Exponential Functions - Lecture 33 Section...
Transcript of Exponential Functions - Lecture 33 Section...
Exponential FunctionsLecture 33Section 4.1
Robb T. Koether
Hampden-Sydney College
Mon, Mar 27, 2017
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 1 / 16
Reminder
ReminderTest #3 is this Friday, March 31.
It will cover Chapter 3.Be there.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 2 / 16
Reminder
ReminderTest #3 is this Friday, March 31.It will cover Chapter 3.
Be there.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 2 / 16
Reminder
ReminderTest #3 is this Friday, March 31.It will cover Chapter 3.Be there.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 2 / 16
Objectives
ObjectivesLearn (or review) the properties of exponential expressions.Learn the properties of exponential functions.Learn about the “natural” base.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 3 / 16
Exponential Expressions and Functions
Definition (Integer Exponential Expression)Let b be a positive real number and let n be a positive integer. Then
bn = b · b · b · · · b︸ ︷︷ ︸n factors
Also, b0 = 1 and
b−n =1bn .
The number b is called the base of the expression. The number n iscalled the exponent.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 4 / 16
Exponential Expressions and Functions
Definition (Rational Exponential Expression)Let b be a positive real number and let n
m be a rational number. Then
bn/m =(
m√
b)n
=m√
bn.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 5 / 16
Properties of Exponential Expressions
Properties of Exponential ExpressionsThe following properties hold for all bases a,b > 0 and all exponentsx , y .
Equality: if b 6= 1, then bx = by if and only if x = y .Products (same base): bxby = bx+y .Products (same exponent): axbx = (ab)x .
Quotients (same base):bx
by = bx−y .
Quotients (same exponent):ax
bx =(a
b
)x.
Powers: (bx)y= bxy .
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 6 / 16
Properties of Exponential Expressions
Properties of Exponential ExpressionsThe following properties hold for all bases a,b > 0 and all exponentsx , y .
Equality: if b 6= 1, then bx = by if and only if x = y .
Products (same base): bxby = bx+y .Products (same exponent): axbx = (ab)x .
Quotients (same base):bx
by = bx−y .
Quotients (same exponent):ax
bx =(a
b
)x.
Powers: (bx)y= bxy .
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 6 / 16
Properties of Exponential Expressions
Properties of Exponential ExpressionsThe following properties hold for all bases a,b > 0 and all exponentsx , y .
Equality: if b 6= 1, then bx = by if and only if x = y .Products (same base): bxby = bx+y .
Products (same exponent): axbx = (ab)x .
Quotients (same base):bx
by = bx−y .
Quotients (same exponent):ax
bx =(a
b
)x.
Powers: (bx)y= bxy .
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 6 / 16
Properties of Exponential Expressions
Properties of Exponential ExpressionsThe following properties hold for all bases a,b > 0 and all exponentsx , y .
Equality: if b 6= 1, then bx = by if and only if x = y .Products (same base): bxby = bx+y .Products (same exponent): axbx = (ab)x .
Quotients (same base):bx
by = bx−y .
Quotients (same exponent):ax
bx =(a
b
)x.
Powers: (bx)y= bxy .
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 6 / 16
Properties of Exponential Expressions
Properties of Exponential ExpressionsThe following properties hold for all bases a,b > 0 and all exponentsx , y .
Equality: if b 6= 1, then bx = by if and only if x = y .Products (same base): bxby = bx+y .Products (same exponent): axbx = (ab)x .
Quotients (same base):bx
by = bx−y .
Quotients (same exponent):ax
bx =(a
b
)x.
Powers: (bx)y= bxy .
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 6 / 16
Properties of Exponential Expressions
Properties of Exponential ExpressionsThe following properties hold for all bases a,b > 0 and all exponentsx , y .
Equality: if b 6= 1, then bx = by if and only if x = y .Products (same base): bxby = bx+y .Products (same exponent): axbx = (ab)x .
Quotients (same base):bx
by = bx−y .
Quotients (same exponent):ax
bx =(a
b
)x.
Powers: (bx)y= bxy .
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 6 / 16
Properties of Exponential Expressions
Properties of Exponential ExpressionsThe following properties hold for all bases a,b > 0 and all exponentsx , y .
Equality: if b 6= 1, then bx = by if and only if x = y .Products (same base): bxby = bx+y .Products (same exponent): axbx = (ab)x .
Quotients (same base):bx
by = bx−y .
Quotients (same exponent):ax
bx =(a
b
)x.
Powers: (bx)y= bxy .
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 6 / 16
Exponential Expressions and Functions
Definition (Exponential Function)An exponential function is a function of the form
f (x) = bx
for some base b > 0.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 7 / 16
Properties of Exponential Functions
Properties of Exponential FunctionsThe following properties hold for all exponential functions.
bx > 0 for all x .The x-axis is a horizontal asymptote of the graph.The point (0,1) lies on the graph.If b > 1, then
limx→+∞
bx = +∞ and limx→−∞
bx = 0.
If b < 1, then
limx→+∞
bx = 0 and limx→−∞
bx = +∞.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 8 / 16
Properties of Exponential Functions
Properties of Exponential FunctionsThe following properties hold for all exponential functions.
bx > 0 for all x .
The x-axis is a horizontal asymptote of the graph.The point (0,1) lies on the graph.If b > 1, then
limx→+∞
bx = +∞ and limx→−∞
bx = 0.
If b < 1, then
limx→+∞
bx = 0 and limx→−∞
bx = +∞.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 8 / 16
Properties of Exponential Functions
Properties of Exponential FunctionsThe following properties hold for all exponential functions.
bx > 0 for all x .The x-axis is a horizontal asymptote of the graph.
The point (0,1) lies on the graph.If b > 1, then
limx→+∞
bx = +∞ and limx→−∞
bx = 0.
If b < 1, then
limx→+∞
bx = 0 and limx→−∞
bx = +∞.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 8 / 16
Properties of Exponential Functions
Properties of Exponential FunctionsThe following properties hold for all exponential functions.
bx > 0 for all x .The x-axis is a horizontal asymptote of the graph.The point (0,1) lies on the graph.
If b > 1, then
limx→+∞
bx = +∞ and limx→−∞
bx = 0.
If b < 1, then
limx→+∞
bx = 0 and limx→−∞
bx = +∞.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 8 / 16
Properties of Exponential Functions
Properties of Exponential FunctionsThe following properties hold for all exponential functions.
bx > 0 for all x .The x-axis is a horizontal asymptote of the graph.The point (0,1) lies on the graph.If b > 1, then
limx→+∞
bx = +∞ and limx→−∞
bx = 0.
If b < 1, then
limx→+∞
bx = 0 and limx→−∞
bx = +∞.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 8 / 16
Properties of Exponential Functions
Properties of Exponential FunctionsThe following properties hold for all exponential functions.
bx > 0 for all x .The x-axis is a horizontal asymptote of the graph.The point (0,1) lies on the graph.If b > 1, then
limx→+∞
bx = +∞ and limx→−∞
bx = 0.
If b < 1, then
limx→+∞
bx = 0 and limx→−∞
bx = +∞.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 8 / 16
Graph of an Exponential Function (b > 1)
The graph of f (x) = 2x
-3 -2 -1 1 2 3
2
4
6
8
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 9 / 16
Graph of an Exponential Function (b < 1)
The graph of f (x) =(
12
)x
-3 -2 -1 1 2 3
2
4
6
8
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 10 / 16
Graph of an Exponential Function (b < 1)
The graph of f (x) =(
12
)x
-3 -2 -1 0 1 2 3
2
4
6
8
10
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 11 / 16
The Natural Base e
The Natural Base eLet f (x) = bx for some base b > 0.What is f ′(x)?
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 12 / 16
The Natural Base e
The Natural Base eIf we apply the definition, we get
f ′(x) =
limh→0
f (x + h)− f (x)h
= limh→0
bx+h − bx
h
= limh→0
bxbh − bx
h
= limh→0
bx(
bh − 1h
)= bx · lim
h→0
bh − 1h
.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 13 / 16
The Natural Base e
The Natural Base eIf we apply the definition, we get
f ′(x) = limh→0
f (x + h)− f (x)h
= limh→0
bx+h − bx
h
= limh→0
bxbh − bx
h
= limh→0
bx(
bh − 1h
)= bx · lim
h→0
bh − 1h
.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 13 / 16
The Natural Base e
The Natural Base eIf we apply the definition, we get
f ′(x) = limh→0
f (x + h)− f (x)h
= limh→0
bx+h − bx
h
= limh→0
bxbh − bx
h
= limh→0
bx(
bh − 1h
)= bx · lim
h→0
bh − 1h
.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 13 / 16
The Natural Base e
The Natural Base eIf we apply the definition, we get
f ′(x) = limh→0
f (x + h)− f (x)h
= limh→0
bx+h − bx
h
= limh→0
bxbh − bx
h
= limh→0
bx(
bh − 1h
)= bx · lim
h→0
bh − 1h
.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 13 / 16
The Natural Base e
The Natural Base eIf we apply the definition, we get
f ′(x) = limh→0
f (x + h)− f (x)h
= limh→0
bx+h − bx
h
= limh→0
bxbh − bx
h
= limh→0
bx(
bh − 1h
)
= bx · limh→0
bh − 1h
.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 13 / 16
The Natural Base e
The Natural Base eIf we apply the definition, we get
f ′(x) = limh→0
f (x + h)− f (x)h
= limh→0
bx+h − bx
h
= limh→0
bxbh − bx
h
= limh→0
bx(
bh − 1h
)= bx · lim
h→0
bh − 1h
.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 13 / 16
The Natural Base e
The Natural Base e
Note that limh→0
bh − 1h
is a constant.
For the right choice of b, that constant will be 1.That choice is approximately 2.71828. . . .We call that number e, the natural base.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 14 / 16
The Natural Base e
The Natural Base e
Note that limh→0
bh − 1h
is a constant.
For the right choice of b, that constant will be 1.
That choice is approximately 2.71828. . . .We call that number e, the natural base.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 14 / 16
The Natural Base e
The Natural Base e
Note that limh→0
bh − 1h
is a constant.
For the right choice of b, that constant will be 1.That choice is approximately 2.71828. . . .
We call that number e, the natural base.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 14 / 16
The Natural Base e
The Natural Base e
Note that limh→0
bh − 1h
is a constant.
For the right choice of b, that constant will be 1.That choice is approximately 2.71828. . . .We call that number e, the natural base.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 14 / 16
Compound Interest
Compound InterestLet P be the present value of an investment, t the duration (in years) ofthe investment, r the annual interest rate, k the number ofcompounding periods per year, and B(t) the future value after t years.Then
B(t) = P(
1 +rk
)kt.
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 15 / 16
Continuous Compounding
Continuous CompoundingIf the interest is compounded continuously, then the future value is
B(t) = Pert .
Robb T. Koether (Hampden-Sydney College) Exponential Functions Mon, Mar 27, 2017 16 / 16