# MATB42H3 Lecture Notes - Lecture 6: Piecewise, Function Composition

## Document Summary

D(cid:0) (t)(cid:1) = provided each i (t) exists, 1 i n. we. 3(t) n(t) will write d(cid:0) (t)(cid:1) as (t) = (cid:0) . If (t) exists, is called a di erentiable path. If is a di erentiable path with continuous derivative, except at. Nitely many places, the image of is called a piecewise smooth curve. (t) is the tangent vector to the curve at the point (t). Hence the tangent line at (t0) is given by. (t0) + (t0), r . Remark: if we think of (t) as representing the position of a parti- cle at time t we can regard (t) as its velocity and (t) as its acceleration. If (t) is velocity, then k (t) k (magnitude of velocity) is speed. And are sometimes used in place of and . De nition: the path integral of f , or the integral of f along the path , denoted z f ds, is de ned by.