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ch.4 - Chapter 4 Partial Dierentiation 4.1 Review on...

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Chapter 4. Partial Differentiation 4.1. Review on functions of one variable Let f : R -→ R be a real-valued function of one variable x R . The graph consists of the points in the Cartesian plane R 2 whose coordinates are the input-output pairs for f . In set notation, the graph is { ( x, f ( x )) T R 2 : x R } . Limits Informally, a function f ( x ) has a limit l at a point a if the value of f ( x ) can be made as close to l as desired, by making x close enough to a . We write lim x a f ( x ) = l to mean that l is the limit of f ( x ) as x a . When computing a limit, we are NOT concerned with what happens to f ( x ) when x = a , but only with what happens to f ( x ) when x is sufficiently close to a . In fact, the limit value does not depend on how the function is defined at a . One-Sided Limits As x approaches a from the right , f ( x ) approaches l - and we write lim x a + f ( x ) = l - . As x approaches a from the left , f ( x ) approaches l + and we write lim x a - f ( x ) = l + . Limits of these forms are called one-sided limits . The limit is a right-hand limit if the approach is from the right. From the left, it is a left-hand limit . If both lim x a - f ( x ) and lim x a + f ( x ) exist, and lim x a - f ( x ) = lim x a + f ( x ) , then lim x a f ( x ) exists. Limits at Infinity There is also a concept for limits as x approaches infinity, that is, as x either grows without bound positively or negatively. Example: lim x + f ( x ) = 1 Continuity A function f ( x ) is continuous at x = c if and only if it meets the following three conditions. 1. f ( c ) exists 2. lim x c f ( x ) exists 3. lim x c f ( x ) = f ( c ) The function f ( x ) = 1 /x is continuous on its natural domain R \ { 0 } . It has a point of discontinuity at x = 0 , as it is not defined there. We also say that f ( x ) has an infinite discontinuity at x = 0 . Every polynomial function P ( x ) = a n x n + a n - 1 x n - 1 + · · · + a 0 is continuous at every point c R , that is, continuous on R , since lim x c P ( x ) = P ( c ) for any c R . Intermediate Value Theorem for Continuous Functions Geometrically, the Intermediate Value Theorem says that any horizontal line y = k crossing the y -axis between the numbers f ( a ) and f ( b ) will cross the curve y = f ( x ) at least once over the interval [ a, b ] . In particular, the theorem tells us that if f ( x ) is continuous, then any interval on which f changes sign contains a root of the function, that is, a solution to the equation f ( x ) = 0 . Differentiation 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

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and the derivative of f ( x ) at x = a is the limit lim h 0 f ( a + h ) - f ( a ) h and we denote this limit by f 0 ( 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 0 ( 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 0 ( a )( x - a ) .
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ch.4 - Chapter 4 Partial Dierentiation 4.1 Review on...

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