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31 - Lecture 2 January 6 2012 1 Calculus with parametric...

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Lecture 2 January 6, 2012 1 Calculus with parametric curves Theorem 1.1. Chain rule Let f : A B and F : B C be given functions so that F ( f ( t )) , t A is defined. Then if f is differentiable at t o A and s o = f ( t o ) and r o = f 0 ( t o ) and F is differentiable at x o B , then d F ( f ( t )) d t = F 0 ( s o ) r o = F 0 ( f ( t o )) f 0 ( t o ) For a proof see your course on single variable calculus. Remark 1.1. The range and domain are important – see the examples for the circle from the previous lecture. Corollary 1.1. If f and F are twice differentiable then d 2 F ( f ( t )) d 2 t o = d F 0 ( f ( t )) f 0 ( t ) d t t = t o = F 0 ( f ( t o )) f ”( t o ) + F ”( f ( t o ))( f 0 ( t o )) 2 Example 1.1. x = t 2 , y = exp( t 2 ) , t [0 , ) d y d x = d y d t d t d x = 2 t exp( t 2 ) × 1 2 t = exp( t 2 ) Theorem 1.2. Integration by substitution Suppose y = f ( t ) t A is contin- uous and x = g ( t ) is differentiable, then R g ( t 2 ) g ( t 1 ) y d x = R t 2 t 1 f ( t ) g 0 ( t )d t. 1
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Example 1.2. Arc length L n X i =1 q δx 2 i + δy 2 i Suppose that distance along the curve is parameterized by a variable t and consider
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