Calculus

Problem 22201

Find $c$ where $f'(c)=0$ and check if $f(x)=e^{-x^{2}}$ has a local extremum at $x=c$ .

See Solution

Problem 22202

Find $c$ where $f'(c)=0$ for $f(x)=e^{-x^{2}}$ and check if $f(x)$ has a local extremum at $x=c$ .

See Solution

Problem 22203

Find where the function $y=(5 x-6)^{\frac{1}{3}}$ is increasing, decreasing, concave up, and concave down using derivatives.

See Solution

Problem 22204

Find where the function $y = (7x - 1)^{\frac{1}{3}}$ is concave up for $x \in \mathbb{R}$ . Answer in interval notation.

See Solution

Problem 22205

Find where the function $y=(2 x-5)^{\frac{1}{3}}$ is increasing, decreasing, concave up, and concave down using derivative tests.

See Solution

Problem 22206

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{2-\cos x}{x^{2}+x}$

See Solution

Problem 22207

Find where the function $y=\frac{x+5}{x}$ is increasing, decreasing, concave up, and concave down using derivative tests.

See Solution

Problem 22208

Graph the function $f(x)=-x^{2}+1$ on $[-1, 1]$ and find its absolute maximum value.

See Solution

Problem 22209

Ein zylindrischer Wasserspeicher hat eine Höhe von 300 cm und einen Durchmesser von 60 cm. Der Wasserstand sinkt mit $h^{\prime}(t)=\frac{1}{54} t-\frac{10}{3}$ . Nach 1,5 Stunden, wie viel Wasser ist verloren gegangen?

See Solution

Problem 22210

Determine where the function $y=16 x e^{-x}$ is increasing, decreasing, concave up, and concave down for $x>0$ .

See Solution

Problem 22211

Find the second derivative of $f(x) = -9 e^{x} \cos x$ .

See Solution

Problem 22212

For a logistic growth model, find $g(N)=\frac{f(N)}{N}=r\left(1-\frac{N}{K}\right)$ , then plot for $r=2$ , $K=8$ . Complete parts (a) to (c).

See Solution

Problem 22213

For a logistic growth model, find $g(N)=\frac{f(N)}{N}=r\left(1-\frac{N}{K}\right)$ , then plot for $r=2$ , $K=9$ .
(a) Plot $g(N)$ and choose the correct answer. (b) Find $g^{\prime}(N)$ for $r=2$ , $K=9$ and determine where $g(N)$ is increasing or decreasing.

See Solution

Problem 22214

Find the inflection points of $f(x)=8 e^{-7 x^{2}}$ for $x \geq 0$ . Provide coordinates as ordered pairs.

See Solution

Problem 22215

Find local maxima and minima, and intervals of increase/decrease for $y=x^{3}-12x+6, x \in \mathbb{R}$ .

See Solution

Problem 22216

Geschwindigkeit und Beschleunigung:
a) Bestimme $\bar{v}(t, z)$ und drücke $v(t)$ aus. b) Bestimme $\bar{a}(t, z)$ und drücke $a(t)$ aus. c) Bei $s(t)=0,5 \cdot t^{2}$ : Berechne $\bar{v}[1, 5]$ und $v(5)$ ; zeige, dass $a(t)$ konstant ist.

See Solution

Problem 22217

Find the derivative $y^{\prime}$ for $y=5 \sqrt{x} \cos \sqrt{x}$ .

See Solution

Problem 22218

For the logistic growth model, find $g'(N)$ for $r=2$ , $K=9$ , and determine where $g(N)$ is increasing/decreasing.

See Solution

Problem 22219

Find $F'(14)$ given $F(x) = f(g(x))$ , $g(14) = 2$ , $g'(14) = 5$ , $f'(14) = 15$ , and $f'(2) = 11$ .

See Solution

Problem 22220

Determine if the series $\sum_{n=1}^{\infty} n^{2}\left(\frac{2}{3}\right)^{n^{2}-n}$ converges.

See Solution

Problem 22221

A balloon inflates at $100 \mathrm{ft}^{3}/\mathrm{min}$ . Find the radius increase rate when the radius is $3 \mathrm{ft}$ .

See Solution

Problem 22222

A kite is at a height of 40 ft, moving horizontally at 3 ft/sec. Find the rate of string release when the string is 50 ft long.

See Solution

Problem 22223

A particle moves along $y=x^{2}$ with $dx/dt=3 \mathrm{~m/s}$ . Find the rate of change of angle $\theta$ when $x=2 \mathrm{~m}$ .

See Solution

Problem 22224

A boat is pulled to a dock with a rope. If the rope is pulled at $2 \mathrm{ft/sec}$ , how fast is the boat moving when $10 \mathrm{ft}$ of rope is left?

See Solution

Problem 22225

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

See Solution

Problem 22226

A snowball with a diameter of $16 \mathrm{ft}$ melts at $0.25 \mathrm{ft}^3/\mathrm{min}$ . Find the radius change rate when the radius is $5 \mathrm{ft}$ .

See Solution

Problem 22227

Evaluate the series $\sum_{n=1}^{\infty}(-1)^{n} \frac{2}{n+4}$ .

See Solution

Problem 22228

Bestimmen Sie die Ableitungsfunktion $\mathrm{f}^{\prime}$ für die folgenden Funktionen: a) $f(x)=\frac{1}{4} x^{4}-2 x^{2}$ , b) $f(x)=-3 x^{2}+4$ , c) $f(x)=3(x-2)^{2}+x$ , d) $f(x)=a x^{3}+b x^{2}+c x+d$ , e) $f(x)=2 \sqrt{x}$ , f) $f(x)=\frac{4}{x}+1$ .

See Solution

Problem 22229

Bestimmen Sie die ersten und zweiten Ableitungen von $f$ : a) $f(x)=2 \cdot e^{x-1}$ , b) $f(x)=k \cdot e^{k \cdot x}$ , c) $f(x)=e^{x}-e^{-x}$ .

See Solution

Problem 22230

Find the critical number(s) of the function $f(t)=3 t^{\frac{3}{4}}-3 t^{\frac{1}{4}}$ .

See Solution

Problem 22231

Evaluate the integral: $\int_{-2}^{5}\left|5 x-x^{2}\right| d x$

See Solution

Problem 22232

Bestimmen Sie die Ableitungsfunktion $f^{\prime}$ für die folgenden Funktionen: a) $f(x)=\frac{1}{4} x^{4}-2 x^{2}$ , b) $f(x)=-3 x^{2}+4$ , c) $f(x)=3(x-2)^{2}+x$ , d) $f(x)=a x^{3}+b x^{2}+c x+d$ , e) $f(x)=2 \sqrt{x}$ , f) $f(x)=\frac{4}{x}+1$ .

See Solution

Problem 22233

Berechne die Steigung des Graphen von $f$ an $x_{0}$ für die Funktionen a) bis f).

See Solution

Problem 22234

A snowball's surface area decreases at $4 \mathrm{~cm}^{2} / \mathrm{min}$ . Find the radius decrease rate when diameter is $39 \mathrm{~cm}$ . Use $S=4 \pi r^{2}$ .

See Solution

Problem 22235

A cube's volume increases at 10 $\mathrm{cm}^{3}/\mathrm{min}$ . Find the surface area increase rate when edge length is $90 \mathrm{cm}$ .

See Solution

Problem 22236

Evaluate the integral: $\int_{0}^{1 / 2} \frac{8 d r}{\sqrt{1-r^{2}}}$

See Solution

Problem 22237

Bestimme den Grenzwert von $f(x)=\frac{1-x}{1-\sqrt{x}}$ für $x \rightarrow 1$ und zeige die Unstetigkeit von $f(x)=\left\{\begin{array}{cc}x & x \leq 0 \\ x-2 & x>0\end{array}\right.$ bei $x_{0}=0$ .

See Solution

Problem 22238

Find the total distance traveled by a particle with velocity $v(t)=2t-2$ from $t=0$ to $t=5$ .

See Solution

Problem 22239

Evaluate the integral: $\int \sqrt{16 x} \sin(1+x^{3/2}) \, dx$

See Solution

Problem 22240

Evaluate the integral: $\int\left(\frac{5-x}{x}\right)^{2} d x$ .

See Solution

Problem 22241

Differentiate the function $f(t)=\frac{\sqrt{t}}{t^{2}-5}$ .

See Solution

Problem 22242

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $x^{5} - y^{5} = 8$ .

See Solution

Problem 22243

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for the equation $e^{x y}-x^{6}+y^{6}=-5$ .

See Solution

Problem 22244

Analyze the function $f(x)=2 x^{3}+3 x^{2}-36 x-10$ for concavity, inflection points, and local extrema. Sketch the curve.

See Solution

Problem 22245

Encontre o valor positivo de $a$ para que a função $f$ seja contínua em $\mathbb{R}$ . Opções: (A) $\ln (5)$ (B) $e^{5}$ (C) 5 (D) $\frac{1}{5}$ .

See Solution

Problem 22246

Encontre o valor de $k \neq 0$ tal que $\lim _{x \rightarrow 0} \frac{\mathrm{e}^{3 x}-\mathrm{e}^{x}}{\mathrm{e}^{k x}-1}=3$ . Opções: (A) 3 (B) $\frac{3}{2}$ (C) $\frac{1}{3}$ (D) $\frac{2}{3}$ .

See Solution

Problem 22247

Bestimme $f^{\prime}(x)$ für: a) $f(x)=3 x^{2}$ , b) $f(x)=5 x^{3}$ .

See Solution

Problem 22248

Find the first derivative with respect to $x$ for these functions: 1) $\frac{d}{dx}\left(\frac{x^{2}+\ln x}{x-1}\right)$ , 2) $\frac{d}{dx}\left(\ln \left(x^{3}-7 x\right)\right)$ , 3) $\frac{d}{dx}\left(3 \cdot 2^{x-4} \cdot x^{-2}\right)$ .

See Solution

Problem 22249

Untersuche die Kurvenschar $f_a(x) = x^2 - (a+1)x + a$ auf Nullstellen, Extremstellen und Flächeninhalte.

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Problem 22250

Bestimme die Punkte von $f(x)=\frac{1}{x^{4}}$ , wo die Tangente die Steigung 4 hat.

See Solution

Problem 22251

b) For funksjonen $f(x)=1-e^{x+1}$ , gjør følgende:
1. Finn nullpunktene.
2. Beskriv asymptoten og dens plassering.
3. Finn tangenten ved $x=-1$ .
4. Skisser grafen til $f$ .

See Solution

Problem 22252

What is the value of \ $600 invested at 8\% interest compounded continuously after 3 years? Round to the nearest cent. Use$ A(t)=P \cdot e^{rt}$. Options: A. \$771.14 B. \$755.83 C. \$744.00 D. \$762.73

See Solution

Problem 22253

Berechne die Ableitung von $f(x)=\frac{x+1}{\sqrt{x}}$ mit der Quotientenregel.

See Solution

Problem 22254

Find the slope of the tangent line to the curve $x^2 - y^2 - 5xy = 25$ at the point where $x=2$ .

See Solution

Problem 22255

Bestimme die Ableitung von $x^{\frac{3}{4}}$ .

See Solution

Problem 22256

Find the limits: $\lim _{x \rightarrow 0} \frac{f(x+1)-f(1)}{x}$ and $\lim _{x \rightarrow 4} \frac{6-6}{x-4}$ .

See Solution

Problem 22257

Find the derivative of the function $\left(\frac{2 x^{2}-1}{x^{2}-9}\right)$ .

See Solution

Problem 22258

Find the critical points of the function $y=6 x^{3}-18 x+1$ . Options: a) $1,-1$ b) $\bigcirc$ c) $0-1$ d) $0 \sqrt{3},-\sqrt{3}$ .

See Solution

Problem 22259

In the Exponential Growth Model with initial quantity 600, how long for it to grow from 1000 to 1500?

See Solution

Problem 22260

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin \left(x^{2}\right)-\ln (1+x)}{x^{4}}$ .

See Solution

Problem 22261

In the Exponential Growth Model with initial quantity 600 growing to 1000 in 10 days, how long to reach 1500?

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Problem 22262

Find local extrema of $f(x)=x^{2}+x-30$ using the first derivative test. Choose the correct option.

See Solution

Problem 22263

Find the extrema of $v(x)=4 x^{3}-3 x^{2}$ for $-\frac{1}{4} \leq x \leq \frac{3}{4}$ .

See Solution

Problem 22264

Calculate the integral from 0 to 2 of $\frac{e^{2 t}-2}{e^{2 t}-4 t+8} dt$ .

See Solution

Problem 22265

What will your BAC be after 18 hours if it starts at $0.36 \mathrm{g} / 100 \mathrm{ml}$ and decreases by one third each hour?

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Problem 22266

Find the general antiderivative of $(2 x-3)^{4}$ . Options: a) $\frac{(2 x-3)^{5}}{5}+c$ b) $0 \frac{2(2 x-3)^{5}}{5}+c$ c) $\frac{(2 x-3)^{5}}{10}+c$ d) $0 \frac{(2 x-3)^{5}}{2}+c$

See Solution

Problem 22267

Find the x-coordinates of the inflection points of $f(x)=\frac{1}{12} x^{4}-2 x^{2}$ . Options: a) none b) $-2, 2$ c) $-2$ d) $2$

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Problem 22268

Find the general antiderivative of $(2 x-3)^{4}$ . Options: a) $\frac{(2 x-3)^{5}}{5}+c$ , b) $\frac{2(2 x-3)^{5}}{5}+c$ , c) $\frac{(2 x-3)^{5}}{10}+c$ , d) $\frac{(2 x-3)^{5}}{2}+c$ .

See Solution

Problem 22269

Find the general antiderivative of $\sin (5 x+2)$ . Choose from the options provided.

See Solution

Problem 22270

Find the general antiderivative of $\sin (5 x+2)$ . Choose the correct option from the following: a) $\frac{\cos (5 x+2)}{5}+c$ b) $0-\cos (5 x+2)+c$ c) $\cos (5 x+2)+c$ d) $\frac{-\cos (5 x+2)}{5}+c$

See Solution

Problem 22271

An artist invests \$1000 at a 2.5% annual interest rate. When will it grow to \$1200? (1 d.p.) Options: A) 8.3 B) 5.9 C) 6.5 D) 4.1 E) 7.3

See Solution

Problem 22272

Find $\lim _{x \rightarrow 3} h(x)$ where $h(x)=\{x^{3}-6x^{2}+9x+18$ if $x<3$ , $5$ if $x=3$ , $\frac{x^{3}-9x}{x-3}$ if $x>3\}$ .

See Solution

Problem 22273

Find the average rate of change of $f(x) = \frac{1}{x-2}$ from $x = -4$ to $x = -3$ . Use the formula $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$ .

See Solution

Problem 22274

Find the general antiderivative of $\frac{6 x^{3}+8 x^{2}}{x^{2}}$ . Options: a) no antiderivative b) $3 x^{2}+8 x+c$ c) $0 \frac{\left(\frac{3 x^{4}}{2}+\frac{8 x^{3}}{3}\right)}{\frac{x^{3}}{3}}+c$ d) $3 x^{2}-8 x+c$

See Solution

Problem 22275

Solve the differential equation $\frac{d y}{d x}=4-5 x$ with initial condition $y=5$ at $x=0$ . Choose the correct option.

See Solution

Problem 22276

Solve the equation $\frac{d^{2} y}{d x^{2}}=4 x^{2}+3 x-1$ for the general solution.

See Solution

Problem 22277

Solve the differential equation $\frac{d y}{d x}=4-5 x$ with initial condition $y=5$ at $x=0$ .

See Solution

Problem 22278

Find the simplified difference quotient for $f(x)=\frac{3}{x^{2}}$ with step-size $h \neq 0$ .

See Solution

Problem 22279

Find the value of $a$ such that the derivative of $f(g(x))$ at $x=1$ equals $\frac{1}{3}$ , where $f(x)=\sqrt{x}$ and $g(x)=a+x^{3}$ .

See Solution

Problem 22280

Evaluate the integral: $\int \frac{3-9 \sin x}{\cos ^{2} x} d x$

See Solution

Problem 22281

Find the value of $b$ such that the derivative of $\frac{u(x)}{v(x)}$ at $x=0$ equals 1, where $u(x)=b x$ and $v(x)=e^{x}$ .

See Solution

Problem 22282

Find the limit of $f(x)=\frac{\sin \left(x^{2}\right)-\ln (1+x)}{x^{4}}$ as $x$ approaches 0. What does it approach?

See Solution

Problem 22283

Evaluate the integral: $\int(8 \sqrt[3]{x}-2 \csc ^{2} x) dx$ (simplify radicals, enclose fractions in parentheses).

See Solution

Problem 22284

Find the simplified difference quotient for $f(x)=\frac{3}{x^{2}}$ with step-size $h \neq 0$ . Options: (A) (B) (C) (D) (E)

See Solution

Problem 22285

Find the value of $a$ such that the derivative of $f(g(x))$ at $x=1$ equals $\frac{1}{3}$ , where $f(x)=\sqrt{x}$ and $g(x)=a+x^{3}$ .

See Solution

Problem 22286

At time $t \geq 0$ , a particle has position $(x(t), y(t))$ and velocity $v(t)=\left(\cos \left(t^{2}\right), e^{0.5 t}\right)$ . Given $(3,5)$ at $t=1$ , find $x(2)$ .

See Solution

Problem 22287

Find the function $f(x)$ given that $f^{\prime}(x)=e^{x}+\frac{1}{1+x^{2}}$ and $f(1)=0$ .

See Solution

Problem 22288

Find the sum of the critical values of the function $f(x)=x^{2} \cdot \ln (x)+3$ .

See Solution

Problem 22289

A car overheats at $280^{\circ} \mathrm{F}$ and cools down to $230^{\circ} \mathrm{F}$ with an outside temp of $80^{\circ} \mathrm{F}$ . If the cooling rate is $r=0.0058$ , how long until you can drive again? Round to the nearest minute. You will have to wait about $\square$ minutes.

See Solution

Problem 22290

A company can sell $x$ solar panels for a profit $P(x)=300,000,000+2,000,000 x+1000 x^{2}-3 x^{3}$ . Maximize $P(x)$ . Options: (A) $\frac{5000(7+\sqrt{12})}{3}$ , (B) $\frac{100(5+\sqrt{15})}{2}$ , (C) $\frac{1000(1+\sqrt{19})}{y^{3}}$ , (D) $\frac{3000(2+\sqrt{19})}{8}$ , (E) $\frac{500(3+\sqrt{18})}{7}$ .

See Solution

Problem 22291

Calculate the integral: $\int(4x+2)(x^{2}+x+1)^{3} \, dx$

See Solution

Problem 22292

Determine local extrema of $y=f(x)$ using $f'(x)$ and $f''(x)$ given. What does the Second Derivative Test yield?

See Solution

Problem 22293

Find the function $f(x)$ given $f^{\prime}(x)=e^{x}+\frac{1}{1+x^{2}}$ and $f(1)=0$ .

See Solution

Problem 22294

Calculate the integral: $\int 10 \cos ^{2} 5 x \, dx$

See Solution

Problem 22295

Evaluate the integral: $\int(4 x+2)(x^{2}+x+1)^{3} \, dx$

See Solution

Problem 22296

Sam invests \ $700 in an account with continuous$ 15\%$ interest. What is the account balance after 8 years?

See Solution

Problem 22297

Calculate the integral from 1 to 4 of the function (3x - 5) with respect to x: $\int_{1}^{4}(3 x-5) d x$ .

See Solution

Problem 22298

Find the average rate of change of the function $f$ on the interval $a \leq x \leq c$ , given $f(a)=b$ and $f(c)=d$ .

See Solution

Problem 22299

Find the integral of $\sin (\cos x) \sin x$ with respect to $x$ : $\int \sin (\cos x) \sin x \, dx$ .

See Solution

Problem 22300

Find the limit: $\lim _{x \rightarrow+\infty} \frac{\frac{\sqrt{x-1}}{x}}{x}$ .

See Solution

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Calculus

Problem 22201

Find ccc where f′(c)=0f'(c)=0f′(c)=0 and check if f(x)=e−x2f(x)=e^{-x^{2}}f(x)=e−x2 has a local extremum at x=cx=cx=c.

Problem 22202

Find ccc where f′(c)=0f'(c)=0f′(c)=0 for f(x)=e−x2f(x)=e^{-x^{2}}f(x)=e−x2 and check if f(x)f(x)f(x) has a local extremum at x=cx=cx=c.

Problem 22203

Find where the function y=(5x−6)13y=(5 x-6)^{\frac{1}{3}}y=(5x−6)31​ is increasing, decreasing, concave up, and concave down using derivatives.

Problem 22204

Find where the function y=(7x−1)13y = (7x - 1)^{\frac{1}{3}}y=(7x−1)31​ is concave up for x∈Rx \in \mathbb{R}x∈R. Answer in interval notation.

Problem 22205

Find where the function y=(2x−5)13y=(2 x-5)^{\frac{1}{3}}y=(2x−5)31​ is increasing, decreasing, concave up, and concave down using derivative tests.

Problem 22206

Evaluate the limit: lim⁡x→∞2−cos⁡xx2+x\lim _{x \rightarrow \infty} \frac{2-\cos x}{x^{2}+x}x→∞lim​x2+x2−cosx​

Problem 22207

Find where the function y=x+5xy=\frac{x+5}{x}y=xx+5​ is increasing, decreasing, concave up, and concave down using derivative tests.

Problem 22208

Graph the function f(x)=−x2+1f(x)=-x^{2}+1f(x)=−x2+1 on [−1,1][-1, 1][−1,1] and find its absolute maximum value.

Problem 22209

Ein zylindrischer Wasserspeicher hat eine Höhe von 300 cm und einen Durchmesser von 60 cm. Der Wasserstand sinkt mit h′(t)=154t−103h^{\prime}(t)=\frac{1}{54} t-\frac{10}{3}h′(t)=541​t−310​. Nach 1,5 Stunden, wie viel Wasser ist verloren gegangen?

Problem 22210

Determine where the function y=16xe−xy=16 x e^{-x}y=16xe−x is increasing, decreasing, concave up, and concave down for x>0x>0x>0.

Problem 22211

Find the second derivative of f(x)=−9excos⁡xf(x) = -9 e^{x} \cos xf(x)=−9excosx.

Problem 22212

For a logistic growth model, find g(N)=f(N)N=r(1−NK)g(N)=\frac{f(N)}{N}=r\left(1-\frac{N}{K}\right)g(N)=Nf(N)​=r(1−KN​), then plot for r=2r=2r=2, K=8K=8K=8. Complete parts (a) to (c).

Problem 22213

Problem 22214

Find the inflection points of f(x)=8e−7x2f(x)=8 e^{-7 x^{2}}f(x)=8e−7x2 for x≥0x \geq 0x≥0. Provide coordinates as ordered pairs.

Problem 22215

Find local maxima and minima, and intervals of increase/decrease for y=x3−12x+6,x∈Ry=x^{3}-12x+6, x \in \mathbb{R}y=x3−12x+6,x∈R.

Problem 22216

Problem 22217

Find the derivative y′y^{\prime}y′ for y=5xcos⁡xy=5 \sqrt{x} \cos \sqrt{x}y=5x​cosx​.

Problem 22218

For the logistic growth model, find g′(N)g'(N)g′(N) for r=2r=2r=2, K=9K=9K=9, and determine where g(N)g(N)g(N) is increasing/decreasing.

Problem 22219

Find F′(14)F'(14)F′(14) given F(x)=f(g(x))F(x) = f(g(x))F(x)=f(g(x)), g(14)=2g(14) = 2g(14)=2, g′(14)=5g'(14) = 5g′(14)=5, f′(14)=15f'(14) = 15f′(14)=15, and f′(2)=11f'(2) = 11f′(2)=11.

Problem 22220

Determine if the series ∑n=1∞n2(23)n2−n\sum_{n=1}^{\infty} n^{2}\left(\frac{2}{3}\right)^{n^{2}-n}∑n=1∞​n2(32​)n2−n converges.

Problem 22221

A balloon inflates at 100ft3/min100 \mathrm{ft}^{3}/\mathrm{min}100ft3/min. Find the radius increase rate when the radius is 3ft3 \mathrm{ft}3ft.

Problem 22222

A kite is at a height of 40 ft, moving horizontally at 3 ft/sec. Find the rate of string release when the string is 50 ft long.

Problem 22223

A particle moves along y=x2y=x^{2}y=x2 with dx/dt=3 m/sdx/dt=3 \mathrm{~m/s}dx/dt=3 m/s. Find the rate of change of angle θ\thetaθ when x=2 mx=2 \mathrm{~m}x=2 m.

Problem 22224

A boat is pulled to a dock with a rope. If the rope is pulled at 2ft/sec2 \mathrm{ft/sec}2ft/sec, how fast is the boat moving when 10ft10 \mathrm{ft}10ft of rope is left?

Problem 22225

Bestimme die Ableitung der Funktionen: a) f(x)=1x5f(x)=\frac{1}{x^{5}}f(x)=x51​, b) f(x)=3x4f(x)=\frac{3}{x^{4}}f(x)=x43​, c) f(x)=−13x6f(x)=-\frac{1}{3 x^{6}}f(x)=−3x61​, d) f(x)=x4−x+1x3f(x)=\frac{x^{4}-x+1}{x^{3}}f(x)=x3x4−x+1​.

Problem 22226

A snowball with a diameter of 16ft16 \mathrm{ft}16ft melts at 0.25ft3/min0.25 \mathrm{ft}^3/\mathrm{min}0.25ft3/min. Find the radius change rate when the radius is 5ft5 \mathrm{ft}5ft.

Problem 22227

Evaluate the series ∑n=1∞(−1)n2n+4\sum_{n=1}^{\infty}(-1)^{n} \frac{2}{n+4}∑n=1∞​(−1)nn+42​.

Problem 22228

Problem 22229

Bestimmen Sie die ersten und zweiten Ableitungen von fff: a) f(x)=2⋅ex−1f(x)=2 \cdot e^{x-1}f(x)=2⋅ex−1, b) f(x)=k⋅ek⋅xf(x)=k \cdot e^{k \cdot x}f(x)=k⋅ek⋅x, c) f(x)=ex−e−xf(x)=e^{x}-e^{-x}f(x)=ex−e−x.

Problem 22230

Find the critical number(s) of the function f(t)=3t34−3t14f(t)=3 t^{\frac{3}{4}}-3 t^{\frac{1}{4}}f(t)=3t43​−3t41​.

Problem 22231

Evaluate the integral: ∫−25∣5x−x2∣dx\int_{-2}^{5}\left|5 x-x^{2}\right| d x∫−25​∣∣​5x−x2∣∣​dx

Problem 22232

Problem 22233

Berechne die Steigung des Graphen von fff an x0x_{0}x0​ für die Funktionen a) bis f).

Problem 22234

A snowball's surface area decreases at 4 cm2/min4 \mathrm{~cm}^{2} / \mathrm{min}4 cm2/min. Find the radius decrease rate when diameter is 39 cm39 \mathrm{~cm}39 cm. Use S=4πr2S=4 \pi r^{2}S=4πr2.

Problem 22235

A cube's volume increases at 10 cm3/min\mathrm{cm}^{3}/\mathrm{min}cm3/min. Find the surface area increase rate when edge length is 90cm90 \mathrm{cm}90cm.

Problem 22236

Evaluate the integral: ∫01/28dr1−r2\int_{0}^{1 / 2} \frac{8 d r}{\sqrt{1-r^{2}}}∫01/2​1−r2​8dr​

Problem 22237

Bestimme den Grenzwert von f(x)=1−x1−xf(x)=\frac{1-x}{1-\sqrt{x}}f(x)=1−x​1−x​ für x→1x \rightarrow 1x→1 und zeige die Unstetigkeit von f(x)={xx≤0x−2x>0f(x)=\left\{\begin{array}{cc}x & x \leq 0 \\ x-2 & x>0\end{array}\right.f(x)={xx−2​x≤0x>0​ bei x0=0x_{0}=0x0​=0.

Problem 22238

Find the total distance traveled by a particle with velocity v(t)=2t−2v(t)=2t-2v(t)=2t−2 from t=0t=0t=0 to t=5t=5t=5.

Problem 22239

Evaluate the integral: ∫16xsin⁡(1+x3/2) dx\int \sqrt{16 x} \sin(1+x^{3/2}) \, dx∫16x​sin(1+x3/2)dx

Problem 22240

Evaluate the integral: ∫(5−xx)2dx\int\left(\frac{5-x}{x}\right)^{2} d x∫(x5−x​)2dx.

Problem 22241

Differentiate the function f(t)=tt2−5f(t)=\frac{\sqrt{t}}{t^{2}-5}f(t)=t2−5t​​.

Problem 22242

Find $c$ where $f'(c)=0$ and check if $f(x)=e^{-x^{2}}$ has a local extremum at $x=c$ .

Find $c$ where $f'(c)=0$ for $f(x)=e^{-x^{2}}$ and check if $f(x)$ has a local extremum at $x=c$ .

Find where the function $y=(5 x-6)^{\frac{1}{3}}$ is increasing, decreasing, concave up, and concave down using derivatives.

Find where the function $y = (7x - 1)^{\frac{1}{3}}$ is concave up for $x \in \mathbb{R}$ . Answer in interval notation.

Find where the function $y=(2 x-5)^{\frac{1}{3}}$ is increasing, decreasing, concave up, and concave down using derivative tests.

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{2-\cos x}{x^{2}+x}$

Find where the function $y=\frac{x+5}{x}$ is increasing, decreasing, concave up, and concave down using derivative tests.

Graph the function $f(x)=-x^{2}+1$ on $[-1, 1]$ and find its absolute maximum value.

Ein zylindrischer Wasserspeicher hat eine Höhe von 300 cm und einen Durchmesser von 60 cm. Der Wasserstand sinkt mit $h^{\prime}(t)=\frac{1}{54} t-\frac{10}{3}$ . Nach 1,5 Stunden, wie viel Wasser ist verloren gegangen?

Determine where the function $y=16 x e^{-x}$ is increasing, decreasing, concave up, and concave down for $x>0$ .

Find the second derivative of $f(x) = -9 e^{x} \cos x$ .

For a logistic growth model, find $g(N)=\frac{f(N)}{N}=r\left(1-\frac{N}{K}\right)$ , then plot for $r=2$ , $K=8$ . Complete parts (a) to (c).

Find the inflection points of $f(x)=8 e^{-7 x^{2}}$ for $x \geq 0$ . Provide coordinates as ordered pairs.

Find local maxima and minima, and intervals of increase/decrease for $y=x^{3}-12x+6, x \in \mathbb{R}$ .

Find the derivative $y^{\prime}$ for $y=5 \sqrt{x} \cos \sqrt{x}$ .

For the logistic growth model, find $g'(N)$ for $r=2$ , $K=9$ , and determine where $g(N)$ is increasing/decreasing.

Find $F'(14)$ given $F(x) = f(g(x))$ , $g(14) = 2$ , $g'(14) = 5$ , $f'(14) = 15$ , and $f'(2) = 11$ .

Determine if the series $\sum_{n=1}^{\infty} n^{2}\left(\frac{2}{3}\right)^{n^{2}-n}$ converges.

A balloon inflates at $100 \mathrm{ft}^{3}/\mathrm{min}$ . Find the radius increase rate when the radius is $3 \mathrm{ft}$ .

A particle moves along $y=x^{2}$ with $dx/dt=3 \mathrm{~m/s}$ . Find the rate of change of angle $\theta$ when $x=2 \mathrm{~m}$ .

A boat is pulled to a dock with a rope. If the rope is pulled at $2 \mathrm{ft/sec}$ , how fast is the boat moving when $10 \mathrm{ft}$ of rope is left?

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

A snowball with a diameter of $16 \mathrm{ft}$ melts at $0.25 \mathrm{ft}^3/\mathrm{min}$ . Find the radius change rate when the radius is $5 \mathrm{ft}$ .

Evaluate the series $\sum_{n=1}^{\infty}(-1)^{n} \frac{2}{n+4}$ .

Bestimmen Sie die ersten und zweiten Ableitungen von $f$ : a) $f(x)=2 \cdot e^{x-1}$ , b) $f(x)=k \cdot e^{k \cdot x}$ , c) $f(x)=e^{x}-e^{-x}$ .

Find the critical number(s) of the function $f(t)=3 t^{\frac{3}{4}}-3 t^{\frac{1}{4}}$ .

Evaluate the integral: $\int_{-2}^{5}\left|5 x-x^{2}\right| d x$

Berechne die Steigung des Graphen von $f$ an $x_{0}$ für die Funktionen a) bis f).

A snowball's surface area decreases at $4 \mathrm{~cm}^{2} / \mathrm{min}$ . Find the radius decrease rate when diameter is $39 \mathrm{~cm}$ . Use $S=4 \pi r^{2}$ .

A cube's volume increases at 10 $\mathrm{cm}^{3}/\mathrm{min}$ . Find the surface area increase rate when edge length is $90 \mathrm{cm}$ .

Evaluate the integral: $\int_{0}^{1 / 2} \frac{8 d r}{\sqrt{1-r^{2}}}$

Bestimme den Grenzwert von $f(x)=\frac{1-x}{1-\sqrt{x}}$ für $x \rightarrow 1$ und zeige die Unstetigkeit von $f(x)=\left\{\begin{array}{cc}x & x \leq 0 \\ x-2 & x>0\end{array}\right.$ bei $x_{0}=0$ .

Find the total distance traveled by a particle with velocity $v(t)=2t-2$ from $t=0$ to $t=5$ .

Evaluate the integral: $\int \sqrt{16 x} \sin(1+x^{3/2}) \, dx$

Evaluate the integral: $\int\left(\frac{5-x}{x}\right)^{2} d x$ .

Differentiate the function $f(t)=\frac{\sqrt{t}}{t^{2}-5}$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $x^{5} - y^{5} = 8$ .

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for the equation $e^{x y}-x^{6}+y^{6}=-5$ .

Analyze the function $f(x)=2 x^{3}+3 x^{2}-36 x-10$ for concavity, inflection points, and local extrema. Sketch the curve.

Encontre o valor positivo de $a$ para que a função $f$ seja contínua em $\mathbb{R}$ . Opções: (A) $\ln (5)$ (B) $e^{5}$ (C) 5 (D) $\frac{1}{5}$ .

Encontre o valor de $k \neq 0$ tal que $\lim _{x \rightarrow 0} \frac{\mathrm{e}^{3 x}-\mathrm{e}^{x}}{\mathrm{e}^{k x}-1}=3$ . Opções: (A) 3 (B) $\frac{3}{2}$ (C) $\frac{1}{3}$ (D) $\frac{2}{3}$ .

Bestimme $f^{\prime}(x)$ für: a) $f(x)=3 x^{2}$ , b) $f(x)=5 x^{3}$ .

Untersuche die Kurvenschar $f_a(x) = x^2 - (a+1)x + a$ auf Nullstellen, Extremstellen und Flächeninhalte.

Bestimme die Punkte von $f(x)=\frac{1}{x^{4}}$ , wo die Tangente die Steigung 4 hat.

b) For funksjonen $f(x)=1-e^{x+1}$ , gjør følgende:
1. Finn nullpunktene.
2. Beskriv asymptoten og dens plassering.
3. Finn tangenten ved $x=-1$ .
4. Skisser grafen til $f$ .

Berechne die Ableitung von $f(x)=\frac{x+1}{\sqrt{x}}$ mit der Quotientenregel.

Find the slope of the tangent line to the curve $x^2 - y^2 - 5xy = 25$ at the point where $x=2$ .

Bestimme die Ableitung von $x^{\frac{3}{4}}$ .

Find the limits: $\lim _{x \rightarrow 0} \frac{f(x+1)-f(1)}{x}$ and $\lim _{x \rightarrow 4} \frac{6-6}{x-4}$ .

Find the derivative of the function $\left(\frac{2 x^{2}-1}{x^{2}-9}\right)$ .

Find the critical points of the function $y=6 x^{3}-18 x+1$ . Options: a) $1,-1$ b) $\bigcirc$ c) $0-1$ d) $0 \sqrt{3},-\sqrt{3}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin \left(x^{2}\right)-\ln (1+x)}{x^{4}}$ .

Find local extrema of $f(x)=x^{2}+x-30$ using the first derivative test. Choose the correct option.

Find the extrema of $v(x)=4 x^{3}-3 x^{2}$ for $-\frac{1}{4} \leq x \leq \frac{3}{4}$ .

Calculate the integral from 0 to 2 of $\frac{e^{2 t}-2}{e^{2 t}-4 t+8} dt$ .

What will your BAC be after 18 hours if it starts at $0.36 \mathrm{g} / 100 \mathrm{ml}$ and decreases by one third each hour?

Find the x-coordinates of the inflection points of $f(x)=\frac{1}{12} x^{4}-2 x^{2}$ . Options: a) none b) $-2, 2$ c) $-2$ d) $2$

Find the general antiderivative of $\sin (5 x+2)$ . Choose from the options provided.

Find $\lim _{x \rightarrow 3} h(x)$ where $h(x)=\{x^{3}-6x^{2}+9x+18$ if $x<3$ , $5$ if $x=3$ , $\frac{x^{3}-9x}{x-3}$ if $x>3\}$ .

Find the average rate of change of $f(x) = \frac{1}{x-2}$ from $x = -4$ to $x = -3$ . Use the formula $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$ .