The cost in dollars of producing x units of a particular camera is C(x) = x^2 - 10000. (10 points) 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 units in your answer. Find the instantaneous rate of change of C with respect to x when x = 100. Include units in your answer
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To find the average rate of change of C with respect to x when the production level is changed from x = 100 to x = 101, we need to calculate the difference in the cost function C(x) between these two values of x.
1. Calculate C(100):
Plug x = 100 into the cost function: C(100) = 100^2 - 10000 = 10000 - 10000 = 0.
2. Calculate C(101):
Plug x = 101 into the cost function: C(101) = 101^2 - 10000 = 10201 - 10000 = 201.
3. Calculate the average rate of change:
The average rate of change (AROC) is given by the formula:
AROC = (C(101) - C(100)) / (101 - 100).
Substituting the values we calculated:
AROC = (201 - 0) / (101 - 100) = 201 / 1 = 201.
Therefore, the average rate of change of C with respect to x when the production level is changed from x = 100 to x = 101 is 201 dollars per unit.
To find the instantaneous rate of change of C with respect to x when x = 100, we need to calculate the derivative of the cost function C(x).
1. Find the derivative of C(x):
Take the derivative of x^2 - 10000 with respect to x:
C'(x) = 2x.
2. Calculate the instantaneous rate of change:
The instantaneous rate of change (IROC) is given by evaluating the derivative function at the specific value of x.
Substituting x = 100 into the derivative: C'(100) = 2(100) = 200.
Therefore, the instantaneous rate of change of C with respect to x when x = 100 is 200 dollars per unit.