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Unformatted text preview: Ch4/MATH1804/YMC/200910 1 Chapter 4. Differentiation and Its Applications 4.1. Review of derivatives • Difference quotient and Derivatives The difference quotient of a function f ( x ) at x = a with increment h is given by f ( a + h ) f ( a ) h and the derivative of f ( x ) at x = a is the limit lim h → f ( a + h ) f ( a ) h and we denote this limit by f ( a ) or d dx f ( x ) x = a . • Geometric interpretation of derivatives The tangent line of the graph of f ( x ) at the point P = ( a,f ( a )) is the line through P whose slope is the limit of the secant slopes as a nearby point Q approaches P . This limit is the derivative f ( a ) of f at a . The equation for the tangent line to f ( x ) at the ( a,f ( a )) is then given by y = f ( a ) + f ( a )( x a ) . • Physical interpretation of derivatives The derivative can also be interpreted as the instantaneous rate of change , for example, the velocity of a moving object at the instant t . • Existence of derivatives The following shows some cases when the derivative fails to exist: (1) The limit lim h → f ( a + h ) f ( a ) h will exist if both the lefthand limit and righthand limit lim h → f ( a + h ) f ( a ) h and lim h → + f ( a + h ) f ( a ) h exist and if they are equal to each other. For example, the absolute value function f ( x ) =  x  is NOT differentiable at x = 0 , since lim h →  h   h = 1 and lim h → +  h   h = 1 . Ch4/MATH1804/YMC/200910 2 (2) The derivative does not exist at x = a if the function is discontinuous at a , for example, the function f ( x ) = ( 1 if x < x if x ≥ In this case, the lefthand limit does not exist at a = 0 : lim h → f (0 + h ) f (0) h = lim h → 1 h =∞ . (3) The derivative does not exist at x = a if the function has a vertical tangent line at x = a (the slope is undefined in this case). For example, the function f ( x ) = x 1 3 has a vertical tangent line at x = 0 and hence is NOT differentiable at x = 0 . • Differentiation rules 1. If f ( x ) is a constant function, that is, f ( x ) = c for some constant c , then d dx f ( x ) = d dx c = 0 . 2. If n is a positive/negative integer, then d dx x n = nx n 1 . 3. If f ( x ) is a differentiable function and c is a constant, then d dx ( cf ( x ) ) = c d dx f ( x ) . 4. If f ( x ) and g ( x ) are differentiable functions, then d dx ( f ( x ) ± g ( x ) ) = d dx f ( x ) ± d dx g ( x ) . 5. If f ( x ) and g ( x ) are differentiable functions, then d dx ( f ( x ) g ( x ) ) = d dx f ( x ) g ( x ) + f ( x ) d dx g ( x ) = f ( x ) g ( x ) + f ( x ) g ( x ) . 6. If f ( x ) and g ( x ) are differentiable functions, then d dx f ( x ) g ( x ) = g ( x ) d dx f ( x ) f ( x ) d dx g ( x ) ( g ( x ) ) 2 = g ( x ) f ( x ) f ( x ) g ( x ) ( g ( x ) ) 2 ....
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 Spring '10
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 Derivative, lim, Convex function

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