- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Power functions and first sketches practice question 2
Let . Close to , which term dominates, and what does the graph look like there?
This is about close to , not large . For a number between and , is a higher power of it bigger or smaller than a lower one?
Try : compare with . Small numbers get smaller when you raise them to a power.
Theory UBC Math 100 §1.2Key results
Worked solution
Strategy
Near zero the roles reverse: the lower power wins, because a number smaller than 1 shrinks each time it is multiplied by itself.
Working
The linear term is six thousand times the quartic term.
The graph follows the dominant term, here a straight line of slope 6 through the origin.
Answer
The answer is The 6x term dominates, and near the origin the graph looks like the line y = 6x..
Check
At x = 0.01, g = 0.06000001, differing from 6x = 0.06 only in the eighth decimal place.
Where this goes wrong
Carrying over the rule for large x. For large x the higher power dominates; close to 0 the lower power does, because there x⁴ is far smaller than x.
Try a similar question
Let . For very large positive , which term dominates, and what is the sign of ?Saved in this browser only.