- Topic
- Curve sketching
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Curve sketching practice question 6
Find the smallest positive period of : the smallest with for every .
Each term repeats on its own cycle. What has to be true of a shift for both terms to come back to their starting values at once?
repeats every and every . The sum repeats only when both do: find the smallest positive number that is a whole number of periods of each.
An exact expression, or a decimal to two decimal places.
Theory UBC Math 100 §7CLP-1 §3.6Key results
Worked solution
Strategy
Each term has its own period. The sum repeats only after a shift that is a whole number of periods of both terms at once.
Working
sin(kx) and cos(kx) have period 2π/k.
List the multiples of each; the first one they share is 2π.
Both angles move on by whole turns. No smaller shift does that for both terms.
Answer
The answer is two pi.
Check
f(0) = 1, while f(2π/3) ≈ 0.13, f(π) = −1 and f(4π/3) ≈ 1.87, so no shift under 2π that repeats one term repeats the sum.
Where this goes wrong
Giving the period of one term, π or 2π/3. A shift by π leaves sin(2x) alone but turns cos(3x) into −cos(3x), so the graph does not repeat until both terms do, at 2π.
Leans on
Graphs of sine and cosine, from Functions, graphs and inverses
Saved in this browser only.