Given the function f (x) = e2x and f (xi) = fi, which of these is the correct 3rd order Taylor seriesexpansion for fi+1 = f (xi + ∆x)?A. fi+1 ≈ e2xi + 4∆xe2xi + 2∆x2e2xi + 43 ∆x3e2xiB. fi+1 ≈ e2xi + 2∆xe2xi + 2∆x2e2xiC. fi+1 ≈ e2xi + 2∆xe2xi + 2∆x2e2xi + 43 ∆x3e2xi
Question
Given the function f (x) = e2x and f (xi) = fi, which of these is the correct 3rd order Taylor seriesexpansion for fi+1 = f (xi + ∆x)?A. fi+1 ≈ e2xi + 4∆xe2xi + 2∆x2e2xi + 43 ∆x3e2xiB. fi+1 ≈ e2xi + 2∆xe2xi + 2∆x2e2xiC. fi+1 ≈ e2xi + 2∆xe2xi + 2∆x2e2xi + 43 ∆x3e2xi
Solution
The Taylor series expansion of a function about a point can be represented as:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...
Given the function f(x) = e^(2x), its derivatives are:
f'(x) = 2e^(2x) f''(x) = 4e^(2x) f'''(x) = 8e^(2x)
Substituting these into the Taylor series expansion gives:
f(x) = e^(2a) + 2e^(2a)(x-a) + 4e^(2a)(x-a)^2/2! + 8e^(2a)(x-a)^3/3!
Simplifying this gives:
f(x) = e^(2a) + 2e^(2a)(x-a) + 2e^(2a)(x-a)^2 + 4/3 e^(2a)(x-a)^3
Now, let's substitute a = xi and x = xi + ∆x:
f(xi + ∆x) = e^(2xi) + 2e^(2xi)∆x + 2e^(2xi)∆x^2 + 4/3 e^(2xi)∆x^3
So, the correct 3rd order Taylor series expansion for fi+1 = f(xi + ∆x) is:
fi+1 ≈ e^(2xi) + 2∆xe^(2xi) + 2∆x^2e^(2xi) + 4/3 ∆x^3e^(2xi)
Therefore, none of the given options A, B, or C is correct.
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