The Tangent Line

1
The Tangent Line

• If F(x) is defined on an open interval containing
  c, and if Lim?x-gt0(F(c?x)-F(c))/ ?x m,
  then the line through (c, F(c)) having slope m is
  called the tangent line to the graph of F(x) at
  the point (c, F(c).

2
The Derivative

• The derivative of F at x is given by
  Lim?x-gt0(F(x?x)-F(x))/ ?x, if the limit exists.
• The Derivative of F(x) is denoted by F(x).
• The symbol dy/dx is also used, where y F(x).
• We sometimes use DxF(x)

3
A definition

• If the limit of the previous slide exists, we say
  that F(x) is differentiable at x.

4
Differentiability and Continuity

• If F(x) is differentiable at c, then it is also
  continuous at c.
• F(x) may be continuous at c and not be
  differentiable at c.