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math16B_suppl_notes_1

math16B_suppl_notes_1 - Math 16B S06 Supplementary Notes 1...

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Math 16B – S06 – Supplementary Notes 1 The Derivative and Local Linear Approximation For a function g ( x ) of one variable, the value g ( a ) of the derivative of g at the point x = a is the slope of the tangent line to the graph of g at the point ( a, g ( a )). Near that point, the tangent line is a “good” approximation to the graph, in the following sense. The equation of the tangent line is y = g ( a ) + g ( a )( x - a ) . The difference (1) R ( x ) = g ( x ) - g ( a ) - g ( a )( x - a ) is the error you get at x when you approximate g by the function whose graph is the tangent line (see Figure 1 . 1). The approximation is good near a in the sense that R ( x ) is small for x near a compared to x - a , vanishingly small in the limit: (2) lim x →∞ R ( x ) x - a = 0 . To see this, simply divide (1), the equality that defines R ( x ), by x - a , to get R ( x ) x - a = g ( x ) - g ( x ) x - a - g ( a ) . As x approaches a , the difference quotient on the right side of the equality approaches g ( a ), so the ratio on the left side tends to 0. Among all straight lines through the point ( a, g ( a )), the tangent line gives the best linear approx- imation near a to the graph of g . For the line with slope c = g ( a ), the error in the approximation at x , divided by x - a , has the limit g ( a ) - c , not 0, as x approaches a . The relation (2), while it says you can expect the error in a certain approximation to be small, does not tell you how small. It does not tell you, for example, how close x should be to a to guarantee that the approximation is accurate to, say, three decimal places.
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