Help on average rate of change and instantaneous rate of change?

Question

The cost in dollars of producing x units of a sofa is c(x)=x^2-10000

A - find the average rate of change of c with respect to x when the production level is changed from x=100 to x=101 include unit in the answer

B- find the instantaneous rate of change of c with respect to x when x=100 include unit in your answer

Also show work so I could know how to do it

Answer 1

A).

The function c(x) = x^2 - 10000.

Average rate of change with respect to x = c ' (x) = (c(b) - c(a))/(b - a).

Where, b = 101 and a = 100.

c ' (x) = (c(101) - c(100))/(101 - 100)

= [101^2 - 10000 - (100^2 - 10000)]/1

= 101^2 - 10000 - 100^2 + 10000

= 101^2 - 100^2

= 10201 - 10000

= 201.

Therefore, the average rate of change of c with respect to x when the production level is changed from x = 100 to 101 is 201.

B).

instantaneous rate of change of c(x) : c ' (x) = 2x.

when x = 100, c ' (x) = 2 * 100 = 200.

Therefore, instantaneous rate of change of c with respect to x when x = 100 is 200 units.