MATH100 December 2013
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q2 (a) • Q2 (b) • Q2 (c) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 • Q6 • Q7 • Q8 (a) • Q8 (b) • Q8 (c) • Q8 (d) • Q9 • Q10 (a) • Q10 (b) • Q11 •
Question 08 (d)
Let . Sketch the graph of and indicate the inflection point(s) on your graph. (The results from parts (a), (b), and (c) may be useful.)
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What are the critical numbers of ? What are the intervals of increase/decrease of ? How do these relate to the extrema of ?
What are the intervals of concavity of ? What is an inflection point, and how does it relate to concavity?
It may be useful to find the x-intercept(s) of . In addition, note the behaviour of near .
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From parts (a), (b), and (c) we know the following about :
- The critical numbers of (where is zero or does not exist) are
- is increasing on and decreasing on
- has a local minimum at and a local maximum at (note the change in the increase/decrease of )
- is concave up on and concave down on
- has an inflection point (changes concavity) at
- and thus has x-intercepts at
- and , thus the graph of is very steep near
Plotting the points of interest gives the following:
Accounting for increase and decrease gives the following rough sketch:
Finally, incorporating concavity leads us to the final graph:
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