Calculus

Problem 27201

Approximate the area under $y=x^{4}-4 x^{3}+8 x+18$ from $x=-2$ to $x=4$ using the Mid-Ordinate rule with step size $2$ .

See Solution

Problem 27202

Approximate the area under $y = x^4 - 4x^3 + 8x + 18$ from $x = -2$ to $x = 4$ using Simpson's rule with step size 1.

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Problem 27203

Integrate $f(x) = x^4 - 4x^3 + 8x + 18$ to find the area from $x = -2$ to $x = 4$ . What is the area?

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Problem 27204

Calculate the average of the function $f(x) = x^4 - 4x^3 + 8x + 18$ from $x = -2$ to $x = 4$ .

See Solution

Problem 27205

Find the value of $x$ for the curve $x^{3}+xy-y^{2}=10$ where there is a vertical tangent line.

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Problem 27206

Is the rate of change of the volume of a sphere, $V$ , constant if the radius $r$ changes at a constant rate, $\frac{d r}{d t}$ ?

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Problem 27207

An astronaut drops a rock on the Moon. Find the average speed, in m/s, of the rock between 5 and 10 seconds using $d(t)=0.8 t^{2}$ .

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Problem 27208

Is $\frac{d V}{r}$ constant if $\frac{d r}{r}$ is constant for a sphere's volume $V$ with radius $r$ ? Explain.

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Problem 27209

Gegeben ist die Funktion $f_{2}(x)=\frac{1}{4} x^{3}-a x^{2}+a^{2} x$ mit $a \in \mathbb{R}^{+}$ .
a) Finde die Nullstellen von $f$ in Abhängigkeit von $a$ und bewerte sie. b) Bestimme die Extremwerte und den Wendepunkt W in Abhängigkeit von $a$ . c) Berechne $a$ , sodass der Flächeninhalt A = 27 FE ist. d) Zeichne $G_{3}$ für $a=3$ im Intervall $[-0,5 ; 7]$ .

See Solution

Problem 27210

Find the derivative $f^{\prime}(x)$ of $f(x)=-2 x^{4}+3 x^{-2}+5 x-4$ . Which option is correct? A, B, C, or D?

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Problem 27211

Find the derivative $y^{\prime}$ of the function $y=\sqrt{1-2x}$ .

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Problem 27212

Graph $f(x)=\left\{\begin{array}{ll} 5-x, & x \neq 2 \\ -3, & x=2 \end{array}\right.$ , find $\lim_{x \rightarrow 2} f(x)$ , $f(2)$ , and check continuity at 2.

See Solution

Problem 27213

Find the volume of the solid with base area $\mathrm{R}$ under $f(x)=3x^3$ , $y=1$ , and the $y$ -axis, with semicircular cross sections.

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Problem 27214

Find the inflection point of the function $f(x)=x^{3}-3 x$ . Choose from: A. $(0,-1)$ B. $(0,0)$ C. $(0,1)$ D. $(0,2)$ .

See Solution

Problem 27215

The function $f(x)=|x+1|$ is not differentiable at which point: A. $x=0$ , B. $x=-1$ , C. $x=1$ , D. $x=2$ ?

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Problem 27216

Find the marginal profit for production of 20 units given demand $p(x)=3x+5x^2$ and average cost $\bar{C}(x)=5+4x$ .

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Problem 27217

Find the function with the instantaneous rate of change near 5 at $x=2$ : a) $f(x)=x^{2}$ , b) $h(x)=x^{2}+0.5 x+1$ , c) $j(x)=1.1^{x}$ , d) $g(x)=x^{2}+1$ . Also, estimate the rate of change for $g(x)=x^{2}-2 x+5$ at $x=3$ .

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Problem 27218

Find the displacement and total distance traveled by a particle with velocity $v(t)=5t-9$ for $0 \leq t \leq 3$ .

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Problem 27219

Find the marginal cost of production when the average cost is $\bar{C}(x)=-2 x^{2}+4+\frac{100}{x}$ at 50 units.

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Problem 27220

Find where the function $f(x)=|x-1|$ is not differentiable: A. $x=0$ , B. $x=-1$ , C. $x=1$ , D. $x=2$ .

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Problem 27221

Find the instantaneous rate of change of $f(x)=130000(1.06)^{x}$ after 5 years. Choices: a) \$8000 b) \$10000 c) \$12000 d) \$14000

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Problem 27222

Find the integral $\int \frac{x^{5}}{x^{6}-3} d x$ using the substitution $u=x^{6}-3$ , and find $d u$ . Rewrite and evaluate the integral.

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Problem 27223

Find the integral $\int \cos(8x) \, dx$ using the substitution $u=8x$ . Determine $du$ and rewrite the integral as $\int(\square) \, du$ . Evaluate it with constant $C$ .

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Problem 27224

Find the average rate of change in height from $t=4$ s to $t=5$ s for $h(t)=-4.9 t^{2}+150$ . Options: a) $24.5$ m/s, b) $-19.6$ m/s, c) $-44.1$ m/s, d) $-24.5$ m/s.

See Solution

Problem 27225

Find the value of $x$ where the function's instantaneous rate of change is negative. Also, estimate the value change rate at $t=5$ for $v(t) = 24000(0.95)^t$ .

See Solution

Problem 27226

Find the inflection point of the function $f(x)=x^{3}-3 x^{2}+1$ . Choose from: A. $(1,-1)$ B. $(1,1)$ C. $(-1,0)$ D. $(1,0)$

See Solution

Problem 27227

Rewrite the integral $\int t^{4} e^{-t^{5}} d t$ using substitution to express it as $-\frac{1}{5} e^{u} d u$ . Find $u$ and $d u$ , then evaluate the integral with constant $C$ .

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Problem 27228

Calculate the total oil leaked in the first hour from the rate $r(t)=65 e^{-0.02 t}$ liters/minute. Round to the nearest liter.

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Problem 27229

A drug with a half-life of 67 min starts at $1.7 \mu \mathrm{g} / \mathrm{mL}$ . What is its concentration after 268 min? Round to 2 sig. figs. $\square \frac{\mu \mathrm{g}}{\mathrm{mL}}$ $\square \times 10^{\square}$

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Problem 27230

A drug with a half-life of 73 min starts at $1.4 \mu \mathrm{g} / \mathrm{mL}$ . What is the concentration after 146 min? Round to 2 sig. digits. $\square \frac{\mu \mathrm{g}}{\mathrm{mL}}$

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Problem 27231

A drug with a half-life of 82 min starts at $0.30 \mu \mathrm{g} / \mathrm{mL}$ . What is its concentration after 246 min? Round to 2 sig. figs. $\square \frac{\mu \mathrm{g}}{\mathrm{mL}}$

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Problem 27232

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-x(x+1)^{2}$ . Choose from the options provided.

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Problem 27233

Find the derivative of $f(x)=2^{(\cos x)}$ . What is $f^{\prime}(x)$ ? A. $2^{(\cos x)}(\sin x) \ln 2$ B. $-2^{(\cos x)}(\sin x) \ln 2$ C. $2^{(\cos x)}(\sin x)$ D. $-2^{(\cos x)}(\sin x)$

See Solution

Problem 27234

A cuboid with a square base has height $2s$ . Volume increases at $24 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ . Find surface area change rate when height is $12 \mathrm{~cm}$ .

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Problem 27235

Find the value of $\frac{f(3+h)-f(3)}{h}$ for the function $f(x)=21-5x$ .

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Problem 27236

Estimate and classify the critical points of the function $f(x) = \sin(x)$ .

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Problem 27237

Calculate the average rate of change for $f(x)=\log _{7} 8 x$ from $x=3$ to $x=9$ . Answer as an integer or decimal to three decimal places.

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Problem 27238

Find the average rate of the function $f(x)=e^{x-2}-4$ between $x=2$ and $x=4$ .

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Problem 27239

Find the average velocity of an object dropped from rest on Planet $X$ over the interval $t=6$ to $t=9$ using $d(t)=20 t^{2}$ .

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Problem 27240

Calculate the average rate of change of $f(x)=\log _{4} 8 x$ from $x=4$ to $x=9$ . Answer as an integer or decimal to three places.

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Problem 27241

Find the integral of the function: $\int \frac{4+2 x}{x^{3}} d x$ .

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Problem 27242

Find the sum: $\sum_{n=1}^{\infty} n^{k}\left(\frac{n}{n+1}\right)^{n^{2}}$ .

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Problem 27243

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

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Problem 27244

Evaluate the series: $\sum_{n=1}^{\infty} \frac{(n!)^{3} \cdot 3^{n}}{(3n)!} \cdot \sin(n)$ .

See Solution

Problem 27245

Find the limit: $\lim _{x \rightarrow 0} \frac{x \sin (x)}{1-\cos (x)}$

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Problem 27246

Find the normalization constant $N$ , expectation values of $L$ , $L^{2}$ , and their variances for $\psi=N e^{-\alpha r^{2}}(x+y) z$ .

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Problem 27247

Find the slope and tangent line equation at point $P$ for (1) $y=x^{2}-4x$ , $P:(1,-3)$ and (2) $y=2-x^{3}$ , $P:(1,1)$ .

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Problem 27248

Berechnen Sie den mittleren Anstieg der Funktion $p(t)=-2 t^{3}+9 t^{2}+15 t+75$ und bestimmen Sie, wann die Änderungsrate maximal ist.

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Problem 27249

Fruchtfliegen wachsen um $35\%$ täglich. Gegeben sind 20 Fliegen. a) Funktionsgleichung aufstellen. b) Fliegenzahl nach 7 Tagen berechnen. c) Zeit bis 2000 Fliegen berechnen.

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Problem 27250

Find the average rate of change of $h(t)=\cot t$ over these intervals: a. $[\pi / 4,3 \pi / 4]$ b. $[\pi / 6, \pi / 2]$

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Problem 27251

Prove that if $y=f(u)$ and $u=g(x)$ , then $\frac{d y}{d x}=\frac{d y}{d u} \times \frac{d u}{d x}$ .

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Problem 27252

Which integral finds the length of the graph $y=\sin (\sqrt{x})$ from $x=a$ to $x=b$ where $0<a<b$ ?
Select one: a. $\int_{a}^{b} \sqrt{x+\cos ^{2}(\sqrt{x})} d x$ b. $\int_{a}^{b} \sqrt{\frac{1+\cos ^{2}(\sqrt{x})}{4 x}} d x$ c. $\int_{a}^{b} \sqrt{\sin ^{2}(\sqrt{x})+\frac{1}{4 x} \cos ^{2}(\sqrt{x})} d x$ d. $\int_{a}^{b} \sqrt{1+\frac{1}{4 x} \cos ^{2}(\sqrt{x})} d x$

See Solution

Problem 27253

Berechne die Ableitung der Funktion $f(x)=-2x^{2}$ an der Stelle $x_{0}=3$ .

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Problem 27254

Finde $k<R$ , so dass $\int_{0}^{k} 2 x^{2} d x=18$ . Skizziere und erkläre die Fragestellung.

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Problem 27255

Bearbeiten Sie die Wasserzuflussgeschwindigkeit $v(t)=5t-t^{2}$ für $t=0$ bis $t=5$ . Berechnen Sie die Menge in 4 Stunden. Analysieren Sie die Gewinnfunktion $f(x)=-20x^{3}+240x^{2}-1200$ für 12 Monate.

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Problem 27256

Ein Lieferant möchte die Produktionskosten überprüfen. Gegeben ist die Kostenfunktion $K(x)=0,75 x^{3}+3,75 x^{2}+13,5 x+30$ . Aktuell produziert er 15 Produkte und erwartet eine $20 \%$ Steigerung.
a) Wie ändern sich die geschätzten Kosten? b) Wie hoch ist die exakte Kostenveränderung?
Die Erlösfunktion ist $E(x)=0,5 x^{3}+7,5 x^{2}+40 x$ .
c) Was ist der Erfolg bei keiner Produktion? d) Wo liegen die Gewinnschwellen? e) Wie groß ist die Fläche unter $K(x)$ von 1-3? f) Bei welcher Menge ist der Gewinn maximal und wie hoch ist der Gesamtgewinn?

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Problem 27257

Bestimme die Ableitung von $f(x)=0,5 x^{2}$ an der Stelle $x_{0}=4$ mit dem Differenzenquotienten.

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Problem 27258

Bestimme die Ableitung von $f(x)=0,5 x^{2}$ bei $x_{0}=4$ und die Ableitungsfunktion von $f(x)=4 x^{2}-5$ .

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Problem 27259

Bestimme die Ableitungsfunktion von $f(x)=4 x^{2}-5$ .

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Problem 27260

Bestimme die Ableitungen der Funktionen: a. $f(x)=7 x^{3}-2 x^{2}$ , b. $g(x)=3 x^{5}+5 x^{3}$ .

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Problem 27261

Finde den Wert von $a$ , sodass der Graph der Funktion $f_{a}(x) = \frac{1}{a} \cdot x^3 - x$ bei $x=3$ einen Extrempunkt hat.

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Problem 27262

Find the limit as $x$ approaches 1 from the left: $\lim _{x \rightarrow 1^{-}}\left(\frac{x-1}{2}-\frac{1}{\ln (x)}\right)$ .

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Problem 27263

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} x \tan ^{-1} \frac{3}{x}$ .

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Problem 27264

Find the $x$ -coordinates of relative extrema and inflection points for $f(x)$ using its derivatives from the table.

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Problem 27265

Evaluate the integral: $\int \frac{d x}{\sqrt{-x^{2}-4 x-3}}$

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Problem 27266

Calculate the integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \sec ^{2} x \, dx$ .

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Problem 27267

Find the particular solution to $\frac{d y}{d x}=x^{2}+1$ that passes through $(3,13)$ .

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Problem 27268

Find the derivative of the integral $\frac{d}{d x}\left[\int_{2}^{x} \frac{\sqrt{t}}{t^{2}+1} d t\right]$ .

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Problem 27269

Find the limit as $x$ approaches 0 for the function $\sqrt{x}$ .

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Problem 27270

Find the second derivative $y^{\prime \prime}$ of the function $y=e^{x} \cos x$ .

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Problem 27271

What does the table suggest about $\lim _{x \rightarrow 2^{-}} f(x)$ ? Options: 2, 6.5, 27, or does not exist?

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Problem 27272

Find the upper sum $S_{b}$ for $f(x)=x^{2}-x$ over $[0,10]$ with 5 subintervals. What is $S_{6}$ ? Options: 340, 420, 260, 500.

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Problem 27273

What does the table indicate about the derivative of $f(x)=\frac{1}{x}$ at $x=2$ ? Options: $f^{\prime}(2) \approx -0.2500$ , $f^{\prime}(2)=0$ , $f^{\prime}(2)=\frac{1}{2}$ , $f^{\prime}(2) \approx -0.2000$ .

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Problem 27274

Find the second derivative of $y=e^{x} \cos x$ . Choices: $-2 e^{x} \cos x$ , $-2 e^{x} \sin x$ , $2 e^{x} \sin x$ , $2 e^{x} \cos x$ .

See Solution

Problem 27275

Find the value of $x$ for which the function $f$ has a relative minimum given $f^{\prime}(x)=3 x^{2}-18 x+15$ .

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Problem 27276

Find the sum of the series: $\sum_{n=0}^{\infty} \left( \frac{1}{125} \right) \left( -5 \right)^n$

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Problem 27277

Find the value of $x$ for the relative minimum of $f$ given $f^{\prime}(x)=3 x^{2}-18 x+15$ . Options: 5, 0, 4, 2.

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Problem 27278

Find the derivative of the integral from 1 to x of $\ln t$ with respect to $x$ . What is the result?

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Problem 27279

Find the derivative of $f(x)=\ln (\sec x)$ . What is $f^{\prime}(x)$ ? Options: $\csc x$ , $\cot x$ , $\tan ^{2} x$ , $\tan x$ .

See Solution

Problem 27280

Find the upper sum $S_{5}$ for the area under $f(x)=x^{2}-x$ from $x=0$ to $x=10$ using 5 subintervals. Options: 420, 340, 500, 260.

See Solution

Problem 27281

Find the limit as $(x, y)$ approaches $(0, 0)$ for $\frac{x^{6}+y^{6}}{x^{2}-y^{2}}$ . Options: a. 0, b. 1, c. 2, d. 3, e. Does not exist.

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Problem 27282

(a) Use the tangent line at $x=1$ to estimate $s(1.1)$ . Is this an over or underestimate? (b) When is the particle at rest? (c) When is speed decreasing for $0<t<6$ ? Given: $s(t) = \frac{1}{3} t^{3} + 3 t^{2} - 8 t + 5$

See Solution

Problem 27283

A 70 kg skateboarder moves at 6 m/s over a hill with a radius of curvature of 9 m. Find:
1. Centripetal acceleration
2. Centripetal net force
3. Normal force

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Problem 27284

Andres wants \$4,700 in 11 years at a 4\% continuous interest rate. How much should he invest?

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Problem 27285

Michael wants to invest for an account to reach \ $100,000 in 14 years at a continuous interest rate of$ 6.7 \%$. How much does he need to invest?

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Problem 27286

Q1 Sketch level curves of $f(x)=4-2x-3y$ for values $-4, 0, 4, 8$ . Q2 Find the domain and sketch $f(x, y)=\ln(2x-3y+1)$ . Q3 Show $f(x, y)=x^{2}-y^{2}+2xy$ satisfies Laplace's equation. Q4 Sketch $f(x, y)=y^{2}$ in 3D. Q5 Find parametric equations for the tangent line at $(3,1,6)$ for $z=2y^{2}x$ with planes $x=3$ and $y=1$ . Q6 Find $dz$ as $(x,y)$ changes from $(2,-1)$ to $(2.01,-0.98)$ for $z=2y^{2}+3x^{2}$ . Q7 Use differentials to find the pressure change in $PV=8.314T$ as volume goes from $20L$ to $20.2L$ and $T$ from $300K$ to $295K$ . Q8 Find slopes of $f(x, y)=1-(x-1)^{2}-(y+2)^{3}$ at $(1,2,1)$ in $x$ and $y$ directions. Q9 Show $u(x, t)=10e^{-t}\sin x$ satisfies the heat equation with $k=1$ .

See Solution

Problem 27287

Compare the volume growth rates of a cube $V(x)=x^3$ and a sphere $V(x)=\frac{4}{3}\pi(0.5x)^3$ over $[1,2]$ .

See Solution

Problem 27288

Find the average rate of change of height $h(t)=-4.9 t^{2}+553$ from $t=4$ to $t=5$ seconds.

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Problem 27289

How long for a \$2500 account at 1\% continuous interest to grow to \$3000? Round to the nearest month.

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Problem 27290

A bowling ball drops from a building and hits the ground at $37.0 \mathrm{~m} / \mathrm{s}$ . Find the building's height.

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Problem 27291

A ring of radius $r = 1.0 \, \mathrm{m}$ has sections with charge densities $\lambda_{1} = +1.0 \, \mathrm{nC/m}$ and $\lambda_{2} = -2.2 \, \mathrm{nC/m}$ . Find the electric potential $V$ at point $P$ , $d = 75 \, \mathrm{cm}$ above the center. Options: a) -25 V, b) 9.0 V, c) 11 V, d) 15 V, e) 34 V.

See Solution

Problem 27292

Find the limit: $\lim _{x \rightarrow \infty} \frac{x-3}{x(x-1)}$ .

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Problem 27293

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x-3}{x(x-1)}$ .

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Problem 27294

Find the max initial velocity $v_{0}$ for a roller coaster at height $h=16 \mathrm{~m}$ and radius $R=26 \mathrm{~m}$ due to gravity. Answer in $\mathrm{m/s}$ .

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Problem 27295

Find values of $a$ and $b$ so that the piecewise function $g(x)$ is differentiable at $x=1$ .

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Problem 27296

Find the rate at which the water level rises when the water is $2 \mathrm{ft}$ deep in a conical tank that is $8 \mathrm{ft}$ high and $12 \mathrm{ft}$ wide, with water flowing in at $15 \mathrm{n}^{3} / \mathrm{min}$ .

See Solution

Problem 27297

Find the velocity of a glass piece falling 33.39 m in 5.34 seconds. Round your answer to the nearest hundredth.

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Problem 27298

Find the values of $A$ for which the function $f(x)=\left\{\begin{array}{l}x^{3}-2, x \leq 2 \\ 8 x+A, x>2\end{array}\right.$ is continuous at $x=2$ .

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Problem 27299

An observer 100 m away sees a balloon rise at 3 m/s.
a) Find the rate of change in distance when the balloon is 105 m high.
b) Find the rate of change of the angle of elevation in radians/sec at that height.

See Solution

Problem 27300

A cone has height $10 \mathrm{~cm}$ and diameter $10 \mathrm{~cm}$ . Water depth $h$ changes at $-\frac{3}{10} \mathrm{~cm/hr}$ .
a) Find volume $V$ when $h=5$ . Include units. b) Find rate of change of volume when $h=5 \mathrm{~cm}$ . Include units.

See Solution

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Calculus

Problem 27201

Approximate the area under y=x4−4x3+8x+18y=x^{4}-4 x^{3}+8 x+18y=x4−4x3+8x+18 from x=−2x=-2x=−2 to x=4x=4x=4 using the Mid-Ordinate rule with step size 222.

Problem 27202

Approximate the area under y=x4−4x3+8x+18y = x^4 - 4x^3 + 8x + 18y=x4−4x3+8x+18 from x=−2x = -2x=−2 to x=4x = 4x=4 using Simpson's rule with step size 1.

Problem 27203

Integrate f(x)=x4−4x3+8x+18f(x) = x^4 - 4x^3 + 8x + 18f(x)=x4−4x3+8x+18 to find the area from x=−2x = -2x=−2 to x=4x = 4x=4. What is the area?

Problem 27204

Calculate the average of the function f(x)=x4−4x3+8x+18f(x) = x^4 - 4x^3 + 8x + 18f(x)=x4−4x3+8x+18 from x=−2x = -2x=−2 to x=4x = 4x=4.

Problem 27205

Find the value of xxx for the curve x3+xy−y2=10x^{3}+xy-y^{2}=10x3+xy−y2=10 where there is a vertical tangent line.

Problem 27206

Is the rate of change of the volume of a sphere, VVV, constant if the radius rrr changes at a constant rate, drdt\frac{d r}{d t}dtdr​?

Problem 27207

An astronaut drops a rock on the Moon. Find the average speed, in m/s, of the rock between 5 and 10 seconds using d(t)=0.8t2d(t)=0.8 t^{2}d(t)=0.8t2.

Problem 27208

Is dVr\frac{d V}{r}rdV​ constant if drr\frac{d r}{r}rdr​ is constant for a sphere's volume VVV with radius rrr? Explain.

Problem 27209

Problem 27210

Find the derivative f′(x)f^{\prime}(x)f′(x) of f(x)=−2x4+3x−2+5x−4f(x)=-2 x^{4}+3 x^{-2}+5 x-4f(x)=−2x4+3x−2+5x−4. Which option is correct? A, B, C, or D?

Problem 27211

Find the derivative y′y^{\prime}y′ of the function y=1−2xy=\sqrt{1-2x}y=1−2x​.

Problem 27212

Graph f(x)={5−x,x≠2−3,x=2f(x)=\left\{\begin{array}{ll} 5-x, & x \neq 2 \\ -3, & x=2 \end{array}\right.f(x)={5−x,−3,​x=2x=2​, find lim⁡x→2f(x)\lim_{x \rightarrow 2} f(x)limx→2​f(x), f(2)f(2)f(2), and check continuity at 2.

Problem 27213

Find the volume of the solid with base area R\mathrm{R}R under f(x)=3x3f(x)=3x^3f(x)=3x3, y=1y=1y=1, and the yyy-axis, with semicircular cross sections.

Problem 27214

Find the inflection point of the function f(x)=x3−3xf(x)=x^{3}-3 xf(x)=x3−3x. Choose from: A. (0,−1)(0,-1)(0,−1) B. (0,0)(0,0)(0,0) C. (0,1)(0,1)(0,1) D. (0,2)(0,2)(0,2).

Problem 27215

The function f(x)=∣x+1∣f(x)=|x+1|f(x)=∣x+1∣ is not differentiable at which point: A. x=0x=0x=0, B. x=−1x=-1x=−1, C. x=1x=1x=1, D. x=2x=2x=2?

Problem 27216

Find the marginal profit for production of 20 units given demand p(x)=3x+5x2p(x)=3x+5x^2p(x)=3x+5x2 and average cost Cˉ(x)=5+4x\bar{C}(x)=5+4xCˉ(x)=5+4x.

Problem 27217

Problem 27218

Find the displacement and total distance traveled by a particle with velocity v(t)=5t−9v(t)=5t-9v(t)=5t−9 for 0≤t≤30 \leq t \leq 30≤t≤3.

Problem 27219

Find the marginal cost of production when the average cost is Cˉ(x)=−2x2+4+100x\bar{C}(x)=-2 x^{2}+4+\frac{100}{x}Cˉ(x)=−2x2+4+x100​ at 50 units.

Problem 27220

Find where the function f(x)=∣x−1∣f(x)=|x-1|f(x)=∣x−1∣ is not differentiable: A. x=0x=0x=0, B. x=−1x=-1x=−1, C. x=1x=1x=1, D. x=2x=2x=2.

Problem 27221

Find the instantaneous rate of change of f(x)=130000(1.06)xf(x)=130000(1.06)^{x}f(x)=130000(1.06)x after 5 years. Choices: a) \$8000 b) \$10000 c) \$12000 d) \$14000

Problem 27222

Find the integral ∫x5x6−3dx\int \frac{x^{5}}{x^{6}-3} d x∫x6−3x5​dx using the substitution u=x6−3u=x^{6}-3u=x6−3, and find dud udu. Rewrite and evaluate the integral.

Problem 27223

Find the integral ∫cos⁡(8x) dx\int \cos(8x) \, dx∫cos(8x)dx using the substitution u=8xu=8xu=8x. Determine dududu and rewrite the integral as ∫(□) du\int(\square) \, du∫(□)du. Evaluate it with constant CCC.

Problem 27224

Find the average rate of change in height from t=4t=4t=4 s to t=5t=5t=5 s for h(t)=−4.9t2+150h(t)=-4.9 t^{2}+150h(t)=−4.9t2+150. Options: a) 24.524.524.5 m/s, b) −19.6-19.6−19.6 m/s, c) −44.1-44.1−44.1 m/s, d) −24.5-24.5−24.5 m/s.

Problem 27225

Find the value of xxx where the function's instantaneous rate of change is negative. Also, estimate the value change rate at t=5t=5t=5 for v(t)=24000(0.95)tv(t) = 24000(0.95)^tv(t)=24000(0.95)t.

Problem 27226

Find the inflection point of the function f(x)=x3−3x2+1f(x)=x^{3}-3 x^{2}+1f(x)=x3−3x2+1. Choose from: A. (1,−1)(1,-1)(1,−1) B. (1,1)(1,1)(1,1) C. (−1,0)(-1,0)(−1,0) D. (1,0)(1,0)(1,0)

Problem 27227

Rewrite the integral ∫t4e−t5dt\int t^{4} e^{-t^{5}} d t∫t4e−t5dt using substitution to express it as −15eudu-\frac{1}{5} e^{u} d u−51​eudu. Find uuu and dud udu, then evaluate the integral with constant CCC.

Problem 27228

Calculate the total oil leaked in the first hour from the rate r(t)=65e−0.02tr(t)=65 e^{-0.02 t}r(t)=65e−0.02t liters/minute. Round to the nearest liter.

Problem 27229

A drug with a half-life of 67 min starts at 1.7μg/mL1.7 \mu \mathrm{g} / \mathrm{mL}1.7μg/mL. What is its concentration after 268 min? Round to 2 sig. figs. □μgmL\square \frac{\mu \mathrm{g}}{\mathrm{mL}}□mLμg​ □×10□\square \times 10^{\square}□×10□

Problem 27230

A drug with a half-life of 73 min starts at 1.4μg/mL1.4 \mu \mathrm{g} / \mathrm{mL}1.4μg/mL. What is the concentration after 146 min? Round to 2 sig. digits. □μgmL\square \frac{\mu \mathrm{g}}{\mathrm{mL}}□mLμg​

Problem 27231

A drug with a half-life of 82 min starts at 0.30μg/mL0.30 \mu \mathrm{g} / \mathrm{mL}0.30μg/mL. What is its concentration after 246 min? Round to 2 sig. figs. □μgmL\square \frac{\mu \mathrm{g}}{\mathrm{mL}}□mLμg​

Problem 27232

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=−x(x+1)2f(x)=-x(x+1)^{2}f(x)=−x(x+1)2. Choose from the options provided.

Problem 27233

Problem 27234

A cuboid with a square base has height 2s2s2s. Volume increases at 24 cm3 s−124 \mathrm{~cm}^{3} \mathrm{~s}^{-1}24 cm3 s−1. Find surface area change rate when height is 12 cm12 \mathrm{~cm}12 cm.

Problem 27235

Find the value of f(3+h)−f(3)h\frac{f(3+h)-f(3)}{h}hf(3+h)−f(3)​ for the function f(x)=21−5xf(x)=21-5xf(x)=21−5x.

Problem 27236

Estimate and classify the critical points of the function f(x)=sin⁡(x)f(x) = \sin(x)f(x)=sin(x).

Problem 27237

Calculate the average rate of change for f(x)=log⁡78xf(x)=\log _{7} 8 xf(x)=log7​8x from x=3x=3x=3 to x=9x=9x=9. Answer as an integer or decimal to three decimal places.

Problem 27238

Find the average rate of the function f(x)=ex−2−4f(x)=e^{x-2}-4f(x)=ex−2−4 between x=2x=2x=2 and x=4x=4x=4.

Problem 27239

Find the average velocity of an object dropped from rest on Planet XXX over the interval t=6t=6t=6 to t=9t=9t=9 using d(t)=20t2d(t)=20 t^{2}d(t)=20t2.

Problem 27240

Calculate the average rate of change of f(x)=log⁡48xf(x)=\log _{4} 8 xf(x)=log4​8x from x=4x=4x=4 to x=9x=9x=9. Answer as an integer or decimal to three places.

Problem 27241

Find the integral of the function: ∫4+2xx3dx\int \frac{4+2 x}{x^{3}} d x∫x34+2x​dx.

Approximate the area under $y=x^{4}-4 x^{3}+8 x+18$ from $x=-2$ to $x=4$ using the Mid-Ordinate rule with step size $2$ .

Approximate the area under $y = x^4 - 4x^3 + 8x + 18$ from $x = -2$ to $x = 4$ using Simpson's rule with step size 1.

Integrate $f(x) = x^4 - 4x^3 + 8x + 18$ to find the area from $x = -2$ to $x = 4$ . What is the area?

Calculate the average of the function $f(x) = x^4 - 4x^3 + 8x + 18$ from $x = -2$ to $x = 4$ .

Find the value of $x$ for the curve $x^{3}+xy-y^{2}=10$ where there is a vertical tangent line.

Is the rate of change of the volume of a sphere, $V$ , constant if the radius $r$ changes at a constant rate, $\frac{d r}{d t}$ ?

An astronaut drops a rock on the Moon. Find the average speed, in m/s, of the rock between 5 and 10 seconds using $d(t)=0.8 t^{2}$ .

Is $\frac{d V}{r}$ constant if $\frac{d r}{r}$ is constant for a sphere's volume $V$ with radius $r$ ? Explain.

Find the derivative $f^{\prime}(x)$ of $f(x)=-2 x^{4}+3 x^{-2}+5 x-4$ . Which option is correct? A, B, C, or D?

Find the derivative $y^{\prime}$ of the function $y=\sqrt{1-2x}$ .

Graph $f(x)=\left\{\begin{array}{ll} 5-x, & x \neq 2 \\ -3, & x=2 \end{array}\right.$ , find $\lim_{x \rightarrow 2} f(x)$ , $f(2)$ , and check continuity at 2.

Find the volume of the solid with base area $\mathrm{R}$ under $f(x)=3x^3$ , $y=1$ , and the $y$ -axis, with semicircular cross sections.

Find the inflection point of the function $f(x)=x^{3}-3 x$ . Choose from: A. $(0,-1)$ B. $(0,0)$ C. $(0,1)$ D. $(0,2)$ .

The function $f(x)=|x+1|$ is not differentiable at which point: A. $x=0$ , B. $x=-1$ , C. $x=1$ , D. $x=2$ ?

Find the marginal profit for production of 20 units given demand $p(x)=3x+5x^2$ and average cost $\bar{C}(x)=5+4x$ .

Find the displacement and total distance traveled by a particle with velocity $v(t)=5t-9$ for $0 \leq t \leq 3$ .

Find the marginal cost of production when the average cost is $\bar{C}(x)=-2 x^{2}+4+\frac{100}{x}$ at 50 units.

Find where the function $f(x)=|x-1|$ is not differentiable: A. $x=0$ , B. $x=-1$ , C. $x=1$ , D. $x=2$ .

Find the instantaneous rate of change of $f(x)=130000(1.06)^{x}$ after 5 years. Choices: a) \$8000 b) \$10000 c) \$12000 d) \$14000

Find the integral $\int \frac{x^{5}}{x^{6}-3} d x$ using the substitution $u=x^{6}-3$ , and find $d u$ . Rewrite and evaluate the integral.

Find the integral $\int \cos(8x) \, dx$ using the substitution $u=8x$ . Determine $du$ and rewrite the integral as $\int(\square) \, du$ . Evaluate it with constant $C$ .

Find the average rate of change in height from $t=4$ s to $t=5$ s for $h(t)=-4.9 t^{2}+150$ . Options: a) $24.5$ m/s, b) $-19.6$ m/s, c) $-44.1$ m/s, d) $-24.5$ m/s.

Find the value of $x$ where the function's instantaneous rate of change is negative. Also, estimate the value change rate at $t=5$ for $v(t) = 24000(0.95)^t$ .

Find the inflection point of the function $f(x)=x^{3}-3 x^{2}+1$ . Choose from: A. $(1,-1)$ B. $(1,1)$ C. $(-1,0)$ D. $(1,0)$

Rewrite the integral $\int t^{4} e^{-t^{5}} d t$ using substitution to express it as $-\frac{1}{5} e^{u} d u$ . Find $u$ and $d u$ , then evaluate the integral with constant $C$ .

Calculate the total oil leaked in the first hour from the rate $r(t)=65 e^{-0.02 t}$ liters/minute. Round to the nearest liter.

A drug with a half-life of 67 min starts at $1.7 \mu \mathrm{g} / \mathrm{mL}$ . What is its concentration after 268 min? Round to 2 sig. figs. $\square \frac{\mu \mathrm{g}}{\mathrm{mL}}$ $\square \times 10^{\square}$

A drug with a half-life of 73 min starts at $1.4 \mu \mathrm{g} / \mathrm{mL}$ . What is the concentration after 146 min? Round to 2 sig. digits. $\square \frac{\mu \mathrm{g}}{\mathrm{mL}}$

A drug with a half-life of 82 min starts at $0.30 \mu \mathrm{g} / \mathrm{mL}$ . What is its concentration after 246 min? Round to 2 sig. figs. $\square \frac{\mu \mathrm{g}}{\mathrm{mL}}$

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-x(x+1)^{2}$ . Choose from the options provided.

A cuboid with a square base has height $2s$ . Volume increases at $24 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ . Find surface area change rate when height is $12 \mathrm{~cm}$ .

Find the value of $\frac{f(3+h)-f(3)}{h}$ for the function $f(x)=21-5x$ .

Estimate and classify the critical points of the function $f(x) = \sin(x)$ .

Calculate the average rate of change for $f(x)=\log _{7} 8 x$ from $x=3$ to $x=9$ . Answer as an integer or decimal to three decimal places.

Find the average rate of the function $f(x)=e^{x-2}-4$ between $x=2$ and $x=4$ .

Find the average velocity of an object dropped from rest on Planet $X$ over the interval $t=6$ to $t=9$ using $d(t)=20 t^{2}$ .

Calculate the average rate of change of $f(x)=\log _{4} 8 x$ from $x=4$ to $x=9$ . Answer as an integer or decimal to three places.

Find the integral of the function: $\int \frac{4+2 x}{x^{3}} d x$ .

Find the sum: $\sum_{n=1}^{\infty} n^{k}\left(\frac{n}{n+1}\right)^{n^{2}}$ .

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

Evaluate the series: $\sum_{n=1}^{\infty} \frac{(n!)^{3} \cdot 3^{n}}{(3n)!} \cdot \sin(n)$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{x \sin (x)}{1-\cos (x)}$

Find the normalization constant $N$ , expectation values of $L$ , $L^{2}$ , and their variances for $\psi=N e^{-\alpha r^{2}}(x+y) z$ .

Find the slope and tangent line equation at point $P$ for (1) $y=x^{2}-4x$ , $P:(1,-3)$ and (2) $y=2-x^{3}$ , $P:(1,1)$ .

Berechnen Sie den mittleren Anstieg der Funktion $p(t)=-2 t^{3}+9 t^{2}+15 t+75$ und bestimmen Sie, wann die Änderungsrate maximal ist.

Fruchtfliegen wachsen um $35\%$ täglich. Gegeben sind 20 Fliegen. a) Funktionsgleichung aufstellen. b) Fliegenzahl nach 7 Tagen berechnen. c) Zeit bis 2000 Fliegen berechnen.

Find the average rate of change of $h(t)=\cot t$ over these intervals: a. $[\pi / 4,3 \pi / 4]$ b. $[\pi / 6, \pi / 2]$

Prove that if $y=f(u)$ and $u=g(x)$ , then $\frac{d y}{d x}=\frac{d y}{d u} \times \frac{d u}{d x}$ .

Berechne die Ableitung der Funktion $f(x)=-2x^{2}$ an der Stelle $x_{0}=3$ .

Finde $k<R$ , so dass $\int_{0}^{k} 2 x^{2} d x=18$ . Skizziere und erkläre die Fragestellung.

Bearbeiten Sie die Wasserzuflussgeschwindigkeit $v(t)=5t-t^{2}$ für $t=0$ bis $t=5$ . Berechnen Sie die Menge in 4 Stunden. Analysieren Sie die Gewinnfunktion $f(x)=-20x^{3}+240x^{2}-1200$ für 12 Monate.

Bestimme die Ableitung von $f(x)=0,5 x^{2}$ an der Stelle $x_{0}=4$ mit dem Differenzenquotienten.

Bestimme die Ableitung von $f(x)=0,5 x^{2}$ bei $x_{0}=4$ und die Ableitungsfunktion von $f(x)=4 x^{2}-5$ .

Bestimme die Ableitungsfunktion von $f(x)=4 x^{2}-5$ .

Bestimme die Ableitungen der Funktionen: a. $f(x)=7 x^{3}-2 x^{2}$ , b. $g(x)=3 x^{5}+5 x^{3}$ .

Finde den Wert von $a$ , sodass der Graph der Funktion $f_{a}(x) = \frac{1}{a} \cdot x^3 - x$ bei $x=3$ einen Extrempunkt hat.

Find the limit as $x$ approaches 1 from the left: $\lim _{x \rightarrow 1^{-}}\left(\frac{x-1}{2}-\frac{1}{\ln (x)}\right)$ .

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} x \tan ^{-1} \frac{3}{x}$ .

Find the $x$ -coordinates of relative extrema and inflection points for $f(x)$ using its derivatives from the table.

Evaluate the integral: $\int \frac{d x}{\sqrt{-x^{2}-4 x-3}}$

Calculate the integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \sec ^{2} x \, dx$ .

Find the particular solution to $\frac{d y}{d x}=x^{2}+1$ that passes through $(3,13)$ .

Find the derivative of the integral $\frac{d}{d x}\left[\int_{2}^{x} \frac{\sqrt{t}}{t^{2}+1} d t\right]$ .

Find the limit as $x$ approaches 0 for the function $\sqrt{x}$ .

Find the second derivative $y^{\prime \prime}$ of the function $y=e^{x} \cos x$ .

What does the table suggest about $\lim _{x \rightarrow 2^{-}} f(x)$ ? Options: 2, 6.5, 27, or does not exist?

Find the upper sum $S_{b}$ for $f(x)=x^{2}-x$ over $[0,10]$ with 5 subintervals. What is $S_{6}$ ? Options: 340, 420, 260, 500.

Find the second derivative of $y=e^{x} \cos x$ . Choices: $-2 e^{x} \cos x$ , $-2 e^{x} \sin x$ , $2 e^{x} \sin x$ , $2 e^{x} \cos x$ .