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# 0401 ch 4 day 1

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• ### 0401 ch 4 day 1

1. 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. 2. 4.1 Exponential Functions
3. 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. 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. 5. 4.1 Exponential Functions x y=b
6. 6. 4.1 Exponential Functions x y=b the base of the exponent
7. 7. 4.1 Exponential Functions x y=b our input variable the base of the exponent
8. 8. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5
9. 9. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain:
10. 10. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain: {x : x ∈° }
11. 11. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain: {x : x ∈° }Range:
12. 12. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain: {x : x ∈° }Range: {y : y > 0}
13. 13. 4.1 Exponential Functions Graph and discuss x x y1 = 2 y2 = 5Domain: {x : x ∈° }Range: {y : y > 0}x-intercept:
14. 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. 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. 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. 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. 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. 19. 4.1 Exponential FunctionsLet’s experiment with various b values by graphing x x x y=8 y = .7 y =1
20. 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. 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. 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. 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. 24. 4.1 Exponential Functions x ⎛ 1 ⎞Graph y1 = 2 and x y2 = ⎜ ⎟ ⎝ 2 ⎠
25. 25. 4.1 Exponential Functions x ⎛ 1 ⎞Graph y1 = 2 and x y2 = ⎜ ⎟ ⎝ 2 ⎠ they are symmetric about the y-axis
26. 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. 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. 28. 4.1 Exponential Functions xGraph y1 = e (the Natural Exponential Function)
29. 29. 4.1 Exponential Functions xGraph y1 = e (the Natural Exponential Function) e is a constant ... irrational ... like π or 2
30. 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. 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. 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. 33. 4.1 Exponential Functions Predict, verify with a graph, then discuss
34. 34. 4.1 Exponential Functions Predict, verify with a graph, then discuss x 1) y = 2 + 3
35. 35. 4.1 Exponential Functions Predict, verify with a graph, then discuss x 1) y = 2 + 3 x 2) y = −e
36. 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. 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. 38. 4.1 Exponential Functions
39. 39. 4.1 Exponential Functions HW #1Striving for success without hard work is liketrying to harvest where you haven’t planted. David Bly