The function is defined on the closed interval .

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

The function

q(x)=\arcsin(x)

is defined on the closed interval

[-1,1]

. (a) Explain why

q

is differentiable on

(-1,1)

but not differentiable at

x=-1

x=1

. Your explanation should address the behavior of the slope of secant lines approaching each endpoint. (b) Evaluate

\displaystyle\lim_{x\to 1^-} q'(x)

and interpret the result in terms of the geometric appearance of the graph near

x=1

. Justify your limit evaluation.