How do you find the smallest value of #n# for which the Taylor Polynomial #p_n(x,c)# to approximate a function #y=f(x)# to within a given error on a given interval #(c-r,c+r)#?

Answer 1
Let #epsilon>0# be an acceptable error.

One way to estimate the error is by

#|f(x)-p_n(x,c)|=|{f^{(n+1)}(z)}/{(n+1)!}(x-c)^{n+1}|#, where #z# in #(c-r,c+r)#.
since #x# in #(c-r,c+r)#,
#\leq |f^{(n+1)}(z)|/{(n+1)!}r^{n+1}#
if we know that the absolute value of all derivatives of $f(x)$ are bounded by some fix value #M#, then
#leq M/{(n+1)!}r^{n+1}#
Find #n# such that
#M/{(n+1)!}r^{n+1} le epsilon#
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Answer 2

To find the smallest value of ( n ) for which the Taylor Polynomial ( p_n(x,c) ) approximates a function ( y = f(x) ) within a given error on a given interval ((c-r, c+r)), you can use the remainder term formula of Taylor's theorem, which is given by:

[ R_n(x) = \frac{f^{(n+1)}(z)}{(n+1)!}(x-c)^{n+1} ]

where ( z ) is some point between ( x ) and ( c ). You want ( R_n(x) ) to be less than or equal to the given error on the interval ((c-r, c+r)). So, you set up the inequality:

[ \left| \frac{f^{(n+1)}(z)}{(n+1)!}(x-c)^{n+1} \right| \leq \text{error} ]

Then, solve for ( n ) by finding the smallest integer that satisfies this inequality.

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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