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Taylor Approximations
Approximate eˣ, ln(1+x), sin x or the logistic function by polynomials of order 1–4 around any point.
Day 23: Derivatives, Partial Derivatives & Taylor Seriesf(x) = ln(1+x)Taylor polynomial of order 1 at a
error at x = a + 0.39
0.06070
f′(a) (slope)
1.0000
the delta-method multiplier
The Taylor polynomial matches the function's value and first few derivatives at the point a. Near a the error shrinks like |x − a|^(order+1); far away the approximation can be poor. Hover to read the error.