Need some help with understanding linear approximations

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SUMMARY

This discussion centers on the concept of linear approximation in calculus, specifically using the function f(x) = x^4 to estimate the value of (2.94)^4. Participants emphasize the importance of selecting a nearby integer, x = 3, and calculating Δx as -0.06. The correct application of the linear approximation formula, f(x + Δx) ≈ f(x) + f'(x)Δx, leads to an approximate value of 74.52 for (2.94)^4. The conversation highlights common misunderstandings and the necessity of grasping both the derivative and the approximation process.

PREREQUISITES
  • Understanding of basic calculus concepts, including derivatives.
  • Familiarity with the linear approximation formula: f(x + Δx) ≈ f(x) + f'(x)Δx.
  • Ability to compute derivatives for polynomial functions, specifically f(x) = x^4.
  • Knowledge of how to select appropriate values for x and Δx in approximation problems.
NEXT STEPS
  • Study the application of linear approximation in various functions beyond polynomials.
  • Learn how to calculate derivatives for different types of functions, including trigonometric and exponential functions.
  • Explore the concept of Taylor series and how it relates to linear approximation.
  • Practice solving linear approximation problems with varying degrees of complexity.
USEFUL FOR

Students of calculus, educators teaching mathematical concepts, and anyone seeking to improve their understanding of approximation techniques in mathematics.

  • #31
LCKurtz said:
Apparently you've got it now, but I would suggest part of your problem is inability to do algebra/arithmetic.
As epenguin said "we all have our blind moments".
 

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