MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Consider the function f(x)=x+1x2+3x+2, defined for all x=−1.
(a) Write an explicit expression for the difference quotient hf(x+h)−f(x), where h=0. Simplify the numerator completely so that no denominators in x or h remain.
(b) Evaluate h→0limhf(x+h)−f(x) to obtain an explicit formula for f′(x) for all x=−1. Show all algebraic steps that justify the limit computation.
(c) Determine the numerical value of f′(2) and interpret this value as the slope of a tangent line on the graph of f at the corresponding point.
(d) Explain why f′(x) exists at x=−1 or does not exist at x=−1, citing the behavior of the limit in part (b).