CALCULUS For Business, Economics, and the Social and ：微积分的商业，经济，和社会.pptVIP

* Solution: The marginal cost is C’(x)=(1/4)x+3. The total revenue is , the marginal revenue is R’(x)=25-2x/3. The cost of producing the ninth unit is the change in cost as x increases from 8 to 9 and can be estimated by the marginal cost C’(8)=8/4+3=$5. The actual cost of producing the ninth unit is C(9)-C(8)=$5.13 which is reasonably well approximated by the marginal cost C’(8) The revenue obtained from the sale of the ninth unit is approximated by the marginal revenue R’(8)=25-2(8)/3=$19.67 The actual revenue obtained from the sale of the ninth unit is R(9)-R(8)=$19.33. * Marginal analysis is an important example of a general Incremental approximation procedure * Based on the fact that since then for small h, the derivative is approximately equal to the difference quotient . We indicate this approximation by writing Or, equivalently, if , then Approximation by Increment If f(x) is differentiable at x=x0 and △x is a small change in x, then * Example 23 During a medical procedure, the size of a roughly tumor is estimated by measuring its diameter and using the formula to compute its volume. If the diameter is measured as 2.5 cm with a maximum error of 2%, how accurate is the volume measurement? Solution: A sphere of radius R and diameter x=2R has volume The error made in computing this volume using the diameter 2.5 when the actual diameter is 2.5+△x is To be continued * The corresponding maximum error in the calculation of volume is Since it follows that Thus, at worst, the calculation of the volume as 8.181 is off by 0.491 , so the actual volume V must satisfy * The percentage change of a quantity expresses the change in that quantity as a percentage of its size prior to the change. In particular, * Example 24 The GDP of a certain country was bi