**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 )( x-a ) = f ± ( a )( x-a ) + f ( a )-f ( a ) = L ( x )-f ( a ) . While L ( x ) approximates f ( x ), dy approximates how f ( x ) has changed from f ( a ). Fig-ure 6.20 illustrates the relationships. a x ............................................................................................................. ................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ←--dx = Δ x--→ ↑ | Δ y |↓ ↑ | dy | ↓ Figure 6.20 Diﬀerentials....

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- Fall '07
- JonathanRogawski
- Math, Calculus, Numerical Analysis, Approximation, Derivative, Linear Approximation, Mathematical analysis, Logarithm