Calculus

Problem 9201

Find the average rate of change of the function $h(x)=-x^{2}+7 x+18$ from $x=1$ to $x=10$ .

See Solution

Problem 9202

Calculate the double integral: $\int_{1}^{4} \int_{0}^{3}(3 x^{2}+y^{2}) \, dx \, dy$ .

See Solution

Problem 9203

Evaluate the double integral $\iint_{R} f(x, y) d A$ for $f(x, y)=x$ over the region $R=[2,3] \times[-3,3]$ .

See Solution

Problem 9204

Differentiate the function $f(x)=\ln \sqrt{5 x^{2}-9 x}$ .

See Solution

Problem 9205

Calculate the double integral $\iint_{\mathbf{R}} x \cos (2 x+y) d A$ for $0 \leq x \leq \frac{\pi}{3}, 0 \leq y \leq \frac{\pi}{4}$ .

See Solution

Problem 9206

Differentiate $g(s)=\ln(\sin^{2}(7s))$ with respect to $s$ .

See Solution

Problem 9207

Differentiate $y=(4 x)^{\ln 4 x}$ using logarithmic differentiation to find $y^{\prime}$ .

See Solution

Problem 9208

Find $\frac{dy}{dx}$ for $y=\ln \left(\frac{(x^{2}+4)^{5}}{\sqrt{8-x}}\right)$ .

See Solution

Problem 9209

Differentiate $y=(\sin 8 x)^{2 x}$ using logarithmic differentiation.

See Solution

Problem 9210

Find the rocket's climbing speed at $t = 4$ sec, given height $h = 3t^2$ ft after liftoff.

See Solution

Problem 9211

A cone-shaped tank has a radius of $3 m$ and height $5 m$ . Water is added at $2 \mathrm{~m}^{3} / \mathrm{min}$ . Find the rise rate of water when it's $2 m$ deep. $\square / m i n$

See Solution

Problem 9212

Find the linearization $L(x)$ of $f(x)=\sqrt{x^{2}+21}$ at $x=-2$ .

See Solution

Problem 9213

Estimate $f(7) - f(4)$ using the Mean Value Theorem given $-3 \leq f^{\prime}(x) \leq 2$ .

See Solution

Problem 9214

Estimate $\sqrt{4.1}$ using a linear approximation.

See Solution

Problem 9215

Find the second derivative of $y=\frac{7 x^{3}}{6}-8$ .

See Solution

Problem 9216

Estimate $\ln(0.91)$ using linear approximation.

See Solution

Problem 9217

Estimate $e^{0.02}$ using linear approximation.

See Solution

Problem 9218

Find the limit: $\lim _{x \rightarrow \infty} \frac{x}{3} \cdot \tan \left(\frac{1}{x}\right)$

See Solution

Problem 9219

An object is dropped from $162 \mathrm{ft}$ . Its height after $t$ seconds is $s=162-16t^{2}$ . Find velocity, speed, and acceleration at time $t$ .

See Solution

Problem 9220

An object is dropped from 162 ft. Find its velocity, speed, acceleration at time $t$ , time to hit ground, and impact velocity.

See Solution

Problem 9221

A stone is thrown down at 88 ft/s from a 135 ft bridge. Find $s(t)$ for height after $t$ seconds and check if it hits water in 2s.

See Solution

Problem 9222

Find the first and second derivatives of $r=\frac{1}{3 s^{2}}-\frac{7}{2 s}$ .

See Solution

Problem 9223

Find the first and second derivatives of $y=\theta^{2}(2 \theta+7)$ .

See Solution

Problem 9224

Find the second and third derivatives of $y=\theta^{2}(2 \theta+7)$ with respect to $\theta$ .

See Solution

Problem 9225

Evaluate $\int \tan^{12}(x) \sec^{2}(x) \, dx$ using an appropriate substitution.

See Solution

Problem 9226

Calculate the integral of -5 with respect to x: $\int -5 \, dx$ .

See Solution

Problem 9227

Calculate the integral $\int \frac{3}{\sqrt{x^{2}+16}} d x$ using a trigonometric substitution.

See Solution

Problem 9228

Find $g^{\prime \prime}(0)$ for the function $g(s)=\frac{e^{s}}{s+1}$ .

See Solution

Problem 9229

Solve the differential equation $y^{\prime}=2 x^{3}-x$ .

See Solution

Problem 9230

Evaluate the integral $\int \sin^{2}(x) \cos^{5}(x) \, dx$ using the identity $\sin^{2}(x) + \cos^{2}(x) = 1$ and substitution.

See Solution

Problem 9231

Find the derivative $\frac{d y}{d x}$ for the equation $y=3 x^{4}-8 x+9$ .

See Solution

Problem 9232

Evaluate the integral $\int \frac{1}{\sqrt{9-(x+2)^{2}}} d x$ using the substitution $x+2=3 \sin (\theta)$ .

See Solution

Problem 9233

Evaluate the integral: $\int x \cdot \sqrt[3]{x^{2}} \, dx$

See Solution

Problem 9234

Find the derivative $\frac{d y}{d x}$ for the equation $y=-6 x^{5}+9 x^{4}-6 x^{3}+x^{2}$ .

See Solution

Problem 9235

Find the derivative $\frac{d y}{d x}$ for the equation $y=3 x^{5}+4 x^{4}+6 x^{3}-8 x-5$ .

See Solution

Problem 9236

Analyze the limits of $f(n)=3(2)^{n}-1$ as $n \to -\infty$ and $n \to \infty$ . What are the results?

See Solution

Problem 9237

Analyze the end behavior of $f(t)=-2\left(\frac{1}{4}\right)^{t}-2$ . Find limits as $t \to -\infty$ and $t \to \infty$ .

See Solution

Problem 9238

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-2 x^{5}-7 x^{3}+8 x$ .

See Solution

Problem 9239

Find the profit function from the marginal profit $\frac{d P}{d x}=-20 x+250$ and initial condition $P(4)=\$ 610$ .

See Solution

Problem 9240

Find the derivative $\frac{d y}{d x}$ for the equation $y=-x^{5}-x^{3}+9 x+7$ .

See Solution

Problem 9241

Find the integral of $\frac{1}{(5 x)^{2}}$ with respect to $x$ .

See Solution

Problem 9242

Find the cost of producing 100 items if the marginal cost is $1.83 - 0.006x$ and one item costs \$560.

See Solution

Problem 9243

Find the cost function given the marginal cost $\frac{d C}{d x}=\frac{\sqrt[4]{x}}{90}+40$ and fixed cost \$2,400.

See Solution

Problem 9244

Evaluate $\frac{d y}{d x}$ for $y=-x^{2}+3$ at $x=-1$ .

See Solution

Problem 9245

Evaluate $f^{\prime}(1)$ for the function $f(x)=2 x^{4}-5 x$ .

See Solution

Problem 9246

Find the derivative of $y$ from the equation $y^{5} \ln (y) - x^{4} \ln (x) = 5$ . Answer: $y'=$

See Solution

Problem 9247

Evaluate $\frac{d y}{d x}$ for $y=-4 x^{5}+3 x^{3}$ at $x=-1$ .

See Solution

Problem 9248

Find the derivative of the function $f(x)=\frac{-8}{\sqrt{6 x^{2}+4}}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 9249

Find the intervals where the function $f(x)=x^{4}-4 x^{3}-3$ is concave up: $(-\infty, 0)$ , $(0,2)$ , $(2, \infty)$ .

See Solution

Problem 9250

Find the derivative $\frac{d y}{d x}$ for the equation $y=-9 x^{4}-5 x^{3}-7$ .

See Solution

Problem 9251

Find the interval(s) where the function $f(x)=\frac{x^{3}}{3}-5 x^{2}+21 x+1$ is decreasing and concave down. Use interval notation.

See Solution

Problem 9252

Find $\frac{d P}{d t}$ for $b=1.8$ , $P=15 \mathrm{kPa}$ , $V=120 \mathrm{~cm}^{3}$ , $\frac{d V}{d t}=60 \mathrm{~cm}^{3}/\mathrm{min}$ . $\frac{d P}{d t}=\square \mathrm{kPa}/\mathrm{min}$

See Solution

Problem 9253

Find local extrema using the First Derivative Test for $f(x)=-8 x^{\frac{5}{3}}+\sqrt[3]{x}$ . List $x$ values or $\varnothing$ .

See Solution

Problem 9254

Find local extrema for $f(x)=-9-\frac{21 x^{2}}{2}-\frac{7 x^{3}}{3}$ using the First Derivative Test.

See Solution

Problem 9255

Water fills a cylindrical pool at 4 ft³/min. Find the height change rate when water is 5 ft deep. Answer in $\mathrm{ft} / \mathrm{min}$ .

See Solution

Problem 9256

Find the local minima of the continuous function $f(x)$ with critical points at $x=-3, 0, 3, 7$ based on $f^{\prime \prime}(x)$ signs.

See Solution

Problem 9257

Find local extrema of $f(x)=(x+4)^{3}(3 x-2)^{2}$ using the First Derivative Test. List $x$ values or $\varnothing$ .

See Solution

Problem 9258

Find the number of items, $n$ , that minimizes costs given the function $C(n)=75 n^{2}-1800 n+60000$ .

See Solution

Problem 9259

Bestimmen Sie die Ableitung von $f(x)=2 x^{4}-12 x^{2}+12$ .

See Solution

Problem 9260

Find how fast the total resistance $R$ increases when $R_{1}=100 \Omega$ , $R_{2}=70 \Omega$ , with rates $0.9$ and $0.7 \Omega/\mathrm{s}$ .

See Solution

Problem 9261

Bestimmen Sie den Wendepunkt und die Wendetangente für die Funktionen: a) $f(x)=0,5 x^{3}-3 x^{2}+5 x$ , b) $f(x)=x^{3}+3 x^{2}+x+2$ , c) $f(x)=-0,5 x^{3}-1,5 x^{2}$ , d) $f(x)=x^{3}+9 x^{2}+7 x-18$ , e) $f(x)=-x^{3}-3 x^{2}+4 x+4$ , f) $f(x)=x^{3}-6 x^{2}+11 x$ .

See Solution

Problem 9262

Bestimme die 1. und 2. Ableitung für die folgenden Funktionen: a) $f(x)=-3 x^{2}+4 x-1$ , b) $f(x)=\frac{2}{x^{2}}+\frac{x^{4}}{4}$ , c) $f(x)=2 \sin x-\ln x$ , d) $f(x)=\frac{1}{8} x^{4}+\frac{3}{2} x^{3}+2 x^{2}+\frac{4}{5}$ , e) $f(x)=(3 x+2)^{5}$ , f) $f(x)=4 x \cdot e^{x}$ .

See Solution

Problem 9263

Gegeben ist $f(x)=3 x^{2}-4 x+1$ .
a) Bestimme die Tangente $t_{1}(x)$ an $f(x)$ bei $P(1, f(1))$ .
b) Finde den Schnittpunkt und Schnittwinkel der Tangenten $t_{1}(x)$ und $t_{2}(x)$ durch $Q(4, 33)$ .
c) Bestimme die Gleichung der Geraden $g(x)$ , die parallel zu $t_{1}(x)$ verläuft und durch $R(1, -1)$ geht.

See Solution

Problem 9264

Finde die Punkte, an denen $k(x)=\frac{5}{3} x^{3}-8 x^{2}-16 x+3$ waagerechte Tangenten hat.

See Solution

Problem 9265

Zeigen Sie, dass $F$ eine Stammfunktion von $f$ ist für die folgenden Funktionen:
a) $f(x)=5 x^{4}+x^{3}$ , $F(x)=x^{5}+\frac{1}{4} x^{4}$ ; b) $f(x)=3 x^{2}-8 x^{3}$ , $F(x)=x^{3}-2 x^{4}-3$ ; c) $f(x)=\cos (x)+1$ , $F(x)=\sin (x)+x$ ; d) $f(x)=\sin \left(\frac{1}{3} x\right)$ , $F(x)=-3 \cos \left(\frac{1}{3} x\right)+\pi$ ; e) $f(x)=\frac{3}{2} x^{2}+4 e^{2 x}$ , $F(x)=\frac{1}{2} x^{3}+2 e^{2 x}$ ; f) $f(x)=(4 x-3)^{2}$ , $F(x)=\frac{1}{12}(4 x-3)^{3}$ .

See Solution

Problem 9266

Gegeben ist die Funktion $y=f_{1}(x)=\left(x+\frac{1}{t}\right)e^{-t x}$ . Führen Sie eine Kurvendiskussion durch und berechnen Sie Flächeninhalte und Volumen.

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Problem 9267

Eine Tangente $t_{3}(x)$ berührt $h(x)=\frac{x^{3}}{2}+2$ in $Q_{1}$ oder $Q_{2}$ . Bestimme die Punkte und $t_{3}(x)$ parallel zu $y=6x+2$ . Finde auch die Normale $i(x)$ bei $S(-2, h(-2))$ .

See Solution

Problem 9268

Find the limit: $\lim _{x \rightarrow-4} \frac{x^{3}+64}{x+4}$ .

See Solution

Problem 9269

Find the derivative $\frac{d y}{d x}$ for the function $y=(2 x-5)(x-3)$ .

See Solution

Problem 9270

Evaluate the integral $\int 2xy \, dy$ .

See Solution

Problem 9271

Find the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-2 x+5}{x+6}$ .

See Solution

Problem 9272

Gegeben ist $f(x)=-x^{2}+9$ . Berechne für $0 \leqq u \leqq 3$ : a) den maximalen Flächeninhalt des Rechtecks, b) den maximalen Umfang.

See Solution

Problem 9273

Bestimme die 1., 2. und 3. Ableitung der Funktion $f(x)=x \cdot (x+3)^{2}$ .

See Solution

Problem 9274

Find local extrema for $f(x)=\frac{(x-5)^{2}}{x+9}$ in $(-9,7)$ using the Second Derivative Test.

See Solution

Problem 9275

Leiten Sie die Funktion $f$ ab: a) $f(x)=x \cdot \sin (3 x)$ b) $f(x)=(2 x-1)^{2} \cdot \sqrt{x}$ c) $f(x)=3 x^{5} \cdot \cos (2 x)$ d) $f(x)=3 x \cdot \sin (4 x-1)$ e) $f(x)=(4-3 x)^{2} \cdot \sin (x)$ f) $f(x)=0,5 x^{2} \cdot \sqrt{4-x}$ g) $f(x)=x^{2} \cdot \cos (1-x)$ h) $f(x)=\sqrt{2 x+3} \cdot x^{2}$ i) $f(x)=(5 x+2)^{7} \cdot \cos (x)$

See Solution

Problem 9276

Solve the differential equation $y'=\sqrt{3} y$ with initial condition $y(0)=12$ . Find $y(t)$ .

See Solution

Problem 9277

Calculate the average rate of change of $f(x)=-3x^{2}-x$ from $x=5$ to $x=6$ . Options: A. -34 B. -2 C. $-\frac{1}{6}$ D. $\frac{1}{2}$

See Solution

Problem 9278

A protozoa population starts with 7 members and grows at a rate of 0.469/day. Find the population size after 8 days.

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Problem 9279

Find the derivative of $f(x)=\ln(6+e)$ and state its domain. What is $f'(x)$ ? Domain:

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Problem 9280

Ergänzen Sie die Ableitungen für die Funktionen: a) $f(x)=x^{2} \cdot \sin (x)$ b) $f(x)=(2 x-3) \cdot \cos (4 x)$ c) $f(x)=\sin (2-x) \cdot \sqrt{3 x-1}$ d) $f(x)=\sqrt{x} \cdot(4 x+1)$ e) $f(x)=(x+2)^{3} \cdot \cos (-3 x)$

See Solution

Problem 9281

Bestimmen Sie die 1. Ableitung der Funktionen: a) $f(x)=x^{z-3}$ , b) $f(x)=\frac{1}{3} x^{3}+x-1$ , c) $f(b)=4 a b$ , d) $f(z)=\frac{4}{z}+\sqrt{z}$ , e) $f(x)=-4 x^{2}+0,5 x+1$ .

See Solution

Problem 9282

Find the arc length parametrization for the curve $\mathbf{p}(t)=2 \sqrt{t} \mathbf{i}+(1-4 \sqrt{t}) \mathbf{j}+(1+6 \sqrt{t}) \mathbf{k}$ , $t \geq 1$ .

See Solution

Problem 9283

A feather drops from 1.40 m on the moon with gravity $1.67 \mathrm{~m} / \mathrm{s}^{2}$ . Find the fall time.

See Solution

Problem 9284

Bestimme den Differenzenquotienten für die Funktionen: a) $f(x)=x^{3}-2x^{2}+1$ bei $P(-1)$ und $Q(2)$ , b) $f(x)=\cos(x)$ bei $A(1)$ und $B(5)$ . Berechne auch Änderungsmaße für $f(x)=2x-1$ im Intervall [3; 7].

See Solution

Problem 9285

Determine the time for a feather to fall 1.40 meters to Earth's surface.

See Solution

Problem 9286

Differentiate $x^{2}+3xy+y^{3}=9$ with respect to $x$ . Which result is correct?

See Solution

Problem 9287

Calculez la dérivée seconde de $y=7 \frac{1}{x^{3}}+7 x^{2}$ en $x=7$ .

See Solution

Problem 9288

Use implicit differentiation on $\sin(x+y) = \cos(xy)$ to find $\frac{dy}{dx}$ . What are the results?

See Solution

Problem 9289

Determine where the function $f(x)$ is increasing, decreasing, and locate its local extrema for $f(x)=-4 x^{2}-32 x-22$ .

See Solution

Problem 9290

Calculate the average rate of change of $f(x)=\frac{1}{3} x^{2}-4$ between $x=0$ and $x=3$ .

See Solution

Problem 9291

Determine if these statements are true or false: I. If you can solve $y$ from $x$ , implicit differentiation can't be used. II. Implicit differentiation finds slopes for curves that aren't functions. III. It's fine if $\frac{d y}{d x}$ involves both $x$ and $y$ .

See Solution

Problem 9292

Determine where $f(x)$ is increasing, decreasing, and identify local extrema for $f(x)=(x-8)e^{-6x}$ .

See Solution

Problem 9293

Identify cases needing implicit differentiation for $\frac{d y}{d x}$ :
1. $y=x^{2} \sin (5 x)$
2. $\sqrt[3]{x^{4}}=y$
3. $y^{5}+2 y^{3}-x^{4}=1$
4. $x^{2} y^{3}=1+y$
5. $y^{2}-1+\sqrt{x+2 y}=0$
6. $y=\sqrt{2+\tan ^{-1}\left(3 x^{2}\right)}$
7. $x y=\sin (x+y)$

See Solution

Problem 9294

Find (A) $f'(x)$ , (B) the partition numbers for $f'$ , and (C) the critical numbers of $f$ for $f(x)=\frac{8}{x+6}$ .

See Solution

Problem 9295

Identify the valid multiple integrals from the options below:
1. $\int_{a=0}^{1} \int_{b=0}^{1} \int_{c=0}^{1}(a b c) d c d a d b$
2. $\int_{y=0}^{1} \int_{x=-1}^{y} \int_{z=-y}^{x^{2}}(x-2 y+z) d z d x d y$
3. $\int_{x=0}^{1} \int_{y=0}^{x} \int_{z=0}^{y} d z d y d x$
4. $\int_{u=0}^{1} \int_{v=0}^{-u} e^{u-v} d u d v$
5. $\int_{x=2}^{-1} \int_{y=-x}^{x} \int_{z=2}^{7}\left(e^{x} y z\right) d z d x d y$
6. $\int_{y=0}^{x} \int_{x=0}^{y} x^{2}+y^{2} d x d y$

See Solution

Problem 9296

Calculate the double integral: $\int_{x=0}^{1} \int_{y=0}^{x} 42 x^{3} y^{2} d y d x$ .

See Solution

Problem 9297

Determine where the function $f(x)=(x-5) e^{-6 x}$ is increasing, decreasing, and locate its local extrema.

See Solution

Problem 9298

Find the derivative of $f(x) = a \sqrt[3]{x}+x^{2}-2$ at $x = -1$ .

See Solution

Problem 9299

Find the slope of the tangent line for $f(x)=5 x^{2}-11 x-9$ at $x=-2$ .

See Solution

Problem 9300

Find the slope of the tangent line for the function $f(x)=-6x-9$ at the point where $x=6$ .

See Solution

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Calculus

Problem 9201

Find the average rate of change of the function h(x)=−x2+7x+18h(x)=-x^{2}+7 x+18h(x)=−x2+7x+18 from x=1x=1x=1 to x=10x=10x=10.

Problem 9202

Calculate the double integral: ∫14∫03(3x2+y2) dx dy\int_{1}^{4} \int_{0}^{3}(3 x^{2}+y^{2}) \, dx \, dy∫14​∫03​(3x2+y2)dxdy.

Problem 9203

Evaluate the double integral ∬Rf(x,y)dA\iint_{R} f(x, y) d A∬R​f(x,y)dA for f(x,y)=xf(x, y)=xf(x,y)=x over the region R=[2,3]×[−3,3]R=[2,3] \times[-3,3]R=[2,3]×[−3,3].

Problem 9204

Differentiate the function f(x)=ln⁡5x2−9xf(x)=\ln \sqrt{5 x^{2}-9 x}f(x)=ln5x2−9x​.

Problem 9205

Calculate the double integral ∬Rxcos⁡(2x+y)dA\iint_{\mathbf{R}} x \cos (2 x+y) d A∬R​xcos(2x+y)dA for 0≤x≤π3,0≤y≤π40 \leq x \leq \frac{\pi}{3}, 0 \leq y \leq \frac{\pi}{4}0≤x≤3π​,0≤y≤4π​.

Problem 9206

Differentiate g(s)=ln⁡(sin⁡2(7s))g(s)=\ln(\sin^{2}(7s))g(s)=ln(sin2(7s)) with respect to sss.

Problem 9207

Differentiate y=(4x)ln⁡4xy=(4 x)^{\ln 4 x}y=(4x)ln4x using logarithmic differentiation to find y′y^{\prime}y′.

Problem 9208

Find dydx\frac{dy}{dx}dxdy​ for y=ln⁡((x2+4)58−x)y=\ln \left(\frac{(x^{2}+4)^{5}}{\sqrt{8-x}}\right)y=ln(8−x​(x2+4)5​).

Problem 9209

Differentiate y=(sin⁡8x)2xy=(\sin 8 x)^{2 x}y=(sin8x)2x using logarithmic differentiation.

Problem 9210

Find the rocket's climbing speed at t=4t = 4t=4 sec, given height h=3t2h = 3t^2h=3t2 ft after liftoff.

Problem 9211

A cone-shaped tank has a radius of 3m3 m3m and height 5m5 m5m. Water is added at 2 m3/min2 \mathrm{~m}^{3} / \mathrm{min}2 m3/min. Find the rise rate of water when it's 2m2 m2m deep. □/min\square / m i n□/min

Problem 9212

Find the linearization L(x)L(x)L(x) of f(x)=x2+21f(x)=\sqrt{x^{2}+21}f(x)=x2+21​ at x=−2x=-2x=−2.

Problem 9213

Estimate f(7)−f(4)f(7) - f(4)f(7)−f(4) using the Mean Value Theorem given −3≤f′(x)≤2-3 \leq f^{\prime}(x) \leq 2−3≤f′(x)≤2.

Problem 9214

Estimate 4.1\sqrt{4.1}4.1​ using a linear approximation.

Problem 9215

Find the second derivative of y=7x36−8y=\frac{7 x^{3}}{6}-8y=67x3​−8.

Problem 9216

Estimate ln⁡(0.91)\ln(0.91)ln(0.91) using linear approximation.

Problem 9217

Estimate e0.02e^{0.02}e0.02 using linear approximation.

Problem 9218

Find the limit: lim⁡x→∞x3⋅tan⁡(1x)\lim _{x \rightarrow \infty} \frac{x}{3} \cdot \tan \left(\frac{1}{x}\right)limx→∞​3x​⋅tan(x1​)

Problem 9219

An object is dropped from 162ft162 \mathrm{ft}162ft. Its height after ttt seconds is s=162−16t2s=162-16t^{2}s=162−16t2. Find velocity, speed, and acceleration at time ttt.

Problem 9220

An object is dropped from 162 ft. Find its velocity, speed, acceleration at time ttt, time to hit ground, and impact velocity.

Problem 9221

A stone is thrown down at 88 ft/s from a 135 ft bridge. Find s(t)s(t)s(t) for height after ttt seconds and check if it hits water in 2s.

Problem 9222

Find the first and second derivatives of r=13s2−72sr=\frac{1}{3 s^{2}}-\frac{7}{2 s}r=3s21​−2s7​.

Problem 9223

Find the first and second derivatives of y=θ2(2θ+7)y=\theta^{2}(2 \theta+7)y=θ2(2θ+7).

Problem 9224

Find the second and third derivatives of y=θ2(2θ+7)y=\theta^{2}(2 \theta+7)y=θ2(2θ+7) with respect to θ\thetaθ.

Problem 9225

Evaluate ∫tan⁡12(x)sec⁡2(x) dx\int \tan^{12}(x) \sec^{2}(x) \, dx∫tan12(x)sec2(x)dx using an appropriate substitution.

Problem 9226

Calculate the integral of -5 with respect to x: ∫−5 dx\int -5 \, dx∫−5dx.

Problem 9227

Calculate the integral ∫3x2+16dx\int \frac{3}{\sqrt{x^{2}+16}} d x∫x2+16​3​dx using a trigonometric substitution.

Problem 9228

Find g′′(0)g^{\prime \prime}(0)g′′(0) for the function g(s)=ess+1g(s)=\frac{e^{s}}{s+1}g(s)=s+1es​.

Problem 9229

Solve the differential equation y′=2x3−xy^{\prime}=2 x^{3}-xy′=2x3−x.

Problem 9230

Evaluate the integral ∫sin⁡2(x)cos⁡5(x) dx\int \sin^{2}(x) \cos^{5}(x) \, dx∫sin2(x)cos5(x)dx using the identity sin⁡2(x)+cos⁡2(x)=1\sin^{2}(x) + \cos^{2}(x) = 1sin2(x)+cos2(x)=1 and substitution.

Problem 9231

Find the derivative dydx\frac{d y}{d x}dxdy​ for the equation y=3x4−8x+9y=3 x^{4}-8 x+9y=3x4−8x+9.

Problem 9232

Evaluate the integral ∫19−(x+2)2dx\int \frac{1}{\sqrt{9-(x+2)^{2}}} d x∫9−(x+2)2​1​dx using the substitution x+2=3sin⁡(θ)x+2=3 \sin (\theta)x+2=3sin(θ).

Problem 9233

Evaluate the integral: ∫x⋅x23 dx\int x \cdot \sqrt[3]{x^{2}} \, dx∫x⋅3x2​dx

Problem 9234

Find the derivative dydx\frac{d y}{d x}dxdy​ for the equation y=−6x5+9x4−6x3+x2y=-6 x^{5}+9 x^{4}-6 x^{3}+x^{2}y=−6x5+9x4−6x3+x2.

Problem 9235

Find the derivative dydx\frac{d y}{d x}dxdy​ for the equation y=3x5+4x4+6x3−8x−5y=3 x^{5}+4 x^{4}+6 x^{3}-8 x-5y=3x5+4x4+6x3−8x−5.

Problem 9236

Analyze the limits of f(n)=3(2)n−1f(n)=3(2)^{n}-1f(n)=3(2)n−1 as n→−∞n \to -\inftyn→−∞ and n→∞n \to \inftyn→∞. What are the results?

Problem 9237

Analyze the end behavior of f(t)=−2(14)t−2f(t)=-2\left(\frac{1}{4}\right)^{t}-2f(t)=−2(41​)t−2. Find limits as t→−∞t \to -\inftyt→−∞ and t→∞t \to \inftyt→∞.

Problem 9238

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=−2x5−7x3+8xf(x)=-2 x^{5}-7 x^{3}+8 xf(x)=−2x5−7x3+8x.

Problem 9239

Find the profit function from the marginal profit dPdx=−20x+250\frac{d P}{d x}=-20 x+250dxdP​=−20x+250 and initial condition P(4)=$610P(4)=\$ 610P(4)=$610.

Problem 9240

Find the average rate of change of the function $h(x)=-x^{2}+7 x+18$ from $x=1$ to $x=10$ .

Calculate the double integral: $\int_{1}^{4} \int_{0}^{3}(3 x^{2}+y^{2}) \, dx \, dy$ .

Evaluate the double integral $\iint_{R} f(x, y) d A$ for $f(x, y)=x$ over the region $R=[2,3] \times[-3,3]$ .

Differentiate the function $f(x)=\ln \sqrt{5 x^{2}-9 x}$ .

Calculate the double integral $\iint_{\mathbf{R}} x \cos (2 x+y) d A$ for $0 \leq x \leq \frac{\pi}{3}, 0 \leq y \leq \frac{\pi}{4}$ .

Differentiate $g(s)=\ln(\sin^{2}(7s))$ with respect to $s$ .

Differentiate $y=(4 x)^{\ln 4 x}$ using logarithmic differentiation to find $y^{\prime}$ .

Find $\frac{dy}{dx}$ for $y=\ln \left(\frac{(x^{2}+4)^{5}}{\sqrt{8-x}}\right)$ .

Differentiate $y=(\sin 8 x)^{2 x}$ using logarithmic differentiation.

Find the rocket's climbing speed at $t = 4$ sec, given height $h = 3t^2$ ft after liftoff.

A cone-shaped tank has a radius of $3 m$ and height $5 m$ . Water is added at $2 \mathrm{~m}^{3} / \mathrm{min}$ . Find the rise rate of water when it's $2 m$ deep. $\square / m i n$

Find the linearization $L(x)$ of $f(x)=\sqrt{x^{2}+21}$ at $x=-2$ .

Estimate $f(7) - f(4)$ using the Mean Value Theorem given $-3 \leq f^{\prime}(x) \leq 2$ .

Estimate $\sqrt{4.1}$ using a linear approximation.

Find the second derivative of $y=\frac{7 x^{3}}{6}-8$ .

Estimate $\ln(0.91)$ using linear approximation.

Estimate $e^{0.02}$ using linear approximation.

Find the limit: $\lim _{x \rightarrow \infty} \frac{x}{3} \cdot \tan \left(\frac{1}{x}\right)$

An object is dropped from $162 \mathrm{ft}$ . Its height after $t$ seconds is $s=162-16t^{2}$ . Find velocity, speed, and acceleration at time $t$ .

An object is dropped from 162 ft. Find its velocity, speed, acceleration at time $t$ , time to hit ground, and impact velocity.

A stone is thrown down at 88 ft/s from a 135 ft bridge. Find $s(t)$ for height after $t$ seconds and check if it hits water in 2s.

Find the first and second derivatives of $r=\frac{1}{3 s^{2}}-\frac{7}{2 s}$ .

Find the first and second derivatives of $y=\theta^{2}(2 \theta+7)$ .

Find the second and third derivatives of $y=\theta^{2}(2 \theta+7)$ with respect to $\theta$ .

Evaluate $\int \tan^{12}(x) \sec^{2}(x) \, dx$ using an appropriate substitution.

Calculate the integral of -5 with respect to x: $\int -5 \, dx$ .

Calculate the integral $\int \frac{3}{\sqrt{x^{2}+16}} d x$ using a trigonometric substitution.

Find $g^{\prime \prime}(0)$ for the function $g(s)=\frac{e^{s}}{s+1}$ .

Solve the differential equation $y^{\prime}=2 x^{3}-x$ .

Evaluate the integral $\int \sin^{2}(x) \cos^{5}(x) \, dx$ using the identity $\sin^{2}(x) + \cos^{2}(x) = 1$ and substitution.

Find the derivative $\frac{d y}{d x}$ for the equation $y=3 x^{4}-8 x+9$ .

Evaluate the integral $\int \frac{1}{\sqrt{9-(x+2)^{2}}} d x$ using the substitution $x+2=3 \sin (\theta)$ .

Evaluate the integral: $\int x \cdot \sqrt[3]{x^{2}} \, dx$

Find the derivative $\frac{d y}{d x}$ for the equation $y=-6 x^{5}+9 x^{4}-6 x^{3}+x^{2}$ .

Find the derivative $\frac{d y}{d x}$ for the equation $y=3 x^{5}+4 x^{4}+6 x^{3}-8 x-5$ .

Analyze the limits of $f(n)=3(2)^{n}-1$ as $n \to -\infty$ and $n \to \infty$ . What are the results?

Analyze the end behavior of $f(t)=-2\left(\frac{1}{4}\right)^{t}-2$ . Find limits as $t \to -\infty$ and $t \to \infty$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-2 x^{5}-7 x^{3}+8 x$ .

Find the profit function from the marginal profit $\frac{d P}{d x}=-20 x+250$ and initial condition $P(4)=\$ 610$ .

Find the derivative $\frac{d y}{d x}$ for the equation $y=-x^{5}-x^{3}+9 x+7$ .

Find the integral of $\frac{1}{(5 x)^{2}}$ with respect to $x$ .

Find the cost of producing 100 items if the marginal cost is $1.83 - 0.006x$ and one item costs \$560.

Find the cost function given the marginal cost $\frac{d C}{d x}=\frac{\sqrt[4]{x}}{90}+40$ and fixed cost \$2,400.

Evaluate $\frac{d y}{d x}$ for $y=-x^{2}+3$ at $x=-1$ .

Evaluate $f^{\prime}(1)$ for the function $f(x)=2 x^{4}-5 x$ .

Find the derivative of $y$ from the equation $y^{5} \ln (y) - x^{4} \ln (x) = 5$ . Answer: $y'=$

Evaluate $\frac{d y}{d x}$ for $y=-4 x^{5}+3 x^{3}$ at $x=-1$ .

Find the derivative of the function $f(x)=\frac{-8}{\sqrt{6 x^{2}+4}}$ . What is $f^{\prime}(x)$ ?

Find the intervals where the function $f(x)=x^{4}-4 x^{3}-3$ is concave up: $(-\infty, 0)$ , $(0,2)$ , $(2, \infty)$ .

Find the derivative $\frac{d y}{d x}$ for the equation $y=-9 x^{4}-5 x^{3}-7$ .

Find the interval(s) where the function $f(x)=\frac{x^{3}}{3}-5 x^{2}+21 x+1$ is decreasing and concave down. Use interval notation.

Find $\frac{d P}{d t}$ for $b=1.8$ , $P=15 \mathrm{kPa}$ , $V=120 \mathrm{~cm}^{3}$ , $\frac{d V}{d t}=60 \mathrm{~cm}^{3}/\mathrm{min}$ . $\frac{d P}{d t}=\square \mathrm{kPa}/\mathrm{min}$

Find local extrema using the First Derivative Test for $f(x)=-8 x^{\frac{5}{3}}+\sqrt[3]{x}$ . List $x$ values or $\varnothing$ .

Find local extrema for $f(x)=-9-\frac{21 x^{2}}{2}-\frac{7 x^{3}}{3}$ using the First Derivative Test.

Water fills a cylindrical pool at 4 ft³/min. Find the height change rate when water is 5 ft deep. Answer in $\mathrm{ft} / \mathrm{min}$ .

Find the local minima of the continuous function $f(x)$ with critical points at $x=-3, 0, 3, 7$ based on $f^{\prime \prime}(x)$ signs.

Find local extrema of $f(x)=(x+4)^{3}(3 x-2)^{2}$ using the First Derivative Test. List $x$ values or $\varnothing$ .

Find the number of items, $n$ , that minimizes costs given the function $C(n)=75 n^{2}-1800 n+60000$ .

Bestimmen Sie die Ableitung von $f(x)=2 x^{4}-12 x^{2}+12$ .

Find how fast the total resistance $R$ increases when $R_{1}=100 \Omega$ , $R_{2}=70 \Omega$ , with rates $0.9$ and $0.7 \Omega/\mathrm{s}$ .

Finde die Punkte, an denen $k(x)=\frac{5}{3} x^{3}-8 x^{2}-16 x+3$ waagerechte Tangenten hat.

Gegeben ist die Funktion $y=f_{1}(x)=\left(x+\frac{1}{t}\right)e^{-t x}$ . Führen Sie eine Kurvendiskussion durch und berechnen Sie Flächeninhalte und Volumen.

Eine Tangente $t_{3}(x)$ berührt $h(x)=\frac{x^{3}}{2}+2$ in $Q_{1}$ oder $Q_{2}$ . Bestimme die Punkte und $t_{3}(x)$ parallel zu $y=6x+2$ . Finde auch die Normale $i(x)$ bei $S(-2, h(-2))$ .

Find the limit: $\lim _{x \rightarrow-4} \frac{x^{3}+64}{x+4}$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=(2 x-5)(x-3)$ .

Evaluate the integral $\int 2xy \, dy$ .

Find the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-2 x+5}{x+6}$ .

Gegeben ist $f(x)=-x^{2}+9$ . Berechne für $0 \leqq u \leqq 3$ : a) den maximalen Flächeninhalt des Rechtecks, b) den maximalen Umfang.

Bestimme die 1., 2. und 3. Ableitung der Funktion $f(x)=x \cdot (x+3)^{2}$ .

Find local extrema for $f(x)=\frac{(x-5)^{2}}{x+9}$ in $(-9,7)$ using the Second Derivative Test.