1) Given the power series \( \sum_{n=1}^{\infty} \frac{2^{n}}{n}(4 x-8)^{n} \), then: (a) the center of the power series is (b) the radius of convergence of the power series is (c) the interval of convergence of the power series is

2. \( \int \sec ^{2}(-2 x) d x= \)

A marshy region used for agricultural drainage has become contaminated with a particular element. It has been determined that flushing the area with clean water will reduce the element for a while, but it will then begin to build up again. A biologist has found that the percent of the element in the soil \( x \) months after the flushing begins is given by the function below. When will the element be reduced to a minimum? What is the minimum percent? \( f(x)=\frac{x^{2}+16}{2 x}, 1 \leq x \leq 12 \) Find \( f^{\prime}(x) \). \( f^{\prime}(x)=\frac{1}{2}-\frac{8}{x^{2}} \) When will the element be reduced to a minimum? The element will be reduced to a minimum after 4 month(s). (Type an integer or a decimal.) What is the minimum percent of the element?

\( \int \frac { ( 16 - 9 x ^ { 2 } ) ^ { 3 / 2 } d x } { x ^ { 6 } } \)

3. \( \int \sec ^{2}(-5 x) d x= \)

\( 7 y ^ { \prime \prime } - 2 \sqrt { 7 } y ^ { \prime } + y = 125 \cos ( \sqrt { 7 } x ) \)