Taylor Series: Find 2nd Degree Series for x^2+y^2=4 at [1, -\sqrt[]{3}]

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SUMMARY

The discussion focuses on deriving the second-degree Taylor series for the equation \(x^2 + y^2 = 4\) at the point \([1, -\sqrt{3}]\). Participants emphasize the need to evaluate the function and its first two derivatives, \(f(1)\), \(f'(1)\), and \(f''(1)\), to construct the polynomial. The formula for the second-degree Taylor polynomial is provided, highlighting the importance of calculating these derivatives accurately to achieve the correct series expansion.

PREREQUISITES
  • Understanding of Taylor series expansion
  • Knowledge of calculus, specifically differentiation
  • Familiarity with implicit functions and their derivatives
  • Basic algebraic manipulation skills
NEXT STEPS
  • Learn how to compute derivatives of implicit functions
  • Study the application of Taylor series in multivariable calculus
  • Explore examples of Taylor series expansions for different functions
  • Practice evaluating Taylor series at various points
USEFUL FOR

Students and educators in calculus, mathematicians interested in series expansions, and anyone seeking to deepen their understanding of Taylor series in the context of implicit functions.

Anewk
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How would I find the second-degree Taylor series for $$x^2+y^2=4$$ at $$[1, -\sqrt[]{3}]$$?
 
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Hello and welcome to MHB! :D

Can you tell us what you've tried so we know where you are stuck or what you may be doing wrong? We can offer better help that way.
 
Anewk said:
How would I find the second-degree Taylor series for $$x^2+y^2=4$$ at $$[1, -\sqrt[]{3}]$$?

The second degree Taylor polynomial for a function $\displaystyle \begin{align*} f(x) \end{align*}$ centred at the point is $\displaystyle \begin{align*} (h, k) \end{align*}$ is $\displaystyle \begin{align*} f(x) = f(h) + f'(h) \, \left( x - h \right) + \frac{f''(h)}{2} \, \left( x - h \right) ^2 \end{align*}$.

Can you evaluate $\displaystyle \begin{align*} f(1), f'(1), f''(1) \end{align*}$?
 

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