MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
The function g(x)=3x+7 is defined for all x≥−37.
(a) Write an expression for g(2+h)−g(2) and form the quotient hg(2+h)−g(2).
(b) Evaluate h→0limhg(2+h)−g(2). Show the algebraic manipulation necessary to remove the radical from the numerator.
(c) Using the value found in part (b), construct the linearization L(x) of g at x=2. Then use L(x) to estimate g(2.02) and state whether your estimate is an overestimate or an underestimate. Explain your reasoning briefly.