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# 3.8.notes - y = f x Δ x-f x Example Suppose that C = 0 02...

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MAC 2233/001-006 Business Calculus, Fall 2011 CRN: 81700–81702, 81705, 81716, 81715 Assistants : Matthew Fleeman, Arbin Rai, Tadesse Zerihun, Vindya Kumari, Junyi Tu, Jing- han Meng On differentials and approximation to Δ y I like to give a brief explanation of the concepts in Section 3.8. Suppose that y = f ( x ) and if x is increased by a small amount Δ x , then this will effect a small change in y . This small change in y is denoted by Δ y and Δ y = f ( x + Δ x ) - f ( x ) . We learned from the § 2.3 on marginals that the derivative of f (for example, dR dx ) gives the approximate change in f per unit change in x . Hence, when Δ x is close to 0, we have Δ y = f ( x + Δ x ) - f ( x ) f ( x x. (1) The differential of x is denoted by dx . It represents a nonzero number, usually a close to 0. Also, the differential of y is defined as dy = f ( x ) dx. (2) Now if dx = Δ x (that is, if we equate dx and Δ x ), then we have from (1) and (2) that the differential of y is an approximation for Δ y , that is, dy Δ y, where Δ
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Unformatted text preview: y = f ( x + Δ x )-f ( x ) . Example . Suppose that C = 0 . 02 x 2 +3 . 05 x +1 . 2 is the cost (in thousand dollars) for recycling x , in thousand pounds, of aluminum. Currently x = 3 . 4. Find the additional cost if x is raised to 3.6. Solution. Here x = 3 . 4 and dx = Δ x = 0 . 2. Thus, Δ C ≈ dC = dC dx dx = (0 . 04 x + 3 . 05) dx = (0 . 04 × 3 . 4 + 3 . 05) × . 2 = 0 . 6372 thousand dollars . That is, to increase production of recycled aluminum (from current level of 3400 pounds) by 200 pounds will incur an approximate cost of \$637.2. Note that the actual additional cost is Δ C = C (3 . 6)-C (3 . 4) = 12 . 4392-11 . 8012 = 0 . 6380 thousand dollars . It is clear that the approximation is easier to calculate. To have an “accurate” approximation, we need Δ x to be small when compared with x ....
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