The document discusses exponential functions and compound interest. It defines exponential growth using an initial population P and growth rate r. It presents the compound interest formula and explains how interest compounds annually, quarterly, monthly, and daily. The formula approaches e as the number of compounding periods increases indefinitely. Continuous compounding results in growth of Pert at the limit.

1. Chapter 4
4.1 Exponential Functions
Day 2
Titus 3:7 so that being justified by his grace we might
become heirs according to the hope of eternal life.

2. 4.1 Exponential Functions
Consider modeling population growth
with exponential functions

3. 4.1 Exponential Functions
Let P = initial population
r = rate of increase per year
0 < r < 1 (a %)

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. 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. 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. 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. 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. 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. 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. 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. 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 )
t
after t years: P (1+ r )

13. 4.1 Exponential Functions
Compound Interest Formula

14. 4.1 Exponential Functions
Compound Interest Formula
nt
⎛ r ⎞
A = P ⎜ 1+ ⎟
⎝ n ⎠

15. 4.1 Exponential Functions
Compound Interest Formula
nt
⎛ r ⎞
A = P ⎜ 1+ ⎟
⎝ n ⎠
A = total amount after t years

16. 4.1 Exponential Functions
Compound Interest Formula
nt
⎛ r ⎞
A = P ⎜ 1+ ⎟
⎝ n ⎠
A = total amount after t years
P = Principal ... original amount

17. 4.1 Exponential Functions
Compound Interest Formula
nt
⎛ r ⎞
A = P ⎜ 1+ ⎟
⎝ n ⎠
A = total amount after t years
P = Principal ... original amount
r = interest rate

18. 4.1 Exponential Functions
Compound Interest Formula
nt
⎛ r ⎞
A = P ⎜ 1+ ⎟
⎝ n ⎠
A = total amount after t years
P = Principal ... original amount
r = interest rate
n = number of times compounded per year

19. 4.1 Exponential Functions
Compound Interest Formula
nt
⎛ r ⎞
A = P ⎜ 1+ ⎟
⎝ n ⎠
A = total amount after t years
P = Principal ... original amount
r = interest rate
n = number of times compounded per year
t = time in years

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. 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. 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. 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. 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. 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. 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. 4.1 Exponential Functions
Consider Continuous Compounding ...
n→∞

28. 4.1 Exponential Functions
Consider Continuous Compounding ...
n→∞
Deposit $1 for 1 year at 100% interest

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. 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. 4.1 Exponential Functions
n(1)
⎛ 1 ⎞
A = 1⎜ 1+ ⎟
⎝ n ⎠
when n = 1,000,000
A = 2.718280469

32. 4.1 Exponential Functions
n(1)
⎛ 1 ⎞
A = 1⎜ 1+ ⎟
⎝ n ⎠
when n = 1,000,000
A = 2.718280469
1
e = 2.718281828

33. 4.1 Exponential Functions
n(1)
⎛ 1 ⎞
A = 1⎜ 1+ ⎟
⎝ n ⎠
when n = 1,000,000
A = 2.718280469
1
e = 2.718281828
as n → ∞, A → e

34. 4.1 Exponential Functions ⎛ r ⎞
nt
A = P ⎜ 1+ ⎟
n(1) ⎝ n ⎠
⎛ 1 ⎞
A = 1⎜ 1+ ⎟
⎝ n ⎠
when n = 1,000,000
A = 2.718280469
1
e = 2.718281828
as n → ∞, A → e

35. 4.1 Exponential Functions ⎛ r ⎞
nt
A = P ⎜ 1+ ⎟
n(1) ⎝ n ⎠
⎛ 1 ⎞ n
A = 1⎜ 1+ ⎟ let m =
⎝ n ⎠ r
when n = 1,000,000
A = 2.718280469
1
e = 2.718281828
as n → ∞, A → e

36. 4.1 Exponential Functions ⎛ r ⎞
nt
A = P ⎜ 1+ ⎟
n(1) ⎝ n ⎠
⎛ 1 ⎞ n
A = 1⎜ 1+ ⎟ let m =
⎝ n ⎠ r
mrt
when n = 1,000,000 ⎛ 1 ⎞
A = P ⎜ 1+ ⎟
A = 2.718280469 ⎝ m ⎠
1
e = 2.718281828
as n → ∞, A → e

37. 4.1 Exponential Functions ⎛ r ⎞
nt
A = P ⎜ 1+ ⎟
n(1) ⎝ n ⎠
⎛ 1 ⎞ n
A = 1⎜ 1+ ⎟ let m =
⎝ n ⎠ r
mrt
when n = 1,000,000 ⎛ 1 ⎞
A = P ⎜ 1+ ⎟
A = 2.718280469 ⎝ m ⎠
m
1 ⎛ 1 ⎞
e = 2.718281828 sub e for ⎜ 1+ ⎟
⎝ m ⎠
as n → ∞, A → e

38. 4.1 Exponential Functions ⎛ r ⎞
nt
A = P ⎜ 1+ ⎟
n(1) ⎝ n ⎠
⎛ 1 ⎞ n
A = 1⎜ 1+ ⎟ let m =
⎝ n ⎠ r
mrt
when n = 1,000,000 ⎛ 1 ⎞
A = P ⎜ 1+ ⎟
A = 2.718280469 ⎝ m ⎠
m
1 ⎛ 1 ⎞
e = 2.718281828 sub e for ⎜ 1+ ⎟
⎝ m ⎠
as n → ∞, A → e rt
A = Pe

39. 4.1 Exponential Functions ⎛ r ⎞
nt
A = P ⎜ 1+ ⎟
n(1) ⎝ n ⎠
⎛ 1 ⎞ n
A = 1⎜ 1+ ⎟ let m =
⎝ n ⎠ r
mrt
when n = 1,000,000 ⎛ 1 ⎞
A = P ⎜ 1+ ⎟
A = 2.718280469 ⎝ m ⎠
m
1 ⎛ 1 ⎞
e = 2.718281828 sub e for ⎜ 1+ ⎟
⎝ m ⎠
as n → ∞, A → e rt
A = Pe
Continuous Compound
Interest formula

40. 4.1 Exponential Functions
rt Continuous Compound
A = Pe Interest formula

41. 4.1 Exponential Functions
rt Continuous Compound
A = Pe Interest formula
How would Jacob have done with
continuous compounding?

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. Chapter 4
HW #2
Don’t go around saying the world owes you a living.
The world owes you nothing. It was here first.
Mark Twain