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Unformatted text preview: 9/13/2010 1 Chain Rule, Plane Curves, and Parametric Equations Calculus I Section 3.5 2 The Chain Rule (Section 3.5) If y=f (u) is a differentiable function of u and u=g(x) is a differentiable function of x , then y=f (g(x)) is a differentiable function of x and or equivalently, In words: The derivative of the outside evaluated at the inside times the derivative of the inside. dx du du dy dx dy ⋅ = [ ] ) ( )) ( ( )) ( ( x g x g f x g f dx d ′ ⋅ ′ = Definition of a Plane Curve If f and g are continuous functions of t on an interval I, then the equations x = f(t) and y=g(t) are called parametric equations and t is called the parameter . The set of points (x,y) obtained as t varies over the interval I is called the graph of the parametric equations. Taken together, the parametric equations and the graph are called a plane curve , denoted by C . Example Compare n x = t 2 + 1, y=2t 2 – 1 n x = 1 – t 2 , y = -1 – 2t 2 Eliminating the Parameter Another way to find out information about...
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