How do you find the smallest value of #n# for which the Taylor series approximates the function #f(x)=e^(2x)# at #c=2# on the interval #0<=x<=1# with an error less than #10^(-6)#?
Upon determining the remainder term,
Since
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To find the smallest value of ( n ) for which the Taylor series approximates the function ( f(x) = e^{2x} ) at ( c = 2 ) on the interval ( 0 \leq x \leq 1 ) with an error less than ( 10^{-6} ), we use the remainder term formula for Taylor series.
The Taylor series expansion for ( f(x) = e^{2x} ) centered at ( c = 2 ) is:
[ f(x) \approx \sum_{k=0}^{n} \frac{f^{(k)}(2)}{k!} (x - 2)^k ]
The remainder term is given by:
[ R_n(x) = \frac{f^{(n+1)}(z)}{(n+1)!} (x - 2)^{n+1} ]
where ( z ) lies between ( 2 ) and ( x ).
We want ( R_n(x) < 10^{-6} ) for ( 0 \leq x \leq 1 ). So, we need to find the maximum value of ( f^{(n+1)}(z) ) on the interval ( 0 \leq x \leq 1 ).
Since ( f(x) = e^{2x} ), ( f^{(n+1)}(x) = 2^{n+1}e^{2x} ), which is maximized at ( x = 1 ).
So, ( f^{(n+1)}(z) \leq 2^{n+1}e^{2} ) for ( 0 \leq z \leq 1 ).
Now, we solve for ( n ) such that:
[ \frac{2^{n+1}e^{2}}{(n+1)!} (1 - 2)^{n+1} < 10^{-6} ]
[ \frac{2^{n+1}e^{2}}{(n+1)!} < 10^{-6} ]
We can use numerical methods or approximation techniques to solve for the smallest ( n ).
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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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