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Values of 2 x For a list of rational numbers approximating 2
x 1.4 1.41 1.414 1.4142 1.41421 1.414213
2 x 2.639 2.65737 2.66475 2.665119 2.665138 2.665143

Bacterial growth

Suppose a particular population of bacteria is known to double in size every 4 hours. If a culture starts with 1000 bacteria, the number of bacteria after 4 hours is n ( 4 ) = 1000 · 2 . The number of bacteria after 8 hours is n ( 8 ) = n ( 4 ) · 2 = 1000 · 2 2 . In general, the number of bacteria after 4 m hours is n ( 4 m ) = 1000 · 2 m . Letting t = 4 m , we see that the number of bacteria after t hours is n ( t ) = 1000 · 2 t / 4 . Find the number of bacteria after 6 hours, 10 hours, and 24 hours.

The number of bacteria after 6 hours is given by n ( 6 ) = 1000 · 2 6 / 4 2828 bacteria. The number of bacteria after 10 hours is given by n ( 10 ) = 1000 · 2 10 / 4 5657 bacteria. The number of bacteria after 24 hours is given by n ( 24 ) = 1000 · 2 6 = 64,000 bacteria.

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Given the exponential function f ( x ) = 100 · 3 x / 2 , evaluate f ( 4 ) and f ( 10 ) .

f ( 4 ) = 900 ; f ( 10 ) = 24 , 300 .

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Go to World Population Balance for another example of exponential population growth.

Graphing exponential functions

For any base b > 0 , b 1 , the exponential function f ( x ) = b x is defined for all real numbers x and b x > 0 . Therefore, the domain of f ( x ) = b x is ( , ) and the range is ( 0 , ) . To graph b x , we note that for b > 1 , b x is increasing on ( , ) and b x as x , whereas b x 0 as x . On the other hand, if 0 < b < 1 , f ( x ) = b x is decreasing on ( , ) and b x 0 as x whereas b x as x ( [link] ).

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 four functions. The first function is “f(x) = 2 to the power of x”, an increasing curved function, which starts slightly above the x axis and begins increasing. The second function is “f(x) = 4 to the power of x”, an increasing curved function, which starts slightly above the x axis and begins increasing rapidly, more rapidly than the first function. The third function is “f(x) = (1/2) to the power of x”, a decreasing curved function with decreases until it gets close to the x axis without touching it. The third function is “f(x) = (1/4) to the power of x”, a decreasing curved function with decreases until it gets close to the x axis without touching it. It decrases at a faster rate than the third function.
If b > 1 , then b x is increasing on ( , ) . If 0 < b < 1 , then b x is decreasing on ( , ) .

Visit this site for more exploration of the graphs of exponential functions.

Note that exponential functions satisfy the general laws of exponents. To remind you of these laws, we state them as rules.

Rule: laws of exponents

For any constants a > 0 , b > 0 , and for all x and y ,

  1. b x · b y = b x + y
  2. b x b y = b x y
  3. ( b x ) y = b x y
  4. ( a b ) x = a x b x
  5. a x b x = ( a b ) x

Using the laws of exponents

Use the laws of exponents to simplify each of the following expressions.

  1. ( 2 x 2 / 3 ) 3 ( 4 x −1 / 3 ) 2
  2. ( x 3 y −1 ) 2 ( x y 2 ) −2
  1. We can simplify as follows:
    ( 2 x 2 / 3 ) 3 ( 4 x −1 / 3 ) 2 = 2 3 ( x 2 / 3 ) 3 4 2 ( x −1 / 3 ) 2 = 8 x 2 16 x −2 / 3 = x 2 x 2 / 3 2 = x 8 / 3 2 .
  2. We can simplify as follows:
    ( x 3 y −1 ) 2 ( x y 2 ) −2 = ( x 3 ) 2 ( y −1 ) 2 x −2 ( y 2 ) −2 = x 6 y −2 x −2 y −4 = x 6 x 2 y −2 y 4 = x 8 y 2 .
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Use the laws of exponents to simplify ( 6 x −3 y 2 ) / ( 12 x −4 y 5 ) .

x / ( 2 y 3 )

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The number e

A special type of exponential function appears frequently in real-world applications. To describe it, consider the following example of exponential growth, which arises from compounding interest in a savings account. Suppose a person invests P dollars in a savings account with an annual interest rate r , compounded annually. The amount of money after 1 year is

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

The amount of money after 2 years is

A ( 2 ) = A ( 1 ) + r A ( 1 ) = P ( 1 + r ) + r P ( 1 + r ) = P ( 1 + r ) 2 .

More generally, the amount after t years is

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

If the money is compounded 2 times per year, the amount of money after half a year is

A ( 1 2 ) = P + ( r 2 ) P = P ( 1 + ( r 2 ) ) .

The amount of money after 1 year is

A ( 1 ) = A ( 1 2 ) + ( r 2 ) A ( 1 2 ) = P ( 1 + r 2 ) + r 2 ( P ( 1 + r 2 ) ) = P ( 1 + r 2 ) 2 .

After t years, the amount of money in the account is

A ( t ) = P ( 1 + r 2 ) 2 t .

More generally, if the money is compounded n times per year, the amount of money in the account after t years is given by the function

Questions & Answers

What is a independent variable
Sifiso Reply
a variable that does not depend on another.
Andrew
solve number one step by step
bil Reply
x-xcosx/sinsq.3x
Hasnain
x-xcosx/sin^23x
Hasnain
how to prove 1-sinx/cos x= cos x/-1+sin x?
Rochel Reply
1-sin x/cos x= cos x/-1+sin x
Rochel
how to prove 1-sun x/cos x= cos x / -1+sin x?
Rochel
how to prove tan^2 x=csc^2 x tan^2 x-1?
Rochel Reply
divide by tan^2 x giving 1=csc^2 x -1/tan^2 x, rewrite as: 1=1/sin^2 x -cos^2 x/sin^2 x, multiply by sin^2 x giving: sin^2 x=1-cos^2x. rewrite as the familiar sin^2 x + cos^2x=1 QED
Barnabas
how to prove sin x - sin x cos^2 x=sin^3x?
Rochel Reply
sin x - sin x cos^2 x sin x (1-cos^2 x) note the identity:sin^2 x + cos^2 x = 1 thus, sin^2 x = 1 - cos^2 x now substitute this into the above: sin x (sin^2 x), now multiply, yielding: sin^3 x Q.E.D.
Andrew
take sin x common. you are left with 1-cos^2x which is sin^2x. multiply back sinx and you get sin^3x.
navin
Left side=sinx-sinx cos^2x =sinx-sinx(1+sin^2x) =sinx-sinx+sin^3x =sin^3x thats proved.
Alif
how to prove tan^2 x/tan^2 x+1= sin^2 x
Rochel
not a bad question
Salim
what is function.
Nawaz Reply
what is polynomial
Nawaz
an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Alif
a term/algebraic expression raised to a non-negative integer power and a multiple of co-efficient,,,,,, T^n where n is a non-negative,,,,, 4x^2
joe
An expression in which power of all the variables are whole number . such as 2x+3 5 is also a polynomial of degree 0 and can be written as 5x^0
Nawaz
what is hyperbolic function
vector Reply
find volume of solid about y axis and y=x^3, x=0,y=1
amisha Reply
3 pi/5
vector
what is the power rule
Vanessa Reply
Is a rule used to find a derivative. For example the derivative of y(x)= a(x)^n is y'(x)= a*n*x^n-1.
Timothy
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
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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