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# 0402 ch 4 day 2

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• ### 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