A water tank is being filled, and the water level [tex]f(t)[/tex] in liters depends on time [tex]t[/tex] in minutes. At minute [tex]t=10[/tex], is the water level changing smoothly without interruption?
To confirm, we check:
* Is [tex]f(10)[/tex] defined? This question implies, was the water level recorded at minute 10?
* Does [tex]\lim _{t \rightarrow 10} f(t)[/tex] exist? This question implies, does the water level approach a specific value as time nears 10?
* Does [tex]\lim _{t \rightarrow 10} f(t)=f(10)[/tex]?
Is the water level changing at [tex]t =10[/tex] seconds discontinuous, continuous and discontinuous, neither continuous nor discontinuous, or continuous?
Answer (1)
To determine if the water level is changing smoothly without interruption at minute t = 10 , we need to assess the continuity of the function f ( t ) . For continuity at a point, three conditions have to be met:
f ( 10 ) must be defined:
This means the water level at minute 10 must have a specific recorded value.
lim t → 10 f ( t ) must exist:
The limit exists if the water level approaches a specific number as time nears 10 from both the left and the right side.
lim t → 10 f ( t ) = f ( 10 ) :
The value that f ( t ) approaches as t gets close to 10 must be the same as f ( 10 ) .
If all these conditions are satisfied, then f ( t ) is continuous at t = 10 . In this case, the water level changes smoothly without interruption.
If any of these conditions are not satisfied, then there is a discontinuity at t = 10 , and the water level change is not smooth.
Based on the provided answer choices, the correct choice is:
Water level changing at t = 10 seconds is continuous
