Calculus

Problem 26401

Find the limit $\lim _{x \rightarrow 1} \frac{x e^{x}-e}{x-1}$ using L'Hopital's rule. Choose (A) $-e$ , (B) $2 e$ , (C) $e$ , or (D) 1.

See Solution

Problem 26402

Find the limit: $\lim _{x \rightarrow+\infty} \frac{2 x^{2}+x-5}{x^{2}-3}$ . Choose from (A) $-\frac{2}{3}$ , (B) 2, (C) 1, (D) $\frac{5}{3}$ .

See Solution

Problem 26403

A plane is at $4 \mathrm{~km}$ altitude and traveling at $500 \mathrm{~m/s}$ .
a) When does the bomb hit the ground? b) What is its final velocity?

See Solution

Problem 26404

Bestimme die Tangentengleichung der Funktion $f(x) = x^{3}$ bei $a = 2$ .

See Solution

Problem 26405

Find the limit as $x$ approaches -6 for the expression $\frac{2 x^{2}+12 x}{x+6}$ .

See Solution

Problem 26406

Find the second derivative of $f(x)=\frac{1}{\sin x}$ . Which is correct: (A) $f^{\prime \prime}(x)=\frac{1-\cos ^{2} x}{\sin ^{3} x}$ , (B) $f^{\prime \prime}(x)=\frac{1+\cos ^{2} x}{\sin ^{3} x}$ , (C) $f^{\prime \prime}(x)=\frac{1+2 \cos ^{2} x}{\sin ^{3} x}$ , or (D) $f^{\prime \prime}(x)=\frac{1-2 \cos ^{2} x}{\sin ^{3} x}$ ?

See Solution

Problem 26407

Find the limit as $x$ approaches 6 from the left for $\frac{2x}{x^2 - 36}$ . What is the value?

See Solution

Problem 26408

Find the derivatives of the following functions:
a) $f(x) = 2x^3 + 5x^2$ , d) $f(x) = 0.25x^4 - x^3$ , 8) $f(x) = \frac{1}{5}x^5 - 3x^3 - 2$ .

See Solution

Problem 26409

Given $f(1) = 3$ , find $g'(3)$ where $g$ is the inverse of $f$ . Options: (A) $\frac{1}{13}$ (B) $\frac{1}{4}$ (C) 1 (D) 4 (E) 13.

See Solution

Problem 26410

Given the table of values for a function $f$ and its derivative, find $g^{\prime}(3)$ where $g$ is the inverse of $f$ . Choices: (A) $\frac{1}{13}$ (B) $\frac{1}{4}$ (C) 1 (D) 4 (E) 13

See Solution

Problem 26411

Find the value of $b$ in the tangent line equation $y=ax+b$ for $y=\sqrt[3]{x}$ at $x=8$ .

See Solution

Problem 26412

Evaluate the limit as $x$ approaches $\frac{\pi}{4}$ : $\lim_{x \to \frac{\pi}{4}} \frac{\tan x - 1}{x - \frac{\pi}{4}}$ .

See Solution

Problem 26413

Find the derivative $h'(2)$ for $h(x) = 16 f(x) - g(x)$ given $f(2) = 5$ , $g(2) = -8$ , $f'(2) = \frac{1}{8}$ , $g'(2) = -5$ .

See Solution

Problem 26414

Find the sum of the series: $\sum_{k=1}^{\infty}\left(-\frac{4}{9}\right)^{k}$ .

See Solution

Problem 26415

Find $dy/dx$ for $\cos(xy) = y - 1$ at $x = \pi/2$ , $y = 1$ . Options: (A) $\frac{-2}{2-\pi}$ (B) $\frac{-2}{2+\pi}$ (C) 0 (D) $\frac{2}{2-\pi}$ (E) $\frac{2}{2+\pi}$ .

See Solution

Problem 26416

Given that $g(x)=f^{-1}(x)$ , find $g^{\prime}(4)$ using values from the table. Options: (A) $-\frac{1}{3}$ , (B) $-\frac{1}{4}$ , (C) $-\frac{3}{100}$ , (D) $\frac{1}{4}$ , (E) $\frac{1}{3}$ .

See Solution

Problem 26417

Find $\frac{d y}{d x}$ for the equation $x^{3}+3 x y+2 y^{3}=17$ . Choose from options (A) to (E).

See Solution

Problem 26418

Find $w'(0)$ for $w(x) = 3 f(x) g(x)$ given $f(0) = 3$ , $g(0) = -2$ , $f'(0) = \frac{1}{2}$ , $g'(0) = 3$ .

See Solution

Problem 26419

Find $g^{\prime}(3)$ for $g(x)=f^{-1}(x)$ given $f(3)=15$ , $f(6)=3$ , $f^{\prime}(3)=-8$ , $f^{\prime}(6)=-2$ .

See Solution

Problem 26420

Find $\frac{d y}{d x}$ for the equation $3 x^{2}+2 x y+y^{2}=1$ . Choose the correct option from (A) to (E).

See Solution

Problem 26421

Is it true or false that the function $f(x)$ defined as
$f(x)=\left\{\begin{array}{ll} \frac{1}{6} x-4, & x<6 \\ -\frac{x}{2}, & x=6 \\ 3-x, & x>6 \end{array}\right.$
satisfies $\lim _{x \rightarrow 6^{-}} f(x)=\lim _{x \rightarrow 6^{+}} f(x)=f(6)$ ?

See Solution

Problem 26422

Find the value of $c$ such that $0 < c < 2$ and $g(x) = -\pi$ given the conditions for $g(x)$ and its derivatives.

See Solution

Problem 26423

Find the derivative of $y=\tan^{-1}(3x)$ at $x=1$ : what is $y'(1)$ ?

See Solution

Problem 26424

Find the derivative of the function $f(x)$ given that $\frac{d}{dx}f(x) = \frac{d}{dx}2^{x}$ .

See Solution

Problem 26425

For $0<c<2$ , what must be true for $g(x)=-\pi$ regarding continuity and differentiability?

See Solution

Problem 26426

Bestimmen Sie die Ableitung der Funktionen: a) $f(x)=x^{-5}$ , b) $f(x)=3x^{-4}$ , c) $f(x)=-\frac{1}{3}x^{-6}$ , d) $f(x)=x^{4}-x+1/x^{3}$ .

See Solution

Problem 26427

Find $\frac{dy}{dx}$ for $e^{xy} - y^2 = e - 4$ at $x = \frac{1}{2}$ , $y = 2$ . Options: (A) $\frac{e}{4}$ , (B) $\frac{e}{2}$ , (C) $\frac{4e}{8-e}$ , (D) $\frac{4e}{4-e}$ , (E) $\frac{8-4e}{e}$ .

See Solution

Problem 26428

Sand forms a pile with volume $V=\frac{r^{3}}{3}$ . If circumference increases at $5\pi$ ft/hr, find $dV/dt$ when circumference is $8\pi$ .

See Solution

Problem 26429

A sphere's cross-sectional area increases at $2 \mathrm{~cm}^{2} / \mathrm{sec}$ . Find the volume change rate when $r=4 \mathrm{~cm}$ .

See Solution

Problem 26430

Ein Superheld schießt aus der Höhe $f(x)=30 e^{-0,1 x}+52$ auf den Schurken in $P(30|52)$ .
a) Bestimme die Projektilfunktion. b) Höhe beim Schuss: $56,06 m$ . c) Finde die Funktion $g(x)=a \cdot b^{x}+60$ für den Weg von $Q(40|52,55)$ nach $R(70|0)$ .

See Solution

Problem 26431

Find the derivative of the inverse function at 5, using $\left(f^{-1}\right)^{\prime}(y) = \frac{1}{f^{\prime}(x)}$ .

See Solution

Problem 26432

Find the rate of change of fuel consumption $F(s)=6 e^{\left(\frac{\cdot}{20}-\frac{s^{2}}{2 m 0}\right)}$ at $s=50$ mph, $ds/dt=20$ mph $^2$ . Choices: (A) 0.215 (B) 4.299 (C) 9.793 (D) 25.793 (E) 515.855.

See Solution

Problem 26433

Find the rate of change of fuel consumption $F(s)=6 e^{\left(\frac{1}{20}-\frac{2^{2}}{2000}\right)}$ at 50 mph with $s' = 20$ mph².

See Solution

Problem 26434

Find $\frac{d y}{d x}$ at the point where $x=-1$ for the equation $x^{3}-x^{2} y+y^{2}=11$ .

See Solution

Problem 26435

Find the rate of change of fuel consumption $F(s)=6 e^{\left(\frac{s}{20}-\frac{s^{2}}{2000}\right)}$ at $s=50$ mph with $ds/dt=20$ mph².

See Solution

Problem 26436

Find $\left(f^{-1}\right)^{\prime}(5)$ given $f(0)=3$ , $f(1)=4$ , $f(2)=5$ , $f'(0)=\frac{1}{2}$ , $f'(1)=\frac{1}{4}$ , $f'(2)=\frac{1}{8}$ .

See Solution

Problem 26437

1. For the bird's height $h(t)$ , explain what $h^{\prime}(4)=2$ means in context.
2. For university enrollment $s(t)$ , explain $s^{\prime}(2.5)=370$ in context.

See Solution

Problem 26438

In a triangle, if $\theta$ increases at 3 radians/min, how fast is $x$ increasing when $x=3$ units? Hypotenuse is 5.

See Solution

Problem 26439

Find the area increase rate when the circumference is $20 \pi$ m, with radius growing at 0.2 m/s. Options: (A) $0.04 \pi \mathrm{m}^{2}/\mathrm{sec}$ or $0.4 \pi \mathrm{m}^{2}/\mathrm{sec}$ .

See Solution

Problem 26440

Finde die Stelle, an der die Steigung der Funktion $f(x)=x^{9}$ gleich 9 ist. Gib nur eine Lösung an.

See Solution

Problem 26441

1. Interpret $h^{\prime}(4)=2$ : At 4 seconds, the bird ascends at 2 meters/second.
2. Interpret $s^{\prime}(2.5)=370$ : At 2.5 years, student enrollment increases by 370/year.
3. Interpret $f^{\prime}(10)=27$ : At 10 minutes, the submarine descends at 27 feet/minute.
4. Interpret $s^{\prime}(7)=10,000$ : On day 7, stock sales increase by 10,000 stocks/day.
5. Interpret $w^{\prime}(20)=-100$ : At 20 seconds, water flow decreases by 100 gallons/second.

See Solution

Problem 26442

1. For $h(t)$ , $h^{\prime}(4)=2$ means the bird ascends at 2 m/s at 4 seconds.
2. For $s(t)$ , $s^{\prime}(2.5)=370$ means enrollment increases by 370 students/year at 2.5 years.
3. For $f(t)$ , interpret $f^{\prime}(10)=27$ as the submarine descends at 27 ft/min at 10 minutes.
4. For $s(d)$ , $s^{\prime}(7)=10,000$ means 10,000 stocks are sold per day at 7 days.

See Solution

Problem 26443

Evaluate the integral: $\int_{0}^{\infty} e^{-x} dx$

See Solution

Problem 26444

Bestimme die Tangentengleichung von $f(x)=x^{6}$ bei $x_{0}=2$ .

See Solution

Problem 26445

Finde die Stelle, an der die Steigung des Graphen von $f(x)=x^{4}$ gleich $32$ ist.

See Solution

Problem 26446

Evaluate the integral: $\int_{0}^{\infty} \frac{2}{x^{2}+4 x+3} d x$

See Solution

Problem 26447

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for $y=\frac{1}{x^{3}}-4 x^{5}+3 x$ . Simplify first!

See Solution

Problem 26448

Calculate the average rate of change of $f(x)=\frac{3}{x+2}$ from $x=1$ to $x=4$ . Options: A. -28 B. $-\frac{1}{6}$ C. -2 D. $\frac{1}{2}$

See Solution

Problem 26449

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for $y=\frac{1}{x^{3}}-4 x^{5}+3 x$ and $\frac{d y}{d x}=y^{3} x^{2}$ .

See Solution

Problem 26450

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for $y=\frac{1}{x^{3}}-4 x^{5}+3 x$ and $\frac{d y}{d x}=y^{3} x^{2}$ .

See Solution

Problem 26451

Determine if the particle with velocity $v(t)=t \sin^{3}(5t)$ at $t=2$ is speeding up or slowing down.

See Solution

Problem 26452

What are the units of the derivative $R'(t)$ if $R(t)$ is in gallons/hour? Options: 1) $-\frac{1}{2}$ 2) $0.653$ 3) $\frac{3}{8}$ 4) $-\frac{3}{4}$

See Solution

Problem 26453

Find the second derivative: 1. For $f(x)=x^{\frac{5}{3}}+x$ , calculate $f^{\prime \prime}(8)$ . 2. For $\frac{d y}{d x}=y^{2}(3-4 x)$ , find $\frac{d^{2} y}{d x^{2}}$ at $\left(1, \frac{1}{2}\right)$ .

See Solution

Problem 26454

Strontium-85 has a half-life of 65 days.
a. How long for radiation to drop to $\frac{1}{4}$ of its original level? b. How long for radiation to drop to $\frac{1}{8}$ of its original level?

See Solution

Problem 26455

How long does it take for an oscillation's amplitude to drop from $80 \mathrm{~cm}$ to $5 \mathrm{~cm}$ in 4 seconds?

See Solution

Problem 26456

A roller coaster starts at $2.5 \mathrm{~m/s}$ at $25 \mathrm{~m}$ . Find its speed at $20 \mathrm{~m}$ , $10 \mathrm{~m}$ , and ground level.

See Solution

Problem 26457

Berechne die Ableitung von $f(x) = x e^{2-x}$ mit der Produktregel. Was ist $f'(x)$ ?

See Solution

Problem 26458

A satellite of mass $m=1371 \mathrm{~kg}$ falls from $R=3.8$ Earth radii. Find its speed before hitting Earth.

See Solution

Problem 26459

Find the maximum acceleration of an object in simple harmonic motion given $x=2.70 \sin(0.413 t + 2.38)$ .

See Solution

Problem 26460

Find the limit: $\lim _{x \rightarrow-2} \frac{x^{2}+5 x+6}{x+2}$

See Solution

Problem 26461

Find the limit: $\lim _{x \rightarrow 2} \frac{\sqrt{x+2}-2}{x-2}$ .

See Solution

Problem 26462

A 0.422 kg mass on a 0.303 m string swings with a 6.96° amplitude. Find the maximum speed of the mass.

See Solution

Problem 26463

Find the intervals where the function $g$ is decreasing, given $f(x)=4-x$ and $g'(x)=f(x) f'(x)(x-2)$ .

See Solution

Problem 26464

Calculate the curve length for $x=5 \sin t + 5 t$ , $y=5 \cos t$ , where $0 \leq t \leq \pi$ .

See Solution

Problem 26465

Find the value of $\frac{d^{2} y}{d x^{2}}$ at $t=1$ where $x=6 t^{2}-3$ and $y=t^{3}$ .

See Solution

Problem 26466

Find $\lim _{x \rightarrow-2^{+}} f(x)$ for the piecewise function: $f(x)=\begin{cases}2 x+7, & x<-2 \\ -\frac{x}{2}-\frac{1}{2}, & x \geq-2\end{cases}$ .

See Solution

Problem 26467

Find the tangent line equation to the curve at $t=\frac{3 \pi}{4}$ for $x=7 \sin t$ , $y=7 \cos t$ .

See Solution

Problem 26468

Find if the water in the gutter is increasing or decreasing at $t=4$ hours using $G(t)=10 \sin \left(\frac{t^{2}}{30}\right)$ and $D(t)=-0.02 t^{3}+0.05 t^{2}+0.87 t$ .

See Solution

Problem 26469

Bestimme die Tangentengleichung der Funktion $f(x) = \frac{1}{x}$ bei $a = -3$ .

See Solution

Problem 26470

Find the tangent line to $f(x)=\sqrt{x+9}$ with slope $\frac{1}{4}$ .

See Solution

Problem 26471

How many years does it take for an investment to double at a continuous growth rate of $14\%$ ? Round to one decimal place.

See Solution

Problem 26472

Find the slope of the curve $r=5-3 \cos \theta$ at $\theta=\frac{\pi}{2}$ .

See Solution

Problem 26473

Determine which statements about a continuous function $f(x)$ on $[a, b]$ are NOT necessarily true: I, II, III.

See Solution

Problem 26474

Given the function $v(t)=t^{3}-5 t^{2}+6 t$ on $[0,5]$ , find when motion is positive/negative, displacement, and distance traveled.

See Solution

Problem 26475

What continuous compounding rate is needed to triple an investment in 7 years? Find the rate as $\square \%$ .

See Solution

Problem 26476

Definieren Sie eine Stammfunktion und nennen Sie drei Beispiele dafür.

See Solution

Problem 26477

Calculate the curve length for $r=3 \sec \theta$ where $0 \leq \theta \leq \frac{\pi}{4}$ .

See Solution

Problem 26478

Determine which statements about a differentiable function $f$ are true: I, II, or III. Options include combinations of these statements.

See Solution

Problem 26479

If $f(x)$ is continuous on $[-3,7]$ and differentiable on $(-3,7)$ with $f(-3)=4$ and $f(7)=2$ , find $c$ such that $f'(c)=1/5$ , $f'(c)=-1/5$ , $f'(c)=5$ , $f'(c)=-5$ , or $f(c)=0$ .

See Solution

Problem 26480

Find the derivative of the integral: $\frac{d}{d x} \int_{3 x}^{4}(\ln t)^{2} d t$ . What is the result?

See Solution

Problem 26481

Find the linearization $L(x)$ of $f(x)=\frac{x}{2x+8}$ at $x=0$ .

See Solution

Problem 26482

Find dy/dt for $y=t^{7}(t^{3}+8)^{5}$ .

See Solution

Problem 26483

Find local min, max, absolute min, max, and inflection points for $y=2x^{3}+15x^{2}+24x$ on $[-6,2]$ . Use first and second derivative tests.

See Solution

Problem 26484

Find the average value of $3x$ over the interval $[1, 5]$ . Options: 36, 9, 4, 18, 3.

See Solution

Problem 26485

Finde die Stammfunktion von $f(x) = \frac{x^2}{5} + 2$ .

See Solution

Problem 26486

Solve the differential equation $\frac{d y}{d x}=\frac{1}{x^{3}}+x$ for $x>0$ with the initial condition $y(3)=2$ .

See Solution

Problem 26487

Given the function $v(t)=t^{3}-5 t^{2}+6 t$ for $t \in [0,5]$ , find when the motion is positive/negative, displacement, and distance.

See Solution

Problem 26488

Finde die Stammfunktion von $f(x) = \frac{6}{x^{2}} + 2$ .

See Solution

Problem 26489

Berechne den Flächeninhalt unter der Funktion $f(x)=\frac{x^{2}}{5}+1$ im Intervall $[0;4]$ .

See Solution

Problem 26490

Berechne den Flächeninhalt der Funktion $f(x)=\frac{3}{2} \sqrt{x}+1$ im Intervall $[1;4]$ .

See Solution

Problem 26491

Berechne den Flächeninhalt von $f(x)=\frac{4}{x^{2}}+1$ im Intervall $[2;3]$ .

See Solution

Problem 26492

A child has a $20 \mathrm{~m}$ string for a kite. The angle with the ground decreases by $5^{\circ}$ /s. Find horizontal speed at $76^{\circ}$ .

See Solution

Problem 26493

Berechne das Wasser im oberen Becken um 12:18, wenn die Durchflussrate durch $f(t)=6 t^{3}-120 t^{2}+240 t+1800$ gegeben ist und zu Beginn $20000 \mathrm{~m}^{3}$ vorhanden sind.

See Solution

Problem 26494

Finde die Stammfunktion von $f(t) = 6t^3 - 120t^2 + 180t + 2400$ .

See Solution

Problem 26495

Calculate the integral from 0 to $\frac{\pi}{2}$ of $x \operatorname{Sin}(2 x)$ .

See Solution

Problem 26496

Finde die Stammfunktion von $f(t) = 36t^3 - 288t^2 + 540t$ .

See Solution

Problem 26497

Find the left and right limits of $f(x)$ as $x$ approaches 1 and state the limit if it exists.
$f(x)=\left\{\begin{array}{ll} |x|-1, & \text { if } x \neq 1 \\ x^{3}, & \text { if } x=1 \end{array}\right.$

See Solution

Problem 26498

Bestimmen Sie die Ableitungen für: a) $f(x)=x^{3}$ , b) $f(x)=x^{5}$ , c) $f(x)=x^{2 n}$ , d) $f(x)=x$ , e) $f(x)=x^{n+4}$ , f) $f(x)=x^{2016}$ .

See Solution

Problem 26499

Use a graphing tool to find the left and right limits of $f(x)$ as $x$ approaches 1. State the limit or DNE.
$f(x)=\left\{\begin{array}{ll} |x|-1, & \text { if } x \neq 1 \\ x^{3}, & \text { if } x=1 \end{array}\right.$
$\lim _{x \rightarrow 1^{-}} f(x)= \\ \lim _{x \rightarrow 1^{+}} f(x)= \\ \lim _{x \rightarrow 1} f(x)=$

See Solution

Problem 26500

Bestimme den Benzinverbrauch eines Autos auf 50 km mit $v(s) = -0,0000006 s^{3} + 0,00006 s^{2} - 0,0017 s + 0,06$ .

See Solution

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Calculus

Problem 26401

Find the limit lim⁡x→1xex−ex−1\lim _{x \rightarrow 1} \frac{x e^{x}-e}{x-1}limx→1​x−1xex−e​ using L'Hopital's rule. Choose (A) −e-e−e, (B) 2e2 e2e, (C) eee, or (D) 1.

Problem 26402

Find the limit: lim⁡x→+∞2x2+x−5x2−3\lim _{x \rightarrow+\infty} \frac{2 x^{2}+x-5}{x^{2}-3}limx→+∞​x2−32x2+x−5​. Choose from (A) −23-\frac{2}{3}−32​, (B) 2, (C) 1, (D) 53\frac{5}{3}35​.

Problem 26403

A plane is at 4 km4 \mathrm{~km}4 km altitude and traveling at 500 m/s500 \mathrm{~m/s}500 m/s. a) When does the bomb hit the ground? b) What is its final velocity?

Problem 26404

Bestimme die Tangentengleichung der Funktion f(x)=x3f(x) = x^{3}f(x)=x3 bei a=2a = 2a=2.

Problem 26405

Find the limit as xxx approaches -6 for the expression 2x2+12xx+6\frac{2 x^{2}+12 x}{x+6}x+62x2+12x​.

Problem 26406

Problem 26407

Find the limit as xxx approaches 6 from the left for 2xx2−36\frac{2x}{x^2 - 36}x2−362x​. What is the value?

Problem 26408

Find the derivatives of the following functions: a) f(x)=2x3+5x2f(x) = 2x^3 + 5x^2f(x)=2x3+5x2, d) f(x)=0.25x4−x3f(x) = 0.25x^4 - x^3f(x)=0.25x4−x3, 8) f(x)=15x5−3x3−2f(x) = \frac{1}{5}x^5 - 3x^3 - 2f(x)=51​x5−3x3−2.

Problem 26409

Given f(1)=3f(1) = 3f(1)=3, find g′(3)g'(3)g′(3) where ggg is the inverse of fff. Options: (A) 113\frac{1}{13}131​ (B) 14\frac{1}{4}41​ (C) 1 (D) 4 (E) 13.

Problem 26410

Given the table of values for a function fff and its derivative, find g′(3)g^{\prime}(3)g′(3) where ggg is the inverse of fff. Choices: (A) 113\frac{1}{13}131​ (B) 14\frac{1}{4}41​ (C) 1 (D) 4 (E) 13

Problem 26411

Find the value of bbb in the tangent line equation y=ax+by=ax+by=ax+b for y=x3y=\sqrt[3]{x}y=3x​ at x=8x=8x=8.

Problem 26412

Evaluate the limit as xxx approaches π4\frac{\pi}{4}4π​: lim⁡x→π4tan⁡x−1x−π4\lim_{x \to \frac{\pi}{4}} \frac{\tan x - 1}{x - \frac{\pi}{4}}limx→4π​​x−4π​tanx−1​.

Problem 26413

Find the derivative h′(2)h'(2)h′(2) for h(x)=16f(x)−g(x)h(x) = 16 f(x) - g(x)h(x)=16f(x)−g(x) given f(2)=5f(2) = 5f(2)=5, g(2)=−8g(2) = -8g(2)=−8, f′(2)=18f'(2) = \frac{1}{8}f′(2)=81​, g′(2)=−5g'(2) = -5g′(2)=−5.

Problem 26414

Find the sum of the series: ∑k=1∞(−49)k\sum_{k=1}^{\infty}\left(-\frac{4}{9}\right)^{k}∑k=1∞​(−94​)k.

Problem 26415

Find dy/dxdy/dxdy/dx for cos⁡(xy)=y−1\cos(xy) = y - 1cos(xy)=y−1 at x=π/2x = \pi/2x=π/2, y=1y = 1y=1. Options: (A) −22−π\frac{-2}{2-\pi}2−π−2​ (B) −22+π\frac{-2}{2+\pi}2+π−2​ (C) 0 (D) 22−π\frac{2}{2-\pi}2−π2​ (E) 22+π\frac{2}{2+\pi}2+π2​.

Problem 26416

Given that g(x)=f−1(x)g(x)=f^{-1}(x)g(x)=f−1(x), find g′(4)g^{\prime}(4)g′(4) using values from the table. Options: (A) −13-\frac{1}{3}−31​, (B) −14-\frac{1}{4}−41​, (C) −3100-\frac{3}{100}−1003​, (D) 14\frac{1}{4}41​, (E) 13\frac{1}{3}31​.

Problem 26417

Find dydx\frac{d y}{d x}dxdy​ for the equation x3+3xy+2y3=17x^{3}+3 x y+2 y^{3}=17x3+3xy+2y3=17. Choose from options (A) to (E).

Problem 26418

Find w′(0)w'(0)w′(0) for w(x)=3f(x)g(x)w(x) = 3 f(x) g(x)w(x)=3f(x)g(x) given f(0)=3f(0) = 3f(0)=3, g(0)=−2g(0) = -2g(0)=−2, f′(0)=12f'(0) = \frac{1}{2}f′(0)=21​, g′(0)=3g'(0) = 3g′(0)=3.

Problem 26419

Find g′(3)g^{\prime}(3)g′(3) for g(x)=f−1(x)g(x)=f^{-1}(x)g(x)=f−1(x) given f(3)=15f(3)=15f(3)=15, f(6)=3f(6)=3f(6)=3, f′(3)=−8f^{\prime}(3)=-8f′(3)=−8, f′(6)=−2f^{\prime}(6)=-2f′(6)=−2.

Problem 26420

Find dydx\frac{d y}{d x}dxdy​ for the equation 3x2+2xy+y2=13 x^{2}+2 x y+y^{2}=13x2+2xy+y2=1. Choose the correct option from (A) to (E).

Problem 26421

Problem 26422

Find the value of ccc such that 0<c<20 < c < 20<c<2 and g(x)=−πg(x) = -\pig(x)=−π given the conditions for g(x)g(x)g(x) and its derivatives.

Problem 26423

Find the derivative of y=tan⁡−1(3x)y=\tan^{-1}(3x)y=tan−1(3x) at x=1x=1x=1: what is y′(1)y'(1)y′(1)?

Problem 26424

Find the derivative of the function f(x)f(x)f(x) given that ddxf(x)=ddx2x\frac{d}{dx}f(x) = \frac{d}{dx}2^{x}dxd​f(x)=dxd​2x.

Problem 26425

For 0<c<20<c<20<c<2, what must be true for g(x)=−πg(x)=-\pig(x)=−π regarding continuity and differentiability?

Problem 26426

Bestimmen Sie die Ableitung der Funktionen: a) f(x)=x−5f(x)=x^{-5}f(x)=x−5, b) f(x)=3x−4f(x)=3x^{-4}f(x)=3x−4, c) f(x)=−13x−6f(x)=-\frac{1}{3}x^{-6}f(x)=−31​x−6, d) f(x)=x4−x+1/x3f(x)=x^{4}-x+1/x^{3}f(x)=x4−x+1/x3.

Problem 26427

Find dydx\frac{dy}{dx}dxdy​ for exy−y2=e−4e^{xy} - y^2 = e - 4exy−y2=e−4 at x=12x = \frac{1}{2}x=21​, y=2y = 2y=2. Options: (A) e4\frac{e}{4}4e​, (B) e2\frac{e}{2}2e​, (C) 4e8−e\frac{4e}{8-e}8−e4e​, (D) 4e4−e\frac{4e}{4-e}4−e4e​, (E) 8−4ee\frac{8-4e}{e}e8−4e​.

Problem 26428

Sand forms a pile with volume V=r33V=\frac{r^{3}}{3}V=3r3​. If circumference increases at 5π5\pi5π ft/hr, find dV/dtdV/dtdV/dt when circumference is 8π8\pi8π.

Problem 26429

A sphere's cross-sectional area increases at 2 cm2/sec2 \mathrm{~cm}^{2} / \mathrm{sec}2 cm2/sec. Find the volume change rate when r=4 cmr=4 \mathrm{~cm}r=4 cm.

Problem 26430

Problem 26431

Find the derivative of the inverse function at 5, using (f−1)′(y)=1f′(x) \left(f^{-1}\right)^{\prime}(y) = \frac{1}{f^{\prime}(x)} (f−1)′(y)=f′(x)1​.

Problem 26432

Find the rate of change of fuel consumption F(s)=6e(⋅20−s22m0)F(s)=6 e^{\left(\frac{\cdot}{20}-\frac{s^{2}}{2 m 0}\right)}F(s)=6e(20⋅​−2m0s2​) at s=50s=50s=50 mph, ds/dt=20ds/dt=20ds/dt=20 mph2^22. Choices: (A) 0.215 (B) 4.299 (C) 9.793 (D) 25.793 (E) 515.855.

Problem 26433

Find the rate of change of fuel consumption F(s)=6e(120−222000)F(s)=6 e^{\left(\frac{1}{20}-\frac{2^{2}}{2000}\right)}F(s)=6e(201​−200022​) at 50 mph with s′=20s' = 20s′=20 mph².

Problem 26434

Find dydx\frac{d y}{d x}dxdy​ at the point where x=−1x=-1x=−1 for the equation x3−x2y+y2=11x^{3}-x^{2} y+y^{2}=11x3−x2y+y2=11.

Problem 26435

Find the rate of change of fuel consumption F(s)=6e(s20−s22000)F(s)=6 e^{\left(\frac{s}{20}-\frac{s^{2}}{2000}\right)}F(s)=6e(20s​−2000s2​) at s=50s=50s=50 mph with ds/dt=20ds/dt=20ds/dt=20 mph².

Problem 26436

Find (f−1)′(5)\left(f^{-1}\right)^{\prime}(5)(f−1)′(5) given f(0)=3f(0)=3f(0)=3, f(1)=4f(1)=4f(1)=4, f(2)=5f(2)=5f(2)=5, f′(0)=12f'(0)=\frac{1}{2}f′(0)=21​, f′(1)=14f'(1)=\frac{1}{4}f′(1)=41​, f′(2)=18f'(2)=\frac{1}{8}f′(2)=81​.

Problem 26437

1. For the bird's height h(t)h(t)h(t), explain what h′(4)=2h^{\prime}(4)=2h′(4)=2 means in context. 2. For university enrollment s(t)s(t)s(t), explain s′(2.5)=370s^{\prime}(2.5)=370s′(2.5)=370 in context.

Problem 26438

In a triangle, if θ\thetaθ increases at 3 radians/min, how fast is xxx increasing when x=3x=3x=3 units? Hypotenuse is 5.

Problem 26439

Find the area increase rate when the circumference is 20π20 \pi20π m, with radius growing at 0.2 m/s. Options: (A) 0.04πm2/sec0.04 \pi \mathrm{m}^{2}/\mathrm{sec}0.04πm2/sec or 0.4πm2/sec0.4 \pi \mathrm{m}^{2}/\mathrm{sec}0.4πm2/sec.

Problem 26440

Finde die Stelle, an der die Steigung der Funktion f(x)=x9f(x)=x^{9}f(x)=x9 gleich 9 ist. Gib nur eine Lösung an.

Problem 26441

Problem 26442

Find the limit $\lim _{x \rightarrow 1} \frac{x e^{x}-e}{x-1}$ using L'Hopital's rule. Choose (A) $-e$ , (B) $2 e$ , (C) $e$ , or (D) 1.

Find the limit: $\lim _{x \rightarrow+\infty} \frac{2 x^{2}+x-5}{x^{2}-3}$ . Choose from (A) $-\frac{2}{3}$ , (B) 2, (C) 1, (D) $\frac{5}{3}$ .

A plane is at $4 \mathrm{~km}$ altitude and traveling at $500 \mathrm{~m/s}$ .
a) When does the bomb hit the ground? b) What is its final velocity?

Bestimme die Tangentengleichung der Funktion $f(x) = x^{3}$ bei $a = 2$ .

Find the limit as $x$ approaches -6 for the expression $\frac{2 x^{2}+12 x}{x+6}$ .

Find the limit as $x$ approaches 6 from the left for $\frac{2x}{x^2 - 36}$ . What is the value?

Find the derivatives of the following functions:
a) $f(x) = 2x^3 + 5x^2$ , d) $f(x) = 0.25x^4 - x^3$ , 8) $f(x) = \frac{1}{5}x^5 - 3x^3 - 2$ .

Given $f(1) = 3$ , find $g'(3)$ where $g$ is the inverse of $f$ . Options: (A) $\frac{1}{13}$ (B) $\frac{1}{4}$ (C) 1 (D) 4 (E) 13.

Given the table of values for a function $f$ and its derivative, find $g^{\prime}(3)$ where $g$ is the inverse of $f$ . Choices: (A) $\frac{1}{13}$ (B) $\frac{1}{4}$ (C) 1 (D) 4 (E) 13

Find the value of $b$ in the tangent line equation $y=ax+b$ for $y=\sqrt[3]{x}$ at $x=8$ .

Evaluate the limit as $x$ approaches $\frac{\pi}{4}$ : $\lim_{x \to \frac{\pi}{4}} \frac{\tan x - 1}{x - \frac{\pi}{4}}$ .

Find the derivative $h'(2)$ for $h(x) = 16 f(x) - g(x)$ given $f(2) = 5$ , $g(2) = -8$ , $f'(2) = \frac{1}{8}$ , $g'(2) = -5$ .

Find the sum of the series: $\sum_{k=1}^{\infty}\left(-\frac{4}{9}\right)^{k}$ .

Find $dy/dx$ for $\cos(xy) = y - 1$ at $x = \pi/2$ , $y = 1$ . Options: (A) $\frac{-2}{2-\pi}$ (B) $\frac{-2}{2+\pi}$ (C) 0 (D) $\frac{2}{2-\pi}$ (E) $\frac{2}{2+\pi}$ .

Given that $g(x)=f^{-1}(x)$ , find $g^{\prime}(4)$ using values from the table. Options: (A) $-\frac{1}{3}$ , (B) $-\frac{1}{4}$ , (C) $-\frac{3}{100}$ , (D) $\frac{1}{4}$ , (E) $\frac{1}{3}$ .

Find $\frac{d y}{d x}$ for the equation $x^{3}+3 x y+2 y^{3}=17$ . Choose from options (A) to (E).

Find $w'(0)$ for $w(x) = 3 f(x) g(x)$ given $f(0) = 3$ , $g(0) = -2$ , $f'(0) = \frac{1}{2}$ , $g'(0) = 3$ .

Find $g^{\prime}(3)$ for $g(x)=f^{-1}(x)$ given $f(3)=15$ , $f(6)=3$ , $f^{\prime}(3)=-8$ , $f^{\prime}(6)=-2$ .

Find $\frac{d y}{d x}$ for the equation $3 x^{2}+2 x y+y^{2}=1$ . Choose the correct option from (A) to (E).

Find the value of $c$ such that $0 < c < 2$ and $g(x) = -\pi$ given the conditions for $g(x)$ and its derivatives.

Find the derivative of $y=\tan^{-1}(3x)$ at $x=1$ : what is $y'(1)$ ?

Find the derivative of the function $f(x)$ given that $\frac{d}{dx}f(x) = \frac{d}{dx}2^{x}$ .

For $0<c<2$ , what must be true for $g(x)=-\pi$ regarding continuity and differentiability?

Bestimmen Sie die Ableitung der Funktionen: a) $f(x)=x^{-5}$ , b) $f(x)=3x^{-4}$ , c) $f(x)=-\frac{1}{3}x^{-6}$ , d) $f(x)=x^{4}-x+1/x^{3}$ .

Find $\frac{dy}{dx}$ for $e^{xy} - y^2 = e - 4$ at $x = \frac{1}{2}$ , $y = 2$ . Options: (A) $\frac{e}{4}$ , (B) $\frac{e}{2}$ , (C) $\frac{4e}{8-e}$ , (D) $\frac{4e}{4-e}$ , (E) $\frac{8-4e}{e}$ .

Sand forms a pile with volume $V=\frac{r^{3}}{3}$ . If circumference increases at $5\pi$ ft/hr, find $dV/dt$ when circumference is $8\pi$ .

A sphere's cross-sectional area increases at $2 \mathrm{~cm}^{2} / \mathrm{sec}$ . Find the volume change rate when $r=4 \mathrm{~cm}$ .

Find the derivative of the inverse function at 5, using $\left(f^{-1}\right)^{\prime}(y) = \frac{1}{f^{\prime}(x)}$ .

Find the rate of change of fuel consumption $F(s)=6 e^{\left(\frac{\cdot}{20}-\frac{s^{2}}{2 m 0}\right)}$ at $s=50$ mph, $ds/dt=20$ mph $^2$ . Choices: (A) 0.215 (B) 4.299 (C) 9.793 (D) 25.793 (E) 515.855.

Find the rate of change of fuel consumption $F(s)=6 e^{\left(\frac{1}{20}-\frac{2^{2}}{2000}\right)}$ at 50 mph with $s' = 20$ mph².

Find $\frac{d y}{d x}$ at the point where $x=-1$ for the equation $x^{3}-x^{2} y+y^{2}=11$ .

Find the rate of change of fuel consumption $F(s)=6 e^{\left(\frac{s}{20}-\frac{s^{2}}{2000}\right)}$ at $s=50$ mph with $ds/dt=20$ mph².

Find $\left(f^{-1}\right)^{\prime}(5)$ given $f(0)=3$ , $f(1)=4$ , $f(2)=5$ , $f'(0)=\frac{1}{2}$ , $f'(1)=\frac{1}{4}$ , $f'(2)=\frac{1}{8}$ .

1. For the bird's height $h(t)$ , explain what $h^{\prime}(4)=2$ means in context.
2. For university enrollment $s(t)$ , explain $s^{\prime}(2.5)=370$ in context.

In a triangle, if $\theta$ increases at 3 radians/min, how fast is $x$ increasing when $x=3$ units? Hypotenuse is 5.

Find the area increase rate when the circumference is $20 \pi$ m, with radius growing at 0.2 m/s. Options: (A) $0.04 \pi \mathrm{m}^{2}/\mathrm{sec}$ or $0.4 \pi \mathrm{m}^{2}/\mathrm{sec}$ .

Finde die Stelle, an der die Steigung der Funktion $f(x)=x^{9}$ gleich 9 ist. Gib nur eine Lösung an.

Evaluate the integral: $\int_{0}^{\infty} e^{-x} dx$

Bestimme die Tangentengleichung von $f(x)=x^{6}$ bei $x_{0}=2$ .

Finde die Stelle, an der die Steigung des Graphen von $f(x)=x^{4}$ gleich $32$ ist.

Evaluate the integral: $\int_{0}^{\infty} \frac{2}{x^{2}+4 x+3} d x$

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for $y=\frac{1}{x^{3}}-4 x^{5}+3 x$ . Simplify first!

Calculate the average rate of change of $f(x)=\frac{3}{x+2}$ from $x=1$ to $x=4$ . Options: A. -28 B. $-\frac{1}{6}$ C. -2 D. $\frac{1}{2}$

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for $y=\frac{1}{x^{3}}-4 x^{5}+3 x$ and $\frac{d y}{d x}=y^{3} x^{2}$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for $y=\frac{1}{x^{3}}-4 x^{5}+3 x$ and $\frac{d y}{d x}=y^{3} x^{2}$ .

Determine if the particle with velocity $v(t)=t \sin^{3}(5t)$ at $t=2$ is speeding up or slowing down.

What are the units of the derivative $R'(t)$ if $R(t)$ is in gallons/hour? Options: 1) $-\frac{1}{2}$ 2) $0.653$ 3) $\frac{3}{8}$ 4) $-\frac{3}{4}$

Find the second derivative: 1. For $f(x)=x^{\frac{5}{3}}+x$ , calculate $f^{\prime \prime}(8)$ . 2. For $\frac{d y}{d x}=y^{2}(3-4 x)$ , find $\frac{d^{2} y}{d x^{2}}$ at $\left(1, \frac{1}{2}\right)$ .

Strontium-85 has a half-life of 65 days.
a. How long for radiation to drop to $\frac{1}{4}$ of its original level? b. How long for radiation to drop to $\frac{1}{8}$ of its original level?

How long does it take for an oscillation's amplitude to drop from $80 \mathrm{~cm}$ to $5 \mathrm{~cm}$ in 4 seconds?

A roller coaster starts at $2.5 \mathrm{~m/s}$ at $25 \mathrm{~m}$ . Find its speed at $20 \mathrm{~m}$ , $10 \mathrm{~m}$ , and ground level.

Berechne die Ableitung von $f(x) = x e^{2-x}$ mit der Produktregel. Was ist $f'(x)$ ?

A satellite of mass $m=1371 \mathrm{~kg}$ falls from $R=3.8$ Earth radii. Find its speed before hitting Earth.

Find the maximum acceleration of an object in simple harmonic motion given $x=2.70 \sin(0.413 t + 2.38)$ .

Find the limit: $\lim _{x \rightarrow-2} \frac{x^{2}+5 x+6}{x+2}$

Find the limit: $\lim _{x \rightarrow 2} \frac{\sqrt{x+2}-2}{x-2}$ .

Find the intervals where the function $g$ is decreasing, given $f(x)=4-x$ and $g'(x)=f(x) f'(x)(x-2)$ .

Calculate the curve length for $x=5 \sin t + 5 t$ , $y=5 \cos t$ , where $0 \leq t \leq \pi$ .

Find the value of $\frac{d^{2} y}{d x^{2}}$ at $t=1$ where $x=6 t^{2}-3$ and $y=t^{3}$ .

Find $\lim _{x \rightarrow-2^{+}} f(x)$ for the piecewise function: $f(x)=\begin{cases}2 x+7, & x<-2 \\ -\frac{x}{2}-\frac{1}{2}, & x \geq-2\end{cases}$ .

Find the tangent line equation to the curve at $t=\frac{3 \pi}{4}$ for $x=7 \sin t$ , $y=7 \cos t$ .

Find if the water in the gutter is increasing or decreasing at $t=4$ hours using $G(t)=10 \sin \left(\frac{t^{2}}{30}\right)$ and $D(t)=-0.02 t^{3}+0.05 t^{2}+0.87 t$ .

Bestimme die Tangentengleichung der Funktion $f(x) = \frac{1}{x}$ bei $a = -3$ .

Find the tangent line to $f(x)=\sqrt{x+9}$ with slope $\frac{1}{4}$ .

How many years does it take for an investment to double at a continuous growth rate of $14\%$ ? Round to one decimal place.

Find the slope of the curve $r=5-3 \cos \theta$ at $\theta=\frac{\pi}{2}$ .

Determine which statements about a continuous function $f(x)$ on $[a, b]$ are NOT necessarily true: I, II, III.

Given the function $v(t)=t^{3}-5 t^{2}+6 t$ on $[0,5]$ , find when motion is positive/negative, displacement, and distance traveled.