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Mean Value Theorem

# Mean Value Theorem - 18.01 Calculus Jason Starr Fall 2005...

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18.01 Calculus Jason Starr Fall 2005 Lecture 8. September 27, 2005 Homework. Problem Set 2 all of Part I and Part II. Practice Problems. Course Reader: 2A-1, 2A-4, 2A-9, 2A-11, 2A-12. 1. Linear approximations. For a differentiable function f ( x ), the linear approximation or linearization of f ( x ) at x = a is the linear function, f ( a ) + f ( a )( x a ) . In a precise sense, this is the best approximation of f ( x ) by a linear function near x = a . For x close to a , the value of f ( x ) is close to the value of the linearization. The notation for this is, f ( x ) f ( a ) + f ( a )( x a ) for x a. Example. The linearization of, f ( x ) = e 3 x sin(2 πx ) + 5 e 3 x cos(2 πx ) , near x = 0 is, f ( x ) 5 (15 2 π ) x for x 0 .

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18.01 Calculus Jason Starr Fall 2005 In particular, for x = 0 . 02, this gives the approximate answer, f (0 . 02) 5 (15 2 π )(0 . 02) 4 . 8. The actual value is approximately 4 . 71. 2. Basic approximations. Some linear approximations occur so often, they should be committed to memory. Each of the following is the linear approximation for x 0, together with the terms in the quadratic and higher approximations. 1 x 1 + + x 2 + x 3 + . . . , 1 x 1 + rx r r (1 + x ) r + x 2 + x 3 + . . . , 2 3 sin( x ) x x 3 / 3! + x 5 / 5! + . . . , cos( x ) 1 x 2 / 2! + x 4 / 4! + . . . , x x 1 x 1 + + x 2 / 2! + x 3 / 3! + . . . , e ln(1 + x ) + x 2 / 2 x 3 / 3 + . . . 3.
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