During a laboratory experiment

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

Problem

During a laboratory experiment, the rate

r

(in milliliters per second) at which a chemical solution drains from a container is modeled by the function

r(t)=\begin{cases} \dfrac{5t}{t-4}, & 0\le t<4,\\[6pt] kt^2-8, & 4\le t<6,\\[6pt] \ln(t-5)+3, & 6\le t\le 10, \end{cases}

where

t

is the time in seconds after the container is opened and

k

is a real constant determined by the chemist. (1) Calculate

\displaystyle\lim_{t\to4^-}r(t)

and

\displaystyle\lim_{t\to4^+}r(t)

in terms of

k

. Interpret the physical meaning of each limit in the context of the draining process. (2) Determine the value of

k

that ensures the model predicts no jump or infinite discontinuity at

t=4

. (3) Even after choosing the value of

k

found in part (b), the model may still fail to be continuous at

t=4

. Describe precisely what additional mathematical condition must be met for continuity at that instant and evaluate whether it is satisfied. (4) Identify every point in the interval

0\le t\le10

where

r

is not continuous. For each such time, classify the discontinuity and justify your classification using appropriate limit calculations. (5) Discuss the practical implications for the laboratory experiment of any discontinuities you identified in part (d). In particular, explain how the chemist might adjust the piecewise definition of

r

to eliminate removable discontinuities while preserving the observed behavior at other times.

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