1. Lesson 2.1
The Derivative and the Tangent Line Problem
Quiz

2. What does it mean to say that a line is tangent to a curve at a point?. P
For a circle, the tangent line at a point P is the line that is perpendicular to the radial line at point P.
For a general curve, however, the problem is more difficult.

3. Essentially, the problem of finding the tangent line at a point P boils down to the problem of finding the slope of the tangent line at point P. You can approximate this slope using a secant line through two points on the curve.
(c+x, f(c+ x) .
(c, f(c)) x y

4. 2.1 The Derivative and the Tangent Line Problem
c=2

5. 2.1 The Derivative and the Tangent Line Problem
The slope of a function is its derivative.

6. 2.1 The Derivative and the Tangent Line Problem

7. 2.1 The Derivative and the Tangent Line Problem

8. 2.1 The Derivative and the Tangent Line Problem

9. 2.1 The Derivative and the Tangent Line Problem

10. 2.1 The Derivative and the Tangent Line Problem

11. 2.1 The Derivative and Tangent Line Problem
AP EXAM

13. Differentiability and continuity
The following alternative limit form of the derivative is useful in investigating the relationship between differentiability and continuity. The derivative of f at c is

14. 2.1 The Derivative and the Tangent Line Problem
Graph by Hand

15. 2.1 The Derivative and the Tangent Line Problem
Vertical Tangent Line
If a function is continuous at a point
c and , then x = c
is a vertical tangent line for the function.

16. 2.1 The Derivative and the Tangent Line Problem
HW 2.1/3,4,5-15odd,16,21,24,25,27-32,33,35,37,
41,45,47,62

17. Common Denominator

18. Common Denominator
Conjugate...

19. Common Denominator

21. 62.

22. Quiz

23. Yea! You finished the lesson!
Now get to work!

24. 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 point (c, f(c)).
lim
x→0
f(c + x) – f(c) x
= m .
(c+x, f(c+ x)
(c, f(c)) x y
(c+x, f(c+ x) .
(c, f(c)) x y .
(c, f(c))