Final answer:
The rates of change when the production output is 600 calculators are: cost at $15,000, revenue at -$23,800,000, and profit at -$23,815,000.
Step-by-step explanation:
The student asked to find the rate of change in cost, revenue, and profit for a company manufacturing calculators. Given the cost function C=70000+30x and the revenue function R=400x−40x², we need to determine these rates when the output x is 600 calculators and the production is increasing at a rate of 500 calculators.
To find the rate of change of cost, we differentiate the cost function with respect to x to get C'(x) = 30. Since the production rate is increasing at 500 calculators, the rate of change in cost is 30 × 500 = 15000 (since C' is constant).
To find the rate of change of revenue, we differentiate the revenue function to get R'(x) = 400 - 80x. When x = 600, R'(600) = 400 - 80(600) = -47600. The rate of change in revenue at 600 calculators is then -47600 × 500 = -23800000.
The profit function is the revenue minus the cost, P(x) = R(x) - C(x). The rate of change in profit is the rate of change in revenue minus the rate of change in cost. Hence, using the previously found rates, the rate of change in profit when the production is 600 calculators would be -23800000 - 15000 = -23815000.