Calculus

Problem 23201

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\sqrt{x^{2}-6x}$ .

See Solution

Problem 23202

A researcher models antibody production as $N^{\prime}(t)=\frac{4,000}{20 t+10}$ . Find $N(t)$ and $N(8)$ in thousand antibodies.

See Solution

Problem 23203

Find the $x$ -coordinates where the tangent line to $f(x)=2 x^{3}+4 x^{2}-8 x+3$ is horizontal.

See Solution

Problem 23204

Find the derivative of g(x) = 3x \cos(x) - 2.

See Solution

Problem 23205

A rock creates a ripple in a pond. If the radius grows at $5 \mathrm{ft} / \mathrm{sec}$ , find the area increase speed at 2 ft radius.

See Solution

Problem 23206

Calculate the indefinite integral: $\int e^{x^{9}} x^{8} d x$ using the substitution $u=x^{9}$ .

See Solution

Problem 23207

Calculate the integral using substitution: $u=x^{2}+1$ . Find $\int\left(x^{2}+1\right)^{6} x d x$ .

See Solution

Problem 23208

Find the maximum value of $y=\frac{2}{3} x^{3}-\frac{1}{2} x^{2}-3 x+1$ on $[0,2]$ .

See Solution

Problem 23209

Find all functions $g$ where $g^{\prime}(x)=\frac{4 x^{2}+x+3}{\sqrt{x}}$ .

See Solution

Problem 23210

Find the consumers' surplus for the demand function $d(x)=700-\frac{1}{2} x$ at demand level $x=400$ .

See Solution

Problem 23211

Calculate the integral $I=\int_{0}^{\pi / 4} \frac{4+3 \cos ^{2} \theta}{\cos ^{2} \theta} d \theta$ .

See Solution

Problem 23212

Evaluate the integral I = ∫(4 + 2y - y²) dy from 0 to 2.

See Solution

Problem 23213

Find the consumers' surplus for the demand function $d(x) = 100 e^{-0.01 x}$ at $x = 300$ . Round to the nearest cent.

See Solution

Problem 23214

Calculate the integral $I=\int_{0}^{\pi / 2} 3 \sin (2 t) d t$ .

See Solution

Problem 23215

Find the gradient of the curve $y=x^{3}-12 x$ at $(1,-11)$ and the coordinates of its turning points.

See Solution

Problem 23216

Approximate $\sqrt[3]{64.1}$ using the tangent line of $f(x)=\sqrt[3]{x}$ at $x=64$ : find $m$ and $b$ for $y=m x+b$ .

See Solution

Problem 23217

Approximate $\frac{1}{0.252}$ using the tangent line of $f(x)=\frac{1}{x}$ at a nearby point.

See Solution

Problem 23218

Approximate $\sqrt{16.1}$ using the tangent line of $f(x)=\sqrt{x}$ at $x=16$ . Find $L(x)=$ . Provide 9 significant figures.

See Solution

Problem 23219

Find the interval where the Intermediate Value Theorem ensures a root for $f(x)=x^{3}+x-3$ . Options: (A) $(-1,0)$ (B) $(0,1)$ (C) $(1,2)$ (D) $(2,3)$ .

See Solution

Problem 23220

Find the acceleration of a particle with velocity $v(t)=-(t+1) \sin \left(\frac{t^{2}}{2}\right)$ at $t=2$ . Is its speed increasing?

See Solution

Problem 23221

Find the limit: $\lim _{x \rightarrow 2^{-}} \frac{9}{x-2}$ . Choose from (A) $-\infty$ , (B) 0, (C) 9, (D) does not exist.

See Solution

Problem 23222

Find the interval where the Intermediate Value Theorem ensures a root for $f(x)=x^{3}+x-3$ . Options: (A) $(-1,0)$ , (B) $(0,1)$ , (C) $(1,2)$ , (D) $(2,3)$ .

See Solution

Problem 23223

A particle's velocity is $v(t)=\frac{10 \sin \left(0.4 t^{2}\right)}{t^{2}-t+3}$ for $0 \leq t \leq 3.5$ . Find acceleration at $t=3$ .

See Solution

Problem 23224

Approximate $\sqrt[3]{8.4}$ using the tangent line of $f(x)=\sqrt[3]{x}$ at $x=8$ . Find $m$ and $b$ for $y=mx+b$ .

See Solution

Problem 23225

Given $f$ values for $3 \leq x \leq 9$ , which could be true: $f'$ negative/increasing, negative/decreasing, positive/increasing, or positive/decreasing?

See Solution

Problem 23226

Approximate $\sqrt{16.2}$ using the tangent line of $f(x)=\sqrt{x}$ at $x=16$ . Find $L(x)=$ . Provide 9 significant figures.

See Solution

Problem 23227

Find the consumers' surplus for the demand function $d(x) = 350 - 0.09 x^{2}$ at $x = 50$ .

See Solution

Problem 23228

Find the $x$ -coordinates of all relative maxima for the function $f(x)=-2 x^{3}-6 x^{2}+18 x-3$ .

See Solution

Problem 23229

Find the interval(s) where the graph of $y=x-x^{3}$ is concave up. Choices: (A) $(-\infty, 0)$ (B) $(0, \infty)$ (C) $\left(-\infty,-\sqrt{\frac{1}{3}}\right) \cup\left(\sqrt{\frac{1}{3}}, \infty\right)$ (D) $\left(-\sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}\right)$

See Solution

Problem 23230

Find $\Delta y$ for $y=4x^2$ at $x=4$ with $\Delta x=0.4$ and find $dy$ with $dx=0.4$ .

See Solution

Problem 23231

Evaluate the integral $\int(2+\sin ^{2} \theta)^{2} \cos \theta d \theta$ using substitution.

See Solution

Problem 23232

Find intervals where the derivative of the function $f(x)=x^{4}-16 x^{3}$ is decreasing.

See Solution

Problem 23233

Find $d y$ for $y=\tan(4x+4)$ at $x=3$ with $d x=0.2$ and $d x=0.4$ .

See Solution

Problem 23234

Given $y=4 \sqrt{x}$ , find $\Delta y$ for $x=5$ and $\Delta x=0.4$ . Also, find $d y$ for $x=5$ and $d x=0.4$ .

See Solution

Problem 23235

Find the tangent line equation for $f(x)=\sqrt[3]{108 x^{3}+108}$ at $x=1$ in $mx+b$ form. Approximate $f(1.1)$ and find the error.

See Solution

Problem 23236

Find $d y$ for $y=\tan(2x+7)$ at $x=1$ with $d x=0.1$ and $d x=0.2$ .

See Solution

Problem 23237

What is the area between the curve $y=x^{2}-4$ and the $x$ -axis from $x=-2$ to $x=2$ ?

See Solution

Problem 23238

Approximate $\sqrt[3]{64.1}$ using the tangent line of $f(x)=\sqrt[3]{x}$ at $x=64$ . Find $m$ and $b$ for $y=mx+b$ .

See Solution

Problem 23239

Graph $y=\frac{1}{2}(x^{3}-3x+7)$ : find domain, symmetries, derivatives, critical points, inflection points, and extremes. Domain: $\square$ .

See Solution

Problem 23240

Find the tangent line of $f(x)=\sqrt[3]{128 x^{3}+384}$ at $x=1$ in the form $L(x)=mx+b$ . Approximate $f(1.1)$ and compute the error $|error| \approx$ (to 5 decimal places).

See Solution

Problem 23241

Find intervals where the function $f(x)=\frac{1}{2} x^{4}+6 x^{3}-48 x^{2}$ is concave down.

See Solution

Problem 23242

Calculate the integral: $\int \sin^{5}(2t) \cos^{2}(2t) \, dt$

See Solution

Problem 23243

Graph the function $y=\frac{1}{2}(x^{3}-3x+7)$ , find derivatives, critical points, concavity, and extreme points. Identify intervals of concavity.

See Solution

Problem 23244

Evaluate the integral: $\int \frac{\ln \left(y^{2}\right)}{3 y} d y$ .

See Solution

Problem 23245

Graph $y=\frac{1}{2}(x^{3}-3x+7)$ : find domain, symmetries, $y'$ , $y''$ , critical points, increasing/decreasing intervals, inflection points, asymptotes, and extreme points. Identify vertical asymptotes.

See Solution

Problem 23246

Calculate the integral: $\int \frac{\sin (4 x)}{(2+\cos (4 x))^{2}} d x$

See Solution

Problem 23247

Find the $x$ -coordinates where the tangent line to $f(x)=2 x^{3}+8 x^{2}-32 x+4$ is horizontal.

See Solution

Problem 23248

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

See Solution

Problem 23249

A rock creates a ripple with radius increasing at $3 \mathrm{ft} / \mathrm{sec}$ . Find area increase speed when radius is 2 ft.

See Solution

Problem 23250

Calculate the integral: $\int \tan^{3}(4x) \sec^{2}(4x) \, dx$

See Solution

Problem 23251

Graph $y=\ln(14-x^{2})$ . Find domain, symmetries, derivatives $y'$ , $y''$ , critical points, behavior, concavity, and extreme points. Domain: $\square$ .

See Solution

Problem 23252

Calculate the integral $I=\int_{0}^{\pi / 5} 2 \sin (5 t) d t$ .

See Solution

Problem 23253

Find functions $g$ where $g^{\prime}(x)=\frac{2 x^{2}+4 x+5}{\sqrt{x}}$ .

See Solution

Problem 23254

Evaluate the integral I = ∫(2 + 6y - y²) dy from 0 to 3.

See Solution

Problem 23255

Calculate the integral $I=\int_{0}^{\pi / 4} \frac{3+\cos ^{2} \theta}{\cos ^{2} \theta} d \theta$ .

See Solution

Problem 23256

Calculate the average rate of change of temperature $T(x)=70+\frac{3}{2} x^{2}$ from 2:00 P.M. to 4:00 P.M. and interpret it.

See Solution

Problem 23257

Find the average rate of change of $g(x)=2 x^{3}-3 x^{2}-2$ between $x=1$ and $x=2$ . Simplify your answer.

See Solution

Problem 23258

Find the intervals where the derivative of the function $f(x)=x^{3}-18 x^{2}+81 x$ is decreasing.

See Solution

Problem 23259

Differentiate $\frac{1}{x^{3}}$ using the first principle.

See Solution

Problem 23260

Traffic harms kids. If $\mathrm{CO}$ decays at $5.5\%$ per meter, find distance for $\mathrm{CO}$ to be a third of 5000 ppm. Also, calculate average $\mathrm{CO}$ from road to 10 meters.

See Solution

Problem 23261

Estimate the area under $f(x)=\frac{9}{x}$ from $x=5$ to $x=9$ using 2 and then 4 rectangles.

See Solution

Problem 23262

Evaluate the triple integral $\int_{2}^{3} \int_{-1}^{4} \int_{1}^{0} (4 x^{2} y - z^{3}) \, dz \, dy \, dx$ .

See Solution

Problem 23263

Gegeben ist die Funktion $f(x)=x^{3}-6 x^{2}+11 x-6$ .
a) Zeigen Sie, dass der Wendepunkt auf der Geraden $y=x-2$ liegt.
b) Nach der Verschiebung hat der Punkt (2/0) die Koordinaten (3/2). Bestimmen Sie die Gleichung der Funktion $h$ .

See Solution

Problem 23264

Bestimmen Sie die Ableitung von $f$ an $x_{0}=2$ für die Funktionen: $f(x)=x^{2}$ , $f(x)=\frac{2}{x}$ , $f(x)=2x^{2}-3$ , $f(x)=x^{4}$ , $f(x)=x^{3}$ , $f(x)=4x-x^{2}$ , $f(x)=-\frac{1}{x}$ , $f(x)=5$ .

See Solution

Problem 23265

Berechnen Sie $\mathrm{f}^{\prime}$ mit der Produktregel für die folgenden Funktionen: a) $t(x)=x^{4} \cdot x^{5}$ b) $f(x)=(2 x^{2}) \cdot(3 x^{4})$ c) $f(x)=(x^{3}+x^{2}) \cdot(x^{2}+x)$ d) $f(x)=\sqrt{x} \cdot \sqrt{x}, x>0$ e) $f(x)=x^{3}, \frac{1}{x}, x \neq 0$ f) $f(x)=(a x^{3}+b x^{2}) \cdot \frac{1}{x^{2}}, x \neq 0$

See Solution

Problem 23266

Given the function $f$ with inflection points at $x=-6, -3.4, 1, 7.2$ , find the relationship between $f(4)$ , $f'(4)$ , and $f''(4)$ .

See Solution

Problem 23267

Berechne die Ableitungsfunktion $W'(t)$ für $W(t)=-0,38 t^{3}+9,12 t^{2}+9,6$ im Bereich $0<t<16$ .

See Solution

Problem 23268

Berechnen Sie die folgenden Integrale: a) $\int_{0}^{2}(3 x^{2}-e^{x}) d x$ b) $\int_{1}^{4}(\sqrt{x}-1) d x$ c) $\int_{0}^{9} \frac{2}{5} \sqrt{x} d x$ d) $\int_{0}^{\pi}(\sin (x)+\cos (x)) d x$ e) $\int_{0}^{2}(2+x)^{3} d x$ f) $\int_{2}^{3}(1+\frac{1}{x^{2}}) d x$ g) $\int_{1}^{5} \frac{3}{x} d x$ h) $\int_{1}^{e}(\frac{1}{x}+\frac{1}{x^{2}}) d x$

See Solution

Problem 23269

Bestimme die allgemeine Formel für die Geschwindigkeit $v(t)$ des Balls mit $h(t)=h_{0}+v_{0} t-5 t^{2}$ .

See Solution

Problem 23270

Finde die allgemeine Stammfunktion von $f(x)=9 x^{7}$ .

See Solution

Problem 23271

Bestimme die allgemeine Stammfunktion von $f(x)=9x^2$ .

See Solution

Problem 23272

Finde die allgemeine Stammfunktion von $f(x)=\frac{4}{3} x^{8}$ .

See Solution

Problem 23273

Ein Ball wird mit $h(t)=1,5+5t-5t^{2}$ geworfen. Berechne $v(t)$ nach 2s, wann $v(t)$ halbiert ist und wann Höhe max. ist.

See Solution

Problem 23274

Bestimme die Ableitungsfunktion $W'(t)$ von $W(t)=-0,38 t^{3}+9,12 t^{2}+9,6$ für $0<t<16$ und erkläre ihre Bedeutung.

See Solution

Problem 23275

Vergleiche die Grenzkosten $K'(x)$ mit den Stückkosten $\frac{K(x)}{x}$ für $K(x)=0,02 x^{3}-0,6 x^{2}+6 x$ .

See Solution

Problem 23276

Bestimme die allgemeine Stammfunktion von $f(x)=16 x^{7}$ .

See Solution

Problem 23277

Finde die Ableitung von $f(x)=\frac{1}{2} x^{2}-2 x+2$ .

See Solution

Problem 23278

Berechne die Partialsumme $S_{N}=\sum_{n=1}^{N}\left(\left(1+\frac{1}{n}\right)^{n}-\left(1+\frac{1}{n+1}\right)^{n+1}\right)$ .

See Solution

Problem 23279

Find the value of $k$ such that $\int_{k}^{5} \frac{1}{x^{2}} d x=\frac{1}{10}$ .

See Solution

Problem 23280

Bestimmen Sie den Wert der Reihe $\sum_{n=1}^{\infty}\left(\left(1+\frac{1}{n}\right)^{n}-\left(1+\frac{1}{n+1}\right)^{n+1}\right)$ .

See Solution

Problem 23281

Bestimme die allgemeine Stammfunktion von $f(x)=\frac{4}{9} x^{4}$ .

See Solution

Problem 23282

Bestimme für welche $q \in \mathbb{R}$ die Reihe $\sum_{n=3}^{\infty}(q^{n+1}-q^{n})$ konvergiert.

See Solution

Problem 23283

Bestimme die Ableitung von $f(x)=3x-\frac{1}{3}x^{3}$ .

See Solution

Problem 23284

Bestimme die Ableitungsfunktion von $f(x)=\frac{1}{2} x^{3}-2 x$ .

See Solution

Problem 23285

Untersuchen Sie die Funktion $f(x)=\ln (x-2)$ für $x>2$ . Zeigen Sie, dass der Flächeninhalt von $D_{2}$ doppelt so groß ist wie der von $D_{1}$ .

See Solution

Problem 23286

Finde eine Stammfunktion für: (I) $f(x)=\frac{4 x}{x^{2}-4}$ und (II) $g(x)=e^{2 x}+e$ .

See Solution

Problem 23287

Funksjonen $f(x)=\ln \left(x^{2}+\frac{1}{4}\right)$ : a) Finn definisjonsmengden. b) Deriver og finn ekstremalpunkter. c) Finn vendepunkt.

See Solution

Problem 23288

A stone is thrown from a $16 \mathrm{~m}$ cliff at $4 \mathrm{~m/s}$ . How far from the base does it land?

See Solution

Problem 23289

Gegeben ist die Produktionsfunktion $x(v)=-0,4 v^{3}+18 v^{2}+24 v$ mit $v \leq 25$ .
a) Finde den Input $v$ , bei dem die Grenzproduktivität maximal ist. b) Beweise, dass im Bereich kein Produktionsmaximum existiert. c) Bestimme den Input $v$ , bei dem der Durchschnittsertrag maximal ist. d) Finde den Input $v$ , bei dem Grenzproduktivität und Durchschnittsertrag gleich sind.

See Solution

Problem 23290

Untersuchen Sie die Monotonie der Funktionen: a) $f(x)=x^{2}+8 x-5$ b) $f(x)=\frac{5}{6} x^{3}+5 x^{2}-2$ c) $f(x)=\frac{1}{4} x^{4}-8 x$ d) $f(x)=\frac{1}{5} x^{5}-\frac{1}{4} x^{4}$ e) $f(x)=\frac{1}{3} x^{3}+2 x^{2}+4 x$ f) $f(x)=\frac{1}{5} x^{5}-\frac{13}{3} x^{3}+36 x$

See Solution

Problem 23291

Find the derivative of $f(x)=\frac{\cos(2x)}{x^{2}}$ .

See Solution

Problem 23292

Find the derivative of $f(x)=\sqrt{3+\tan ^{2} x}$ .

See Solution

Problem 23293

Find the derivative of $f(x)=\frac{-1}{\tan x}$ . Choose the correct option for $f^{\prime}(x)$ .

See Solution

Problem 23294

Find the derivative of $f(x)=x \sin x+\cos x$ .

See Solution

Problem 23295

A sandbag is dropped from a balloon at $340 \mathrm{~m}$ height, ascending at $12 \mathrm{~m/s}$ .
a) Find the max height of the sandbag. b) Determine its position and velocity after $5.0 \mathrm{~s}$ . c) How long to reach the ground after being dropped?

See Solution

Problem 23296

Alice throws a rock at $6.5 \mathrm{~m/s}$ from a $27.5 \mathrm{~m}$ bridge. What is its speed when it hits the river?

See Solution

Problem 23297

Bestimmen Sie die Ableitung von $f(x)=(5 x-2) \cdot e^{-x}$ .

See Solution

Problem 23298

Prove that the sequence $a_{n}=(-1)^{n}\left(\frac{n+\cos n \pi}{2 n}\right)$ is not convergent.

See Solution

Problem 23299

Find non-constant sequences $a_n$ and $b_n$ such that: (a) $\lim_{n \to \infty} a_n = 3$ , $b_n \neq 0$ , $\lim_{n \to \infty} (a_n b_n^2) = 0$ . (b) $\lim_{n \to \infty} a_n = 5$ , $|b_n| \neq |a_n|$ , $\lim_{n \to \infty} (a_n + (-1)^n b_n) = 0$ .

See Solution

Problem 23300

Find the average rate of change of $g(x)=-x^{2}+2x+3$ between $x=3$ and $x=8$ . Simplify your answer.

See Solution

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Calculus

Problem 23201

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=x2−6xf(x)=\sqrt{x^{2}-6x}f(x)=x2−6x​.

Problem 23202

A researcher models antibody production as N′(t)=4,00020t+10N^{\prime}(t)=\frac{4,000}{20 t+10}N′(t)=20t+104,000​. Find N(t)N(t)N(t) and N(8)N(8)N(8) in thousand antibodies.

Problem 23203

Find the xxx-coordinates where the tangent line to f(x)=2x3+4x2−8x+3f(x)=2 x^{3}+4 x^{2}-8 x+3f(x)=2x3+4x2−8x+3 is horizontal.

Problem 23204

Find the derivative of g(x) = 3x \cos(x) - 2.

Problem 23205

A rock creates a ripple in a pond. If the radius grows at 5ft/sec5 \mathrm{ft} / \mathrm{sec}5ft/sec, find the area increase speed at 2 ft radius.

Problem 23206

Calculate the indefinite integral: ∫ex9x8dx\int e^{x^{9}} x^{8} d x∫ex9x8dx using the substitution u=x9u=x^{9}u=x9.

Problem 23207

Calculate the integral using substitution: u=x2+1u=x^{2}+1u=x2+1. Find ∫(x2+1)6xdx\int\left(x^{2}+1\right)^{6} x d x∫(x2+1)6xdx.

Problem 23208

Find the maximum value of y=23x3−12x2−3x+1y=\frac{2}{3} x^{3}-\frac{1}{2} x^{2}-3 x+1y=32​x3−21​x2−3x+1 on [0,2][0,2][0,2].

Problem 23209

Find all functions ggg where g′(x)=4x2+x+3xg^{\prime}(x)=\frac{4 x^{2}+x+3}{\sqrt{x}}g′(x)=x​4x2+x+3​.

Problem 23210

Find the consumers' surplus for the demand function d(x)=700−12xd(x)=700-\frac{1}{2} xd(x)=700−21​x at demand level x=400x=400x=400.

Problem 23211

Calculate the integral I=∫0π/44+3cos⁡2θcos⁡2θdθI=\int_{0}^{\pi / 4} \frac{4+3 \cos ^{2} \theta}{\cos ^{2} \theta} d \thetaI=∫0π/4​cos2θ4+3cos2θ​dθ.

Problem 23212

Evaluate the integral I = ∫(4 + 2y - y²) dy from 0 to 2.

Problem 23213

Find the consumers' surplus for the demand function d(x)=100e−0.01xd(x) = 100 e^{-0.01 x}d(x)=100e−0.01x at x=300x = 300x=300. Round to the nearest cent.

Problem 23214

Calculate the integral I=∫0π/23sin⁡(2t)dtI=\int_{0}^{\pi / 2} 3 \sin (2 t) d tI=∫0π/2​3sin(2t)dt.

Problem 23215

Find the gradient of the curve y=x3−12xy=x^{3}-12 xy=x3−12x at (1,−11)(1,-11)(1,−11) and the coordinates of its turning points.

Problem 23216

Approximate 64.13\sqrt[3]{64.1}364.1​ using the tangent line of f(x)=x3f(x)=\sqrt[3]{x}f(x)=3x​ at x=64x=64x=64: find mmm and bbb for y=mx+by=m x+by=mx+b.

Problem 23217

Approximate 10.252\frac{1}{0.252}0.2521​ using the tangent line of f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ at a nearby point.

Problem 23218

Approximate 16.1\sqrt{16.1}16.1​ using the tangent line of f(x)=xf(x)=\sqrt{x}f(x)=x​ at x=16x=16x=16. Find L(x)=L(x)=L(x)=. Provide 9 significant figures.

Problem 23219

Find the interval where the Intermediate Value Theorem ensures a root for f(x)=x3+x−3f(x)=x^{3}+x-3f(x)=x3+x−3. Options: (A) (−1,0)(-1,0)(−1,0) (B) (0,1)(0,1)(0,1) (C) (1,2)(1,2)(1,2) (D) (2,3)(2,3)(2,3).

Problem 23220

Find the acceleration of a particle with velocity v(t)=−(t+1)sin⁡(t22)v(t)=-(t+1) \sin \left(\frac{t^{2}}{2}\right)v(t)=−(t+1)sin(2t2​) at t=2t=2t=2. Is its speed increasing?

Problem 23221

Find the limit: lim⁡x→2−9x−2\lim _{x \rightarrow 2^{-}} \frac{9}{x-2}limx→2−​x−29​. Choose from (A) −∞-\infty−∞, (B) 0, (C) 9, (D) does not exist.

Problem 23222

Find the interval where the Intermediate Value Theorem ensures a root for f(x)=x3+x−3f(x)=x^{3}+x-3f(x)=x3+x−3. Options: (A) (−1,0)(-1,0)(−1,0), (B) (0,1)(0,1)(0,1), (C) (1,2)(1,2)(1,2), (D) (2,3)(2,3)(2,3).

Problem 23223

A particle's velocity is v(t)=10sin⁡(0.4t2)t2−t+3v(t)=\frac{10 \sin \left(0.4 t^{2}\right)}{t^{2}-t+3}v(t)=t2−t+310sin(0.4t2)​ for 0≤t≤3.50 \leq t \leq 3.50≤t≤3.5. Find acceleration at t=3t=3t=3.

Problem 23224

Approximate 8.43\sqrt[3]{8.4}38.4​ using the tangent line of f(x)=x3f(x)=\sqrt[3]{x}f(x)=3x​ at x=8x=8x=8. Find mmm and bbb for y=mx+by=mx+by=mx+b.

Problem 23225

Given fff values for 3≤x≤93 \leq x \leq 93≤x≤9, which could be true: f′f'f′ negative/increasing, negative/decreasing, positive/increasing, or positive/decreasing?

Problem 23226

Approximate 16.2\sqrt{16.2}16.2​ using the tangent line of f(x)=xf(x)=\sqrt{x}f(x)=x​ at x=16x=16x=16. Find L(x)=L(x)=L(x)=. Provide 9 significant figures.

Problem 23227

Find the consumers' surplus for the demand function d(x)=350−0.09x2d(x) = 350 - 0.09 x^{2}d(x)=350−0.09x2 at x=50x = 50x=50.

Problem 23228

Find the xxx-coordinates of all relative maxima for the function f(x)=−2x3−6x2+18x−3f(x)=-2 x^{3}-6 x^{2}+18 x-3f(x)=−2x3−6x2+18x−3.

Problem 23229

Problem 23230

Find Δy\Delta yΔy for y=4x2y=4x^2y=4x2 at x=4x=4x=4 with Δx=0.4\Delta x=0.4Δx=0.4 and find dydydy with dx=0.4dx=0.4dx=0.4.

Problem 23231

Evaluate the integral ∫(2+sin⁡2θ)2cos⁡θdθ\int(2+\sin ^{2} \theta)^{2} \cos \theta d \theta∫(2+sin2θ)2cosθdθ using substitution.

Problem 23232

Find intervals where the derivative of the function f(x)=x4−16x3f(x)=x^{4}-16 x^{3}f(x)=x4−16x3 is decreasing.

Problem 23233

Find dyd ydy for y=tan⁡(4x+4)y=\tan(4x+4)y=tan(4x+4) at x=3x=3x=3 with dx=0.2d x=0.2dx=0.2 and dx=0.4d x=0.4dx=0.4.

Problem 23234

Given y=4xy=4 \sqrt{x}y=4x​, find Δy\Delta yΔy for x=5x=5x=5 and Δx=0.4\Delta x=0.4Δx=0.4. Also, find dyd ydy for x=5x=5x=5 and dx=0.4d x=0.4dx=0.4.

Problem 23235

Find the tangent line equation for f(x)=108x3+1083f(x)=\sqrt[3]{108 x^{3}+108}f(x)=3108x3+108​ at x=1x=1x=1 in mx+bmx+bmx+b form. Approximate f(1.1)f(1.1)f(1.1) and find the error.

Problem 23236

Find dyd ydy for y=tan⁡(2x+7)y=\tan(2x+7)y=tan(2x+7) at x=1x=1x=1 with dx=0.1d x=0.1dx=0.1 and dx=0.2d x=0.2dx=0.2.

Problem 23237

What is the area between the curve y=x2−4y=x^{2}-4y=x2−4 and the xxx-axis from x=−2x=-2x=−2 to x=2x=2x=2?

Problem 23238

Approximate 64.13\sqrt[3]{64.1}364.1​ using the tangent line of f(x)=x3f(x)=\sqrt[3]{x}f(x)=3x​ at x=64x=64x=64. Find mmm and bbb for y=mx+by=mx+by=mx+b.

Problem 23239

Graph y=12(x3−3x+7)y=\frac{1}{2}(x^{3}-3x+7)y=21​(x3−3x+7): find domain, symmetries, derivatives, critical points, inflection points, and extremes. Domain: □\square□.

Problem 23240

Find the tangent line of f(x)=128x3+3843f(x)=\sqrt[3]{128 x^{3}+384}f(x)=3128x3+384​ at x=1x=1x=1 in the form L(x)=mx+bL(x)=mx+bL(x)=mx+b. Approximate f(1.1)f(1.1)f(1.1) and compute the error ∣error∣≈|error| \approx∣error∣≈ (to 5 decimal places).

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\sqrt{x^{2}-6x}$ .

A researcher models antibody production as $N^{\prime}(t)=\frac{4,000}{20 t+10}$ . Find $N(t)$ and $N(8)$ in thousand antibodies.

Find the $x$ -coordinates where the tangent line to $f(x)=2 x^{3}+4 x^{2}-8 x+3$ is horizontal.

A rock creates a ripple in a pond. If the radius grows at $5 \mathrm{ft} / \mathrm{sec}$ , find the area increase speed at 2 ft radius.

Calculate the indefinite integral: $\int e^{x^{9}} x^{8} d x$ using the substitution $u=x^{9}$ .

Calculate the integral using substitution: $u=x^{2}+1$ . Find $\int\left(x^{2}+1\right)^{6} x d x$ .

Find the maximum value of $y=\frac{2}{3} x^{3}-\frac{1}{2} x^{2}-3 x+1$ on $[0,2]$ .

Find all functions $g$ where $g^{\prime}(x)=\frac{4 x^{2}+x+3}{\sqrt{x}}$ .

Find the consumers' surplus for the demand function $d(x)=700-\frac{1}{2} x$ at demand level $x=400$ .

Calculate the integral $I=\int_{0}^{\pi / 4} \frac{4+3 \cos ^{2} \theta}{\cos ^{2} \theta} d \theta$ .

Find the consumers' surplus for the demand function $d(x) = 100 e^{-0.01 x}$ at $x = 300$ . Round to the nearest cent.

Calculate the integral $I=\int_{0}^{\pi / 2} 3 \sin (2 t) d t$ .

Find the gradient of the curve $y=x^{3}-12 x$ at $(1,-11)$ and the coordinates of its turning points.

Approximate $\sqrt[3]{64.1}$ using the tangent line of $f(x)=\sqrt[3]{x}$ at $x=64$ : find $m$ and $b$ for $y=m x+b$ .

Approximate $\frac{1}{0.252}$ using the tangent line of $f(x)=\frac{1}{x}$ at a nearby point.

Approximate $\sqrt{16.1}$ using the tangent line of $f(x)=\sqrt{x}$ at $x=16$ . Find $L(x)=$ . Provide 9 significant figures.

Find the interval where the Intermediate Value Theorem ensures a root for $f(x)=x^{3}+x-3$ . Options: (A) $(-1,0)$ (B) $(0,1)$ (C) $(1,2)$ (D) $(2,3)$ .

Find the acceleration of a particle with velocity $v(t)=-(t+1) \sin \left(\frac{t^{2}}{2}\right)$ at $t=2$ . Is its speed increasing?

Find the limit: $\lim _{x \rightarrow 2^{-}} \frac{9}{x-2}$ . Choose from (A) $-\infty$ , (B) 0, (C) 9, (D) does not exist.

Find the interval where the Intermediate Value Theorem ensures a root for $f(x)=x^{3}+x-3$ . Options: (A) $(-1,0)$ , (B) $(0,1)$ , (C) $(1,2)$ , (D) $(2,3)$ .

A particle's velocity is $v(t)=\frac{10 \sin \left(0.4 t^{2}\right)}{t^{2}-t+3}$ for $0 \leq t \leq 3.5$ . Find acceleration at $t=3$ .

Approximate $\sqrt[3]{8.4}$ using the tangent line of $f(x)=\sqrt[3]{x}$ at $x=8$ . Find $m$ and $b$ for $y=mx+b$ .

Given $f$ values for $3 \leq x \leq 9$ , which could be true: $f'$ negative/increasing, negative/decreasing, positive/increasing, or positive/decreasing?

Approximate $\sqrt{16.2}$ using the tangent line of $f(x)=\sqrt{x}$ at $x=16$ . Find $L(x)=$ . Provide 9 significant figures.

Find the consumers' surplus for the demand function $d(x) = 350 - 0.09 x^{2}$ at $x = 50$ .

Find the $x$ -coordinates of all relative maxima for the function $f(x)=-2 x^{3}-6 x^{2}+18 x-3$ .

Find $\Delta y$ for $y=4x^2$ at $x=4$ with $\Delta x=0.4$ and find $dy$ with $dx=0.4$ .

Evaluate the integral $\int(2+\sin ^{2} \theta)^{2} \cos \theta d \theta$ using substitution.

Find intervals where the derivative of the function $f(x)=x^{4}-16 x^{3}$ is decreasing.

Find $d y$ for $y=\tan(4x+4)$ at $x=3$ with $d x=0.2$ and $d x=0.4$ .

Given $y=4 \sqrt{x}$ , find $\Delta y$ for $x=5$ and $\Delta x=0.4$ . Also, find $d y$ for $x=5$ and $d x=0.4$ .

Find the tangent line equation for $f(x)=\sqrt[3]{108 x^{3}+108}$ at $x=1$ in $mx+b$ form. Approximate $f(1.1)$ and find the error.

Find $d y$ for $y=\tan(2x+7)$ at $x=1$ with $d x=0.1$ and $d x=0.2$ .

What is the area between the curve $y=x^{2}-4$ and the $x$ -axis from $x=-2$ to $x=2$ ?

Approximate $\sqrt[3]{64.1}$ using the tangent line of $f(x)=\sqrt[3]{x}$ at $x=64$ . Find $m$ and $b$ for $y=mx+b$ .

Graph $y=\frac{1}{2}(x^{3}-3x+7)$ : find domain, symmetries, derivatives, critical points, inflection points, and extremes. Domain: $\square$ .

Find the tangent line of $f(x)=\sqrt[3]{128 x^{3}+384}$ at $x=1$ in the form $L(x)=mx+b$ . Approximate $f(1.1)$ and compute the error $|error| \approx$ (to 5 decimal places).

Find intervals where the function $f(x)=\frac{1}{2} x^{4}+6 x^{3}-48 x^{2}$ is concave down.

Calculate the integral: $\int \sin^{5}(2t) \cos^{2}(2t) \, dt$

Graph the function $y=\frac{1}{2}(x^{3}-3x+7)$ , find derivatives, critical points, concavity, and extreme points. Identify intervals of concavity.

Evaluate the integral: $\int \frac{\ln \left(y^{2}\right)}{3 y} d y$ .

Graph $y=\frac{1}{2}(x^{3}-3x+7)$ : find domain, symmetries, $y'$ , $y''$ , critical points, increasing/decreasing intervals, inflection points, asymptotes, and extreme points. Identify vertical asymptotes.

Calculate the integral: $\int \frac{\sin (4 x)}{(2+\cos (4 x))^{2}} d x$

Find the $x$ -coordinates where the tangent line to $f(x)=2 x^{3}+8 x^{2}-32 x+4$ is horizontal.

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

A rock creates a ripple with radius increasing at $3 \mathrm{ft} / \mathrm{sec}$ . Find area increase speed when radius is 2 ft.

Calculate the integral: $\int \tan^{3}(4x) \sec^{2}(4x) \, dx$

Graph $y=\ln(14-x^{2})$ . Find domain, symmetries, derivatives $y'$ , $y''$ , critical points, behavior, concavity, and extreme points. Domain: $\square$ .

Calculate the integral $I=\int_{0}^{\pi / 5} 2 \sin (5 t) d t$ .

Find functions $g$ where $g^{\prime}(x)=\frac{2 x^{2}+4 x+5}{\sqrt{x}}$ .

Calculate the integral $I=\int_{0}^{\pi / 4} \frac{3+\cos ^{2} \theta}{\cos ^{2} \theta} d \theta$ .

Calculate the average rate of change of temperature $T(x)=70+\frac{3}{2} x^{2}$ from 2:00 P.M. to 4:00 P.M. and interpret it.

Find the average rate of change of $g(x)=2 x^{3}-3 x^{2}-2$ between $x=1$ and $x=2$ . Simplify your answer.

Find the intervals where the derivative of the function $f(x)=x^{3}-18 x^{2}+81 x$ is decreasing.

Differentiate $\frac{1}{x^{3}}$ using the first principle.

Traffic harms kids. If $\mathrm{CO}$ decays at $5.5\%$ per meter, find distance for $\mathrm{CO}$ to be a third of 5000 ppm. Also, calculate average $\mathrm{CO}$ from road to 10 meters.

Estimate the area under $f(x)=\frac{9}{x}$ from $x=5$ to $x=9$ using 2 and then 4 rectangles.

Evaluate the triple integral $\int_{2}^{3} \int_{-1}^{4} \int_{1}^{0} (4 x^{2} y - z^{3}) \, dz \, dy \, dx$ .

Gegeben ist die Funktion $f(x)=x^{3}-6 x^{2}+11 x-6$ .
a) Zeigen Sie, dass der Wendepunkt auf der Geraden $y=x-2$ liegt.
b) Nach der Verschiebung hat der Punkt (2/0) die Koordinaten (3/2). Bestimmen Sie die Gleichung der Funktion $h$ .

Given the function $f$ with inflection points at $x=-6, -3.4, 1, 7.2$ , find the relationship between $f(4)$ , $f'(4)$ , and $f''(4)$ .

Berechne die Ableitungsfunktion $W'(t)$ für $W(t)=-0,38 t^{3}+9,12 t^{2}+9,6$ im Bereich $0<t<16$ .

Bestimme die allgemeine Formel für die Geschwindigkeit $v(t)$ des Balls mit $h(t)=h_{0}+v_{0} t-5 t^{2}$ .

Finde die allgemeine Stammfunktion von $f(x)=9 x^{7}$ .

Bestimme die allgemeine Stammfunktion von $f(x)=9x^2$ .

Finde die allgemeine Stammfunktion von $f(x)=\frac{4}{3} x^{8}$ .

Ein Ball wird mit $h(t)=1,5+5t-5t^{2}$ geworfen. Berechne $v(t)$ nach 2s, wann $v(t)$ halbiert ist und wann Höhe max. ist.

Bestimme die Ableitungsfunktion $W'(t)$ von $W(t)=-0,38 t^{3}+9,12 t^{2}+9,6$ für $0<t<16$ und erkläre ihre Bedeutung.