How do you use the Taylor Remainder term to estimate the error in approximating a function #y=f(x)# on a given interval #(cr,c+r)#?
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To estimate the error in approximating a function ( y = f(x) ) on an interval ( (c  r, c + r) ) using the Taylor Remainder term, you would typically follow these steps:

Find the Taylor series expansion of ( f(x) ) centered at ( x = c ). This involves finding the function's derivatives at ( x = c ).

Write down the ( n )th degree Taylor polynomial ( P_n(x) ) using the Taylor series expansion up to the ( n )th term.

The Taylor Remainder term, denoted by ( R_n(x) ), represents the error between the actual function ( f(x) ) and the Taylor polynomial ( P_n(x) ). It is given by the formula:
[ R_n(x) = f(x)  P_n(x) ]
 To estimate the error on the interval ( (c  r, c + r) ), evaluate the Taylor Remainder term ( R_n(x) ) at the endpoints of the interval, i.e., ( c  r ) and ( c + r ). This gives you the maximum possible error within the interval.
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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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