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8.2.12. Problem. Let Ta:xx+a. The mapping Ta is called translation by a. Note that it is not linear (unless, of course, a=0). We say that functions f and g in Fa are tangent at a if the functions foTa and goTa are tangent at zero.

(a) Let f(x)=3x2+10x+13 and g(x)=-20x-15. Show that f and g are tangent at -2.

(b) Develop a theory for the relationship ('tangency at a“ which generalizes our work on ""tangency at 0“ .

8.2.13. Problem. Each of the following is an abbreviated version of a proposition. Formulate precisely and prove.

(a) C0+OC0.

(b) C0+0C0.

(c) O+0O.

8.2.14. Problem. Suppose that fg. Then the following hold.

(a) If g is continuous at 0, so is f.

(b) If g belongs to O, so does f.

(c) If g belongs to 0, so does f.


One often hears that differentiation of a smooth function f at a point a in its domain is the process of finding the best ""linear approximation“ to f at a. This informal assertion is not quite correct. For example, as we know from beginning calculus, the tangent line at x=1 to the curve y=4+x2 is the line y=2x+3, which is not a linear function since it does not pass through the origin. To rectify this rather minor shortcoming we first translate the graph of the function f so that the point (a,f(a)) goes to the origin, and then find the best linear approximation at the origin. The operation of translation is carried out by a somewhat notorious acquaintance from beginning calculus y. The source of its notoriety is two‐fold: first, in many texts it is inadequately defined; and second, the notation y fails to alert the reader to the fact that under consideration is a function of two variables. We will be careful on both counts.

8.3.1. Definition. Let fFa. Define the function fa by


for all h such that a+h is in the domain of f. Notice that since f is defined in a neighborhood of a, the function fa is defined in a neighborhood of 0; that is, fa belongs to F0. Notice also that fa(0)=0.

8.3.2. Problem. Let f(x)=cosx for 0x2π.

(a) Sketch the graph of the function f.

(b) Sketch the graph of the function fπ.

8.3.3. Proposition. If fFa and αR, then


Proof. To show that two functions are equal show that they agree at each point in their domain. Here





for every h in the domain of fa.