Recall: we first started studying derivatives because we were interested in finding tangent lines to curves. The tangent line is an approximation to the curve and may be very useful in applications: when we can"t evaluate the value of a function at a particular point, when we want to simplify an equation. Definition: the approximation at the point approximation or tangent line approximation of function whose graph is this tangent line is called the linearization of ax and is given by. )( xf axaf af xf is called the linear ax . Notes: what we"re doing is making sure that the values of ax match, as do the values of the slopes there. )(xl is a function of x: the approximation is better the closer x is to a, higher order approximations are possible. xl af axaf xq af axaf. Example: find a linearization of xf sin x at. Exercise: find a linear approximation to xf ln x around.
