Calculus

Problem 13201

Find the function $f(x)$ such that $f^{\prime}(x)=\frac{5}{x}$ and it passes through the point $(1,2)$ .

See Solution

Problem 13202

Evaluate the integral I₁ = ∫₀² 3√(4 - x²) dx using known areas.

See Solution

Problem 13203

Express the limit as a definite integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(2 x_{i}^{} \sin x_{i}^{}\right) \Delta x$ for $n$ subintervals in $[1,7]$ .

See Solution

Problem 13204

Untersuchen Sie die Funktion $f_{a}(x)=x^{3}-3 a^{2} x+2 a^{3}$ auf Extrema und Wendepunkte.

See Solution

Problem 13205

Find the value of the definite integral $I=\int_{a}^{b} f(x) d x$ given the Riemann sum $\sum_{i=1}^{n} f\left(x_{i}^{*}\right) \Delta x_{i}=\frac{3 n^{2}-6 n+5}{6 n^{2}}$ .

See Solution

Problem 13206

Find the marginal-cost function and its values for $q=225$ and $q=450$ given $\bar{c}=\frac{2250 e^{\frac{q}{450}}}{q}$ .

See Solution

Problem 13207

Finde die Stelle, an der die Funktion $f(x)=\frac{1}{2} x^{2}$ den Steigungswinkel $\alpha=45^{\circ}$ hat.

See Solution

Problem 13208

Finde die Stelle, an der die Funktion $f(x)=2 \sqrt{x}$ den Steigungswinkel $a=30^{\circ}$ hat.

See Solution

Problem 13209

Untersuche die Funktion $f_{a}(x)=x^{3}-3 a^{2} x+2 a^{3}$ auf Extrema und Wendepunkte und zeige, dass $x=-2 a$ eine Nullstelle ist.

See Solution

Problem 13210

A balloon's radius decreases at $15 \mathrm{~cm/min}$ . Find the volume change rate when volume is $972 \pi \mathrm{cm}^{3}$ . Volume formula: $V=\frac{4}{3} \pi r^{3}$ .

See Solution

Problem 13211

Leite die Funktion $f$ ab und beantworte folgende Fragen:
1. Produktregel: a) $f(x)=x^{2} \cdot \sin (x)$ b) $f(x)=(3 x+2) \cdot \sqrt{x}$ c) $f(x)=\frac{4}{x} \cdot \cos (x)$
2. Kettenregel: a) $f(x)=2 \cdot(3 x-1)^{2}$ b) $f(x)=\sqrt{1-8 x}$ c) $f(x)=\frac{3}{(2 x+3)^{2}}$
3. Produkt- und Kettenregel: a) $f(x)=3 x \cdot \sin (4 x-1)$ b) $f(x)=\sqrt{0,5 x} \cdot(2 x+1)$ c) $f(x)=\frac{1}{5-2 x} \cdot \cos (x)$
4. Gegeben: $f(x)=-x \cdot(2 x+1)$ a) Leite ab und vereinfache. b) Bestimme die Steigung im Punkt $P(1 \mid f(1))$ und die Tangentengleichung. c) Gibt es waagrechte Tangenten? d) Wo ist die Tangente parallel zu $g: y=x+3$ ? Bestimme die Tangentengleichungen.

See Solution

Problem 13212

Find the marginal-cost function for the average cost $\bar{c}=\frac{5500 e^{\frac{q}{550}}}{q}$ at $q=275$ and $q=550$ .

See Solution

Problem 13213

Simplify the difference quotient for the function $f(x)=\frac{2}{3x-1}$ .

See Solution

Problem 13214

Untersuchen Sie die Funktion $f_{a}(x)=x^{3}-3 a^{2} x+2 a^{3}$ auf Extrema, Nullstellen und Symmetrie. Zeichnen Sie die Graphen.

See Solution

Problem 13215

Find if the limit $\lim _{n \rightarrow \infty} \frac{1}{n^{2}}\left\{\sum_{k=1}^{n}(3+2 k)\right\}$ exists and its value.

See Solution

Problem 13216

Find the limit of the sum as $n \to \infty$ : $\sum_{k=1}^{n}\left(\frac{6}{n^{3}} k^{2}-\frac{4}{n^{2}} k+\frac{3}{n}\right)$ .

See Solution

Problem 13217

Calculate these indefinite integrals: (a) $\int x^{6} dx$ , (b) $\int x^{8/11} dx$ , (c) $\int x^{-3} \sqrt{x} dx$ .

See Solution

Problem 13218

Consider the quadratic $f(x) = ax^2 + bx + c$ .
(a) Show that the average rate of change on $[r, s]$ is $a(r+s) + b$ .
(b) Is there a point in $[r, s]$ where the average rate equals the instantaneous rate? Explain.

See Solution

Problem 13219

Find the general antiderivative of the function $\left(\frac{3}{x^{4}}-\frac{6}{x^{6}}+12\right)$ .

See Solution

Problem 13220

Find the general antiderivative of the function $\frac{1}{6 \sqrt{u}}$ . Antiderivative $=$ $+C$ .

See Solution

Problem 13221

Find $h^{\prime}(1)$ for $h(x)=x f(x) \cos (x)$ given $f(1)=7$ and $f^{\prime}(1)=6$ . Round to three decimals.

See Solution

Problem 13222

Untersuchen Sie die Funktion $f_{a}(x)=x^{3}-3 a^{2} x+2 a^{3}$ auf Extrema, Nullstellen und Symmetrie.

See Solution

Problem 13223

Given $f(4)=10$ and $f^{\prime}(4)=9$ , find $(f h)^{\prime}(4)$ where $h(x)=\sqrt{x}$ . Round to two decimal places.

See Solution

Problem 13224

Find the antiderivative $C(x)$ where $C^{\prime}(x)=3 x^{2}-5 x$ and $C(0)=8000.$

See Solution

Problem 13225

Find the limit: $\lim _{x \rightarrow 0} \frac{f(x)}{\sin (3 x)}$ if $\lim _{x \rightarrow 0} f(x)=0$ and $\lim _{x \rightarrow 0} f^{\prime}(x)=2$ .

See Solution

Problem 13226

Find the antiderivative $C(x)$ such that $C'(x)=6x^2-7x$ and $C(0)=2000.$

See Solution

Problem 13227

Bei einer Verkehrszählung wurde die Funktion $f(x)=\frac{1}{10^{10}} x^{5}-\frac{1}{130000} x^{3}+\frac{1}{6} x+6$ verwendet.
a) Bestimmen Sie die Uhrzeit mit der höchsten Verkehrsdichte. b) Berechnen Sie die Fahrzeuge zwischen 6 und 9 Uhr. c) Ermitteln Sie den durchschnittlichen Fahrzeugstrom zwischen 6 und 8 Uhr.

See Solution

Problem 13228

Find the antiderivative $x(t)$ for $\frac{d x}{d t}=2 e^{t}-8$ with the condition $x(0)=6$ .

See Solution

Problem 13229

An oil tank drains oil with volume $V(t)=0.2(25-t)^{3}$ for $t \in[0,25]$ .
a) Find initial oil volume. b) Calculate average volume change in the first and last 10 minutes.

See Solution

Problem 13230

Given $f(x)$ with $f^{\prime \prime}(-1)=f^{\prime \prime}(0)=f^{\prime \prime}(1)=0$ and specific concavity, which is true?

See Solution

Problem 13231

Evaluate the integral $\int \frac{23}{(7 x+11)^{9}} dx$ using the substitution $u=7 x+11$ .

See Solution

Problem 13232

Evaluate the integral $\int \frac{23}{(7 x+11)^{9}} d x$ using the substitution $u=7 x+11$ .

See Solution

Problem 13233

Evaluate the integral $\int x e^{x^2} dx$ . Include the constant $+\mathrm{C}$ .

See Solution

Problem 13234

Find the limit: $\lim _{x \rightarrow 0^{+}}(f(x)+1)^{1 / \sin (2 x)}$ given $\lim _{x \rightarrow 0^{+}} f(x)=0$ and $\lim _{x \rightarrow 0^{+}} f^{\prime}(x)=2$ .

See Solution

Problem 13235

Evaluate the integral: $\int 4 \sin ^{6}(x) \cos (x) d x$

See Solution

Problem 13236

Find the derivative of $A(x)=\left(\frac{1}{3} x+2\right)^{2}$ .

See Solution

Problem 13237

Find the average and instantaneous rates of change for the boat's distance function $d(t)=0.002 t^{3}+0.05 t^{2}+0.3 t$ at $t=10$ . Interpret both results.

See Solution

Problem 13238

A person 1.7 m tall walks away from a 2.55 m pole at 2 m/s. Find the speed of the tip of his shadow when 8 m from the pole.

See Solution

Problem 13239

Two cars are moving towards an intersection. Car A is 300 m east at 60 km/h, and Car B is 400 m south at 75 km/h. Find the rate of distance change.

See Solution

Problem 13240

Bestimme die Stammfunktion von $\int_{-1}^{1} e^{e} \cdot (x \cdot e^{x}) \, dx =$

See Solution

Problem 13241

Berechne den Flächeninhalt des Segels, das von $f(x)=x+6-\frac{32}{x^{2}}$ , der $x$ -Achse und der Geraden $g$ gebildet wird.

See Solution

Problem 13242

Bestimme die Stammfunktion von $\int_{-1}^{1}\left[e: e^{x}-\left(e \cdot e^{x}\right)\right] d x$ .

See Solution

Problem 13243

Find the correct assertion from the Taylor polynomial $T_{3}(x)=2+(x+\pi)-3(x+\pi)^{2}-(x+\pi)^{3}$ .

See Solution

Problem 13244

Berechne den Flächeninhalt zwischen $f(x)=2 \sqrt{x}$ und $g(x)=2x-4$ für $x \geq 0$ .

See Solution

Problem 13245

Determine the intervals where the function $f(x)=\frac{x^{2}}{2}-2 x+(3+2 x) \ln x+c$ is concave up.

See Solution

Problem 13246

Two cars start from the same point: one goes east at $75 \mathrm{~km/h}$ , the other north at $40 \mathrm{~km/h}$ . Find the distance increase rate after 1 hour.

See Solution

Problem 13247

Given Boyle's Law $PV=C$ , find the rate of volume decrease when $V=450 \mathrm{~cm}^{3}$ , $P=160 \mathrm{kPa}$ , and $\frac{dP}{dt}=20 \frac{\mathrm{kPa}}{\min}$ .

See Solution

Problem 13248

A cylinder with diameter 12 cm leaks water at 0.5 cm/min. Find the leak rate in cm³/min when water is 4 cm high.

See Solution

Problem 13249

Calculate the area between the lines $y=x+4$ and $y=9x^{2}+x$ from $x=1$ to $x=2$ .

See Solution

Problem 13250

Insect population $\mathrm{P}(t)=700 e^{0.08 t}$ : (a) Find $\mathrm{P}(0)$ . (b) What is the growth rate? (c) Find $\mathrm{P}(10)$ . (d) When is $\mathrm{P}=980$ ? (e) When does $\mathrm{P}$ double?

See Solution

Problem 13251

A trebuchet launches a projectile from $47 \mathrm{ft}$ at $40 \mathrm{ft/s}$ . Find the time to reach $60 \mathrm{ft}$ and return.

See Solution

Problem 13252

Differentiate $f(x)=\frac{2}{x}-1$ and check if it's differentiable at $x=1$ .

See Solution

Problem 13253

Bestimmen Sie die Steigung von $K_{r}$ bei $x=-1$ und den Schnittpunkten von $f(x)=3 x^{2}-5 x$ mit der $\mathrm{x}$ -Achse.

See Solution

Problem 13254

Is the tangent line to $f(x)=x e^{-x}$ at $x=1$ above or below the graph? Explain why.

See Solution

Problem 13255

Berechne die Integrale für die Funktion $f$ und gib die Formeln an. a) $\int_{1}^{4} x^{2} d x$ b) $\int_{3}^{5} x d x$ c) $\int_{7}^{8} 3 d x$

See Solution

Problem 13256

Find the function $S(t)$ for monthly sales declining at $S^{\prime}(t)=-10 t^{\frac{2}{3}}$ from 920 to 500 computers.

See Solution

Problem 13257

Un gérant veut construire un enclos rectangulaire de 48 m². Trois côtés coûtent 18 \$/m² et un côté 80 \$/m². Trouver les dimensions minimales.

See Solution

Problem 13258

Bestimme die Integrationsgrenze $b$ für a) $\int_{0}^{b} x \, dx = 2$ und b) $\int_{0}^{b} x^{2} \, dx = 9$ .

See Solution

Problem 13259

Bestimme die Grenze $b$ für $\int_{0}^{b} x^{2} d x=5$ .

See Solution

Problem 13260

Find the first and second derivatives of $f(x) = \sqrt{2}x - 2\cos(x)$ and verify if $f''(x) = 25x$ is correct.

See Solution

Problem 13261

Find the derivative of $f(x)=\frac{4x^2}{x^2}-\frac{5x^2}{x^2}-\frac{2}{x^2}$ .

See Solution

Problem 13262

A cylinder with diameter 12 cm leaks water at 0.5 cm/min. Find the leak rate in cm³/min when water is 4 cm high.

See Solution

Problem 13263

Given $f(3)=8$ , $f'(3)=5$ , $g(3)=9$ , $g'(3)=7$ , find $\left.\frac{d}{d x}\left(\frac{f(x)}{g(x)}\right)\right|_{x=3}$ . Round to 2 decimals.

See Solution

Problem 13264

Find the domain of $f(x)=x^{2} \ln x$ , critical points, intervals of increase/decrease, and inflection point.

See Solution

Problem 13265

Given $f(5)=2$ , $f'(5)=8$ , $g(5)=9$ , and $g'(5)=3$ , find $\left.\frac{d}{d x}(2 f(x)-3 g(x)+4)\right|_{x=5}$ . Round to 2 decimal places.

See Solution

Problem 13266

Given $f(-2)=4$ , $f'(-2)=10$ , $g(-2)=5$ , $g'(-2)=7$ , find $\left.\frac{d}{d x}(f(x) \cdot g(x))\right|_{x=-2}$ .

See Solution

Problem 13267

Find which statement is true based on the Taylor polynomial $T_{3}(x)=2+(x+\pi)-3(x+\pi)^{2}-(x+\pi)^{3}$ .

See Solution

Problem 13268

Find values of $a$ and $b$ for $f(x)=a x^{3}+x^{2}+b x-2$ to have a max at $x=3$ and min at $x=2$ .

See Solution

Problem 13269

Determine where the function $f(x)=\frac{x^{2}}{2}-2 x+(3+2 x) \ln x+c$ is concave up: $(0.5,3)$ , $(0,1)$ , $(1, \infty)$ , $(0,0.5)$ and $(1, \infty)$ , or never.

See Solution

Problem 13270

Find the integral of $\frac{2 x^{2}}{1+x^{2}} \, dx$ .

See Solution

Problem 13271

Finde den Punkt auf dem Graphen von $h(x)=4 x^{3}-3 x^{2}+5 x-7$ , wo die Steigung $m=65$ ist.

See Solution

Problem 13272

Find the critical points of $f(x)=\frac{(2-7 x)}{8^{x}}$ . Provide exact answers, stating if there are no critical points.

See Solution

Problem 13273

A firm has a price of \ $5.00 and a production function$ Q=f(L, K)=20 \times L^{0.25} \times K^{0.5} $. Find the profit-maximizing inputs$ (\bar{L}, \bar{K})$.

See Solution

Problem 13274

Evaluate the integral: $\int\left(\frac{3}{x}+e^{3 x}\right) d x=?$

See Solution

Problem 13275

Two cars move from the same point: one east at $72 \mathrm{~km/h}$ , the other north at $30 \mathrm{~km/h}$ . Find the distance increase rate after 5 hours.

See Solution

Problem 13276

How many years to triple an investment with continuous compounding at $8.2\%$ ?

See Solution

Problem 13277

Oil leaks from a tanker at $r=18 e^{3 t}$ gallons/hour. Find the integral for 3 hours and estimate using a left sum with 3 parts.

See Solution

Problem 13278

Bestimmen Sie die Ableitung der Funktion $f(x)=\frac{1}{4} e^{x}-2,5 x$ .

See Solution

Problem 13279

Find the derivative of $f(x)=\frac{2 x^{3}-4 x^{2}+3}{x}$ .

See Solution

Problem 13280

Find the derivative of $f(x) = \sin(\arccos(x))$ using two methods and verify they match.

See Solution

Problem 13281

Determine if the series $\sum_{n=1}^{\infty} \frac{(-1)^{n} \ln(n^{5})}{n^{2}}$ converges absolutely, conditionally, or diverges.

See Solution

Problem 13282

Find the $x$ of the critical point for $f(x)=\frac{x^{2}}{x^{2}+2}$ with $f^{\prime}(x)=\frac{4 x}{(x^{2}+2)^{2}}$ . Identify intervals of increase/decrease and classify the critical point.

See Solution

Problem 13283

Find local min and max of $f$ where $f^{\prime}(x)=(x^{2}-8)e^{x}$ and concavity in the interval ( $\square$ ).

See Solution

Problem 13284

Ein Würfel hat Kanten von $60 \mathrm{~cm}$ . Kanten verkürzen sich um $2 \mathrm{~cm}$ /Minute. Was bedeuten $f(t)=(60-2 t)^{3}$ und $f^{\prime}(2)$ ?

See Solution

Problem 13285

Find the derivative of $c(x)=\frac{0.5 x}{100-x}$ at $x=80$ , $x=90$ , $x=95$ , and $x=99$ . Calculate $c'(80)$ , $c'(90)$ , $c'(95)$ , $c'(99)$ .

See Solution

Problem 13286

Find local min and max of $f(x)$ given $f'(x)$ has critical points at $x=-2, 4$ and concavity conditions.

See Solution

Problem 13287

Find the limit of the cost function $\bar{c}(x)=2.2+\frac{2500}{x}$ as $x$ approaches infinity.

See Solution

Problem 13288

Compute $f(x)=x^{2}+1$ for $x=1.9, 1.99, 1.999, 2.001, 2.01, 2.1$ to estimate $\lim _{x \rightarrow 2} f(x)$ .

See Solution

Problem 13289

Analyze the series $\sum_{n=2}^{\infty} \frac{1}{n 2^{n}(\ln (n))^{2}}$ for convergence using three tests: Alternating Series, Comparison, and Ratio.

See Solution

Problem 13290

A firm produces output $Q$ with $Q=f(L, K)=100+30 \times \ln L+20 \times \ln K$ . Given $w=\$ 6$ , $r=\$ 10$ , $P=\$ 4$ , which statement is true? A. Hessian is 0.375 B. Inputs $(8,20)$ C. MPP of labor is 30 D. Max output $\overline{Q}=100$ E. MPP value is 120.

See Solution

Problem 13291

Find the $x$ values for local min and max of $f(x)$ , and the interval where $f(x)$ is concave up, given $f'(x)=\frac{\cos x}{\sin x-2}$ and $f''(x)=\frac{-1+2 \sin x}{(\sin x-2)^{2}}$ for $x \in [0, 2\pi]$ .

See Solution

Problem 13292

A firm produces output $Q$ using $L$ and $K$ with $Q=f(L, K)=100+30 \times \ln L+20 \times \ln K$ . Given $w=\$ 6$ , $r=\$ 10$ , and $P=\$ 4$ , determine which statement is true about profit maximization.

See Solution

Problem 13293

Gegeben ist die Funktion $f(x)=-x^{4}+2 x^{2}$ .
a) Zeige, dass $f^{\prime}(a)=-4 \sqrt{2}$ für Nullstellen a gilt. Was ist die Steigung an weiteren Nullstellen?
b) Finde alle Stellen mit Steigung Null und beschreibe die Punkte.

See Solution

Problem 13294

Identify the region with area given by $\text { area }=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{5 n} \cos \left(\frac{i \pi}{5 n}\right)$ without calculating the limit.

See Solution

Problem 13295

Estimate where the function with two "upside-down U" shapes is increasing or decreasing based on its graph.

See Solution

Problem 13296

Find the limit: $\lim _{x \rightarrow \infty} \frac{6 x^{4}-7 x^{2}+1}{3 x^{4}+x^{3}+x^{2}+x+1}$ .

See Solution

Problem 13297

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

See Solution

Problem 13298

Find $h^{\prime}(1)$ for $h(x)=x f(x) \cos (x)$ given $f(1)=7$ and $f^{\prime}(1)=7$ . Round to three decimal places.

See Solution

Problem 13299

Find $g^{\prime}(6)$ for $g(x)=\frac{f(x)}{1+f(x)}$ given $f(6)=7$ and $f^{\prime}(6)=5$ . Round to three decimal places.

See Solution

Problem 13300

Find the derivative of $f(x)=\int_{3}^{x} \frac{1}{1+t^{3}} dt$ . What is $f'(x)$ ?

See Solution

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Calculus

Problem 13201

Find the function f(x)f(x)f(x) such that f′(x)=5xf^{\prime}(x)=\frac{5}{x}f′(x)=x5​ and it passes through the point (1,2)(1,2)(1,2).

Problem 13202

Evaluate the integral I₁ = ∫₀² 3√(4 - x²) dx using known areas.

Problem 13203

Express the limit as a definite integral: lim⁡n→∞∑i=1n(2xi∗sin⁡xi∗)Δx\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(2 x_{i}^{*} \sin x_{i}^{*}\right) \Delta xn→∞lim​i=1∑n​(2xi∗​sinxi∗​)Δx for nnn subintervals in [1,7][1,7][1,7].

Problem 13204

Untersuchen Sie die Funktion fa(x)=x3−3a2x+2a3f_{a}(x)=x^{3}-3 a^{2} x+2 a^{3}fa​(x)=x3−3a2x+2a3 auf Extrema und Wendepunkte.

Problem 13205

Find the value of the definite integral I=∫abf(x)dxI=\int_{a}^{b} f(x) d xI=∫ab​f(x)dx given the Riemann sum ∑i=1nf(xi∗)Δxi=3n2−6n+56n2\sum_{i=1}^{n} f\left(x_{i}^{*}\right) \Delta x_{i}=\frac{3 n^{2}-6 n+5}{6 n^{2}}∑i=1n​f(xi∗​)Δxi​=6n23n2−6n+5​.

Problem 13206

Find the marginal-cost function and its values for q=225q=225q=225 and q=450q=450q=450 given cˉ=2250eq450q\bar{c}=\frac{2250 e^{\frac{q}{450}}}{q}cˉ=q2250e450q​​.

Problem 13207

Finde die Stelle, an der die Funktion f(x)=12x2f(x)=\frac{1}{2} x^{2}f(x)=21​x2 den Steigungswinkel α=45∘\alpha=45^{\circ}α=45∘ hat.

Problem 13208

Finde die Stelle, an der die Funktion f(x)=2xf(x)=2 \sqrt{x}f(x)=2x​ den Steigungswinkel a=30∘a=30^{\circ}a=30∘ hat.

Problem 13209

Untersuche die Funktion fa(x)=x3−3a2x+2a3f_{a}(x)=x^{3}-3 a^{2} x+2 a^{3}fa​(x)=x3−3a2x+2a3 auf Extrema und Wendepunkte und zeige, dass x=−2ax=-2 ax=−2a eine Nullstelle ist.

Problem 13210

A balloon's radius decreases at 15 cm/min15 \mathrm{~cm/min}15 cm/min. Find the volume change rate when volume is 972πcm3972 \pi \mathrm{cm}^{3}972πcm3. Volume formula: V=43πr3V=\frac{4}{3} \pi r^{3}V=34​πr3.

Problem 13211

Problem 13212

Find the marginal-cost function for the average cost cˉ=5500eq550q\bar{c}=\frac{5500 e^{\frac{q}{550}}}{q}cˉ=q5500e550q​​ at q=275q=275q=275 and q=550q=550q=550.

Problem 13213

Simplify the difference quotient for the function f(x)=23x−1f(x)=\frac{2}{3x-1}f(x)=3x−12​.

Problem 13214

Untersuchen Sie die Funktion fa(x)=x3−3a2x+2a3f_{a}(x)=x^{3}-3 a^{2} x+2 a^{3}fa​(x)=x3−3a2x+2a3 auf Extrema, Nullstellen und Symmetrie. Zeichnen Sie die Graphen.

Problem 13215

Find if the limit lim⁡n→∞1n2{∑k=1n(3+2k)}\lim _{n \rightarrow \infty} \frac{1}{n^{2}}\left\{\sum_{k=1}^{n}(3+2 k)\right\}limn→∞​n21​{∑k=1n​(3+2k)} exists and its value.

Problem 13216

Find the limit of the sum as n→∞n \to \inftyn→∞: ∑k=1n(6n3k2−4n2k+3n)\sum_{k=1}^{n}\left(\frac{6}{n^{3}} k^{2}-\frac{4}{n^{2}} k+\frac{3}{n}\right)∑k=1n​(n36​k2−n24​k+n3​).

Problem 13217

Calculate these indefinite integrals: (a) ∫x6dx\int x^{6} dx∫x6dx, (b) ∫x8/11dx\int x^{8/11} dx∫x8/11dx, (c) ∫x−3xdx\int x^{-3} \sqrt{x} dx∫x−3x​dx.

Problem 13218

Consider the quadratic f(x)=ax2+bx+cf(x) = ax^2 + bx + cf(x)=ax2+bx+c. (a) Show that the average rate of change on [r,s][r, s][r,s] is a(r+s)+ba(r+s) + ba(r+s)+b. (b) Is there a point in [r,s][r, s][r,s] where the average rate equals the instantaneous rate? Explain.

Problem 13219

Find the general antiderivative of the function (3x4−6x6+12)\left(\frac{3}{x^{4}}-\frac{6}{x^{6}}+12\right)(x43​−x66​+12).

Problem 13220

Find the general antiderivative of the function 16u\frac{1}{6 \sqrt{u}}6u​1​. Antiderivative === +C+C+C.

Problem 13221

Find h′(1)h^{\prime}(1)h′(1) for h(x)=xf(x)cos⁡(x)h(x)=x f(x) \cos (x)h(x)=xf(x)cos(x) given f(1)=7f(1)=7f(1)=7 and f′(1)=6f^{\prime}(1)=6f′(1)=6. Round to three decimals.

Problem 13222

Untersuchen Sie die Funktion fa(x)=x3−3a2x+2a3f_{a}(x)=x^{3}-3 a^{2} x+2 a^{3}fa​(x)=x3−3a2x+2a3 auf Extrema, Nullstellen und Symmetrie.

Problem 13223

Given f(4)=10f(4)=10f(4)=10 and f′(4)=9f^{\prime}(4)=9f′(4)=9, find (fh)′(4)(f h)^{\prime}(4)(fh)′(4) where h(x)=xh(x)=\sqrt{x}h(x)=x​. Round to two decimal places.

Problem 13224

Find the antiderivative C(x)C(x)C(x) where C′(x)=3x2−5xC^{\prime}(x)=3 x^{2}-5 xC′(x)=3x2−5x and C(0)=8000.C(0)=8000. C(0)=8000.

Problem 13225

Find the limit: lim⁡x→0f(x)sin⁡(3x)\lim _{x \rightarrow 0} \frac{f(x)}{\sin (3 x)}limx→0​sin(3x)f(x)​ if lim⁡x→0f(x)=0\lim _{x \rightarrow 0} f(x)=0limx→0​f(x)=0 and lim⁡x→0f′(x)=2\lim _{x \rightarrow 0} f^{\prime}(x)=2limx→0​f′(x)=2.

Problem 13226

Find the antiderivative C(x)C(x)C(x) such that C′(x)=6x2−7xC'(x)=6x^2-7xC′(x)=6x2−7x and C(0)=2000.C(0)=2000. C(0)=2000.

Problem 13227

Problem 13228

Find the antiderivative x(t)x(t)x(t) for dxdt=2et−8\frac{d x}{d t}=2 e^{t}-8dtdx​=2et−8 with the condition x(0)=6x(0)=6x(0)=6.

Problem 13229

An oil tank drains oil with volume V(t)=0.2(25−t)3V(t)=0.2(25-t)^{3}V(t)=0.2(25−t)3 for t∈[0,25]t \in[0,25]t∈[0,25]. a) Find initial oil volume. b) Calculate average volume change in the first and last 10 minutes.

Problem 13230

Given f(x)f(x)f(x) with f′′(−1)=f′′(0)=f′′(1)=0f^{\prime \prime}(-1)=f^{\prime \prime}(0)=f^{\prime \prime}(1)=0f′′(−1)=f′′(0)=f′′(1)=0 and specific concavity, which is true?

Problem 13231

Evaluate the integral ∫23(7x+11)9dx\int \frac{23}{(7 x+11)^{9}} dx∫(7x+11)923​dx using the substitution u=7x+11u=7 x+11u=7x+11.

Problem 13232

Evaluate the integral ∫23(7x+11)9dx\int \frac{23}{(7 x+11)^{9}} d x∫(7x+11)923​dx using the substitution u=7x+11u=7 x+11u=7x+11.

Problem 13233

Evaluate the integral ∫xex2dx\int x e^{x^2} dx∫xex2dx. Include the constant +C+\mathrm{C}+C.

Problem 13234

Find the limit: lim⁡x→0+(f(x)+1)1/sin⁡(2x)\lim _{x \rightarrow 0^{+}}(f(x)+1)^{1 / \sin (2 x)}limx→0+​(f(x)+1)1/sin(2x) given lim⁡x→0+f(x)=0\lim _{x \rightarrow 0^{+}} f(x)=0limx→0+​f(x)=0 and lim⁡x→0+f′(x)=2\lim _{x \rightarrow 0^{+}} f^{\prime}(x)=2limx→0+​f′(x)=2.

Problem 13235

Evaluate the integral: ∫4sin⁡6(x)cos⁡(x)dx\int 4 \sin ^{6}(x) \cos (x) d x∫4sin6(x)cos(x)dx

Problem 13236

Find the derivative of A(x)=(13x+2)2A(x)=\left(\frac{1}{3} x+2\right)^{2}A(x)=(31​x+2)2.

Problem 13237

Find the average and instantaneous rates of change for the boat's distance function d(t)=0.002t3+0.05t2+0.3td(t)=0.002 t^{3}+0.05 t^{2}+0.3 td(t)=0.002t3+0.05t2+0.3t at t=10t=10t=10. Interpret both results.

Problem 13238

A person 1.7 m tall walks away from a 2.55 m pole at 2 m/s. Find the speed of the tip of his shadow when 8 m from the pole.

Problem 13239

Two cars are moving towards an intersection. Car A is 300 m east at 60 km/h, and Car B is 400 m south at 75 km/h. Find the rate of distance change.

Problem 13240

Bestimme die Stammfunktion von ∫−11ee⋅(x⋅ex) dx=\int_{-1}^{1} e^{e} \cdot (x \cdot e^{x}) \, dx =∫−11​ee⋅(x⋅ex)dx=

Problem 13241

Find the function $f(x)$ such that $f^{\prime}(x)=\frac{5}{x}$ and it passes through the point $(1,2)$ .

Express the limit as a definite integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(2 x_{i}^{} \sin x_{i}^{}\right) \Delta x$ for $n$ subintervals in $[1,7]$ .

Untersuchen Sie die Funktion $f_{a}(x)=x^{3}-3 a^{2} x+2 a^{3}$ auf Extrema und Wendepunkte.

Find the value of the definite integral $I=\int_{a}^{b} f(x) d x$ given the Riemann sum $\sum_{i=1}^{n} f\left(x_{i}^{*}\right) \Delta x_{i}=\frac{3 n^{2}-6 n+5}{6 n^{2}}$ .

Find the marginal-cost function and its values for $q=225$ and $q=450$ given $\bar{c}=\frac{2250 e^{\frac{q}{450}}}{q}$ .

Finde die Stelle, an der die Funktion $f(x)=\frac{1}{2} x^{2}$ den Steigungswinkel $\alpha=45^{\circ}$ hat.

Finde die Stelle, an der die Funktion $f(x)=2 \sqrt{x}$ den Steigungswinkel $a=30^{\circ}$ hat.

Untersuche die Funktion $f_{a}(x)=x^{3}-3 a^{2} x+2 a^{3}$ auf Extrema und Wendepunkte und zeige, dass $x=-2 a$ eine Nullstelle ist.

A balloon's radius decreases at $15 \mathrm{~cm/min}$ . Find the volume change rate when volume is $972 \pi \mathrm{cm}^{3}$ . Volume formula: $V=\frac{4}{3} \pi r^{3}$ .

Find the marginal-cost function for the average cost $\bar{c}=\frac{5500 e^{\frac{q}{550}}}{q}$ at $q=275$ and $q=550$ .

Simplify the difference quotient for the function $f(x)=\frac{2}{3x-1}$ .

Untersuchen Sie die Funktion $f_{a}(x)=x^{3}-3 a^{2} x+2 a^{3}$ auf Extrema, Nullstellen und Symmetrie. Zeichnen Sie die Graphen.

Find if the limit $\lim _{n \rightarrow \infty} \frac{1}{n^{2}}\left\{\sum_{k=1}^{n}(3+2 k)\right\}$ exists and its value.

Find the limit of the sum as $n \to \infty$ : $\sum_{k=1}^{n}\left(\frac{6}{n^{3}} k^{2}-\frac{4}{n^{2}} k+\frac{3}{n}\right)$ .

Calculate these indefinite integrals: (a) $\int x^{6} dx$ , (b) $\int x^{8/11} dx$ , (c) $\int x^{-3} \sqrt{x} dx$ .

Consider the quadratic $f(x) = ax^2 + bx + c$ .
(a) Show that the average rate of change on $[r, s]$ is $a(r+s) + b$ .
(b) Is there a point in $[r, s]$ where the average rate equals the instantaneous rate? Explain.

Find the general antiderivative of the function $\left(\frac{3}{x^{4}}-\frac{6}{x^{6}}+12\right)$ .

Find the general antiderivative of the function $\frac{1}{6 \sqrt{u}}$ . Antiderivative $=$ $+C$ .

Find $h^{\prime}(1)$ for $h(x)=x f(x) \cos (x)$ given $f(1)=7$ and $f^{\prime}(1)=6$ . Round to three decimals.

Untersuchen Sie die Funktion $f_{a}(x)=x^{3}-3 a^{2} x+2 a^{3}$ auf Extrema, Nullstellen und Symmetrie.

Given $f(4)=10$ and $f^{\prime}(4)=9$ , find $(f h)^{\prime}(4)$ where $h(x)=\sqrt{x}$ . Round to two decimal places.

Find the antiderivative $C(x)$ where $C^{\prime}(x)=3 x^{2}-5 x$ and $C(0)=8000.$

Find the limit: $\lim _{x \rightarrow 0} \frac{f(x)}{\sin (3 x)}$ if $\lim _{x \rightarrow 0} f(x)=0$ and $\lim _{x \rightarrow 0} f^{\prime}(x)=2$ .

Find the antiderivative $C(x)$ such that $C'(x)=6x^2-7x$ and $C(0)=2000.$

Find the antiderivative $x(t)$ for $\frac{d x}{d t}=2 e^{t}-8$ with the condition $x(0)=6$ .

An oil tank drains oil with volume $V(t)=0.2(25-t)^{3}$ for $t \in[0,25]$ .
a) Find initial oil volume. b) Calculate average volume change in the first and last 10 minutes.

Given $f(x)$ with $f^{\prime \prime}(-1)=f^{\prime \prime}(0)=f^{\prime \prime}(1)=0$ and specific concavity, which is true?

Evaluate the integral $\int \frac{23}{(7 x+11)^{9}} dx$ using the substitution $u=7 x+11$ .

Evaluate the integral $\int \frac{23}{(7 x+11)^{9}} d x$ using the substitution $u=7 x+11$ .

Evaluate the integral $\int x e^{x^2} dx$ . Include the constant $+\mathrm{C}$ .

Find the limit: $\lim _{x \rightarrow 0^{+}}(f(x)+1)^{1 / \sin (2 x)}$ given $\lim _{x \rightarrow 0^{+}} f(x)=0$ and $\lim _{x \rightarrow 0^{+}} f^{\prime}(x)=2$ .

Evaluate the integral: $\int 4 \sin ^{6}(x) \cos (x) d x$

Find the derivative of $A(x)=\left(\frac{1}{3} x+2\right)^{2}$ .

Find the average and instantaneous rates of change for the boat's distance function $d(t)=0.002 t^{3}+0.05 t^{2}+0.3 t$ at $t=10$ . Interpret both results.

Bestimme die Stammfunktion von $\int_{-1}^{1} e^{e} \cdot (x \cdot e^{x}) \, dx =$

Berechne den Flächeninhalt des Segels, das von $f(x)=x+6-\frac{32}{x^{2}}$ , der $x$ -Achse und der Geraden $g$ gebildet wird.

Bestimme die Stammfunktion von $\int_{-1}^{1}\left[e: e^{x}-\left(e \cdot e^{x}\right)\right] d x$ .

Find the correct assertion from the Taylor polynomial $T_{3}(x)=2+(x+\pi)-3(x+\pi)^{2}-(x+\pi)^{3}$ .

Berechne den Flächeninhalt zwischen $f(x)=2 \sqrt{x}$ und $g(x)=2x-4$ für $x \geq 0$ .

Determine the intervals where the function $f(x)=\frac{x^{2}}{2}-2 x+(3+2 x) \ln x+c$ is concave up.

Two cars start from the same point: one goes east at $75 \mathrm{~km/h}$ , the other north at $40 \mathrm{~km/h}$ . Find the distance increase rate after 1 hour.

Given Boyle's Law $PV=C$ , find the rate of volume decrease when $V=450 \mathrm{~cm}^{3}$ , $P=160 \mathrm{kPa}$ , and $\frac{dP}{dt}=20 \frac{\mathrm{kPa}}{\min}$ .

Calculate the area between the lines $y=x+4$ and $y=9x^{2}+x$ from $x=1$ to $x=2$ .

Insect population $\mathrm{P}(t)=700 e^{0.08 t}$ : (a) Find $\mathrm{P}(0)$ . (b) What is the growth rate? (c) Find $\mathrm{P}(10)$ . (d) When is $\mathrm{P}=980$ ? (e) When does $\mathrm{P}$ double?

A trebuchet launches a projectile from $47 \mathrm{ft}$ at $40 \mathrm{ft/s}$ . Find the time to reach $60 \mathrm{ft}$ and return.

Differentiate $f(x)=\frac{2}{x}-1$ and check if it's differentiable at $x=1$ .

Bestimmen Sie die Steigung von $K_{r}$ bei $x=-1$ und den Schnittpunkten von $f(x)=3 x^{2}-5 x$ mit der $\mathrm{x}$ -Achse.

Is the tangent line to $f(x)=x e^{-x}$ at $x=1$ above or below the graph? Explain why.

Berechne die Integrale für die Funktion $f$ und gib die Formeln an. a) $\int_{1}^{4} x^{2} d x$ b) $\int_{3}^{5} x d x$ c) $\int_{7}^{8} 3 d x$

Find the function $S(t)$ for monthly sales declining at $S^{\prime}(t)=-10 t^{\frac{2}{3}}$ from 920 to 500 computers.

Bestimme die Integrationsgrenze $b$ für a) $\int_{0}^{b} x \, dx = 2$ und b) $\int_{0}^{b} x^{2} \, dx = 9$ .

Bestimme die Grenze $b$ für $\int_{0}^{b} x^{2} d x=5$ .

Find the first and second derivatives of $f(x) = \sqrt{2}x - 2\cos(x)$ and verify if $f''(x) = 25x$ is correct.

Find the derivative of $f(x)=\frac{4x^2}{x^2}-\frac{5x^2}{x^2}-\frac{2}{x^2}$ .

Given $f(3)=8$ , $f'(3)=5$ , $g(3)=9$ , $g'(3)=7$ , find $\left.\frac{d}{d x}\left(\frac{f(x)}{g(x)}\right)\right|_{x=3}$ . Round to 2 decimals.

Find the domain of $f(x)=x^{2} \ln x$ , critical points, intervals of increase/decrease, and inflection point.

Given $f(5)=2$ , $f'(5)=8$ , $g(5)=9$ , and $g'(5)=3$ , find $\left.\frac{d}{d x}(2 f(x)-3 g(x)+4)\right|_{x=5}$ . Round to 2 decimal places.

Given $f(-2)=4$ , $f'(-2)=10$ , $g(-2)=5$ , $g'(-2)=7$ , find $\left.\frac{d}{d x}(f(x) \cdot g(x))\right|_{x=-2}$ .

Find which statement is true based on the Taylor polynomial $T_{3}(x)=2+(x+\pi)-3(x+\pi)^{2}-(x+\pi)^{3}$ .

Find values of $a$ and $b$ for $f(x)=a x^{3}+x^{2}+b x-2$ to have a max at $x=3$ and min at $x=2$ .

Determine where the function $f(x)=\frac{x^{2}}{2}-2 x+(3+2 x) \ln x+c$ is concave up: $(0.5,3)$ , $(0,1)$ , $(1, \infty)$ , $(0,0.5)$ and $(1, \infty)$ , or never.

Find the integral of $\frac{2 x^{2}}{1+x^{2}} \, dx$ .

Finde den Punkt auf dem Graphen von $h(x)=4 x^{3}-3 x^{2}+5 x-7$ , wo die Steigung $m=65$ ist.

Find the critical points of $f(x)=\frac{(2-7 x)}{8^{x}}$ . Provide exact answers, stating if there are no critical points.

A firm has a price of \ $5.00 and a production function$ Q=f(L, K)=20 \times L^{0.25} \times K^{0.5} $. Find the profit-maximizing inputs$ (\bar{L}, \bar{K})$.

Evaluate the integral: $\int\left(\frac{3}{x}+e^{3 x}\right) d x=?$