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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

what is the power rule
Vanessa Reply
how do i deal with infinity in limits?
Itumeleng Reply
Add the functions f(x)=7x-x g(x)=5-x
Julius Reply
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
Cabdalla Reply
difference btwn domain co- domain and range
Cabdalla
x
Verna
The set of inputs of a function. x goes in the function, y comes out.
Verna
where u from verna
Arfan
If you differentiate then answer is not x
Raymond
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
Champro
what is functions
mahin Reply
give different types of functions.
Paul
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
riyad Reply
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz
why do we need to study functions?
abigail Reply
to understand how to model one variable as a direct relationship to another variable
Andrew
integrate the root of 1+x²
Rodgers Reply
use the substitution t=1+x. dt=dx √(1+x)dx = √tdt = t^1/2 dt integral is then = t^(1/2 + 1) / (1/2 + 1) + C = (2/3) t^(3/2) + C substitute back t=1+x = (2/3) (1+x)^(3/2) + C
navin
find the nth differential coefficient of cosx.cos2x.cos3x
Sudhanayaki Reply
determine the inverse(one-to-one function) of f(x)=x(cube)+4 and draw the graph if the function and its inverse
Crystal Reply
f(x) = x^3 + 4, to find inverse switch x and you and isolate y: x = y^3 + 4 x -4 = y^3 (x-4)^1/3 = y = f^-1(x)
Andrew
in the example exercise how does it go from -4 +- squareroot(8)/-4 to -4 +- 2squareroot(2)/-4 what is the process of pulling out the factor like that?
Robert Reply
can you please post the question again here so I can see what your talking about
Andrew
√(8) =√(4x2) =√4 x √2 2 √2 hope this helps. from the surds theory a^c x b^c = (ab)^c
Barnabas
564356
Myong
can you determine whether f(x)=x(cube) +4 is a one to one function
Crystal
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
can you show the steps from going from 3/(x-2)= y to x= 3/y +2 I'm confused as to how y ends up as the divisor
Robert Reply
step 1: take reciprocal of both sides (x-2)/3 = 1/y step 2: multiply both sides by 3 x-2 = 3/y step 3: add 2 to both sides x = 3/y + 2 ps nice farcry 3 background!
Andrew
first you cross multiply and get y(x-2)=3 then apply distribution and the left side of the equation such as yx-2y=3 then you add 2y in both sides of the equation and get yx=3+2y and last divide both sides of the equation by y and you get x=3/y+2
Ioana
Multiply both sides by (x-2) to get 3=y(x-2) Then you can divide both sides by y (it's just a multiplied term now) to get 3/y = (x-2). Since the parentheses aren't doing anything for the right side, you can drop them, and add the 2 to both sides to get 3/y + 2 = x
Melin
thank you ladies and gentlemen I appreciate the help!
Robert
keep practicing and asking questions, practice makes perfect! and be aware that are often different paths to the same answer, so the more you familiarize yourself with these multiple different approaches, the less confused you'll be.
Andrew
please how do I learn integration
aliyu Reply
they are simply "anti-derivatives". so you should first learn how to take derivatives of any given function before going into taking integrals of any given function.
Andrew
best way to learn is always to look into a few basic examples of different kinds of functions, and then if you have any further questions, be sure to state specifically which step in the solution you are not understanding.
Andrew
example 1) say f'(x) = x, f(x) = ? well there is a rule called the 'power rule' which states that if f'(x) = x^n, then f(x) = x^(n+1)/(n+1) so in this case, f(x) = x^2/2
Andrew
great noticeable direction
Isaac
limit x tend to infinite xcos(π/2x)*sin(π/4x)
Abhijeet Reply
can you give me a problem for function. a trigonometric one
geovanni Reply
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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