Estimation is over or under estimate

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SUMMARY

The discussion centers on determining whether the approximation of the area under the curve defined by the differential equation y'' = -y^2 is an overestimation or underestimation. Participants emphasize the importance of analyzing the behavior of the function y(x) over the interval [1,5] to ascertain whether it is increasing or decreasing. The Riemann sum method is suggested for approximating the area, specifically using right endpoint evaluation with n=8. Understanding the first derivative, y', is crucial for this analysis.

PREREQUISITES
  • Understanding of differential equations, specifically y'' = -y^2
  • Knowledge of Riemann sums and their application in area approximation
  • Familiarity with the concepts of increasing and decreasing functions
  • Ability to compute first derivatives and interpret their significance
NEXT STEPS
  • Learn how to derive and analyze the first derivative of a function
  • Study the application of Riemann sums in approximating areas under curves
  • Explore the characteristics of functions defined by differential equations
  • Investigate the implications of concavity and inflection points in relation to area estimation
USEFUL FOR

Students studying calculus, particularly those focusing on differential equations and numerical methods for area approximation, as well as educators seeking to clarify concepts related to Riemann sums and function behavior.

dohsan
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Homework Statement


If y''=-y^2, is this approximationg an over-estimation or an underestimation of the exact area?


Homework Equations





The Attempt at a Solution

 
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Area of what? y''=(-y^2) is a differential equation. What does that have to do with an area?
 
I think this problem has to do with this equation... bc my teacher wrote it as a new problem and it confused me..

Write the Reimann sum that will approximate the area under the graph of the function y=f(x) that is continuous and positive over the interval [1,5] with n=8 using the right endpoint evaluation.
 
dohsan said:
I think this problem has to do with this equation... bc my teacher wrote it as a new problem and it confused me..

Write the Reimann sum that will approximate the area under the graph of the function y=f(x) that is continuous and positive over the interval [1,5] with n=8 using the right endpoint evaluation.

Ok, so to decide whether it is an overestimate or underestimate without actually solving the differential equation you need to at least decide whether y(x) is decreasing or increasing on [1,5]. Can you figure out how to do that?
 
Well, I believe you find y' which if f'(x), but not knowing what f is... it's quite confusing on how to find inc or dec on [1,5]. I just know if f'(x) > 0 .. it's inc.
 

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