A twice–differentiable function is known to be analytic at
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
A twice–differentiable function f is known to be analytic at x=2. A third–degree Taylor polynomial for f about x=2 is
P3(x)=5−4(x−2)+7(x−2)2−3(x−2)3.
Answer the following questions.
(a) Write the coefficient of the (x−2)3 term in P3(x).
(b) Describe, in complete sentences, the relationship between that coefficient and the third derivative of f at x=2. Be explicit about any factorial factors that arise.
(c) Determine the exact value of f′′′(2).