- Topic
- Continuity
- Difficulty
- Challenging
- Suggested time
- 6 min
- Marks
- 4
Continuity practice question 23
The function below is continuous everywhere. Find .
There are two joins, at and , and each gives one equation. Only one of them involves alone: start there.
At : . At : .
An integer.
Theory UBC Math 100 §2.3CLP-1 §1.6Key results
Worked solution
Strategy
Continuity at each join is one equation. The join at 3 pins down b on its own; the join at 1 then gives a.
Working
Middle piece from the left must meet the third piece at 3.
First piece from the left must meet the middle piece at 1.
Combine.
Answer
The answer is a is −6, b is −4, so a + b is −10.
Check
With a = −6 and b = −4 the pieces meet at height −3 when x = 1 and at height 5 when x = 3.
Where this goes wrong
Solving the join at x = 1 first with b still unknown and guessing a value. The system is triangular: the join at 3 determines b, and only then can the join at 1 determine a.
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