Calculus

Problem 7301

Find the max and min of $g(\theta)=5 \theta-7 \sin (\theta)$ on $[0, \pi]$ . Min value =, Max value =.

See Solution

Problem 7302

A 6-ft person walks from a 10-ft lamppost at 3 ft/sec. Find the shadow's tip speed when the person is 10 ft away.

See Solution

Problem 7303

Differentiate $d(r^{2}v) = C$ using the product rule, where $r$ is radius, $v$ is velocity, and $C$ is constant.

See Solution

Problem 7304

Find local extrema of $f(x)=2x^3-30x^2+126x+3$ . Identify $x$ values for min/max and their corresponding $f(x)$ values.

See Solution

Problem 7305

Bestimmen Sie die Ableitung der Funktion $f(x) = \frac{1}{3} x^{6} + x^{4} - x$ .

See Solution

Problem 7306

Bestimmen Sie die Ableitung von $f(x) = 3(-2x+5)^{3}$ .

See Solution

Problem 7307

Find the velocity of a ball dropped from $210 \mathrm{~m}$ when it hits the ground using $g=9.8 \mathrm{~m/s}^2$ . Round to three decimal places. $v \approx$

See Solution

Problem 7308

Bestimme die erste Ableitung der Funktion $f(x)=\frac{9}{x^{2}}$ .

See Solution

Problem 7309

Bestimmen Sie die Ableitung von $f(x)=\sqrt{-4 x+2}$ .

See Solution

Problem 7310

Find the critical number $A$ for the function $f(x)=-3x^2+2x-8$ . Is it a local min, max, or neither? Answer: LMIN, LMAX, or NEITHER.

See Solution

Problem 7311

Find $\theta^{\prime}(t)$ , the rate of change of the angle between clock hands, at 9 o'clock in $\mathrm{rad} / \mathrm{min}$ .

See Solution

Problem 7312

A cone with height 16 ft and radius 5 ft leaks water at 10 ft³/min. Find the water depth change rate when it's 10 ft high.

See Solution

Problem 7313

Determine the critical numbers of $f(x)=-12 x^{5}-45 x^{4}+20 x^{3}+1$ and classify them with a graph.

See Solution

Problem 7314

Determine the absolute min and max of $f(x)=2x^3 + 12x^2 - 126x + 1$ for $-7 \leq x \leq 4$ .

See Solution

Problem 7315

Find the rate of change of the base of a triangle when the altitude is 12 cm and area is 83 cm², given specific rates.

See Solution

Problem 7316

Show that the derivative of $a^{x}$ is $a^{x} \ln(a)$ , where $a$ is a constant.

See Solution

Problem 7317

Plot the line segment between $f(2)$ and $f(5)$ , then find all $c$ in $[2, 5]$ satisfying the Mean Value Theorem for $f$ .

See Solution

Problem 7318

Finde die Tangente an $f(x)=2 x^{-2}$ bei $x_0=3$ .

See Solution

Problem 7319

Bestimmen Sie die Tangente an $f(x)=x^{4}$ bei $P_{0}(0,5)$ .

See Solution

Problem 7320

Given the growth law $\frac{d M}{d t}=C M^{3 / 4}$ , find the growth rate at $M=175 \mathrm{~kg}$ and the mass needed to double growth from $M=0.2 \mathrm{~kg}$ .

See Solution

Problem 7321

Estimate the increase in metabolic rate $P$ when body mass increases from 78 kg to 79 kg using $P=73.3 m^{3/4}$ . Round to three decimal places.

See Solution

Problem 7322

Find the tangent line equation for $f(x)=16 e^{x} \cos^{2}(x)$ at $x=\frac{\pi}{4}$ . Express as $y$ and $x$ .

See Solution

Problem 7323

Estimate the increase in metabolic rate $P$ when body mass $m$ changes from 78 kg to 79 kg using $P=73.3 m^{3/4}$ .

See Solution

Problem 7324

Find the second and third derivatives of $y=5 e^{t} \sin(t)$ .
$y^{\prime \prime}=$ $y^{\prime \prime \prime}=$

See Solution

Problem 7325

Find $\frac{d B}{d I}$ for $B=k I^{2/3}$ and $\frac{d H}{d W}$ for $H=k W^{3/2}$ .

See Solution

Problem 7326

Jan pedals on a stationary bike, modeled by $h(t)=-0.17 \sin (t)+0.25$ . Find the vertical velocity $v$ at $t=3 \pi$ .

See Solution

Problem 7327

Finde Werte für a, damit der Graph $G_{f_{a}}$ rechtsgekrümmt ist für: a) $f_{a}(x)=a \cdot(2 x-3)^{4}$ , b) $f_{a}(x)=\sqrt{a^{2} x}(x \geq 0)$ , c) $f_{a}(x)=\frac{a}{x^{2}}(x \neq 0)$ . Gib ein Gegenbeispiel für: $f^{\prime}$ streng monoton wachsend $\Rightarrow f$ streng monoton wachsend.

See Solution

Problem 7328

Find the second derivative of $16 \cos^2(t)$ . Calculate $\frac{d^{2}}{d t^{2}} 16 \cos ^{2}(t)$ .

See Solution

Problem 7329

A hiker walks 8 miles in 4 hours. Find her instantaneous velocity at 2.1 hours using $s(t)=\frac{3}{32} t^{4}-t^{3}+3 t^{2}$ . Round to three decimal places.

See Solution

Problem 7330

Bestimmen Sie die Tangente an $f(x)=\frac{1}{x}-x^{3}$ im Punkt P0 ( $5|f(5)$ ).

See Solution

Problem 7331

Bestimmen Sie die Tangentengleichung von $f(x)=3 \cdot e^{2 x+2}$ am Punkt (-1/3).

See Solution

Problem 7332

Bestimme die Tangente an $f(x)=2 x^{3}-3 x^{-2}$ im Punkt $P_0(2 | f(2))$ .

See Solution

Problem 7333

Berechne die Fläche unter $f(x)=-x \cdot \ln(0,1x)$ von 0 bis 10 mit Produktintegration. Finde $g'(x)$ , $g(x)$ , $h(x)$ , $h'(x)$ und berechne $\int_{0}^{10}(-x \cdot \ln(0,1 x)) dx$ . Bestimme schließlich $2 \cdot \int_{0}^{10}(-x \cdot \ln(0,1 x)) dx$ .

See Solution

Problem 7334

Bestimme $g^{\prime}(x)$ und $h(x)$ für die folgenden Integrale: a) $\int_{a}^{b}(\cos (x) \cdot x) d x$ b) $\int_{a}^{b}\left(x^{2} \cdot \ln (x)\right) d x$ c) $\int_{a}^{b}\left(-e^{2 x} \cdot x\right) d x$ d) $\int_{a}^{b}(3 x \cdot \sin (2 x-3)) d x$

See Solution

Problem 7335

Gegeben ist die Funktion $f_{a}(x)=a \cdot e^{a+x}$ . a) Zeigen Sie, dass die Tangente $t_{a}$ die Gleichung $\dot{y}=a \cdot e^{a-1} \cdot x+2 \cdot a \cdot e^{a-1}$ hat. b) Bestimmen Sie den Flächeninhalt des Dreiecks, das $t_{a}$ und die Achsen einschließt.

See Solution

Problem 7336

Zeichne die Tangente von $f(x)=-2 x^{-1}$ bei $x=-1$ und schätze die Steigung in diesem Punkt.

See Solution

Problem 7337

Approximate the zero of the differentiable function $f$ using its tangent line at $x=-2$ given $f(-2)=1$ and $f'(-2)=5$ . What is the approximation? D) -2.4

See Solution

Problem 7338

Calcule o limite da indeterminação quando $x$ tende a $-\infty$ : $\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}-1}}{2 x+3}$

See Solution

Problem 7339

Ein Auto beschleunigt mit $a$ in \left.\frac{m}{s^{2}}\right. a) Geschwindigkeit nach 2s und 5s bei $a=3$ . b) Zeit für 0 auf $50$ und $100 \frac{\mathrm{km}}{\mathrm{h}}$ bei $a=3$ . c) Geschwindigkeit für allgemeines $a$ .

See Solution

Problem 7340

Find the IROC of $3 x^{3}+2 x^{2}-19 x+6$ at the point $(3,12)$ .

See Solution

Problem 7341

Bestimmen Sie die Extrem- und Wendepunkte der Funktion $f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-6 x+1$ .

See Solution

Problem 7342

Find the limit: $\lim _{x \rightarrow 2} \frac{3 x-6}{4-x^{2}}$ .

See Solution

Problem 7343

Find the maximum slope of the tangent line to the graph of $f(x)=3x^2-2x^3$ . Options: A) $\frac{3}{2}$ B) $\frac{3}{4}$ C) $\frac{9}{2}$ D) $\frac{1}{2}$

See Solution

Problem 7344

Find the value of the integral $\int \frac{d x}{2+\cos 2 x}$ and identify the correct expression among the options.

See Solution

Problem 7345

Bestimmen Sie den Grenzwert der Folge $a_{n}=2+\frac{1}{n}$ und zeigen Sie die Schritte zur Berechnung.

See Solution

Problem 7346

Gegeben ist die Funktion $f(x)=\frac{1}{9}(3 x+2)^{3}$ . Finde die Steigung bei $P(2 \mid f(2))$ , prüfe waagerechte Tangenten und Punkte mit $45^{\circ}$ Steigungswinkel.

See Solution

Problem 7347

Bestimmen Sie den Grenzwert der Sequenz $a_{n}=\frac{3-2 n}{2 n}$ und erläutern Sie den Lösungsweg.

See Solution

Problem 7348

Find the critical points (min, max, inflection) of $g(z)=32(e^{2z}+4e^{-3z})^8$ .

See Solution

Problem 7349

Find the critical points of $F(t)=\frac{1}{3} t^{3}-3.5 t^{2}+10 t+21$ and classify the smaller one, $t_1$ .

See Solution

Problem 7350

Find the smaller critical point, $t_1$ , of $F(t)=\frac{1}{3} t^{3}-6.5 t^{2}+30 t+26$ and classify it as max, min, or inflection.

See Solution

Problem 7351

Find the smaller critical point $t_1$ of $F(t) = \frac{1}{3} t^{3}-5 t^{2}+16 t+21$ and classify it as max, min, or inflection.

See Solution

Problem 7352

Find the value of $z$ where the function $g(z)=28\left(e^{4 z}+3 e^{-5 z}\right)^{4}$ has a min, max, or inflection point.

See Solution

Problem 7353

Déterminez quels énoncés sur la fonction $f$ sont vrais parmi A à G concernant les dérivées et la continuité.

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Problem 7354

Soit $u(x)$ et $v(x)$ des fonctions dérivables. Quels énoncés suivants sont vrais ? A. Aucun vrai. B. $\nabla\left(k^{2}\right)^{\prime}=2 k$ . C. $(k \cdot u(x))^{\prime}=k^{\prime} \cdot u^{\prime}(x)$ . D. $(u(x) \cdot v(x))^{\prime}=u^{\prime}(x) v^{\prime}(x)$ . E. $\frac{d}{d x}\left(\frac{u(x)}{2}\right)=\frac{u^{\prime}(x)}{2}$ . F. $v(u(x)+v(x))^{\prime}=u^{\prime}(x)+v^{\prime}(x)$ .

See Solution

Problem 7355

Quel est le taux de variation instantané de la fonction $f(x)=x^{3}-5 x^{2}+4 x-6$ en $x=3$ ?

See Solution

Problem 7356

Determine which statements about the slopes of tangent and normal lines to a function $f$ are true.

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Problem 7357

A 15 ft ladder's base moves away from a wall at 2 ft/sec. Find the top's downward rate when the base is 12 ft from the wall. Rate: Blank 1: Blank 2:

See Solution

Problem 7358

A balloon falls at 5 m/s from 40 m away. Find the rate of change of the angle of elevation when it's 30 m high.

See Solution

Problem 7359

Find the second derivative of the function $f(x)=-6 x^{3}-8 x^{2}+8 x$ .

See Solution

Problem 7360

Find the intervals where the function $f(x)=-5 x^{3}+75 x^{2}+5 x+6$ is concave up or down.

See Solution

Problem 7361

Evaluate $f^{\prime \prime}(7)$ , $f^{\prime \prime}(-9)$ , and $f^{\prime \prime}(-10)$ for $f(x)=-6x^{3}-8x^{2}+8x$ .

See Solution

Problem 7362

How far does an egg fall if it hits the ground at $7.0 \mathrm{~m} / \mathrm{s}$ ?

See Solution

Problem 7363

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{x+\sin x}{x}$ using a table of values.

See Solution

Problem 7364

Find the second derivative of the function $f(x)=\sqrt{7x+4}$ .

See Solution

Problem 7365

Find the second derivative of the function $f(x)=\sqrt{8x+4}$ .

See Solution

Problem 7366

Find $x$ such that $f''(x) = -6x^2 + 4 = 0$ . Are there any constraints on $x$ ?

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Problem 7367

Find points of inflection for the function $f(x)=-5 x^{3}+75 x^{2}+5 x+6$ . Provide answers as $(x, y)$ -pairs.

See Solution

Problem 7368

Estimate the volume error of a cube with side length $9 \mathrm{~cm}$ and accuracy $0.04 \mathrm{~cm}$ using differentials.

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Problem 7369

Find the differential $d y$ for the function $y=4 \sin (2 x)$ .

See Solution

Problem 7370

Graph the differential for $f(x)=-\frac{x^{2}}{3}+3x$ at $x=5$ and $\Delta x=-1.4$ . Find $dy$ and $\Delta y$ , rounded to two decimals.

See Solution

Problem 7371

Find the point of diminishing returns for the sales function $S(x)=92+44.4 x^{2}-0.4 x^{3}$ , where $x$ is in thousands.

See Solution

Problem 7372

Find $f^{\prime \prime}(4)$ , $f^{\prime \prime}(5)$ , and $f^{\prime \prime}(-1)$ for $f(x)=\sqrt{3x+5}$ .

See Solution

Problem 7373

Find the differential $d y$ for the function $y=\sqrt{10-3 x}$ .

See Solution

Problem 7374

Find the second derivative of the function $f(x)=-6 x^{2}-6 \sqrt[3]{x}-7$ .

See Solution

Problem 7375

Find the point of diminishing returns for the sales function $S(x)=118+42x^{2}-0.5x^{3}$ , where $0 \leq x \leq 56$ .

See Solution

Problem 7376

Graph the function $f(x)=-\frac{x^{2}}{3}+3 x$ on Desmos. With $x=5$ and $\Delta x=-1.4$ , find $d y$ and $\Delta y$ .

See Solution

Problem 7377

Find the derivative $\frac{d}{d x} f(g(x))$ for $f(x)=x^{7}$ and $g(x)=8 x-6$ . Calculate $f^{\prime}(x)$ and $g^{\prime}(x)$ .

See Solution

Problem 7378

Find $d y$ for $y=\sqrt{2 x-3}$ and evaluate at $x=2$ , $d x=0.1$ . Provide an exact answer.

See Solution

Problem 7379

Find $d y$ for $y=\sqrt{4 x-1}$ and evaluate at $x=3$ , $d x=0.3$ . (Provide an exact answer.)

See Solution

Problem 7380

Find the derivative of the composition $f(g(x))$ where $f(x)=\sqrt{x}$ and $g(x)=x^{2}+8$ .

See Solution

Problem 7381

Find $d y$ for $y=\sqrt{x+4}$ and evaluate at $x=3$ , $d x=0.3$ . (Provide an exact answer.)

See Solution

Problem 7382

Find the limit: $\lim _{x \rightarrow 9} \frac{3-\sqrt{x}}{x-9}$ .

See Solution

Problem 7383

Find $\frac{d}{d x} f(g(x))$ for $f(x)=\frac{7}{x}$ and $g(x)=6-x^{2}$ . First, compute $f^{\prime}(x)$ and $g^{\prime}(x)$ .

See Solution

Problem 7384

Find $d y$ for $y=\sqrt{7 x-9}$ and evaluate at $x=3$ , $d x=-0.1$ .

See Solution

Problem 7385

Find the volume change $d V$ when a cube's side increases from $6 \mathrm{~cm}$ to $6.4 \mathrm{~cm}$ . Round to the nearest tenth. $d V=\square \mathrm{cm}^{3}$

See Solution

Problem 7386

Find the first three derivatives of $f(x)=\cos(3x^{2})$ .

See Solution

Problem 7387

Find the volume change $d V$ when a cube's side goes from $14 \mathrm{~cm}$ to $14.2 \mathrm{~cm}$ . Round to the nearest tenth.

See Solution

Problem 7388

Find the derivative of $f(x) = 6(-\sin(3x^2))$ .

See Solution

Problem 7389

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $x^{2} y = x^{3} + 9 y$ .

See Solution

Problem 7390

Find the volume change $d V$ of a cube when side length changes from $10 \mathrm{~cm}$ to $10.2 \mathrm{~cm}$ .

See Solution

Problem 7391

Find the intervals of $x$ where $\frac{d^{2} y}{d x^{2}}>0$ , given $\frac{d y}{d x}=5 x^{3}+8 x^{2}+11$ .

See Solution

Problem 7392

Find an equation for $y$ given $\frac{d y}{d x}=\frac{11}{2}\left(a x^{2}+\sqrt[3]{a x}\right)^{\frac{1}{2}}\left(2 a x+\frac{a}{3 \sqrt[3]{(a x)^{2}}}\right)$ .

See Solution

Problem 7393

Find the original function $y$ from the derivative $\frac{d y}{d x}=\frac{11}{2}\left(a x^{2}+\sqrt[3]{a x}\right)^{\frac{1}{2}}\left(2 a x+\frac{a}{3 \sqrt[3]{(a x)^{2}}}\right)$ .

See Solution

Problem 7394

Find a possible equation for $y$ given $\frac{d y}{d x}=\frac{11}{2}\left(a x^{2}+\sqrt[3]{a x}\right)^{\frac{1}{2}}\left(2 a x+\frac{a}{3 \sqrt[3]{(a x)^{2}}}\right)$ .

See Solution

Problem 7395

Find the per capita growth rate of the Beverton-Holt model $g(N)=\frac{6 N}{1+\frac{7 N}{45}}$ using $\frac{1}{N} \cdot \frac{d g}{d N}$ .

See Solution

Problem 7396

Gegeben ist die Funktion $f(x)=\frac{1}{9}(3 x+2)^{3}$ .
a) Finde die Steigung von $f$ bei $P(2 \mid f(2))$ . b) Hat der Graph waagerechte Tangenten? c) Finde Punkte mit einem Steigungswinkel von $45^{\circ}$ .

See Solution

Problem 7397

Find where the function $f(x)=5 \sin^{2} x$ is increasing and decreasing on $[-\pi, \pi]$ . $f^{\prime}(x)=10 \sin x \cos x$ . Use interval notation.

See Solution

Problem 7398

Find the derivative of the function $f(x)=\cos(5x-1)\sin(5x+3)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 7399

Find the derivative $\frac{d y}{d x}$ for $y=\frac{e^{x^{5}}}{\sqrt{1-x^{3}}$.

See Solution

Problem 7400

Find the derivative of $f(x)=\sin^{3} x$ and calculate $f^{\prime}(1)$ , rounding to the nearest hundredth.

See Solution

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Calculus

Problem 7301

Find the max and min of g(θ)=5θ−7sin⁡(θ)g(\theta)=5 \theta-7 \sin (\theta)g(θ)=5θ−7sin(θ) on [0,π][0, \pi][0,π]. Min value =, Max value =.

Problem 7302

A 6-ft person walks from a 10-ft lamppost at 3 ft/sec. Find the shadow's tip speed when the person is 10 ft away.

Problem 7303

Differentiate d(r2v)=Cd(r^{2}v) = Cd(r2v)=C using the product rule, where rrr is radius, vvv is velocity, and CCC is constant.

Problem 7304

Find local extrema of f(x)=2x3−30x2+126x+3f(x)=2x^3-30x^2+126x+3f(x)=2x3−30x2+126x+3. Identify xxx values for min/max and their corresponding f(x)f(x)f(x) values.

Problem 7305

Bestimmen Sie die Ableitung der Funktion f(x)=13x6+x4−xf(x) = \frac{1}{3} x^{6} + x^{4} - xf(x)=31​x6+x4−x.

Problem 7306

Bestimmen Sie die Ableitung von f(x)=3(−2x+5)3f(x) = 3(-2x+5)^{3}f(x)=3(−2x+5)3.

Problem 7307

Find the velocity of a ball dropped from 210 m210 \mathrm{~m}210 m when it hits the ground using g=9.8 m/s2g=9.8 \mathrm{~m/s}^2g=9.8 m/s2. Round to three decimal places. v≈ v \approx v≈

Problem 7308

Bestimme die erste Ableitung der Funktion f(x)=9x2f(x)=\frac{9}{x^{2}}f(x)=x29​.

Problem 7309

Bestimmen Sie die Ableitung von f(x)=−4x+2f(x)=\sqrt{-4 x+2}f(x)=−4x+2​.

Problem 7310

Find the critical number AAA for the function f(x)=−3x2+2x−8f(x)=-3x^2+2x-8f(x)=−3x2+2x−8. Is it a local min, max, or neither? Answer: LMIN, LMAX, or NEITHER.

Problem 7311

Find θ′(t)\theta^{\prime}(t)θ′(t), the rate of change of the angle between clock hands, at 9 o'clock in rad/min\mathrm{rad} / \mathrm{min}rad/min.

Problem 7312

A cone with height 16 ft and radius 5 ft leaks water at 10 ft³/min. Find the water depth change rate when it's 10 ft high.

Problem 7313

Determine the critical numbers of f(x)=−12x5−45x4+20x3+1f(x)=-12 x^{5}-45 x^{4}+20 x^{3}+1f(x)=−12x5−45x4+20x3+1 and classify them with a graph.

Problem 7314

Determine the absolute min and max of f(x)=2x3+12x2−126x+1f(x)=2x^3 + 12x^2 - 126x + 1f(x)=2x3+12x2−126x+1 for −7≤x≤4-7 \leq x \leq 4−7≤x≤4.

Problem 7315

Find the rate of change of the base of a triangle when the altitude is 12 cm and area is 83 cm², given specific rates.

Problem 7316

Show that the derivative of axa^{x}ax is axln⁡(a)a^{x} \ln(a)axln(a), where aaa is a constant.

Problem 7317

Plot the line segment between f(2)f(2)f(2) and f(5)f(5)f(5), then find all ccc in [2,5][2, 5][2,5] satisfying the Mean Value Theorem for fff.

Problem 7318

Finde die Tangente an f(x)=2x−2f(x)=2 x^{-2}f(x)=2x−2 bei x0=3x_0=3x0​=3.

Problem 7319

Bestimmen Sie die Tangente an f(x)=x4f(x)=x^{4}f(x)=x4 bei P0(0,5)P_{0}(0,5)P0​(0,5).

Problem 7320

Given the growth law dMdt=CM3/4\frac{d M}{d t}=C M^{3 / 4}dtdM​=CM3/4, find the growth rate at M=175 kgM=175 \mathrm{~kg}M=175 kg and the mass needed to double growth from M=0.2 kgM=0.2 \mathrm{~kg}M=0.2 kg.

Problem 7321

Estimate the increase in metabolic rate PPP when body mass increases from 78 kg to 79 kg using P=73.3m3/4P=73.3 m^{3/4}P=73.3m3/4. Round to three decimal places.

Problem 7322

Find the tangent line equation for f(x)=16excos⁡2(x)f(x)=16 e^{x} \cos^{2}(x)f(x)=16excos2(x) at x=π4x=\frac{\pi}{4}x=4π​. Express as yyy and xxx.

Problem 7323

Estimate the increase in metabolic rate PPP when body mass mmm changes from 78 kg to 79 kg using P=73.3m3/4P=73.3 m^{3/4}P=73.3m3/4.

Problem 7324

Find the second and third derivatives of y=5etsin⁡(t)y=5 e^{t} \sin(t)y=5etsin(t). y′′= y^{\prime \prime}= y′′= y′′′= y^{\prime \prime \prime}= y′′′=

Problem 7325

Find dBdI\frac{d B}{d I}dIdB​ for B=kI2/3B=k I^{2/3}B=kI2/3 and dHdW\frac{d H}{d W}dWdH​ for H=kW3/2H=k W^{3/2}H=kW3/2.

Problem 7326

Jan pedals on a stationary bike, modeled by h(t)=−0.17sin⁡(t)+0.25h(t)=-0.17 \sin (t)+0.25h(t)=−0.17sin(t)+0.25. Find the vertical velocity vvv at t=3πt=3 \pit=3π.

Problem 7327

Problem 7328

Find the second derivative of 16cos⁡2(t)16 \cos^2(t)16cos2(t). Calculate d2dt216cos⁡2(t)\frac{d^{2}}{d t^{2}} 16 \cos ^{2}(t)dt2d2​16cos2(t).

Problem 7329

A hiker walks 8 miles in 4 hours. Find her instantaneous velocity at 2.1 hours using s(t)=332t4−t3+3t2s(t)=\frac{3}{32} t^{4}-t^{3}+3 t^{2}s(t)=323​t4−t3+3t2. Round to three decimal places.

Problem 7330

Bestimmen Sie die Tangente an f(x)=1x−x3f(x)=\frac{1}{x}-x^{3}f(x)=x1​−x3 im Punkt P0 (5∣f(5)5|f(5)5∣f(5)).

Problem 7331

Bestimmen Sie die Tangentengleichung von f(x)=3⋅e2x+2f(x)=3 \cdot e^{2 x+2}f(x)=3⋅e2x+2 am Punkt (-1/3).

Problem 7332

Bestimme die Tangente an f(x)=2x3−3x−2f(x)=2 x^{3}-3 x^{-2}f(x)=2x3−3x−2 im Punkt P0(2∣f(2))P_0(2 | f(2))P0​(2∣f(2)).

Problem 7333

Problem 7334

Problem 7335

Problem 7336

Zeichne die Tangente von f(x)=−2x−1f(x)=-2 x^{-1}f(x)=−2x−1 bei x=−1x=-1x=−1 und schätze die Steigung in diesem Punkt.

Problem 7337

Approximate the zero of the differentiable function fff using its tangent line at x=−2x=-2x=−2 given f(−2)=1f(-2)=1f(−2)=1 and f′(−2)=5f'(-2)=5f′(−2)=5. What is the approximation? D) -2.4

Problem 7338

Calcule o limite da indeterminação quando xxx tende a −∞-\infty−∞: lim⁡x→−∞x2−12x+3\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}-1}}{2 x+3}x→−∞lim​2x+3x2−1​​

Problem 7339

Ein Auto beschleunigt mit aaa in \left.\frac{m}{s^{2}}\right. a) Geschwindigkeit nach 2s und 5s bei a=3a=3a=3. b) Zeit für 0 auf 505050 und 100kmh100 \frac{\mathrm{km}}{\mathrm{h}}100hkm​ bei a=3a=3a=3. c) Geschwindigkeit für allgemeines aaa.

Problem 7340

Find the IROC of 3x3+2x2−19x+63 x^{3}+2 x^{2}-19 x+63x3+2x2−19x+6 at the point (3,12)(3,12)(3,12).

Problem 7341

Bestimmen Sie die Extrem- und Wendepunkte der Funktion f(x)=13x3+12x2−6x+1f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-6 x+1f(x)=31​x3+21​x2−6x+1.

Problem 7342

Find the max and min of $g(\theta)=5 \theta-7 \sin (\theta)$ on $[0, \pi]$ . Min value =, Max value =.

Differentiate $d(r^{2}v) = C$ using the product rule, where $r$ is radius, $v$ is velocity, and $C$ is constant.

Find local extrema of $f(x)=2x^3-30x^2+126x+3$ . Identify $x$ values for min/max and their corresponding $f(x)$ values.

Bestimmen Sie die Ableitung der Funktion $f(x) = \frac{1}{3} x^{6} + x^{4} - x$ .

Bestimmen Sie die Ableitung von $f(x) = 3(-2x+5)^{3}$ .

Find the velocity of a ball dropped from $210 \mathrm{~m}$ when it hits the ground using $g=9.8 \mathrm{~m/s}^2$ . Round to three decimal places. $v \approx$

Bestimme die erste Ableitung der Funktion $f(x)=\frac{9}{x^{2}}$ .

Bestimmen Sie die Ableitung von $f(x)=\sqrt{-4 x+2}$ .

Find the critical number $A$ for the function $f(x)=-3x^2+2x-8$ . Is it a local min, max, or neither? Answer: LMIN, LMAX, or NEITHER.

Find $\theta^{\prime}(t)$ , the rate of change of the angle between clock hands, at 9 o'clock in $\mathrm{rad} / \mathrm{min}$ .

Determine the critical numbers of $f(x)=-12 x^{5}-45 x^{4}+20 x^{3}+1$ and classify them with a graph.

Determine the absolute min and max of $f(x)=2x^3 + 12x^2 - 126x + 1$ for $-7 \leq x \leq 4$ .

Show that the derivative of $a^{x}$ is $a^{x} \ln(a)$ , where $a$ is a constant.

Plot the line segment between $f(2)$ and $f(5)$ , then find all $c$ in $[2, 5]$ satisfying the Mean Value Theorem for $f$ .

Finde die Tangente an $f(x)=2 x^{-2}$ bei $x_0=3$ .

Bestimmen Sie die Tangente an $f(x)=x^{4}$ bei $P_{0}(0,5)$ .

Given the growth law $\frac{d M}{d t}=C M^{3 / 4}$ , find the growth rate at $M=175 \mathrm{~kg}$ and the mass needed to double growth from $M=0.2 \mathrm{~kg}$ .

Estimate the increase in metabolic rate $P$ when body mass increases from 78 kg to 79 kg using $P=73.3 m^{3/4}$ . Round to three decimal places.

Find the tangent line equation for $f(x)=16 e^{x} \cos^{2}(x)$ at $x=\frac{\pi}{4}$ . Express as $y$ and $x$ .

Estimate the increase in metabolic rate $P$ when body mass $m$ changes from 78 kg to 79 kg using $P=73.3 m^{3/4}$ .

Find the second and third derivatives of $y=5 e^{t} \sin(t)$ .
$y^{\prime \prime}=$ $y^{\prime \prime \prime}=$

Find $\frac{d B}{d I}$ for $B=k I^{2/3}$ and $\frac{d H}{d W}$ for $H=k W^{3/2}$ .

Jan pedals on a stationary bike, modeled by $h(t)=-0.17 \sin (t)+0.25$ . Find the vertical velocity $v$ at $t=3 \pi$ .

Find the second derivative of $16 \cos^2(t)$ . Calculate $\frac{d^{2}}{d t^{2}} 16 \cos ^{2}(t)$ .

A hiker walks 8 miles in 4 hours. Find her instantaneous velocity at 2.1 hours using $s(t)=\frac{3}{32} t^{4}-t^{3}+3 t^{2}$ . Round to three decimal places.

Bestimmen Sie die Tangente an $f(x)=\frac{1}{x}-x^{3}$ im Punkt P0 ( $5|f(5)$ ).

Bestimmen Sie die Tangentengleichung von $f(x)=3 \cdot e^{2 x+2}$ am Punkt (-1/3).

Bestimme die Tangente an $f(x)=2 x^{3}-3 x^{-2}$ im Punkt $P_0(2 | f(2))$ .

Zeichne die Tangente von $f(x)=-2 x^{-1}$ bei $x=-1$ und schätze die Steigung in diesem Punkt.

Approximate the zero of the differentiable function $f$ using its tangent line at $x=-2$ given $f(-2)=1$ and $f'(-2)=5$ . What is the approximation? D) -2.4

Calcule o limite da indeterminação quando $x$ tende a $-\infty$ : $\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}-1}}{2 x+3}$

Ein Auto beschleunigt mit $a$ in \left.\frac{m}{s^{2}}\right. a) Geschwindigkeit nach 2s und 5s bei $a=3$ . b) Zeit für 0 auf $50$ und $100 \frac{\mathrm{km}}{\mathrm{h}}$ bei $a=3$ . c) Geschwindigkeit für allgemeines $a$ .

Find the IROC of $3 x^{3}+2 x^{2}-19 x+6$ at the point $(3,12)$ .

Bestimmen Sie die Extrem- und Wendepunkte der Funktion $f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-6 x+1$ .

Find the limit: $\lim _{x \rightarrow 2} \frac{3 x-6}{4-x^{2}}$ .

Find the maximum slope of the tangent line to the graph of $f(x)=3x^2-2x^3$ . Options: A) $\frac{3}{2}$ B) $\frac{3}{4}$ C) $\frac{9}{2}$ D) $\frac{1}{2}$

Find the value of the integral $\int \frac{d x}{2+\cos 2 x}$ and identify the correct expression among the options.

Bestimmen Sie den Grenzwert der Folge $a_{n}=2+\frac{1}{n}$ und zeigen Sie die Schritte zur Berechnung.

Gegeben ist die Funktion $f(x)=\frac{1}{9}(3 x+2)^{3}$ . Finde die Steigung bei $P(2 \mid f(2))$ , prüfe waagerechte Tangenten und Punkte mit $45^{\circ}$ Steigungswinkel.

Bestimmen Sie den Grenzwert der Sequenz $a_{n}=\frac{3-2 n}{2 n}$ und erläutern Sie den Lösungsweg.

Find the critical points (min, max, inflection) of $g(z)=32(e^{2z}+4e^{-3z})^8$ .

Find the critical points of $F(t)=\frac{1}{3} t^{3}-3.5 t^{2}+10 t+21$ and classify the smaller one, $t_1$ .

Find the smaller critical point, $t_1$ , of $F(t)=\frac{1}{3} t^{3}-6.5 t^{2}+30 t+26$ and classify it as max, min, or inflection.

Find the smaller critical point $t_1$ of $F(t) = \frac{1}{3} t^{3}-5 t^{2}+16 t+21$ and classify it as max, min, or inflection.

Find the value of $z$ where the function $g(z)=28\left(e^{4 z}+3 e^{-5 z}\right)^{4}$ has a min, max, or inflection point.

Déterminez quels énoncés sur la fonction $f$ sont vrais parmi A à G concernant les dérivées et la continuité.

Quel est le taux de variation instantané de la fonction $f(x)=x^{3}-5 x^{2}+4 x-6$ en $x=3$ ?

Determine which statements about the slopes of tangent and normal lines to a function $f$ are true.

Find the second derivative of the function $f(x)=-6 x^{3}-8 x^{2}+8 x$ .

Find the intervals where the function $f(x)=-5 x^{3}+75 x^{2}+5 x+6$ is concave up or down.

Evaluate $f^{\prime \prime}(7)$ , $f^{\prime \prime}(-9)$ , and $f^{\prime \prime}(-10)$ for $f(x)=-6x^{3}-8x^{2}+8x$ .

How far does an egg fall if it hits the ground at $7.0 \mathrm{~m} / \mathrm{s}$ ?

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{x+\sin x}{x}$ using a table of values.

Find the second derivative of the function $f(x)=\sqrt{7x+4}$ .

Find the second derivative of the function $f(x)=\sqrt{8x+4}$ .

Find $x$ such that $f''(x) = -6x^2 + 4 = 0$ . Are there any constraints on $x$ ?

Find points of inflection for the function $f(x)=-5 x^{3}+75 x^{2}+5 x+6$ . Provide answers as $(x, y)$ -pairs.

Estimate the volume error of a cube with side length $9 \mathrm{~cm}$ and accuracy $0.04 \mathrm{~cm}$ using differentials.

Find the differential $d y$ for the function $y=4 \sin (2 x)$ .

Graph the differential for $f(x)=-\frac{x^{2}}{3}+3x$ at $x=5$ and $\Delta x=-1.4$ . Find $dy$ and $\Delta y$ , rounded to two decimals.

Find the point of diminishing returns for the sales function $S(x)=92+44.4 x^{2}-0.4 x^{3}$ , where $x$ is in thousands.

Find $f^{\prime \prime}(4)$ , $f^{\prime \prime}(5)$ , and $f^{\prime \prime}(-1)$ for $f(x)=\sqrt{3x+5}$ .

Find the differential $d y$ for the function $y=\sqrt{10-3 x}$ .

Find the second derivative of the function $f(x)=-6 x^{2}-6 \sqrt[3]{x}-7$ .