Consider the function \( f(x)=4 x-3 \sin x \) on the interval \( \left[\frac{\pi}{2}, 2 \pi\right] \). Find the absolute minimum value of \( f(x) \) over the given interval. A) \( \cos ^{-1}\left(\frac{3}{4}\right) \) B) \( 8 \pi \) C) \( 2 \pi \) D) \( 2 \pi-3 \)

7. c) \( \frac{1}{1+x}-\frac{2}{x^{2}-1} \)

Use a Riemann sum to approximate the area under the graph of \( f(x)=4 e^{-x} \) on the interval \( 0 \leq x \leq 1 \) using \( n=5 \) subintervals with the selected points as the right end points. The area is approximately \( \square \) units. (Round to six decimal places as needed.)

A national fast food company has just analyzed the effect of the latest sales promotion. It has determined that the sales decay for its milkshakes is given by \( S=285,000 e^{-0.7 x} \), where \( S \) is the weekly sales and \( x \) is the number of weeks that have passes since the end of the sales promotion. a) Determine the sales 2 weeks after the end of the promotion: Round answer to 2 decimal places. Sales \( =\$ \) b) If no new campaign is started, how many weeks after the end of the promotion will sales drop below 30,000 ? Round answer to 1 decimal place. Num. weeks =

Find the average rate of change of \( h(x)=-2 x^{2}-5 x \) from \( x=-4 \) to \( x=1 \). Simplify your answer as much as possible.

2. \( y=\frac{3^{\sqrt{x}}}{\lg (3-x)} \)