Sketch this curve (two multipled root functions)

In summary, we discussed the steps for sketching a curve and used them to sketch the curve defined by g(x) = x^{1/3}*(x+3)^{2/3}. We found the x-intercept at -3 and the y-intercept at (0,0). We also calculated the first derivative and solved for the extreme value at x = -1. We encountered a divide by zero error in the first derivative, which means the slope is vertical. The second derivative was more complicated and requires further understanding.
  • #1
aeromat
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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...
 
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  • #2
hi aeromat! :smile:
aeromat said:
?: What does it mean when you have a divide by zero error in the first derivative?

it means the "slope" is "vertical" :wink:
 

Related to Sketch this curve (two multipled root functions)

1. What is a "multipled root function"?

A multipled root function is a mathematical equation that contains multiple instances of the square root symbol (√). Examples include functions such as √x and √(x+1).

2. How do I sketch a curve with multipled root functions?

To sketch a curve with multipled root functions, you can start by plotting points on a graph and connecting them with a smooth curve. Alternatively, you can use a graphing calculator or software to generate a graph of the function.

3. What does a multipled root function graph look like?

A multipled root function graph usually has a curved shape, with the curve becoming steeper as the value of x increases. The graph may also have an asymptote (a line that the graph approaches but never touches) at x=0.

4. How do I find the domain and range of a multipled root function?

The domain of a multipled root function is all the possible values of x that can be plugged into the function. To find the domain, you can set the expression inside the square root symbol (the radicand) greater than or equal to 0 and solve for x. The range of a multipled root function is all the possible values of y that the function can output. The range can be found by considering the behavior of the function as x approaches positive and negative infinity.

5. What are some real-world applications of multipled root functions?

Multipled root functions can be used to model various real-world situations, such as the growth of a population or the decay of radioactive substances. They can also be used in engineering and physics to describe the relationship between two variables, such as displacement and time in a moving object.

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