1. Assignment 4 Section 3.1 The Derivative and Tangent Line Problem

2. The Basic Question is… • How do you find the equation of a line that is tangent to a function y=f(x) at an arbitrary point P? • To find the equation of a line you need: a point and a slope

3. How do you find the slope when the line is a tangent line?

4. First, we approximate with the secant line.

5. How do we make the approximation better? • Choose h smaller… • And smaller… • And smaller… • And smaller… • How close to zero can it get? • Infinitely

6. Definition of slope of the tangent line If f(x) is defined on an open interval (a,b) then the slope of the tangent line to the graph of y=f(x) at an arbitrary point (x,f(x)) is given by:

7. Example: • #6—Find the slope of the tangent line to the graph of the function at the given point. • (-2, -2)

8. The limit that is the slope of the tangent line is actually much more.. • 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 the limit exists, is a function of x.

9. Notations for derivative

10. Find the derivative by the limit process. #20 #24

11. Find an equation of the tangent line to th graph of f at the given point. • #26 • ( - 3, 4)

12. #34 Find an equation of the line that is tangent to the graph of f and parallel to the given line.

13. Sketch the graph of f’ #46

14. What destroys the derivative at a point? • Cusps • Corners • Vertical tangents

15. And…Points of Discontinuity Fact: If a function is differentiable at x=c, then f is continuous at x=c