0402 ch 4 day 2

307 views

Published on

Published in: Education, Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
307
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
0
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • 0402 ch 4 day 2

    1. 1. Chapter 4 4.1 Exponential Functions Day 2Titus 3:7 so that being justified by his grace we mightbecome heirs according to the hope of eternal life.
    2. 2. 4.1 Exponential FunctionsConsider modeling population growthwith exponential functions
    3. 3. 4.1 Exponential Functions Let P = initial population r = rate of increase per year 0 < r < 1 (a %)
    4. 4. 4.1 Exponential Functions Let P = initial population r = rate of increase per year 0 < r < 1 (a %)after 1 year: P + rP or P (1+ r )
    5. 5. 4.1 Exponential Functions Let P = initial population r = rate of increase per year 0 < r < 1 (a %)after 1 year: P + rP or P (1+ r )after 2 years: P (1+ r ) + r ⋅ P (1+ r )
    6. 6. 4.1 Exponential Functions Let P = initial population r = rate of increase per year 0 < r < 1 (a %)after 1 year: P + rP or P (1+ r )after 2 years: P (1+ r ) + r ⋅ P (1+ r )
    7. 7. 4.1 Exponential Functions Let P = initial population r = rate of increase per year 0 < r < 1 (a %)after 1 year: P + rP or P (1+ r )after 2 years: P (1+ r ) + r ⋅ P (1+ r ) 2 P (1+ r ) (1+ r ) or P (1+ r )
    8. 8. 4.1 Exponential Functions Let P = initial population r = rate of increase per year 0 < r < 1 (a %)after 1 year: P + rP or P (1+ r )after 2 years: P (1+ r ) + r ⋅ P (1+ r ) 2 P (1+ r ) (1+ r ) or P (1+ r )after 3 years: 2 P (1+ r ) + r ⋅ P (1+ r ) 2
    9. 9. 4.1 Exponential Functions Let P = initial population r = rate of increase per year 0 < r < 1 (a %)after 1 year: P + rP or P (1+ r )after 2 years: P (1+ r ) + r ⋅ P (1+ r ) 2 P (1+ r ) (1+ r ) or P (1+ r )after 3 years: 2 P (1+ r ) + r ⋅ P (1+ r ) 2
    10. 10. 4.1 Exponential Functions Let P = initial population r = rate of increase per year 0 < r < 1 (a %)after 1 year: P + rP or P (1+ r )after 2 years: P (1+ r ) + r ⋅ P (1+ r ) 2 P (1+ r ) (1+ r ) or P (1+ r )after 3 years: 2 P (1+ r ) + r ⋅ P (1+ r ) 2 2 3 P (1+ r ) (1+ r ) or P (1+ r )
    11. 11. 4.1 Exponential Functions Let P = initial population r = rate of increase per year 0 < r < 1 (a %)after 1 year: P + rP or P (1+ r )after 2 years: P (1+ r ) + r ⋅ P (1+ r ) 2 P (1+ r ) (1+ r ) or P (1+ r )after 3 years: 2 P (1+ r ) + r ⋅ P (1+ r ) 2 2 3 P (1+ r ) (1+ r ) or P (1+ r )after t years:
    12. 12. 4.1 Exponential Functions Let P = initial population r = rate of increase per year 0 < r < 1 (a %)after 1 year: P + rP or P (1+ r )after 2 years: P (1+ r ) + r ⋅ P (1+ r ) 2 P (1+ r ) (1+ r ) or P (1+ r )after 3 years: 2 P (1+ r ) + r ⋅ P (1+ r ) 2 2 3 P (1+ r ) (1+ r ) or P (1+ r ) tafter t years: P (1+ r )
    13. 13. 4.1 Exponential Functions Compound Interest Formula
    14. 14. 4.1 Exponential Functions Compound Interest Formula nt ⎛ r ⎞ A = P ⎜ 1+ ⎟ ⎝ n ⎠
    15. 15. 4.1 Exponential Functions Compound Interest Formula nt ⎛ r ⎞ A = P ⎜ 1+ ⎟ ⎝ n ⎠A = total amount after t years
    16. 16. 4.1 Exponential Functions Compound Interest Formula nt ⎛ r ⎞ A = P ⎜ 1+ ⎟ ⎝ n ⎠A = total amount after t yearsP = Principal ... original amount
    17. 17. 4.1 Exponential Functions Compound Interest Formula nt ⎛ r ⎞ A = P ⎜ 1+ ⎟ ⎝ n ⎠A = total amount after t yearsP = Principal ... original amountr = interest rate
    18. 18. 4.1 Exponential Functions Compound Interest Formula nt ⎛ r ⎞ A = P ⎜ 1+ ⎟ ⎝ n ⎠A = total amount after t yearsP = Principal ... original amountr = interest raten = number of times compounded per year
    19. 19. 4.1 Exponential Functions Compound Interest Formula nt ⎛ r ⎞ A = P ⎜ 1+ ⎟ ⎝ n ⎠A = total amount after t yearsP = Principal ... original amountr = interest raten = number of times compounded per yeart = time in years
    20. 20. 4.1 Exponential Functions Jacob invests $4000 at 4.5% interest for ten years. Compare how much Jacob would have if interest is compounded a) annually, b) quarterly, c) monthly, d) daily.
    21. 21. 4.1 Exponential Functions Jacob invests $4000 at 4.5% interest for ten years. Compare how much Jacob would have if interest is compounded a) annually, b) quarterly, c) monthly, d) daily. 1(10) ⎛ .045 ⎞ a) annual: 4000 ⎜ 1+ ⎟ $ = 6211.88 ⎝ 1 ⎠
    22. 22. 4.1 Exponential Functions Jacob invests $4000 at 4.5% interest for ten years. Compare how much Jacob would have if interest is compounded a) annually, b) quarterly, c) monthly, d) daily. 1(10) ⎛ .045 ⎞ a) annual: 4000 ⎜ 1+ ⎟ $ = 6211.88 ⎝ 1 ⎠ 4(10) ⎛ .045 ⎞ b) quarterly: 4000 ⎜ 1+ ⎟ $ = 6257.51 ⎝ 4 ⎠
    23. 23. 4.1 Exponential Functions Jacob invests $4000 at 4.5% interest for ten years. Compare how much Jacob would have if interest is compounded a) annually, b) quarterly, c) monthly, d) daily. 12(10) ⎛ .045 ⎞ c) monthly: 4000 ⎜ 1+ ⎟ $ = 6267.97 ⎝ 12 ⎠
    24. 24. 4.1 Exponential Functions Jacob invests $4000 at 4.5% interest for ten years. Compare how much Jacob would have if interest is compounded a) annually, b) quarterly, c) monthly, d) daily. 12(10) ⎛ .045 ⎞ c) monthly: 4000 ⎜ 1+ ⎟ $ = 6267.97 ⎝ 12 ⎠ 365(10) ⎛ .045 ⎞ d) daily: 4000 ⎜ 1+ ⎟ $ = 6273.07 ⎝ 365 ⎠
    25. 25. 4.1 Exponential Functions Jacob invests $4000 at 4.5% interest for ten years. Compare how much Jacob would have if interest is compounded a) annually, b) quarterly, c) monthly, d) daily. a) $6211.88 b) $6257.51 c) $6267.97 d) $6273.07
    26. 26. 4.1 Exponential Functions Jacob invests $4000 at 4.5% interest for ten years. Compare how much Jacob would have if interest is compounded a) annually, b) quarterly, c) monthly, d) daily. a) $6211.88 as the number of times b) $6257.51 compounded increases, c) $6267.97 the greater the amount. d) $6273.07
    27. 27. 4.1 Exponential Functions Consider Continuous Compounding ... n→∞
    28. 28. 4.1 Exponential Functions Consider Continuous Compounding ... n→∞ Deposit $1 for 1 year at 100% interest
    29. 29. 4.1 Exponential Functions Consider Continuous Compounding ... n→∞ Deposit $1 for 1 year at 100% interest n(1) ⎛ 1 ⎞ A = 1⎜ 1+ ⎟ ⎝ n ⎠
    30. 30. 4.1 Exponential Functions Consider Continuous Compounding ... n→∞ Deposit $1 for 1 year at 100% interest n(1) ⎛ 1 ⎞ A = 1⎜ 1+ ⎟ ⎝ n ⎠ Find A for n = 12 n = 365 n = 1,000,000
    31. 31. 4.1 Exponential Functions n(1) ⎛ 1 ⎞A = 1⎜ 1+ ⎟ ⎝ n ⎠when n = 1,000,000A = 2.718280469
    32. 32. 4.1 Exponential Functions n(1) ⎛ 1 ⎞A = 1⎜ 1+ ⎟ ⎝ n ⎠when n = 1,000,000A = 2.7182804691e = 2.718281828
    33. 33. 4.1 Exponential Functions n(1) ⎛ 1 ⎞A = 1⎜ 1+ ⎟ ⎝ n ⎠when n = 1,000,000A = 2.7182804691e = 2.718281828as n → ∞, A → e
    34. 34. 4.1 Exponential Functions ⎛ r ⎞ nt A = P ⎜ 1+ ⎟ n(1) ⎝ n ⎠ ⎛ 1 ⎞A = 1⎜ 1+ ⎟ ⎝ n ⎠when n = 1,000,000A = 2.7182804691e = 2.718281828as n → ∞, A → e
    35. 35. 4.1 Exponential Functions ⎛ r ⎞ nt A = P ⎜ 1+ ⎟ n(1) ⎝ n ⎠ ⎛ 1 ⎞ nA = 1⎜ 1+ ⎟ let m = ⎝ n ⎠ rwhen n = 1,000,000A = 2.7182804691e = 2.718281828as n → ∞, A → e
    36. 36. 4.1 Exponential Functions ⎛ r ⎞ nt A = P ⎜ 1+ ⎟ n(1) ⎝ n ⎠ ⎛ 1 ⎞ nA = 1⎜ 1+ ⎟ let m = ⎝ n ⎠ r mrtwhen n = 1,000,000 ⎛ 1 ⎞ A = P ⎜ 1+ ⎟A = 2.718280469 ⎝ m ⎠1e = 2.718281828as n → ∞, A → e
    37. 37. 4.1 Exponential Functions ⎛ r ⎞ nt A = P ⎜ 1+ ⎟ n(1) ⎝ n ⎠ ⎛ 1 ⎞ nA = 1⎜ 1+ ⎟ let m = ⎝ n ⎠ r mrtwhen n = 1,000,000 ⎛ 1 ⎞ A = P ⎜ 1+ ⎟A = 2.718280469 ⎝ m ⎠ m1 ⎛ 1 ⎞e = 2.718281828 sub e for ⎜ 1+ ⎟ ⎝ m ⎠as n → ∞, A → e
    38. 38. 4.1 Exponential Functions ⎛ r ⎞ nt A = P ⎜ 1+ ⎟ n(1) ⎝ n ⎠ ⎛ 1 ⎞ nA = 1⎜ 1+ ⎟ let m = ⎝ n ⎠ r mrtwhen n = 1,000,000 ⎛ 1 ⎞ A = P ⎜ 1+ ⎟A = 2.718280469 ⎝ m ⎠ m1 ⎛ 1 ⎞e = 2.718281828 sub e for ⎜ 1+ ⎟ ⎝ m ⎠as n → ∞, A → e rt A = Pe
    39. 39. 4.1 Exponential Functions ⎛ r ⎞ nt A = P ⎜ 1+ ⎟ n(1) ⎝ n ⎠ ⎛ 1 ⎞ nA = 1⎜ 1+ ⎟ let m = ⎝ n ⎠ r mrtwhen n = 1,000,000 ⎛ 1 ⎞ A = P ⎜ 1+ ⎟A = 2.718280469 ⎝ m ⎠ m1 ⎛ 1 ⎞e = 2.718281828 sub e for ⎜ 1+ ⎟ ⎝ m ⎠as n → ∞, A → e rt A = Pe Continuous Compound Interest formula
    40. 40. 4.1 Exponential Functions rt Continuous Compound A = Pe Interest formula
    41. 41. 4.1 Exponential Functions rt Continuous Compound A = Pe Interest formula How would Jacob have done with continuous compounding?
    42. 42. 4.1 Exponential Functions rt Continuous Compound A = Pe Interest formula How would Jacob have done with continuous compounding? .045(10) $ 4000 ⋅ e = 6273.25
    43. 43. Chapter 4 HW #2Don’t go around saying the world owes you a living.The world owes you nothing. It was here first. Mark Twain

    ×