Sketch this curve (two multipled root functions)

  • Thread starter aeromat
  • Start date
  • #1
114
0

Homework Statement


Sketch the curve defined by [tex]g(x) = x^{1/3}*(x+3)^{2/3}[/tex]


Homework Equations


First Derivative
Second Derivative


The Attempt at a Solution


I attempted to use the algorithm for curve sketching:
1) Find the intercepts
2) First Derivative; look for extrema and points where f'(x) DNE
3) Second Derivative; look for each region's concavity according to point of inflection found, and where f''(x) DNE
4) Sketch the curve

First, I made the function y = 0
[tex]0 = (x)^{1/3}*(x+3)^{2/3}[/tex]
[tex]0 = (x+3)^{2/3}[/tex]
[tex]0^3 = ((x+3)^{2/3})^3[/tex]
[tex]\sqrt{0} = \sqrt{(x+3)^2}[/tex]
[tex]0 = (x + 3)[/tex]
[tex]-3 = x[/tex]
Therefore, x-intercept at x = -3.

[tex]y = (0)^{1/3}*(3)^{2/3}[/tex]
[tex]y = 0[/tex]
Therefore, the graph crosses the origin; (0,0)

First Derivative:
[tex]g'(x) = a'(b) + a(b)'[/tex]
[tex]a = (x)^{1/3}[/tex]
[tex]a' = \frac{1}{3}*(x)^{-2/3}[/tex]
[tex]b = (x+3)^{2/3}[/tex]
[tex]b' = \frac{2}{3}*(x+3)^{-1/3}[/tex]

The extreme value retrieved from the first derivative:
[tex]0 = (x+1)[/tex]
[tex]-1 = x [/tex]


Simplifying for the first derivative, and I get:
[tex]g'(x) = \frac{x+1}{x^{2/3}*(x+3)^{1/3}}[/tex]

?: What does it mean when you have a divide by zero error in the first derivative?

The second derivative is much more complicated. I finished it but I first need to understand as to what do I have to do when I get it in the first place...
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,832
251
hi aeromat! :smile:
?: What does it mean when you have a divide by zero error in the first derivative?
it means the "slope" is "vertical" :wink:
 

Related Threads on Sketch this curve (two multipled root functions)

  • Last Post
Replies
2
Views
738
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
717
  • Last Post
Replies
2
Views
899
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
3
Views
733
  • Last Post
Replies
4
Views
472
Top