Mean Value Theorems & Taylor-Maclaurin Series
“Rolle's, Lagrange's, Cauchy's MVT, Taylor polynomial approximations, and remainder formulas.”
1.Mean Value Theorems and Polynomial Approximations
§ 2.1.Mean Value Theorems connect average rate of change over an interval to instantaneous rate of change at an interior point.
Geometric Meaning of LMVT: The secant line connecting (a, f(a)) and (b, f(b)) is parallel to the tangent line to the curve at some interior point (c, f(c)).
Root Isolation Corollary of Rolle's Theorem: Between any two real roots of a differentiable function f(x) = 0, there exists at least one real root of f'(x) = 0. Therefore, if f'(x) has at most k real roots, f(x) can have at most k + 1 real roots.
Taylor Series Expansion: Represents smooth functions as infinite polynomials:
f(x) = f(a) + f'(a)(x - a) + [f''(a) / 2!](x - a)^2 + ... + [f^{(n)}(a) / n!](x - a)^n + ...
If |f'(x)| <= M for all x in [a, b], LMVT guarantees |f(b) - f(a)| <= M * |b - a|. GATE frequently tests this inequality to bound function values without integrating.
2. Worked Problem: Finding the Point 'c' in LMVT
If multiple roots for c arise, discard any values outside the open interval (a, b).
Problem [Original Practice Problem · Modeled on GATE Pattern]:
Verify Lagrange's Mean Value Theorem for f(x) = x^3 - 3x on the interval [0, 2], and find the value of c in (0, 2) that satisfies the theorem.
Step 1: Verify hypotheses:
f(x) is a polynomial, hence continuous on [0, 2] and differentiable on (0, 2).
Step 2: Compute f(a), f(b), and the average slope:
f(0) = 0^3 - 3(0) = 0
f(2) = 2^3 - 3(2) = 8 - 6 = 2
Average rate of change = [f(2) - f(0)] / (2 - 0) = [2 - 0] / 2 = 1.
Step 3: Differentiate f(x) and equate f'(c) to average slope:
f'(x) = 3x^2 - 3
f'(c) = 3c^2 - 3 = 1
3c^2 = 4 => c^2 = 4/3 => c = +/- 2 / sqrt(3).
Step 4: Check domain restriction c in (0, 2):
c = +2 / sqrt(3) ≈ 2 / 1.732 ≈ 1.155 in (0, 2).
c = -2 / sqrt(3) is rejected since -1.155 is not in (0, 2).
Conclusion: c = 2 / sqrt(3).
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