What Happens at the Cusp on the Original Graph When Viewing the Derivative?

In summary, if the original graph has a cusp, the derivative is not defined at the x-value of the cusp, resulting in an asymptote. However, if the derivative graph has a cusp, it means that the second derivative does not exist. A perfect example of this is the function y=|x|, which does not have a derivative at x=0. This can be seen in the graph of its derivative, f ', which also has a cusp at x=0.
  • #1
sarahr
13
0
If the original graph, f, has a cusp, obviously the derivative is not defined at the x-value of the cusp (resulting in an asymptote).

but, what if you are viewing a graph of the derivative, f ', and it has a cusp.. what is going on at the x-value of the cusp on the original graph, f ?
 
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  • #2
If the derivative graph has a cusp, that means that the second derivative does not exist. Think about y= |x| which does not have a derivative at x= 0. If we let f '(x)= |x| = (x if x>= 0 and -x if x< 0) and integrate we get
f(x)= ((1/2)x^2 if x>=0 and -(1/2)x^2 if x< 0). What does its graph look like around x= 0?
 
  • #3
perfect example! thankss
 

FAQ: What Happens at the Cusp on the Original Graph When Viewing the Derivative?

What is a graph of derivatives?

A graph of derivatives is a visual representation of the rate of change of a function at different points. It shows the slope of the function at each point, which is the value of the derivative.

Why are graphs of derivatives important?

Graphs of derivatives are important because they allow us to understand the behavior of a function and its rate of change. They can also help us find critical points, extrema, and inflection points.

How do you interpret a graph of derivatives?

In a graph of derivatives, the x-axis represents the independent variable and the y-axis represents the derivative of the function. The slope of the graph at any point represents the rate of change of the function at that point. A positive slope indicates an increasing function, while a negative slope indicates a decreasing function.

What is the relationship between a function and its derivative?

The derivative of a function is the slope of the function at a given point. This means that the derivative tells us how the function is changing at that point. The derivative is also used to find the maximum and minimum values of a function.

Can you use a graph of derivatives to find the original function?

Yes, it is possible to use a graph of derivatives to find the original function. The original function can be found by integrating the derivative function. However, the constant of integration must also be taken into account, as it is lost when finding the derivative.

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