Cálculo preguntas y respuestas

Aufgabe 3 (4 Punkte). Die R-wertigen Folgen \( \left(a_{n}\right) \) und \( \left(b_{n}\right) \) seien gegeben durch \( \quad\left(a_{n}\right)=(0,1,2,1,0,1,2,1,0,1,2,1, \ldots), \quad b_{n}=(2,1,1,0,2,1,1,0,2,1,1,0, \ldots) \) Bestimmen Sie \( \quad \liminf _{n \rightarrow \infty} a_{n}+\liminf _{n \rightarrow \infty} b_{n}, \quad \liminf _{n \rightarrow \infty}\left(a_{n}+b_{n}\right), \quad \limsup _{n \rightarrow \infty}\left(a_{n}+b_{n}\right), \limsup _{n \rightarrow \infty} a_{n}+\limsup _{n \rightarrow \infty} b_{n} \). Aufgabe \( 4(3 \) Punkte \( ) \). 1. Zeigen Sie, dass \( \sum_{k=0}^{n} \frac{k}{2^{k}}=2-\frac{n+2}{2^{n}} \) für alle \( n \in \mathbb{N}_{0} \) gilt. 2. Bestimmen Sie den Grenzwert \( \sum_{k=0}^{\infty} \frac{k}{2^{k}} \). 3. Zeigen Sie mit Hilfe des Leibniz-Kriteriums, dass die Reihe \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{\sqrt{2 n+1}} \) konvergiert.

Question 5 (1 point) Find the maximum value of \( f(x)=-2 x^{2}+7 x-3 \), if your answer fis a decimal, give it to 3 decimal places Blank 1: Question 6 (1 point)

Homing pigeons avoid flying over water. Suppose a homing pigeon is released on an island at point C, which is 11 mi directly out in the water from a point B on shore. Point B is 24 mi downshore from the pigeon's home loft at point A. Assume that a pigeon flying over water uses energy at a rate 1.33 times the rate over land. Toward what point S downshore from A should the pigeon fly in order to minimize the total energy required to get to the home loft at A?

Homing pigeons avoid flying over water. Suppose a homing pigeon is released on an island at point \( C \), which is 11 mi directly out in the water from a point \( B \) on shore. Point \( B \) is 24 mi downshore from the pigeon's home loft at point \( A \). Assume that a pigeon flying over water uses energy at a rate 1.33 times the rate over land. Toward what point \( S \) downshore from A should the pigeon fly in order to minimize the total energy required to get to the home loft at A?

A power line is to be constructed from a power station at point \( A \) to an island at point C , which is 4 mi directly out in the water from a point B on the shore. Point \( B \) is 8 mi downshore from the power station at \( A \). It costs \( \$ 2800 \) per mile to lay the power line under water and \( \$ 2000 \) per mile to lay the line under ground. At what point S downshore from A should the line come to the shore in order to minimize cost? Note that S could very well be B or A . (Hint. The length of CS is \( \sqrt{16+\mathrm{x}^{2}} \).)

a) \( f(x)=\tan (3 x) \) 8) Mediante derivación implicita, obtenga la derivada con respecto a \( x \) de la finción \( \begin{array}{ll}\text { a) } 3 x^{5} y^{3}+2 x^{4}=x^{2} y^{2}+1 & \text { b) } x^{3}-y^{4}+3=\operatorname{sen}\left(x^{3} y\right)\end{array} \)

An open-top cylindrical container is to have a volume \( 2197 \mathrm{~cm}^{3} \). What dimensions (radius and height) will minimize the surface area?

2. Llene la casilla en blanco con la letra \( \mathbf{F} \) (Falso) o \( \mathbf{V} \) (Verdadero) según sea el caso. Justifique. (a) (4 puntos) \( \square \) Si \( f(x)=\int_{-2}^{x} \frac{1}{t+3} d t \), donde \( x>-2 \), entonces \( f^{\prime}(7)=1 / 10 \) (b) (4 puntos) \( \square \lim _{n \rightarrow \infty} \sum_{i=1}^{n} \sin \left(\frac{2 i}{n}\right) \frac{2}{n}=\int_{0}^{2} \sin x d x \) (c) (2 puntos) \( \square \int_{-\pi}^{\pi} \sin ^{13} x d x=0 \)

Вычислите неправильный интеграл: \( \int_{1}^{\infty} \frac{1}{x^{2}} \, dx \).

Evaluate the integral \( \int (3x^2) \cdot e^{x^3} \, dx \) using substitution.