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 can calculate the difference in C(x) values at these two points and divide it by the difference in x values.
1. Calculate C(100):
C(100) = (100)^2 - 10000 = 10000 - 10000 = 0
2. Calculate C(101):
C(101) = (101)^2 - 10000 = 10201 - 10000 = 201
3. Calculate the difference in C(x) values:
ΔC = C(101) - C(100) = 201 - 0 = 201
4. Calculate the difference in x values:
Δx = 101 - 100 = 1
5. Calculate the average rate of change:
Average rate of change = ΔC / Δx = 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 can find the derivative of the function C(x) = x^2 - 10000 and substitute x = 100 into it.
1. Take the derivative of C(x):
C'(x) = d/dx(x^2 - 10000)
= 2x
2. Substitute x = 100 into C'(x):
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.