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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

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
state and prove L hospital rule
Krishna Reply
I want to know about hospital rule
Faysal
If you tell me how can I Know about engineering math 1( sugh as any lecture or tutorial)
Faysal
I don't know either i am also new,first year college ,taking computer engineer,and.trying to advance learning
Amor
if you want some help on l hospital rule ask me
Jawad
it's spelled hopital
Connor
hi
BERNANDINO
you are correct Connor Angeli, the L'Hospital was the old one but the modern way to say is L 'Hôpital.
Leo
I had no clue this was an online app
Connor
Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In 2012 (t=0), total online holiday sales were $42.3 billion, whereas in 2013 they were $48.1 billion. Find a linear function S that estimates the total online holiday sales in the year t . Interpret the slope of the graph of S . Use part a. to predict the year when online shopping during Christmas will reach $60 billion?
Nguyen Reply
what is the derivative of x= Arc sin (x)^1/2
morfling Reply
y^2 = arcsin(x)
Pitior
x = sin (y^2)
Pitior
differentiate implicitly
Pitior
then solve for dy/dx
Pitior
thank you it was very helpful
morfling
questions solve y=sin x
Obi Reply
Solve it for what?
Tim
you have to apply the function arcsin in both sides and you get arcsin y = acrsin (sin x) the the function arcsin and function sin cancel each other so the ecuation becomes arcsin y = x you can also write x= arcsin y
Ioana
what is the question ? what is the answer?
Suman
there is an equation that should be solve for x
Ioana
ok solve it
Suman
are you saying y is of sin(x) y=sin(x)/sin of both sides to solve for x... therefore y/sin =x
Tyron
or solve for sin(x) via the unit circle
Tyron
what is unit circle
Suman
a circle whose radius is 1.
Darnell
the unit circle is covered in pre cal...and or trigonometry. it is the multipcation table of upper level mathematics.
Tyron
what is function?
Ryan Reply
A set of points in which every x value (domain) corresponds to exactly one y value (range)
Tim
what is lim (x,y)~(0,0) (x/y)
NIKI Reply
limited of x,y at 0,0 is nt defined
Alswell
But using L'Hopitals rule is x=1 is defined
Alswell
Could U explain better boss?
emmanuel
value of (x/y) as (x,y) tends to (0,0) also whats the value of (x+y)/(x^2+y^2) as (x,y) tends to (0,0)
NIKI
can we apply l hospitals rule for function of two variables
NIKI
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
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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