1. 2-1: The Derivative Objectives: Explore the tangent line problem
Define the derivative
Discuss the relationship between differentiability and continuity
©2002 Roy L. Gover (

2. Example Average speed is 50 mph measured over 10 hours
500
distance in miles 10
time in hours

3. Example Average speed is 35.7 mph measured over 7 hours
distance in miles
250 7
time in hours

4. Important Idea
The average speed over a time period is the slope of the line connecting the beginning and end of the time period.

5. Example Average speed is 24 mph measured over 5 hours
distance in miles
120 5
time in hours

6. Try This
Describe in words how you could find the speed at exactly the 5th hour.
distance in miles 5
time in hours

7. Solution
The instantaneous velocity at exactly the 5th hour is the slope of the line tangent to the velocity function at t=5.

8. Example What is the slope of the tangent line at t=5? 5
distance in miles 5
time in hours

9. Solution
42.9 mph
distance in miles
300 mi.
7 hrs. 5

10. Try This What is the instantaneous velocity at 8 hours? 70 mph
Distance in miles
70 mph

11. Important Idea
The instantaneous velocity at a point, or any other rate of change, is the slope of the tangent line at the point

12. Important Idea
The tangent line problem is: in general, how do you find the slope of the tangent line at a point?

13. Analysis

14. Analysis
As , the slope of the secant linethe slope of the tangent line at x

15. Analysis
As , the slope of the secant linethe slope of the tangent line at x

16. Analysis
As , the slope of the secant linethe slope of the tangent line at x

17. Analysis
As , the slope of the secant linethe slope of the tangent line at x
is very, very close to 0

18. Analysis
Slope of secant line:
AP Exam: instead of

19. Analysis
The slope of the secant line becomes the slope of the tangent line:

20. Example
Find the slope of the line tangent to the following graph at the point (2,1):
Why did we not use the point (2,1) in the solution?

22. Example
Find the slope of the line tangent to the following graph at the point (2,5):

24. Try This
Find the slope of the line tangent to the following graph at the point (3,17):
m=12

25. Try This
Find the slope of the line tangent to the following graph at the point (1,1):
Hint: rationalize the numerator

26. Try This
Estimate the slope of the line tangent to the following graph at the point (3,1):
(3,1)

27. Try This
Estimate the slope of the line tangent to the following graph at the indicated point:

28. Try This
Estimate the slope of the line tangent to the following graph at the right end point

29. Try This
Estimate the slope of the line tangent to the following graph at the left end point

30. Try This
Estimate the slope of the line tangent to the following graph at the point (0,0)

31. Definition
The slope of the tangent line, if it exists, is:
sometimes:

32. Definition
The slope of the tangent line at a point is called the derivative of f(x) and the derivative is:
if the limit exists.

33. Important Idea
Differentiation at a point implies continuity at the point, however,…
continuity at a point does not imply differentiability at a point

34. f(x) is differentiable at (3,1) implies f(x) is continuous at (3,1)
Example
f(x)
f(x) is differentiable at (3,1) implies f(x) is continuous at (3,1)
(3,1)

35. Example
f(x) continuous at (3,1) does not imply f(x) is differentiable at (3,1)
f(x)
(3,1)

36. Definition An alternative form of the derivative at a point c is:
providing the limit exists

37. Example
Use the alternative form to find f’(x) at x=2:

38. Example
Use your knowledge of derivatives and algebra to find the equation of the line tangent to
at (2,8).

39. Warm-Up
Use your knowledge of derivatives and algebra to find the equation of the line tangent to
at (-3,4).

41. Warm-Up
at (-3,4).

42. Lesson Close
A derivative is…

43. Assignment
Page 103
Problems 3, 4, odd,
71, 73