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A ( t ) = P ( 1 + r n ) n t .

What happens as n ? To answer this question, we let m = n / r and write

( 1 + r n ) n t = ( 1 + 1 m ) m r t ,

and examine the behavior of ( 1 + 1 / m ) m as m , using a table of values ( [link] ).

Values of ( 1 + 1 m ) m As m
m 10 100 1000 10,000 100,000 1,000,000
( 1 + 1 m ) m 2.5937 2.7048 2.71692 2.71815 2.718268 2.718280

Looking at this table, it appears that ( 1 + 1 / m ) m is approaching a number between 2.7 and 2.8 as m . In fact, ( 1 + 1 / m ) m does approach some number as m . We call this number e    . To six decimal places of accuracy,

e 2.718282 .

The letter e was first used to represent this number by the Swiss mathematician Leonhard Euler during the 1720s. Although Euler did not discover the number, he showed many important connections between e and logarithmic functions. We still use the notation e today to honor Euler’s work because it appears in many areas of mathematics and because we can use it in many practical applications.

Returning to our savings account example, we can conclude that if a person puts P dollars in an account at an annual interest rate r , compounded continuously, then A ( t ) = P e r t . This function may be familiar. Since functions involving base e arise often in applications, we call the function f ( x ) = e x the natural exponential function    . Not only is this function interesting because of the definition of the number e , but also, as discussed next, its graph has an important property.

Since e > 1 , we know e x is increasing on ( , ) . In [link] , we show a graph of f ( x ) = e x along with a tangent line to the graph of at x = 0 . We give a precise definition of tangent line in the next chapter; but, informally, we say a tangent line to a graph of f at x = a is a line that passes through the point ( a , f ( a ) ) and has the same “slope” as f at that point . The function f ( x ) = e x is the only exponential function b x with tangent line at x = 0 that has a slope of 1. As we see later in the text, having this property makes the natural exponential function the most simple exponential function to use in many instances.

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from 0 to 4. The graph is of the function “f(x) = e to power of x”, an increasing curved function that starts slightly above the x axis. The y intercept is at the point (0, 1). At this point, a line is drawn tangent to the function. This line has the label “slope = 1”.
The graph of f ( x ) = e x has a tangent line with slope 1 at x = 0 .

Compounding interest

Suppose $ 500 is invested in an account at an annual interest rate of r = 5.5 % , compounded continuously.

  1. Let t denote the number of years after the initial investment and A ( t ) denote the amount of money in the account at time t . Find a formula for A ( t ) .
  2. Find the amount of money in the account after 10 years and after 20 years.
  1. If P dollars are invested in an account at an annual interest rate r , compounded continuously, then A ( t ) = P e r t . Here P = $ 500 and r = 0.055 . Therefore, A ( t ) = 500 e 0.055 t .
  2. After 10 years, the amount of money in the account is
    A ( 10 ) = 500 e 0.055 · 10 = 500 e 0.55 $ 866.63 .

    After 20 years, the amount of money in the account is
    A ( 20 ) = 500 e 0.055 · 20 = 500 e 1.1 $ 1 , 502.08 .
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If $ 750 is invested in an account at an annual interest rate of 4 % , compounded continuously, find a formula for the amount of money in the account after t years. Find the amount of money after 30 years.

A ( t ) = 750 e 0.04 t . After 30 years, there will be approximately $ 2 , 490.09 .

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Logarithmic functions

Using our understanding of exponential functions, we can discuss their inverses, which are the logarithmic functions. These come in handy when we need to consider any phenomenon that varies over a wide range of values, such as pH in chemistry or decibels in sound levels.

Questions & Answers

why n does not equal -1
K.kupar Reply
ask a complete question if you want a complete answer.
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
Darnell Reply
proof the formula integration of udv=uv-integration of vdu.?
Bg Reply
Find derivative (2x^3+6xy-4y^2)^2
Rasheed Reply
no x=2 is not a function, as there is nothing that's changing.
Vivek Reply
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
Joys Reply
y=800
Gift
800
Bg
how do u factor the numerator?
Drew Reply
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
volume between cone z=√(x^2+y^2) and plane z=2
Kranthi Reply
answer please?
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Lerato Reply
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
Amna Reply
what is a function? f(x)
Jeremy Reply
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
jon Reply
is x=2 a function?
The
What is limit
MaHeSh Reply
it's the value a function will take while approaching a particular value
Dan
don ger it
Jeremy
what is a limit?
Dlamini
it is the value the function approaches as the input approaches that value.
Andrew
Thanx
Dlamini
Its' complex a limit It's a metrical and topological natural question... approaching means nothing in math
Antonio
is x=2 a function?
The
3y^2*y' + 2xy^3 + 3y^2y'x^2 = 0 sub in x = 2, and y = 1, isolate y'
Andrew Reply
what is implicit of y³+x²y³=5 at (2,1)
Estelita Reply
tel mi about a function. what is it?
Jeremy
A function it's a law, that for each value in the domaon associate a single one in the codomain
Antonio
function is a something which another thing depends upon to take place. Example A son depends on his father. meaning here is the father is function of the son. let the father be y and the son be x. the we say F(X)=Y.
Bg
yes the son on his father
pascal
a function is equivalent to a machine. this machine makes x to create y. thus, y is dependent upon x to be produced. note x is an independent variable
moe
x or y those not matter is just to represent.
Bg
Practice Key Terms 7

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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