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Transcript

  • 1. Chapter 4 Exponential & Logarithmic FunctionsProverbs 24:14 Know that wisdom is such to yoursoul; if you find it, there will be a future, and yourhope will not be cut off.
  • 2. 4.1 Exponential Functions
  • 3. 4.1 Exponential FunctionsA function that can be expressed inthe form y = b , b > 0 and b ≠ 1 is xcalled an exponential function.
  • 4. 4.1 Exponential FunctionsA function that can be expressed inthe form y = b , b > 0 and b ≠ 1 is xcalled an exponential function.These functions are used to modelsituations such as growth and decay.
  • 5. 4.1 Exponential Functions x y=b
  • 6. 4.1 Exponential Functions x y=b the base of the exponent
  • 7. 4.1 Exponential Functions x y=b our input variable the base of the exponent
  • 8. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5
  • 9. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain:
  • 10. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain: {x : x ∈° }
  • 11. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain: {x : x ∈° }Range:
  • 12. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain: {x : x ∈° }Range: {y : y > 0}
  • 13. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain: {x : x ∈° }Range: {y : y > 0}x-intercept:
  • 14. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain: {x : x ∈° }Range: {y : y > 0}x-intercept: none y = 0 is a H.A.
  • 15. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain: {x : x ∈° }Range: {y : y > 0}x-intercept: none y = 0 is a H.A.y-intercept:
  • 16. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain: {x : x ∈° }Range: {y : y > 0}x-intercept: none y = 0 is a H.A.y-intercept: ( 0,1)
  • 17. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain: {x : x ∈° }Range: {y : y > 0}x-intercept: none y = 0 is a H.A.y-intercept: ( 0,1)increasing
  • 18. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain: {x : x ∈° }Range: {y : y > 0}x-intercept: none y = 0 is a H.A.y-intercept: ( 0,1)increasing1 to 1
  • 19. 4.1 Exponential FunctionsLet’s experiment with various b values by graphing x x x y=8 y = .7 y =1
  • 20. 4.1 Exponential FunctionsLet’s experiment with various b values by graphing x x x y=8 y = .7 y =1 as b → ∞ ... grows faster
  • 21. 4.1 Exponential FunctionsLet’s experiment with various b values by graphing x x x y=8 y = .7 y =1 as b → ∞ ... grows faster b >1 ... increasing
  • 22. 4.1 Exponential FunctionsLet’s experiment with various b values by graphing x x x y=8 y = .7 y =1 as b → ∞ ... grows faster b >1 ... increasing 0 < b <1 ... decreasing
  • 23. 4.1 Exponential FunctionsLet’s experiment with various b values by graphing x x x y=8 y = .7 y =1 as b → ∞ ... grows faster b >1 ... increasing 0 < b <1 ... decreasing b =1 ... constant (not exponential)
  • 24. 4.1 Exponential Functions x ⎛ 1 ⎞Graph y1 = 2 and x y2 = ⎜ ⎟ ⎝ 2 ⎠
  • 25. 4.1 Exponential Functions x ⎛ 1 ⎞Graph y1 = 2 and x y2 = ⎜ ⎟ ⎝ 2 ⎠ they are symmetric about the y-axis
  • 26. 4.1 Exponential Functions x ⎛ 1 ⎞Graph y1 = 2 and x y2 = ⎜ ⎟ ⎝ 2 ⎠ they are symmetric about the y-axis both have a y-intercept of ( 0,1)
  • 27. 4.1 Exponential Functions x ⎛ 1 ⎞Graph y1 = 2 and x y2 = ⎜ ⎟ ⎝ 2 ⎠ they are symmetric about the y-axis both have a y-intercept of ( 0,1) x −x 1 ⎛ 1 ⎞ note : 2 = x = ⎜ ⎟ 2 ⎝ 2 ⎠
  • 28. 4.1 Exponential Functions xGraph y1 = e (the Natural Exponential Function)
  • 29. 4.1 Exponential Functions xGraph y1 = e (the Natural Exponential Function) e is a constant ... irrational ... like π or 2
  • 30. 4.1 Exponential Functions xGraph y1 = e (the Natural Exponential Function) e is a constant ... irrational ... like π or 2 n ⎛ 1 ⎞ e = ⎜ 1+ ⎟ as n → ∞ ⎝ n ⎠
  • 31. 4.1 Exponential Functions xGraph y1 = e (the Natural Exponential Function) e is a constant ... irrational ... like π or 2 n ⎛ 1 ⎞ e = ⎜ 1+ ⎟ as n → ∞ ⎝ n ⎠ 1 do e to see that e ≈ 2.7183
  • 32. 4.1 Exponential Functions xGraph y1 = e (the Natural Exponential Function) e is a constant ... irrational ... like π or 2 n ⎛ 1 ⎞ e = ⎜ 1+ ⎟ as n → ∞ ⎝ n ⎠ 1 do e to see that e ≈ 2.7183 e is the base of natural logarithms (more on this later)
  • 33. 4.1 Exponential Functions Predict, verify with a graph, then discuss
  • 34. 4.1 Exponential Functions Predict, verify with a graph, then discuss x 1) y = 2 + 3
  • 35. 4.1 Exponential Functions Predict, verify with a graph, then discuss x 1) y = 2 + 3 x 2) y = −e
  • 36. 4.1 Exponential Functions Predict, verify with a graph, then discuss x 1) y = 2 + 3 x 2) y = −e x ⎛ 1 ⎞ 3) y = ⎜ ⎟ − 4 ⎝ 3 ⎠
  • 37. 4.1 Exponential Functions Predict, verify with a graph, then discuss x 1) y = 2 + 3 x 2) y = −e x ⎛ 1 ⎞ 3) y = ⎜ ⎟ − 4 ⎝ 3 ⎠ Be sure to read Example 8 carefully ... do it on your calculator!
  • 38. 4.1 Exponential Functions
  • 39. 4.1 Exponential Functions HW #1Striving for success without hard work is liketrying to harvest where you haven’t planted. David Bly