An electrical circuit switches instantaneously from a 3 volt battery to a 23 volt battery 6 seconds after being turned on. Sketch on a sheet of paper a graph the battery voltage against time. Then fill in the formulas below for the function represented by your graph.
For t< , V(t)=
For t≥ , V(t)= .
At what point or points is your function discontinuous?
t=
Oh, I'm so excited! Time to draw a graph! *grabs a sheet of paper* Okay, here we go!
^
23 | ______
| |
| |
| |
3 |___|_________________________
0 6
Ah, beautiful, isn't it? Now, let me fill in the formulas for this masterpiece of mine.
For t < 6, V(t) = 3 volts (because it's before the switcheroo)
For t ≥ 6, V(t) = 23 volts (after the sudden voltage increase)
And now, for the grand finale, the point(s) of discontinuity! Drumroll, please...
*taps desk with fingers* The function is discontinuous at t = 6! That's when the voltage unexpectedly jumps from 3 to 23 volts in a blink of an eye.
Ta-da! I hope you find my electrical circus act entertaining! *winks*
To sketch the graph of the battery voltage against time, we need to consider the change in voltage at the specific times mentioned.
First, we know that initially (for t < 6 seconds), the circuit is connected to a 3-volt battery. Therefore, the graph will be a horizontal line at y = 3 for this duration.
Next, at t = 6 seconds, the circuit switches instantaneously to a 23-volt battery. So, at t = 6 seconds, the graph will have a vertical line connecting y = 3 to y = 23.
To fill in the formulas, we can represent the graph as follows:
For t < 6, V(t) = 3 (constant value of 3 volts)
For t ≥ 6, V(t) = 23 (constant value of 23 volts)
Hence, the function V(t) is discontinuous at t = 6 seconds, where there is an abrupt change in voltage.
To sketch a graph of the battery voltage against time, you would start with the initial voltage of 3 volts and maintain it until the switch at 6 seconds. Then, the voltage instantaneously changes to 23 volts at that point and remains constant afterwards.
The graph would look like a horizontal line at 3 volts for t < 6 seconds, and then a sudden vertical jump to a horizontal line at 23 volts for t ≥ 6 seconds.
Now, let's fill in the formulas for the function represented by the graph:
For t < 6, V(t) = 3 volts (since the voltage remains constant at 3 volts before the switch).
For t ≥ 6, V(t) = 23 volts (since the voltage remains constant at 23 volts after the switch).
The function is discontinuous at t = 6 seconds since there is an instantaneous jump in the voltage value at that specific point in time.
6, 23
0, 3
7