224CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION3.2Linear and Affine FunctionsSo far we have seen the derivative in two settings. For a real-valuedfunctionf(x) of one variable, the derivativef′(x0) at a pointx0firstcomes up as a number, which turns out to be the slope of the tangent line.This in turn is the line which best approximates the graphy=f(x) nearthe point, in the sense that it is the graph of the polynomial of degree one,Tx0f=f(x0) +f′(x0) (x−x0), which hasfirst-order contactwith thecurve at the point (x0,f(x0)):vextendsinglevextendsinglef(x)−Tx0f(x)vextendsinglevextendsingle=o(|x−x0|)orvextendsinglevextendsinglef(x)−Tx0f(x)vextendsinglevextendsingle|x−x0|→0 asx→x0.If we look back at the construction of the derivativevectorp′(t) of a vector-valuedfunction−→p:R→R3in§2.3, we see a similar phenomenon:vectorp′(t0) =−→v(t0)is the direction vector for a parametrization of the tangent line, and theresulting vector-valued function,Tt0
This is the end of the preview.
access the rest of the document.