121616949-math.146

# 121616949-math.146 - 132 Chapter 6 Applications of the...

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132 Chapter 6 Applications of the Derivative With modern calculators and computing software it may not appear necessary to use linear approximations. But in fact they are quite useful. In cases requiring an explicit numerical approximation, they allow us to get a quick rough estimate which can be used as a “reality check” on a more complex calculation. In some complex calculations involving functions, the linear approximation makes an otherwise intractable calculation possible, without serious loss of accuracy. EXAMPLE 6.23 Consider the trigonometric function sin x . Its linear approximation at x = 0 is simply L ( x ) = x . When x is small this is quite a good approximation and is used frequently by engineers and scientists to simplify some calculations. DEFINITION 6.24 Let y = f ( x ) be a diﬀerentiable function. We deﬁne a new in- dependent variable dx , and a new dependent variable dy = f ± ( x ) dx . Notice that dy is a function both of x (since f ± ( x ) is a function of x ) and of dx . We say that dx and dy are diﬀerentials. Let Δ x = x - a and Δ
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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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