1. 2.1 Tangent Line Problem

2. Tangent Line Problem
The tangent line can be found by finding the slope of the secant line through the point of tangency and a point on the curve
Point A is the point of tangency 2

3. Tangent Line Problem How to find slope of a curve at a point?
Secant Line
Tangent Line x
x + Δx

4. Setting up a limit!
Slope of the
Tangent Line x
x + Δx

5. 1.) Find slope of the secant linex
x + Δx

6. Conclusion: x
x + Δx
Called the difference quotient

7. Definition of the Derivative
For a function f(x) the average rate of change along the function is given by:
Which is called the derivative of f

8. Notation of the Derivative
The derivative of a function at x is
given by:
**Provided the limit exists
Notation:

9. 2.) Find the slope of the tangent line to the curve at (2,6)
First, find the Slope at any point

10. Terminology
Differentiation (Differentiate) – the process of finding the derivative
Differentiable – when a functions derivative exists at x

11. When Derivatives Fail
Cusp or sharp corner:
cusp

12. When Derivatives Fail
Vertical asymptotes:

13. When Derivatives Fail
3. Jumps or Breaks

14. When Derivatives Fail
4. Removable discontinuity

15. When Derivatives Fail
5. Vertical tangents

16. 3.) Differentiate (if possible)

17. 4.) Differentiate (if possible)

18. 5.) Differentiate if possible

19. 6.) Find the derivative of

20. HOMEWORK Page 104 # 5 – 21 (odd), 61 and 62,
For (all). Find where f(x) is not differentiable and state the type of discontinuity