(Solved): please answer all parts 3. The following table shows the year 2000 Federal Tax Rate Schedule for sin ...

3. The following table shows the year 2000 Federal Tax Rate Schedule for single filers: In a previous problem set, you expressed the dollar amount of tax \( T \) owed by a single person as a piecewise-defined function of the person's yearly taxable income \( m \). \[ T(m)=\left\{\begin{array}{ll} 0.15 m & \text { if } 0 \leq m \leq 26,250 \\ 3,937+0.28(m-26,250) & \text { if } 26,250288,350 \end{array}\right. \] (a) Compute \( T(63,549), T(63,550) \) and \( T(63,551) \). Does the amount of taxes owed change very much for these three possible incomes? (a) Compute \( T(63,549), T(63,550) \) and \( T(63,551) \). Does the amount of taxes owed change very much for these three possible incomes? (b) Suppose you make \( \$ 63,550 \) a year and pay taxes according to this schedule. Calculate you taxes owed as well as the left, right and twosided limit of \( T(m) \) as \( m \) approaches \( \$ 63,550 \). That is, compute each of \[ \lim _{m \rightarrow 63,550^{-}} T(m) \quad \lim _{m \rightarrow 63,50^{+}} T(m) \quad \lim _{m \rightarrow 63,550} T(m) \quad T(63,550) \] (c) What do your computations in part (b) tell you about the continuity of \( T \) at \( m=63,350 \) ? You should use the definition of continuity in your answer. (d) Show that \( T \) is continuous at \( m=26,250 \). (e) Explain in your own words why continuity is a desirable property when designing any tax schedule \( T \).