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Why Do We Need Derivatives? Bernd Schr oder logo1 Bernd Schr - PowerPoint PPT Presentation

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Introduction Velocities Tangent Lines Rates of Change Why Do We Need Derivatives? Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives? Introduction

  1. Introduction Velocities Tangent Lines Rates of Change Driving From Kansas To Virginia 1. Distance: 1 , 300 mi , driving time: 3 days 2. v avg = distance traveled = 1 , 300 mi ≈ 433 . 33 mi day, time needed 3 days useful to plan where to book hotels along the way, etc. 3. Average velocity is always v avg = s t 4. Suppose after driving 200 miles in the first four hours (you’re in Missouri), a highway patrolman stops you and reveals that you’ve been going 65 mph in an area in which the speed limit is 55 mph . v = 200 mi 4 hrs = 50 mph ??? 5. Even sensible average velocity can look strange: 433 . 33 mi day = 433 . 33 mi 24 hr ≈ 18 . 06mi hr 6. We need a way to define instantaneous velocity. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  2. Introduction Velocities Tangent Lines Rates of Change Tangent Lines logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  3. Introduction Velocities Tangent Lines Rates of Change Tangent Lines logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  4. Introduction Velocities Tangent Lines Rates of Change Tangent Lines logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  5. Introduction Velocities Tangent Lines Rates of Change Tangent Lines logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  6. Introduction Velocities Tangent Lines Rates of Change Tangent Lines logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  7. Introduction Velocities Tangent Lines Rates of Change Tangent Lines logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  8. Introduction Velocities Tangent Lines Rates of Change Tangent Lines logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  9. Introduction Velocities Tangent Lines Rates of Change Tangent Lines logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  10. Introduction Velocities Tangent Lines Rates of Change Tangent Lines logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  11. Introduction Velocities Tangent Lines Rates of Change Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  12. Introduction Velocities Tangent Lines Rates of Change Definition. Let s be a position function. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  13. Introduction Velocities Tangent Lines Rates of Change Definition. Let s be a position function. We define the average velocity in the interval [ a , b ] by v avg : = s ( b ) − s ( a ) . b − a logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  14. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  15. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] s ( b ) − s ( 2 ) b b − 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  16. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] s ( b ) − s ( 2 ) b b − 2 2.1 -20.09 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  17. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] s ( b ) − s ( 2 ) b b − 2 2.1 -20.09 2.01 -19.649 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  18. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] s ( b ) − s ( 2 ) b b − 2 2.1 -20.09 2.01 -19.649 2.001 -19.6049 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  19. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] s ( b ) − s ( 2 ) b b − 2 2.1 -20.09 2.01 -19.649 2.001 -19.6049 2.0001 -19.6005 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  20. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] s ( b ) − s ( 2 ) b b − 2 2.1 -20.09 2.01 -19.649 2.001 -19.6049 2.0001 -19.6005 2 undefined logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  21. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] s ( b ) − s ( 2 ) b b − 2 2.1 -20.09 2.01 -19.649 2.001 -19.6049 2.0001 -19.6005 2 undefined logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  22. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] s ( b ) − s ( 2 ) b b − 2 2.1 -20.09 2.01 -19.649 2.001 -19.6049 2.0001 -19.6005 2 undefined 1.9 -19.11 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  23. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] s ( b ) − s ( 2 ) b b − 2 2.1 -20.09 2.01 -19.649 2.001 -19.6049 2.0001 -19.6005 2 undefined 1.99 -19.551 1.9 -19.11 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  24. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] s ( b ) − s ( 2 ) b b − 2 2.1 -20.09 2.01 -19.649 2.001 -19.6049 2.0001 -19.6005 2 undefined 1.999 -19.5951 1.99 -19.551 1.9 -19.11 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  25. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] s ( b ) − s ( 2 ) b b − 2 2.1 -20.09 2.01 -19.649 2.001 -19.6049 2.0001 -19.6005 2 undefined 1.9999 -19.5995 1.999 -19.5951 1.99 -19.551 1.9 -19.11 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  26. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] s ( b ) − s ( 2 ) b b − 2 2.1 -20.09 2.01 -19.649 2.001 -19.6049 2.0001 -19.6005 2 undefined ( − 19 . 6 “feels right”) 1.9999 -19.5995 1.999 -19.5951 1.99 -19.551 1.9 -19.11 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  27. Introduction Velocities Tangent Lines Rates of Change Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  28. Introduction Velocities Tangent Lines Rates of Change Definition. Let s be a position function. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  29. Introduction Velocities Tangent Lines Rates of Change Definition. Let s be a position function. We define the instantaneous velocity at a by s ( b ) − s ( a ) v inst : = lim . b − a b → a logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  30. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  31. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] . s ( b ) − s ( 2 ) lim b − 2 b → 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  32. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] . − 4 . 9 b 2 + 20 − − 4 . 9 · 2 2 + 20 � � s ( b ) − s ( 2 ) = lim lim b − 2 b − 2 b → 2 b → 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  33. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] . − 4 . 9 b 2 + 20 − − 4 . 9 · 2 2 + 20 � � s ( b ) − s ( 2 ) = lim lim b − 2 b − 2 b → 2 b → 2 − 4 . 9 b 2 + 4 . 9 · 2 2 = lim b − 2 b → 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  34. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] . − 4 . 9 b 2 + 20 − − 4 . 9 · 2 2 + 20 � � s ( b ) − s ( 2 ) = lim lim b − 2 b − 2 b → 2 b → 2 − 4 . 9 b 2 + 4 . 9 · 2 2 = lim b − 2 b → 2 � b 2 − 2 2 � − 4 . 9 = lim b − 2 b → 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  35. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] . − 4 . 9 b 2 + 20 − − 4 . 9 · 2 2 + 20 � � s ( b ) − s ( 2 ) = lim lim b − 2 b − 2 b → 2 b → 2 − 4 . 9 b 2 + 4 . 9 · 2 2 = lim b − 2 b → 2 � b 2 − 2 2 � − 4 . 9 = lim b − 2 b → 2 − 4 . 9 ( b + 2 )( b − 2 ) = lim b − 2 b → 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  36. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] . − 4 . 9 b 2 + 20 − − 4 . 9 · 2 2 + 20 � � s ( b ) − s ( 2 ) = lim lim b − 2 b − 2 b → 2 b → 2 − 4 . 9 b 2 + 4 . 9 · 2 2 = lim b − 2 b → 2 � b 2 − 2 2 � − 4 . 9 = lim b − 2 b → 2 − 4 . 9 ( b + 2 )( b − 2 ) = lim b − 2 b → 2 = b → 2 − 4 . 9 ( b + 2 ) lim logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  37. Introduction Velocities Tangent Lines Rates of Change Find the Instantaneous Velocity of a Coin with Vertical Position s ( t ) = − 4 . 9 t 2 + 20 [ m ] at Time t = 2 [ s ] . − 4 . 9 b 2 + 20 − − 4 . 9 · 2 2 + 20 � � s ( b ) − s ( 2 ) = lim lim b − 2 b − 2 b → 2 b → 2 − 4 . 9 b 2 + 4 . 9 · 2 2 = lim b − 2 b → 2 � b 2 − 2 2 � − 4 . 9 = lim b − 2 b → 2 − 4 . 9 ( b + 2 )( b − 2 ) = lim b − 2 b → 2 = b → 2 − 4 . 9 ( b + 2 ) = − 19 . 6 lim logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  38. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  39. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  40. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  41. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  42. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  43. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  44. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  45. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  46. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  47. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  48. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  49. Introduction Velocities Tangent Lines Rates of Change Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  50. Introduction Velocities Tangent Lines Rates of Change Definition. Let f be a function. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  51. Introduction Velocities Tangent Lines Rates of Change Definition. Let f be a function. The secant line of f through � � � � a , f ( a ) and b , f ( b ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  52. Introduction Velocities Tangent Lines Rates of Change Definition. Let f be a function. The secant line of f through � � � � a , f ( a ) and b , f ( b ) is the unique straight line that goes � � � � a , f ( a ) b , f ( b ) through the points and . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  53. Introduction Velocities Tangent Lines Rates of Change Definition. Let f be a function. The secant line of f through � � � � a , f ( a ) and b , f ( b ) is the unique straight line that goes � � � � a , f ( a ) b , f ( b ) through the points and . Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  54. Introduction Velocities Tangent Lines Rates of Change Definition. Let f be a function. The secant line of f through � � � � a , f ( a ) and b , f ( b ) is the unique straight line that goes � � � � a , f ( a ) b , f ( b ) through the points and . Definition. Let f be a function. The tangent line of f at � � a , f ( a ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  55. Introduction Velocities Tangent Lines Rates of Change Definition. Let f be a function. The secant line of f through � � � � a , f ( a ) and b , f ( b ) is the unique straight line that goes � � � � a , f ( a ) b , f ( b ) through the points and . Definition. Let f be a function. The tangent line of f at � � a , f ( a ) (if it exists) is the unique line that goes through � � a , f ( a ) whose slope is f ( b ) − f ( a ) lim . b − a b → a logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  56. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  57. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 f ( b ) − f ( − 1 ) lim b − ( − 1 ) b →− 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  58. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 b 3 − 4 b + 1 − 4 f ( b ) − f ( − 1 ) = lim lim b − ( − 1 ) b + 1 b →− 1 b →− 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  59. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 b 3 − 4 b + 1 − 4 b 3 − 4 b − 3 f ( b ) − f ( − 1 ) = = lim lim lim b − ( − 1 ) b + 1 b + 1 b →− 1 b →− 1 b →− 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  60. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 b 3 − 4 b + 1 − 4 b 3 − 4 b − 3 f ( b ) − f ( − 1 ) = = lim lim lim b − ( − 1 ) b + 1 b + 1 b →− 1 b →− 1 b →− 1 ( − 1 + h ) 3 − 4 ( − 1 + h ) − 3 = lim ( − 1 + h )+ 1 h → 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  61. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 b 3 − 4 b + 1 − 4 b 3 − 4 b − 3 f ( b ) − f ( − 1 ) = = lim lim lim b − ( − 1 ) b + 1 b + 1 b →− 1 b →− 1 b →− 1 ( − 1 + h ) 3 − 4 ( − 1 + h ) − 3 = lim ( − 1 + h )+ 1 h → 0 − 1 + 3 h − 3 h 2 + h 3 + 4 − 4 h − 3 = lim h h → 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  62. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 b 3 − 4 b + 1 − 4 b 3 − 4 b − 3 f ( b ) − f ( − 1 ) = = lim lim lim b − ( − 1 ) b + 1 b + 1 b →− 1 b →− 1 b →− 1 ( − 1 + h ) 3 − 4 ( − 1 + h ) − 3 = lim ( − 1 + h )+ 1 h → 0 − 1 + 3 h − 3 h 2 + h 3 + 4 − 4 h − 3 = lim h h → 0 h 3 − 3 h 2 − h = lim h h → 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  63. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 b 3 − 4 b + 1 − 4 b 3 − 4 b − 3 f ( b ) − f ( − 1 ) = = lim lim lim b − ( − 1 ) b + 1 b + 1 b →− 1 b →− 1 b →− 1 ( − 1 + h ) 3 − 4 ( − 1 + h ) − 3 = lim ( − 1 + h )+ 1 h → 0 − 1 + 3 h − 3 h 2 + h 3 + 4 − 4 h − 3 = lim h h → 0 � h 2 − 3 h − 1 � h 3 − 3 h 2 − h h = = lim lim h h h → 0 h → 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  64. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 b 3 − 4 b + 1 − 4 b 3 − 4 b − 3 f ( b ) − f ( − 1 ) = = lim lim lim b − ( − 1 ) b + 1 b + 1 b →− 1 b →− 1 b →− 1 ( − 1 + h ) 3 − 4 ( − 1 + h ) − 3 = lim ( − 1 + h )+ 1 h → 0 − 1 + 3 h − 3 h 2 + h 3 + 4 − 4 h − 3 = lim h h → 0 � h 2 − 3 h − 1 � h 3 − 3 h 2 − h h = = lim lim h h h → 0 h → 0 h → 0 h 2 − 3 h − 1 = lim logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  65. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 b 3 − 4 b + 1 − 4 b 3 − 4 b − 3 f ( b ) − f ( − 1 ) = = lim lim lim b − ( − 1 ) b + 1 b + 1 b →− 1 b →− 1 b →− 1 ( − 1 + h ) 3 − 4 ( − 1 + h ) − 3 = lim ( − 1 + h )+ 1 h → 0 − 1 + 3 h − 3 h 2 + h 3 + 4 − 4 h − 3 = lim h h → 0 � h 2 − 3 h − 1 � h 3 − 3 h 2 − h h = = lim lim h h h → 0 h → 0 h → 0 h 2 − 3 h − 1 = − 1 = lim logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  66. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  67. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 Slope: − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  68. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 Slope: − 1, point: ( − 1 , 4 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  69. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 Slope: − 1, point: ( − 1 , 4 ) . = mx + b y logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  70. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 Slope: − 1, point: ( − 1 , 4 ) . = mx + b y 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  71. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 Slope: − 1, point: ( − 1 , 4 ) . = mx + b y = ( − 1 ) 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  72. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 Slope: − 1, point: ( − 1 , 4 ) . = mx + b y = ( − 1 )( − 1 ) 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  73. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 Slope: − 1, point: ( − 1 , 4 ) . = mx + b y = ( − 1 )( − 1 )+ b 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  74. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 Slope: − 1, point: ( − 1 , 4 ) . = mx + b y = ( − 1 )( − 1 )+ b 4 = b 3 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  75. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 Slope: − 1, point: ( − 1 , 4 ) . = mx + b y = ( − 1 )( − 1 )+ b 4 = b 3 = − x + 3 y logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  76. Introduction Velocities Tangent Lines Rates of Change Compute the Equation of the Tangent Line of f ( x ) = x 3 − 4 x + 1 at a = − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  77. Introduction Velocities Tangent Lines Rates of Change Estimate the Slope of the Tangent Line at x = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  78. Introduction Velocities Tangent Lines Rates of Change Estimate the Slope of the Tangent Line at x = 1 ✻ 5 4 3 2 1 ✲ − 3 − 2 − 1 1 2 3 − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  79. Introduction Velocities Tangent Lines Rates of Change Estimate the Slope of the Tangent Line at x = 1 ✻ 5 4 3 2 1 ✲ − 3 − 2 − 1 1 2 3 − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  80. Introduction Velocities Tangent Lines Rates of Change Estimate the Slope of the Tangent Line at x = 1 ✻ 5 4 3 slope ≈ 2 2 1 ✲ − 3 − 2 − 1 1 2 3 − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

  81. Introduction Velocities Tangent Lines Rates of Change Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Why Do We Need Derivatives?

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