A lake-level monitoring station records the height (in feet above a fixed reference)…

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

Problem

A lake-level monitoring station records the height

L(t)

(in feet above a fixed reference) of the water surface at time

t

, measured in hours after noon on a particular day. The data satisfy the following conditions: •

L(12)=4.2

, •

L(18)=3.6

, •

L

is continuous for

12\le t\le18

. Additional spot measurements give

L(14)=4.4

and

L(16)=3.8

. (a) Prove that there exists a time

t_1

with

12<t_1<18

such that

L(t_1)=4.0

. (b) Use the values at

t=14

and

t=18

to identify a specific sub-interval of

[12,18]

on which at least one time

t_2

satisfies

L(t_2)=3.9

. Explain your choice of interval. (c) Given the four measurements, explain whether the value

L(t)=4.0

can occur more than once on

[12,18]

. Provide a complete argument.