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

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Unformatted text preview: 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 \ { } . 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 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 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 ....
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This note was uploaded on 12/20/2011 for the course MATH 1813 taught by Professor Wu during the Fall '11 term at HKU.

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

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