precw_5.pdf - THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MATH1010 University Mathematics(Fall 2019 Pre-Coursework Exercise 5 If you

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THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MATH1010 University Mathematics (Fall 2019) Pre-Coursework Exercise 5 If you find any typos/mistakes, please email us at [email protected]

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3 Here are a few useful observations and notes on the topic of limits: Let Δ f = f ( a + Δ x ) - f ( a ). The Linear Approximation is the estimate Δ f f 0 ( a x (for Δ x small) . Differential notation: dx is the change in x , df = f 0 ( a ) dx , Δ y = f ( a + dx ) - f ( a ) . In this notation, the Linear Approximation reads Δ y dy, (for Δ x small) . The linearization of f ( x ) at x = a is the function L ( x ) = f 0 ( a )( x - a ) + f ( a ) . The Linear Approximation is equivalent to the approximation f ( x ) L ( x ) , (for x close to a ) . The error in the Linear Approximation is the quantity Error = | Δ f - f 0 ( a x | . In many cases, the percentage error is more important than the error itself: Percentage error = error actual value × 100% . Extreme Values on an Interval Definition 1. Let f ( x ) be a function on an interval I and let a I. We say that f ( a ) is the Absolute minimum of f ( x ) on I if f ( a ) f ( x ) for all x I .

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