Calculus

Problem 28301

Find the first five terms of the sequence $\left[\frac{(-1)^{n+1}}{(n-6)^{2}}\right]_{n=1}^{5}$ and check for convergence.

See Solution

Problem 28302

Determine if the sequence converges or diverges. If convergent, find the limit: $\lim _{n \rightarrow \infty} \frac{14(2^{n})+10}{7(2^{n})}$

See Solution

Problem 28303

Determine if the sequence $a_{n}=\frac{n^{3}+\sin (7 n+15)}{n^{7}+15}$ converges or diverges, and find the limit if it converges.

See Solution

Problem 28304

Find the limit of the sequence $c_{n}=\ln \left(\frac{4 n-7}{3 n+4}\right)$ as $n \to \infty$ , or state DIV if it diverges.

See Solution

Problem 28305

Find the smallest number $M$ such that:
(a) $\left|a_{n}-1\right| \leq 0.001$ for $n \geq M$ : $M=\square$
(b) $\left|a_{n}-1\right| \leq 0.00001$ for $n \geq M$ : $M=\square$
(c) Prove $\lim_{n \rightarrow \infty} a_{n}=1$ by finding $M$ such that $\left|a_{n}-1\right|<t$ for all $n>M$ : $M=\square(\text{ in terms of } t)$

See Solution

Problem 28306

Find the limits: m. $\lim _{x \rightarrow 2} \frac{x^{2}-4}{3 x^{2}-7 x+2}$ , g. $\lim _{x \rightarrow 7} \frac{\sqrt{x+2}-3}{x-7}$ , o. $\lim _{x \rightarrow 2} \frac{x^{2}-2 x}{2 x^{2}-7 x+6}$ , p. $\lim _{x \rightarrow 5} \frac{2 x}{x-5}$ .

See Solution

Problem 28307

Die Funktion w beschreibt die Strecke einer Rennrodlerin im Eiskanal. Beantworten Sie die Fragen zu Geschwindigkeit und Strecke:
a) Bestimmen Sie $w^{\prime}(30)$ in $\frac{m}{s}$ . b) Wenn $w(20)=600$ , finden Sie $w(40)$ . c) Zeigen Sie, dass es zwischen 20 s und 40 s einen Zeitpunkt t mit $w^{\prime}(t)>35$ gibt. d) Was kann man über $w(30)$ sagen, wenn $w(20)=600$ ?

See Solution

Problem 28308

How fast will Fay Wray's shoe be moving when it falls from the 321 m tall Empire State Building?

See Solution

Problem 28309

Finde die erste Ableitung von $f(x) = \frac{\sqrt{2}}{x}$ .

See Solution

Problem 28310

Bestimmen Sie die Ableitung von $f(x)=\sqrt{\frac{2}{x}}$ .

See Solution

Problem 28311

Bestimme die erste Ableitung von $f(x)=\sqrt{\frac{2}{x}}$ mit der Potenz- und Faktorregel.

See Solution

Problem 28312

A particle is dropped from 3 m high with an initial speed of $9.5 \mathrm{~m} \mathrm{~s}^{-1}$ . Find its time of flight.

See Solution

Problem 28313

Given the sequence $a_n = \frac{1}{2^n}$ , find:
a. First five terms: $\square$ . b. First five partial sums: $\square$ . c. Partial sum $S_n$ : $\square$ in terms of $n$ . d. Total sum: $\sum_{n=1}^{\infty} \frac{1}{2^n} = \lim_{n \rightarrow \infty}(\square) = \square$ . e. True statements: A, B, C, D, E, F.

See Solution

Problem 28314

4) A tornado's wind speed is given by $S(d)=93 \log d+65$ .
a) Find the average speed change from $d=100$ to $d=1000$ .
b) Estimate the instantaneous speed change at $d=100$ using $h=0.001$ .

See Solution

Problem 28315

As $n$ approaches infinity, determine limits for $r$ in $\lim_{n \to \infty} r^{n+1}=0$ and sum series $\lim_{n \to \infty} \frac{1-r^{n+1}}{1-r}$ .

See Solution

Problem 28316

Identify the divergent infinite series from the following: I: 24-12+6-3+... II: -0.2+1-5+25-... III: 64+32+16+8+... IV: $\frac{1}{9}-\frac{1}{3}+1-3+...$

See Solution

Problem 28317

Find the partial sums $S_{2}, S_{4}$ , and $S_{6}$ for the series $3+\frac{3}{2^{2}}+\frac{3}{3^{2}}+\cdots$ .

See Solution

Problem 28318

Consider the sequence $a_n = \frac{1}{2^n}$ :
a. List the first five terms. b. List the first five partial sums. c. Find $S_n = \frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2^n}$ in terms of $n$ . d. Find the limit $\sum_{n=1}^{\infty} \frac{1}{2^n}$ . e. True statements: A. converges to 0, B. converges to 1, C. limit of partial sums is 0, D. limit is 1, E. series converges to 0, F. series converges to 1.

See Solution

Problem 28319

Check if the series converges and find its sum: $\sum_{k=1}^{\infty} \frac{2^{k+3}}{5^{k+1}}=\square$ (Enter DNE if no sum exists.)

See Solution

Problem 28320

Find the displacement of a particle with velocity $v=2 t^{6}+4 t^{4}+\frac{13}{2} t^{2}$ from $t=0$ to $t=2$ , given $s=7$ at $t=0$ . Round to 3 significant figures.

See Solution

Problem 28321

Find $\lim _{x \rightarrow 3} h(x)$ for the piecewise function $h(x)=\left\{\begin{array}{cc}x^{2}-2 x+1 & , x \neq 3 \\ 7 & , x=3\end{array}\right.$ .

See Solution

Problem 28322

Water is added to a tank with radius $5 \mathrm{~m}$ and height $10 \mathrm{~m}$ at $100 \mathrm{~L/min}$ . Find $\frac{d h}{d t}$ when water is $6 \mathrm{~m}$ deep.

See Solution

Problem 28323

How much would Friedrich Schiller owe in 2008 if charged 200 euros/year with 1% interest since 1805? Round to two decimal places.

See Solution

Problem 28324

Find the derivative $\frac{d y}{d x}$ for $y=\frac{x^{5}}{3-x}$ .

See Solution

Problem 28325

Bestimme die Tangentengleichung an $f(x)=e^{x}+1$ im Punkt $P(0 \mid f(0))$ und berechne den eingeschlossenen Flächeninhalt.

See Solution

Problem 28326

Find the function $f(x)$ such that $f^{\prime \prime}(x) = -\frac{1}{4} \sin \left(\frac{x}{2}\right)$ .

See Solution

Problem 28327

Berechne die Steigung von $f$ an $x_{0}$ : a) $f(x)=x^{2}$ , $x_{0}=-1$ ; b) $f(x)=a x^{2}$ , $x_{0}=1$ .

See Solution

Problem 28328

Find the general solution of $\frac{d^{2} x}{d t^{2}}+2 \frac{d x}{d t}=e^{-t}$ with $x(0)=0$ , $x'(0)=0$ . Choices: a, b, c, d, e.

See Solution

Problem 28329

Calculate the work $W$ done by the force $\vec{F}=(4 \hat{i}+2 \hat{j}-k) N$ from $A(1,1,0)$ to $B(1,-1,-2)$ .

See Solution

Problem 28330

1 Bedeutung der Ableitung im Sachzusammenhang a) Bestimme $f^{\prime}(5)$ für $f(t)=2,5 t^{2}$ (Strecke in m) und erkläre die Bedeutung. b) Berechne $f^{\prime}(6)$ für $f(t)=-0,3 t^{3}+6 t^{2}-t+10$ (Höhe in cm) und erläutere die Bedeutung.

See Solution

Problem 28331

Determine if the series $\sum_{n=1}^{\infty} \frac{\arctan (n)}{n^{2}}$ converges or diverges.

See Solution

Problem 28332

Find the radius of convergence for the series $\sum_{n=1}^{\infty} \frac{6^{n} n !}{n^{n}} x^{n}$ .

See Solution

Problem 28333

Differentiate $f(x) = x e^{x}$ three times and find its inflection point(s).

See Solution

Problem 28334

Find the horizontal asymptote of the curve $y=\frac{\sqrt{4 x^{6}+2 x^{2}}}{10+x^{3}}$ . Choices: a. $y=2$ \& $y=-2$ , b. $y=-2$ , c. $y=2$ , d. $y=0$ , e. none.

See Solution

Problem 28335

Calculate the limit: $\lim _{x \rightarrow-\infty} e^{x} \sin \left(x^{2}\right)$ . State if it doesn't exist.

See Solution

Problem 28336

Determine if the series $\sum_{n=0}^{\infty} \frac{n^{50}}{3^{n}}$ converges or diverges.

See Solution

Problem 28337

Find the value of $\mathrm{c}$ for the mean value theorem for $f(x)=x+\frac{1}{x}$ on $[1,3]$ . Options: a. 0 b. 2 c. $\sqrt{3}$ d. $\{\sqrt{3}, -\sqrt{3}\}$ e. None

See Solution

Problem 28338

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{\tan (7 x)}$ . Does it exist?

See Solution

Problem 28339

Calculate the average rate of change of wind speed $S(d)=93 \log d+65$ from $d=100$ to $d=1000$ . Use $m=\frac{S(1000)-S(100)}{900}$ .

See Solution

Problem 28340

1. Sprawdź, czy funkcja $f=\ln$ jest jednostajnie ciągła i lipschitzowska na dziedzinach $(a ;+\infty)$ i $(0 ;+\infty)$ .
2. Udowodnij, że ciągła i okresowa funkcja $f$ z $T_{n} \to 0$ jest stała. Czy ciągłość jest kluczowa?

See Solution

Problem 28341

Find the number of real roots of the continuous function $f$ in the interval $(-3,2)$ given $f(-3)=2, f(-1)=-2, f(0)=-1, f(2)=5$ . Options: a. Exactly two b. at least 3 c. at least two d. at most two.

See Solution

Problem 28342

Find $\left(f^{-1}\right)^{\prime}(-1)$ for $f(x)=(3 x-1)+\log _{3}(1+x^{2})$ . Options: a. $\frac{1}{2}$ b. $\frac{1}{3}$ c. Else d. does not exist e. $3-\frac{1}{\ln 3}$ f. $3+\frac{1}{\ln 3}$ g. $\frac{\ln 3}{3^{*} \ln 3-1}$ .

See Solution

Problem 28343

Calculate the limit: $\lim _{x \rightarrow-1^{-}} e^{\frac{-1}{1-x^{2}}}$ . Does it exist?

See Solution

Problem 28344

Find the inflection point of the graph of $f(x)=\sinh (2-x)+1$ . Choose from: a. $(2,1)$ b. $(-1,-2)$ c. $(1,2)$ d. $(-2,-1)$ .

See Solution

Problem 28345

Find the limit: $\lim _{h \rightarrow 0} \frac{\left(1+2(a+h)^{2}\right)^{3}-\left(1+2 a^{2}\right)^{3}}{h}$ . Choices: a, b, c, d.

See Solution

Problem 28346

Calculate the integral: $\int \sqrt{1+\sin ^{2} x} \, dx$

See Solution

Problem 28347

The limit as $x$ approaches $3$ from the right of $[6 - 3x]$ equals $-3$ . True or False?

See Solution

Problem 28348

Find $g^{\prime}(3)$ given $f(3)=6$ , $f^{\prime}(3)=3$ , and $g(x)=\frac{f^{2}(x)}{x}$ . Choices: a. 9 b. 4 c. 5 d. 8

See Solution

Problem 28349

If $f^{\prime}(x)=0$ for all $x$ , is $f$ constant on its domain? True or False?

See Solution

Problem 28350

Evaluate the integral: $\int \cos x \sqrt{1+\sin ^{2} x} \, dx$

See Solution

Problem 28351

Bestimme die Wachstumsgeschwindigkeit der Pilzkultur mit $f(t)=7,5 e^{0,3 t}$ nach 5 Wochen und 3 Tagen. Finde den Zeitpunkt für 10 Millionen/Woche und die durchschnittliche Geschwindigkeit in der ersten Woche.

See Solution

Problem 28352

Find the $n^{\text{th}}$ derivative of $f(x)=-\cosh 2x - \sinh 2x$ at $x=0$ : $\stackrel{(n)}{f}(0)=$ a. $(-1)^{n-1}2^{n}$ b. $-2^{n}$ c. 0 d. $(-1)^{n}2^{n}$ e. $2^{n}$

See Solution

Problem 28353

Which limit is not an indeterminate form given $\lim _{x \rightarrow a} f(x)=\infty$ , $\lim _{x \rightarrow a} g(x)=1$ , $\lim _{x \rightarrow a} h(x)=0^{-}$ ? a. $\lim _{x \rightarrow a}[g(x)]^{f(x)}$ b. $\lim _{x \rightarrow a}\left[f(x)-\frac{1}{h(x)}\right]$ c. $\lim _{x \rightarrow a}[f(x)]^{h(x)}$ d. $\lim _{x \rightarrow a}\left[f(x)+\frac{1}{h(x)}\right]$

See Solution

Problem 28354

Find the marginal cost $M C(x)$ , revenue $R(x)$ , and marginal revenue $M R(x)$ for producing $x$ drills. Then, calculate $R^{\prime}(1100)$ and $R^{\prime}(4000)$ in dollars per drill, and interpret the results. Finally, find profit $P(x)$ and marginal profit $M P(x)$ , and calculate $P^{\prime}(1100)$ .

See Solution

Problem 28355

Estimate the wind speed change rate at $d=100$ miles using $S(d)=93 \log d + 65$ and $h=0.001$ .

See Solution

Problem 28356

A bacteria culture grows as $N(t)=3000\left(1+\frac{4t}{t^{2}+100}\right)$ .
(a) Find the growth rate $N^{\prime}(t)=\frac{3000\left(400-4t^{2}\right)}{\left(t^{2}+100\right)^{2}}$ .
(b) Calculate $N^{\prime}(0)$ , $N^{\prime}(10)$ , $N^{\prime}(20)$ , and $N^{\prime}(30)$ .
(c) Interpret these results.
(d) Find $N^{\prime \prime}(0)$ , $N^{\prime \prime}(10)$ , $N^{\prime \prime}(20)$ , and $N^{\prime \prime}(30)$ . Interpret these results.

See Solution

Problem 28357

Find the velocity $v(t)$ and acceleration $a(t)=\frac{2 t(t^{2}-3)}{(t^{2}+1)^{3}}$ for $s(t)=\frac{t}{1+t^{2}}$ . Determine when the particle is slowing down or speeding up.

See Solution

Problem 28358

Evaluate the integral: $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 2 \pi \cos x \sqrt{1+\sin ^{2} x} \, dx$ .

See Solution

Problem 28359

Find the value of $\int_{0}^{\frac{\pi}{6}}(\cos x+\sec x)^{2} d x$ .

See Solution

Problem 28360

Given the function $f(x)=\frac{-5}{x-1}$ and $a=3$ , find the slope of the tangent line and its equation at $x=a$ .

See Solution

Problem 28361

Find the area between the curves $y=5 \sqrt{x}$ and $y=5 x^{2}$ .

See Solution

Problem 28362

Find the annual deposit needed to save \$100,000 in 13 years at a continuous 10% interest rate. Round to the nearest dollar.

See Solution

Problem 28363

Berechne die Fläche unter der Funktion $f(x)=x^2$ im Intervall $I=[-1; 2]$ mit dem Integral $\int_{-1}^{2} x^{2} dx$ .

See Solution

Problem 28364

Find the minima, maxima, intercepts, and intervals of increase/decrease for $f(x) = x^{3} - 2x^{2} - 4x + 8$ .

See Solution

Problem 28365

Calculate the present value (PV) needed now for a future value (FV) of \ $100,000 in 13 years at 10\% interest, using$ FV = PV \cdot e^{rt}$.

See Solution

Problem 28366

Find the minima, maxima, and intervals of increase/decrease for the function $f(x)=x^3-2x^2-4x+8$ .

See Solution

Problem 28367

Find the limit: $\lim _{x \rightarrow \pi} \frac{\cos x+\sin (2 x)+1}{x^{2}-\pi^{2}}$ . Choose (A) $\frac{1}{2 \pi}$ , (B) $\frac{1}{\pi}$ , (C) 1, or (D) nonexistent.

See Solution

Problem 28368

Find the unique value of $x$ where the tangents of the curves $y \cdot 0.9 \ln 3 - 3^{0.9 x} = 0$ and $5^{0.4 + 0.3 x} = y \cdot 0.3 \ln 5$ are parallel.

See Solution

Problem 28369

Check if the function defined by $f(x) = \left\{\begin{array}{l}8 x-7 \text { if } x \geq 1 \\ 7 x-8 \text { if } x<1\end{array}\right.$ is differentiable at $x=1$ .

See Solution

Problem 28370

Test if the function $f(x)=\left\{\begin{array}{l}3 x-2 \text { when } x \geq 1 \\ 2 x-3 \text { when } x<1\end{array}\right.$ is differentiable at $x=1$ .

See Solution

Problem 28371

Test if the function $f(x) = 3x - 2$ for $x \geq 1$ and $f(x) = 2x - 3$ for $x < 1$ is differentiable at $x = 1$ .

See Solution

Problem 28372

Evaluate the double integral: $\int_{5}^{6} \int_{\sqrt{x}}^{x} \frac{y}{x} dy dx$

See Solution

Problem 28373

Evaluate the double integral: $\int_{4}^{5} \int_{\sqrt{x}}^{x} \frac{y}{x} \, dy \, dx$

See Solution

Problem 28374

Berechne die Fläche der Funktion $f(x)=x^2-9$ im Intervall $I=[-4; 3]$ mit dem Integral: $\int_{-4}^{-3} x^{2}-9 \, dx + \int_{-3}^{3} x^{2}-9 \, dx$

See Solution

Problem 28375

Evaluate the double integral: $\int_{0}^{9} \int_{0}^{y} \sqrt{3x+y} \, dx \, dy$

See Solution

Problem 28376

Berechne die Fläche unter der Funktion $f(x)=x^2-9$ im Intervall $I=[-4; 3]$ mit dem Integral $\int_{-2}^{1} x^{2}-9 \, dx$ .

See Solution

Problem 28377

Find the limit: $\lim _{n \rightarrow \infty} \frac{(e-1) \sum_{k=0}^{\infty} \frac{1}{e^{k}}}{\left(1+\frac{2}{n}\right)^{n}}$

See Solution

Problem 28378

Bestimme das Krümmungsverhalten der Funktion $f$ aus der zweiten Ableitung $f^{\prime \prime}(x)=k x+d$ mit $k<0$ .

See Solution

Problem 28379

Find $\frac{d f^{-1}}{d x}$ for $f(x)=\frac{1}{(1-x)^{2}}$ at $x=\frac{1}{4}$ , where $x>1$ .

See Solution

Problem 28380

Evaluate the double integral: $\int_{0}^{1} \int_{0}^{y} \sqrt{3x+y} \, dx \, dy$

See Solution

Problem 28381

Find the length of the curve $f(x)=\frac{1}{3}(x^{2}+2)^{\frac{3}{2}}$ for $0 \leq x \leq 2$ .

See Solution

Problem 28382

Find the length of the curve $f(x)=\frac{-1}{3}(x^{2}+2)^{2}$ for $0 \leq x \leq 2$ .

See Solution

Problem 28383

Evaluate the integral $\int_{4}^{-3} (x^{2}-9) \, dx$ .

See Solution

Problem 28384

Pionen haben eine Halbwertszeit von $T_{1 / 2}=18 \cdot 10^{-9} \mathrm{~s}$ . Berechnen Sie die Halbwertszeit im Laborsystem und die Laufzeit für $100 \mathrm{~m}$ . Bestimmen Sie den verbleibenden Prozentsatz der Pionen.

See Solution

Problem 28385

Zeigen Sie, dass die Funktionen a) $f(x)=\frac{1}{2} x^{2}+1$ im $(0 ; \infty)$ , b) $f(x)=-x^{2}+4$ im $(-\infty ; 0)$ , c) $f(x)=2 x$ im $(-\infty ; \infty)$ und d) $f(x)=x^{3}-3 x$ im $(1 ; \infty)$ streng monoton wachsen.

See Solution

Problem 28386

Find the limit: $\lim _{x \rightarrow \infty} 2 x^{3}\left(e^{-\frac{1}{x^{3}}}-1\right)$ .

See Solution

Problem 28387

Approximate the area between the $x$ -axis and $g(x)$ from $x=10$ to $x=16$ using a trapezoidal sum with 3 subdivisions.

See Solution

Problem 28388

Approximate the area under $h(x)$ from $x=3$ to $x=11$ using a trapezoidal sum with 4 subdivisions.

See Solution

Problem 28389

Berechne die Fläche zwischen der Funktion $f(x)=x^2-9$ und der x-Achse im Intervall $I=[-2; 1]$ mit dem Integral $\int_{-2}^{1} (x^2-9) \, dx$ .

See Solution

Problem 28390

Find the mixed partial derivative $f_{y x}(x, y)$ for the function $f(x, y)=7 x y^{3}+8 x^{2}+9 y^{3}$ .

See Solution

Problem 28391

Approximate the area between the $x$ -axis and $g(x)=2^{x}$ from $x=-2$ to $x=2$ using a trapezoidal sum with 4 parts.

See Solution

Problem 28392

Approximate the area under $g(x)$ from $x=-1$ to $x=5$ using a trapezoidal sum with 3 equal parts.

See Solution

Problem 28393

Check if the Mean Value Theorem applies to $f(x) = \sin x$ on the interval $[0, \pi]$ .

See Solution

Problem 28394

Find the partial derivative $f_{y}(x, y)$ for the function $f(x, y)=2 x+4 x^{2} y^{2}+7 y^{2}$ .

See Solution

Problem 28395

Find the slope of the tangent line to the surface $z=4 x^{2} y-x y^{3}$ at $y=-2$ and point $(1, -2, 0)$ .

See Solution

Problem 28396

Approximate the area under $g(x)=x^{2}$ from $x=1$ to $x=4$ using a midpoint Riemann sum with 3 subdivisions.

See Solution

Problem 28397

Approximate the area between $h(x) = x^2 + 2$ and the x-axis from $x=0.5$ to $x=3.5$ using a midpoint Riemann sum with 3 subdivisions.

See Solution

Problem 28398

Approximate the area under $g(x)$ from $x=2$ to $x=6$ using the trapezoidal sum $T(4)$ with 4 subdivisions.

See Solution

Problem 28399

Find $f_{x}(x, y)$ for $f(x, y)=e^{(3 x+4 y)}$ . Choose the correct option from the given choices.

See Solution

Problem 28400

Approximate area under $f(x)=(x-3)^{2}$ from $x=0$ to $x=6$ using a trapezoidal sum with 3 subdivisions.

See Solution

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Calculus

Problem 28301

Find the first five terms of the sequence [(−1)n+1(n−6)2]n=15\left[\frac{(-1)^{n+1}}{(n-6)^{2}}\right]_{n=1}^{5}[(n−6)2(−1)n+1​]n=15​ and check for convergence.

Problem 28302

Determine if the sequence converges or diverges. If convergent, find the limit: lim⁡n→∞14(2n)+107(2n)\lim _{n \rightarrow \infty} \frac{14(2^{n})+10}{7(2^{n})}n→∞lim​7(2n)14(2n)+10​

Problem 28303

Determine if the sequence an=n3+sin⁡(7n+15)n7+15a_{n}=\frac{n^{3}+\sin (7 n+15)}{n^{7}+15}an​=n7+15n3+sin(7n+15)​ converges or diverges, and find the limit if it converges.

Problem 28304

Find the limit of the sequence cn=ln⁡(4n−73n+4)c_{n}=\ln \left(\frac{4 n-7}{3 n+4}\right)cn​=ln(3n+44n−7​) as n→∞n \to \inftyn→∞, or state DIV if it diverges.

Problem 28305

Problem 28306

Problem 28307

Problem 28308

How fast will Fay Wray's shoe be moving when it falls from the 321 m tall Empire State Building?

Problem 28309

Finde die erste Ableitung von f(x)=2xf(x) = \frac{\sqrt{2}}{x}f(x)=x2​​.

Problem 28310

Bestimmen Sie die Ableitung von f(x)=2xf(x)=\sqrt{\frac{2}{x}}f(x)=x2​​.

Problem 28311

Bestimme die erste Ableitung von f(x)=2xf(x)=\sqrt{\frac{2}{x}}f(x)=x2​​ mit der Potenz- und Faktorregel.

Problem 28312

A particle is dropped from 3 m high with an initial speed of 9.5 m s−19.5 \mathrm{~m} \mathrm{~s}^{-1}9.5 m s−1. Find its time of flight.

Problem 28313

Problem 28314

4) A tornado's wind speed is given by S(d)=93log⁡d+65S(d)=93 \log d+65S(d)=93logd+65. a) Find the average speed change from d=100d=100d=100 to d=1000d=1000d=1000. b) Estimate the instantaneous speed change at d=100d=100d=100 using h=0.001h=0.001h=0.001.

Problem 28315

As nnn approaches infinity, determine limits for rrr in lim⁡n→∞rn+1=0\lim_{n \to \infty} r^{n+1}=0limn→∞​rn+1=0 and sum series lim⁡n→∞1−rn+11−r\lim_{n \to \infty} \frac{1-r^{n+1}}{1-r}limn→∞​1−r1−rn+1​.

Problem 28316

Identify the divergent infinite series from the following: I: 24-12+6-3+... II: -0.2+1-5+25-... III: 64+32+16+8+... IV: 19−13+1−3+...\frac{1}{9}-\frac{1}{3}+1-3+...91​−31​+1−3+...

Problem 28317

Find the partial sums S2,S4S_{2}, S_{4}S2​,S4​, and S6S_{6}S6​ for the series 3+322+332+⋯3+\frac{3}{2^{2}}+\frac{3}{3^{2}}+\cdots3+223​+323​+⋯.

Problem 28318

Problem 28319

Check if the series converges and find its sum: ∑k=1∞2k+35k+1=□\sum_{k=1}^{\infty} \frac{2^{k+3}}{5^{k+1}}=\squarek=1∑∞​5k+12k+3​=□ (Enter DNE if no sum exists.)

Problem 28320

Find the displacement of a particle with velocity v=2t6+4t4+132t2v=2 t^{6}+4 t^{4}+\frac{13}{2} t^{2}v=2t6+4t4+213​t2 from t=0t=0t=0 to t=2t=2t=2, given s=7s=7s=7 at t=0t=0t=0. Round to 3 significant figures.

Problem 28321

Find lim⁡x→3h(x)\lim _{x \rightarrow 3} h(x)limx→3​h(x) for the piecewise function h(x)={x2−2x+1,x≠37,x=3h(x)=\left\{\begin{array}{cc}x^{2}-2 x+1 & , x \neq 3 \\ 7 & , x=3\end{array}\right.h(x)={x2−2x+17​,x=3,x=3​.

Problem 28322

Water is added to a tank with radius 5 m5 \mathrm{~m}5 m and height 10 m10 \mathrm{~m}10 m at 100 L/min100 \mathrm{~L/min}100 L/min. Find dhdt\frac{d h}{d t}dtdh​ when water is 6 m6 \mathrm{~m}6 m deep.

Problem 28323

How much would Friedrich Schiller owe in 2008 if charged 200 euros/year with 1% interest since 1805? Round to two decimal places.

Problem 28324

Find the derivative dydx\frac{d y}{d x}dxdy​ for y=x53−xy=\frac{x^{5}}{3-x}y=3−xx5​.

Problem 28325

Bestimme die Tangentengleichung an f(x)=ex+1f(x)=e^{x}+1f(x)=ex+1 im Punkt P(0∣f(0))P(0 \mid f(0))P(0∣f(0)) und berechne den eingeschlossenen Flächeninhalt.

Problem 28326

Find the function f(x)f(x)f(x) such that f′′(x)=−14sin⁡(x2)f^{\prime \prime}(x) = -\frac{1}{4} \sin \left(\frac{x}{2}\right)f′′(x)=−41​sin(2x​).

Problem 28327

Berechne die Steigung von fff an x0x_{0}x0​: a) f(x)=x2f(x)=x^{2}f(x)=x2, x0=−1x_{0}=-1x0​=−1; b) f(x)=ax2f(x)=a x^{2}f(x)=ax2, x0=1x_{0}=1x0​=1.

Problem 28328

Find the general solution of d2xdt2+2dxdt=e−t\frac{d^{2} x}{d t^{2}}+2 \frac{d x}{d t}=e^{-t}dt2d2x​+2dtdx​=e−t with x(0)=0x(0)=0x(0)=0, x′(0)=0x'(0)=0x′(0)=0. Choices: a, b, c, d, e.

Problem 28329

Calculate the work WWW done by the force F⃗=(4i^+2j^−k)N\vec{F}=(4 \hat{i}+2 \hat{j}-k) NF=(4i^+2j^​−k)N from A(1,1,0)A(1,1,0)A(1,1,0) to B(1,−1,−2)B(1,-1,-2)B(1,−1,−2).

Problem 28330

Problem 28331

Determine if the series ∑n=1∞arctan⁡(n)n2\sum_{n=1}^{\infty} \frac{\arctan (n)}{n^{2}}n=1∑∞​n2arctan(n)​ converges or diverges.

Problem 28332

Find the radius of convergence for the series ∑n=1∞6nn!nnxn\sum_{n=1}^{\infty} \frac{6^{n} n !}{n^{n}} x^{n}∑n=1∞​nn6nn!​xn.

Problem 28333

Differentiate f(x)=xexf(x) = x e^{x}f(x)=xex three times and find its inflection point(s).

Problem 28334

Find the horizontal asymptote of the curve y=4x6+2x210+x3y=\frac{\sqrt{4 x^{6}+2 x^{2}}}{10+x^{3}}y=10+x34x6+2x2​​. Choices: a. y=2y=2y=2 \& y=−2y=-2y=−2, b. y=−2y=-2y=−2, c. y=2y=2y=2, d. y=0y=0y=0, e. none.

Problem 28335

Calculate the limit: lim⁡x→−∞exsin⁡(x2)\lim _{x \rightarrow-\infty} e^{x} \sin \left(x^{2}\right)limx→−∞​exsin(x2). State if it doesn't exist.

Problem 28336

Determine if the series ∑n=0∞n503n\sum_{n=0}^{\infty} \frac{n^{50}}{3^{n}}n=0∑∞​3nn50​ converges or diverges.

Problem 28337

Find the value of c\mathrm{c}c for the mean value theorem for f(x)=x+1xf(x)=x+\frac{1}{x}f(x)=x+x1​ on [1,3][1,3][1,3]. Options: a. 0 b. 2 c. 3\sqrt{3}3​ d. {3,−3}\{\sqrt{3}, -\sqrt{3}\}{3​,−3​} e. None

Problem 28338

Calculate the limit: lim⁡x→0sin⁡(3x)tan⁡(7x)\lim _{x \rightarrow 0} \frac{\sin (3 x)}{\tan (7 x)}limx→0​tan(7x)sin(3x)​. Does it exist?

Problem 28339

Calculate the average rate of change of wind speed S(d)=93log⁡d+65S(d)=93 \log d+65S(d)=93logd+65 from d=100d=100d=100 to d=1000d=1000d=1000. Use m=S(1000)−S(100)900m=\frac{S(1000)-S(100)}{900}m=900S(1000)−S(100)​.

Problem 28340

1. Sprawdź, czy funkcja f=ln⁡f=\lnf=ln jest jednostajnie ciągła i lipschitzowska na dziedzinach (a;+∞)(a ;+\infty)(a;+∞) i (0;+∞)(0 ;+\infty)(0;+∞).2. Udowodnij, że ciągła i okresowa funkcja fff z Tn→0T_{n} \to 0Tn​→0 jest stała. Czy ciągłość jest kluczowa?

Problem 28341

Find the number of real roots of the continuous function fff in the interval (−3,2)(-3,2)(−3,2) given f(−3)=2,f(−1)=−2,f(0)=−1,f(2)=5f(-3)=2, f(-1)=-2, f(0)=-1, f(2)=5f(−3)=2,f(−1)=−2,f(0)=−1,f(2)=5. Options: a. Exactly two b. at least 3 c. at least two d. at most two.

Problem 28342

Problem 28343

Calculate the limit: lim⁡x→−1−e−11−x2\lim _{x \rightarrow-1^{-}} e^{\frac{-1}{1-x^{2}}}limx→−1−​e1−x2−1​. Does it exist?

Find the first five terms of the sequence $\left[\frac{(-1)^{n+1}}{(n-6)^{2}}\right]_{n=1}^{5}$ and check for convergence.

Determine if the sequence converges or diverges. If convergent, find the limit: $\lim _{n \rightarrow \infty} \frac{14(2^{n})+10}{7(2^{n})}$

Determine if the sequence $a_{n}=\frac{n^{3}+\sin (7 n+15)}{n^{7}+15}$ converges or diverges, and find the limit if it converges.

Find the limit of the sequence $c_{n}=\ln \left(\frac{4 n-7}{3 n+4}\right)$ as $n \to \infty$ , or state DIV if it diverges.

Finde die erste Ableitung von $f(x) = \frac{\sqrt{2}}{x}$ .

Bestimmen Sie die Ableitung von $f(x)=\sqrt{\frac{2}{x}}$ .

Bestimme die erste Ableitung von $f(x)=\sqrt{\frac{2}{x}}$ mit der Potenz- und Faktorregel.

A particle is dropped from 3 m high with an initial speed of $9.5 \mathrm{~m} \mathrm{~s}^{-1}$ . Find its time of flight.

4) A tornado's wind speed is given by $S(d)=93 \log d+65$ .
a) Find the average speed change from $d=100$ to $d=1000$ .
b) Estimate the instantaneous speed change at $d=100$ using $h=0.001$ .

As $n$ approaches infinity, determine limits for $r$ in $\lim_{n \to \infty} r^{n+1}=0$ and sum series $\lim_{n \to \infty} \frac{1-r^{n+1}}{1-r}$ .

Identify the divergent infinite series from the following: I: 24-12+6-3+... II: -0.2+1-5+25-... III: 64+32+16+8+... IV: $\frac{1}{9}-\frac{1}{3}+1-3+...$

Find the partial sums $S_{2}, S_{4}$ , and $S_{6}$ for the series $3+\frac{3}{2^{2}}+\frac{3}{3^{2}}+\cdots$ .

Check if the series converges and find its sum: $\sum_{k=1}^{\infty} \frac{2^{k+3}}{5^{k+1}}=\square$ (Enter DNE if no sum exists.)

Find the displacement of a particle with velocity $v=2 t^{6}+4 t^{4}+\frac{13}{2} t^{2}$ from $t=0$ to $t=2$ , given $s=7$ at $t=0$ . Round to 3 significant figures.

Find $\lim _{x \rightarrow 3} h(x)$ for the piecewise function $h(x)=\left\{\begin{array}{cc}x^{2}-2 x+1 & , x \neq 3 \\ 7 & , x=3\end{array}\right.$ .

Water is added to a tank with radius $5 \mathrm{~m}$ and height $10 \mathrm{~m}$ at $100 \mathrm{~L/min}$ . Find $\frac{d h}{d t}$ when water is $6 \mathrm{~m}$ deep.

Find the derivative $\frac{d y}{d x}$ for $y=\frac{x^{5}}{3-x}$ .

Bestimme die Tangentengleichung an $f(x)=e^{x}+1$ im Punkt $P(0 \mid f(0))$ und berechne den eingeschlossenen Flächeninhalt.

Find the function $f(x)$ such that $f^{\prime \prime}(x) = -\frac{1}{4} \sin \left(\frac{x}{2}\right)$ .

Berechne die Steigung von $f$ an $x_{0}$ : a) $f(x)=x^{2}$ , $x_{0}=-1$ ; b) $f(x)=a x^{2}$ , $x_{0}=1$ .

Find the general solution of $\frac{d^{2} x}{d t^{2}}+2 \frac{d x}{d t}=e^{-t}$ with $x(0)=0$ , $x'(0)=0$ . Choices: a, b, c, d, e.

Calculate the work $W$ done by the force $\vec{F}=(4 \hat{i}+2 \hat{j}-k) N$ from $A(1,1,0)$ to $B(1,-1,-2)$ .

Determine if the series $\sum_{n=1}^{\infty} \frac{\arctan (n)}{n^{2}}$ converges or diverges.

Find the radius of convergence for the series $\sum_{n=1}^{\infty} \frac{6^{n} n !}{n^{n}} x^{n}$ .

Differentiate $f(x) = x e^{x}$ three times and find its inflection point(s).

Find the horizontal asymptote of the curve $y=\frac{\sqrt{4 x^{6}+2 x^{2}}}{10+x^{3}}$ . Choices: a. $y=2$ \& $y=-2$ , b. $y=-2$ , c. $y=2$ , d. $y=0$ , e. none.

Calculate the limit: $\lim _{x \rightarrow-\infty} e^{x} \sin \left(x^{2}\right)$ . State if it doesn't exist.

Determine if the series $\sum_{n=0}^{\infty} \frac{n^{50}}{3^{n}}$ converges or diverges.

Find the value of $\mathrm{c}$ for the mean value theorem for $f(x)=x+\frac{1}{x}$ on $[1,3]$ . Options: a. 0 b. 2 c. $\sqrt{3}$ d. $\{\sqrt{3}, -\sqrt{3}\}$ e. None

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{\tan (7 x)}$ . Does it exist?

Calculate the average rate of change of wind speed $S(d)=93 \log d+65$ from $d=100$ to $d=1000$ . Use $m=\frac{S(1000)-S(100)}{900}$ .

1. Sprawdź, czy funkcja $f=\ln$ jest jednostajnie ciągła i lipschitzowska na dziedzinach $(a ;+\infty)$ i $(0 ;+\infty)$ .
2. Udowodnij, że ciągła i okresowa funkcja $f$ z $T_{n} \to 0$ jest stała. Czy ciągłość jest kluczowa?

Find the number of real roots of the continuous function $f$ in the interval $(-3,2)$ given $f(-3)=2, f(-1)=-2, f(0)=-1, f(2)=5$ . Options: a. Exactly two b. at least 3 c. at least two d. at most two.

Calculate the limit: $\lim _{x \rightarrow-1^{-}} e^{\frac{-1}{1-x^{2}}}$ . Does it exist?

Find the inflection point of the graph of $f(x)=\sinh (2-x)+1$ . Choose from: a. $(2,1)$ b. $(-1,-2)$ c. $(1,2)$ d. $(-2,-1)$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{\left(1+2(a+h)^{2}\right)^{3}-\left(1+2 a^{2}\right)^{3}}{h}$ . Choices: a, b, c, d.

Calculate the integral: $\int \sqrt{1+\sin ^{2} x} \, dx$

The limit as $x$ approaches $3$ from the right of $[6 - 3x]$ equals $-3$ . True or False?

Find $g^{\prime}(3)$ given $f(3)=6$ , $f^{\prime}(3)=3$ , and $g(x)=\frac{f^{2}(x)}{x}$ . Choices: a. 9 b. 4 c. 5 d. 8

If $f^{\prime}(x)=0$ for all $x$ , is $f$ constant on its domain? True or False?

Evaluate the integral: $\int \cos x \sqrt{1+\sin ^{2} x} \, dx$

Bestimme die Wachstumsgeschwindigkeit der Pilzkultur mit $f(t)=7,5 e^{0,3 t}$ nach 5 Wochen und 3 Tagen. Finde den Zeitpunkt für 10 Millionen/Woche und die durchschnittliche Geschwindigkeit in der ersten Woche.

Find the $n^{\text{th}}$ derivative of $f(x)=-\cosh 2x - \sinh 2x$ at $x=0$ : $\stackrel{(n)}{f}(0)=$ a. $(-1)^{n-1}2^{n}$ b. $-2^{n}$ c. 0 d. $(-1)^{n}2^{n}$ e. $2^{n}$

Estimate the wind speed change rate at $d=100$ miles using $S(d)=93 \log d + 65$ and $h=0.001$ .

Find the velocity $v(t)$ and acceleration $a(t)=\frac{2 t(t^{2}-3)}{(t^{2}+1)^{3}}$ for $s(t)=\frac{t}{1+t^{2}}$ . Determine when the particle is slowing down or speeding up.

Evaluate the integral: $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 2 \pi \cos x \sqrt{1+\sin ^{2} x} \, dx$ .

Find the value of $\int_{0}^{\frac{\pi}{6}}(\cos x+\sec x)^{2} d x$ .

Given the function $f(x)=\frac{-5}{x-1}$ and $a=3$ , find the slope of the tangent line and its equation at $x=a$ .

Find the area between the curves $y=5 \sqrt{x}$ and $y=5 x^{2}$ .

Berechne die Fläche unter der Funktion $f(x)=x^2$ im Intervall $I=[-1; 2]$ mit dem Integral $\int_{-1}^{2} x^{2} dx$ .

Find the minima, maxima, intercepts, and intervals of increase/decrease for $f(x) = x^{3} - 2x^{2} - 4x + 8$ .

Calculate the present value (PV) needed now for a future value (FV) of \ $100,000 in 13 years at 10\% interest, using$ FV = PV \cdot e^{rt}$.

Find the minima, maxima, and intervals of increase/decrease for the function $f(x)=x^3-2x^2-4x+8$ .

Find the limit: $\lim _{x \rightarrow \pi} \frac{\cos x+\sin (2 x)+1}{x^{2}-\pi^{2}}$ . Choose (A) $\frac{1}{2 \pi}$ , (B) $\frac{1}{\pi}$ , (C) 1, or (D) nonexistent.

Find the unique value of $x$ where the tangents of the curves $y \cdot 0.9 \ln 3 - 3^{0.9 x} = 0$ and $5^{0.4 + 0.3 x} = y \cdot 0.3 \ln 5$ are parallel.

Check if the function defined by $f(x) = \left\{\begin{array}{l}8 x-7 \text { if } x \geq 1 \\ 7 x-8 \text { if } x<1\end{array}\right.$ is differentiable at $x=1$ .

Test if the function $f(x)=\left\{\begin{array}{l}3 x-2 \text { when } x \geq 1 \\ 2 x-3 \text { when } x<1\end{array}\right.$ is differentiable at $x=1$ .

Test if the function $f(x) = 3x - 2$ for $x \geq 1$ and $f(x) = 2x - 3$ for $x < 1$ is differentiable at $x = 1$ .

Evaluate the double integral: $\int_{5}^{6} \int_{\sqrt{x}}^{x} \frac{y}{x} dy dx$

Evaluate the double integral: $\int_{4}^{5} \int_{\sqrt{x}}^{x} \frac{y}{x} \, dy \, dx$

Berechne die Fläche der Funktion $f(x)=x^2-9$ im Intervall $I=[-4; 3]$ mit dem Integral: $\int_{-4}^{-3} x^{2}-9 \, dx + \int_{-3}^{3} x^{2}-9 \, dx$