Basic Cal Lesson 3 Slope of a Tangent Line

Basic Calculus

1. Basic Calculus Science, Technology, Engineering, and Mathematics Lesson 4.1 Slope and Equation of a Tangent Line

2. 2 When a ball is thrown upward, it follows a parabolic path. How do we find the average velocity of the ball for a certain period of time?

3. 3 How about its velocity at a specific time? For example, what is its velocity at exactly 1 second?

4. 4 This is called instantaneous velocity . To solve for instantaneous velocity, we need to understand slopes of tangent lines .

5. 5 How do we determine the slope of a tangent line?

6. 6 Illustrate the tangent line to the graph of a function at a given point (STEM_BC11D-IIIe-1).

7. 7 Find the slope of the tangent line to a curve. Determine the equation of the tangent line.

8. 8 Tangent Line a line that “just touches” the curve and has the same direction as the graph at the point of tangency Secant Line a line that cuts through the curve Slope of the Tangent Line

9. 9 Slope of the Tangent Line Slope of a Line Given two points on the line, and , the slope of the line is given by  

10. 10 Slope of the Tangent Line Example: What is the slope of the secant line to the function which passes through the points and ?  

11. 11 Slope of the Tangent Line Determine the slope of each secant line of which passes through and each point on the table.   Slope of secant line Slope of secant line

12. 12 Slope of the Tangent Line Notice that the given points approach the point .  

13. 13 Slope of the Tangent Line Determine the slope of each secant line of which passes through and each point on the table.   Slope of secant line 2.5 2.25 2.2 2.15 2.1 2.05 2.001 Slope of secant line 2.5 2.25 2.2 2.15 2.1 2.05 2.001

14. 14 Slope of the Tangent Line The points approach What values do the slopes of the secant lines approach?   Slope of secant line 2.5 2.25 2.2 2.15 2.1 2.05 2.001 Slope of secant line 2.5 2.25 2.2 2.15 2.1 2.05 2.001

15. 15 Slope of the Tangent Line As the points approach , what value do the slopes of the secant lines approach?   Slope of secant line 2.5 2.25 2.2 2.15 2.1 2.05 2.001 Slope of secant line 2.5 2.25 2.2 2.15 2.1 2.05 2.001

16. 16 Slope of the Tangent Line The slope of the tangent line of at the point is the limit of the slopes of the secant lines as the points approach .   Slope of tangent line = 2

17. 17 Slope of the Tangent Line        

18. 18 Equation 1: Slope of the Tangent Line The slope of the tangent line to the graph of the function at the point is given by provided that this limit exists.  

19. 19 Determine the slope of the tangent line to the function at the point .  

20. 20 4 Determine the slope of the tangent line to the function at the point .  

21. 21 21 Determine the slope of the tangent line to the function at the point .  

22. 22 Determine the slope of the tangent line to the function at the point .  

23. 23   Determine the slope of the tangent line to the function at the point .  

24. 24 24 Determine the slope of the tangent line to the function at the point .  

25. 25 Determine the slope of the tangent line to the function at the point .  

26. 26 –1 Determine the slope of the tangent line to the function at the point .  

27. 27 27 Determine the slope of the tangent line of the function at the point .  

28. 28 Slope of the Tangent Line Equation 1: Slope of the Tangent Line Let be a random point on the curve of such that is units away from , i.e. .  

29. 29 Slope of the Tangent Line Thus, we have . Consequently, as approaches , approaches .  

30. 30 Equation 2: Slope of the Tangent Line The slope of the tangent line to the graph of the function at the point is given by provided that this limit exists.  

31. 31 Determine the slope of the tangent line to the function at the point using Equation 2.  

32. 32 4 Determine the slope of the tangent line to the function at the point using Equation 2.  

33. 33 33 Determine the slope of the tangent line to the function at the point using Equation 2.  

34. 34 Determine the slope of the tangent line to the function at the point .  

35. 35   Determine the slope of the tangent line to the function at the point .  

36. 36 36 Determine the slope of the tangent line to the function at the point using Equation 2.  

37. 37 Determine the slope of the tangent line of the function at the point . using Equation 2.  

38. 38 –1 Determine the slope of the tangent line of the function at the point . using Equation 2.  

39. 39 39 Determine the slope of the tangent line to the function at the point using Equation 2.  

40. 40 To solve for the equation of a line we can use the point-slope form where is the slope and is a point on the line.   Equation of the Tangent Line

41. 41 After this, we use the slope-intercept form to express the equation of the line.   Equation of the Tangent Line

42. 42 In the equation of a line , what values are being represented by and ?  

43. 43 Equation of the Tangent Line From the previous example the tangent line to at has a slope of .  

44. 44 Equation of the Tangent Line Equation of the tangent line:  

45. 45 Determine the equation of the tangent line to the function at the point given that its slope is equal to 4.  

46. 46   Determine the equation of the tangent line to the function at the point given that its slope is equal to 4.  

47. 47 47 Determine the equation of the tangent line to the function at the point given that its slope is equal to .  

48. 48 Determine the equation of the tangent line to the function at the point .  

49. 49   Determine the equation of the tangent line to the function at the point .  

50. 50 50 Determine the equation of the tangent line to the function at the point .  

51. 51 Determine the equation of the tangent line to the function at the point given that its slope is equal to .  

52. 52   Determine the equation of the tangent line to the function at the point given that its slope is equal to .  

53. 53 53 Determine the equation of the tangent line to the function at the point given the its slope is equal to .  

54. 54 For each item, write two expressions for the slope of the tangent line to the given function at the given point of tangency. No need to simplify the expression. 1. ; 2. ; 3. ; 4. ; 5. ;  

55. 55 For each item, find the slope of the tangent line at the given point. Then, determine the equation of the tangent line in the form .   1. ; 2. ; 3. ; 4. ; 5. ;  

56. 56 Equation 1: Slope of the Tangent Line The slope of the tangent line to the function at the point is given by .  

57. 57 Equation 2: Slope of the Tangent Line An alternative formula for the slope of the tangent line to the function at the point is where .  

58. 58 Steps in Determining the Equation of a Tangent Line Use the formula to find the slope of the tangent line to the curve at the point of tangency . Substitute the slope and the point of tangency to the point-slope form of the equation of a line: .  

59. 59 Concept Formula Description Equation 1: Slope of the tangent line to a curve This formula gives the slope of the tangent line of the graph of the function at the point . Concept Formula Description Equation 1: Slope of the tangent line to a curve  

60. 60 Concept Formula Description Equation 2: Slope of the tangent line to a curve This formula gives the slope of the tangent line of the graph of the function at the point where . Concept Formula Description Equation 2: Slope of the tangent line to a curve  

61. 61 Concept Formula Description Equation of the Tangent Line This formula gives the equation of the tangent line to the function at the point where is the slope of the tangent line and is the point of tangency. Concept Formula Description Equation of the Tangent Line  

62. 62 62 A ball is thrown upward in the air. Its height in feet is given by where is the time in seconds. What is the instantaneous velocity of the ball at time seconds?  

63. 63 Slides 2 to 3 : Bouncing ball strobe edit.jpg by MichaelMaggs is licensed under CC BY-SA 3.0 via Wikipedia . Edwards, C.H., and David E. Penney.  Calculus: Early Transcendentals . 7th ed. Upper Saddle River, New Jersey: Pearson/Prentice Hall, 2008.   Larson, Ron H., and Bruce H. Edwards. Calculus. 9th ed. Belmont, CA: Cengage Learning, 2010.   Leithold , Louis.  The Calculus 7 . New York: HarperCollins College Publ., 1997.   Smith, Robert T., and Roland B. Milton.  Calculus . New York: McGraw Hill, 2012.   Stewart, James. Calculus . Massachusetts: Cengage Learning, 2016.