exponential functions - lecture 33 section...
TRANSCRIPT
![Page 1: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/1.jpg)
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
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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
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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
![Page 4: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/4.jpg)
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
![Page 5: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/5.jpg)
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
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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
![Page 7: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/7.jpg)
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
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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
![Page 9: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/9.jpg)
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
![Page 10: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/10.jpg)
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
![Page 11: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/11.jpg)
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
![Page 12: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/12.jpg)
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
![Page 13: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/13.jpg)
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
![Page 14: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/14.jpg)
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
![Page 15: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/15.jpg)
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
![Page 16: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/16.jpg)
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
![Page 17: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/17.jpg)
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
![Page 18: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/18.jpg)
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
![Page 19: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/19.jpg)
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
![Page 20: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/20.jpg)
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
![Page 21: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/21.jpg)
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
![Page 22: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/22.jpg)
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
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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
![Page 24: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/24.jpg)
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
![Page 25: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/25.jpg)
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
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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
![Page 27: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/27.jpg)
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
![Page 28: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/28.jpg)
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
![Page 29: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/29.jpg)
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
![Page 30: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/30.jpg)
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
![Page 31: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/31.jpg)
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
![Page 32: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/32.jpg)
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
![Page 33: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/33.jpg)
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
![Page 34: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/34.jpg)
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
![Page 35: Exponential Functions - Lecture 33 Section 4people.hsc.edu/faculty-staff/robbk/Math140/Lectures...Exponential Functions Lecture 33 Section 4.1 Robb T. Koether Hampden-Sydney College](https://reader036.vdocuments.us/reader036/viewer/2022062605/5fccd4c835f04601a140bd26/html5/thumbnails/35.jpg)
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.
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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.
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Continuous Compounding
Continuous CompoundingIf the interest is compounded continuously, then the future value is
B(t) = Pert .
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