1a) Build the formula, \( A(t) \), for the value of the account after \( t \) years if \( \$ 2 \), is invested at a rate of \( 100 \% \) compounded continuously, \( A(t)=2000 e^{t} \) 1b) Find the derivative, \( A^{\prime}(t) \).

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. \( \int_{2}^{7} 6 g(x) d x=30 \) \( \int_{2}^{7}[g(x)-f(x)] d x=-1 \) (Simplify your answer) (Simplify your answer) \( \int_{2}^{7}[5 g(x)-f(x)] d x=\square \) (Simplify your answer.)

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

\( e^{(4+i)x} \)

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

\( \int _ { 1 } ^ { 2 } \int _ { - 1 } ^ { 2 } ( 12 x y ^ { 2 } - 8 y ^ { 3 } ) d y d \)