Chapter 12, Problem 1CE

To determine

To calculate: The $Z^{\prime }\left(m\right)$.

Answer to Problem 1CE

Solution:

The derivative of the function is:

$\underset{\_}{Z^{\prime }\left(m\right)=-\frac{C_1}{m^2}+\frac{D{C}_3}{2}}$.

Explanation of Solution

Given:

The mathematical model for determining the total cost per month of a training program is provided by

$Z\left(m\right)=\frac{C_1}{m}+Dt{C}_2+D{C}_3\left(\frac{m-1}{2}\right)$

Here,

$D$ is demand for trainees per month;

$C_1$ is a fixed cost of training a batch of trainees;

$C_2$ is the variable cost of training per trainee per month;

$C_3$ is salary paid monthly to a trainee who has not yet been given a job after training;

$m$ time interval in months between successive batches of trainees;

$t$ is the length of the training program in months;

$Z\left(m\right)$ is the total monthly cost of the program.

Formula used:

Power rule differentiation,

$\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$

Calculation:

Consider the provided function,

$Z\left(m\right)=\frac{C_1}{m}+Dt{C}_2+D{C}_3\frac{\left(m-1\right)}{2}$.

Expand the terms,

$Z\left(m\right)=\frac{C_1}{m}+Dt{C}_2+\frac{D{C}_{3}m}{2}-\frac{D{C}_3}{2}$.

As $Z^{\prime }\left(m\right)$ is the derivative of the function, take the derivative of the function.

Use derivative formulae in the above function which is:

$\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$

And,

$\frac{d}{dx}\left(c\right)=0$

Where $c$ constant value and its derivative is zero.

In the above function $Dt{C}_2$ and $\frac{D{C}_3}{2}$ is a constant value so its derivative is zero.

Then,

$\begin{array}{ccc}\hfill \frac{d}{dm}\left(\frac{C_1}{m}+Dt{C}_2+\frac{D{C}_{3}m}{2}-\frac{D{C}_3}{2}\right)& =& -\frac{C_1}{m^2}+0+\frac{D{C}_3}{2}-0\hfill \\ \hfill & =& -C_1{m}^{-2}+\frac{D{C}_3}{2}\hfill \end{array}$

Then,

$Z^{\prime }\left(m\right)=-C_1{m}^{-2}+\frac{D{C}_3}{2}$