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4.5: Linear Approximations, Differentials and Newton’s Method

4.5: Linear Approximations, Differentials and Newton’s Method. Greg Kelly, Hanford High School, Richland, Washington. We call the equation of the tangent the linearization of the function.

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4.5: Linear Approximations, Differentials and Newton’s Method

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  1. 4.5: Linear Approximations, Differentials and Newton’s Method Greg Kelly, Hanford High School, Richland, Washington

  2. We call the equation of the tangent the linearization of the function. For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point.

  3. is the standard linear approximation of f at a. Start with the point/slope equation: linearization of f at a The linearization is the equation of the tangent line, and you can use the old formulas if you like.

  4. Differentials: When we first started to talk about derivatives, we said that becomes when the change in x and change in y become very small. dy can be considered a very small change in y. dx can be considered a very small change in x.

  5. Let be a differentiable function. The differential is an independent variable. The differential is:

  6. Example: Consider a circle of radius 10. If the radius increases by 0.1, approximately how much will the area change? very small change in r very small change in A (approximate change in area)

  7. (approximate change in area) Compare to actual change: New area: Old area:

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