Final answer:
To approximate the change in profit for a one-unit increase in sales using differentials, calculate the derivative of the profit function at the current sales level and multiply by one. For a function P(x) = -0.2x^2 + 50x - 80 at x = 50, the approximate change is $30. This approximation can be compared with the actual change by evaluating the profit function at x = 50 and x = 51.
Step-by-step explanation:
To approximate the change in profit using differentials for a one-unit increase in sales or production, one would typically take the derivative of the profit function with respect to the number of units sold (x) and then multiply by the change in x. However, there seems to be a typo in the profit function provided. Assuming the correct function is P(x) = -0.2x^2 + 50x - 80, and we're approximating the change in profit at x = 50 units, the differential dP can be calculated by finding P'(x) and then using the increment dx = 1 for the one-unit increase.
The derivative P'(x) = dP/dx = -0.4x + 50. At x = 50, P'(50) = -0.4(50) + 50 = -20 + 50 = 30. Thus, the approximate change in profit for a one-unit increase is dP = P'(50) × dx = 30 × 1 = $30.
To compare this with the actual change in profit, we'll compute P(51) - P(50). P(51) = -0.2(51)^2 + 50(51) - 80 and P(50) = -0.2(50)^2 + 50(50) - 80. The actual change in profit is therefore P(51) - P(50), which can be calculated as an exact value and compared with the approximate change of $30 obtained using differentials.