MATH 150: NOTES FOR CHAPTERS 1 AND 2
Trig handouts:
1.3 Handout on Trig Identities (Part 1; I will hand this out in class.)
1.3 Handout on Trig Identities (Part 2; I will hand this out in class.)
My Precalculus (141) web site:
Math 141 (Precalculus)
Algebra: Preliminaries, Chapter 1 especially
Trig: Chapters 4 and 5 especially
Great Precalculus Review (site)
Trig review with Java applets to play with! (site)
Trig Review (site)
Look under "Trigonometry": http://oakroadsystems.com/math/
• SOME COMMENTS ON ASYMPTOTES AND HOLES: (REVISED 2/20/08)
Consider the graph of a function represented by y = f(x) in the standard xy-plane.
• We have a vertical asymptote (VA) at x=a if and only if f(x) approaches infinity or -infinity as x approaches a from the left or the right.
If f is a rational function, where f(x) is written as (polynomial)/(polynomial), then real zeros of the denominator correspond to VAs or holes (removable discontinuities). If c is a real zero of the denominator, and the limit form when x approaches c is of the form (nonzero real number)/0, then we have a VA at x = c. If, however, the limit form is 0/0, and the factors of the form (x-c) are completely canceled out in the denominator in the simplification process, then there is a removable discontinuity at x = c.
• We have a horizontal asymptote (HA) at y=L if and only if f(x) approaches L as x approaches infinity or -infinity. The graph of a rational function can have at most one HA; if there is a "long-run" limit in one direction, it must also be the long-run limit in the other direction.
• WHEN DOES f HAVE A JUMP DISCONTINUITY AT c? (2.5, A):
This happens if and only if the corresponding left-hand and right-hand limits exist, but they are unequal.
• NOTES ON IVT (2.5, D)
First sentence should end with "on [a,b]."
Chapters 1 and 2 Notes: 61 pages