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Unformatted text preview: Calculus refresher Disclaimer: I claim no original content on this document, which is mostly a summaryrewrite of what any standard college calculus book offers. (Here Ive used Calculus by Dennis Zill.) I consider this as a brief refresher of the barebones calculus requirements we will be using during the course. No exercises are offered: any calculus book provides an insane amount of practice problems. 1. Rules of differentiation: Basics In what follows, let f ( x ) be a onevariable function, and denote its derivative by f ( x ). The standard differentiation rules are the following: Theorem 1.1 (The Power Rule, 1) . Let n be a positive integer. Then d dx [ x n ] = nx n 1 . (1) Example 1.2. The derivative of y = x 4 is given by dy dx = 4 x 4 1 = 4 x 3 . Theorem 1.3 (Derivative of a constant function) . If f ( x ) = k and k is a constant, then f ( x ) = 0 . Theorem 1.4 (Derivative of a constant multiple of a function) . If c is any constant and f is a differentiable function, then d dx [ cf ( x )] = cf ( x ) . (2) Example 1.5. Following theorems 1.1 and 1.4, the derivative of y = 3 x 5 is given by: dy dx = 3 d dx x 5 = 3(5 x 4 ) = 15 x 4 . Theorem 1.6 (The Sum Rule) . Let f and g be two differentiable functions. Then d dx [ f ( x ) + g ( x )] = f ( x ) + g ( x ) . (3) Example 1.7. From theorems 1.1. and 1.6, the derivative of y = x 4 + x 3 is: dy dx = d dx x 4 + d dx x 3 = 4 x 3 + 3 x 2 . Theorem 1.8 (The Product Rule) . If f and g are differentiable functions, then d dx [ f ( x ) g ( x )] = f ( x ) g ( x ) + g ( x ) f ( x ) . (4) Example 1.9. The differential of y = ( x 3 2 x 2 + 4)(8 x 2 + 5 x ) is: dy dx = ( x 3 2 x 2 + 4) d dx (8 x 2 + 5 x ) + (8 x 2 + 5 x ) d dx ( x 3 2 x 2 + 4) = ( x 3 2 x 2 + 4)(16 x + 5) + (8 x 2 + 5 x )(3 x 2 4 x ) . Theorem 1.10 (The Quotient Rule) . If f and g are differentiable functions, then d dx " f ( x ) g ( x ) # = g ( x ) f ( x ) f ( x ) g ( x ) [ g ( x )] 2 . (5) 1 2 Example 1.11. The differential of y = 3 x 2 1 2 x 3 +5 x 2 +7 is: dy dx = (2 x 3 + 5 x 2 + 7) d dx (3 x 2 1) (3 x 2 1) d dx (2 x 3 + 5 x 2 + 7) (2 x 3 + 5 x 2 + 7) 2 = (2 x 3 + 5 x 2 + 7) (6...
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This note was uploaded on 12/10/2010 for the course ECON 3101 taught by Professor Staff during the Spring '08 term at Minnesota.
 Spring '08
 Staff
 Economics

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