A company has determined the demand curve for their product is \( q=\sqrt{5000-p^{2}} \), where \( p \) is the price in dollars, and \( q \) is the quantity in millions. If weather conditions are driving the price up \( \$ 2 \) a week, find the rate at which demand is changing when the price is \( \$ 40 \). (Round your answer to the nearest hundredth.) A. Decreasing by 1.37 million items per week B. Increasing by 1.37 million items per week C. Decreasing by 3.17 million items per week D. Increasing by 3.17 million items per week E. None of the answers

77-78 Use l'Hospital's Rule to evaluate the limit. 77. \( \lim _{x \rightarrow 0} \frac{1}{x^{2}} \int_{0}^{x} \frac{2 t}{\sqrt{t^{3}+1}} d t \)

Aufgabe 2 (3 Punkte). (a) Zeigen Sie mit 7.21 und 7.28 , dass für alle \( q \in \mathbb{C} \) mit \( |q|<1 \) gilt \[ \sum_{n=0}^{\infty}(n+1) q^{n}=\frac{1}{(1-q)^{2}} \] (b) Für \( n \in \mathbb{N}_{0} \) sei \( a_{n}=b_{n}=(-1)^{n} / \sqrt{n+1} \). Zeigen Sie, dass das Cauchyprodukt \( \sum_{n=0}^{\infty} c_{n} \) der konvergenten Reihen \( \sum_{n=0}^{\infty} a_{n} \) und \( \sum_{n=0}^{\infty} b_{n} \) divergiert (Hinweis: Zeigen Sie, dass \( \left(c_{n}\right) \) keine Nullfolge ist und benutzen Sie dann 7.11\( ) \).

\( \int _{}^{}\frac{1}{e^{x^{2}}}e^{\int _{}^{}2 d x} d x \)

Calculate the Riemann sum for the function \( f(x) = x^2 \) on the interval [1, 3] using 4 subintervals with left endpoints.

\( \int _{}^{}\frac{1}{e^{x^{2}}}e^{\int _{}^{}2 d x} d x \)