Engineering Mathematics I

Rolle's Theorem and Mean Value Theorems

• Applicable to functions continuous on [a, b] and differentiable on (a, b).
• Rolle's TheoremIf f(a) = f(b), then there exists c in (a, b) such thatf'(c) = 0.
• Cauchy's Mean Value TheoremFor functions f and g:(f'(c))/ (g'(c)) = (f(b)-f(a))/(g(b)-g(a)).
• Lagrange's Mean Value Theorem (special case):f'(c) = (f(b)-f(a))/(b-a).

Diagram: mvt-graph.png

Taylor and Maclaurin Theorems

• Taylor Series for f(x) about a:f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + Rn
• Maclaurin Series is Taylor expansion at a = 0.
• Remainder term ensures accuracy:Rn = f^(n+1)(ξ)(x-a)^(n+1)/(n+1)!.
• Used for approximations in engineering computations.

Diagram: taylor-expansion.png

Indeterminate Forms

• Forms: 0/0, ∞/∞, 0 × ∞, ∞ − ∞, 1^∞, 0^0 and ∞^0.
• Usually resolved using L Hospital rule:lim f/g = lim f'/g'when both numerator and denominator approach 0 or ∞.

Concavity, Convexity and Points of Inflexion

• If f''(x) &rt; 0 the curve is convex. If f''(x) < 0 the curve is concave.
• Point of inflexion occurs where f''(x) = 0 and concavity changes.
• Used to classify graph behaviour and optimization.

Diagram: concavity-inflexion.png

Asymptotes and Curvature

• Asymptotes are straight lines approached by the curve at large values of x.
• Curvature k:k = |f''(x)| / [1 + (f'(x))^2]^(3/2)
• Radius of curvature R:R = 1/k.

Diagram: curvature-graph.png