Differential Calculus: Applications II
Differential Calculus: Applications II tests the mean value theorems, series expansions, and evaluation of indeterminate limits. The exam often asks you to verify Rolle's theorem on a given interval or to use Taylor/Maclaurin expansions.
Theorems and tools
Rolle's theorem: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there is some c in (a,b) with f'(c)=0. Lagrange's mean value theorem: under continuity on [a,b] and differentiability on (a,b), there is c in (a,b) with f'(c)=[f(b)−f(a)]/(b−a). Taylor's expansion about a is f(x)=f(a)+f'(a)(x−a)+f''(a)(x−a)²/2!+…; Maclaurin is the case a=0. L'Hôpital's rule applies to 0/0 or ∞/∞ forms: differentiate numerator and denominator separately, then take the limit, provided the new limit exists.
Worked example
Verify Rolle's theorem for f(x)=x²−4 on [−2,2]. The function is a polynomial, hence continuous and differentiable everywhere. Also f(−2)=0=f(2). Solve f'(x)=2x=0, giving c=0. Since 0 lies in (−2,2), the conclusion holds.
Traps
Rolle's theorem requires f(a)=f(b); MVT does not. The point c must be strictly inside the interval. L'Hôpital's rule is not valid for forms such as 1/0 or ∞−∞ unless they are converted to 0/0 or ∞/∞. Differentiating a quotient as a whole is wrong: differentiate numerator and denominator separately. Check differentiability before invoking any mean value theorem.