A disease has hit a city. The percentage of the population infected \( t \) days after the disease arrives is approximated by \( p(t)=11 t e^{-t / 9} \) for \( 0 \leq t \leq 45 \). (a) After how many days is the percentage of infected people a maximum? (b) What is the maximum percent of the population infected?

Suppose \( \int_{2}^{4} f(x) d x=-8 \int_{2}^{7} f(x) d x=6 \), and \( \int_{2}^{7} g(x) d x=5 \) Evaluate the following integrals \( \int_{7}^{2} g(x) d x=\square \) \( \int_{2}^{7} 6 g(x) d x=\square \) (Simplify your answer.) \( \int_{2}^{7}[g(x)-f(x)] d x=\square \) (Simplify your answer )

(a) \( \sum_{n=1}^{\infty} 2\left(\frac{4}{3}\right)^{n-1} \)

Complete the following steps for the given function interval, and value of \( n \). a. Sketch the graph of the function on the given interval. b. Calculate \( \Delta x \) and the grid points \( x_{0} x_{1} \quad x_{n} \) c. Illustrate the left and right Riemann sums, and determine which Riemann sum underestimates and which sum overestimates the area under the curve d. Calculate the left and right Riemann sums. \[ f(x)=\cos (x-5 \pi / 8) \text { on }[\pi / 2 \pi] n=4 \]

\( \int _ { 0 } ^ { 3 \pi / 2 } x \sin x d x = \square \)

b) \( \lim _{x \rightarrow \frac{\pi}{4}} \frac{\sqrt{2} \sin x-1}{2 \cos ^{2} x-1} \)