Let for all real numbers .

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Let

g(x)=(x-2)^{\frac{2}{3}}

for all real numbers

x

. (a) Determine the domain of

g

and state whether

g

is continuous at

x=2

. Justify your response. (b) Calculate

g'(x)

for

x\neq 2

by any valid differentiation method. (c) Evaluate

\displaystyle \lim_{x\to 2}g'(x)

. Show each algebraic manipulation that leads to your conclusion. (d) Based on your work in parts (a)–(c), determine whether

g

has a tangent line at

x=2

. If so, describe its slope; if not, characterize the graphical feature present at that point and support your answer with appropriate reasoning.

Related theory

Advanced Differentiation
Derivative Foundations

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