Week 10 Calculus I Outline of Notes

by S. Gramlich

(updated 12/11/06)

I) 4.3 & 4.5 Curve Sketching, pp. 317-318

A. Domain (sec 1.1, restrictions)

B. X & Y Intercepts (Appendix B)

C. Symmetry (sec 1.1, f(-x)=f(x) => even & symm wrt y-axis,

f(-x)=-f(x) => odd & symm wrt origin)

D. Asymptotes

Vertical (sec 2.2)

Horizontal (sec 2.6, div by hi pwr of den & use HA thm)

Slant (sec 4.5, if degree of num is +1> degree of den,

then use long division & quotient is SA)

E. Intervals of Incr/Decr (sec 4.3)

I/D Test, p. 296; f'>0, incr; f'<0, decr

1. find f'(c)=0 & solve for c

2. setup intervals wrt c & use test points (tp) to id incr/decr

F. Local Max & Min

1st Deriv Test, p. 297

from I/D Test: f' goes + to - => max; f' goes - to + => min;

no change => no max/min

OR 2nd Deriv Test, p. 301

f"(c) > 0, min; f"(c) < 0, max; f"(c)=0, inconclusive

G. Concavity & Points of Inflection

Concavity, p. 299, Def & Fig 6;

Concave upward=holds water up, Concave downward=spills water down

Concavity Test, p. 300; f">0, up; f"<0, down

1. solve f"(p)=0

2. setup intervals wrt p & use tp to id up/down

Points of Inflection, p. 299, Fig 7

p. 300; from Conc test: concave up to down & vice versa

H. Sketch using all information from A-G

4.5, p. 322, Example 6 & 4.3, p. 298, Example 3

II) 4.4 Indeterminate Forms

(A) if lim _{(x -> a)} f(x)/g(x) = 0/0 or ∞/∞,

then use L'Hospital's Rule & Limit Law #5 (sec 2.3, p. 104): lim (x -> a) f'(x)/g'(x)

4.4, Example 2, p. 309

(B) if indeterminate form 0*∞,

then rewrite fg as f/ (1/g) to change form to 0/0 or ∞/∞ & use L'Hospital's Rule

4.4, Example 6, p. 311

(C) if indeterminate form ∞ - ∞,

then convert difference to a quotient 0/0 or ∞/∞ & use L'Hospital's Rule

(D) if indeterminate form 0⁰, ∞⁰, 1^∞,

then use logarithmic differentiation or e^{ln x}=x (sec 1.6, p.68) & L'Hospital's Rule

4.4, Example 8, p. 312