1. The Derivative and the Tangent Line Problem
Section 2.1

2. After this lesson, you should be able to:
find the slope of the tangent line to a curve at a point
use the limit definition of a derivative to find the derivative of a function
understand the relationship between differentiability and continuity

3. Tangent Line
A line is tangent to a curve at a point P if the line is perpendicular to the radial line at point P.
Note: Although tangent lines do not intersect a circle, they may cross through point P on a curve, depending on the curve. P

4. The Tangent Line Problem
Find a tangent line to the graph of f at P.
Why would we want a tangent line??? f
Remember, the closer you zoom in on point P, the more the graph of the function and the tangent line at P resemble each other. Since finding the slope of a line is easier than a curve, we like to use the slope of the tangent line to describe the slope of a curve at a point since they are the same at a particular point.
A tangent line at P shares the same point and slope as point P. To write an equation of any line, you just need a point and a slope. Since you already have the point P, you only need to find the slope. P

5. Definition of a Tangent
Let Δx shrink from the left

6. Definition of a Tangent Line with Slope m

7. The Derivative of a Function
Differentiation- the limit process is used to define the slope of a tangent line.
Really a fancy slope formula… change in y divided by the change in x.
Definition of Derivative: (provided the limit exists,)
This is a major part of calculus and we will differentiate until the cows come home!
Also,
= slope of the line tangent to the graph of f at (x, f(x)).
= instantaneous rate of change of f(x) with respect to x.

8. Definition of the Derivative of a Function

9. Notations For Derivative
Let
If the limit exists at x, then we say that f is differentiable at x.

10. Note:
dx does not mean d times x !
dy does not mean d times y !

11. Note: does not mean ! does not mean !
(except when it is convenient to think of it as division.)
does not mean !
(except when it is convenient to think of it as division.)

12. Note: does not mean times !
(except when it is convenient to treat it that way.)

13. The derivative is the slope of the original function.
The derivative is defined at the end points of a function on a closed interval.

15. A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. p

16. Theorem 2.1 Differentiability Implies Continiuty

17. The Slope of the Graph of a Line
Example: Find the slope of the graph of
at the point (2, 5).

18. The Slope of the Graph of a Line
Example: Find the slope of the graph of
at the point (2, 5).

19. The Slope of the Graph of a Line
Example: Find the slope of the graph of
at the point (2, 5).

20. The Slope of the Graph of a Line
Example: Find the slope of the graph of
at the point (2, 5).

21. The Slope of the Graph of a Line
Example: Find the slope of the graph of
at the point (2, 5).

22. The Slope of the Graph of a Line
Example: Find the slope of the graph of
at the point (2, 5).

23. The Slope of the Graph of a Non-Linear Function
Example: Given , find f ’(x) and the equation of the tangent lines at:
a) x = 1
b) x = -2
a) x = 1:

24. The Slope of the Graph of a Non-Linear Function
Example: Given , find f ’(x) and the equation of the tangent lines at:
a) x = 1
b) x = -2
a) x = 1:

25. The Slope of the Graph of a Non-Linear Function
Example: Given , find f ’(x) and the equation of the tangent lines at:
a) x = 1
b) x = -2
a) x = 1:

26. The Slope of the Graph of a Non-Linear Function
Example: Given , find f ’(x) and the equation of the tangent lines at:
a) x = 1
b) x = -2
a) x = 1:

27. The Slope of the Graph of a Non-Linear Function
Example: Given , find f ’(x) and the equation of the tangent lines at:
a) x = 1
b) x = -2
a) x = 1:

28. The Slope of the Graph of a Non-Linear Function
Example: Given , find f ’(x) and the equation of the tangent lines at:
a) x = 1
a) x = 1:
b) x = -2

29. The Slope of the Graph of a Non-Linear Function
Example: Given , find f ’(x) and the equation of the tangent line at:
b) x = -2

30. The Slope of the Graph of a Non-Linear Function
Example: Find f ’(x) and the equation of the tangent line at x = 2 if

31. The Slope of the Graph of a Non-Linear Function
Example: Find f ’(x) and the equation of the tangent line at x = 2 if

32. The Slope of the Graph of a Non-Linear Function
Example: Find f ’(x) and the equation of the tangent line at x = 2 if

33. The Slope of the Graph of a Non-Linear Function
Example: Find f ’(x) and the equation of the tangent line at x = 2 if

34. Example-Continued
If x = 2, the slope is, -¼. So, y = 1/4x + b. Going back to the original equation of y = 1/x, we see if x = 2, y = 1/2. So:

35. Derivative
Example: Find the derivative of f(x) = 2x3 – 3x.

36. Derivative
Example: Find the derivative of f(x) = 2x3 – 3x.

37. Derivative
Example: Find the derivative of f(x) = 2x3 – 3x.

38. Derivative
Example: Find the derivative of f(x) = 2x3 – 3x.

39. Derivative
Example: Find the derivative of f(x) = 2x3 – 3x.

40. Derivative
Example: Find for

41. Derivative
Example: Find for

42. Derivative
Example: Find for

43. Derivative
Example: Find for
THIS IS A HUGE RULE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

44. Example-Continued Let’s work a little more with this example…
Find the slope of the graph of f at the points (1, 1) and (4, 2). What happens at (0, 0)?

45. Example-Continued
Let’s graph tangent lines with our calculator…we’ll draw the tangent line at x = 1.
Graph the function on your calculator. 1 3
Select 5: Tangent(
Type the x value, which in this case is 1, and then hit  4
(I changed my window) 2
Now, hit  DRAW
Here’s the equation of the tangent line…notice the slope…it’s approximately what we found

46. Differentiability Implies Continuity
If f is differentiable at x, then f is continuous at x.
Some things which destroy differentiability:
A discontinuity (a hole or break or asymptote)
A sharp corner (ex. f(x)= |x| when x = 0)
A vertical tangent line (ex: when x = 0)

47. 2.1 Differentiation Using Limits of Difference Quotients
Where a Function is Not Differentiable:
1) A function f(x) is not differentiable at a point x = a, if there is a “corner” at a.

48. 2.1 Differentiation Using Limits of Difference Quotients
Where a Function is Not Differentiable:
2) A function f (x) is not differentiable at a point
x = a, if there is a vertical tangent at a.

49. 3. Find the slope of the tangent line to at x = 2.
This function has a sharp turn at x = 2.
Therefore the slope of the tangent line at x = 2 does not exist.
Functions are not differentiable at
Discontinuities
Sharp turns
Vertical tangents

50. 2.1 Differentiation Using Limits of Difference Quotients
Where a Function is Not Differentiable:
3) A function f(x) is not differentiable at a point x = a, if it is not continuous at a.
Example: g(x) is not
continuous at –2,
so g(x) is not
differentiable at x = –2.

51. 4. Find any values where is not differentiable.
This function has a V.A. at x = 3.
Therefore the derivative at x = 3 does not exist.
Theorem:
If f is differentiable at x = c,
then it must also be continuous at x = c.

52. Example
Find an equation of the line that is tangent to the graph of f and parallel to the given line.
f(x) = x Line: 3x – y – 4 = 0

53. Example
Find an equation of the line that is tangent to the graph of f and parallel to the given line.
f(x) = x Line: 3x – y – 4 = 0
Taking my word for it, the derivative of the function is
This is 3 when x is

54. Definition of Derivative
The derivative is the formula which gives the slope of the tangent line at any point x for f(x)
Note: the limit must exist
no hole
no jump
no pole
no sharp corner
A derivative is a limit !