How do you find the largest interval #(cr,c+r)# on which the Taylor Polynomial #p_n(x,c)# approximates a function #y=f(x)# to within a given error?
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To find the largest interval ((c  r, c + r)) on which the Taylor Polynomial (p_n(x, c)) approximates a function (y = f(x)) to within a given error, you can use the Taylor Remainder Theorem or Taylor's Inequality.

Using Taylor Remainder Theorem: Taylor Remainder Theorem states that if (f) has (n+1) continuous derivatives on an interval containing (c) and (f^{(n+1)}(x)) exists on ((c  r, c + r)), then for each (x) in ((c  r, c + r)), there exists a number (z) between (x) and (c) such that: [ R_n(x) = \frac{f^{(n+1)}(z)}{(n+1)!}(x  c)^{n+1} ] Here, ( R_n(x) ) is the remainder term of the (n)th degree Taylor polynomial. You can choose (r) such that the error (R_n(x)) is less than or equal to the desired error.

Using Taylor's Inequality: Taylor's Inequality provides an upper bound on the error of the Taylor polynomial approximation. It states that if (f) has (n+1) continuous derivatives on an interval containing (c) and (f^{(n+1)}(x)) exists on ((c  r, c + r)), then for each (x) in ((c  r, c + r)): [ f(x)  p_n(x, c) \leq \frac{M}{(n+1)!}x  c^{n+1} ] where ( M ) is an upper bound for ( f^{(n+1)}(x) ) on ((c  r, c + r)). You can choose ( r ) such that ( \frac{M}{(n+1)!}x  c^{n+1} ) is less than or equal to the desired error.
In both cases, you may need to analyze the behavior of the function and its derivatives to find suitable values for (r).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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