1. The Tangent Line Problem “And I dare say that this is not only the most useful and general problem in geometry that I know, but even that I ever desire to know.” - René Descartes

2. Differentiation The process in which a secant line approaches the tangent line of a given point on a curve.

3. Difference Quotient It all starts with the slope…

4. Difference Quotient This is a big deal!!! Slope = Rate of Change This is “miles per hour”, “feet per squared second”, “dollars per hour”, etc.

5. Practice Ex 1) Find the slope of the graph of f(x) = 2x – 3 at the point (2, 1).

6. Practice Ex 2) Find the slopes of the tangent lines to the graph of f(x) = x 2 + 1 at the points (0, 1) and (-1, 2)

7. Vertical Tangent Lines The definition of a tangent line does not cover the possibility of a vertical tangent line. So, if f is continuous at c and the vertical line x = c passing through (c, f(c)) is a vertical tangent line to the graph of f.

8. Derivative The derivative is the function for the instantaneous rate of change of another function. In other words, it’s the equation that finds the slope at a given point.

9. Derivative Notation

10. Practice Ex 3) Find the derivative of f(x) = x 3 + 2x

11. Practice Ex 4) Find f’(x) for. Then find the slope of the graph of f at the points (1, 1) and (4, 2). Discuss the behavior of f at (0, 0).

12. Alternate Form of the Derivative This form is useful in investigating the relationship between differentiability and continuity. This forms REQUIRES that the Left and Right Side Limits exist and are equal. The one-sided limits are called derivatives from the left and from the right.

13. Alternate Form of the Derivative Differentiability implies continuity!!! The converse is NOT true. Continuity does not imply differentiability.