Consider the function h(x)= cases 9-x^2 , & x 1

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Consider the function

h(x)=\begin{cases} \sqrt{9-x^2}, & x\le1,\\ c(1-x)+\sqrt{8}, & x>1, \end{cases}

where

c

is a real constant. The graph of

h

is intended to represent the continuous outline of a sculpture from

x=-3

x=4

. (1) Part 1: (a) Without performing any algebraic manipulation, describe qualitatively the shape of each piece of

h

. Your description should reference basic graph features such as intercepts, end behavior within the stated domain, and general curvature. (2) Part 2: (b) Determine the value of

c

that makes

h

continuous at

x=1

. Show the limit comparison that justifies your answer. (3) Part 3: (c) Explain, in complete sentences, why matching only the function values at

x=1

is insufficient to guarantee overall continuity if the limits from the left and right were not equal. Relate your explanation to the formal definition of continuity at a point.

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