Chapter 2.6, Problem 45E

i.

To determine

The average rate of change of C with respect to x when the production level is changed from from $x=100\text{ to }x=105$ .

Explanation of Solution

Given information:

The cost (in dollars) of production of x units of a certain commodity is

  $C(x)=5000+10x+0.5x^2$

Calculations:

Here the cost of producing $\text{100 unit }$ is

  $\begin{array}{l}C(100)=5000+10\times 100+0.5\times {100}^2\\ C(100)=5000+1000+5000=11000\end{array}$

Here the cost of producing $\text{101 unit }$ is

  $C(101)=5000+10\times 101+0.5\times {101}^2=11110.5$

Here the cost of producing $\text{102 unit }$ is

  $C(102)=5000+10\times 102+0.5\times {102}^2=11222$

Here the cost of producing $\text{103 unit }$ is

  $C(103)=5000+10\times 103+0.5\times {103}^2=11334.5$

Here the cost of producing $\text{104 unit }$ is

  $C(104)=5000+10\times 104+0.5\times {104}^2=11448$

Here the cost of producing $\text{105 unit }$ is

  $C(105)=5000+10\times 105+0.5\times {105}^2=11562.5$

Now, Rate of change of production cost from $x=100\text{ to }x=101$ is $11110.5-11000=110.5$

Rate of change of production cost from $x=101\text{ to }x=102$ is $11222-11110.5=111.5$

Rate of change of production cost from $x=102\text{ to }x=103$ is $11334.5-11222=112.5$

Rate of change of production cost from $x=103\text{ to }x=104$ is $11448-11334.5=113.5$

Rate of change of production cost from $x=104\text{ to }x=105$ is $11562.5-11448=114.5$

Now the average rate of change of production cost(C) from $x=100\text{ to }x=105$ is $\frac{110.5+111.5+112.5+113.5+114.5}{5}=\frac{562.5}{5}=112.5$

ii.

To determine

The average rate of change of C with respect to x when the production level is changed from from $x=100\text{ to }x=101$

Explanation of Solution

Given information:

The cost (in dollars) of production of x units of a certain commodity is

  $C(x)=5000+10x+0.5x^2$

Calculations:

Here ,The cost of producing $\text{100 unit }$ is

  $\begin{array}{l}C(100)=5000+10\times 100+0.5\times {100}^2\\ C(100)=5000+1000+5000=11000\end{array}$

The cost of producing $\text{101 unit }$ is

  $C(101)=5000+10\times 101+0.5\times {101}^2=11110.5$

Now,

Rate of change of production cost from $x=100\text{ to }x=101$ is $11110.5-11000=110.5$

iii.

To determine

The instantaneous rate of change of C with respect to x when $x=100$

Explanation of Solution

Given information:

The cost (in dollars) of production of x units of a certain commodity is

  $C(x)=5000+10x+0.5x^2$

Calculations:

Here, the cost of producing $\text{100 unit }$ is,

  $\begin{array}{l}C(100)=5000+10\times 100+0.5\times {100}^2\\ C(100)=5000+1000+5000=11000\end{array}$

Here the cost of producing $\text{99 unit }$ is

  $C(99)=5000+10\times 99+0.5\times {99}^2=10890.5$

Now,

The instantaneous rate of change of C with respect to x when $x=100$ is $11000-10890.5=109.5$