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graph/analyze a function of a rational /w complex root?!?
The function is y=x^2/(x^2+3)
First and Second Derivatives
Chart to find intervals of increase/decrease and concavity.
1) Domain
{XeR}
2) Intercepts
If x=0, y=0
(0,0) is the only intercept.
3) Symmetry
(-x,y) = (x,y) therefore, even.
4) Vertical Asymptotes
My question here is, since the "asymptote" would technically be the sqroot of 3i, in my class we don't use complex numbers/roots so would I simply answer this as "does not exist" or saw it equals 0?
5) Horizontal Asymptote
lim x->+- infinity 1
y=1
6) Slant Asymptotes
none
7) First Derivative
dy/dx = 6x/(x^2+3)^2 (quotient rule)
8) Second Derivative
d2y/dx2 = -18(x+1)(x-1)/(x^2+3)^3 (quotient rule + chain rule)
9) Critical numbers
Set first deriv equal to 0, critical number is x=0
10) Intervals of Increase/Decrease
This is where I start having trouble. Now I know that in order to find the values of increase and decrease, you need to use your critical numbers and restrictions. The method given to use to find the intervals is to plug it into a chart and see at which intervals is the graph positive or negative. However, my question is, what would the intervals I test be? Since my restriction is a complex root and we don't work with those in our class; what would I use for my intervals? Simply (-infinity, 0), (0, infinity) ? and my values I would be testing for are 6x, and (x^2+3)^2 ?
11) Concavity
Set second deriv = to 0. Would my x values be +-1 and again, would I use anything for restriction, use 0 as an interval ? or?
12) Intervals of Concavity
This, I also have a similar issue to finding just like intervals of increase/decrease, what intervals would I be using? Do I include anything in intervals that relates to restrictions? or do I simply use (-infinity,-1),(-1,1) and (1,infinity) ? and the binomials/intervals I would be testing these in are, -(x+1), -(x-1), and (x^2+3)^3?
Any help would GREATLY be appreciated, I can't figure out this root issue.
Homework Statement
The function is y=x^2/(x^2+3)
Homework Equations
First and Second Derivatives
Chart to find intervals of increase/decrease and concavity.
The Attempt at a Solution
1) Domain
{XeR}
2) Intercepts
If x=0, y=0
(0,0) is the only intercept.
3) Symmetry
(-x,y) = (x,y) therefore, even.
4) Vertical Asymptotes
My question here is, since the "asymptote" would technically be the sqroot of 3i, in my class we don't use complex numbers/roots so would I simply answer this as "does not exist" or saw it equals 0?
5) Horizontal Asymptote
lim x->+- infinity 1
y=1
6) Slant Asymptotes
none
7) First Derivative
dy/dx = 6x/(x^2+3)^2 (quotient rule)
8) Second Derivative
d2y/dx2 = -18(x+1)(x-1)/(x^2+3)^3 (quotient rule + chain rule)
9) Critical numbers
Set first deriv equal to 0, critical number is x=0
10) Intervals of Increase/Decrease
This is where I start having trouble. Now I know that in order to find the values of increase and decrease, you need to use your critical numbers and restrictions. The method given to use to find the intervals is to plug it into a chart and see at which intervals is the graph positive or negative. However, my question is, what would the intervals I test be? Since my restriction is a complex root and we don't work with those in our class; what would I use for my intervals? Simply (-infinity, 0), (0, infinity) ? and my values I would be testing for are 6x, and (x^2+3)^2 ?
11) Concavity
Set second deriv = to 0. Would my x values be +-1 and again, would I use anything for restriction, use 0 as an interval ? or?
12) Intervals of Concavity
This, I also have a similar issue to finding just like intervals of increase/decrease, what intervals would I be using? Do I include anything in intervals that relates to restrictions? or do I simply use (-infinity,-1),(-1,1) and (1,infinity) ? and the binomials/intervals I would be testing these in are, -(x+1), -(x-1), and (x^2+3)^3?
Any help would GREATLY be appreciated, I can't figure out this root issue.