Quadratic vs linear function

  • #1
Mod note: Changed thread title from "exponential vs linear function" to "quadratic vs linear function.
basic question, won't give the full question as i don't need it solved for me, but rather need a conceptual overview of HOW to solve so i can do it myself


given two functions
A f(n)= c * n
B f(n)= c * n^2

i need to figure out when equation b is greater than equation a for the same values of n (when it is more efficent)

i tried some simple things that got me nowhere and i need some help

i found the point of intersection of the two functions but that doesn't help as the exponential function still grows faster

i tried differentiating and finding when the rate of change is equal (for n), that value was 40500
i'm not sure how that helps me
 
Last edited by a moderator:

Answers and Replies

  • #2
member 587159
basic question, won't give the full question as i don't need it solved for me, but rather need a conceptual overview of HOW to solve so i can do it myself

given two functions
A f(n)= c * n
B f(n)= c * n^2

i need to figure out when equation b is greater than equation a for the same values of n (when it is more efficent)

i tried some simple things that got me nowhere and i need some help

i found the point of intersection of the two functions but that doesn't help as the exponential function still grows faster

i tried differentiating and finding when the rate of change is equal (for n), that value was 40500
i'm not sure how that helps me
I don't see an exponential function?
 
  • #3
i meant function B, where the degree of the function =2

sorry if i said that wrong
 
  • #4
i assumed this would be a precalc question, if i'm incorrect in that regard, please move this to the appropriate subforum
 
  • #5
member 587159
Well, define:

f(n) = cn
g(n) =cn²

You can use: f>g <=> f-g>0
I think you should be able to continue now on your own.
 
Last edited by a moderator:
  • #6
34,688
6,394
Mod note: Changed thread title from "exponential vs linear function" to "quadratic vs linear function.
basic question, won't give the full question as i don't need it solved for me, but rather need a conceptual overview of HOW to solve so i can do it myself


given two functions
A f(n)= c * n
B f(n)= c * n^2

i need to figure out when equation b is greater than equation a for the same values of n (when it is more efficent)

i tried some simple things that got me nowhere and i need some help

i found the point of intersection of the two functions but that doesn't help as the exponential function still grows faster
As already noted by another member, neither of these function is an exponential function. The second function is a quadratic function. BTW, you shouldn't give both functions the same name: f.
quicksilver123 said:
i tried differentiating and finding when the rate of change is equal (for n), that value was 40500
The graphs of the two functions intersect at (1, c). The slope of the first function is always c. The slope of the second function is 2cn, so that should give you an idea of how the graphs are different.
quicksilver123 said:
i'm not sure how that helps me
I don't see how 40500 plays a role here at all. You don't need concepts from calculus to answer your question. Just a graph of the two functions would give you an idea of where cn2 > cn.
 
  • #7
ok i done goofed
here's a different and more accurate set of functions

f(x)=0.002n^2
g(x)=700n

i need to find out when g(x) is larger than f(x) for identical values of x


i tried to be intentionally vague to avoid having the question outright answered but i didn't state things properly. my bad.
 
  • #8
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,358
1,031
ok i done goofed
here's a different and more accurate set of functions

f(x)=0.002n^2
g(x)=700n

i need to find out when g(x) is larger than f(x) for identical values of x

i tried to be intentionally vague to avoid having the question outright answered but i didn't state things properly. my bad.
I suppose you mean:
f(x)=0.002x2
g(x)=700x
 
  • #9
member 587159
ok i done goofed
here's a different and more accurate set of functions

f(x)=0.002n^2
g(x)=700n

i need to find out when g(x) is larger than f(x) for identical values of x


i tried to be intentionally vague to avoid having the question outright answered but i didn't state things properly. my bad.
1) g(x) is a function in x. This implies that what you have written is a constant function. Probably, you meant g(n)
2) In the original question, you did not mention that the c in f(n) and g(n) were different.

Now, let's get started.

We have 2 continuous functions f and g.

You want to know the n values of f(n) and g(n) for which: f(n) > g(n)

Thus: f(n) > g(n) <=> f(n) -g(n) > 0
Try to use this.
 
  • #10
It's when g(n) is bigger than f(n), so when the linear is bigger than the quadratic, the linear has a greater c value.

I can use the above method for trial and error but I was wondering if there was a method to generate an exact value of x at which g(x) is greater than f(x).
Hence my misguided attempt to find the derivative. Please ignore the values of the derivative that I have in my first post.

If I was to grind it out via trial and error there would be no method and thus no reason for my thread.
 
  • #11
Sorry if my previous sounded ungrateful, I just want to know how to compute this without using trial and error in order to learn something.
 
  • #12
35,268
11,535
You can solve f(x)=g(x) for x. For values smaller than that, the linear function is larger, for values larger than that the quadratic function is larger (assuming both have a positive coefficient).

Alternatively, simplify f(x)<g(x) to get the region where f(x) is smaller than g(x) - nearly the same calculation, but needs more care in some aspects.
 
  • #14
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,358
1,031
It's when g(n) is bigger than f(n), so when the linear is bigger than the quadratic, the linear has a greater c value.

I can use the above method for trial and error but I was wondering if there was a method to generate an exact value of x at which g(x) is greater than f(x).
Hence my misguided attempt to find the derivative. Please ignore the values of the derivative that I have in my first post.

If I was to grind it out via trial and error there would be no method and thus no reason for my thread.
Actually, trial and error is a method. In this case, it might be good as a method for showing how some other methods work. However, ...

In post #5, @Math_QED gave the following suggestion.
...

Thus: f(n) > g(n) <=> f(n) -g(n) > 0
Try to use this.
That can lead to the answer, even though you have corrected the problem as follows:
i need to find out when g(x) is larger than f(x) for identical values of x.
So, if g(x) > f(x), then 0 > f(x) - g(x) .

In other words, f(x) - g(x) < 0 .

There are several ways to solve this.

A way I like is to recognize that the function f(x) - g(x) is continuous. So as x increases, f(x) - g(x) Cannot change sign without passing through zero.

So solve for f(x) - g(x) = 0 . Then check at "test points".

By the way: You are at the point of having 100 posts on PF.
  1. Congratulations.
  2. You should learn to use the reply feature so readers know which post you are replying to.
  3. You should learn to use the superscript & subscript features. X2 X2 in the darker blue banner above the message composing box.
  4. Also in that banner the ∑ (Sigma) character will give you access to many useful symbols.
 
  • #15
is there a button to quote/reply to a post or do i have to use code?

i'm familiar with using sub/superscript
 
  • #16
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,358
1,031
is there a button to quote/reply to a post or do i have to use code?

i'm familiar with using sub/superscript
It's in the lower right hand corner of the following image.

upload_2016-5-28_21-20-7.png
 
  • #17
Svein
Science Advisor
Insights Author
2,129
687
Mod note: Changed thread title from "exponential vs linear function" to "quadratic vs linear function.
basic question, won't give the full question as i don't need it solved for me, but rather need a conceptual overview of HOW to solve so i can do it myself


given two functions
A f(n)= c * n
B f(n)= c * n^2

i need to figure out when equation b is greater than equation a for the same values of n (when it is more efficent)

i tried some simple things that got me nowhere and i need some help

i found the point of intersection of the two functions but that doesn't help as the exponential function still grows faster

i tried differentiating and finding when the rate of change is equal (for n), that value was 40500
i'm not sure how that helps me
Assuming n≠0 and a(n) = c⋅n; b(n) = c⋅n2, then b(n) > a(n) ⇔ b(n)/a(n) >1. Since b(n)/a(n) = n (continue yourself).
 

Related Threads on Quadratic vs linear function

Replies
6
Views
664
  • Last Post
Replies
3
Views
660
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
13
Views
3K
  • Last Post
Replies
9
Views
4K
  • Last Post
Replies
1
Views
1K
Replies
3
Views
504
  • Last Post
Replies
1
Views
6K
  • Last Post
Replies
1
Views
1K
Replies
1
Views
7K
Top