MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
A student states: “For any polynomial R(x) of degree d, the nth derivative R(n)(x) is zero whenever n>d, and when n≤d the coefficient of the highest-degree term in R(n)(x) always contains a factorial factor.” Evaluate this statement.
(a) Using derivative properties, explain rigorously why R(n)(x) is identically zero for all n>d.
(b) For n≤d, describe how repeated differentiation affects the coefficient of the leading term of R(x), and clarify the role of factorial factors in your explanation.
(c) Let S(x)=7x3−2x+5. Compute S′(x), S′′(x), and S(3)(x). Discuss how your results support or contradict the student’s claim.