2224-Sec12_3-HWT

2224-Sec12_3-HWT - Mat h 22 24 Multiva ri a ble C alc Sec ....

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Math 2224 Multivariable Calc – Sec. 12.3: Partial Derivatives I. Review from math 1205 A. First Derivative 1. Def n : The derivative of the function f with respect to the variable x is the function f’ whose value at x is ! f ( x ) = lim h " 0 f x + h ( ) - f x ( ) h provided the limit exists. 2. Notation All of the following are ways of representing the derivative dy dx , ! y , ! f ( x ), df dx , d dx f x ( ) ( ) , D x f , ! y 3. Graphically, the derivative at a point is the slope of the tangent line at that point. 4. Physically, the derivative at a point is the velocity at that point. B. Second Derivative 1. Def n : If f is a differentiable function, the second derivative is the derivative of the first derivative , i.e. d dx dy dx ! " # $ . The second derivative is the rate of change of the first derivative wrt x or you can interpret the second derivative as the rate of change of the rate of change. 2. Notation d 2 y dx 2 , !! y , !! f ( x ), d 2 f dx 2 , d dx ! f x ( ) ( ) , D x 2 f , D x 2 y II. Partial Derivatives A. A partial derivative is a rate of change or slope of a cross-sectional model (vertical slice of the graph) of a multivariable function. That is, it is the slope of the tangent plane to a point on the surface of the given function. B. Definitions of Partial Derivatives
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2224-Sec12_3-HWT - Mat h 22 24 Multiva ri a ble C alc Sec ....

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