Q. 88 - 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 $\$288,350$ a year and pay taxes according to the given formula.

(a)  Calculate the value of T(288,350) and the limit of $T\left(m\right)$ as $m$ approaches $288,350$ from the left and from the right.

(b)  Use part $\left(a\right)$ to argue that the function T(m) is continuous at $m=288,350$. What does this mean in real-world terms?

Step-by-Step Solution

Answer

Ans:  (a) $T\left(m\right)=91857$

         (b)This means that when you change tax brackets, the higher tax rate only applies to the amount over $288350$.

Step 1. Given information.

given, $m=288350$

Step 2. Put m = 288350 in the function to get the value of T ( 288350 ) as below:

$\begin{array}{r}T\left(m\right)=35787+0.36(m-132600)\\ T\left(288350\right)=35787+0.36(288350-132600)\\ =35787+0.36\left(155750\right)\\ =35787+56070\\ =91857\end{array}$

Step 3. Use part ( a ) to argue that the function T ( m ) is continuous at m = 288350 .

The value for $T\left(288350\right)$ matches up with the value of the next piece $91857+0.396(m-288350)$ when $m=288350$. This means that when you change tax brackets, the higher tax rate only applies to the amount over $288350$ .