13_summaryLinearApprxDifferentials.ppt

13_summaryLinearApprxDifferentials.ppt - Linear...

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1 Linear Approximations and Differentials It might be easy to calculate a value f ( a ) of a function, but difficult (or even impossible) to compute nearby values of f . So we settle for the easily computed values of the linear function L whose graph is the tangent line of f at ( a , f ( a )). (See Figure 1.) Figure 1
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2 Linear Approximations and Differentials In other words, we use the tangent line at ( a , f ( a )) as an approximation to the curve y = f ( x ) when x is near a . An equation of this tangent line is and the approximation is called the linear approximation or tangent line approximation of f at a .
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3 Linear Approximations and Differentials The linear function whose graph is this tangent line, that is, is called the linearization of f at a .
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4 Example 1 Find the linearization of the function at a = 1 and use it to approximate the numbers and . Are these approximations overestimates or underestimates? Solution: The derivative of is and so we have
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5 Example 1 – Solution Putting these values into Equation 2, we see that the linearization is The corresponding linear approximation is cont’d
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6 Example 1 – Solution In particular, we have The linear approximation is illustrated in Figure 2.
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