Estimated Instantaneous Rate Of Change

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SUMMARY

The discussion centers on finding the estimated instantaneous rate of change (EIRC) for the function f(x) = x³ - 2x² + x. Participants identified critical points where the EIRC is zero, specifically at x = 0, x = 1, and x = 1/3. The conversation emphasizes the distinction between the function's value being zero and the instantaneous rate of change being zero, highlighting the necessity of differentiation to analyze these rates accurately. The importance of visual aids in understanding the graph's behavior was also noted.

PREREQUISITES
  • Understanding of calculus concepts, specifically differentiation.
  • Familiarity with polynomial functions and their graphs.
  • Ability to identify critical points and intervals of increase/decrease.
  • Knowledge of how to interpret instantaneous rates of change.
NEXT STEPS
  • Study the principles of differentiation in calculus.
  • Learn how to find critical points and analyze their significance.
  • Explore graphing techniques for polynomial functions.
  • Investigate the relationship between a function's value and its derivative.
USEFUL FOR

Students studying calculus, particularly those learning about differentiation and instantaneous rates of change, as well as educators seeking to enhance their teaching methods with visual aids.

Veronica_Oles
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Homework Statement


You have to graph f(x) = x3 - 2x2 + x and find where the instantaneous rate of change is positive negative and zero.

Homework Equations

The Attempt at a Solution


I factored this and found the zero's to be x= 0 and x = 1. With all this info given I graphed it and realized the EIRC is zero at 1 however it's also at 1/3 but I am unsure of how to obtain 1/3?
 
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The question does not ask for the point where "##f(x)##" is positive, negative and zero, rather it asks for where "the instantaneous rate of change" is positive, negative and zero.
If ##f(x)## is zero, it does not mean that the instantaneous rate of change is also zero. For example, consider the graph of ##y=x^3-2x^2+1##
Untitled.png

Have you learned about differentiation?
 
arpon said:
The question does not ask for the point where "##f(x)##" is positive, negative and zero, rather it asks for where "the instantaneous rate of change" is positive, negative and zero.
If ##f(x)## is zero, it does not mean that the instantaneous rate of change is also zero. For example, consider the graph of ##y=x^3-2x^2+1##
View attachment 107689
Have you learned about differentiation?
I understand it now. Thank you for the great visual it helped a lot. As for writing the intervals for where it is positive, negative and zero the teacher said to estimate the values of x.
 

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