Type a function and a point x = a to get the first five terms of its Taylor series about that point.
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From the article Taylor Series — Approximating a Function with a Polynomial.
f(x) = ex. Every derivative of ex is ex again, and e0 = 1, so every coefficient f(n)(0) equals 1:
f(0) = 1, f′(0) = 1, f″(0) = 1, f‴(0) = 1, …
Substituting into the Maclaurin formula:
ex = 1 + x + x²/2! + x³/3! + x⁴/4! + x⁵/5! + …
Sanity check: differentiating term by term turns each term into the one before it, so the series is its own derivative — as ex must be.
For f(x) = sin x the derivatives cycle in fours:
f(x) = sin x → f(0) = 0
f′(x) = cos x → f′(0) = 1
f″(x) = −sin x → f″(0) = 0
f‴(x) = −cos x → f‴(0) = −1
f⁗(x) = sin x → f⁗(0) = 0 (and the cycle repeats)
Only the odd powers survive, with signs alternating:
sin x = x − x³/3! + x⁵/5! − x⁷/7! + …
For f(x) = cos x the same cycle starts one step earlier: f(0) = 1, f′(0) = 0, f″(0) = −1, f‴(0) = 0, f⁗(0) = 1. Only the even powers survive:
cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + …
Both require x in radians, because the derivative of sin x is cos x only when x is measured in radians.
Notice that the odd-power terms of ex look like the sine series and the even-power terms like the cosine series. That is no coincidence: putting ix in place of x in the exponential series gives Euler's relation eix = cos x + i sin x.
Worked in full in Taylor Series — Approximating a Function with a Polynomial.
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