SOLUTION: Hello all I would like your help one more time. I've got most of what my teacher asked us to do but I am stuck on the following the task is to: Evaluate the functions for the va

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Question 150910: Hello all I would like your help one more time. I've got most of what my teacher asked us to do but I am stuck on the following the task is to:
Evaluate the functions for the values of x given as 1, 2, 4, 8, and 16. Describe the differences in the rate at which each function changes with increasing values of x. f(x) = x2 - 3x + 2
f(x) = 2x3 + 7x2 - x - 1
f(x) = 10x
f(x) = ln x (I don't know if this is (one n x) or the letter (l)n x, whatever that means)
Any help would be GREATLY appreciated!!
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
# 1





Let's find the function value when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Let's find the function value when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Let's find the function value when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Let's find the function value when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Let's find the function value when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Now let's make a table of the values we just found.





Jump to Top of Page

Table of Values:





xy
10


20


46


842


16210






# 2




Let's find the function value when :


Start with the given equation.


Plug in .


Cube to get .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Let's find the function value when :


Start with the given equation.


Plug in .


Cube to get .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Let's find the function value when :


Start with the given equation.


Plug in .


Cube to get .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Let's find the function value when :


Start with the given equation.


Plug in .


Cube to get .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Let's find the function value when :


Start with the given equation.


Plug in .


Cube to get .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Now let's make a table of the values we just found.



Table of Values:





xy
17


241


4235


81463


169967






# 3




Let's find the function value when :


Start with the given equation.


Plug in .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Let's find the function value when :


Start with the given equation.


Plug in .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Let's find the function value when :


Start with the given equation.


Plug in .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Let's find the function value when :


Start with the given equation.


Plug in .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Let's find the function value when :


Start with the given equation.


Plug in .


Multiply and to get .


Combine like terms.


So if , then .

-------------

Now let's make a table of the values we just found.


Table of Values:





xy
110


220


440


880


16160







# 4

Note: ln(x) also looks like LN(x) (to pronounce it, simply read off the letters "L" "N")

This is the natural log of x. So it is a logarithmic function.

Let's find the y value when


Start with the given equation.


Plug in .


Take the natural log of 1 to get 0


So when , .


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Take the natural log of 2 to get 0.693


So when , .



----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Take the natural log of 4 to get 1.386


So when , .

-------------------------------

Let's find the y value when


Start with the given equation.


Plug in .


Take the natural log of 8 to get 2.079


So when , .


-------------------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Take the natural log of 16 to get 2.773


So when , .


Now let's make a table of the values we just found.



Table of Values:





xy
10


20.693


41.386


82.079


162.773





===========================================================

So to compare the rates, simply compare the tables:

Table for 




xy
10


20


46


842


16210



Table for 





xy
17


241


4235


81463


169967


Table for 





xy
110


220


440


880


16160


Table for 





xy
10


20.693


41.386


82.079


162.773



From the tables, the functions can be sorted from slowest growing to fastest growing like this:

, , , and



So is the slowest growing function and is the fastest growing function.



If all of that is confusing, simply graph all of the functions to get

Graph of (red), (green), (blue), and (purple)

From the graph, we can see that the order from slowest growing to fastest growing is

, , , and

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