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Section	3.1
Exponential	Functions

  V63.0121.034, Calculus	I



     October	19, 2009




                        .    .   .   .   .   .
Outline



  Definition	of	exponential	functions


  Properties	of	exponential	Functions


  The	number e and	the	natural	exponential	function
     Compound	Interest
     The	number e
     A limit




                                            .   .     .   .   .   .
Definition
If a is	a	real	number	and n is	a	positive	whole	number, then

                        an = a · a · · · · · a
                                  n factors




                                                 .   .   .   .   .   .
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




                                                  .   .   .   .   .   .
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.




                                          .   .      .   .   .   .
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




                                                              .     .      .     .     .   .
Let’s	be	conventional



      The	desire	that	these	properties	remain	true	gives	us
      conventions	for ax when x is	not	a	positive	whole	number.




                                            .    .   .    .   .   .
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
                                         !




                                                    .   .   .   .   .   .
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.




                                                      .   .   .   .   .   .
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).




                                                      .   .   .   .   .   .
Conventions	for	negative	exponents

   If n ≥ 0, we	want

                       an · a−n = an+(−n) = a0 = 1
                               !




                                               .     .   .   .   .   .
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




                                                   .   .   .   .   .   .
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


                                                   .   .   .    .   .      .
Conventions	for	fractional	exponents


   If q is	a	positive	integer, we	want

                             (a1/q )q = a1 = a
                                     !




                                                 .   .   .   .   .   .
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.




                                                     .   .   .   .     .   .
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




                                                     .   .   .   .     .   .
Conventions	for	irrational	powers



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




                                               .   .   .   .   .   .
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




                                                 .   .   .   .   .   .
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 .




                                                  .   .    .   .    .      .
Graphs	of	various	exponential	functions
                          y
                          .




                           .                              x
                                                          .

                                   .      .   .   .   .       .
Graphs	of	various	exponential	functions
                          y
                          .




                                                          . = 1x
                                                          y

                           .                              x
                                                          .

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




                                                               . = 1x
                                                               y

                           .                                   x
                                                               .

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




                                                           . = 1x
                                                           y

                           .                               x
                                                           .

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




                                                           . = 1x
                                                           y

                           .                               x
                                                           .

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




                                                            . = 1x
                                                            y

                           .                                x
                                                            .

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




                                                            . = 1x
                                                            y

                           .                                x
                                                            .

                                    .     .   .    .   .        .
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
                                                            .

                                    .     .   .    .   .        .
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
                                                                  .

                                         .   .    .      .   .        .
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
                                                                     .

                                            .   .    .      .   .        .
Outline



  Definition	of	exponential	functions


  Properties	of	exponential	Functions


  The	number e and	the	natural	exponential	function
     Compound	Interest
     The	number e
     A limit




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




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

Solution
               √
               3          √
    8 2 /3 =       82 =
                          3
                              64 = 4




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

Solution
            √3   √
    82/3 = 82 = 64 = 4
                 3

        (√ )2
            8 = 22 = 4.
          3
    Or,




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

Solution
            √3   √
    82/3 = 82 = 64 = 4
                 3

        (√ )2
            8 = 22 = 4.
          3
    Or,


Example √
                 8
Simplify:       1/2
            2




                          .   .   .   .   .   .
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



                          .   .   .   .   .   .
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→−∞




                                           .       .        .   .   .   .
Outline



  Definition	of	exponential	functions


  Properties	of	exponential	Functions


  The	number e and	the	natural	exponential	function
     Compound	Interest
     The	number e
     A limit




                                            .   .     .   .   .   .
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?




                                             .   .   .    .     .   .
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




                                             .   .   .    .     .   .
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




                                             .   .   .    .     .   .
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 .



                                             .   .   .    .     .   .
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?




                                              .   .   .    .     .   .
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,




                                              .   .   .    .     .   .
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 !




                                              .   .   .    .     .   .
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




                                              .   .   .    .     .   .
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 .



                                              .   .   .    .     .   .
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?




                                          .   .   .    .   .     .
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




                                          .   .   .    .   .     .
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?




                                             .    .   .    .   .      .
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




                                              .   .   .    .   .      .
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?




                                             .    .   .    .   .    .
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




                                                 .   .   .   .   .   .
The	magic	number



  Definition
                             (          )n
                                    1
                   e = lim       1+
                      n→∞           n




                                             .   .   .   .   .   .
The	magic	number



  Definition
                                 (          )n
                                        1
                       e = lim       1+
                           n→∞          n

  So	now	continuously-compounded	interest	can	be	expressed	as

                           B(t) = Pert .




                                                 .   .   .   .   .   .
Existence	of e
See	Appendix	B




                         (           )n
                                1
                 n           1+
                                n
                 1       2
                 2       2.25




                 .   .       .   .        .   .
Existence	of e
See	Appendix	B




                         (           )n
                                1
                 n           1+
                                n
                 1       2
                 2       2.25
                 3       2.37037




                 .   .       .   .        .   .
Existence	of e
See	Appendix	B




                          (           )n
                                 1
                 n            1+
                                 n
                 1        2
                 2        2.25
                 3        2.37037
                 10       2.59374




                 .    .       .   .        .   .
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




                 .     .       .   .        .   .
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




                 .      .       .   .        .   .
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




                 .      .       .   .        .   .
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




                                      .      .       .   .        .   .
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




                                      .      .       .   .        .   .
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




                                      .      .       .   .        .   .
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

                                 .    .   .   .       .   .
A limit

   Question
              eh − 1
   What	is lim       ?
          h→0    h




                         .   .   .   .   .   .
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




                                                      .   .   .   .   .   .
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

                                                      .   .   .   .   .   .

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Lesson 14: Exponential Functions

  • 1. Section 3.1 Exponential Functions V63.0121.034, Calculus I October 19, 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 . . . . . . .
  • 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 . . . . . .