- Topic
- Continuity
- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
Continuity practice question 16
A drug is given by a steady infusion that stops after 2 hours. Its concentration in the blood, in mg/L, hours after the infusion starts is modelled below, where is a constant. The concentration does not jump when the infusion stops. Find , and then enter the concentration at .
No jump is a condition at one instant. Which instant, and what must the two pieces do there?
No jump at means is continuous there: the limit from the right, , must equal .
With , . Here is the natural logarithm, so .
An exact number, or a decimal to two places.
Theory UBC Math 100 §2.3CLP-1 §1.6Key results
Worked solution
Strategy
Each piece is continuous on its own, so continuity is one equation at t = 2, which fixes A. Then evaluate the second piece, simplifying the exponential with the rules of logarithms.
Working
At t = 2 itself the first piece applies.
Just after the infusion stops the second piece applies.
No jump: the limit equals the value.
Halve the exponent, then split it: ea + b = ea · eb.
e · e−1 = 1, and e−log 3 = 1/3 because log is the natural logarithm.
Answer
The answer is A is 12e, and the concentration is 4 mg/L.
Check
After t = 2 the level halves every 2 log 2 ≈ 1.39 h, and 2 log 3 ≈ 2.20 h is between one and two halvings, so 4 mg/L, between 6 and 3, fits.
Where this goes wrong
Taking A = 12, as though the exponential piece started at t = 0. Continuity matches the pieces at t = 2, where e−t/2 is e−1, so A must be 12e.
Leans on
Exponentials and logarithms, from Functions, graphs and inverses
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