Exponential functions change addition into multiplication. Different bases for exponentials produce different functions but they share similar characteristics. One base--a number we call e--is an especially good one.

1. Section 3.1
Exponential Functions
V63.0121.027, Calculus I
October 20, 2009
. . . . . .

2. Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential function
Compound Interest
The number e
A limit
. . . . . .

3. Definition
If a is a real number and n is a positive whole number, then
an = a · a · · · · · a
n factors
. . . . . .

4. Definition
If a is a real number and n is a positive whole number, then
an = a · a · · · · · a
n factors
Examples
23 = 2 · 2 · 2 = 8
34 = 3 · 3 · 3 · 3 = 81
(−1)5 = (−1)(−1)(−1)(−1)(−1) = −1
. . . . . .

5. Fact
If a is a real number, then
ax+y = ax ay
ax
ax−y = y
a
(ax )y = axy
(ab)x = ax bx
whenever all exponents are positive whole numbers.
. . . . . .

6. Fact
If a is a real number, then
ax+y = ax ay
ax
ax−y = y
a
(ax )y = axy
(ab)x = ax bx
whenever all exponents are positive whole numbers.
Proof.
Check for yourself:
a x +y = a · a · · · · · a = a · a · · · · · a · a · a · · · · · a = a x a y
x + y factors x factors y factors
. . . . . .

7. Let’s be conventional
The desire that these properties remain true gives us
conventions for ax when x is not a positive whole number.
. . . . . .

8. Let’s be conventional
The desire that these properties remain true gives us
conventions for ax when x is not a positive whole number.
For example:
a n = a n +0 = a n a 0
!
. . . . . .

9. Let’s be conventional
The desire that these properties remain true gives us
conventions for ax when x is not a positive whole number.
For example:
a n = a n +0 = a n a 0
!
Definition
If a ̸= 0, we define a0 = 1.
. . . . . .

10. Let’s be conventional
The desire that these properties remain true gives us
conventions for ax when x is not a positive whole number.
For example:
a n = a n +0 = a n a 0
!
Definition
If a ̸= 0, we define a0 = 1.
Notice 00 remains undefined (as a limit form, it’s
indeterminate).
. . . . . .

11. Conventions for negative exponents
If n ≥ 0, we want
an · a−n = an+(−n) = a0 = 1
!
. . . . . .

12. Conventions for negative exponents
If n ≥ 0, we want
an · a−n = an+(−n) = a0 = 1
!
Definition
1
If n is a positive integer, we define a−n = .
an
. . . . . .

13. Conventions for negative exponents
If n ≥ 0, we want
an · a−n = an+(−n) = a0 = 1
!
Definition
1
If n is a positive integer, we define a−n = .
an
Fact
1
The convention that a−n = “works” for negative n as well.
an
am
If m and n are any integers, then am−n = n .
a
. . . . . .

14. Conventions for fractional exponents
If q is a positive integer, we want
(a1/q )q = a1 = a
!
. . . . . .

15. Conventions for fractional exponents
If q is a positive integer, we want
(a1/q )q = a1 = a
!
Definition √
If q is a positive integer, we define a1/q = q
a. We must have
a ≥ 0 if q is even.
. . . . . .

16. Conventions for fractional exponents
If q is a positive integer, we want
(a1/q )q = a1 = a
!
Definition √
If q is a positive integer, we define a1/q = q
a. We must have
a ≥ 0 if q is even.
Fact
Now we can say ap/q = (a1/q )p without ambiguity
. . . . . .

17. Conventions for irrational powers
So ax is well-defined if x is rational.
What about irrational powers?
. . . . . .

18. Conventions for irrational powers
So ax is well-defined if x is rational.
What about irrational powers?
Definition
Let a > 0. Then
ax = lim ar
r→x
r rational
. . . . . .

19. Conventions for irrational powers
So ax is well-defined if x is rational.
What about irrational powers?
Definition
Let a > 0. Then
ax = lim ar
r→x
r rational
In other words, to approximate ax for irrational x, take r close to x
but rational and compute ar .
. . . . . .

20. Graphs of various exponential functions
y
.
. x
.
. . . . . .

21. Graphs of various exponential functions
y
.
. = 1x
y
. x
.
. . . . . .

22. Graphs of various exponential functions
y
.
. = 2x
y
. = 1x
y
. x
.
. . . . . .

23. Graphs of various exponential functions
y
.
. = 3x. = 2x
y y
. = 1x
y
. x
.
. . . . . .

24. Graphs of various exponential functions
y
.
. = 10x= 3x. = 2x
y y
. y
. = 1x
y
. x
.
. . . . . .

25. Graphs of various exponential functions
y
.
. = 10x= 3x. = 2x
y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .

26. Graphs of various exponential functions
y
.
. = (1/2)x
y . = 10x= 3x. = 2x
y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .

27. Graphs of various exponential functions
y
.
y y
. = x
. = (1/2)x (1/3) . = 10x= 3x. = 2x
y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .

28. Graphs of various exponential functions
y
.
y y
. = x
. = (1/2)x (1/3) . = (1/10)x. = 10x= 3x. = 2x
y y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .

29. Graphs of various exponential functions
y
.
yy = 213)x
. . = ((//2)x (1/3)x
y
. = . = (1/10)x. = 10x= 3x. = 2x
y y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .

30. Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential function
Compound Interest
The number e
A limit
. . . . . .

31. Properties of exponential Functions
Theorem
If a > 0 and a ̸= 1, then f(x) = ax is a continuous function with
domain R and range (0, ∞). In particular, ax > 0 for all x. If
a, b > 0 and x, y ∈ R, then
ax+y = ax ay
ax
ax−y = y
a
(ax )y = axy
(ab)x = ax bx
Proof.
This is true for positive integer exponents by natural
definition
Our conventional definitions make these true for rational
exponents
Our limit definition make these for irrational exponents, too
. . . . . .

32. Properties of exponential Functions
Theorem
If a > 0 and a ̸= 1, then f(x) = ax is a continuous function with
domain R and range (0, ∞). In particular, ax > 0 for all x. If
a, b > 0 and x, y ∈ R, then
ax+y = ax ay
ax
ax−y = y negative exponents mean reciprocals.
a
(ax )y = axy
(ab)x = ax bx
Proof.
This is true for positive integer exponents by natural
definition
Our conventional definitions make these true for rational
exponents
Our limit definition make these for irrational exponents, too
. . . . . .

33. Properties of exponential Functions
Theorem
If a > 0 and a ̸= 1, then f(x) = ax is a continuous function with
domain R and range (0, ∞). In particular, ax > 0 for all x. If
a, b > 0 and x, y ∈ R, then
ax+y = ax ay
ax
ax−y = y negative exponents mean reciprocals.
a
(ax )y = axy fractional exponents mean roots
(ab)x = ax bx
Proof.
This is true for positive integer exponents by natural
definition
Our conventional definitions make these true for rational
exponents
Our limit definition make these for irrational exponents, too
. . . . . .

34. Example
Simplify: 82/3
. . . . . .

35. Example
Simplify: 82/3
Solution
√
3 √
8 2 /3 = 82 =
3
64 = 4
. . . . . .

36. Example
Simplify: 82/3
Solution
√3 √
82/3 = 82 = 64 = 4
3
(√ )2
8 = 22 = 4.
3
Or,
. . . . . .

37. Example
Simplify: 82/3
Solution
√3 √
82/3 = 82 = 64 = 4
3
(√ )2
8 = 22 = 4.
3
Or,
Example √
8
Simplify: 1/2
2
. . . . . .

38. Example
Simplify: 82/3
Solution
√3 √
82/3 = 82 = 64 = 4
3
(√ )2
8 = 22 = 4.
3
Or,
Example √
8
Simplify: 1/2
2
Answer
2
. . . . . .

39. Fact (Limits of exponential
functions) y
.
. = (= 2()1/3)x3)x
y . 1/=x(2/
y
y . . = (. /10)10x = 2x. =
y y = x . 3x y
y y
1 . =
If a > 1, then
lim ax = ∞ and
x→∞
lim ax = 0
x→−∞
If 0 < a < 1, then
lim ax = 0 and y
. =
x→∞
lim ax = ∞ . x
.
x→−∞
. . . . . .

40. Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential function
Compound Interest
The number e
A limit
. . . . . .

41. Compounded Interest
Question
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have
After one year?
After two years?
after t years?
. . . . . .

42. Compounded Interest
Question
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100 + 10% = $110
. . . . . .

43. Compounded Interest
Question
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100 + 10% = $110
$110 + 10% = $110 + $11 = $121
. . . . . .

44. Compounded Interest
Question
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100 + 10% = $110
$110 + 10% = $110 + $11 = $121
$100(1.1)t .
. . . . . .

45. Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
. . . . . .

46. Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100(1.025)4 = $110.38,
. . . . . .

47. Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !
. . . . . .

48. Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !
$100(1.025)8 = $121.84
. . . . . .

49. Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100(1.025)4 = $110.38, not $100(1.1)4 !
$100(1.025)8 = $121.84
$100(1.025)4t .
. . . . . .

50. Compounded Interest: monthly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded twelve times a year. How much do you have after t
years?
. . . . . .

51. Compounded Interest: monthly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded twelve times a year. How much do you have after t
years?
Answer
$100(1 + 10%/12)12t
. . . . . .

52. Compounded Interest: general
Question
Suppose you save P at interest rate r, with interest compounded n
times a year. How much do you have after t years?
. . . . . .

53. Compounded Interest: general
Question
Suppose you save P at interest rate r, with interest compounded n
times a year. How much do you have after t years?
Answer
( r )nt
B(t) = P 1 +
n
. . . . . .

54. Compounded Interest: continuous
Question
Suppose you save P at interest rate r, with interest compounded
every instant. How much do you have after t years?
. . . . . .

55. Compounded Interest: continuous
Question
Suppose you save P at interest rate r, with interest compounded
every instant. How much do you have after t years?
Answer
( ( )
r )nt 1 rnt
B(t) = lim P 1 + = lim P 1 +
n→∞ n n→∞ n
[ ( )n ]rt
1
= P lim 1 +
n→∞ n
independent of P, r, or t
. . . . . .

56. The magic number
Definition
( )n
1
e = lim 1+
n→∞ n
. . . . . .

57. The magic number
Definition
( )n
1
e = lim 1+
n→∞ n
So now continuously-compounded interest can be expressed as
B(t) = Pert .
. . . . . .

58. Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
. . . . . .

59. Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037
. . . . . .

60. Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
. . . . . .

61. Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
. . . . . .

62. Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
. . . . . .

63. Existence of e
See Appendix B
( )n
1
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
106 2.71828
. . . . . .

64. Existence of e
See Appendix B
( )n
1
We can experimentally n 1+
n
verify that this number
1 2
exists and is
2 2.25
e ≈ 2.718281828459045 . . . 3 2.37037
10 2.59374
100 2.70481
1000 2.71692
106 2.71828
. . . . . .

65. Existence of e
See Appendix B
( )n
1
We can experimentally n 1+
n
verify that this number
1 2
exists and is
2 2.25
e ≈ 2.718281828459045 . . . 3 2.37037
10 2.59374
e is irrational 100 2.70481
1000 2.71692
106 2.71828
. . . . . .

66. Existence of e
See Appendix B
( )n
1
We can experimentally n 1+
n
verify that this number
1 2
exists and is
2 2.25
e ≈ 2.718281828459045 . . . 3 2.37037
10 2.59374
e is irrational 100 2.70481
e is transcendental 1000 2.71692
106 2.71828
. . . . . .

67. Meet the Mathematician: Leonhard Euler
Born in Switzerland,
lived in Prussia
(Germany) and Russia
Eyesight trouble all his
life, blind from 1766
onward
Hundreds of
contributions to
calculus, number theory,
graph theory, fluid
mechanics, optics, and
astronomy
Leonhard Paul Euler
Swiss, 1707–1783
. . . . . .

68. A limit
Question
eh − 1
What is lim ?
h→0 h
. . . . . .

69. A limit
Question
eh − 1
What is lim ?
h→0 h
Answer
If h is small enough, e ≈ (1 + h)1/h . So
eh − 1
≈1
h
. . . . . .

70. A limit
Question
eh − 1
What is lim ?
h→0 h
Answer
If h is small enough, e ≈ (1 + h)1/h . So
eh − 1
≈1
h
eh − 1
In fact, lim = 1.
h→0 h
This can be used to characterize e:
2h − 1 3h − 1
lim = 0.693 · · · < 1 and lim = 1.099 · · · < 1
h→0 h h→0 h
. . . . . .