2.1 The Derivative and the Tangent Line Problem.

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Presentation transcript:

2.1 The Derivative and the Tangent Line Problem

Find the slope of the graph of f(x) at the point (2,1).

Find the slopes of the tangent lines to the graph of f(x) at the point (0,1) and (-1,2).

Notation for derivatives:

Find the derivative of f(x).

Find f’(x) for f(X). Then find the slope of the graph of f at the points (1,1) and (4,2).

Find the derivative with respect to t for the function y.

Alternate form of derivative:

A function with a sharp turn f(x) is continuous, but… The limits from both sides do not equal. So, f is not differentiable at x=2 and the graph of f does not have a tangent line at the point (2,0).

Graph with a vertical tangent line

The following statements summarize the relationship between continuity and differentiability. 1.If a function is differentiable at x=c, then it is continuous at x=c. So, differentiability implies continuity. 2.It is possible for a function to be continuous at x=c and not be differentiable at x=c. So, continuity does not imply differentiability.