Taylor Series Expansions & Single-Variable Extrema
“Taylor & Maclaurin expansions, truncation error, critical points, and first/second derivative classification tests.”
1.Polynomial Approximations & Local Landscape Curvature
§ 1.1.Taylor's Theorem: Expresses any infinitely differentiable function as a polynomial series around expansion center a.
Truncation Error (Lagrange Remainder):
R_n(x) = [ f^{(n+1)}(xi) / (n + 1)! ] * (x - a)^{n+1} for some xi in (a, x).
In numerical optimization, the second-order Taylor polynomial f(x) ≈ f(x_k) + f'(x_k) Delta x + (1/2) f''(x_k) (Delta x)^2 forms the exact quadratic surrogate minimized by Newton's method.
Single-Variable Extrema:
A point c is a critical point if f'(c) = 0 or f'(c) does not exist.
By Fermat's Theorem, interior local extrema can occur ONLY at critical points.
2. Worked Practice Problem: 3rd-Order Taylor Approximation & Error
Remember the alternating signs in ln(1 + u) = u - u^2/2 + u^3/3 - ... and apply the chain rule.
Problem: Find the 3rd-degree Taylor polynomial P_3(x) for f(x) = ln(1 + 2x) centered at a = 0. Use P_3(x) to approximate ln(1.2), and find the absolute error compared to the exact value.
Step 1: Compute derivatives of f(x) at a = 0:
f(x) = ln(1 + 2x) => f(0) = ln(1) = 0.
f'(x) = 2 / (1 + 2x) => f'(0) = 2.
f''(x) = -4 / (1 + 2x)^2 => f''(0) = -4.
f'''(x) = 16 / (1 + 2x)^3=> f'''(0) = 16.
Step 2: Construct Taylor polynomial P_3(x):
P_3(x) = f(0) + f'(0)x + [ f''(0) / 2! ]x^2 + [ f'''(0) / 3! ]*x^3
P_3(x) = 0 + 2x + (-4 / 2)x^2 + (16 / 6)*x^3
P_3(x) = 2x - 2x^2 + (8 / 3)*x^3.
Step 3: Approximate ln(1.2) by setting 1 + 2x = 1.2 => 2x = 0.2 => x = 0.1:
P_3(0.1) = 2(0.1) - 2(0.1)^2 + (8 / 3)(0.1)^3
P_3(0.1) = 0.2 - 2(0.01) + (8 / 3)(0.001) = 0.2 - 0.02 + 0.002667 = 0.182667.
Step 4: Compare with exact value:
Exact ln(1.2) ≈ 0.182322.
Absolute Error = |0.182667 - 0.182322| ≈ 0.000345 (3.45 x 10^-4).
Conclusion: P_3(x) = 2x - 2x^2 + (8/3)x^3; Approximation = 0.18267 with error < 0.00035.
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