Linearization_lec22 (1).ppt - Linear Approximations...

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Linear Approximations
2 Let’s start with the function x y at x = 4 because we know what is 4 Mysterious radiation has caused all calculators to stop working. It just so happens that at this time, you need to find out the square root of 5. The best you are going to be able to do now is approximate it. But how? First of all, what is the tangent line at x = 4? 2 ) 4 ( 4 1 x y Now graph both the function and its tangent line… When we make a graph of this process, you will get a better sense of why we are doing this.
3 2 ) 4 ( 4 1 x y Notice that for numbers close to 4, the tangent line is very close to the curve itself… So lets try plugging 5 into the tangent line equation and see what we get… Use the tangent line to the function x y at x = 4 to approximate 5
4 2 ) 4 ( 4 1 x y So let’s plug 5 into the tangent line to get… As it turns out… (4, 2) (5, 2.25) ) 5 , 5 ( 25 . 2 2 ) 4 5 ( 4 1 y 2.23607 5 2.23607) (5, Notice how close the point on the line is to the curve… So in this case, our approximation will be a very good one as long as we use a number close to 4. Use the tangent line to the function x y at x = 4 to approximate 5
5 For any function f ( x ), the tangent is a close approximation of the function for some small distance from the tangent point. y x 0 x a f x f a We call the equation of the tangent the linearization of the function.
6 The linearization is the equation of the tangent line, and you can use the old formulas if you like. Start with the point/slope equation: 1 1 y y m x x 1 x a 1 y f a m f a   y f a f a x a   y f a f a x a   L x f a f a x a linearization of f at a f x L x is the standard linear approximation of f at a.
7 Remember: The linearization is just the equation of the tangent line. The use of the term L ( x ) is to make it known that you are using the tangent line to make a linear approximation of the function in question.
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