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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

f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)
Naufal Reply
fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5
diron
pweding paturo nsa calculus?
jimmy
how to use fundamental theorem to solve exponential
JULIA Reply
find the bounded area of the parabola y^2=4x and y=16x
Omar Reply
what is absolute value means?
Geo Reply
Chicken nuggets
Hugh
🐔
MM
🐔🦃 nuggets
MM
(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
Ismael
find integration of loge x
Game Reply
find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
Godwin Reply
how does this work
Brad Reply
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
Cosmos Reply
log tan (x/4+x/2)
Rohan
please answer
Rohan
y=(x^2 + 3x).(eipix)
Claudia
is this a answer
Ismael
A Function F(X)=Sinx+cosx is odd or even?
WIZARD Reply
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
Yuliana Reply
apa itu?
fauzi
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Gerald Reply
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Adegoke
Which kind of pdf do you want bro?
Aftab
hi
Abdul
can I get calculus in pdf
Abdul
explain for me
Usman
How to use it to slove fraction
Tricia Reply
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
Im not in college but this will still help
nothing
how en where can u apply it
Migos
how can we scatch a parabola graph
Dever Reply
Ok
Endalkachew
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
v=(x,y) وu=(x,y ) ∂u/∂x* ∂x/∂u +∂v/∂x*∂x/∂v=1
log tan (x/4+x/2)
Rohan
what is the procedures in solving number 1?
Vier 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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