Section 4.1 Increasing and decreasing functions
A function f(x) is increasing over an interval I, if for 2 values of x, x1 , < x2
, then f(x1 )< f(x2)
A function f(x) is decreasing over an interval I , if for 2 values of x, x₁ < x2, then f(x2)> f(x2)
For a function f(x) that is defined, continuous and differentiable on an interval I :
f(x) is increasing if f'(x)>0 over I
f(x) is decreasing if f'(x)<0 over I
Section 4.2 Critical points, Local maxima and Local minima
The critical point determines a local minimum if f(c)<f(x) for all values of x near c and a local maximum if f(c) > f(x) for all values of x near c.
The local minimum and local maximum are called local extrema.
Section 4.3 Vertical and Horizontal Asymptotes
A rational function f(x)=\(\frac{p(x)}{q(x)}\)has a vertical asymptote at x=c if q(c)=0 and p(c)≠0 and one of the following is true .
lim(x-> c⁻) f(x)= +∞ , lim(x-> c⁻) f(x)= -∞
lim(x-> c⁺) f(x)= +∞ , lim(x-> c⁺) f(x)= -∞
The Reciprocal Function and Limits at infinity
lim(x-> +∞) \(\frac{1}{x}\) = 0 and lim(x-> -∞) \(\frac{1}{x}\)= 0
Horizontal Asymptotes and Limits at Infinity
If lim(x-> +∞) f(x)=L or lim(x-> -∞) f(x)=L, the line y=L is a horizontal asymptote to the graph y=f(x)
When in a rational function f(x)=\(\frac{p(x)}{q(x)}\), the degree of p(x) is greater than degree of q(x), then there will be an oblique or slant asymptote.
Section 4.4 Concavity and points of inflection
The graph of a function f(x) is concave up on an interval I if f'(x) is increasing on the interval. The graph of the function f(x) is concave down on an interval if f'(x) is decreasing on the interval I.
A point of inflection is a point on the graph of f(x) where the function changes from concave up to concave down, or vice versa. f”(c)=0 or is undefined if is a point (c, f(c)) of inflection on the graph of f(x).
Section 4.5 An Algorithm for curve sketching
Sketching the Graph of a Polynomial or Rational Function
The second derivative can also be used to identify local maxima and minima.