intervallen: a) $f(x)=\frac{1}{2} x^{2}+1$ , $I=[1 ; 3]$ ; b) $f(x)=\frac{4}{x}$ , $I=[2 ; 8]$ ; c) $f(x)=2x$ , $I=[0 ; 1]$ ; d) $f(x)=2^{x}$ , $I=[1 ; 3]$ .

See Solution

Problem 26385

Die Funktion $p(t)=0,2 t e^{-0,0625 t}+5$ modelliert Plastikmüll in Mio. Tonnen. Überprüfen Sie: a) $p(0)=5$ , b) $p(10)-p(5) \approx 0,34$ , c) $p^{\prime}(16)=0$ , $p^{\prime \prime}(16)<0$ .

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Problem 26386

Find the limit: $\lim _{x \rightarrow 0} e^{\frac{1}{x}}$

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Problem 26387

Sophie hat Fieber, beschrieben durch $f(t)=1,5 \cdot t \cdot e^{-0,5 t+1}+37$ . Beantworte: a) Max. Temp., b) stärkster Rückgang, c) langfristige Temp., d) $f(6,5)-f(4,5)=-1$ interpretieren, e) Rückkehr zur Ausgangstemperatur nach Medikament.

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Problem 26388

Bestimme die Stammfunktion von $f(x)=\frac{1}{10^{10}} x^{5}-\frac{1}{130000} x^{3}+\frac{1}{6} x+6$ .

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Problem 26389

Find the limit as $x$ approaches 0 for the expression $e^{-\frac{1}{x^{2}}$.

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Problem 26390

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin \alpha x}{b x}$ .

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Problem 26391

Find the second derivative of $f(x)=\frac{x^{3}-4 x^{2}+6}{x^{2}}$ .

See Solution

Problem 26392

Find the derivative of $f(x) = \frac{x}{\sqrt{x^{4} + 1}}$ and explain each step.

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Problem 26393

Find the limit: $\lim _{x \rightarrow 1} \frac{e^{1-x}-x}{x-1}$ using L'Hopital's rule.

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Problem 26394

Find the value of $a$ such that $\lim _{x \rightarrow-\infty} \frac{\sqrt{a x^{2}+b x+c}}{d-a x}=2$ .

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Problem 26395

Find the second derivative $f^{\prime \prime}(1)$ for the function $f(x)=(x-1)|x-1|$ . Options: (A) 2 (B) -2 (C) 0 (D) undefined

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Problem 26396

Evaluate the limit as $x$ approaches 0: $\lim _{x \rightarrow 0} \frac{\sqrt{x+36}-6}{x}$

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Problem 26397

Bestimmen Sie die Grenzwerte von $f(x)=\frac{1-2x}{x+2}$ für $x \rightarrow \infty$ und $x \rightarrow -\infty$ sowie $\lim_{x \rightarrow 4} \frac{2x^2-32}{x-4}$ .

See Solution

Problem 26398

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

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Problem 26399

Find $g^{\prime}(2)$ for $g(x)=f^{-1}(x)$ where $f(x)=x^{3}+x$ and $g(2)=1$ . Options: (A) $\frac{1}{13}$ (B) $\frac{1}{4}$ (C) $\frac{7}{4}$ (D) 4 (E) 13

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Problem 26400

Find the limit: $\lim _{x \rightarrow e} \frac{\ln x-1}{\frac{e}{x}-1}$ using L'Hopital's rule. Options: (A) -1, (B) 1, (C) $e$ , (D) $-e$ .

See Solution

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Calculus

Problem 26301

Find the derivative of f(x)=(x+5x2+2)2f(x) = \left(\frac{x+5}{x^{2}+2}\right)^{2}f(x)=(x2+2x+5​)2 and explain each step.

Problem 26302

Find the integral for the volume of a triangle revolving about the xxx-axis given y=3−xy=3-xy=3−x. Choices include various integrals.

Problem 26303

You have two machines with data rates f(t)f(t)f(t) and g(t)g(t)g(t). For sss hours, express data processed by the first machine as a definite integral: ∫0sf(t) dt\int_0^s f(t) \, dt∫0s​f(t)dt.

Problem 26304

Problem 26305

Find the derivative of the function y=18xy=\frac{1}{8^{x}}y=8x1​.

Problem 26306

Calculate the integral ∫−15(3x2−7) dx\int_{-1}^{5}(3 x^{2}-7) \, dx∫−15​(3x2−7)dx.

Problem 26307

Find the slope of the tangent line to y=xexy=x e^{x}y=xex at x=ln⁡5x=\ln 5x=ln5. Options: (A) 5ln⁡55 \ln 55ln5 (B) 5ln⁡5+55 \ln 5+55ln5+5 (C) e5(ln⁡5)+e5e^{5}(\ln 5)+e^{5}e5(ln5)+e5 (D) 5+2ln⁡5e5+\frac{2 \ln 5}{e}5+e2ln5​

Problem 26308

You have two machines with rates f(t)=9−tf(t)=9-tf(t)=9−t and g(t)=2t+1g(t)=2t+1g(t)=2t+1. How long to use the first machine to maximize data processed in 6 hours?

Problem 26309

Find the values of rrr for which y=erxy=e^{r x}y=erx solves the equation 2y′′+y′−y=02 y^{\prime \prime}+y^{\prime}-y=02y′′+y′−y=0.

Problem 26310

Berechnen Sie die Anzahl der Erkrankten am 10. Tag mit der Funktion f(x)=−1250x3+110x2f(x)=-\frac{1}{250} x^{3}+\frac{1}{10} x^{2}f(x)=−2501​x3+101​x2. Bestimmen Sie das Ende der Epidemie, den Definitions- und Wertebereich, Grenzwerte und die Symmetrie.

Problem 26311

Berechne die Ableitung von f(x)=3x2+12x−8f(x)=3 x^{2}+12 x-8f(x)=3x2+12x−8 bei x0=−6x_{0}=-6x0​=−6 als Grenzwert.

Problem 26312

Bestimme die Ableitung von f(x)=x2−8xf(x)=x^{2}-8xf(x)=x2−8x bei x0=3x_{0}=3x0​=3 als Grenzwert von lim⁡h→0f(3+h)−f(3)h\lim_{h \to 0} \frac{f(3+h)-f(3)}{h}limh→0​hf(3+h)−f(3)​.

Problem 26313

Find dydx\frac{dy}{dx}dxdy​ when x=2x=2x=2 for the equation x2y+y2+4=0x^{2} y+y^{2}+4=0x2y+y2+4=0.

Problem 26314

Determine the left and right end behavior of the function f(x)=x3−5x2+3x−1x4−7x+2f(x)=\frac{x^{3}-5 x^{2}+3 x-1}{x^{4}-7 x+2}f(x)=x4−7x+2x3−5x2+3x−1​.

Problem 26315

Bestimme die Ableitung von f(x)=4(x−7)2+8f(x)=4(x-7)^{2}+8f(x)=4(x−7)2+8 an der Stelle x0=6x_0=6x0​=6 als Grenzwert.

Problem 26316

Find the xxx values for local minima in a quartic function (answer as x=a,b,cx=a, b, cx=a,b,c). Are they −2,−1,0,1,2-2, -1, 0, 1, 2−2,−1,0,1,2?

Problem 26317

Given velocity data, find the object's acceleration, distance in 4s, and distance from 4s to 7s. Choices included.

Problem 26318

A woman 1.5 m tall runs away from a 3 m pole at 1.8 m/s. Find the speed of her shadow's tip when she's 15 m from the pole.

Problem 26319

Find the derivative of f(x)=x2f(x)=x^{2}f(x)=x2 at x=−4x=-4x=−4 using first principles.

Problem 26320

Find the derivative of the following functions with respect to x\mathrm{x}x: i) f(x)=xf(x)=xf(x)=x, ii) f(x)=x3f(x)=x^{3}f(x)=x3, iii) f(x)=x4f(x)=x^{4}f(x)=x4, iv) f(x)=x5f(x)=x^{5}f(x)=x5.

Problem 26321

Find the derivative of f(x)=x2f(x)=x^{2}f(x)=x2 using first principles. Calculate f′(−11)f'(-11)f′(−11), f′(0)f'(0)f′(0), and f′(5)f'(5)f′(5).

Problem 26322

Find g′(x)g'(x)g′(x) for g(x)=(6x3+4x2−3)(4x2+2x+7)g(x)=(6x^{3}+4x^{2}-3)(4x^{2}+2x+7)g(x)=(6x3+4x2−3)(4x2+2x+7).

Problem 26323

Find the derivative of the function f(x)=7x2−4x+2f(x)=7 x^{2}-4 x+2f(x)=7x2−4x+2, i.e., compute f′(x)f^{\prime}(x)f′(x).

Problem 26324

Find the derivative of f(x)=x39f(x)=x^{39}f(x)=x39 and generalize it for f(x)=xnf(x)=x^{n}f(x)=xn, where nnn is a positive integer.

Problem 26325

Find the derivatives: a) f(x)=x13f(x)=x^{13}f(x)=x13, b) g(x)=x−5g(x)=x^{-5}g(x)=x−5. Calculate f′(x)f^{\prime}(x)f′(x) and g′(x)g^{\prime}(x)g′(x).

Problem 26326

How many times larger is a cell population growing at 100%100\%100% per day today compared to yesterday? Options: 2, 2.25, 2.5, 2.75.

Problem 26327

Find dydx\frac{d y}{d x}dxdy​ at x=−2x=-2x=−2 for y=x4(3x+7)3y=x^{4}(3 x+7)^{3}y=x4(3x+7)3 and at x=−1x=-1x=−1 for y=(2x+1)5(3x+2)4y=(2 x+1)^{5}(3 x+2)^{4}y=(2x+1)5(3x+2)4.

Problem 26328

Find the derivative h′(x)h'(x)h′(x) for h(x)=(x2+3)(4−2x3)h(x)=(x^{2}+3)(4-2x^{3})h(x)=(x2+3)(4−2x3).

Problem 26329

Find dydx\frac{d y}{d x}dxdy​ for x=−2x=-2x=−2 in y=x4(3x+7)3y=x^{4}(3 x+7)^{3}y=x4(3x+7)3 and for x=−1x=-1x=−1 in y=(2x+1)5(3x+2)4y=(2 x+1)^{5}(3 x+2)^{4}y=(2x+1)5(3x+2)4.

Problem 26330

Calculate the average rate of change of f(x)=tan⁡(3x)f(x)=\tan(3x)f(x)=tan(3x) from 0 to π9\frac{\pi}{9}9π​.

Problem 26331

Find the second derivative of these functions: a) y=x3−2x2y=x^{3}-2 x^{2}y=x3−2x2, b) y=x5y=x^{5}y=x5, c) y=x−8y=x^{-8}y=x−8, d) y=xy=\sqrt{x}y=x​.

Problem 26332

Find f′(g(−1))f^{\prime}(g(-1))f′(g(−1)) where f(x)=2x2+x+3f(x)=2 x^{2}+x+3f(x)=2x2+x+3 and g(x)=4x2+5x−12g(x)=4 x^{2}+5 x-12g(x)=4x2+5x−12.

Problem 26333

i) Find the 9th derivative of y=x9y=x^{9}y=x9. ii) Find the 11th derivative of y=13x11y=13 x^{11}y=13x11. e) Use implicit differentiation on y2−xy=3y^{2}-x y=3y2−xy=3.

Problem 26334

A particle moves with s(t)=t3−15t2+63t−49s(t)=t^{3}-15 t^{2}+63 t-49s(t)=t3−15t2+63t−49. Find velocity, acceleration, critical points, and speeding/slowing intervals.

Problem 26335

Find the max/min of f(x)=x3−x2f(x)=x^{3}-x^{2}f(x)=x3−x2 for x∈[−2,1]x \in [-2,1]x∈[−2,1]. Set f′(x)=0f'(x)=0f′(x)=0 to locate critical points.

Problem 26336

Find the motion of a particle at time t=1t=1t=1 given y(t)=t3−4t2+4t+3y(t)=t^{3}-4t^{2}+4t+3y(t)=t3−4t2+4t+3. Choose the correct statement about its movement.

Problem 26337

Solve dydx=2−y\frac{d y}{d x}=2-ydxdy​=2−y with y=1y=1y=1 at x=1x=1x=1. Find y=y=y= (A) 2−ex−12-e^{x-1}2−ex−1 (B) 2−e1−x2-e^{1-x}2−e1−x (C) 2−e−x2-e^{-x}2−e−x (D) 2+e−x2+e^{-x}2+e−x.

Problem 26338

Problem 26339

Find interval where f′(c)=2f'(c)=2f′(c)=2 using values from f(0)=8f(0)=8f(0)=8, f(4)=0f(4)=0f(4)=0, f(8)=2f(8)=2f(8)=2, f(12)=10f(12)=10f(12)=10, f(16)=1f(16)=1f(16)=1. Options: (A) (0,4) (B) (4,8) (C) (8,12) (D) (12,16).

Problem 26340

At time t=0t=0t=0, a tank fills with water at a rate of 1W(i)1 W(i)1W(i) ft/hr. What does W′(2)>3W^{\prime}(2)>3W′(2)>3 mean?

Problem 26341

Find the derivative of $f(x) = \left(\frac{x+5}{x^{2}+2}\right)^{2}$ and explain each step.

Find the integral for the volume of a triangle revolving about the $x$ -axis given $y=3-x$ . Choices include various integrals.

You have two machines with data rates $f(t)$ and $g(t)$ . For $s$ hours, express data processed by the first machine as a definite integral: $\int_0^s f(t) \, dt$ .

Find the derivative of the function $y=\frac{1}{8^{x}}$ .

Calculate the integral $\int_{-1}^{5}(3 x^{2}-7) \, dx$ .

Find the slope of the tangent line to $y=x e^{x}$ at $x=\ln 5$ . Options: (A) $5 \ln 5$ (B) $5 \ln 5+5$ (C) $e^{5}(\ln 5)+e^{5}$ (D) $5+\frac{2 \ln 5}{e}$

You have two machines with rates $f(t)=9-t$ and $g(t)=2t+1$ . How long to use the first machine to maximize data processed in 6 hours?

Find the values of $r$ for which $y=e^{r x}$ solves the equation $2 y^{\prime \prime}+y^{\prime}-y=0$ .

Berechnen Sie die Anzahl der Erkrankten am 10. Tag mit der Funktion $f(x)=-\frac{1}{250} x^{3}+\frac{1}{10} x^{2}$ . Bestimmen Sie das Ende der Epidemie, den Definitions- und Wertebereich, Grenzwerte und die Symmetrie.

Berechne die Ableitung von $f(x)=3 x^{2}+12 x-8$ bei $x_{0}=-6$ als Grenzwert.

Bestimme die Ableitung von $f(x)=x^{2}-8x$ bei $x_{0}=3$ als Grenzwert von $\lim_{h \to 0} \frac{f(3+h)-f(3)}{h}$ .

Find $\frac{dy}{dx}$ when $x=2$ for the equation $x^{2} y+y^{2}+4=0$ .

Determine the left and right end behavior of the function $f(x)=\frac{x^{3}-5 x^{2}+3 x-1}{x^{4}-7 x+2}$ .

Bestimme die Ableitung von $f(x)=4(x-7)^{2}+8$ an der Stelle $x_0=6$ als Grenzwert.

Find the $x$ values for local minima in a quartic function (answer as $x=a, b, c$ ). Are they $-2, -1, 0, 1, 2$ ?

Find the derivative of $f(x)=x^{2}$ at $x=-4$ using first principles.

Find the derivative of the following functions with respect to $\mathrm{x}$ : i) $f(x)=x$ , ii) $f(x)=x^{3}$ , iii) $f(x)=x^{4}$ , iv) $f(x)=x^{5}$ .

Find the derivative of $f(x)=x^{2}$ using first principles. Calculate $f'(-11)$ , $f'(0)$ , and $f'(5)$ .

Find $g'(x)$ for $g(x)=(6x^{3}+4x^{2}-3)(4x^{2}+2x+7)$ .

Find the derivative of the function $f(x)=7 x^{2}-4 x+2$ , i.e., compute $f^{\prime}(x)$ .

Find the derivative of $f(x)=x^{39}$ and generalize it for $f(x)=x^{n}$ , where $n$ is a positive integer.

Find the derivatives: a) $f(x)=x^{13}$ , b) $g(x)=x^{-5}$ . Calculate $f^{\prime}(x)$ and $g^{\prime}(x)$ .

How many times larger is a cell population growing at $100\%$ per day today compared to yesterday? Options: 2, 2.25, 2.5, 2.75.

Find $\frac{d y}{d x}$ at $x=-2$ for $y=x^{4}(3 x+7)^{3}$ and at $x=-1$ for $y=(2 x+1)^{5}(3 x+2)^{4}$ .

Find the derivative $h'(x)$ for $h(x)=(x^{2}+3)(4-2x^{3})$ .

Find $\frac{d y}{d x}$ for $x=-2$ in $y=x^{4}(3 x+7)^{3}$ and for $x=-1$ in $y=(2 x+1)^{5}(3 x+2)^{4}$ .

Calculate the average rate of change of $f(x)=\tan(3x)$ from 0 to $\frac{\pi}{9}$ .

Find the second derivative of these functions: a) $y=x^{3}-2 x^{2}$ , b) $y=x^{5}$ , c) $y=x^{-8}$ , d) $y=\sqrt{x}$ .

Find $f^{\prime}(g(-1))$ where $f(x)=2 x^{2}+x+3$ and $g(x)=4 x^{2}+5 x-12$ .

i) Find the 9th derivative of $y=x^{9}$ . ii) Find the 11th derivative of $y=13 x^{11}$ . e) Use implicit differentiation on $y^{2}-x y=3$ .

A particle moves with $s(t)=t^{3}-15 t^{2}+63 t-49$ . Find velocity, acceleration, critical points, and speeding/slowing intervals.

Find the max/min of $f(x)=x^{3}-x^{2}$ for $x \in [-2,1]$ . Set $f'(x)=0$ to locate critical points.

Find the motion of a particle at time $t=1$ given $y(t)=t^{3}-4t^{2}+4t+3$ . Choose the correct statement about its movement.

Solve $\frac{d y}{d x}=2-y$ with $y=1$ at $x=1$ . Find $y=$ (A) $2-e^{x-1}$ (B) $2-e^{1-x}$ (C) $2-e^{-x}$ (D) $2+e^{-x}$ .

Find interval where $f'(c)=2$ using values from $f(0)=8$ , $f(4)=0$ , $f(8)=2$ , $f(12)=10$ , $f(16)=1$ . Options: (A) (0,4) (B) (4,8) (C) (8,12) (D) (12,16).

At time $t=0$ , a tank fills with water at a rate of $1 W(i)$ ft/hr. What does $W^{\prime}(2)>3$ mean?

Find the value of $\int_{0}^{12} f(x) d x$ given $\int_{0}^{17} f(x) d x=8$ , $\int_{17}^{20} f(x) d x=-3$ , $\int_{13}^{20} f(x) d x=7$ .

Find $\int_{-2}^{4} \frac{f(x)}{8} d x$ if the average value of $f$ on $[-2,4]$ is 12. Options: (A) $\frac{3}{2}$ (B) 3 (C) 9 (D) 72.

Find the solution to $\frac{d y}{d x}=\frac{2 y}{2 x+1}$ with $y(0)=e$ for $x>-\frac{1}{2}$ . Options: (A) (B) (C) (D)

Find $\frac{d y}{d x}$ at the point $(2,1)$ for the equation $x^{2}+x y-3 y=3$ . Choices: (A) 5, (B) 4, (C) $\frac{7}{3}$ , (D) 2.

Determine the max and min values of $f(x)=\frac{x^{2}-9}{x^{2}+9}$ on the interval $[-5,5]$ .

Find $c$ from the Mean Value Theorem for $f(x)=x \sqrt{x+5}$ on $[-5,0]$ and calculate $f^{\prime}(c)$ .

Find the max and min values of $f(x)=x \sqrt{64-x^{2}}$ on $[-8,8]$ . a) Max b) Min

Find the max and min of $f(x)=10 \cos (x)+5 \sin (2 x)$ on $[0, \frac{\pi}{2}]$ .

Calculate the limit of $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=6 x^{2}+x+3$ .

Find $c$ guaranteed by the Mean Value Theorem for $f(x)=3 \sqrt[4]{x}$ on $[1,16]$ .

Evaluate the Riemann sum for $f(x)=2 x^{2}-5$ on $[-1,2]$ using 3 rectangles with midpoints for height.

Find all numbers $c$ in $[-1,2]$ for $f(x)=2 x^{3}-2 x^{2}+3 x-3$ that satisfy the Mean Value Theorem.

Find local extrema of $f(x)=3 x^{3}-9 x^{2}-216 x+7$ for $x \in (-\infty, \infty)$ . List min, max values or DNE.

Given the function $f(x)=4 x^{3}+9 x^{2}-12 x+4$ , find: a) intervals where $f$ is increasing, b) intervals where $f$ is decreasing, c) local maxima as ordered pairs.

Find the derivative of $y=e^{kx}$ and determine the values of $k$ for which this holds true.

Find the derivative $f^{\prime}(x)$ and values of $x$ where the tangent line of $f(x)=\frac{x}{(7x-8)^{6}}$ is horizontal.

Find $h'(1)$ and $p'(3)$ for $h(x)=f(g(x))$ and $p(x)=g(f(x))$ using the given function values and derivatives.

Given $f(x)=4 x^{3}+15 x^{2}-18 x+2$ , find intervals for increasing, decreasing, local maxima, minima, and concavity.

Find the next approximation $x_{2}$ using Newton's method for $f(x)=4-2x+\sin(x)$ with $x_{1}=3.5$ , rounded to 3 decimals.

Use Newton's method to approximate solutions for $3x^{3} + 3x - 5 = 0$ with initial guess $x_{1} = 1$ . Find $x_{2}$ and $x_{3}$ . Round to 3 decimal places.

Estimate the area under $f(x) = 1 + 2^{\frac{x}{6}}$ from $a = 0$ to $b = 8$ using rectangles (4 left, 4 right, 8 left, 8 right). Round to 3 decimal places.

Estimate the area under $f(x) = 7 - \frac{x^2}{36}$ from $x = 0$ to $x = 12$ using 6 rectangles. Round to 3 decimal places.

Estimate the area under $f(x) = 4x - \ln(x)$ from $x = 1$ to $x = 5$ using 4 rectangles. Round to three decimal places.

Evaluate the Riemann sum for $f(x)=4 x^{2}-3$ on $[-1,2]$ using 3 rectangles with midpoints for height.

Determine the max and min of $f(x)=8 \cos (x)+4 \sin (2 x)$ on $[0, \frac{\pi}{2}]$ .

Find the max and min of $f(x)=10 \cos (x)+5 \sin (2 x)$ on $[0, \frac{\pi}{2}]$ .

Find function $f$ such that $f^{\prime \prime}(x)=\sin x+\cos x$ , $f^{\prime}(0)=2$ , and $f(0)=9$ .

Estimate the area under $f(x) = \sqrt{2x}$ from $x = 0$ to $x = 4$ using 4 rectangles (left, right, midpoints). Round to 3 decimal places. Is each estimate an underestimate or overestimate?

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

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\sqrt{14 x}$ , where $h \neq 0$ . Simplify fully.

Find the limit as $x$ approaches 1 from the left: $\lim _{x \rightarrow 1^{-}} \frac{x^{2}+4 x-5}{x-1}$ .