Before the midterm
Letting the higher power rule near the origin
The habit: Close to 0 the smaller exponent dominates; the larger one takes over only far away.
6 practice questions are built to catch it, in 1 topic.
See it fail in a figure: A sum near 0, in the power functions and first sketches lesson
What it looks like
A question from power functions and first sketches, with the answer this slip gives beside the right one.
Let . Close to , which term dominates, and what does the graph look like there?
- With the slip
- The x4 term dominates: near the origin the graph is flat and bowl-shaped.
- Worked through
- The 6x term dominates: near the origin the graph looks like the line y = 6x.
At x = 0.1, x4 = 0.0001 while 6x = 0.6, six thousand times larger. Close to 0 the term in x is the bigger one, so it sets the shape there.
Where it comes up
- Power functions and first sketches6 questions
Practise against it
Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.
- 01Let , with in radians. Which term dominates close to , and which dominates for large ?
Power functions and first sketchesIntroductory
- 02The graph of one of these power functions is shown, with the points and marked. Which function is it?
Power functions and first sketchesIntroductory
- 03The graph of is shown, where and are nonzero constants. What are the signs of and ?
Power functions and first sketchesIntroductory
- 04Which description matches the graph of ?
Power functions and first sketchesStandard
- 05Let , where is a nonzero constant. Which statement about the graph is correct for both signs of ?
Power functions and first sketchesCapstone