alaina's third holiday blog

First, i'm going to talk about chapter 3 section 5.
limits at infinity.
-to find a horizontal asymptote, you take the limit as x=>infinity
y=ans is an asymptote

degree of top=degree of bottom -> coefficient of highest exponent
degree of top>degree of bottom ->-infinity or infinity
degree of top0

Ex 1: 2x+5/(3x^2+1)
lim x->infinity 2x+5/(3x^2+1)
lim x->infinity is 0
0 is a horizontal asymptote.

Chapter 3 section 6
Curve sketching

Steps
1. identify the domain and range
2. find the x and y intercepts, identify the vertical and horizontal asymptotes
3. do the first derivative test
4. do the second derivative test
5. plot the information and sketch

Ex: y=2(x^2-9)(x^2-4)
1. domain: x^2-4=0; x=+/- 2
domain: (-infinity, -2)u(-2,2)u(2,infinity)
range: find the horizontal asymptotes (limits approaching infinity) and set up intervals y=2; (-infinity,2)u(2,infinity)

2. vertal asymptotes: x=+/-2
horizontal asymptotes: y=2
x intercepts:(3,0), (-3,0)
y intercepts:(0,9/2)

3. a)use quotient rule
f'=20x/(x^2=4)^2=0
20x=0
x=0
b)x=+/-2
c)(-infinity,-2)u(-2,0)u(0,2)u(2,infinity)
d)f'(-3)=-ve; f'(-1)=-ve;f'(1)=+ve;f'(3)=+ve
e)decreasing; increasin; decreasing
f)concave down; concave up; concave down.
x=-2,2 are points of inflection