Q. 79 - Calculus

Question

As a vacuum cleaner salesman, Alex earns a salary of \(8,500 a year, whether he sells any vacuum cleaners or

not. In addition, for every 30 vacuum cleaners he sells, he earns a \)1,500 commission.

(a) Construct a piecewise-defined function M(v) that describes the amount of money M that Alex will make in a year if he sells v vacuum cleaners over the course of the year. Assume he sells between 0 and 90 vacuum cleaners in a year.

(b) Check that your function makes sense by using it to calculate M(0), M(30), M(59), M(61), and M(90). Then sketch a graph of M(v) on the interval 0 ≤ v ≤ 90.

(c) The piecewise-defined function M(v) is not continuous. List all the values at which M(v) fails to be continuous, and use the definition of continuity to support your answers.

Step-by-Step Solution

Answer

Part (a)  $M\left(v\right)=\left\{\begin{array}{l}8,500 0\le v\le 30\\ 10,00 30\le v\le 60\\ 11,500 60\le v\le 90\\ 13,000 \mathrm{v}=90\end{array}\right.$

part (b) $M\left(0\right)=\$8,500, M\left(30\right)=\$10,000, M\left(61\right)=\$11,500, M\left(90\right)=\$13,000$

part (c) $v=30,60, \mathrm{and} 60$.

Part (a) Step 1. Given information.

Consider the given information,

Alex earns a salary of $8,500 a year, whether he sells any vacuum cleaners or not. In addition, for every 30 vacuum cleaners he sells, he earns a $1,500 commission.

Part (a) Step 2. Write the function.

Now, construct a piecewise-defined function M(v), as Alex is getting a $1500 commission for every 30 vacuum cleaners he sells. So, write the function.

$M\left(v\right)=\left\{\begin{array}{l}8,500 0\le v\le 30\\ 10,00 30\le v\le 60\\ 11,500 60\le v\le 90\\ 13,000 \mathrm{v}=90\end{array}\right.$

Part (b) Step 2. Find all the values.

Consider the obtained function in part (a).

$M\left(v\right)=\left\{\begin{array}{l}8,500 0\le v\le 30\\ 10,00 30\le v\le 60\\ 11,500 60\le v\le 90\\ 13,000 \mathrm{v}=90\end{array}\right.$

M(0), M(30), M(59), M(61), and M(90). 

$$\begin{gathered} M\left(0\right)=\$8,500 \\ M\left(30\right)=\$10,000 \\ M\left(61\right)=\$11,500 \\ M\left(90\right)=\$13,000 \end{gathered}$$

Now, sketch a graph of M(v) on the interval 0 ≤ v ≤ 90.

Part (c) Step 1. Explanation.

Consider the given information,

The piecewise function M(v) is not continuous at the values v=30, 60, and 90.

By the definition of continuous function, the function is said to be continuous at a point,

If the left-hand limit is equal to the right-hand limit.

If $\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right)$, then the function is said to be continuous at point a.

Since the given function is piecewise continuous, at the points 30, 60, and 90, the left-hand limit and the right-hand do not exist in the same interval.