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We are now in a position to discuss the two central problems of calculus as mentioned in
Section . In this chapter we will take up the problem of finding tangent lines; in Chapter
4 we will consider the problem of finding areas. We choose this order only because the
work we do in solving the tangent line problem in this chapter will be of use, through the
Fundamental Theorem of Calculus, in solving area problems in the next.
We begin with some preliminary notation and terminology. If f is a function with
domain contained in the set A and range contained√ in the set B, then we may denote this
fact by writing f : A → B. For example, if g(t) = 1 − t2 and R denotes the set of real
numbers, then the statements g : R → R, g : [−1, 1] → R, and g : [−1, 1] →[0, 1] are all
correct. We will work exclusively with functions of the form f : R → R until Chapter 7,
where we will introduce functions of the form f : R → C and f : C → C, where C denotes
the set plex numbers.
We call a function f : R → R linear if there is a constant m such that f(x) = mx for
all values of x. Graphically, linear functions are functions whose graphs are straight lines
passing through the origin. We call a function f : R → R affine if there are constants
m and b such that f(