Slope and Derivative

# Slope and Derivative - 18.01 Calculus Jason Starr Fall 2005...

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18.01 Calculus Jason Starr Fall 2005 Lecture 2. September 9, 2005 Homework. Problem Set 1 Part I: (f)–(h); Part II: Problems 3. Practice Problems. Course Reader: 1C-2, 1C-3, 1C-4, 1D-3, 1D-5. 1. Tangent lines to graphs. For y = f ( x ), the equation of the secant line through ( x 0 , f ( x 0 )) and ( x 0 + Δ x, f ( x 0 + Δ x )) is, y = f ( x 0 + Δ x ) f ( x 0 ) ( x x 0 ) + f ( x 0 ) . Δ x In the limit, the equation of the tangent line through ( x 0 , f ( x 0 )) is, y = f ( x 0 )( x x 0 ) + y 0 . Example. For the parabola y = x 2 , the derivative is, y ( x 0 ) = 2 x 0 . The equation of the tangent line is, y = 2 x 0 ( x x 0 ) = 2 x 0 x x 2 0 . For instance, the equation of the tangent line through (2 , 4) is, y = 4 x 4. 2 Given a point ( x, y ), what are all points ( x 0 , x 0 ) on the parabola whose tangent line contains ( x, y )? To solve, consider x and y as constants and solve for x 0 . For instance, if ( x, y ) = (1 , 3), this gives, 2 ( 3) = 2 x 0 (1) x 0 , or, 2 x 0 2 x 0 3 = 0 .

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## This note was uploaded on 06/03/2008 for the course MATH B6A taught by Professor Moretti during the Spring '08 term at Bakersfield College.

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Slope and Derivative - 18.01 Calculus Jason Starr Fall 2005...

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