lecture7-09[1] - Math 18.02(Spring 2009 Lecture 7 Partial...

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Math 18.02 (Spring 2009): Lecture 7 Partial derivatives. Tangent plane approximation. February 20 Today: Partial derivatives. Tangent plane approximation. Last time: Velocity & acceleration. Derivatives of · and × product. Kepler’s second law. Reading Material: From Simmos 19.2, 19.3 and 19.4. From the Lecture Notes TA. 2 Partial Derivatives We first recall the concept of derivative for a function: Given a scalar valued function y = f ( x ) we denote the derivative of f at x 0 as f ( x 0 ) = d dx f ( x 0 ) = dy dx x 0 = rate of change of y when one changes x at x 0 = slope of tangent line to curve at the point ( x 0 , f ( x 0 )) 1
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On the xy-plane the equation of the tangent line to the graph of y = f ( x ) (curve) at the point P 0 = ( x 0 , y 0 ) where y 0 = f ( x 0 ) is ( y - y 0 ) ( deviat. of y from y 0 ) = dy dx x 0 ( x - x 0 ) ( deviat. of x from x 0 ) Partial derivatives: Given a two variable function z = f ( x, y ) we define the partial derivative of f with respect to x ∂z ∂x = dz dx but hold y (the other variable) constant. = rate of change of z when one changes only x the partial derivative of f with respect to y ∂z ∂y = dz dy but hold x (the other variable) constant.
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  • Fall '08
  • Auroux
  • Derivative, tangent plane approximation

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