2.1 The Derivative and the Tangent Line Problem Main Ideas Find the slope of the tangent line to a curve at a point. Use the limit definition to find the derivative of a function. Understand the relationship between differentiability and continuity.

Calculus was developed during the seventeenth century because of four major problems that mathematicians of the time could not solve. 1.Tangent line- to a curve 2.Velocity and acceleration 3.Minimum and Maximum 4.Area-under a curve * Each problem involves the idea of a limit.

Who were the mathematicians? Pierre de Fermat, Rene Descartes, Christian Huygens, Isaac Barrow, Isaac Newton, Gottfried Leibniz Definition of a tangent line A line sharing a common point with a curve or surface and being the closest linear approximation of the curve or surface at that point.

Tangent Line Problem Given a function f and a point P on its graph Find the equation of the tangent line to the graph at point P.

How can you estimate the slope of the tangent line? Secant lines

Definition of Tangent line If f is defined on an open interval containing c, and if the limit exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)).

“You have now arrived at a crucial point in the study of calculus. “ p99 Definition of the Derivative of a function The derivative of f at x is given by provided the limit exists. For all x for which this limit exists, f’ is a function of x.

Other notations used to find the derivative of a function Note that f’ is a function of f. This “new” function f ’ gives the slope of the tangent line to the entire graph of f or just one point depending on which version of the definition you use.

When does a limit fail to exist at a given point?  Jump at a point  Asymptote at a point

 Sharp turn at a point (this includes a cusp)  Vertical tangent line at a point