Question

The cost (in dollars) of producing x units of a certain commodity is C(x)=500+10x+0.05x^2


I) find the average rate of change of C with respect x when the production level is from x=100 to x=105 units

ii) find the instantaneous rate of change of C with respect to x=100

Answer 1

Given $C(x)=500+10x+0.05x^2$

i) The average rate of change is the change in $y$ value over the change in $x$ value for two distinct points, hence if $x$ changes from 100 to 105, then you may evaluate the average rate of change of $C(x)$  such that: $\frac{C(105)-C(100)}{105 - 100 }$ .

Now, $C(105)-C(100) = (500 + 10(105) + 0.05 (105)^2) - (500 + 10(100) + 0.05 (100)^2) = (1050 - 1000) + (551.25 - 500) = 101.25$

So, Average rate of change of $C(x)$ from 100 to 105 is $\frac{101.25}{5} = 20.25$.

ii)  The instantaneous rate of change of $C$ with respect to x=100 is $C'(x)|_{x=100}$ .

Now, given $C(x)=500+10x+0.05x^2$

$\implies C'(x) = 10 + 0.1 x$

The instantaneous rate of change of $C$ with respect to 100 is $C'(100) = 10 + 0.1(100) = 20$ .