Q. 87 - Calculus

Question

You constructed a piecewise-defined function from the $2000$ Federal Tax Rate Schedule that you will use in the next two problems. Specifically, you found that a person who makes m dollars a year will pay $T\left(m\right)$ dollars in tax, given by the function

$\begin{array}{r}0.15m,\text{ if }0\le m\le 26,250\\ 3,937+0.28(m-26,250),\text{ if }26,250<m\le 63,550\\ 14,381+0.31(m-63,550),\text{ if }63,550<m\le 132,600\\ 35,787+0.36(m-132,600),\text{ if }132,600<m\le 288,350\\ 91,857+0.396(m-288,350),\text{ if }m>288,350\end{array}$

Suppose you make $\$63,550$ a year and pay tax according to the given formula.

(a)  Calculate the value of T(63,550) and the limit of $T\left(m\right)$as m approaches 63,550 from the left and from the right.

(b)  Use part $\left(a\right)$ to argue that the function $T\left(m\right)$ is continuous at $m=63,550$. What does this mean in real-world terms?

Step-by-Step Solution

Answer

Ans:   (a) $T\left(m\right)=14381$

          (b)This means that when you change tax brackets, the higher tax rate only applies to the amount over $63550$ .

Step 1. Given information.

given, $m=63550$ 

Step 2. Put m = 63550 in the function to get the value of T ( 63550 ) as below:

$\begin{array}{r}T\left(m\right)=3937+0.28(m-26250)\\ T\left(63550\right)=3937+0.28(63550-26250)\\ =3937+0.28\left(37300\right)\\ =3937+10444\\ =14381\end{array}$

Step 3. Use part ( a ) to argue that the function T ( m ) is continuous at m = 63 , 550 .

The value for $T\left(63550\right)$ matches up with the value of the next piece $14381+0.31(m-63550)$when $m=63550$. This means that when you change tax brackets, the higher tax rate only applies to the amount over $63550$ .