10-1. To find the marginal revenue function, we need to take the derivative of the revenue function R(x)=250x−0.01x2 R(x) = 250x - 0.01x^2 R(x)=250x−0.01x2 with respect to x x x.
10-2. To find R(72)−R(71) R(72) - R(71) R(72)−R(71), we calculate the revenue at 72 units and subtract the revenue at 71 units. R′(72) R'(72) R′(72) is the derivative of the revenue function evaluated at x=72 x = 72 x=72, which represents the approximate change in revenue for an additional unit sold at 72 units.
10-3. The profit function P(x) P(x) P(x) is the difference between the revenue function R(x) R(x) R(x) and the cost function C(x) C(x) C(x). The marginal profit function is the derivative of the profit function with respect to x x x.
To find the marginal revenue function, we differentiate the revenue function R(x)=−0.01x2+250x R(x) = -0.01x^2 + 250x R(x)=−0.01x2+250x with respect to x x x: R′(x)=250−0.02x R'(x) = 250 - 0.02x R′(x)=250−0.02x
Next, we compute the revenue at x=72 x = 72 x=72 and x=71 x = 71 x=71: R(72)≈17948.16,R(71)≈17699.59 R(72) \approx 17948.16, \quad R(71) \approx 17699.59 R(72)≈17948.16,R(71)≈17699.59 Thus, the difference in revenue is: R(72)−R(71)≈248.57 R(72) - R(71) \approx 248.57 R(72)−R(71)≈248.57 We also find the marginal revenue at x=72 x = 72 x=72: R′(72)≈248.56 R'(72) \approx 248.56 R′(72)≈248.56
The profit function P(x) P(x) P(x) is given by the difference between revenue and cost: P(x)=R(x)−C(x)=(−0.01x2+250x)−(200x+10000)=−0.01x2+50x−10000 P(x) = R(x) - C(x) = (-0.01x^2 + 250x) - (200x + 10000) = -0.01x^2 + 50x - 10000 P(x)=R(x)−C(x)=(−0.01x2+250x)−(200x+10000)=−0.01x2+50x−10000 The marginal profit function is the derivative of the profit function: P′(x)=50−0.02x P'(x) = 50 - 0.02x P′(x)=50−0.02x
R′(x)=250−0.02x,R(72)−R(71)≈248.57,R′(72)≈248.56 \boxed{R'(x) = 250 - 0.02x, \quad R(72) - R(71) \approx 248.57, \quad R'(72) \approx 248.56} R′(x)=250−0.02x,R(72)−R(71)≈248.57,R′(72)≈248.56
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