Unformatted text preview: y = f ( x )f ( a ). If x is near a then Δ x is small. If we set dx = Δ x then dy = f ± ( a ) dx ≈ Δ y Δ x Δ x = Δ y. Thus, dy can be used to approximate Δ y , the actual change in the function f between a and x . This is exactly the approximation given by the tangent line: dy = f ± ( a )( xa ) = f ± ( a )( xa ) + f ( a )f ( a ) = L ( x )f ( a ) . While L ( x ) approximates f ( x ), dy approximates how f ( x ) has changed from f ( a ). Figure 6.20 illustrates the relationships. a x ............................................................................................................. ................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ←dx = Δ x→ ↑  Δ y ↓ ↑  dy  ↓ Figure 6.20 Diﬀerentials....
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 Spring '07
 JonathanRogawski
 Math, Calculus, Numerical Analysis, Approximation, Derivative, Linear Approximation, Mathematical analysis, Logarithm

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