- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Power functions and first sketches practice question 1
Let . For very large positive , which term dominates, and what is the sign of ?
For large , what decides which term is bigger: the coefficient in front, or the power of ? Then look at the sign the bigger term carries.
Try : compare the size of with the size of .
Theory UBC Math 100 §1.2Key results
Worked solution
Strategy
For large x the higher power wins, however big the coefficient on the lower power. Decide which term is bigger, then read off its sign.
Working
Already at x = 10 the fifth power is more than three hundred times the square.
The graph follows its dominant term.
A negative multiple of an odd power is negative when x is positive.
Answer
The answer is The x to the fifth term dominates, and f(x) is negative for large positive x..
Check
f(x) = x²(3 − x³) is negative once x³ > 3, that is for every x above ∛3 ≈ 1.44.
Where this goes wrong
Letting the coefficient 3 decide. A coefficient only scales a term; for large enough x the higher power always overtakes, whatever the coefficients are.
Try a similar question
Let . Close to , which term dominates, and what does the graph look like there?Saved in this browser only.