Calculus

Problem 24201

Calculate the integral $\int_{-10}^{10} \sqrt[3]{16 x^{5}-4 x^{3}-3 x} d x$ and choose the correct value: 0, -1, 1, None, or $1/4$ .

See Solution

Problem 24202

Determine the series for convergence of $\sum_{n=1}^{\infty} \frac{n}{5 n^{4}+5 n^{3}+2 n}$ . Choose from:
1. $\sum_{n=1}^{\infty} \frac{1}{n^{3}}$
2. $\sum_{n=1}^{\infty} \frac{1}{n^{4}}$
3. $\sum_{n=1}^{\infty} \frac{1}{n^{5}}$
4. $\sum_{n=1}^{\infty} \frac{1}{n}$

See Solution

Problem 24203

A seller estimates ice cream sales $n$ with temperature $T^{\circ}C$ : $n=-\frac{1}{50} T^{3}+T^{2}+50$ , $0 \leq T \leq 30$ .
a) Find $n$ at $T=20$ . b) Find the rate of sales increase in terms of $T$ . c) Find $T$ for max sales increase. d) Explain its significance on the $n$ vs. $T$ graph.

See Solution

Problem 24204

Find where the function $f(x)=\sqrt[3]{x^{3}-3 x}$ has vertical tangent lines: $x=0, x=-3$ ; $x=1, x=-1$ ; none; $x=3, x=-3$ ; $x=0, x=3$ .

See Solution

Problem 24205

Determine if the series $1+\frac{1}{(1)(1)^{1 / 4}}+\frac{1}{(2)(2)^{1 / 4}}+\cdots$ converges or diverges.

See Solution

Problem 24206

Find the average rate of change of $f(x)$ on $[1,7]$ given rates of 8 on $[1,3]$ and 11 on $[3,7]$ .

See Solution

Problem 24207

Find the limit as $x$ approaches 3 of $3 f^{2}(x) + \frac{\sqrt{x^{2}+7}-4}{2-\sqrt{x+1}} - 4$ given $f(3) = 2$ . Choices: 17, 5, 4, 1, None.

See Solution

Problem 24208

Calculate $\int_{1}^{6}\left(2 f(x)+\frac{6}{x^{2}}\right) d x$ given $\int_{1}^{3}\left(\frac{f(x)}{2}-2 x\right) d x=2$ and $\int_{6}^{3} 4 f(x) d x=20$ . Choices: 1, 35, -15, 55, None.

See Solution

Problem 24209

Evaluate the integral $\int_{4}^{4} 2 \sqrt{x^{3}+5} d x$ . What is the result?

See Solution

Problem 24210

Find the limit: $\lim _{x \rightarrow 4^{-}} \frac{x+1}{(x-4)^{2}}$ . What is its value? 0, $-\infty$ , 1, or $\infty$ ?

See Solution

Problem 24211

Find the average value of the function $f(x)=6 x^{2}+4 x-5$ on the interval $[1,2]$ . Choices: 15, 30, 3, $15 / 2$ .

See Solution

Problem 24212

Calculate $\int_{1}^{6}\left(2 f(x)+\frac{6}{x^{2}}\right) d x$ given the integrals $\int_{1}^{3}\left(\frac{f(x)}{2}-2 x\right) d x=2$ and $\int_{6}^{3} 4 f(x) d x=20$ . Options: 15, None, 55, 35, $-15$ .

See Solution

Problem 24213

Find the absolute maximum of $f(x)=x^{4}+4x+5$ on the interval $[0,1]$ . Choices: 1, -4, 12, 10.

See Solution

Problem 24214

Find the value of $\lim _{x \rightarrow 3}\left(3 f^{2}(x)+\frac{\sqrt{x^{2}+7}-4}{2-\sqrt{x+1}}+8\right)$ if $f(3)=2$ . Choices: 17, 1, 5, Does Not Exist, None.

See Solution

Problem 24215

Find the tangent line to $(x+y)^{3}-5 x+7 y=10$ at the intersection with $2-x-y=0$ . Which equation is correct?

See Solution

Problem 24216

Find where the function $f(x)=\sqrt[3]{x^{2}-1}$ has a vertical tangent line: $x=0$ , $x=3$ , $x=-3$ , $x=1$ , $x=-1$ , or none?

See Solution

Problem 24217

Calculate $\int_{0}^{4} \frac{2 x}{\sqrt{x^{2}+9}} d x$ . What is the result? 4, 8, 1, or 2?

See Solution

Problem 24218

Express the integral $\int_{1}^{3} \frac{1}{1+x} d x$ as a limit of a Riemann sum.

See Solution

Problem 24219

Find the critical numbers of the function $f(x)=x^{3}+x^{2}-5x-5$ .

See Solution

Problem 24220

Find $G^{\prime}(0)$ for the function $G(x)=\int_{x^{2}+1}^{3 x+4}(t^{2}+2)^{2} dt$ .

See Solution

Problem 24221

Calculate the integral $\int_{-8}^{8} \sqrt{64-x^{2}} d x$ . What is the result? Options: $-12 / 3$ , $32 \pi$ , $64 \pi$ , $12 \pi$ .

See Solution

Problem 24222

Find the inflection point of the function $f(x)=x+\frac{4}{x}$ from the options: $x=0$ , $x=-1$ , $x=1$ , or None.

See Solution

Problem 24223

Find $\lim _{x \rightarrow 1} \frac{|2 x-1|-|4-5 x|}{x-1}$ . What is the limit value? Choices: $-3$ , 3, Does Not Exist, $-1$ .

See Solution

Problem 24224

Evaluate the integral $\int_{1}^{1} 6 \sqrt[5]{x^{11}-4} d x$ . What is the result? Choices: -1, 1/3, 0, 1.

See Solution

Problem 24225

Calculate $\int_{-2}^{2} \sqrt{4-x^{2}} d x$ . What is the result: $12 \pi$ , $6 \pi$ , $2 \pi$ , or $4 \pi$ ?

See Solution

Problem 24226

Verify the inequality: $\int_{3}^{4} x \, dx \leq \int_{3}^{4} x^{2} \, dx$ without calculating the integrals.

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Problem 24227

Calculate the integral $\int_{-2}^{2} \sqrt[5]{x^{5}+x^{3}} d x$ and choose the correct value: 0, $-1$ , $1/3$ , or none.

See Solution

Problem 24228

Evaluate the integral $\int_{0}^{4} \frac{2 x}{\sqrt{x^{2}+9}} d x$ . What is the result?

See Solution

Problem 24229

Find the value of $c$ that satisfies the Intermediate Value Theorem for $f(x)=x^{3}+1$ on $[-1,2]$ with $w=2$ .

See Solution

Problem 24230

Find the bounds $m$ and $n$ for the integral $I=\int_{0}^{\pi} \frac{6}{3-2 \cos ^{2}(x)} d x$ .

See Solution

Problem 24231

Gegeben ist die Funktion $f(x)=(10 x-5) \cdot e^{2 x}$ . Bestimmen Sie Nullstellen, Extrempunkte, Wendepunkt und Fläche für $x=-2$ .

See Solution

Problem 24232

Find the limit: $\lim _{x \rightarrow 3^{+}} \frac{|x-3|}{\sqrt{x-3}}$ . What is the result?

See Solution

Problem 24233

Find the values of $a$ for the continuous function $f(x)$ defined as:
$f(x)=\left\{\begin{array}{ll} \frac{x^{3}-1}{x-1} & , \text { if } x \neq 1 \\ 2 a^{2}-5 & , \text { if } x=1 \end{array}\right.$
Options: $1, -1$ ; 2 only; None; -2 only; $2, -2$ .

See Solution

Problem 24234

Find the average rate of change of $f(x)$ on $[1,7]$ given rates of 8 on $[1,3]$ and 11 on $[3,7]$ . Choices: 1, 5, None, 19, 10.

See Solution

Problem 24235

Find $G'(1)$ for $G(x)=\int_{x^{2}+1}^{2x+4}(t^{2}+2)^{2} dt$ . Choose the correct value from the options.

See Solution

Problem 24236

Find the general antiderivative of these functions: a. $f(x)=\frac{\sec(x)+\cos(x)}{\cos(x)}$ ; b. $f(x)=\frac{a}{a+x}$ .

See Solution

Problem 24237

Evaluate the integral $I=\int_{0}^{\pi} \frac{6}{3-2 \cos ^{2}(x)} d x$ to find $m$ and $n$ such that $m \leq I \leq n$ . Options: $\pi$ , $6\pi$ ; $2\pi$ , $6\pi$ ; $2\pi$ , $3\pi$ ; None; -2, 12.

See Solution

Problem 24238

Determine the intervals where the function $f(x)=\frac{x^{2}+1}{x}$ is concave upward: $(-\infty, 0)$ , $R$ , $(0, \infty)$ , $(-\infty-2)$ .

See Solution

Problem 24239

Evaluate the integral $\int_{-2}^{2} \sqrt[5]{x^{5}+x^{3}} d x$ . What is the result? Choices: $1/3$ , 0, 1, $-1$ , None.

See Solution

Problem 24240

Solve the following differential equations: a. $\frac{d y}{d s}=\cos (2 \pi s), 0 \leq s \leq 1$ b. $\frac{d y}{d x}=\frac{2}{x^{3}}-x^{3}, \mathrm{x}>0$ c. $\frac{d y}{d t}=e^{\frac{-t}{2}}, t \geq 0$

See Solution

Problem 24241

Hausaufgabe 21: Betrachten Sie die Differentialgleichung $y^{\prime \prime}-\frac{1}{x} y^{\prime}-\frac{1}{x^{2}} y=\ln (x)$ .
(a) Für welches $\alpha$ ist $g(x)=x^{\alpha}$ eine Lösung der homogenen Gleichung? (b) Bestimmen Sie ein Fundamentalsystem der homogenen Gleichung. (c) Finden Sie eine partikuläre Lösung $f_{\mathrm{p}}(x)$ der inhomogenen Gleichung. (d) Bestimmen Sie die Lösung des Anfangswertproblems: $y^{\prime \prime}-\frac{1}{x} y^{\prime}-\frac{1}{x^{2}} y=\ln (x)$ mit $y(1)=0$ , $y^{\prime}(1)=0$ .

See Solution

Problem 24242

Berechne den Differenzenquotienten von $f(x)=-7 x^{2}+1$ für die Intervalle $[0, \frac{1}{2}]$ , $[0, \frac{1}{4}]$ , $[0, \frac{1}{16}]$ . Stelle eine Vermutung für $[0, \frac{1}{n}]$ auf, wenn $n$ groß ist.

See Solution

Problem 24243

Find the derivative $f^{\prime}(x)$ for these functions: a. $f(x)=\int_{1}^{x}\left(6-\frac{t^{3}}{5}\right) dt$ b. $f(x)=\int_{2}^{x} \sqrt{1+e t^{2}} dt$ c. $f(x)=\int_{3}^{x} \cos^{2}(t^{2}-1) dt$

See Solution

Problem 24244

Bestimmen Sie das charakteristische Polynom $p(X)$ der Differentialgleichung und die Lösungen der inhomogenen Gleichung.

See Solution

Problem 24245

Evaluate the integral $\int_{-10}^{10} \sqrt[3]{16 x^{5}-4 x^{3}-3 x} d x$ . What is the result? Options: 0, -1, 1, None, $1 / 4$ .

See Solution

Problem 24246

Calculate the integral $\int_{0}^{4} \frac{4 x}{\sqrt{x^{2}+9}} d x$ . What is the result? Options: 2, 16, 4, 8.

See Solution

Problem 24247

Find $G'(0)$ for the function $G(x)=\int_{x^{2}+1}^{3 x+4}(t^{2}+2)^{2} dt$ . Choices: 972, 324, 0, 648.

See Solution

Problem 24248

Find the volume of the vase formed by revolving $y=2+\sin (x)$ from $x=0$ to $x=5$ around the x-axis in cubic inches.

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Problem 24249

A 10-ft ladder leans against a house. When the base is 8 ft away and moving at 2 ft/sec, how fast is the top sliding down?

See Solution

Problem 24250

Find the absolute maximum of $f(x)=x^{4}+4x+5$ on the interval $[0,1]$ .

See Solution

Problem 24251

Find the critical numbers of the function $f(x)=x^{3}+x^{2}-5x-5$ .

See Solution

Problem 24252

Evaluate the integral $I=\int_{0}^{\pi} \frac{6}{3-2 \cos ^{2}(x)} d x$ and find the bounds $m$ and $n$ for $m \leq I \leq n$ .

See Solution

Problem 24253

Classify the following theorems: a) Intermediate Value Theorem, b) Mean Value Theorem, c) Extreme Value Theorem.

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Problem 24254

Determine where the function $f(x)=x^{4}+4 x+5$ is increasing: $(-\infty,-6]$ , $[-1, \infty)$ , $(-\infty,-1]$ , or $R$ .

See Solution

Problem 24255

Find the limit: $\lim _{x \rightarrow 1} \frac{|2 x-1|-|4-5 x|}{x-1}$ . What is the value?

See Solution

Problem 24256

Hausaufgabe 21: Betrachte die Differentialgleichung $y^{\prime \prime}-\frac{1}{x} y^{\prime}-\frac{1}{x^{2}} y=\ln (x)$ .
(a) Für welches $\alpha$ ist $g(x)=x^{\alpha}$ eine Lösung der homogenen Gleichung? (b) Finde ein Fundamentalsystem der homogenen Gleichung. (c) Bestimme eine partikuläre Lösung $f_{\mathrm{p}}(x)$ . (d) Löse das Anfangswertproblem $y^{\prime \prime}-\frac{1}{x} y^{\prime}-\frac{1}{x^{2}} y=\ln (x)$ mit $y(1)=0$ , $y^{\prime}(1)=0$ .

See Solution

Problem 24257

Find the average value of the function $f(x)=6 x^{2}+4 x-5$ on the interval $[1,2]$ .

See Solution

Problem 24258

Find the derivative $\frac{d}{dx} f(g(x))$ for $f(x)=(x+2)^{2}$ and $g(x)=\cos(2x)$ at $x=\pi$ .

See Solution

Problem 24259

Find values of $a$ for the function defined as $f(x)=\frac{x^{3}-1}{x-1}$ if $x \neq 1$ and $f(1)=2a^{2}-5$ . Options: $1,-1$ ; $2$ only; none; $-2$ only; $2,-2$ .

See Solution

Problem 24260

Find the Linearization $L(x)$ of $f(x)=\left(1-\frac{1}{2+x}\right)^{2 / 3}$ at $x=0$ .

See Solution

Problem 24261

Find $k > 0$ such that $\int_{0}^{k} \frac{x}{x^{2}+4} dx = \frac{1}{2} \ln (4)$ . Choices: A. 0 B. $\sqrt{2}$ C. 2 D. $\sqrt{12}$ E. $\frac{1}{2} \tan (\ln (\sqrt{2})$ .

See Solution

Problem 24262

Evaluate the integral: $\int \frac{7-98 x}{\sqrt{81-49 x^{2}}} d x$

See Solution

Problem 24263

Gegeben ist die Funktion $f(x)=x^{2}-3 x$ . Skizzieren Sie den Graphen für $-1 \leq x \leq 4$ und berechnen Sie Steigung und Winkel bei $x_0=2$ .

See Solution

Problem 24264

Where does the function $f(x)=\sqrt[3]{x^{2}-1}$ have a vertical tangent line? Choices: $x=0$ , $x=\pm3$ , $x=\pm1$ , None.

See Solution

Problem 24265

Find the limit as $x$ approaches 3 of $3f^2(x) + \frac{\sqrt{x^2+7}-4}{2-\sqrt{x+1}} + 8$ given $f(3)=2$ . Choices: 17, 1, 5, DNE, None.

See Solution

Problem 24266

Calculate the integral $\int_{1}^{2} \frac{x^{2}-x-5}{x+2} dx$ . Choose the correct answer from the options given.

See Solution

Problem 24267

A tank has 50 liters of oil at $t=4$ hours. Use a right Riemann sum with $R(t)$ values to find oil at $t=15$ hours. Options: A. 64.9 B. 68.2 C. 114.9 D. 116.6 E. 118.2

See Solution

Problem 24268

Gegeben ist die Funktion $f(x)=x^{2}-3 x$ . Skizzieren Sie den Graphen für $-1 \leq x \leq 4$ und bestimmen Sie Steigung und Winkel bei $x_{0}=2$ .

See Solution

Problem 24269

A 10-ft ladder leans against a house. When the base is 8 ft away and moving at 2 ft/sec, how fast is the top sliding down?

See Solution

Problem 24270

Find the average value of the function $f(x)=6 x^{2}+4 x-5$ on the interval $[1,2]$ . Options: 15, 30, 3, $15 / 2$ .

See Solution

Problem 24271

Find the limit as $\theta$ approaches 0 from the positive side for $(2 - \cot \theta)$ . What is it?

See Solution

Problem 24272

If $\sqrt{x y}=1$ , find $\frac{d y}{d x}$ . Choose the correct option from the list.

See Solution

Problem 24273

Find the average rate of change of $f(x)=3(2)^{2x}$ from $x=0$ to $x=2$ . Options: a) 45, b) 225, c) 75, d) 5.5.

See Solution

Problem 24274

Find the instantaneous rate of change of $f(x)=3 x^{4}+2 x-1$ at $x=-1$ . Options: a) -10, b) 10, c) 0, d) -14.

See Solution

Problem 24275

Identify the FALSE statement about average rate of change from the options provided.

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Problem 24276

Estimate the instantaneous rate of change of $f(x)=\frac{x}{x-2}$ at $x=2$ . Options: a) 20000 b) 2 c) 0 d) Not continuous.

See Solution

Problem 24277

Estimate the rate of change of $f(x)=\cos (x-\pi)+4$ at $x=0$ . Choose: a) 4, b) 1, c) -1, d) 0.

See Solution

Problem 24278

Estimate the instantaneous rate of change of a function at a specific $x$ -value using slopes of secant or tangent lines.

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Problem 24279

Materialprobe wird erhitzt: $f(t)=70-50 \cdot e^{-0,2 t}$ . Bestimme Graphen, maximale Erwärmungsgeschwindigkeit, Durchschnittstemperatur, Zeit für Hälfte der Endtemperatur und Zeit bis zur Halbierung der Anfangsgeschwindigkeit.

See Solution

Problem 24280

Gegeben ist die Funktion $f(x)=\frac{3}{2} x-\frac{1}{2} x^{2}$ .
a) Berechne die Steigung bei $x_{0}=1$ . b) Finde die Tangentengleichung bei $x_{0}=1$ . c) Bestimme den Winkel $\gamma$ an der $y$ -Achse. d) Finde $x_{1}$ für einen Steigungswinkel von $60^{\circ}$ .

See Solution

Problem 24281

Berechnen Sie mit der Funktion $f(x)=-\frac{1}{250} x^{3}+\frac{1}{10} x^{2}$ die EHEC-Erkrankten am Tag 10, das Ende der Epidemie, und die Symmetrie.

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Problem 24282

A 250 kg roller coaster starts at a 20 m hill. What is its speed at the bottom, ignoring friction?

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Problem 24283

Estimate the area under $f(x)=x^{3}$ from $x=1$ to $x=2$ using 2 and then 4 rectangles at midpoints.

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Problem 24284

Estimate the area under $f(x)=8-x^{2}+2x$ from $x=-2$ to $x=4$ using lower/upper sums with 2 or 4 rectangles.

See Solution

Problem 24285

Estimate the area under $f(x)=3 x^{2}$ from $x=0$ to $x=10$ using lower and upper sums with 2 and 4 rectangles.

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Problem 24286

If $\lim _{x \rightarrow 2} \frac{f(x)}{(x-2)^{2}}=\frac{\pi}{2}$ , find $\lim _{x \rightarrow 2} \frac{f(x)}{x-2}$ .

See Solution

Problem 24287

Berechne den Wirkstoffgehalt nach 6 Stunden, wenn $60 \mathrm{mg}$ eingenommen und $40 \mathrm{mg}$ nach 1 Stunde nachweisbar sind.

See Solution

Problem 24288

Find the rate of change of $f(x)=\frac{x}{x-3}$ for $-1 \leq x \leq 4$ . Round your answer to two decimal places.

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Problem 24289

Leia funktsiooni $y=f(x)$ puutuja tõusunurk, kui $y=4x-x^{2}$ ja $x_{0}=-2$ . Vastus täisarvuna.

See Solution

Problem 24290

Approximate the integral $\int_{2}^{14}(x^{2}-5) \, dx$ using 4 rectangles.

See Solution

Problem 24291

Find the instantaneous rate of change of $f(x)=\frac{x}{x-2}$ at $x=-1$ . Round your answer to two decimal places.

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Problem 24292

Zeigen Sie, dass $(k+2)! \geq 2^{k}$ für alle $k \in \mathbb{N}_{0}$ und berechnen Sie $\lim_{m \to \infty} m \sum_{k=2}^{\infty} \frac{1}{m^{k} k!}$ .

See Solution

Problem 24293

Evaluate the integral $\int_{-2}^{5} \frac{d x}{\sqrt[3]{x^{2}+6 x+9}}$ .

See Solution

Problem 24294

Find the instantaneous rate of change of $f(x)=3x+2$ at $x=-2$ . What is the value?

See Solution

Problem 24295

Estimate the rate of change of $f(x)=\log x$ at $x=100$ . Options: 0.00043, 2, 0.0043, 0.000087.

See Solution

Problem 24296

A $1.84 \mathrm{~kg}$ fish falls from $213 \mathrm{~m}$ while the pelican flies at $5.68 \mathrm{~m/s}$ . Find the fish's impact speed.

See Solution

Problem 24297

Vergleiche die Temperaturänderungen der Funktionen $f(x)=x^{2}+25$ und $g(x)=250-\frac{226}{x^{2}+1}$ in den ersten 15 Minuten.

See Solution

Problem 24298

Select functions with a positive instantaneous rate of change at $x=0$ : $y=\log _{2}(x)$ , $y=\left(\frac{1}{2}\right)^{x}$ , $y=\tan x$ , $y=2^{x}$ .

See Solution

Problem 24299

La suite $w_{n}=\frac{2}{n^{2}}$ converge vers 0. Que se passe-t-il pour les valeurs de $w_{n}$ dans un intervalle $[A;+\infty[$ ?

See Solution

Problem 24300

Find the rate of change of $y=\tan x$ at $x=0$ . Use $\frac{f(x+h)-f(x)}{h}$ to estimate it. What is the result?

See Solution

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Calculus

Problem 24201

Calculate the integral ∫−101016x5−4x3−3x3dx\int_{-10}^{10} \sqrt[3]{16 x^{5}-4 x^{3}-3 x} d x∫−1010​316x5−4x3−3x​dx and choose the correct value: 0, -1, 1, None, or 1/41/41/4.

Problem 24202

Problem 24203

Problem 24204

Find where the function f(x)=x3−3x3f(x)=\sqrt[3]{x^{3}-3 x}f(x)=3x3−3x​ has vertical tangent lines: x=0,x=−3x=0, x=-3x=0,x=−3; x=1,x=−1x=1, x=-1x=1,x=−1; none; x=3,x=−3x=3, x=-3x=3,x=−3; x=0,x=3x=0, x=3x=0,x=3.

Problem 24205

Determine if the series 1+1(1)(1)1/4+1(2)(2)1/4+⋯1+\frac{1}{(1)(1)^{1 / 4}}+\frac{1}{(2)(2)^{1 / 4}}+\cdots1+(1)(1)1/41​+(2)(2)1/41​+⋯ converges or diverges.

Problem 24206

Find the average rate of change of f(x)f(x)f(x) on [1,7][1,7][1,7] given rates of 8 on [1,3][1,3][1,3] and 11 on [3,7][3,7][3,7].

Problem 24207

Find the limit as xxx approaches 3 of 3f2(x)+x2+7−42−x+1−43 f^{2}(x) + \frac{\sqrt{x^{2}+7}-4}{2-\sqrt{x+1}} - 43f2(x)+2−x+1​x2+7​−4​−4 given f(3)=2f(3) = 2f(3)=2. Choices: 17, 5, 4, 1, None.

Problem 24208

Problem 24209

Evaluate the integral ∫442x3+5dx\int_{4}^{4} 2 \sqrt{x^{3}+5} d x∫44​2x3+5​dx. What is the result?

Problem 24210

Find the limit: lim⁡x→4−x+1(x−4)2\lim _{x \rightarrow 4^{-}} \frac{x+1}{(x-4)^{2}}limx→4−​(x−4)2x+1​. What is its value? 0, −∞-\infty−∞, 1, or ∞\infty∞?

Problem 24211

Find the average value of the function f(x)=6x2+4x−5f(x)=6 x^{2}+4 x-5f(x)=6x2+4x−5 on the interval [1,2][1,2][1,2]. Choices: 15, 30, 3, 15/215 / 215/2.

Problem 24212

Problem 24213

Find the absolute maximum of f(x)=x4+4x+5f(x)=x^{4}+4x+5f(x)=x4+4x+5 on the interval [0,1][0,1][0,1]. Choices: 1, -4, 12, 10.

Problem 24214

Find the value of lim⁡x→3(3f2(x)+x2+7−42−x+1+8)\lim _{x \rightarrow 3}\left(3 f^{2}(x)+\frac{\sqrt{x^{2}+7}-4}{2-\sqrt{x+1}}+8\right)limx→3​(3f2(x)+2−x+1​x2+7​−4​+8) if f(3)=2f(3)=2f(3)=2. Choices: 17, 1, 5, Does Not Exist, None.

Problem 24215

Find the tangent line to (x+y)3−5x+7y=10(x+y)^{3}-5 x+7 y=10(x+y)3−5x+7y=10 at the intersection with 2−x−y=02-x-y=02−x−y=0. Which equation is correct?

Problem 24216

Find where the function f(x)=x2−13f(x)=\sqrt[3]{x^{2}-1}f(x)=3x2−1​ has a vertical tangent line: x=0x=0x=0, x=3x=3x=3, x=−3x=-3x=−3, x=1x=1x=1, x=−1x=-1x=−1, or none?

Problem 24217

Calculate ∫042xx2+9dx\int_{0}^{4} \frac{2 x}{\sqrt{x^{2}+9}} d x∫04​x2+9​2x​dx. What is the result? 4, 8, 1, or 2?

Problem 24218

Express the integral ∫1311+xdx\int_{1}^{3} \frac{1}{1+x} d x∫13​1+x1​dx as a limit of a Riemann sum.

Problem 24219

Find the critical numbers of the function f(x)=x3+x2−5x−5f(x)=x^{3}+x^{2}-5x-5f(x)=x3+x2−5x−5.

Problem 24220

Find G′(0)G^{\prime}(0)G′(0) for the function G(x)=∫x2+13x+4(t2+2)2dtG(x)=\int_{x^{2}+1}^{3 x+4}(t^{2}+2)^{2} dtG(x)=∫x2+13x+4​(t2+2)2dt.

Problem 24221

Calculate the integral ∫−8864−x2dx\int_{-8}^{8} \sqrt{64-x^{2}} d x∫−88​64−x2​dx. What is the result? Options: −12/3-12 / 3−12/3, 32π32 \pi32π, 64π64 \pi64π, 12π12 \pi12π.

Problem 24222

Find the inflection point of the function f(x)=x+4xf(x)=x+\frac{4}{x}f(x)=x+x4​ from the options: x=0x=0x=0, x=−1x=-1x=−1, x=1x=1x=1, or None.

Problem 24223

Find lim⁡x→1∣2x−1∣−∣4−5x∣x−1\lim _{x \rightarrow 1} \frac{|2 x-1|-|4-5 x|}{x-1}limx→1​x−1∣2x−1∣−∣4−5x∣​. What is the limit value? Choices: −3-3−3, 3, Does Not Exist, −1-1−1.

Problem 24224

Evaluate the integral ∫116x11−45dx\int_{1}^{1} 6 \sqrt[5]{x^{11}-4} d x∫11​65x11−4​dx. What is the result? Choices: -1, 1/3, 0, 1.

Problem 24225

Calculate ∫−224−x2dx\int_{-2}^{2} \sqrt{4-x^{2}} d x∫−22​4−x2​dx. What is the result: 12π12 \pi12π, 6π6 \pi6π, 2π2 \pi2π, or 4π4 \pi4π?

Problem 24226

Verify the inequality: ∫34x dx≤∫34x2 dx\int_{3}^{4} x \, dx \leq \int_{3}^{4} x^{2} \, dx∫34​xdx≤∫34​x2dx without calculating the integrals.

Problem 24227

Calculate the integral ∫−22x5+x35dx\int_{-2}^{2} \sqrt[5]{x^{5}+x^{3}} d x∫−22​5x5+x3​dx and choose the correct value: 0, −1-1−1, 1/31/31/3, or none.

Problem 24228

Evaluate the integral ∫042xx2+9dx\int_{0}^{4} \frac{2 x}{\sqrt{x^{2}+9}} d x∫04​x2+9​2x​dx. What is the result?

Problem 24229

Find the value of ccc that satisfies the Intermediate Value Theorem for f(x)=x3+1f(x)=x^{3}+1f(x)=x3+1 on [−1,2][-1,2][−1,2] with w=2w=2w=2.

Problem 24230

Find the bounds mmm and nnn for the integral I=∫0π63−2cos⁡2(x)dxI=\int_{0}^{\pi} \frac{6}{3-2 \cos ^{2}(x)} d xI=∫0π​3−2cos2(x)6​dx.

Problem 24231

Gegeben ist die Funktion f(x)=(10x−5)⋅e2xf(x)=(10 x-5) \cdot e^{2 x}f(x)=(10x−5)⋅e2x. Bestimmen Sie Nullstellen, Extrempunkte, Wendepunkt und Fläche für x=−2x=-2x=−2.

Problem 24232

Find the limit: lim⁡x→3+∣x−3∣x−3\lim _{x \rightarrow 3^{+}} \frac{|x-3|}{\sqrt{x-3}}limx→3+​x−3​∣x−3∣​. What is the result?

Problem 24233

Problem 24234

Find the average rate of change of f(x)f(x)f(x) on [1,7][1,7][1,7] given rates of 8 on [1,3][1,3][1,3] and 11 on [3,7][3,7][3,7]. Choices: 1, 5, None, 19, 10.

Problem 24235

Find G′(1)G'(1)G′(1) for G(x)=∫x2+12x+4(t2+2)2dtG(x)=\int_{x^{2}+1}^{2x+4}(t^{2}+2)^{2} dtG(x)=∫x2+12x+4​(t2+2)2dt. Choose the correct value from the options.

Problem 24236

Find the general antiderivative of these functions: a. f(x)=sec⁡(x)+cos⁡(x)cos⁡(x)f(x)=\frac{\sec(x)+\cos(x)}{\cos(x)}f(x)=cos(x)sec(x)+cos(x)​; b. f(x)=aa+xf(x)=\frac{a}{a+x}f(x)=a+xa​.

Problem 24237

Evaluate the integral I=∫0π63−2cos⁡2(x)dxI=\int_{0}^{\pi} \frac{6}{3-2 \cos ^{2}(x)} d xI=∫0π​3−2cos2(x)6​dx to find mmm and nnn such that m≤I≤nm \leq I \leq nm≤I≤n. Options: π\piπ, 6π6\pi6π; 2π2\pi2π, 6π6\pi6π; 2π2\pi2π, 3π3\pi3π; None; -2, 12.

Problem 24238

Determine the intervals where the function f(x)=x2+1xf(x)=\frac{x^{2}+1}{x}f(x)=xx2+1​ is concave upward: (−∞,0)(-\infty, 0)(−∞,0), RRR, (0,∞)(0, \infty)(0,∞), (−∞−2)(-\infty-2)(−∞−2).

Problem 24239

Evaluate the integral ∫−22x5+x35dx\int_{-2}^{2} \sqrt[5]{x^{5}+x^{3}} d x∫−22​5x5+x3​dx. What is the result? Choices: 1/31/31/3, 0, 1, −1-1−1, None.

Problem 24240

Problem 24241

Problem 24242

Berechne den Differenzenquotienten von f(x)=−7x2+1f(x)=-7 x^{2}+1f(x)=−7x2+1 für die Intervalle [0,12][0, \frac{1}{2}][0,21​], [0,14][0, \frac{1}{4}][0,41​], [0,116][0, \frac{1}{16}][0,161​]. Stelle eine Vermutung für [0,1n][0, \frac{1}{n}][0,n1​] auf, wenn nnn groß ist.

Problem 24243

Problem 24244

Calculate the integral $\int_{-10}^{10} \sqrt[3]{16 x^{5}-4 x^{3}-3 x} d x$ and choose the correct value: 0, -1, 1, None, or $1/4$ .

Find where the function $f(x)=\sqrt[3]{x^{3}-3 x}$ has vertical tangent lines: $x=0, x=-3$ ; $x=1, x=-1$ ; none; $x=3, x=-3$ ; $x=0, x=3$ .

Determine if the series $1+\frac{1}{(1)(1)^{1 / 4}}+\frac{1}{(2)(2)^{1 / 4}}+\cdots$ converges or diverges.

Find the average rate of change of $f(x)$ on $[1,7]$ given rates of 8 on $[1,3]$ and 11 on $[3,7]$ .

Find the limit as $x$ approaches 3 of $3 f^{2}(x) + \frac{\sqrt{x^{2}+7}-4}{2-\sqrt{x+1}} - 4$ given $f(3) = 2$ . Choices: 17, 5, 4, 1, None.

Evaluate the integral $\int_{4}^{4} 2 \sqrt{x^{3}+5} d x$ . What is the result?

Find the limit: $\lim _{x \rightarrow 4^{-}} \frac{x+1}{(x-4)^{2}}$ . What is its value? 0, $-\infty$ , 1, or $\infty$ ?

Find the average value of the function $f(x)=6 x^{2}+4 x-5$ on the interval $[1,2]$ . Choices: 15, 30, 3, $15 / 2$ .

Find the absolute maximum of $f(x)=x^{4}+4x+5$ on the interval $[0,1]$ . Choices: 1, -4, 12, 10.

Find the value of $\lim _{x \rightarrow 3}\left(3 f^{2}(x)+\frac{\sqrt{x^{2}+7}-4}{2-\sqrt{x+1}}+8\right)$ if $f(3)=2$ . Choices: 17, 1, 5, Does Not Exist, None.

Find the tangent line to $(x+y)^{3}-5 x+7 y=10$ at the intersection with $2-x-y=0$ . Which equation is correct?

Find where the function $f(x)=\sqrt[3]{x^{2}-1}$ has a vertical tangent line: $x=0$ , $x=3$ , $x=-3$ , $x=1$ , $x=-1$ , or none?

Calculate $\int_{0}^{4} \frac{2 x}{\sqrt{x^{2}+9}} d x$ . What is the result? 4, 8, 1, or 2?

Express the integral $\int_{1}^{3} \frac{1}{1+x} d x$ as a limit of a Riemann sum.

Find the critical numbers of the function $f(x)=x^{3}+x^{2}-5x-5$ .

Find $G^{\prime}(0)$ for the function $G(x)=\int_{x^{2}+1}^{3 x+4}(t^{2}+2)^{2} dt$ .

Calculate the integral $\int_{-8}^{8} \sqrt{64-x^{2}} d x$ . What is the result? Options: $-12 / 3$ , $32 \pi$ , $64 \pi$ , $12 \pi$ .

Find the inflection point of the function $f(x)=x+\frac{4}{x}$ from the options: $x=0$ , $x=-1$ , $x=1$ , or None.

Find $\lim _{x \rightarrow 1} \frac{|2 x-1|-|4-5 x|}{x-1}$ . What is the limit value? Choices: $-3$ , 3, Does Not Exist, $-1$ .

Evaluate the integral $\int_{1}^{1} 6 \sqrt[5]{x^{11}-4} d x$ . What is the result? Choices: -1, 1/3, 0, 1.

Calculate $\int_{-2}^{2} \sqrt{4-x^{2}} d x$ . What is the result: $12 \pi$ , $6 \pi$ , $2 \pi$ , or $4 \pi$ ?

Verify the inequality: $\int_{3}^{4} x \, dx \leq \int_{3}^{4} x^{2} \, dx$ without calculating the integrals.

Calculate the integral $\int_{-2}^{2} \sqrt[5]{x^{5}+x^{3}} d x$ and choose the correct value: 0, $-1$ , $1/3$ , or none.

Evaluate the integral $\int_{0}^{4} \frac{2 x}{\sqrt{x^{2}+9}} d x$ . What is the result?

Find the value of $c$ that satisfies the Intermediate Value Theorem for $f(x)=x^{3}+1$ on $[-1,2]$ with $w=2$ .

Find the bounds $m$ and $n$ for the integral $I=\int_{0}^{\pi} \frac{6}{3-2 \cos ^{2}(x)} d x$ .

Gegeben ist die Funktion $f(x)=(10 x-5) \cdot e^{2 x}$ . Bestimmen Sie Nullstellen, Extrempunkte, Wendepunkt und Fläche für $x=-2$ .

Find the limit: $\lim _{x \rightarrow 3^{+}} \frac{|x-3|}{\sqrt{x-3}}$ . What is the result?

Find the average rate of change of $f(x)$ on $[1,7]$ given rates of 8 on $[1,3]$ and 11 on $[3,7]$ . Choices: 1, 5, None, 19, 10.

Find $G'(1)$ for $G(x)=\int_{x^{2}+1}^{2x+4}(t^{2}+2)^{2} dt$ . Choose the correct value from the options.

Find the general antiderivative of these functions: a. $f(x)=\frac{\sec(x)+\cos(x)}{\cos(x)}$ ; b. $f(x)=\frac{a}{a+x}$ .

Evaluate the integral $I=\int_{0}^{\pi} \frac{6}{3-2 \cos ^{2}(x)} d x$ to find $m$ and $n$ such that $m \leq I \leq n$ . Options: $\pi$ , $6\pi$ ; $2\pi$ , $6\pi$ ; $2\pi$ , $3\pi$ ; None; -2, 12.

Determine the intervals where the function $f(x)=\frac{x^{2}+1}{x}$ is concave upward: $(-\infty, 0)$ , $R$ , $(0, \infty)$ , $(-\infty-2)$ .

Evaluate the integral $\int_{-2}^{2} \sqrt[5]{x^{5}+x^{3}} d x$ . What is the result? Choices: $1/3$ , 0, 1, $-1$ , None.

Berechne den Differenzenquotienten von $f(x)=-7 x^{2}+1$ für die Intervalle $[0, \frac{1}{2}]$ , $[0, \frac{1}{4}]$ , $[0, \frac{1}{16}]$ . Stelle eine Vermutung für $[0, \frac{1}{n}]$ auf, wenn $n$ groß ist.

Bestimmen Sie das charakteristische Polynom $p(X)$ der Differentialgleichung und die Lösungen der inhomogenen Gleichung.

Evaluate the integral $\int_{-10}^{10} \sqrt[3]{16 x^{5}-4 x^{3}-3 x} d x$ . What is the result? Options: 0, -1, 1, None, $1 / 4$ .

Calculate the integral $\int_{0}^{4} \frac{4 x}{\sqrt{x^{2}+9}} d x$ . What is the result? Options: 2, 16, 4, 8.

Find $G'(0)$ for the function $G(x)=\int_{x^{2}+1}^{3 x+4}(t^{2}+2)^{2} dt$ . Choices: 972, 324, 0, 648.

Find the volume of the vase formed by revolving $y=2+\sin (x)$ from $x=0$ to $x=5$ around the x-axis in cubic inches.

Find the absolute maximum of $f(x)=x^{4}+4x+5$ on the interval $[0,1]$ .

Find the critical numbers of the function $f(x)=x^{3}+x^{2}-5x-5$ .

Evaluate the integral $I=\int_{0}^{\pi} \frac{6}{3-2 \cos ^{2}(x)} d x$ and find the bounds $m$ and $n$ for $m \leq I \leq n$ .

Determine where the function $f(x)=x^{4}+4 x+5$ is increasing: $(-\infty,-6]$ , $[-1, \infty)$ , $(-\infty,-1]$ , or $R$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{|2 x-1|-|4-5 x|}{x-1}$ . What is the value?

Find the average value of the function $f(x)=6 x^{2}+4 x-5$ on the interval $[1,2]$ .

Find the derivative $\frac{d}{dx} f(g(x))$ for $f(x)=(x+2)^{2}$ and $g(x)=\cos(2x)$ at $x=\pi$ .

Find values of $a$ for the function defined as $f(x)=\frac{x^{3}-1}{x-1}$ if $x \neq 1$ and $f(1)=2a^{2}-5$ . Options: $1,-1$ ; $2$ only; none; $-2$ only; $2,-2$ .

Find the Linearization $L(x)$ of $f(x)=\left(1-\frac{1}{2+x}\right)^{2 / 3}$ at $x=0$ .

Find $k > 0$ such that $\int_{0}^{k} \frac{x}{x^{2}+4} dx = \frac{1}{2} \ln (4)$ . Choices: A. 0 B. $\sqrt{2}$ C. 2 D. $\sqrt{12}$ E. $\frac{1}{2} \tan (\ln (\sqrt{2})$ .

Evaluate the integral: $\int \frac{7-98 x}{\sqrt{81-49 x^{2}}} d x$

Gegeben ist die Funktion $f(x)=x^{2}-3 x$ . Skizzieren Sie den Graphen für $-1 \leq x \leq 4$ und berechnen Sie Steigung und Winkel bei $x_0=2$ .

Where does the function $f(x)=\sqrt[3]{x^{2}-1}$ have a vertical tangent line? Choices: $x=0$ , $x=\pm3$ , $x=\pm1$ , None.

Find the limit as $x$ approaches 3 of $3f^2(x) + \frac{\sqrt{x^2+7}-4}{2-\sqrt{x+1}} + 8$ given $f(3)=2$ . Choices: 17, 1, 5, DNE, None.

Calculate the integral $\int_{1}^{2} \frac{x^{2}-x-5}{x+2} dx$ . Choose the correct answer from the options given.

A tank has 50 liters of oil at $t=4$ hours. Use a right Riemann sum with $R(t)$ values to find oil at $t=15$ hours. Options: A. 64.9 B. 68.2 C. 114.9 D. 116.6 E. 118.2

Gegeben ist die Funktion $f(x)=x^{2}-3 x$ . Skizzieren Sie den Graphen für $-1 \leq x \leq 4$ und bestimmen Sie Steigung und Winkel bei $x_{0}=2$ .

Find the average value of the function $f(x)=6 x^{2}+4 x-5$ on the interval $[1,2]$ . Options: 15, 30, 3, $15 / 2$ .

Find the limit as $\theta$ approaches 0 from the positive side for $(2 - \cot \theta)$ . What is it?

If $\sqrt{x y}=1$ , find $\frac{d y}{d x}$ . Choose the correct option from the list.

Find the average rate of change of $f(x)=3(2)^{2x}$ from $x=0$ to $x=2$ . Options: a) 45, b) 225, c) 75, d) 5.5.

Find the instantaneous rate of change of $f(x)=3 x^{4}+2 x-1$ at $x=-1$ . Options: a) -10, b) 10, c) 0, d) -14.